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Full text of "Physics for college students"

Collej & 

PHYSICS 



I 



FOR 



COLLEGE STUDENTS 



BY 

HENRY S. CARHART, Sc.D., LL.D. 

/f 

PROFESSOR EMERITUS OF PHYSICS, UNIVERSITY OF MICHIGAN 



ALLYN AND BACON 
Boston ano Ctjtcago 



CONTENTS 

MECHANICS 
Chapter I. Introduction 

PAGE 

I. Matter and Energy 1 

II. Properties of Matter 3 

III. Physical Measurements 9 

Chapter II. Kinematics 

I. Rectilinear Motion . . . . . . . .15 

II. Curvilinear Motion ........ 21 

III. Periodic Motion 23 

Chapter III. Dynamics 

I. Newton's Laws of Motion 36 

II. Gravitation 43 

III. Work and Energy 50 

IV. Friction 60 

V. Motion of Rotation 65 

VI. Equilibrium 80 

VII. Machines 85 

VIII. Elasticity . . . . . . . . . .97 

Chapter IV. Mechanics of Fluids 

I. Molecular Phenomena in Liquids 103 

II. Pressure in Fluids 114 

III. Bodies immersed in Liquids 120 

IV. Density and Specific Gravity 123 

V. Fluids in Motion . . . . . . . .127 

VI. Properties of Gases 130 

VII. Pressure of the Atmosphere 135 

VIII. Instruments depending on the Pressure of the Air . . 141 

SOUND 
Chapter V. Waves 

I. Wave Motion 150 

II. Water Waves 153 

III. Propagation and Reflection of Waves . . . , . 156 



vi CONTENTS 

Chapter VI. Production and Transmission of Sound 

PAGE 

I. Nature and Motion of Sound . ... . . . 162 

II. Velocity of Sound &.'-. 166 

Chapter VII. Physical Basis of Music 

I. Musical Intervals and Scales . . . . . .179 

II. Transverse Vibration of Strings . . . . .186 

III. Resonance , . . 192 

IV. Organ Pipes ; V-' ; . . 195 

V. Quality . . . . 204 

VI. Interference and Beats . '. .M/^.a-... . . 206 

VII. Vibration of Rods, Plates, and Bells . . , , . v . . 212 



LIGHT 

Chapter VIII. Nature and Propagation of Light 

I. General Phenomena . . . ,,.,.,. .,. ., , * ; - . 217 

II. Velocity of Light . . . . . " . o 223 

III. Reflection of Light ' . 229 

IV. Refraction of Light -. . 239 

V. Lenses ,.246 

VI. Dispersion 256 

Chapter IX. Light as a Wave Motion 

I. Interference and Diffraction . . ;* . . 267 

II. Emission and Absorption of Radiation . . , . 279 

Chapter X. Sensations of Color . . . . . . * .293 

Chapter XI. Polarized Light 

I. Double Refraction 302 

II. Polarization by Reflection 308 

III. Polarization by Double Refraction 311 

IV. Colors by Polarized Light . . . . . . .322 

Chapter XII. Optical Instruments 

' I. The Eye 328 

II. 'Microscopes and Telescopes . ... . . . 334 



CONTENTS vii 

HEAT 
Chapter XIII. Nature and Effects of Heat 

PAGE 

I. Heat and Temperature . . . . . . . 340 

II. Thermometry -344 

III. Expansion ......... 351 

IV. Measurement of Heat 362 

V. Change of State .367 

VI. Liquefaction of Gases . 385 

Chapter XIV. Transmission and Radiation of Heat 

I. Conduction 392 

II. Convection 396 

III. Radiation and Absorption ...... 400 

Chapter XV . Thermodynamics 

I. Mechanical Equivalent of Heat 409 

II. Carnot's Cycle 413 

MAGNETISM AND ELECTRICITY 
Chapter XVI. Magnets and Magnetic Fields 

I. Properties of Magnets 425 

II. The Magnetic Field 430 

III. Theory of Magnetism 433 

IV. Terrestrial Magnetism . 435 



Chapter XVII. Electrostatics 

I. Electric Charges .... 
II. Electrostatic Induction 
III. Electric Potential .... 
IV. Electric Capacity and Condensers 
V. Atmospheric Electricity 


. 441 
. 450 
. . . .454 

. . .460 
470 



Chapter XVIII. Electric Currents 



II. Electrolysis 
III. Ohm's Law and its Applications .... 
IV. Thermal Effects of a Current 
V. Electric Lighting 
VI. Thermoelectromotive Force . . . . . 


. 485 
. 494 
. 503 
. 508 
. 511 



viii CONTENTS 

Chapter XIX. Electromagnetism 

PAGK 

I. Magnetic Relations of a Current . . **-* 516 

II. Galvanometers 522 

III. Electrodynamics >.>v ^ . 525 

IV. Electromagnets . 530 

V. Magnetic Induction . . . . . ^ N* . . 533 

VI. The Magnetic Circuit .... - a . . 538 

Chapter XX. Electromagnetic Induction 

I. Faraday's Experiments 546 

II. Self-induction . . . . . . . . '" ' . 554 

III. The Induction Coil .556 

IV. The Telephone . . .- 559 

Chapter XXI. Dynamo-Electric Machines 

I. Direct Current Machines . . .... . . 563 

II. Alternating Current Machines ...... 572 

Chapter XXII. Electric Oscillations and Waves 

I. Electric Oscillations i . 583 

II. Electromagnetic Waves . . . . . . 588 

III. Some Relations between Light and Magnetism . . 592 

Chapter XXIII. Passage of Electricity through Gases 

I. Discharges in Partial Vacua . . . . " t . 595 

II. lonization of Gases ' ' 603 

III. Radioactivity .609 



COLLEGE PHYSICS 

MECHANICS 

CHAPTER I 

INTRODUCTORY 
I. MATTER AND ENERGY 

1. Reality of Physical Phenomena. In studying the facts 
of nature we assume the objective reality of physical phe- 
nomena and the existence of external objects apart from the 
mind of the observer. While we become acquainted with 
the physical universe solely through our senses, it will prob- 
ably be admitted that every form of matter, such as a pebble, 
a drop of dew, and the oxygen of the air, has objective exist- 
ence. The supreme test of physical reality is the fact that 
the material world remains unchanged in quantity in what- 
ever way it is measured. From this point of view, only two 
classes of things or entities are found in the physical world, 
matter and energy. 

2. Matter. The ultimate nature and structure of matter 
are not known with any degree of certainty. The discovery 
of radium, and its spontaneous disintegration into substances 
of simpler constitution, such as helium, and of the genera- 
tion of heat by its disintegration, have profoundly modified 
the older views of the constitution of matter. Matter is 
often denned as something occupying space, but it is much 
better described by its properties. 

A limited portion of matter is a body, and different kinds 
of matter, having distinct properties, are called substances. 

1 



2 MECHANICS 

A copper coin, a drop of rain, the air in a pneumatic tire, are 
bodies. Copper, water, and air are substances, since each 
has properties distinguishing it from all others. 

3. Energy. There is every reason to believe that the mo- 
tion of a body cannot be altered unless motion is imparted to 
it by another body or system of bodies by some such method, 
for example, as collision. It is in accordance with the usual 
mode of expression in such cases to say that the second body 
does work on the first. Energy may be defined as the capacity 
for doing work, where work may be considered as the act of 
changing the motion, the relative position, or the chemical 
and physical constitution of another body against a resistance 
opposing the change. Thus, work is done in winding a 
clock by coiling a spring against the reaction of bending, or 
by lifting a weight against the resistance of gravity. The 
coiled spring or the lifted weight then possesses energy, and 
it may in turn do work by giving motion to the pendulum 
and keeping it swinging against friction and the resistance 
of the air. 

Energy, like matter, has the property of conservation, and 
must be regarded as having real existence. It may be trans- 
formed and passed on in an endless series of changes, but it 
remains unchanged in amount. 

4. Definition of Physics. Physics is the science which 
treats of the related phenomena of matter and energy. It 
must not be assumed that physics is sharply differentiated 
from the other physical sciences by well-defined boundaries. 
A branch of science is classified in accordance with its chief 
aims. Chemistry, for example, endeavors to ascertain how 
the so-called elements enter into the composition of bodies, 
and the laws governing those changes in matter which affect 
its properties and identity. The combustion of carbon, the 
rusting of iron, the conversion of limestone into lime by 
heat, the fermentation of wine, are all changes in the con- 
stitution of the individual molecules of the several bodies. 



PROPERTIES OF MATTER 3 

They are therefore chemical changes. At the same time 
all of them involve changes in the associated energy, and 
this fact gives them a physical character. The old dis- 
tinctions between physics and chemistry are highly artifi- 
cial, and the discovery of new phenomena has the effect of 
making those distinctions obsolete. In fact, so obscure is 
now the border line between them that a new subdivision, 
called physical chemistry, which lies partly in the one field 
and partly in the other, is recognized as a distinct branch 
of science. 

II. PROPERTIES OF MATTER 

5. General and Special Properties. The properties of mat- 
ter are the qualities which serve to describe it and to define 
it provisionally. These properties are either general, that is, 
common to all kinds of matter in whatever state it may 
exist; or special, those distinctive of some kinds of matter 
and conspicuously absent in others. A property shared by 
all kinds of matter alike is extension, which means that every 
body occupies space or has dimensions. Another general 
property is impenetrability, which means that two bodies 
cannot occupy the same space at the same time. 

On the other hand, a piece of clear glass lets light pass 
through it, or is transparent; while a piece of sheet iron 
does not transmit light, or is opaque. A watch spring 
recovers its shape after bending and is elastic; while a strip 
of lead possesses this property in so slight a degree that it 
is classed as inelastic. Extension and impenetrability are 
general properties of matter, while transparency and elas- 
ticity are special properties. 

6. Inertia. The most general and characteristic property 
exhibited by all matter is inertia. In fact, inertia is the 
only property inherent in matter which has to do with mo- 
tion. Inertia is the persistence of matter in its state of 
either rest or motion, and its resistance to any attempt to 



MECHANICS 




Fig. I 



change that state. If a moving body stops, its arrest is 
always due to some influence outside of itself ; and if a body 
at rest be set in motion, the motion must be imparted to it 
by some other body. 

Many familiar phenomena are 
due to inertia. When a fireman 
shovels coal into a furnace, he 
suddenly arrests the motion of 
the shovel and leaves the coal to 
move forward into the furnace 
by its inertia. Suspend a heavy 
body and strike it a blow with a 
light hammer; the hammer re- 
bounds as if the heavy body were 
fixed in space by its inertia. A 
smooth cloth may be snatched 
from under a dish almost with- 
out disturbing it. The persist- 
ence with which a spinning top 
maintains the direction of its axis of rotation is due to its inertia. The 
violent jar to a water pipe when a faucet is suddenly closed is accounted 
for by the inertia of the stream. Tall columns 
and the detached portions of chimneys are some- 
times twisted around by sudden gyratory earth- 
quake movements (Fig. 1). The sudden circular 
motion of the earth under the column leaves it 
behind, and the slower return motion carries the 
column with it. 

Suspend a heavy weight by a string, as in Fig- 
ure 2, using the same kind of string above and 
below the weight. A steady pull on the string 
at B will break the upper string because it car- 
ries the greater load. A sudden downward pull on 
B, however, will cause the lower string to break. 
On account of the inertia of the weight, the 
lower string breaks before the pull reaches the 
upper one. 

7. Mass. The mass of a body is the measure of its iner- 
tia, that is, of the resistance which a body offers to motion 
or change of motion. This resistance to motion must be 




PROPERTIES OF MATTER 5 

carefully distinguished from friction, resistance of the air, 
or any other opposition to motion or change of motion ex- 
cept the inherent resistance of inertia. Mass must not be 
confused with weight ( 55), because mass is not dependent 
on gravity. The mass of a meteoric body is the same when 
flying through space as when it strikes the earth and embeds 
itself in the ground. If it could reach the center of the 
earth, its weight would be zero; at the surface of the sun, 
it would weigh nearly twenty-eight times as much as at 
the earth's surface; but its mass would be the same every- 
where. 

8. Porosity. Sandstone, hardened plaster of Paris, un- 
glazed porcelain, and many other bodies absorb a consider- 
able quantity of water without appreciable change in volume. 
The water fills the interspaces which are visible either to 
the naked eye or under a microscope. These interspaces, or 
pores, whether visible or invisible, are not a part of the space 
occupied by the body, and therefore interpenetration is pos- 
sible. All matter is probably porous in structure, and the 
corresponding property is called porosity. 

We may regard all bodies as built up of a number of very small masses 
called molecules. In gases these molecules are separated by relatively 
large spaces, while even in solids and liquids the molecules are not in 
actual contact. Hence the possibility of interpenetration and compres- 
sion. If, for example, 50 cm. 3 of alcohol be mixed with 50 cm. 3 of water, 
the volume of the mixture will not be 100 cm. 3 , but only about 97 cm. 3 . 
Interpenetration takes place to a certain extent. In a famous experi- 
ment of the Florentine Academicians, a hollow sphere of heavily gilded 
silver was filled with water and subjected to pressure. The water exuded 
through the pores of the silver and gold and stood in beads on the sur- 
face. Francis Bacon observed the same phenomenon with a lead sphere. 

9. Elasticity. Stretch a rubber band, bend a bow or a 
knitting needle, compress a tennis ball, twist a steel wire. 
In each case the form or volume has been changed, or both, 
and the body has been strained. A strain is either a change 
of shape or a change of volume due to a distorting force, 



6 MECHANICS 

called a stress. The property of recovery from a strain when 
the stress is removed is called elasticity. When a body re- 
covers its form after release from a force of distortion, it has 
elasticity of form ; it has elasticity of volume when the tempo- 
rary distortion is one of volume. 

10. Plasticity. The inability of a body to recover from 
distortion produced by a stress is called plasticity. In so far 
as bodies are not elastic they are plastic, and even elastic 
bodies are plastic beyond the limits where they cease to be 
perfectly elastic. Plastic bodies require force to change their 
shape, but the continued application of a stress to maintain 
the change is not necessary. Bodies are classed as elastic if 
they have a large limit of elasticity, and plastic if their limit 
of elasticity is small. A bar of lead will vibrate to a certain 
extent if struck, showing that it is somewhat elastic; but its 
elastic limit is so small that it is classed as a plastic body. 

11. Cohesion. The particles or molecules of a body are 
held together by cohesion. Cohesion unites the molecules 
together throughout the mass, whether the molecules be like 
or unlike. Adhesion unites bodies by their adjacent surfaces. 
Cohesion not only holds together the molecules to form a vis- 
ible mass, but it resists a force tending to break or crush a 
body. Welding consists in bringing clean metallic surfaces 
into intimate molecular contact by heating and hammering 
so that they cohere. When a clean glass rod is dipped into 
water and then withdrawn, a drop adheres to its surface. 
Two freshly cut surfaces of lead will adhere if brought into 
close contact by pressure. 

12. Tenacity. Tenacity is the resistance which a body 
offers to being torn asunder. It is determined by finding 
the weight necessary to break it in the form of a round wire. 

Tenacity diminishes with the duration of the pull. A smaller force 
applied for a long time will often break a wire which would not be 
broken by a larger force of short duration. 



PROPERTIES OF MATTER 7 

Lead has the least tenacity of all solid metals, and cast steel the great- 
est ; yet the latter is exceeded by fibers of silk and cotton. Single fibers 
of cotton can support millions of times their own weight. The tenacity 
of wood is greater along the fibers than across them. 

13. Ductility and Malleability. Ductility is the property 
of a substance which permits it to be drawn into wires or 
filaments. Platinum is the most ductile of all metals. 
Wollaston drew a wire of it only 0.00003 of an inch in 
diameter. A mile of this wire would have weighed only 
1.25 grains. 

Some substances become highly ductile only at high temperatures. 
Molten glass has been spun into a thread less than 0.0001 of an inch in 
diameter; a mile of it would weigh only a third of a grain. Professor 
Boys has drawn fibers from white-hot quartz not more than 0.00001 inch 
in diameter. Such quartz threads have a tenacity approaching that of 
steel. 

Malleability is a modification of ductility which permits 
some metals to be hammered or rolled into thin sheets. 

Pure gold is more malleable than any other substance. It has been 
hammered between skins to a thickness of only 0.00003 inch. Other 
metals possess this property, but to a less degree. Zinc is malleable when 
heated to a temperature of from 100 to 150 C. and it can then be drawn 
into wire or rolled into sheets. At 210 C. it again becomes brittle and 
can be crushed to powder. Nickel at red heat can be worked like 
wrought iron. 

Articles cast from pig iron may be made somewhat malleable by heat- 
ing for several days in contact with an agent, such as hematite, which 
removes some of the carbon from the cast iron. 

14. Hardness. Hardness is the resistance offered by a 
body to scratching or abrasion. It is a relative term only. 
Diamond is the hardest of all bodies because it scratches all 
others and is not scratched by any. It can be ground only 
by its own powder. Diamonds subjected to great hydraulic 
pressure between mild steel plates completely embed them- 
selves in the metal. Hard bodies, such as emery and car- 
borundum, are used as polishing powders and for grinding 
metals. 



8 MECHANICS 

Steel becomes very hard and brittle when suddenly cooled from a high 
temperature. The process of tempering consists in the subsequent grad- 
ual reheating till the hardness is diminished to the required extent, as 
indicated by the color of a polished surface. Cutting instruments are 
made of tempered steel. An alloy of four parts copper and one part tin 
is ductile and malleable when rapidly cooled, but hard and brittle when 
cooled slowly. 

15. States of Matter. Matter exists in three distinct states 
of molecular aggregation, the solid, the liquid, and the 
gaseous. All three are exemplified by water, which may be 
in the condition of ice, water, or water vapor. The differ- 
ence is not one dependent entirely on temperature. All three 
forms may exist in contact at the same temperature as ice, 
ice-cold water, and vapor of water. 

A solid has independent form and volume, and resists any 
stress tending to alter its shape or size. 

A liquid has volume but no shape of its own. It is mobile 
and conforms to the shape of the containing vessel. It offers 
but slight resistance to any stress except one tending to 
change its volume. 

A gas has the distinctive property of indefinite expansion. 
It has neither independent form nor independent volume, but 
completely fills the vessel containing it. 

The term fluid applies both to liquids and gases. 

While solids, liquids, and gases may be distinguished as above, there 
are nevertheless some substances which are neither wholly in the one state 
nor the other. Sealing wax passes imperceptibly by the aid of heat from 
the solid state to the liquid. Shoemaker's wax so far resembles a solid 
that it will break into fragments under the blow of a hammer, and yet 
under long-continued stress it flows like a liquid and can be molded at 
will. 

In addition there is " the critical state," in which the properties exhib- 
ited by a substance do not determine conclusively whether it is a liquid 
or a gas. 

16. The Constitution of Matter. The theory that matter is 
not infinitely divisible is one of great antiquity. According 
to this theory, matter is made up of indivisible parts called 






PHYSICAL MEASUREMENTS 9 

atoms. The terra molecule has long been applied to the 
smallest subdivision of a substance which can exist by itself 
and still exhibit the properties of that substance. Thus a 
molecule of water is composed of a group of atoms, one of 
oxygen and two of hydrogen. When this group is broken 
up, the parts are distinct atoms of oxygen and hydrogen gas, 
and not water. This theory of the constitution of matter has 
prevailed for a very long period, and has served useful pur- 
poses in science. 

If this theory were the ultimate truth, then the molecule 
would not be divisible without loss of physical and chemical 
properties, and the atom of one kind of matter could not be 
resolved into atoms of any other kind. The so-called "ele- 
ments " of chemistry would then be the ultimate constituents 
of matter. 

The phenomena of electrical discharges through rarefied 
gases have shown that a molecule may have dislodged from 
it very minute particles, or carriers of negative electricity, 
called electrons ; and that these Electrons have a mass only 
about Y^Vo t ^ ia ^ ^ the hydrogen atom. 

Moreover, the investigation of radium has le.d to the dis- 
covery that it disintegrates spontaneously into simpler and 
lighter atoms of other elements, notably helium. Hence it 
now appears entirely plausible that there is but one kind of 
" stuff " in the physical universe, perhaps hydrogen, and that 
all other kinds are fashioned from this ultimate element. 
While therefore atoms and molecules still exist, the concepts 
attaching to these terms are now very different from those 
which have prevailed for a long time. 



III. PHYSICAL MEASUREMENTS 

17. Units. Every physical quantity considered in modern 
Physics has a definite magnitude, and to measure it a 
certain fixed amount of the same kind of physical quantity 
is employed as the unit of measurement. Thus, to measure 



10 MECHANICS 

a length, some standard length, such as a foot, is taken as the 
unit, and the process of measurement consists in finding 
how many times this unit is contained in the given length. 

The expression for the magnitude of any physical quantity 
always consists of two parts. One of these is the name 
of a certain quantity of the same kind as the quantity to 
be measured, which is taken as the standard or unit; the 
other is merely numerical and expresses the number of 
times the unit must be applied to make up the quantity 
measured. For example, (150) (feet), (50) (grams), (30) 
(seconds). 

Further, the numerical parts of two expressions for the 
same quantity in different units are inversely as the magni- 
tudes of the units employed. Thus, a certain smokestack 
is either 150 feet high, or 50 yards high ; and since the yard 
is three times the foot, the numeric in yards is one third as 
great as in feet. 

Since every physical quantity must be measured in terms 
of a unit of its own kindffchere are as many units as there 
are different kinds of physical quantities to be measured. 

18. Fundamental and Derived Units. It has been found 
that nearly all the units for physical measurement may be 
denned in terms of three others, which are arbitrarily 
chosen. The three generally employed for the purpose are 
the units of length, mass, and time. These are called funda- 
mental units to distinguish them from all others, which are 
called derived units. The system almost universally used in 
physical investigations employs the centimeter for the unit 
of length, the gram for the unit of mass, and the second for 
the unit of time. This system is accordingly known as 
the c. g. s. (centimeter-gram-second) system. The centi- 
meter, the gram, and the second are then the fundamental 
units; the square centimeter as the unit of area, the cubic 
centimeter as the unit of volume, and the centimeter per 
second as the unit of velocity are examples of derived units. 



PHYSICAL MEASUREMENTS 11 

19. Units of Length. In the metric system the standard 
unit of length is the meter (m.). It is the distance between 
two transverse lines on a bar of platinum-iridium, at C. 
(centigrade scale), constructed by the International Metric 
Commission and preserved in the vaults of the International 
Bureau of Weights and Measures at Sevres, a suburb of 
Paris. This is called the international prototype. The centi- 
meter (cm.) is the one hundredth part of the meter. The 
only multiple of the meter in common use is the kilometer 
(km.), equal to 1000 meters. It is the unit employed on 
the continent of Europe for such distances as we express 
in miles. One kilometer is equal to 0.6214 mile. 

The United States possesses two national prototype stand- 
ard meters constructed by the International Commission 
and preserved at the Bureau of Standards in Washington. 

By Act of Congress in 1866, the use of the metric system of 
weights and measures became lawful in the United States, 
and the weights and measures in common use were defined 
in terms of those of the metric system. By this same act 
the legal value of the yard in the United States is f ff f of a 
meter; conversely the meter is equal to 39.37 inches. The 
inch is 2.540 cm. 

The unit of length in the English system for the United 
States is the yard, defined as above. In Great Britain it 
is the distance between the transverse lines in two gold 
plugs in a certain bronze bar at 62 F., preserved in the 
Standards' office, Westminster. One third of the yard is 
the/00, and one thirty-sixth is the inch. 

20. Units of Mass. The unit of mass in the metric 
system is the kilogram. It is the mass of the international 
prototype kilogram preserved with the international pro- 
totype meter at Sevres. The gram (gm.) is the one 
thousandth part of the kilogram. The kilogram (kgm.), 
was originally designed to represent the mass of a cubic 
decimeter (liter) of pure water at 4 C., the temperature of 



12 MECHANICS 

the greatest density of water. For practical purposes this 
is the mass of a kilogram; and the gram is the mass of a 
cubic centimeter of water at the same temperature. 

Two of the national prototypes, constructed of platinum- 
iridium by the International Metric Commission, are the 
standard kilograms for the United States. They are also 
preserved at the Bureau of Standards in Washington. 

The standard unit of mass in the English system is the 
avoirdupois pound. The ton of 2000 pounds is the chief 
multiple in the United States; its submultiples are the ounce 
and the grain. The avoirdupois pound is equal to 16 ounces, 
and to 7000 grains. The coinage of the United States is 
regulated by the " troy pound of the mint," containing 5760 
grains. By the law of May 16, 1866, the weight of the 5- 
cent nickel- copper piece was fixed at 5 grams; and by the 
law of February 12, 1873, the weight of the silver half dol- 
lar was fixed at 12.5 grams. 

In accordance with the International Postal Convention, the metric 
system of weights was " adopted for international postal relations to the 
exclusion of every other system." The revised statutes of the United 
States for 1872 contain the clause that " fifteen grams (of the metric 
system) shall be the equivalent for postal purposes of one half ounce 
avoirdupois." The interchange of mail by all civilized countries repre- 
sents the most extensive use of a uniform system of weights in the world. 

21. The Unit of Time. The unit of time in universal use 
is the second of mean solar time. An apparent solar, or sun- 
dial, day is the interval between two successive transits of 
the sun's center across the meridian of any place. But the 
apparent solar day varies in length from day to day by reason 
of the varying speed of the earth in its orbit and the incli- 
nation of its axis to the orbit. Hence the average length 
of all the apparent days throughout the year is taken as the 
length of a mean solar day. This day is divided into 86,400 
equal parts, each of which is a second of mean solar time. 

Mean solar or clock time agrees with sundial time on April 15, June 
14, September 1, and December 24. At other times the difference be- 



PHYSICAL MEASUREMENTS 



13 



tween noon as indicated by a mean-time clock and a sundial is called the 
equation of time. It is the correction which must be applied to apparent 
time to get mean time. The maximum value of this equation of time may 
amount to plus 14 min. 32 sec. and minus 16 min. 18 sec., the dates and 



+10" 


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+ 5"' 


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\ 








/ 


7 J 


F 


M 


A 


< 


j. 


/ J 


A 


S v 


\ 


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-10" 1 




















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-15'" 



Fig. 3 

amounts varying slightly from year to year. The curve in Figure 3 rep- 
resents the equation of time for the year. The positive ordinates of the 
curve mean that the mean-time clock is ahead of the transit of the sun. 
The astronomical unit of time is the sidereal day. It is the interval 
between two successive transits of a star across the meridian. Since the 
diameter of the earth's orbit is small compared with the distance of a 
fixed star, the line joining the earth and a star always remains parallel to 
itself. Hence the sidereal day represents the period of the rotation of 
the earth on its axis. A sidereal day is equal to 23 hours, 56 minutes, 
4.09 seconds of mean solar time; that is, the sidereal day is nearly 4 
minutes shorter than the mean solar day. 

22. Units of Angular Measure. The unit of angular measure 
commonly used in numerical calculations is the degree of arc. 
The circumference of a circle corresponds to 360 degrees, and 
a right angle to 90 degrees. A degree is divided into 60 min- 
utes of arc, and a minute into 60 seconds of arc. The 
degree of arc is a relic of an ancient sexagesimal system, 
consisting of multiples of 60. 

The other system of measuring angles is the circular meas- 
ure, in which the unit is the radian. The radian is the angle 
subtended by an arc equal in length to the radius of the circle. 
If the length of the arc is 8 and the radius of the circle r, the 




14 MECHANICS 

angle subtended by this arc is s/r. If the radius is unity, 
then the length of the arc measures the subtended angle in 
radians. Hence in circular measure 7r/2 is equivalent to 90, 
TT to 180, and 2 TT to 360. Since 

2 TT radians = 360, 1 radian =360/27r= 57. 2958. 

23. Trigonometrical Functions. The expression of physical rela- 
tions quantitatively is greatly facilitated by the use of a few simple 

trigonometrical relations. It will be 
convenient to define them here for 
students unfamiliar with the elements 
of plane trigonometry. 
~~C Let ABC (Fig. 4) be a right tri- 

angle, and let its sides be denoted by 
the letters a, &, c. Then the following terms may be defined as ratios : 

sine = - ; cosine 6 = - ; tangent = -. 
c c b 

It follows that a = c sin 6 ; b = c cos ; a = b tan 0. 

Also a 2 + 6 2 = c 2 (sin 2 + cos 2 0). And from the right triangle 
a 2 + 6 a = c 2 . 
Therefore, sin 2 + cos 2 - 1 . 

Also from the first definitions, 

5| = = tanft 
cos0 b 

When the angle 6 is very small, the side a may be identified with 
the arc subtending the angle, and the side b with the radius c. 

Then 8in0 = 2 = ; tan0 = - = tf; cos = 5=1.. 

c b c 

For small angles, the angle in radians is equal to either the sine or 
the tangent of the angle, and the cosine is unity. These .relations will 
be found of considerable utility in simplifying expressions involving 
these trigonometrical functions. 



CHAPTER II 

KINEMATICS 

I. RECTILINEAR MOTION 

24. Motion of a Material Particle. A material particle 
is an ideal body assumed to be without sensible dimensions. 
The limiting size of a material particle is relative only. 
In some astronomical problems, the earth and the planets 
may be treated as material particles, since their dimensions 
are negligible on a scale representing their distances from 
the sun ; on the other hand, in studying the phenomena of 
electric discharges through rarefied gases, it may not be 
permissible to regard the molecule even as a material particle. 

If a .material particle has relatively no sensible dimensions, 
its position may be denoted by a geometrical point. Now 
when such a particle is displaced from point to point, it 
must occupy in succession every point along the path of 
displacement, and time must be consumed in the operation. 
Motion is the process of change in the position of a material 
particle, considered as taking place during a definite interval 
of time. 

Kinematics is the science of the motion of a material 
particle without reference to its cause or to any physical 
actions that produce changes of motion. 

25. Types of Motion. In motion of translation any line 
joining two points in a body maintains the same direction ; 
that is to say, its successive positions are all parallel to one 
another. In addition to motion of translation, a body may 
spin or rotate about a definite line in the body, and this 

15 



16 MECHANICS 

motion is called rotatory motion. A material particle is 
capable of motion of translation only ; or, at least, its rota- 
tion is of no dynamical significance. In general the motion 
of an extended body is a combination of a rotation and a 
translation. 

Neglecting the curvature of the earth, a steamship sailing in a 
straight line is an example of motion of translation. ^The motion of 
the armature and pulley of a stationary electric motor is one of pure 
rotation. The wheel of a moving locomotive and a ball rolling along 
the floor of a bowling alley combine motion of rotation with motion of 
translation. 

The translatory motion of a material particle may be 
either rectilinear or curvilinear. The present section is 
restricted to the subject of rectilinear motion, or motion along 
a straight line. 

26. Velocity Uniform and Variable. Velocity is the time 
rate of motion of a body. By the time rate of motion is 
meant the distance traversed in a given time divided by that 
time. If a point moves over equal spaces in equal succes- 
sive time intervals, its motion is uniform and its velocity 
constant. The motion of a star across the field of view of a 
fixed telescope is an instance of uniform motion. The or >eed 
of a railway train may be constant for a considera jle 
distance. It may, for example, travel 88 feet for each 
second of an entire minute, or at the rate of a mile a minute. 

When a body traverses unequal spaces in successive equal 
periods, its motion is variable. The motfon of a falling 
body is variable, for it moves faster and faster as it descends. 
The velocity at any instant in variable motion is the distance 
the body would move in the next unit of time if at that 
instant its motion were to become uniform without other 
change. For example, the velocity of a shell as it leaves 
the muzzle of a gun is the distance it would pass over in 
the next second if it should continue to move uniformly 
without disturbance. The velocity of a falling body at any 
instant is the distance it would fall during the next second 



RECTILINEAR MOTION 17 

from that instant, if the attraction of the earth and the 
resistance of the air could be withdrawn. 

27. Formulae for Uniform Motion. Let v be the constant 
velocity of a body moving with uniform motion. Then, if 
the space s is passed over in t units of time, the velocity will 
be given by the relation s 

= t' 

From the same relation we have s = vt and t = -. 

v 

Even though the motion is not uniform, if the space 8 is 
passed over in time , the mean or average velocity is still 
given by equation (1). If both the space and the time be 
reduced to indefinitely small quantities, then the mean 
velocity becomes the actual velocity for the instant. 

The practical unit of velocity in the c. g. s. system is the 
velocity of one centimeter per second. 

28. Acceleration. Acceleration is the time rate of change of 
velocity. A case of special interest is one in which the 
change in velocity is the same from second to second. The 
motion is then one of uniform acceleration. If the velocity 
increases, the acceleration is positive; if it decreases, it is 
negative. When a heavy body falls, its gain in velocity per 
second is 9.8 m. for every second it falls. Its acceleration is, 
therefore, 9.8 m. per second per second; in other words, an 
increase in velocity of 9.8 m. per second is acquired in a 
second of time. This acceleration is the same as an increase 
in velocity of 588 m. per second acquired in a minute of 
time. 

If a railway train should start from rest and increase its speed one 
foot a second for a whole minute, its velocity at the end of the minute 
would be 60 feet a second. Since it would acquire in one second a 
velocity of one foot a second, and in one minute a velocity of 60 feet a 
second, its acceleration would be either one foot per second per second, 
or 60 feet per second per minute. Acceleration is expressed in terms 
of the fundamental units of length and time, the first power of a length 
and the negative second power of a time. 



18 MECHANICS 

Let VQ be the initial velocity at the instant from which 
the time is counted, v the final velocity at the time , and a 
the acceleration. Then by definition 

(2) 
From this equation v = t> 4- at> (3) 

The practical unit of acceleration is the acceleration of 
one centimeter per second per second in the c. g. s. system, 
or one foot per second per second in the English system. 

29. Formulae for Uniformly Accelerated Motion. Since the 
gain in velocity is constant, the average velocity is one half 
the sum of the initial and final velocities, and the space s 
passed over is the product of this average velocity and the 

(4) 

Substitute for v its value from equation (3), and 

n . . 

(5) 



Multiply together (2) and (4), and 



as = 




or v 2 = v 2 + 2 as. (6) 



If the initial velocity is zero, (3), (5), and (6) become 

v = at, (7) 

8 = i^ 2 , (8) 

v* = 2 as. (9) 

30. Scalar and Vector Quantities. A scalar quantity is one 
having magnitude only. A complete specification of scalar 
quantities is made by giving their numerical value in terms 
of the proper unit. Thus, 100' cm. 3 of water, 10 kgm. of 



RECTILINEAR MOTION 



sugar, 50 minutes of time, are all completely expressed scalar 
quantities. Volume, mass, time, density, energy, etc., are 
scalar quantities. 

A vector quantity has not only magnitude but direction, 
and a vector quantity is not completely expressed unless its 
direction is given as well as its magnitude. The difference 
between scalar and vector quantities becomes apparent when- 
ever it is necessary to add together two or more vectors. 
Thus, if a steamship is propelled by its screw 20 miles an 
hour, and at the same time is driven by the wind 5 miles an 
hour, the distance actually traveled in an hour is indeter- 
minate unless both directions are given in which the ship is 
driven. If one railway porter pulls on a truck with a force 
of 200 units and another with a force of 150 units, the total 
force applied to move the truck is indeterminate, both in 
magnitude and direction, unless the directions of both pulls 
are given. Displacement, velocity, acceleration, momentum, 
force, etc., are vector quantities. 

The addition of scalar quantities is effected by simple 
arithmetic ; the addition of vector quantities involves their 
directions and is effected by geometrical operations. It is of 
great advantage to represent vector quantities by straight 
lines, the length of the line denoting the magnitude of the 
vector, and the direction in which it is drawn, the direction 
of the vector. 

31. Addition of Vectors. 

Let two vector quantities, 
P and (), be represented by 
the lines AB and BO (Fig. 
5). A particle may suffer, 
for example, two successive displacements, the first repre- 
sented by the line AB, and the second by the line BC. 
Then the resultant displacement R, or the single displace- 
ment which would leave the point in the same position as the 
two successive displacements, is represented by the line AC. 




Fig. 5 



20 MECHANICS 

Or further, suppose the point to have two velocities at the 
same time, one represented by AB, and the other by AE. 
The actual resultant velocity is represented by the diagonal 
AO\ for if the point moves uniformly along the line AB in 
one second, and at the same time AB is carried parallel to 
itself the distance BO, the point B moving to (7, then the 
actual path of the moving point is A Q. Hence the vector 
sum of P and Q is represented by the third side R of the 
triangle ABO. The triangle is half of the corresponding 
parallelogram. Hence the following parallelogram law : 

If two vectors, which are applied to a material particle at 
the same time, are represented in magnitude and direction 
by two adjacent sides of a parallelogram, then their vector 
sum* will be represented in magnitude and direction by the 
diagonal of the parallelogram drawn through tl%e inter- 
section of these two sides. 

The magnitude of the resultant vector may be calculated 
from Figure 5 as follows : Let 6 be the angle between the vec- 
tors, that is, the angle through which Q must be turned to 
coincide with P. Then 



or W = (P + Q cos tf) 2 + 2 sin 2 0. 

= P 2 + 2 PQ cos + Q 2 ( 23). (10) 

This formula applies to all values of 0. Thus, if 6 = 0, 
cos 9 = 1, and R = P + Q ; if = 90, cos = 0, and IP = 
P 2 + #2. if Q = 180, cos 9 = - 1, and R = P - Q. 

32. Resolution of Vectors. A vector may be resolved into 
components in any given directions. This operation is the 
reverse of finding the vector sum. The given vector is 
assumed to be replaced by two or more component vectors, 
which are so chosen that the actual vector is the resultant of 
these assumed vectors. 

The component vector in any given direction is its value 
as a vector in that direction. The most common case is 



CURVILINEAR MOTION 21 

the resolution of a vector into two components in two direc- 
tions at right angles to each other. In most cases it suffices 
to find the component vector in the direction in which the 
attention for the time being is directed ; the other compo- 
nent at right angles to the first is obviously without effect 
in the directfon considered. 

To illustrate : Let it be required to resolve a velocity of 
8 m. a second into two rectangular components. 

First. Suppose one of the components is to be 3 m. per 
second ; find the other component. The problem is to con- 
struct a rectangle with a diagonal of 8 and one side 3, to 
find the adjacent side. w 

Draw BA (Fig. 6) 3 units in length, 
and at A draw AC perpendicular to 
BA. With B as a center and with a 
radius of 8 units, draw the arc mn 



cutting AC at 0. Complete the rec- Fj g- 6 

tangle ABDO. Then BA and BD are the two components 
of BO, and BD is the one required. Its value may be found 
from the right triangle BDO. 



= V8 2 - 8 2 =7.416. 

Second. It is required to find the component BD in a 
direction making an angle with the vector BO. From 23, 



If BO and the direction angle 6 are given, the problem is 
easily solved numerically. In general, a component vector in 
any direction {BD) is found by multiplying the given vector 
{BO) by the cosine of the direction angle (0). 

II. CURVILINEAR MOTION 

33. Uniform Circular Motion. Up to this point a velocity 
has been assumed to vary in magnitude only, and therefore 
the acceleration has been confined to the direction of motion. 



99 



MECHANICS 



B 



Fig. 7 




But a velocity may vary in direction as well as in magnitude. 
If a particle has a uniform motion in a straight line, its 
acceleration is zero in every direction. If its velocity 
changes in magnitude only, then its acceleration is positive 
or negative along the line of motion. But if the direction 
of motion changes, then the particle has at least a component 
acceleration at right angles to its path, and its motion is 
curvilinear. 

For example, if a particle moves uniformly along AB 
(Fig. 7), while from B to (7 it describes a curved path, then 

between these two points there 
is an acceleration normal to the 
path. 

In uniform circular motion, 
the velocity of the particle 
along the circumference of the 
circle is constant ; but since its 
rate of deflection from the tangent to the circle is constant, 
the acceleration is constant and is directed everywhere toward 
the center of the circle. If it were not toward the center, it 
could be resolved into two components, one toward the center 
and the other along a tangent to the circle ; the latter would 
mean a change of velocity in the circle itself. But the 
velocity in the circle is uniform, and there is therefore no 
tangential component of acceleration, or the acceleration is 
directed wholly toward the center. In 
other words it is centripetal. 

34. Centripetal Acceleration. Let ABO 
(Fig. 8) be the circle in which the point 
revolves, and AB the very small portion of 
the circular path described in the time t. 
Denote the length of the arc AB by s. 
Then, since the motion is uniform, s = vt. 

AB is the diagonal of a very small parallelogram AEBT), 
and AE is the distance through which the moving point 




PERIODIC MOTION 23 

is deflected toward the center while traversing the small 
distance AB. Since the acceleration is constant, we have, 
from equation (8) for uniformly accelerated motion, 

AE= I at 2 . 

The two right triangles ABE and ABO are similar, and 
therefore AE : AB = AB : A O. 

Whence AB 2 = AE x AC. 

Substitute in this equation the values of AB and AE 
above, and for AC the diameter 2 r, and the equation 

becomes 2 2 .. 2 

v 2 t 2 \ at 2 x 2 r = at 2 r ; 

v 2 
whence a = . 0-1) 



If ^ be the period of rotation of the point, 

and , = 



Substitute this value of # 2 in (11), and finally 



III. PERIODIC MOTION 

35. Definition of Periodic Motion. When a body goes 
through the same series of movements at regularly recurring 
intervals, its motion is said to be periodic. Thus the motion 
of the earth in its orbit around the sun is periodic. Its 
velocity is not uniform, but at intervals of a year it returns 
to the same value in the same direction. If the motion 
returns periodically to the same value, and in addition is pe- 
riodically reversed in direction, it is then vibratory or oscil- 
latory. The motion of a pendulum, of a violin string, or of 
the prong of a tuning fork, is vibratory. 



24 



MECHANICS 



36. Simple Harmonic Motion. Simple harmonic motion is 
the projection of uniform motion in a circle, either on a 

diameter or on a line in the plane 
of the circle. 

Suspend a ball by a long thread 
and set it swinging in a small hori- 
zontal circle (Fig. 9). It will 
travel round and round with uni- 
form speed, the thread describing 
the surface of a cone. Place a 
white screen back of the ball ; 
stand a few feet away, and with 
the eye on a level with the ball, 
watch the projection of the ball 
on the screen. The eye discerns 
the motion to the right and left 
of the line of sight, but not the 
motion toward the observer and 
away from him. The apparent 
motion of the ball is simple har- 
monic, or like the motion of a simple pendulum. 

Let the circle of Figure 10 represent the path of the ball, 
and ABCD, etc., its projection on the screen. When the ball 
moves over the arc adg, it appears to 
an observer far to the left of the figure 
to be moving from A through B, C, 
etc., to 6r, where it momentarily 
comes to rest. It then starts back 
toward A, at first very slowly, but 
with increasing velocity till' it passes 
D. Its velocity then diminishes, and 
at A is again zero and a reversal of 
the motion takes place. At a and g the actual motion of the 
ball is all in the line of sight, away from the observer at a 
and toward him at g. At k and d the ball is moving across 
the line of sight and the projected motion at D is the fastest. 




Fig. 9 




Fig. 10 



D 



PERIODIC MOTION 



25 



When a point vibrates to and fro along a straight line, as 
AGr, in such a manner that its position at any moment is the 
same as the projection on that line of a point moving uniformly 
in a circle whose diameter is the length of the straight line, 
its motion is simple harmonic. 

The circle adgk is called the circle of reference. Its radius 
is the amplitude of vibration. 

The period of the motion is the time of a double oscil- 
lation, or the time of a complete revolution of the auxiliary 
point around the circle of reference. The reciprocal of the 
period, that is, the number of vibrations per second, is called 
the frequency. For example, if the period is -^ of a second, 
the frequency is 10 double vibrations per second. 

Motion from left to right is positive, and from right to 
left negative. Displacement to the right of the middle point 
is positive, and to the left negative. 

The phase is the fraction of a period which has elapsed 
since the particle last passed through the middle point of its 
path in the positive direction. 

37. The Harmonic Curve. The harmonic curve, or " curve 
of sines," is the result of combining a simple harmonic motion 
(s. h. m.) with a uniform motion at right angles to it. In 

ABCDEFGH I JKLMNOPQRS 




\ 



Fig. II 

Figure 11 the equal distances between the vertical lines A, B, 
0, etc., represent the uniform motion along a horizontal line. 
The circle is the circle of reference with a radius equal to the 
amplitude of vibration. Its circumference is divided into 
any convenient number of equal parts (some multiple of four 



26 



MECHANICS 



is best), and through the points of division are drawn the 
horizontal lines cutting the vertical ones. The horizontal 
lines divide the verticals into spaces traversed by the par- 
ticle in successive twelfths of a period. 

If the particle is in the line A for the horizontal motion, 
and in the horizontal line through 8 on the circle of ref- 
erence for the s. h. m. motion, it must be at the intersection 
of the two at a. After one twelfth of a period it will be at 
6, etc. The desired curve is found by drawing a smooth 
curve through the successive intersections of the two sets 
of lines. 

Such a curve may be drawn experimentally by causing a large tuning 
fork to inscribe its vibrations on smoked paper fastened around a drum, 
which is rotated with uniform angular velocity, while a light tracing 
point attached to the fork inscribes a sine curve. 

It may also be drawn in a very simple way by means of a long flat 
strip of clear wood, securely mounted horizontally by one end so as 
to vibrate in a horizontal plane, and carrying at the other end a 
small camel's-hair brush saturated with ink. A long strip of paper 
attached to a narrow board is drawn as uniformly as possible against 
a guide parallel to the vibrating wood strip, and the brush marks the 
harmonic curve. The zero line is drawn in the same manner with the 
brush at rest. 

38. Acceleration in Simple Harmonic Motion. Let the aux- 
iliary point be at B (Fig. 12) in the circle of reference and 

moving toward Y. The corre- 
sponding displacement of the point 
executing simple harmonic motion 
along the diameter through X is 
0(7; denote it by x. The only 
acceleration present is the cen- 
tripetal acceleration a along the 
radius BO. The problem is to 
find the component of this accel- 
Fig - l2 eration in the direction of the 

harmonic motion along OX. Denote this component by a x . 
Acceleration is a vector, and by 32, the component of a 




PERIODIC MOTION 27 

vector in any direction is found by multiplying by the cosine 
of the direction angle 6. Therefore 



= a cos 



e. 



Q x TT a 4 7T 2 

But cos 6 - Hence a x = x ^ ' x - 

Since both a and r are constants, the acceleration along OX is 
proportional to the displacement x, and opposite in sense. 

The acceleration in simple harmonic motion is always di- 
rected toward the middle point of the path of the moving 
point ; and the proportionality of the acceleration to the dis- 
placement is the distinguishing characteristic of simple har- 
monic motion. 

Similarly the acceleration parallel to OY is y. The 

two accelerations differ in phase by a quarter of a period, for 
when a x is a maximum, a v is zero, and conversely. Obviously 
uniform circular motion is composed of two simple harmonic 
motions at right angles to each other, of the same period and 
amplitude, and differing in phase by a quarter of a period. 
If the amplitudes are not equal, the resulting motion is in 
an ellipse. 

To illustrate the composition at right angles of two oscillations of the 
same period, suspend a steel ball by a long, fine fishline so that the ball 
just clears the surface of a table. Set it swinging north and south, and 
strike it with a block of wood in an east and west line as it crosses the 
middle point of its swing. The resultant oscillation will be in a diagonal 
line between the north-south and east-west directions. The two oscilla- 
tions combined are in the same phase. 

Start the ball as before and strike it at right angles to its path as it 
reaches the extreme limit of its swing. If the blow is rightly gauged, 
the resulting motion will be sensibly circular. The two motions combined 
are nearly simple harmonic, of the same period and amplitude, and they 
differ in phase by a quarter of a period. 

39. Velocity in Simple Harmonic Motion. The accelera- 
tion in simple harmonic motion is greatest when the dis- 
placement is greatest, or at either limit of the swing. It 
declines from that point to the middle position, where it 



28 MECHANICS 


becomes zero. The velocity, on the other hand, is zero at 

either limit of the excursion of the oscillating particle and 
increases from that point to a maximum at the median 
position. The point starts from rest with the greatest 
acceleration, or rate of change of velocity, and its velocity 
increases all the way till it passes the middle point, but 
at a constantly decreasing rate. The gain in velocity for 
each equal increment of time is less and less as the point 
approaches the middle of its excursion, but becomes zero 
only as the point passes this median position, after which 
the velocity decreases at a constantly increasing rate up 
to rest at the other limit of displacement. 

Acceleration and velocity in simple harmonic motion 
stand in such a relation that when one is greatest the other 
is least. At either limit of motion, the acceleration is the 
same as the centripetal acceleration in the circle of reference, 
while the velocity is zero ; at the middle point of the motion, 
the velocity is the same as in the circle of reference, while 
the acceleration is zero. If, as in the last article, the 
acceleration is proportional to cos 6, the velocity is propor- 
tional to sin'0, since the sine and cosine of an angle are 
related as described above for acceleration and velocity in 
simple harmonic motion. 

40. Composition of Two Simple Harmonic Motions in the 
Same Direction. A. When the periods are the same: The 
simplest case of the composition of two s. h. m.'s along the same 
line occurs when the two motions have the same period. They 
may differ in phase and amplitude, but their resultant will 
always be simple harmonic and of the same period as that 
of the component motions. Their composition is readily 
effected graphically by means of the harmonic curve of 37. 

Let ABCDE (Fig. 13) be the harmonic curve correspond- 
ing to one s. h. m., the amplitude being IB and the period 
AE\ and let abode represent the other s. h. m. of amplitude 
Cc and of the same period as the first. Further, the second 



PERIODIC MOTION 



29 



s. h. m. is one quarter of a period behind the first in phase. 
Then the resulting displacements will be found by adding 
together the corresponding displacements of the two s. h. m.'s 
with their proper signs. Thus the resultant displacement at 






7 



\ 



A 









v 



Fig. 13 



the instant denoted by the point Gr is the sum of Q-p and GrP 
or GQ. At C it is Co, since the displacement of the curve 
ABCDE is zero. At H the two displacements are equal 
and of opposite sign; the resultant is therefore zero. The 





s* 

/ 




\ 


































/' 


/^ 


1 

E^ 


^ 














,~- 


^ 




"-x 














^v 


/- 






N 


V 

c ^\ 


c 




e 


/ 


**/ 


/- 


\ 


/ 


N N 


\ 


F 


/ 


^*~^ 


x 


^ 


A 








^ 


^ 


d^ 


/ 


/ 


/ 


E 




\ 


N *>^_ 


X 

_^,0S 


X 




__ - 


-- 


r/ 












\ 


^ 


D 


1> 
















^c 




^s 




s 













































Fig. 14 

dotted curve is the resultant due to the composition of the 
other two. It is harmonic, of the same period as the compo- 
nent curves, and intermediate in phase between them. 

-B. When the periods are not the same. If the periods of 
the two s. h. m.'s combined are not the same, the resultant 



30 MECHANICS 

curve will be periodic, but not harmonic. Moreover, its 
period will be longer than that of either of the combined 
s. h. rn.'s. Let ABCDE (Fig. 14) be the harmonic curve 
corresponding to one of the s. h. m.'s and Abode to another 
of different period and amplitude. The period of the first 
is AE and that of the second is Ae. The resulting displace- 
ments are obtained in the same manner as before by adding 
corresponding ordinates with their proper signs. The dotted 
curve is the result. Its ordinates are everywhere equal to 
the algebraic sum of the corresponding ordinates of the two 
component harmonic curves. The figure shows only half 
of the complete periodic curve. The other half is similar 
to this, taken in the reverse order, and all ordinates of 
opposite sign. 

The same harmonic curves are combined in Figure 15, but 
the scale is smaller, so that four periods of the one and five 



Fig. 15 

of the other are included in the diagram. This gives the 
complete resultant periodic curve. Its period is four times 
that of the slower oscillation and five times that of the 
quicker one. 

The composition of two s. h. m.'s of very nearly the same period in 
the same direction gives rise to a periodic curve which is everywhere 
very nearly a sine curve in form, but its amplitude increases and 
decreases periodically. The maxima occur when the component vibra- 
tions are the same in phase, and the minima when they are opposite in 
phase. From maximum to maximum one vibration gains exactly one 
period on the other. If, for example, the frequencies are as 24 to 25, 
a maximum occurs at every 25th period of the quicker vibration. 

The waxing and waning of the resultant amplitude when two s. h. m.'s 
of nearly equal periods are combined in the same direction explain the 
familiar phenomenon of beats in music. This will be illustrated in the 
part on Sound. 



PERIODIC MOTION- 







41. Composition of Two Simple Harmonic Motions at Eight 
Angles. The rectangular composition of two s. h. m.'s is 
easily effected graphically by means of two circles of refer- 
ence. Their periods may be the same, or they may be in any 
simple ratio. 

A. Periods equal. Let the radii of the two half circles 
ADG- and adg (Fig. 16) be the relative amplitudes of the 
two s. h. m.'s. Di- 

J_j ,/V 

vide the two cir- 
cumferences into 
the same number of 
equal parts, and 
through the points 
of division draw 
parallel horizontal 
and vertical lines 
respectively. The 
rectangle, whose 
sides are LM and 
LN, will contain all the figures 
resulting from the composition of 
the two s. h. m.'s with any phase 
difference. 

The spaces between the horizontal 
lines represent the distance traveled 
in equal fractions of a period for the s. h. m. of larger ampli- 
tude, OA ; and those between the vertical lines represent the 
distances traveled in the same equal intervals by the s. h. m. 
of smaller amplitude, oa. The intersection 0' of the lines 
through the centers of the circles corresponds to a phase 
difference of zero between the two component motions. 
Each point of intersection along O'j corresponds in this figure 
to a phase difference of one twelfth of a period. Thus, if i is 
one point of the resultant curve, the difference of phase is 
one sixth of a period ; if j is on the resultant curve, the phase 
difference is one quarter of a period. 




o 

Fig. 16 



32 



MECHANICS 



If we start with no phase difference, the moving point 
will be at 0' for both the horizontal and vertical motions. 
After one twelfth of a period, it will have advanced one 
division to the right and one upward, and the point satis- 
fying both these conditions is the opposite corner k of 
the small rectangle. Continuing in this way, it will be 
found that the point traces the diagonal MN of the large 
rectangle. 

If the phase difference is one quarter of a period, the 
starting point is j. One space to the left and one up gives 
I for the position after one twelfth of a period. Continuing 
in this way and passing a smooth curve through the suc- 
cessive diagonal corners of the small rectangles, the result- 
ant is the large ellipse, whose major and minor axes are the 
diameters of the two circles of reference. If at the same 
time the two amplitudes are equal to each other, the ellipse 
,, becomes a circle 

( 38). 

If the phase dif- 
ference is one 
twelfth of a period, 
the resultant mo- 
tion is in the 
smaller ellipse. If 
the periods of the 
two s. h. m.'s are 
not exactly equal, 
the resultant curve 
will pass through all 
the possible ellipses 
in succession be- 
tween the two diag- 
onal straight lines 
of the large rec- 
tangle as limits, the large ellipse being the intermediate form. 
During the passage from one of these diagonals over to the 



o 




X 



X 




f 



Fig. 17 






PERIODIC MOTION 



33 



other and back again, one of the component motions has 
gained a complete vibration on the other. 

B. Periods one to two. In Figure 17 the larger circle has 
twice as many divisions as the smaller one. If the spaces 
between the vertical and horizontal lines denote the dis- 
tances traveled in equal successive intervals of time for the 
two motions respec- A 
tively, then the pe- 
riods of the s. h. m.'s 
combined are as one 
to two. 

Wit]i no phase 
difference, the re- 
sulting curve is the 
figure 8 (the lemnis- 
cate), traced in the 

same manner as 

t/ 

already described. 
If the phase difference is a quarter 
of a period, the resultant curve is 
the parabola through f. For a 
phase difference of three quarters 
of a period, the curve will be a 
parabola with its vertex on the opposite side of the rectangle. 
If the periods are not rigorously as one to two, the two pa- 
rabolas are the limiting forms of the resultant curve, and the 
figure 8 is intermediate between them. 

0. Periods two to three. For this case the number of equal 
divisions of the circles of reference are as two to three 
(Fig. 18). By changing the phase difference one eighth of 
a period, the resultant curve passes from the full line figure 
to the dotted one. 

If in tracing the curve we 0ount the passages across the 
large rectangle, both vertically and horizontally, we can al- 
ways determine the ratio of the periods combined. Thus, 
the full line curve of Figure 18 is completed by two passages 




34 MECHANICS 

across vertically, and by three passages across horizontally. 
The two periods combined are therefore as two to three. 

Problems 

1. A railway train has a speed of 60 mi. an hour. What is its speed 
in feet per second ? 

2. A body moving uniformly in a circular path of 15 m. radius makes 
10 revolutions in 5 sec. Find the speed per second. 

3. If the radius of the earth is 4000 mi., what is the linear velocity 
per second of a point on the equator due to the earth's rotation on its 
axis? 

4. A railway train 430 m. long passes over a bridge 150 m. long at a 
speed of 45 km. an hour. How long a time does the train take to pass 
completely over the bridge ? 

5. A body starting from rest passes over a distance of 256 m. in 4 sec. 
What is its acceleration ? 

6. A body starts with an initial velocity of 300 m. per second. If it 
comes to rest in 1 min. 2.5 sec., find the uniform negative acceleration. 

7. A body has an initial velocity of 6 m. per second. Find its velocity 
at the end of 3 and 6 sec. respectively, if a equals 9.8 m. per second 
per second. 

8. A train running at 60 rni. an hour is stopped with uniform 
retardation in 44 sec. by the application of brakes. What is the 
retardation per second per second ? 

9. How far will a ball roll before coming to rest if its motion is 
uniformly retarded, its initial velocity being 15 m. per second and the 
duration of its motion 10 sec. ? 

10. A body is projected upward with any initial velocity, and t and t' 
denote the times during which it is respectively above and below the 
middle point of its path. Find the ratio t/t'. 

11. What is the final velocity of a body which passes over a distance 
of 144 m. in 1 min. with uniform acceleration: (a) when the initial 
velocity is zero ; (6) when the initial velocity is 10 cm. per second ? 

12. What must be the initial vertical velocity of a ball to return 
to its starting point in 8 sec. ? (The acceleration of gravity is 980 cm. 
per second per second.) 

13. What acceleration per minute per minute must a body have 
to acquire in 20 min. a speed of 20 mi. per hour ? 



PERIODIC MOTION 35 

14. At what angle with the shore must a boat be steered in order 
to reach a point on the other shore directly opposite, if the actual 
velocity of the boat directly across is 8 mi. an hour and that of the 
stream 4 mi. an hour? 

15. A locomotive driving wheel is 2 m. in diameter ; if it makes 
200 revolutions per minute, what is the average linear velocity of a 
point on the periphery? What is its greatest velocity? What is its 
least? 

16. How far will a body move along a horizontal plane from rest in 
-30 sec., if it has an acceleration of 3000 cm. per second per second? 

17. What is the acceleration of gravity g where a body falls 485 cm. 
in the first second ? 

. 18. A body moves along a horizontal plane with an acceleration of 
360 m. per minute per second. How far will it travel in the fourth 
second ? 

19. A body slides down a smooth inclined plane and has a velocity 
of 10 m. per second at the end of four seconds. How far will it slide 
in the next four seconds ? 

20. A body moves uniformly around a circle 40 cm. in diameter 
at the rate of 24 revolutions per minute. Compute the acceleration 
toward the center. 

21. A body moves around a circle with uniform velocity once a 
second and its centripetal acceleration is 1974 cm. per second per second. 
What is the diameter of the circle? 

22. A vector drawn east has a length of 30 cm. and one drawn north- 
east a length of 50 cm. Find their vector sum. 

23. An elastic rod is clamped at one end; when the other end is 
pulled aside 1 cm. and then released, it starts with an acceleration 
of 4 cm. per second per second. What is the period of its vibration? 

24. In the last problem what is the greatest velocity of the free end 
of the vibrating rod ? 

25. A man weighing 75 kgm. stands on the platform of an automatic 
weighing machine which is placed on the floor of an elevator. What 
will be the indicated weight of the man when the elevator starts to 
descend with an acceleration of 100 cm. per second per second, if the 
acceleration of gravity is 980 cm. per second per second? 



CHAPTER III 

DYNAMICS 
I. NEWTON'S LAWS OF MOTION 

42. Dynamics Defined. Up to this point the motion of a 
body has been considered in the abstract ; and although the 
motion has been assumed to vary in certain definite ways, no 
inquiry has been made into the cause of these variations. A 
further step must now be taken by making this inquiry. 
That branch of Mechanics which studies the effects of force 
in producing motion or change of motion of definite masses 
of matter is called Dynamics. 

43. Momentum. Before proceeding to a discussion of 
Newton's laws of motion, which outline the relations between 
force and motion, it is necessary to define two terms associ- 
ated with these laws. One of them is momentum. It is the 
product of the mass and the linear velocity of a moving body. 

Momentum = mass x linear velocity, or M mv. (13) 

In the e.g. s. system, the unit of momentum is the momen- 
tum of a mass- of 1 gm. moving with the velocity of 1 cm. 
per second. 

The change in the momentum of a body is proportional to 
the change in its velocity, since its mass is a fixed quantity. 
Hence the rate of change of momentum is proportional to the 
rate of change of velocity, that is, to the acceleration. 

44. Impulse. In estimating the effect of a force, the time 
during which it acts and its magnitude are of equal impor- 
tance. This effect is doubled if the magnitude of the force 
is doubled, or if the time it continues to act is doubled. 



NEWTON'S LAWS OF MOTION 37 

Impulse is the product of the magnitude of a force and the time 
it continues to act. 

Suppose a ball of 15 gm. mass fired frqm a rifle with a velocity of 
40,000 cm. a second. Its momentum would be 600,000 units. If a truck 
weighing 300 kgm. moves at the rate of 2 cm. a second, its momentum is 
also 600,000 units. The ball acquires its momentum in a fraction of a 
second, while force may have been applied to the truck for perhaps thirty 
seconds to give to it the same momentum. In some sense the effect of 
the force in giving motion to the ball is the same as that of the force re- 
quired to give the equivalent motion to the truck, because their momenta 
are equal. This equivalence is expressed by the term impulse, which is 
the same in the two cases. 

Forces of very short duration, like the blow of a hammer, were for- 
merly called impulsive forces, and their effect an impulse. The term im- 
pulse is now used in the more general sense as the product of a force and 
the duration of its action. 

45. Laws of Motion. Newton's laws of motion are to be 
regarded as physical axioms, incapable of rigorous experi- 
mental proof. They are axiomatic to those who have suf- 
ficient knowledge of physical phenomena to interpret their 
relations. The laws of motion must be considered as resting 
on convictions drawn from observation and experiment in the 
domain of physics and astronomy. 

The most powerful argument for the validity of the laws of 
motion rests on the fact that their application to the solution 
of problems in mechanics leads to results which always agree 
with those of observation. The time of a coming eclipse, for 
example, is calculated by assuming the truth of Newton's 
laws ; and the remarkable agreement between the calculated 
time and that subsequently observed confirms the laws. 

These laws as enunciated by Newton are : 

I. Every body continues in its state of rest or of uniform 
motion in a straight line, except in so far as it may be com- 
pelled by ijnpressed force to change that state. 

II. Change of motion is proportional to the impressed 
force, and takes place in the direction in which the force 
acts. 



38 MECHANICS 

III. To every action there is always an equal and con* 
trary reaction ; or the mutual actions of two bodies are al- 
ways equal and in opposite directions. 

By " change of motion " we are to understand change of mo- 
mentum, and by " impressed force ," impulse. 

46. Discussion of the First Law. This law is known as the 
law of inertia, since it states that a body persists in its con- 
dition, either of rest or uniform motion, unless it is compelled 
to change that state by the intervention of an external force. 
It is further true that a body offers resistance to any such 
change in proportion to its mass. Hence the term mass is now 
commonly used to denote the measure of the body's inertia. 

From this law we derive a definition of force, for the law 
asserts that force is the sole cause of change of motion. 

47. Discussion of the Second Law. The first law teaches 
that a change of momentum is due to impressed force. The 
second law points out what the measure of this force is. 
Maxwell restated it so as to make it read as follows : " The 
change of momentum of a body is numerically equal to the im- 
pulse which produces it, and is in the same direction." By a 
proper choice of units, impulse may be placed equal to the 
change of momentum produced, or 

Ft = mv. (14) 

Hence I'= =ma. (15) 

L 

The initial velocity at the instant when the force begins to 
act is here assumed to be zero, and the final velocity at the 
conclusion of the force action is v ; mv is therefore the change 
in momentum, and v/t is the acceleration. 

Force is then measured by the rate of change of momentum, 
or by the product of the mass and the acceleration produced. 

Not only is the change of momentum the measure of the 
force producing it, but it always takes place in the direction 
in which the force acts. Both are vector quantities and their 






NEWTON'S LAWS OF MOTION 39 

directions are the same. The composition and resolution of 
forces are therefore effected in the same manner as those of 
other vector quantities ( 31, 32). 

48. Units of Force. Two systems of measuring force are 
in common use, the gravitational and the absolute. The latter 
is usually in the c. g. s. system. The gravitational unit of 
force is the weight of a standard mass, as the pound of force, 
or the kilogram of force. Gravitational units are not strictly 
constant, but vary with the place on the earth's surface 
( 59). They are not suitable, therefore, for precise scien- 
tific measurements. 

The absolute unit of force in the c. g. s. system is the dyne. 
The dyne is the force which produces an acceleration of one 
centimeter per second per second in a mass of one gram. 

The dyne is invariable in value. The earth's attraction 
for a gram mass in New York is approximately 980 dynes, 
since at that place gravity will impart to a gram an acceler- 
ation of 980 centimeters per second per second. A dyne is 
therefore ^J-^- of a gram of force, and the numerical value 
of any force expressed in dynes is 980 times as great as in 
grams of force ( 52). Conversely, to convert dynes into 
grams of force, divide by the acceleration of gravity, 980. 
One kilogram of force is equivalent to 980,000 dynes. 

49. Graphical Representation of a Force. A force as a vec- 
tor quantity has both direction and magnitude ; in addition 
it is often necessary to know its point of application. These 
three particulars may be represented by a straight line drawn 
through the point of application of the force, in the direction 
in which the force acts, and as many units in length as there 
are units of force. If a line 1 cm. long stands for a force of 

1 dyne, a line 4 cm. long, in A B 

the direction AB (Fig. 19), Fig. 19 

will represent a force of 4 dynes acting in the direction from 
A toward B. Any point on the line AB may be used to 
indicate the point at which the force acts. 



40 MECHANICS 

Further, if it is desired to represent graphically the fact 
that the two forces act on a body at the same time, for 
example, one a force of 3 kgm. horizontally, and the other a 
force of 2 kgm. vertically, two lines 
are drawn from the point of applica- 
tion of the forces A (Fig. 20), one 3 
units long toward the right, and the 
other 2 units long on the same scale 
~~ ~*B toward the top of the page. The two 

Fig - 20 lines, AB and AC, represent the two 

forces in point of application, direction, and magnitude. 

50. How a Force is Measured. A method of measuring a 
force, based on the relation F = ma, consists in measuring 
the mass moved by direct counterpoise (weighing) and 
observing the acceleration imparted to it by the force to be 
measured. But there are serious practical diffi- 
culties in measuring the acceleration. 

The simplest method of measuring a force is by 
the use of an instrument known as a draw scale or 
spring balance, and in another form as a dyna- 
mometer. It consists essentially of a spring, to the 
free end of which is attached a pointer, arranged 
to move in front of a scale graduated in equal 
parts (Fig. 21). The form of the steel spring 
used is quite independent of the general principle 
that if two forces produce equal distortions of Ijie 
spring, the forces themselves are equal to each 
other. If a weight of 20 pounds be hung on the 
spring and the position of the pointer be marked, 
then any other 20 pounds of force, whatever its 
origin, and in whatever direction applied, will stretch the 
spring to the same point. If a man by lifting stretches a 
spring two inches, and if a weight of 400 pounds stretches 
the spring to the same extent, then the man lifts with a force 
of 400 pounds of force. 




NEWTON'S LAWS OF MOTION 41 

The dynamometer, or spring balance, may be graduated in 
pounds of force, in kilograms or grams of force, or in dynes. 

51. Discussion of the Third Law. The essential meaning 
of the third law of motion is that all action between two 
bodies is mutual. It is a stress, and a stress always has two 
aspects, or is a two-sided phenomenon. The word action in 
Newton's third law is used to denote one aspect of a stress, 
and the word reaction the other. 

Considered only with respect to one portion of a system 
of bodies a stress is called action; with respect to the 
remainder of the system it is called reaction. The third 
law states that these two phases of a stress are always equal 
and in opposite directions. 

The stress in a stretched cord pulls the two things to 
which it is attached equally in opposite directions; the 
stress in a compressed rubber buffer exerts an equal push 
both ways. The former is a tension; the latter, & pressure. 

The third law includes also what is sometimes called the 
conservation of momentum, and it may be expressed in modern 
phraseology as follows : 

In every action between two bodies, the momentum 
gained by tlie one is equal to that lost by the other, or the 
momenta in opposite directions are the same. 

When a bullet is fired from a gun, the momentum of the 
gun in one direction is equal to that of the bullet in the 
other. The 'velocities are not equal, but are inversely as 
the masses of the two. 

52. Centripetal and Centrifugal Forces. The acceleration 
in uniform circular motion has already been shown to be 
equal to v 2 /r ( 34). But force is the product of mass and 
acceleration (F=ma, 47); hence centripetal force is equal 
to mv 2 /r. It is the constant force in uniform circular motion 
that deflects a body from a rectilinear path, and compels 
it to move in a circle. 



42 MECHANICS 

But by the third law of motion action and reaction are 
equal ; hence the centripetal force has opposed to it the 
equal and opposite centrifugal force. The latter is the 
resistance which a body offers, on account of its inertia, 
to deflection from a straight path. We have then for either 
centripetal or centrifugal force the expression, 



If m is in grams, v in centimeters per second, and r in 
centimeters, F is expressed in dynes. If it is desired to 
express F in gravitational units, divide the value obtained 
for .Fby the acceleration of gravity ( 48). The result will 
be in grams of force or pounds of force, according to the 
units employed in equation (16). 

To illustrate : If a mass of 500 gm. is made to revolve in a horizontal 
circle, whose radius is 1 m., and with a velocity of 3 m. per second, the 
centrifugal force is 

j, = 600x800; =4g x 101 d and *5xlO* = 4592 Q( foree _ 

100 980 

Again, if a body having a mass of 5 pounds move in a circle of 10 
feet radius with a velocity of 25 feet a second, 

F= 5 x 252 = 9.72 pounds of force. 
10 x 32.15 

(The acceleration of gravity in feet per second per second is 32.15.) 

53. Illustrations of Centrifugal Force. The stress, whose two 
aspects are centripetal and centrifugal force, exists in the medium or 
parts of the structure connecting the revolving member with the center. 
If a cord be just strong enough to sustain a weight of x gm. of matter, 
then it can sustain a stretching force of 980 x dynes. If this cord be used 
to whirl a mass of m gm. in a horizontal circle, it will snap unless the 
velocity is such that mv 2 /r is less than 980 x dynes. 

The stress in the cord consists on the one side of the centripetal force 
required to deflect the mass m from the tangent to the circle ; on the 
other it is the centrifugal force of reaction which the body m exerts 
through the cord on the center. 



GRAVITATION 43 

Water adhering to the surface of a grindstone leaves the stone just as 
soon as the centripetal force, increasing with the velocity, is greater than 
the adhesion of the water. 

An automobile rounding a curve at high speed is subject to strong 
centrifugal forces, which act through the tires. The centripetal force 
consists solely of friction between the tires and the ground. If this fric- 
tion is insufficient, as on wet or icy ground, " skidding " ensues. In any 
case, rapid driving around curves brings to bear great lateral stresses on 
the tires. 

Centrifugal machines are used in chemical laboratories to separate 
crystals from the mother liquors, in sugar refineries to separate sugar 
crystals from the syrup, and in dyeworks and laundries to dry yarn and 
cloth rapidly. Honey is extracted from the comb in a similar way. 
When light and heavy particles in a mixture are whirled, the heavier 
ones tend toward the outside, are left behind in the rotation. Thus, the 
fat globules of milk, constituting the cream, are lighter than the liquid 
of the emulsion. Hence, when fresh milk is whirled in a dairy separator, 
the cream and the milk form distinct layers and are collected in separate 
chambers. 

In Watt's steam engine governor, the balls open outward with in- 
creasing speed, thereby actuating a suitable train of mechanism, which 
throttles the steam at the inlet valve. 

When a spherical vessel, containing some mercury and water, is rap- 
idly whirled on its axis, both the mercury and the water rise and spread 
in separate bands as far from the axis of rotation as possible, the mer- 
cury outside. 

The centrifugal force on a body may easily exceed its weight. If 
mi? 2 /980 r exceeds the weight of the body in grams, it will revolve in a 
vertical circle. Thus, a small open can, partly filled with water, may be 
whirled around in a vertical circle with an attached string without spill- 
ing the water. The " centrifugal railway " is similarly explained. 



II. GRAVITATION 

54. Free Fall of Bodies. The erroneous notion of the 
early philosophers that heavy bodies fall faster than light 
ones was first corrected by Galileo. He dropped various 
bodies from the top of the leaning tower of Pisa, and found 
that they fell to the ground in nearty the same time, what- 
ever their size or weight. The slight difference he observed 
he rightly ascribed to the resistance of the air. 



44 



MECHANICS 



Frictional air resistance is well illustrated by the " guinea 
and feather tube" (Fig. 22). If a small coin and either a 
feather or a pith ball are placed in the 
tube, the coin will fall to the bottom 
first when the tube is quickly inverted. 
But if the air is exhausted by a good 
air pump, the lighter object will fall as 
fast as the heavier one. In a perfect 
vacuum, all bodies at the same place on 
the earth's surface would show the same 
downward acceleration. 




Fig. 22 



The friction of the air against the surface of 
bodies moving through it leads to a limiting 
velocity. A cloud floats, not because it is lighter 
than the atmosphere, for it is actually heavier, 
but because the surface friction is so large in 
comparison with the weight of the minute 
drops of water, that the limiting 
velocity of fall is very small. 

When a small stream of 
water flows over a high preci- 
pice, it is broken into fine spray 
and falls slowly. Such is the 

explanation of the Staubbach fall at Lauterbrunnen in 
Switzerland. The precipice is 300 m. high, and the fall 
viewed from, the face resembles a magnificent transparent 
veil, kept in movement by currents of air. 

In a vacuum water falls like a solid. The " water ham- 
mer " (Fig. 23) illustrates this fact. In filling the tube the 
water is boiled till all the air is expelled just before the 
tube is sealed in a blowpipe flame. When such a tube is 
suddenly inverted, the water falls like a solid and strikes 
the glass with a metallic ring. The same phenomenon may 
be observed when steam condenses in the cold pipes of a 
steam-heating system. The condensation produces a par- 
tial vacuum, and the water under steam pressure flows into it with a water 
hammer effect. 

'55. Weight. All the experimental evidence goes to show 
that every mass of matter at any given place on the earth's 




Fig. 23 



GRAVITATION 45 

surface will attain, when falling freely in a vacuum, the same 
velocity in a second. The force due to the earth's attraction, 
called gravity, is then proportional to the mass. The force 
with which the earth attracts a body is known in science as 
weight. The acceleration of gravity is denoted by g. What- 
ever may be the local value of the acceleration of gravity, the 
equation of force, F = ma ( 47) takes the form for gravity 
in the absolute system, TP (17} 

where TFis weight, m mass, and g the acceleration of gravity. 
If the acceleration g varies from place to place, the local 
weight of a given mass varies in the same proportion. 

56. Direction of Gravity. The path described by a body 
falling freely is a vertical line. A line or plane perpendicular 
to it is said to be horizontal. The direction of the vertical 
at any point, which is nearly the direction in which gravity 
acts, may be determined by suspending a weight by a cord 
passing through the point. The cord suspending the weight 
is called a plumb line. The direction of the plumb line is 
perpendicular to the surface of still water. 

A beautiful experimental demonstration of this fact consists in sus- 
pending a small weight by a thread so that the weight hangs under the 
surface of darkened water. The image of the thread in the water as a 
mirror may be distinctly seen, and it is exactly in line with the thread 
itself. But the image of a straight line in a plane mirror coincides 
in direction with the line itself only when the line is perpendicular to 
the mirror. Hence the thread as a plumb line is perpendicular to the 
surface of the darkened water. 

Vertical lines drawn through neighboring points may be 
considered parallel without sensible error, for vertical lines 
100 ft. apart make with each other an angle of only one second 
of arc ; one second of arc is the angle subtended by a pinhead 
at the distance of about a quarter of a mile. At the poles of 
the earth and at the equator, the direction of gravity is that 
of the plumb line ; elsewhere there is a slight deviation from 
this line on account of the rotation of the earth on its axis. 



46 MECHANICS 

57. Newton's Law of Gravitation. The famous astronomer 
Kepler discovered the laws of planetary motion, but he 
left untouched the forces which determine the motion. It 
remained for Sir Isaac Newton to give a dynamical explana- 
tion of Kepler's laws, and to show that a stress between 
each planet and the sun, directly proportional to the product 
of their masses and inversely proportional to the square of 
the distance between them, would account for the planetary 
motions according to Kepler's laws. The law of universal 
gravitation enunciated by Newton is : 

Every portion of matter attracts every other portion, and 
the stress between them is proportional to tlie product of their 
masses and inversely proportional to the square of the dis- 
tance between them. 

The law expressed in symbols is 

__ v*jj^ *-*=. < 18 > 

where m and m f are the two attracting masses, d the 
distance between them, and Gr a proportionality factor or 
the constant of gravitation. For spherical bodies the dis- 
tance d in the law of gravitation is the distance between 
their centers. It is readily shown* that the attraction 
at any external point due to a sphere, either uniform in 
density ( 154) or made up of concentric layers each uniform 
throughout, is the same as if the mass of the sphere were 
collected at its center. By the attraction at a point is 
meant the attraction on unit mass at the point. 

58. Law of Gravitation applied to the Moon. Given the 

acceleration of gravity at the surface of the earth, the law of inverse 
squares gives the acceleration produced by the gravitational attraction 
of the earth at the distance of the moon. Let g' be this latter value, 
and let R be the distance of the moon, and r the radius of the earth; 
then y . g> f>2 . r z % 

Whence g' = g. 

* University Physics, Part II, Art. 121. 



GRAVITATION 47 

If the law of gravitation is true, g' should be equal to the centripetal 
acceleration of the moon in its orbit, for the attraction of the earth for 
the moon is the only force present to keep the moon in its orbit. 

From equation (12) the centripetal acceleration of the moon in its 
orbit is given by the equation 

4 7T R 

-fr 

where T is the period of the moon, or the lunar month. Then if the 
law of gravitation, is true, g' should be equal to a. 

The following are the necessary data for the approximate calculation 
of g f and a : 

R = 240,000 miles, 
r = 4000 miles, 

T= 27 da. 8hr. = 2,361,600 sec., 
g 32.2 ft. per second per second. 

gf = 32.2 ( 4000 ) 2 = 0.00894 ft. per second per second. 



Also = 4^x240,000x5280 = ^^ gecond d 

(2,361,600)2 

The agreement is good for the approximate data used, and it shows 
that the moon is attracted by the earth with a force which follows the 
law of universal gravitation. 

59. Law of Weight. Since the earth is flattened at the 
poles, it follows that the acceleration of gravity, and the 
weight of any body, increase in going from the equator 
toward the poles. If the earth were a uniform sphere and 
stationary, the value of g would be the same all over its 
surface. But the value of g varies from point to point 
on the earth's surface, even at sea level, both because the 
earth is not a sphere and because it rotates on its axis. The 
value of g at the equator is 978.1 and at the poles 983.1, both 
in centimeters per second per second. At New York it is a 
little over 980 cm. per second per second, or 32.15 ft. per 
second per second. 

The diminution of gravity at the equator on account of the rotation 
of the earth on its axis is easily calculated. The equatorial radius of 



48 MECHANICS 

the earth is 6378 x 10 5 cm. The period of rotation is a sidereal day 
of 86,164 sec. Hence 

fl = 4 7T 2 x 6378 x 



(86,164)5 



Since the centripetal acceleration varies inversely as the square of the 
period of revolution, it would equal g at the equator if the period of 
rotation of the earth were reduced to one seventeenth of a day. The 
apparent acceleration of gravity at the equator would then be reduced 
to zero, or bodies there would have no weight. 

60. Laws of Falling Bodies. The most familiar and im- 
portant example of uniformly accelerated motion, the formu- 
lae for which have already been given in 29, are pre- 
sented by falling bodies. Since the acceleration g is sensibly 
constant for small distances above the earth's surface, the 
formulae of 29 are directly applicable by substituting 
for a the specific acceleration g. Equations (7), (8), and 
(9) then become ^ = ^ (19) 



(20) 
i> 2 =2#s. (-21) 

If in (20) t is made one second, 8 = | g ; or the space de- 
scribed in the first second, when the body starts from rest, is 
half the value of acceleration of gravity. A body falls 
490 cm. the first second ; the velocity attained is 980 cm. 
per second, and the acceleration is 980 cm. per second per 
second. 

The following laws are embodied in the above equations : 

" I. The velocity attained by a falling body is proportional 
to the time of falling. 

II. The space described is proportional to the square of the 
time. 

III. The acceleration is twice the space through which a 
body falls in the -first second. 



GRAVITATION 49 

61. Projection Upward. When a body is thrown vertically 
upward, the acceleration is negative, and its velocity de- 
creases each second by g units (980 cm. or 32.15 ft.). The 
time of ascent to the highest point will be the time taken to 
bring the body to rest. If the velocity lost is g units per second, 
the time required to lose v units of velocity will be the quo- 
tient of v divided by ^, or 

t = -. (22) 

9 

If, for example, the velocity of projection vertically up- 
ward is 1960 cm. a second, the time of ascent, neglecting 
atmospheric resistance, is J ^J Q -, or 2 seconds. This is the 
the same as the time of descent again to the starting point. 

62. Center of Gravity. A body is conceived to be com- 
posed of an indefinitely large number of parts, each of which 
is acted on by gravity. For bodies of ordinary size these 
forces of gravity are parallel and proportional to the masses 
of the several small parts. The point of application of their 
resultant is the center of gravity of the body. This point is 
also called the center of mass and the center of inertia. 

If the body is of uniform density throughout, the position 
of its center of gravity depends on its geometrical figure 
only. Thus, the center of gravity (1) of a straight line is 
its middle point ; (2) of a circle or ring, its center ; (3) of a 
sphere or a spherical shell, its center ; (4) of a parallelogram, 
the intersection of its diagonals ; (5) of a cylinder, the mid- 
dle point of its axis. 

It is necessary to guard against the idea that the force of 
gravity on a body acts at its center of gravity. Gravity acts 
on all the constituent particles of a body ; but its effect is 
generally the same as if the resultant, that is, the entire 
weight of the body, acted at its center of gravity. 

63. Center of Gravity of a Triangle. A single example will 
serve to illustrate the geometrical method of finding the cen- 
ter of gravity of a figure uniform throughout. 




50 MECHANICS 

Let ABC (Fig. 24) be a uniform thin triangle. Draw a 
line from A to D, the middle point of the base. The triangle 

may be conceived to be made up of 
lines parallel to its base BC ; and 
since the bisector AD passes through 
the middle point of all these lines, 
the center of gravity of each line lies 
on AD, and therefore the center of 
gravity of the whole triangle also lies 
on it. 

Let CE be another bisector drawn 
from C to the middle point of the side opposite. For reasons 
similar to those above, the center of gravity of the triangle 
must lie also on the line CE. It must therefore be at the 
intersection of the two bisectors. 

The triangles DEGr and ACG are similar. But DE is 
one half of AC; therefore DGr is one half of AGr, or one 
third of AD. The center of gravity of a triangle is therefore 
on a line drawn from the apex to the middle point of the 
base and two thirds of the distance down. 

III. WORK AND ENERGY 

64. Work Defined. When a force acts on a body and pro- 
duces displacement in the direction in which it acts, the force 
is said to do mechanical work. 

Examples: Gravity does work on the weight of a pile driver, caus- 
ing it to descend ; steam exerts pressure on the piston of a steam engine, 
imparts motion against resistance, and does work ; a horse does work 
in pulling a wagon up an inclined roadway ; the electric current, by 
means of a motor, does work when it drives an air compressor and forces 
air into a compression tank. 

Unless the point of application of the force has a com- 
ponent motion in the direction of the force, no work is done. 
Thus, gravity does no work on a vessel moving over the 
level surface of the sea, because its motion is at right angles 
to the direction of the force of gravity ; the pillars support- 



WORK AND ENERGY 51 

ing a pediment over a portico do no work, though they 
support a weight and exert a force. The forces are balanced 
and there is no motion. 

The measure of mechanical work is the product of the 
force and displacement of its point of application in the di- 
rection of the force, or 

W= Fs. (23) 

Since force is the product of mass and acceleration ( 47), 
W= mas. (24) 

When the displacement produced makes an angle a with 
the direction of the force, 

W= Fs cos a. (25) 

This expression may be interpreted to mean either, (a) the 
product of the force (J 7 ) and the component of the displace- 
ment in the direction of the force (s cos a), or () the pro- 
duct of the displacement (s) and the component of the force in 
the direction of the displacement (.Fcos a). 

65. Units of Work. Three units of work are in common 

use: 

1. The foot pound, or the work done by a pound of force 
working through a distance of one foot. This unit is in 
general use among English-speaking engineers. 

2. The kilogram meter, or the work done by a kilogram of 
force working through a distance of one meter. This is the 
gravitational unit of work in the metric system. 

3. The erg, or the work done by a force of one dyne 
working through a distance of one centimeter. The erg is 
the absolute unit in the c. g. s. system and is invariable. 

Gravity gives to the gram a velocity of about 980 cm. a 
second. It is therefore equal to 980 dynes ; and if a gram 
mass be lifted vertically one centimeter, the work done 
against gravity is one gram centimeter, or 980 ergs. 

The mass of a "nickel" is 5 gm. The work done in lifting 
it vertically two meters is the continued product of 5, 200, 



52 MECHANICS 

and 980, or 980,000 ergs. The erg is therefore a very small 
unit, and it is more convenient to use a multiple for practical 
measurements. The multiple in common use is the joule. 

Its value is 

1 joule = 107 ergs = 10,000,000 ergs. 

Expressed in this larger unit, the work done in lifting the 
"nickel" is 0.098 joule. 

66. Graphical Representation of Work. Since work is the 
product of force and length, work may be represented nu- 
merically by an area. When the force is 
constant in value, the work done may be 
denoted by the area of a rectangle, one 
side of which is as many units in length 
~X as there are units of force, while the 
adjacent side is numerically equal to the 
displacement in the direction of the force (Fig. 25). 

If the force increases uniformly from zero to a final value 
jF 7 , then the work done is the product of the mean value of 
the force and the displacement. It may be represented by 
the area of a right triangle (Fig. 26), in which the base is 
the displacement in the direction in which the force acts, and 
the altitude the final value of F\ for the work then equals 
J Fs cos a, which is the expression for 
the area of the triangle. 

In many cases, as in the cylinder 
of a steam engine, the force, which is 
now the pressure of the steam, varies 
according to some law less simple. 
If p is the pressure per unit area of 

the piston, and A its area, then the whole pressure on the 
piston is P = pA. Let now the piston move through a very 
small distance x, so small that the pressure may be considered 
constant during this motion ; then the work done by the 
steam during expansion is 

w =pAx. 



cos a 



F 



-X 



WORK AND ENERGY 53 

But Ax is the increase in the volume of the steam, which 
may be denoted by v. Then 



or the element of work done during a very small movement of 
the piston is equal to the product of the pressure per unit 
area and the small change in volume. 

The whole work done during any considerable expansion of the steam 
may be represented by the area of the figure ABba (Fig. 27), in which 
the ordinates of the curve AB are the succes- y 
sive pressures, and the abscissas are the corre- 
sponding volumes of the steam. Take any 
small element of the area, as Aa. Then the 
length of this strip is the instantaneous pres- 
sure p, and its width is the indefinitely small 
change in volume. Its area is therefore the 




element w of work done, and the sum of all 6 a 

Fig 27 

such elements is the entire work done by the 

steam during the expansion from the volume denoted by the point b to 
that denoted by the point a. The sum is the area ABb'a. This principle 
is the one employed in the steam indicator diagram. 

67. Power. Power is the time rate of doing work. The 
unit of power commonly used by American and British en- 
gineers is the horse power; it is the rate of doing work equal 
to 33,000 foot pounds per minute, or 550 foot pounds per 
second. 

In the c. g. s. system the unit of power is the watt. It is 
the work done at the rate of one joule (10 7 ergs) per second. 
A kilowatt is 1000 watts. 

To convert a horse power into watts, multiply 550 by the 
ratio between a foot and a centimeter, then by the ratio be- 
tween a pound and a gram, and the product is in gram centi- 
meters. To find the equivalent in ergs multiply by 980. 
Then 550 x 30.4T97 x 453.59 x 980 = 746 x 10 7 ergs per sec- 
ond, or 746 watts. 

One horse power is therefore equivalent to 746 watts. A 
kilowatt (K. W.) is nearly one and one third horsepowers. 



54 MECHANICS 

To convert kilowatts into horse powers, add one-third ; to 
convert horse powers into kilowatts, subtract one fourth. 
For example, 90 K. W. equals 120 H. P., and 200 H. P. 
equals 150 K. W. 

The capacities of electric generators are now universally 
expressed in kilowatts ; electric motors and steam engines are 
also commonly rated in the terms of the same unit of power. 

The horse power, was originally determined by James Watt, and 
the average work per horse turned out to be 22,000 foot pounds per min- 
ute. For some reason Watt added fifty per cent and made the horse power 
33,000 foot pounds per minute. (Report of the Electrical Conference, 
Philadelphia, 1884.) If he had adopted the value found of 22,000 foot 
pounds per minute, one horse power would have been almost precisely 
one half kilowatt. 

68. Energy. Experience teaches that under certain well- 
defined circumstances bodies possess the capacity for doing 
work. Thus a body of water in an elevated position, air under 
pressure in a tank, steam under more than atmospheric pres- 
sure in a steam boiler, are all able to do work by means of an 
appropriate motor mechanism. In general, a body or sys- 
tem upon which work has been done is found to have an in- 
creased capacity for doing work. It is then said to possess 
more energy than before. Energy is the capacity for doing 
work. It is therefore measured in the same units as work. 

When a steam engine lifts the weight of a pile driver, it does work 
on it against the force of gravity. In its new position relative to the 
earth the weight itself has acquired the ability to do work ; for when it 
is released, it descends, overcomes the resistance offered by the pile, and 
.forces it into the ground. 

Work may be done on a storage battery by means of a steam engine 
and a dynamo machine. The charged battery has then conferred on it 
the capacity for doing work, because it is capable of furnishing an elec- 
tric current to run a motor. It may circulate the air by a fan, drive a 
printing press, run a street car, propel an electric launch, or operate the 
machinery of a factory. 

Consider examples of a different character. Work is done on a can- 
non ball by means of the pressure of the gases arising from the explosion 
of the powder. The ball acquires a high speed; and, as a result, it now 



WORK AND ENERGY 55 

possesses the capacity of overcoming resistance. By virtue of its mass 
and its motion it may demolish fortifications, or pierce the steel plates 
of a battleship. 

When steam does work on the piston of an engine, the heavy flywheel 
is made to revolve on its axis. Work is done on it in giving it motion of 
rotation. When the steam is shut off, the engine continues to revolve 
and does work by means of the rotatory effort of the massive flywheel. 

In all these cases while the acting agent is doing work on the body, 
energy is transferred from it to the body or system on which the work is 
done ; and the body or system of bodies which has acquired the capacity 
for doing work is said to possess energy. 

69. Potential Energy. Cases abound in which energy is 
stored in mechanical displacements, or in chemical and 
physical changes in a body. The first two illustrations of 
the last section belong to this class. In the air gun, a mass 
of air is compressed into a small volume by doing work on 
it, and it tends to recover its original dimensions ; if per- 
mitted to do so, it may be made to restore the work done on 
it by propelling a bullet. When a clock is wound by coil- 
ing a spring or lifting a weight, work is done on the system 
and energy is stored. This energy is recovered when the 
system returns slowly to its unstrained condition by the un- 
coiling of the spring, or the fall of the weight. The work 
done in bending a bow is quickly restored in imparting mo- 
tion to the arrow as the bow is relieved from the stress. 

In all such cases of the storage of energy a stress is always 
present. The compressed air pushes outward in the air 
gun ; the spring strives to uncoil in the clock ; the bent bow 
tends to unbend ; and the electric pressure of the charged stor- 
age battery is ready to produce a current as soon as the 
circuit is closed. Hence the energy thus acquired is called 
energy of stress, or more commonly potential energy. The 
energy ' of an elevated mass, of bending, twisting, defor- 
mation, of chemical separation, and of stress in the ether in a 
magnetic field are all cases of potential energy. 

70. Kinetic Energy. Bodies have capacity for doing work 
also in consequence of their motion. In the last two illus- 



56 MECHANICS 

trations of 68, the immediate and obvious effect of doing 
work on the body is to set it moving, but in reality energy 
is imparted to it. The energy which it acquires is called 
kinetic energy, or energy of motion. 

Whenever a meteoric body, flying through space, enters the earth's 
atmosphere, its energy of motion is converted into heat by friction with 
the air, and the heat generated raises its temperature till it glows like a 
star. It may even burn up or become impalpable powder. 

The invisible molecular motions of bodies constituting heat are in- 
cluded under kinetic energy no less than their visible motions. Heat is 
a form of kinetic energy. 

Kinetic energy must not be confused with force. A mov- 
ing mass of matter carries with it a definite quantity of 
energy, but it exerts no force until it encounters resistance 
or opposition. Energy is then transferred to the resisting 
or opposing body ; force is exerted only during this transfer. 

71. Kinetic Energy in Terms of Mass and Velocity. Suppose 
a force F to act on a body of mass m for an interval of time t ; 
then the measure of the effect is the impulse Ft. By the 
second law of motion impulse equals the momentum imparted. 
Assuming that the body m starts from rest and acquires in 
time t a velocity v, the momentum produced is mv. Hence 

k 

Ft mv. (a) 

A constant force gives rise to uniformly accelerated motion, 
and in this type of motion the mean velocity is half the sum of 
the initial and final velocities, or, in this case, \ v. The mean 
velocity is also the space traversed divided by the time, 

o 

or - Hence s - ,,. 

t - = %v. (5) 

Multiply (a) and (5) together, member by member, and 
the result is J^W. (26) 

But Fs measures the work done by the force F on the mass 
m to give to it the velocity v, while working through the 



WORK AND ENERGY 57 

space s ; and as the kinetic energy acquired by the body is 
measured .by the work done on it, it follows that the energy 
of the mass m moving with the velocity v is |- mv 2 . 

If m is expressed in grams and v in centimeters per second, the result 
is in ergs. To reduce to gram centimeters, divide by the value of g in 
centimeters per second per second, 980. If m is expressed in pounds and 
v in feet per second, to obtain the energy in foot pounds, divide by the 
value of tj in feet per second per second, 32.15. 

72. Energy changes Form. The energy of a body may 
change from potential to kinetic, and conversely. Suppose 
work has been done on a weight of mass m gm. sufficient to 
lift it to a height of h cm. against gravity. It then possesses 
potential energy equal to mgh ergs. If it is allowed to fall, 
it loses potential energy and gains energy of motion. After 
it has fallen a distance s, its velocity is given by the equation 
v 2 =2gs. Its kinetic energy is then ^ mv 2 = mgs. But its 
potential energy has been reduced to mg (h s), since h s 
is now its height above the point from which it was lifted. 
The sum of mgs and mg(h s) is mgh, the original potential 
energy. Whatever, then, the weight gains in kinetic energy 
as it falls, it loses in energy of the potential form. When 
the weight reaches the ground, the velocity acquired is given 
by the relation v 2 = 2 gh, and therefore | mv 2 = mgh, or the 
energy of the weight is now all kinetic and is the same as the 
potential energy possessed by the weight at the elevation h. 
During the fall the potential energy is continuously con- 
verted into the kinetic form, but in such a way that the sum 
of the two is a constant. 

When a pendulum is drawn aside, it is lifted and acquires 
potential energy. As it descends toward its position of 
equilibrium it acquires velocity, and at the lowest point of 
its path, its energy is all energy of motion. Its energy then 
gradually returns to the potential form, and at the extremity 
of the swing on the other side it is again all potential. If 
the pendulum could swing without friction and resistance of 



58 MECHANICS 

the air, this process of conversion of energy from the one 
form into the other and back again would continue indefi- 
nitely without loss. This would constitute a form of per- 
petual motion, but it would be one in which no energy is 
given out to other bodies. It is not the form sought after by 
the deluded. 

These forms of potential and kinetic energy of a mass of matter, 
such as a pendulum bob, are not the only ones assumed by energy. 
When the lifted weight falls and reaches the ground, its motion may 
be suddenly arrested, and its energy of visible mechanical motion 
disappears. What becomes of it? It is found that both the weight 
and the ground are warmed, and by a wide induction from similar 
cases, it is learned that heat is a form of energy. The same quantity of 
work, if entirely spent in producing heat, will always produce the same 
amount. 

When a ball strikes a target, heat is generated, and the ball may be 
partly fused. A flash of light is often noted and sound is produced. 
The molecular motions constituting heat, light, and sound represent 
kinetic energy. 

The heat and light produced by combustion are forms of kinetic 
energy, derived by transformation from the potential energy of chemical 
separation and chemical affinity. 

When a storage battery is charged, the energy is stored as the poten- 
tial energy of chemical separation. When it discharges, the kinetic 
energy of the current flowing is derived by transformation from the 
potential form. 

73. Conservation of Energy. The principle of the conser- 
vation of energy is a generalization from an extensive range 
of observations. It has been found that if a system of 
bodies within a given boundary, through which energy 
is not allowed to pass, has a certain amount of energy, 
this amount remains constant, whatever actions take place 
between the parts of the system, and whatever forms this 
energy may assume. 

If this system is brought into relation with other bodies, 
so as to form a larger system, in which the energy ij differ- 
ently distributed, then the entire energy of the two systems 
within the larger boundary remains invariable in amount. 



WORK AND ENERGY 59 

So if the boundary is extended to include the whole physical 
universe, the doctrine of the conservation of energy amounts 
to the assertion that the quantity of energy in the universe 
is fixed and invariable. 

It will be seen that this principle denies the possibility of 
any form of " perpetual motion " machine by which mechani- 
cal work can be done continuously without supplying the 
machine with equivalent energy in some other form. Every 
machine or device that works without interruption must 
receive from without at least as much energy as it expends, 
either continuously or by periodic additions. 

74. Availability of Energy. While the quantity of energy 
in the universe remains unaltered, its availability for the 
operations of nature and for purposes useful to man is not 
a constant. 

Whenever a transformation of energy occurs, especially 
from the kinetic to the potential form, some of it is inevita- 
bly wasted through conversion into heat. This conversion, 
or really degradation of energy, occurs through friction, 
radiation, the heating of electric conductors, or other analo- 
gous modes. This heat is gradually diffused, and diffused 
heat is no longer available as useful energy to do work. 

Not only this, but all the processes of nature exhibit 
changes of energy on the way from the more available to the 
less available form. Hence the quantity of available energy 
is never increased, but is always diminished in every physi- 
cal process, and it therefore tends toward zero. 

The constant dissipation of energy associated with every physical 
phenomenon leads to two interesting conclusions. In the first place, 
if we permit ouselves to inquire into the past history of energy, we shall 
inevitably arrive at a period in the past when none of it had been dissi- 
pated or had become unavailable. Before this period no physical phe- 
nomenon, like those with which we are acquainted, could have occurred, 
for every such phenomenon is attended with dissipation of energy. 

In the second place, unless some new order intervenes, of which we 
have no conception, we are forced to contemplate a moment in the 



60 MECHANICS 

distant future when all energy will be in the unavailable form of equally 
diffused heat, and the whole physical universe will have run down like 
the weights of a clock. 

IV. FRICTION 

75. Kinds of Friction. A body in motion relative to 
another body, with which it is in contact, or to a medium 
through which it moves, is always subject to retarding 
forces tending to bring it to rest. This action is called 
friction. It may take several forms : 

1. Sliding friction. This occurs when the surface of one 
body slides along that of another. Sliding friction opposes 
the motion of many parts of machines, such as the sliding 
motion of the crosshead of a steam engine between its 
guides, or its shaft sliding round and round in its bearings. 

2. Rolling friction. Whenever a wheel or a cylinder rolls 
on a plane surface, the resistance to motion at the line of 
contact is called rolling friction. Rolling friction is proba- 
bly due to the yielding of the wheel and the surface on 
which it rolls. The effect is much the same as if the wheel 
were continually climbing a slight ascent. 

3. Fluid friction. The air offers resistance to the pas- 
sage of a body through it ; a pipe offers resistance to the 
flow of steam or water through it ; water offers resistance 
to the motion of a boat, or the flight of a torpedo through 

it. These resistances are 
called fluid friction. 

76. Coefficient of Sliding 
Friction. Suppose a body 
M (Fig. 28) on a hori- 
zontal plane AB. Let a 
force F act on it parallel 
to the face of the plane. Also let P be the force vertically 
downward, pressing the surfaces of M and AB together. 
P may be the weight of the body M, or it may have any 
other value or source. The resultant of F and P is R' ; and 




FRICTION 61 

if M remains at rest, the reaction of the plane AB must 
be equal and opposite to R', making an angle with the 
normal N. 

The body M will not begin to slide unless F is a certain 
fraction of P. This fraction is called the coefficient of statical 
friction. It is usually denoted by /*. Then 

F=^P. s (27) 

The coefficient /-t has to be determined experimentally. 

The force F encounters a f rictional resistance which has a 
maxium value pP. If F is less than /*P, sliding does not 
occur; but as F is increased, the f rictional resistance also 
increases up to the value /-iP, when slipping begins. It can- 
not increase beyond this limiting value, and is slightly 
reduced after the body begins to slip. 

The coefficient /-t depends on the nature of the surfaces in 
contact, on their condition as to smoothness or roughness, and 
on the presence or absence of lubricants. It is independent 
of the area of the surfaces in contact, and of the pressure P. 
The whole frictional resistance is therefore proportional to 
the pressure P. But as this remains constant while the area 
of contact is diminished, the frictional resistance per unit 
area is directly as the pressure. 

77. The Limiting Angle. If F in Figure 28 is the value of 
the frictional resistance just as slipping begins, then the angle 
6 between R and N is the limiting angle of friction. It is the 
angle at which the reaction of the plane is inclined to the ver- 
tical. Since the horizontal component of R' is F and its vertical 

F 

component P, we have tan 6 = = n ; that is, the tangent of 

the limiting angle is equal to the coefficient of friction. If 
there were no friction, any force applied to M, deviating ever 
so little from the vertical, would produce slipping; but since 
there is friction, to produce sliding the force applied must 
make an angle with the normal to the surface of contact 



62 



MECHANICS 



somewhat larger than the limiting angle. Any pressure, 
however great, so long as its direction lies within the cone 
described by rotating the line R about the normal N as an 
axis, is incapable of making the body slip. 

Friction continues after the sliding motion has begun and 
opposes motion; but its magnitude is somewhat less than the 
friction of rest at the moment the slipping begins. 

78. Angle of Repose. Let a body M (Fig. 29) be placed 
on a plane surface, and let this surface be tilted until the 
body is on the point of beginning to slide. Let 6 be the 
angle of elevation of the plane. Then the weight of M equal 

to mg acts vertically down- 
ward, and the friction F is 
directed upward parallel to the 
face of the plane. 

Resolve mg into a normal 
component GrD, which is offset 
by the reaction of the plane, 
and a component DE, which tends to make the body M slide 
down the plane. The friction F must exactly balance this 
latter component at the instant when slipping begins. The 
angle 6 is then called the angle of repose. It is the same as 
the limiting angle. For the pressure P is the component 
GiD equal to mg cos 0. The component DE, representing 
F, is equal to mgsinQ. Hence 




- 2i =: ^ 
mg cos 6 P 



or IT 

The angle of repose is the angle at which heaps of sand, 
grain, or even fruit and shot, adjust themselves when dumped 
and allowed to find their own position of rest under sliding 
friction. 

79. Rolling Friction. When a cylindrical body rolls over 
a surface, the frictional resistance to motion is less than for 



FRICTION 63 

sliding. This fact explains the conspicuous advantage of 
wheeled vehicles over sledges. Rolling friction is equivalent 
to a small force acting at the circumference of the cylinder 
and bearing a small ratio to the pressure of the cylinder on 
the surface. 

This ratio is affected by the relative yielding of the sur- 
faces in contact. Osborne Reynolds has shown that an iron 
wheel rolling on india rubber raises before it a little hum- 
mock. This hummock tends to recover its form and to 
drive the wheel backwards. The rolling friction of iron 
on india rubber was found to be ten times as great as of 
iron on iron. 

Conversely, a rubber tire (automobile) is visibly deformed 
when rolling on a hard surface. The part in contact with 
the hard surface changes length, and hence there is some 
slip between the two. This action means a thrust on the 
contact layer of both substances. In the case of iron rails, it 
results in the constant peeling off of thin scales of iron. 

When the motive power is applied to rotate the wheel, as in the drivers 
of a locomotive or the rear wheels of an automobile, both rolling and 
slipping may be present. The friction between the driving wheels of a 
locomotive and the rails acts forward, while the rail itself is pulled back- 
ward on the ties and ground. This friction constitutes the external 
force pulling the train forward. The bottom of the drivers is forced 
backward by the pressure of the steam on the pistons, and friction on the 
rails resists this motion. On the other hand, the railway coaches are 
pulled forward ; friction opposes this motion and causes the bottom of 
the wheels to turn backward. The friction between the drivers and the 
rails is therefore necessary to motion forward. 
Without it, the wheels of the locomotive would 
simply spin around without moving the train. 
This is what sometimes happens when the rails 
are wet or covered with ice. 

If the brakes are set too hard, the friction 
between the carriage wheels and the rails may 
be insufficient to turn them against the brake 
friction. The wheels are then set and the 

breaking action is as great as possible. In this way a railway carriage 
may actually slide down a heavy grade without a wheel turning. 




64 



MECHANICS 



The friction of a round solid rolling on a smooth surface is always 
less than when it slides. Advantage is taken of this fact to reduce the 
friction of bearings. A ball bearing (Fig. 30) substitutes the rolling 
friction between balls and rings for the sliding friction between a shaft 
and its journal. 

80. Loss of Energy Due to Friction. Friction acts in gen- 
eral as a resistance opposing motion. Whenever a displace- 
ment takes place against frictional resistance work must be 
done. The energy equivalent to this work is converted into 
heat, which is gradually diffused among neighboring bodies, 
and the energy so transformed is no longer available to do 
work. Friction, therefore, decreases the efficiency of ma- 
chinery by wasting energy. 

81. Friction Dynamometer. An extreme case of the absorp- 
tion of energy by friction is presented by the friction dyna- 
mometer. It is a device for measuring the 
power of a steam or a gas engine, or of an 
electric motor. 

In Figure 31, A is a pulley fixed to the 
revolving shaft of the engine or motor. It 
has a channeled rim, the bottom of the chan- 
nel being flat. Around it is wrapped a cord 
with one end attached to a spring balance S, 
which measures the tension in that end of 
the cord. A tension represented by the 
weight IT suspended on the other end keeps 
the cord straight and determines the load. When the pulley 
is running, the work done is all absorbed by the friction 
between the stretched cord and pulley. 

Let r be the radius of the circle at the axis of the cord. 
The force of friction worked against is the tension F, indi- 
cated by the spring balance S, less the weight TT, since this 
weight acts in the same direction as the pulley turns. The 
distance worked through in one revolution is 2 TTT. If the 
speed is n revolutions per second, the distance worked through 




Fig. 31 



MOTION OF ROTATION 65 

by the pulley in a second is 2 irnr. The work done per sec- 
ond, or the power, is then 



If F and IF are in pounds and r is in feet, the horse power is 
obtained by dividing by 550. If F and W are in dynes 
(gm. x 980) and r is expressed in centimeters, the result 
will be found in watts by dividing by 10 7 , or in kilowatts by 
dividing by 10 10 . 

The quantity 2 TTU is the angular velocity co ( 87), because 
it is the angle in radians described per second. (F W)r 
is called the torque, T. The power is therefore equal to Ta>, 
or the product of the torque and angular velocity. 

V. MOTION OF ROTATION 

82. Rotation about a Fixed Axis. An unbalanced force ap- 
plied to a rigid body will in general produce both motion of 
translation and motion of rotation, unless it is directed through 
the body's center of mass, when the motion will be one of 
pure translation ( 25). In order, then, to study motion of 
rotation by itself, we may assume that the body has in it a 
fixed line, so that the only motion possible for it is rotation 
about this line as an axis. Under these conditions a single 
unbalanced force produces rotation only. An example is 
the flywheel of an engine, or a door swinging on its hinges. 
All points in the body then describe circles about the fixed 
axis as a center, and all have the same angular velocity. 

83. Moment of a Force. The effect of a force F in pro- 
ducing rotation is dependent not only on the magnitude of 
the force, but also on the distance of its line of action from 
the axis of rotation. It is obvious that a smaller force is re- 
quired to close a door when applied at right angles to the door 
at the knob, than when applied near the hinge. Also, an 
increase in the rotatory effect on a flywheel may be secured 
either by increasing the force, or by lengthening the crank. 



66 MECHANICS 

The effectiveness of a force in producing rotation depends on 
two quantities : (a) the magnitude of the force, and (&) the 
shortest distance between the axis and the line of action of 
the force. 

The measure of this effectiveness is the product of the 
force and the perpendicular distance between its line of ac- 
tion and the axis of rotation. This product is called the 
moment of the force. When the force is applied to turn a 
shaft, the moment is usually called the torque. 

Let M (Fig. 32) be a rigid body which may rotate about 
an axis through perpendicular to the plane of the figure. 
Then the moment of the force F is not so 
great if it is applied at B in the direction 
'< AB, as if the same force is applied at B in 
the direction OB. The moment for the first 
M P\4. direction is F x OA, and for the second, 

Fx OB. 

A moment is considered positive if it 
tends to produce rotation in one direction, 

and negative if in the other. It is im mate- 
Fig. 32 

rial which direction is considered positive 

if the same direction remains positive throughout any one 
problem. 

84. The Moment of the Resultant equals the Algebraic Sum 
of the Moments of the Components. If the resultant of any 
number of forces is a single force producing the same effect 
as the component forces acting conjointly, then the moment 
of this resultant about any point should equal the algebraic 
sum of the moments of the several components about the 
same point; otherwise the resultant could riot replace the 
components in producing rotation. (Note the special case 
of a couple, 86.) 

The principle applied to a parallelogram of forces may be 
demonstrated as follows: Let two forces P and Q be repre- 
sented by the lines AP and AQ, and their resultant R by the 




MOTION OF ROTATION 67 

diagonal AR of the parallelogram (Fig. 33). Let be any 
point in the plane of the figure as the point about which the 
moments are to be taken. 
It is called the center of 
moments. 

The area of the triangle 
AOQ is half the moment 
of the force Q ; for the 
moment of Q is the prod- 

uct of the lines AQ and Fig - 33 

Ji, while the area of the triangle is half this product. In the 
same way it may be shown that the moment of P is twice 
the area of A OP, and the moment of R is twice the area of 
A OR. We are to prove then that the area AOR equals the 
sum of the areas A OP and AOQ. 

Area AOQ = Area APR + Area OPR, 

since the three triangles have equal bases AQ and PR, and 
the altitude h of the first is equal to the sum of the altitudes 
AJ and h 2 of the other two. Also, since the whole is equal 
to the sum of its parts, 

Area AOR = Area A OP + Area APR + Area OPR. 

Hence, substituting for the last two areas their equivalent 
area AOQ, Afea A QR = Area A0 p + Area A Q % 



Therefore the moment of the resultant R is equal to the sum 
of the moments of the components P and Q. 

If the point 0, about which the moments are taken, lies on 
the line denoting the direction of the resultant, the moment 
of the resultant for that point is zero, and the algebraic sum 
of the moments of all the component forces is zero ; or the 
sum of the positive moments in one direction is equal to the^ 
sum of the negative moments in the other. 

85. Parallel Forces. An important case for the application 
of moments is that of parallel forces. 



68 MECHANICS 

The resultant of several parallel forces is equal to their 
algebraic sum and acts in a direction parallel to them ( 31). 

Let FV Fy F& etc., be any 
}< -- fq~->! number of parallel forces (Fig. 
34), and OX any line perpen- 
2 dicular to their directions. 

Also let x v x v x y etc., be the 
distances from any point on 
Flg * OX to the forces respectively. 

Then by the principle of moments 

RX= F lXl 



or X = l l 2 2 3 3 ' = ^^--^ ^^8^ 

XTi^rTiITi -VCT* V^ ^ 

-^1 +^^+^3+ ** 



R is the resultant and X is its distance from 0. (The 
sign of summation 2 is read "the sum of such terms as.") 

Specifically, if there are only two forces, F 1 and jP 2 , acting 
in the same direction, let the point be on the resultant itself. 

Then X is zero and A -r 

= I 

= . 
Ji ^ 

Now since x l + % is the distance d between the parallel 
forces, it is obvious that the resultant divides' this line into 
parts inversely as the forces. 

If the parallel forces are in Opposite directions, and F l is 
the greater, the resultant is F l I* 2 . Take the point on _F 2 . 
Then 



But F^F^ is necessarily less than F v and X is therefore 
greater than d ; i. e. the resultant lies outside the forces and 
on the side of the greater. 

Parallel forces may be illustrated by means of a graduated bar balanced 
on a knife-edge at (Fig. 35). The weight of the bar itself does not 




MOTION OF ROTATION 69 

produce rotation if its center of gravity coincides with the knife-edge. A 
weight placed on the right side produces a rotation clockwise ; one on the 
left, rotation counter-clockwise. 

The two weights, W l and W 2 , ' A 
placed at distances 5 and 8 units 

respectively from 0, will balance x 

each other if their moments about 
are the same, i.e. if 

W l x 5 = W 2 x 8, 
or if W 2 = 5/8 x W r F^. 35 

Also, any number of weights on one side will balance any number on 
the other if the sum of the moments of those on one side is equal to the 
sum of the moments of those on the other. Whenever the bar is balanced 
the resultant of all the weights passes through the knife-edge ; hence the 
moment of the resultant about this point is zero, and the sum of all the 
moments in one direction must equal the sum of all those in the other. 

86. Couples. Two equal parallel forces acting in opposite 
directions constitute a couple. The resultant of a couple is 
zero. It does not produce motion of translation nor accel- 
eration of the center of mass. 

A couple is simply a rotator, and its moment is equal to 
the product of one of the forces and the perpendicular dis- 
tance between them, Fd. The perpendicular distance d is 
called the arm of the couple. The moment of the couple is 
therefore the same about any point in its plane. 

All couples in the same plane are equivalent to one another 
if their moments are equal, because they produce the same 
rotational effect about any point in their plane. A couple 
may then be supposed moved to any part of its plane. 

The resultant of any number of couples in one plane is 
another couple in the same plane. The moment of the 
resultant couple must be equal to the algebraic sum of 
the moments of the component couples. 

87. Angular Velocity. Let be the angle through which 
a body turns in t seconds. Then the mean angular velocity 
during the t seconds is 0/t. The angular velocity of a body is 



70 MECHANICS 

the time rate of angular displacement. If the angle is ex- 
pressed in radians and the time in seconds, the quotient 6/t is 
in radians per second. 

Since 2 TT radians are described in one revolution, if a body 
makes n revolutions per second, the angular velocity will 

be 9 _ 

-Jm-^r (29) 




where ^is the period of rotation. 

If Figure 36 is a wheel or a cylinder rotating about an 

axis through and making n revolutions per second, its angu- 
lar velocity is 2 TTH radians per second. Any 
particle m at distance r from the axis describes 
a circular path of which the circumference is 
2 Trr. Since the particle traces this path n 
times a second, its linear velocity v is 27rrn units 
Fig 36 a second. But 2 irn is the angular velocity CD 

of the wheel in radians per second. Therefore 

v = ra>. (30) 

88. Angular Acceleration. When a,n unbalanced torque is 
applied to rotate a body, such as a flywheel or a cylinder, for 
example, the effect is to change the angular velocity. There 
is then an angular acceleration of the body and a linear 
acceleration of every particle in it. Angular acceleration is 
the time rate of change of angular velocity 

If the angular velocity changes from an initial value o> to 
a value o> in t seconds, the mean angular acceleration a is 

(31) 

in ^adians per second per second. But linear acceleration a 
is the rate of change of linear velocity. Therefore 

rco ro) n a> ay, 




and a = ro, (32) 



MOTION OF ROTATION 71 

or the linear acceleration of any particle in a rotating body 
is equal to the product of its angular acceleration and its 
radius of rotation. Linear velocity equals the radius times 
angular velocity; linear acceleration equals the radius times 
angular acceleration. 

89. Kinetic Energy of Rotation. Moment of Inertia. A par- 
ticle of mass m (Fig. 36), situated at a distance r from axis of 
rotation through 0, has, let us suppose, an angular velocity o>. 
Its linear velocity v is rco and its kinetic energy ( J mv 2 ) is 



The kinetic energy of the entire rotating body is the sum 
of such expression as this for all the particles composing the 
body, or -nr -m 



But since the angular velocity co is the same for every par- 
ticle of the body, it may be placed outside the sign of sum- 

mation 2, and we have JTr , 2V ,^ Q . 

TF=|&> 2 2wr 2 . (33) 

The quantity 2 wr 2 is called the moment of inertia of the body 
about the axis through 0. It measures the importance of 
the body's inertia with respect to rotation. The work done 
on the body to give it an angular velocity co about any axis 
is therefore proportional to its moment of inertia about the 
same axis. With a given angular velocity, the energy of 
rotation of- a body depends not only on its mass, but also on 
the manner in which that mass is disposed about the axis. 
The moment of inertia will hereafter be represented by the 
letter K. 

90. Torque and Moment of Inertia. Referring again to 
Figure 36, if the particle m has a linear acceleration a directed 
along the tangent to the circle in which the small mass m 
revolves, the expression for the tangential force to produce 
this acceleration is by equation (15) ma. The moment of 
this force about is mra. But since a equals ra, the mo- 



72 MECHANICS 

ment of the force on the particle m is mr 2 a. The sum of 
the moments of the forces on all the particles composing the 
body is then 2wr 2 . But since the angular acceleration a 
is the same for all the particles, the expression for the turn- 
ing moment or torque may be written 



Fb = aZmr 2 = aK, (34) 

where F is the force applied to produce rotation against the 
resistance due to the body's inertia, and b is its lever arm, or 
its perpendicular distance from the axis of rotation. 

From this expression it is obvious that the moment of 
inertia may be denned as the torque required to produce unit 
angular acceleration; that is, an angular acceleration of one 
radian per second per second. 

91. Moment of Inertia of a Uniform Rod. As an illustration 

of the method of calculating the moment of inertia, let us 

suppose the uniform rod (Fig 37) to be mounted so as to 

m rotate about an axis 

through one end 0, the 

axis being perpendicu- 

lar to the length of the rod. Let the rod be divided 
into a very large number n equal lengths, and let the mass 
of each small section be m. Then the whole mass M of the 
rod is mn. The distances of the several elements of the rod 
from are 1, 2, 3, etc., to n in terms of the equal divisions 
of the rod as the unit of length. The moment of inertia of 
the rod is the sum of the moments of inertia of the parts, or 



Whence JT= * (n + 1) 



6 
Since now n is indefinitely large, this expression becomes 



MOTION OF ROTATION 



73 



But since n is the number of units in the length of the 
rod, it may be replaced by Z, and then finally 



In a similar way it may readily be shown that the moment 
of inertia of a uniform rod about an axis perpendicular to its 
length and passing through its middle point is 



TABLE OF MOMENTS OF INERTIA 



SOLIDS OF MASS M 



POSITION OF Axis OF KOTATION 



VALUE OF K 



Circle of radius r 

Cylinder of radius r 
Hollow cylinder, inner 

radius r v outer r 2 
Rectangle, length a, 

width b 

Sphere of radius r 



Through center perpendicular to 

plane 
Axis of cylinder 

Axis of cylinder 

Right angles to plane and through 
center of figure 

Through center of sphere 



$ Mr* 



92. The Simple Pendulum. The simple pendulum is an 
ideal pendulum consisting of a heavy particle suspended by 
a thread without weight. An approximation to a simple 
pendulum may be made by suspending a small dense sphere 
from a fixed point by means of a fine thread. If the small 
sphere is slightly displaced and then released, it will oscillate 
back and forth about its position of rest. Its excursions 
become gradually smaller, but if the arc traversed is small, 
the period of its swing remains unchanged. This charac- 
teristic of pendular motion first attracted the attention of 
Galileo, who noted it in the oscillations of a bronze chandelier 
suspended by a long rope from the roof of the cathedral in 



74 



MECHANICS 



or double oscillation. 



Pisa. This " lamp of Galileo " may still be seen in the same 
place. 

A single vibration is the motion from N (Fig. 38) to either 
B or E and back again ; a complete or double vibration is the 
motion from Nto B, across to E, and then back again to N. 
The period of a complete oscillation is the interval between 
two successive passages of the pendulum bob through N 
in the same direction ; that is, it is the period of a complete 

The period of a single oscillation 
is half that of a double oscillation. 
The amplitude is the arc BN or NE. 

93. Law of the Pendulum. When 
the amplitude does not exceed 2 or 
3, the period of the simple pendulum 
depends on the length of thread I 
and the acceleration of gravity g 
only. 

Let AN be a simple pendulum, 
with a small mass m at the bottom, in 
its position of rest (Fig. 38). Let it 
be deflected to the position AB. 
The acceleration due to gravity g, 
represented by the vector BG-, may be resolved into two 
components; one, BD, in the direction of the thread, and 
the other tangential. The former is annulled by the con- 
straint of the thread; the latter is 

BO = g sin 0, 

6 denoting the angle BAN, or its equal BCrO. Then if is 
small, sin 6 may be put equal to 6 ( 23). , Moreover, 6 = x/l, 
where x is the arc or displacement BN. Therefore 

BC=-ge=- 9 x. 

But g/l is a constant. Hence the acceleration of B along 
the arc is proportional to the displacement x from the middle 







Fig. 38 



MOTION OF ROTATION 75 

point JV. This relation is characteristic of simple har- 
monic motion. The motion of the pendulum is therefore 
simple harmonic to the same approximation that sin 6 = 6. 

The acceleration a x in any direction x for simple harmonic 
motion is ( 38) 

X 47T 2 r X 4-7T 2 



Putting this general expression for the acceleration in 
simple harmonic motion equal to the particular one for the 
simple pendulum, we have 



g 



Solving for T, T = 2 



This is the period of a complete or double swing. For a 
single oscillation, 

(35) 

The following are the laws for a simple pendulum : 

I. The period of vibration is independent of the ampli- 
tude, if the latter is small. 

II. The period of vibration is proportional to the square 
root of the length. 

III. The period of vibration is inversely proportional to 
the square root of the acceleration of gravity. 

94. The Physical Pendulum. It is physically impossible 
to realize a simple pendulum with its mass all at a single 
point. Such a pendulum has only an imaginary existence ; 
any real pendulum which does not conform to it is called a 
compound or physical pendulum. It is possible, however, to 
find the length of the corresponding simple pendulum, which 
will perform its oscillations in the same time as the physical 



76 



MECHANICS 



pendulum. This operation is called finding the length of the 

equivalent simple pendulum. 

Let the mass of the physical pendulum be M, and let its 

center of inertia be at a distance h from the axis A, about 
which it oscillates (Fig. 39). The weight of 
the pendulum is Mg, and this is the force the 
moment of which produces rotation about the 
axis A. The lever arm of this force with 
reference to A as the origin of the moments is 
h sin 0, and the turning moment or torque is 

Mgli sin 0. 

But when is small, this moment is approxi- 
mately MgW. By 90 this moment may also 

be expressed in terms of angular acceleration 

4<j 
and the moment of inertia as aK, or 6K 

J12 

( 88, 93.) In both expressions for the moment is the 
small displacement of the center of inertia of the pendulum. 
Equating the two expressions for the moment of the force, 




Solving for T, 



For a single swing 



7T 



Mgh 



(36) 



Comparing this expression with equation (35), it will be 
seen that the physical pendulum swings as if it were a simple 
pendulum whose length is K/Mli. This is therefore the 
length I of the equivalent simple pendulum. 

95. The Center of Oscillation. .Let AB (Fig. 40) be a bar 
suspended so as to swing freely about a horizontal axis 
through C. is the center of suspension. Let the center of 



MOTION OF ROTATION 



77 



Fig. 40 



mass be at 6r. Such a physical pendulum has a period of 
vibration equal to that of a simple pendulum, the length of 
which is I = K/Mh. Lay off the distance I on the line CG- 
produced, so that OD=l. The point D is called the center 
of oscillation. The length of the equivalent simple pendulum 
is therefore the distance between the centers of sus- A 
pension and oscillation. It follows that if the whole 
mass of the physical pendulum were concentrated at 
the center of oscillation, its period as a simple pen- 
dulum would be the same as that of the actual 
physical pendulum. 

One of the interesting properties of the center of oscillation 
is the following : If the pendulum is struck a blow at its center 
of oscillation, it will be set swinging around its center of suspen- 
sion, and the blow will not produce any pressure or shock on the 
axle or knife-edge on which the pendulum is supported. For this reason 
the center of oscillation is said to coincide with the center of percussion. 
A baseball bat swung in the hands has a center of percussion, and it 
should strike the ball at this point to avoid jarring the hands. If the bat 
is struck higher up or lower down, a distinct " sting " will be felt. If a 
thin strip of wood about a meter long is held between the thumb and 
forefinger near one end, and a blow is struck on the flat side with a soft 
mallet, a point may be found where the blow will not throw the wood 

strip into rapid vibrations or 
shivers, but will only set it 
swinging like a pendulum. 
This point is the center of 
percussion or the center of 
oscillation. 

The position of the center 

of percussion may be deter- 

[ mined experimentally by 
means of a device shown in 
Figure 41. A heavy bar is 
loosely pivoted on a slender axle at S and rests on a blunt edge at P. 
The position of P is adjustable toward or away from S. If the bar be 
lifted and allowed to drop on the edge P, the slender pivot at S will 
break unless P is near the center of percussion of the bar. If P is too 
near S, the pivoted end will be thrust upwards by the blow ; if it is too 
far away, the thrust will be downwards. 



Fig. 41 



78 MECHANICS 

96. The Two Centers Interchangeable. Pluyghens, a cele- 
brated Dutch physicist, discovered that the centers of sus- 
pension and oscillation are interchangeable ; that is, the 
period of vibration of the pendulum is the same whether 
it swings from the one as an axis or the other. This dis- 
covery led to the invention by Captain Kater of a pendulum 
with two parallel axes of suspension, and with weights which 
can be adjusted until the pendulum has the same period of 
vibration about the two axes. The distance between 
them is then the length I of the equivalent simple 
pendulum. 

One form of this pendulum is shown in Figure 42. 
At a and b are the two knife-edges turned toward each 
other. W and V are adjustable weights. When the 
weights have been adjusted so that the period is the 
same about a as about 5, the distance between a and b 
is the length I of the equivalent simple pendulum. 

97. Accelerations of Gravity Compared by Means of the 
Pendulum. When the adjustments of a Kater's pen- 
dulum have been finally completed, the length ?, 
together with the observed period of vibration T, when 
inserted in equation (35), will give the value of g at 
the place of observation. 

Moreover, if the period of this same pendulum is 
observed at different points on the earth's surface, the 
corresponding accelerations of gravity may be com- 
pared, since equation (35) shows that the period is 
inversely as the square root of g ; whence 



g 

98. Length of Seconds Pendulum. If g is known, 
the length of a pendulum beating seconds may be found 
by placing T equal to unity in equation (35) and solv- 
ing for I. Thus, at New York g equals 980.19 cm. /see. 2 . 



MOTION OF ROTATION 



79 



Therefore, 
Whence 



1= 



, I 

'980.19' 



99. Utility of the Pendulum. The discovery of Galileo 
suggested an obvious use of the pendulum as a timekeeper. 
In the common clock the oscillations of the pendulum regu- 
late the motion of the hands. The train of wheels is kept 
in motion by a weight or a spring, and the regulation is 
effected by means of the escapement (Fig. 43). The pendu- 
lum rod, passing between the prongs of a fork, communicates 
its motion to an axis carrying the escapement, which termi- 
nates in two pallets. These pallets engage alternately with 
the teeth of the escapement wheel, one tooth of 

the wheel escaping from the pallet every double 
vibration of the pendulum. The escapement 
wheel is a part of the train of the clock; 
and as the pendulum controls the escapement 
it also controls the motion of the hands. 

100. Compensation for Temperature. Since 
the vibration period is affected by changes 
in the length of the pendulum, a common 
clock with an uncompensated pendulum suf- 
fers a change of rate with a change of tem- 
perature, losing time in hot weather and 
gaming in cold. A correction may be made 
by raising or lowering the bob by means of 
the running nut on which it rests. 

Astronomical clocks for precise measurements have 
compensated pendulums, which adjust themselves au- 
tomatically when the temperature changes. The mer- 
curial pendulum, commonly used for this purpose, has 
in it a mass of mercury, which expands upward while Fig. 43 

the pendulum rod expands downward, thus effecting a compensation. 
In one form the mercury in glass tubes forms the pendulum bob ; in 
another the bob is lens-shaped, and the mercury partly fills a steel tube 
carrying the bob, similar to the one shown in the figure. 




80 MECHANICS 

VI. EQUILIBRIUM 

101. .Definition of Equilibrium. When the forces acting 
on a material particle have no resultant, that is, when their 
vector sum is zero, they produce no acceleration and are 
in equilibrium. It does not follow that the particle is at 
rest because the forces acting on it are in equilibrium. 
Equilibrium does not mean that the velocity of the particle 
is zero, but that its acceleration is zero. Rest means zero 
velocity ; equilibrium, zero acceleration. If the particle is 
at rest, it will remain at rest ; if in motion, it will continue 
to move without change when the forces acting on it are 
balanced, or are in equilibrium. 

102. Conditions for the Equilibrium of a Particle. A parti- 
cle cannot, be in equilibrium if it is acted on by a single 
force only. 

Two forces must fulfill the following conditions for equi- 
librium : they must be equal in magnitude ; they must be 
opposite in direction. These two conditions are sufficient 
for a material particle because it is incapable of rotation, 
or at least its rotation has a vanishingly small dynamical 
significance. 

For three forces the condition of equilibrium is that each 
force must be equal and opposite to the resultant of the other 
two ; for we may replace two of the forces by their resultant, 
and the problem will then be reduced to the equilibrium of 
two forces. 

This condition may be resolved into two others. It fol- 
lows, first, that the three forces must lie in the same plane ; 
for the resultant of two of them, as P and Q for example, 
lies in the plane of P and Q; and then the third force must 
lie in the same plane, or it could not be opposite to the 
resultant of P and Q for equilibrium. 

Again, it follows that the three forces must be so related to 
one another that they can be represented in magnitude and 
direction by the three sides of a triangle, taken in order the 



EQUILIBRIUM 81 

same way round (Fig. 44). If the three forces Q, P, and R 
can be represented by the sides AB, BC, and CA of the tri- 
angle, then R is equal and opposite to R' , the resultant of P 
and , and there is equilibrium. 

In a similar manner it may be shown that the condition for 
the equilibrium of any number of forces acting on a particle 
is that each force must be equal and opposite to the resultant 
of all the others. 

If the forces then all lie in the same plane, they must have 
such relative magnitudes that they may be represented by 




the sides of a closed polygon taken in order around the 
figure. If they do not all lie in the same plane, the alge- 
braic sum of their components in each of three rectangular 
directions must be zero. 

103. Conditions for the Equilibrium of a Rigid Body. The 

forces applied to a body of sensible dimensions may not all 
pass through a single point. The necessary condition for 
equilibrium is therefore that the forces must produce neither 
translatory nor rotatory acceleration, for such a system of 
forces may produce either motion of translation, or motion of 
rotation, or both. A single force applied to a rigid body 
produces both translatory and rotatory motion, unless its 
direction passes through the center of inertia of the body. 

The first condition of equilibrium is, therefore, that the 
vector sum of all the forces shall be zero; that is, that the 
algebraic sum of the components of all the forces taken in 
three rectangular directions shall be severally equal to zero. 
There will then be no rectilinear acceleration in any direc- 
tion. 



82 MECHANICS 

Again, that the forces may not produce motion of rotation 
around any axis, it is necessary that the algebraic sum of 
their moments about any three non-coincident axes shall be 
zero. The three, axes are usually taken at right angles to 
one another. 

If the forces all lie in one plane, it is sufficient for equi- 
librium that their vector sum be zero, and that the algebraic 
sum of their moments about any point in the plane be zero. 

104. Equilibrium under Gravity. It is convenient to divide 
this topic into three divisions : 

A. On a horizontal plane. The weight of the body is then 
equal and opposite to the reaction of the plane. Therefore 
the vertical line through the center of gravity of the body 
must fall within its base of support. If this vertical falls 
outside the base, the weight of the body and the reaction of 
the base form a couple, and the body will overturn. 

If the plane on which the body rests is moving up or 
down with uniform velocity, like the floor of an elevator, 
there is no acceleration, and the body is in equilibrium, though 
not at rest. When the elevator starts to ascend, it has accel- 
eration ; the pressure of the body on the floor is then greater 
than its weight. When the elevator starts to descend, it has 
an acceleration downward, and the pressure of the body on its 
floor is less than its weight Mg. If the elevator should start 
with an acceleration equal to g downward, there would be no 

pressure of the body on the floor 
of the elevator. 

B. On an inclined plane. If 
the body rests by friction on an 
inclined plane (Fig. 45), then the 
component OA of its weight W 

down the plane is equal and oppo- 

F| g- 45 site to the sliding friction F. The 

other component OB of its weight perpendicular to the plane 
is in equilibrium with the reaction R of the plane. On an 





EQUILIBRIUM 83 

inclined plane, therefore, the weight of a body is in equilib- 
rium with the sliding friction F and the reaction R of the 
plane. Strictly, the component OA down the plane and the 
friction F form a couple, and the farther the center of gravity 
of the body from the plane, the greater the moment of this 
couple ; but there will be equilibrium so long as the vertical 
through does not fall outside the base of the body, and the 
angle of elevation of the plane is less than 
the limiting angle of friction. 

C. About a horizontal axis. If the body 
is free to turn about a horizontal axis, it 
can be in equilibrium only when the vertical 
line through its center of gravity passes 
through this axis. Let the body whose cen- 
ter of gravity is at G- be supported by an 
axis through B (Fig. 46). Let the vertical 
line GrE represent its weight. When the 
line GrE does not pass through B, the mo- 
ment of the weight about the axis through B is the product 
of GrE and BC, and the body rotates clockwise. As the 
body swings, shortening the line BC, the point Gr approaches 
the vertical through B\ and when the point C coincides with 
B, the moment becomes zero. The body is then in equilib- 
rium both for rotation and translation. 

105. Stable, Unstable, and Neutral Equilibrium. Consider 
the case of a body suspended from an axis about which it can 
turn freely, as in Figure 46. When the body is turned so 
that its center of gravity Gr is in a vertical line through B, 
either above or below, it is in equilibrium because the turning 
moment is zero. In the second position, with Gr vertically 
below B, if we suppose the body slightly displaced about the 
axis, as in the figure, the moment of the weight of the body 
about the axis of suspension tends to decrease the displace- 
ment and to bring the body back again to the position of 
equilibrium. The equilibrium is then said to be stable. 




84 MECHANICS 

If on the contrary the center of gravity Gr be vertically 
above the axis B, and the body be slightly displaced, the 
turning moment will tend to increase the displacement. 
Hence the equilibrium in which the center of gravity of the 
body is vertically above the axis of suspension is unstable. 

If the axis of suspension passes through the center of 
gravity of the body, a displacement of the body does not 
bring into operation a couple tending either to increase or 
decrease it, and the equilibrium is neutral. 

The three kinds of equilibrium may be illustrated also by 
a body resting on a horizontal plane. Let a cone (Fig. 47) 
rest on a hori- 
zontal plane in 
the position A. It 
is in stable equi- 
librium. In the 
position B it is 
in unstable equi- 

Fig 47 

librium, which is 

a physically impossible equilibrium. In the position (7, its 
equilibrium is neutral for a rolling displacement. A ball on 
a horizontal plane is in neutral equilibrium for any point on 
its surface in contact with the plane. 

The three kinds of equilibrium for a body are therefore 
these : stable, for any displacement which causes its center of 
gravity to rise ; unstable, for any displacement which causes 
its center of gravity to fall ; neutral, for any displacement 
which does not change the height of its center of gravity. 

In general, if a body in neutral or unstable equilibrium is 
slightly displaced, it has no tendency to return, and conse- 
quently will not rock about its position of neutral or unstable 
equilibrium. An oscillation, then, is always a motion around 
or through a position of stable equilibrium. 

106. Stability. The most useful measure of the stability 
of a body is the work necessary to overturn it ; that is, it is 



MACHINES 



85 



the product of its weight and the difference between the 
distances A C and AD in Figure 48. In the diagrams C is the 





\ 



Fig 48 

center of gravity, A the point about which the body is turned, 
and BD the height through which the body is lifted to bring 
it to a position of unstable equilibrium. 

A brick has less stability when standing on end on a level surface 
than when lying on edge; and it has less stability on edge than when 
lying 011 its broad side. 

Suppose a short cylinder of wood loaded on one side, so that its center 
of gravity falls at G (Fig. 49) . This cylinder may be placed on a plane, 




Fig. 49 

slightly inclined, in such a position that it will roll up the plane into a 
position of stable equilibrium. 

If the cylinder be rolled along a horizontal plane, its center of gravity 
will describe a curve with crests and hollows, similar to the one shown in 
the figure. Every hollow corresponds to a position of stable equilibrium, 
and every crest to one of unstable equilibrium. If the path were a straight 
line, the equilibrium would be neutral. 



VII. MACHINES 

107. Definition of Machines. A machine is a device de- 
signed to change the direction or the magnitude of the forces 
required to accomplish some useful purpose, or to transform 
and transfer energy. For example, by the use of a simple 
pulley to change the direction of the force, a body may be 
lifted while the force is applied in a downward direction. 



86 MECHANICS 

Again, by means of a bicycle a man propels himself forward, or even 
up an incline, by a downward and backward thrust of his feet on the 
pedals. 

In the approaches to the Saint Gothard tunnel through tne Alps, an 
ascent insurmountable directly is made possible by extending the ele- 
vation to be made over a long inclined plane, like the thread of a screw, 
cut partly in the face of the mountain and partly in circular tunnels 
through the rock. In this manner the force required to lift the train is 
reduced to the capacity of the locomotive. The effect of this engineering 
device is to reduce the rate of doing the work of lifting the train. 

A steam engine is a machine designed to transform the 
heat energy applied to it into useful mechanical work; a 
dynamo-electric machine transforms mechanical energy into 
the energy of an electric current. A water wheel transforms 
the potential and kinetic energy of falling water into 
mechanical energy represented by the shaft of the wheel 
turning with a definite torque. 

A complex machine, consisting of a train of mechanism, 
may comprise a series of simple or elementary machines, 
or mechanical powers. These are : the lever, pulley, wheel 
and axle, inclined plane, and screw. All complex machines 
are mechanically only combinations of two or more simple 
machines. 

108. Mechanical Advantage. All machines designed to 
transform or transfer energy have the common characteristic 
that the force applied at one part to produce motion enables 
work to be done at another part against resistance. The 
force applied is called the effort, and the force worked against, 
the resistance. The problem in simple machines is reduced 
to finding the ratio of the resistance to the effort. This ratio 
is taken as the measure of the mechanical advantage of the 
machine. 

It is evident that the advantage derived from the use of 
machines is not all mechanical ; for in many cases the 
advantages gained in other respects may more than com- 
pensate for a loss of technical mechanical advantage. 



MACHINES 87 

In elementary discussions it is customary to neglect fric- 
tion and to assume that the parts of a machine are rigid and 
without weight. 

109. General Law of Machines. Every machine must con- 
form to the principle of the conservation of energy; that is, 
the work done by the effort must equal the work done in over- 
coming the resistance, except that some of the energy may be 
dissipated as heat or may not appear in mechanical form. A 
machine can never produce an increase in the amount of 
energy applied. 

Denote the effort by^F and the resistance by R, and let d 
and D denote the distances through which they work respec- 
tively. Then from the law of conservation of energy, 

Fd = ED, (37) 

or the effort multiplied by the displacement in its direction 
is equal to the resistance multiplied by the displacement 
directly against it. 

110. Efficiency If all wasteful resistance could be elimi- 
nated from a machine, it would waste no energy and its effi- 
ciency would be unity. But in practice there is always 
present some wasteful resistance due to friction, rigidity of 
cords, etc. The work done is, therefore, always partly useful 
and partly wasteful. The efficiency of a machine is the ratio 
of the useful work done by it to the total work done on it; 
it is the output divided by the input of energy. Efficiency 
is always a proper fraction and it is expressed as a percentage. 
An efficiency of 90 per cent means that the energy recovered 
is 90 per cent of the energy put into the machine. A machine 
that will do either useful or useless work continuously, with- 
out receiving a continuous or intermittent supply of energy 
from without, is clearly an impossibility. 

111. The Lever. A lever is a rigid bar turning about a 
fixed axis called the fulcrum. The perpendicular distances 
between the fulcrum and the lines of action of the effort and 



88 MECHANICS 

the resistance are called the arms of the lever. A straight 
lever has its two arms in the same straight line. 

Several cases arise according to the relative positions of 
the forces with respect to the fulcrum. If the fulcrum 
is between the effort and the resistance, the lever is of 
the first kind ; if the resistance is between the effort and 
F the fulcrum, the lever is of the 

second kind ; and, finally, if the 
effort is between the resistance 
and the fulcrum, the lever is 
f of the third kind. The three 
kinds of lever are represented 



p | j by the three diagrams of Figure 

50 in order. 



A crowbar is a lever of the first 
kind. A pair of scissors consists of 

W Fi 50 two levers of the first kind joined 

together. A nutcracker is a double 

lever of the second kind. A pair of spring shears, used for shearing sheep 
or clipping grass, is a double lever of the third kind. So is an ordinary 
pair of tongs. The forearm is also an example of a lever of the third 
kind ; the fulcrum is at the elbow, the resistance is at the hand, and the 
effort is applied by the biceps muscle between the two. 




Fig. 51 

The steelyard (Fig. 51) is a lever of the first kind with unequal arms. 
It. is a form of balance in which the body to be weighed is suspended 
from the shorter arm of the lever, and a counterpoise is caused to slide 
along the longer arm to produce equilibrium. The center of gravity of 



MACHINES 89 

the steelyard is at the fulcrum. The body weighed is heavier than the 
counterpoises; the divisions of the beam are equidistant and indicate 
the weight. 

112. Mechanical Advantage of the Lever. In the three dia- 
grams of Figure 50, the arms of the lever are AF and BF. 
Let P be the effort and W the resistance (a weight or other 
force). The most direct and simple way to obtain the rela- 
tion between the effort and resistance is -to take moments 
around the fulcrum as the origin. If the lever is to be in 
equilibrium, these moments must be equal and opposite. 
Hence 

PxAF=Wx 



Therefore, the mechanical advantage of the lever equals the 
inverse ratio of the arms. 

If it is desired to take into account the weight of the 
lever, the moment of this weight, considered as acting at 
the center of gravity of the bar, must be added to either 
the moment of the effort or of the resistance, according 
as the weight acts to produce rotation in the direction of 
the one or the other. 

In this discussion it has been tacitly assumed that the 
forces act at right angles to the length of the bar. If they 
do not, then the lever arms are the distances 
measured perpendicularly from the fulcrum 
to the directions of the two forces. 

113. The Wheel and Axle. This simple 
machine may be considered as a continuous 
lever. It consists of two cylinders of dif- 
erent diameters turning together on the 
same shaft. The center of the shaft is at 
O (Fig. 52). A rope is wound around 
each cylinder, right-handedly around one and in the opposite 
direction around the other* When the cylinders are turned, 
one rope unwinds and the other winds up. 




90 



MECHANICS 



If the radii of the large and the small cylinders are R and 
r respectively, during one revolution the distance worked 
through by the effort P is 2 irR, and the distance moved 

against the resistance W is 

2 irr. Hence 

2-TT^x P=%7rr X W. 

Whence K=*. 
P r 

The weight P may be 
replaced by any effort P 
applied to the circumference 
of the wheel, and the weight 
W by any resistance W at the 
circumference of the axle. 
. The mechanical advantage is 
the ratio of the radius of the 
wheel to that of the axle. 

^-^^^ammmmw^ 

114. Application. The derrick 

is a form of wheel and axle much used for lifting heavy weights in con- 
struction work. The essential parts are shown in Figure 53. The der- 
rick may be considered as two 
sets of wheel and axle in series. 
The axle of the first set works 
on the wheel of the second by 
means of the spur gears. The 
cranks of the first set answer 
the same purpose as a wheel. 
The mechanical advantage is 
the product of the radii of the 
wheels divided by the product 
of the radii of the axles. 

In the capstan (Fig. 54) 
handspikes inserted in holes 
at the top are used instead of a wheel ; the rope by which the work 
is done is wrapped around the body of the capstan as an axle. 

115. The Pulley. The pulley is a wheel, called a sheave, 
free to turn about an axle in a frame called a block. The 





Fig. 54 



MACHINES 



91 



Fig. 55 



effort and the resistance are connected by means of a rope, 

which lies in a groove cut in the circumference of the wheel. 

The object in using the wheel instead of a ^ 

fixed cylinder to change the direction of the 

rope is to reduce friction. 

In the simple movable pulley of Figure 

55 the effort, or tension in the rope, is half 

the weight or other resistance. But the 

lower sheave is supported by two branches 

of the rope. If the weight is lifted, it 

rises half as fast as the free part of the 

rope travels. The mechanical advantage of 

such a pulley with one movable block is 

obviously equal to two. 

The most useful pulley consists of two 

blocks, each with several sheaves turning on 
the same axle. One of the blocks is fixed, 
while to the other is attached the weight or 
other resistance (Fig. 56). 

The principle involved in determining the 
mechanical advantage of a system of pulleys 
is the transmission of the same tension to all 
parts of the rope. In reality the rope is 
stiff and there is friction at the axles. The 
effect of this rigidity and friction is a dimi- 
nution in the tension of the rope as it passes 
a pulley. If this diminution is a fixed ratio, 
allowance can be made for it. 

When there is one continuous rope passing 
around the two pulleys, as in Figure 56, it 
is obvious that the weight is sustained by 
the several parts of the rope, the tension in 
each part being the effort P applied at the 
free end, neglecting rigidity and friction. 

If there are n lengths of the rope between the fixed and 

movable blocks, the sum of the tensions supporting the 




Fig. 56 



92 MECHANICS 

weight or resistance is nP ; that is, 

W= nP, and n. 

When a single rope is used, the mechanical advantage of a 
system of pulleys is therefore equal to the number of times the 
rope passes between the two blocks. 

116. The Inclined Plane. Suppose a body resting on an in- 
clined plane without friction. The weight of the body acts 
vertically downward, while the reaction of the plane is per- 
pendicular to its surface ; so that a third force must be ap- 
plied to maintain the body in equilibrium on the incline. 
Two principal cases occur : first, when the force is applied 

parallel to the face of the 
plane ; second, when it is 
applied parallel to the base 
of the plane. 

A. The most convenient 
method of obtaining the rela- 
tion between the force F 
(Fig. 57) and the weight W 
is to apply the principle of work. Suppose D to move under 
the influence of the force F from A to C without accelera- 
tion up the plane. Then the work done by the force F is 
F x A C. The work done on the body D against gravity in 
lifting it through the vertical height BC is Wx BC, and 

7= WxBC, 




or 



W 
F 



AC 
BC 



The mechanical advantage when the force is applied parallel 
to the face of the plane is the ratio of the length of the plane to 
its -height. 

B. When the force is applied parallel to the base of the 
plane, the component of the displacement in the direction of 



MACHINES 



93 



the force, when the body D moves from A to O (Fig. 58), 
is the base of the plane AB. Therefore the work done by F 
is F x AB. The work done on the weight W against gravity 
is the same as in the first case. 
Hence /X_N__^^ 

U2^ 



or 



W 
F' 



AB 
BO 



w 

Fig. 58 



The mechanical advantage 
when the force is applied parallel to the base of the plane is the 
ratio of the length of the base of the plane to its height. 

117. The Screw. The screw is a cylinder on the surface 
of which is a uniform spiral called the thread. The faces of 
the thread are inclined planes. If a long triangular strip of 

paper be wound on a cylinder such 
as a pencil, with the base of the tri- 
angle perpendicular to the axis of 
the cylinder, the hypotenuse of the 
triangle will form the thread of a 
screw (Fig. 59). The distance s 
between successive turns of the 
thread is called the pitch of the 
screw. The screw works in a block called a nut, on the inner 
surface of which is a spiral groove. This groove is the exact 
counterpart of the thread (Fig 60). When either the screw 
or the nut makes a complete turn, 
the relative motion of the two paral- 
lel to the axis of the screw is the 
distance s. 

The screw is usually combined Fig - 60 

either with a lever or its equivalent, a wheel. The mechani- 
cal advantage may then be found most readily by applying 
the principle of work expressed in the general law of 
machines ( 109). If s is the pitch of the screw, and I is the 



Fig. 59 




94 



MECHANICS 



lever arm or radius of the wheel, then when the effort P 
makes a complete turn, the equation of a work is 



or 



Hence, the mechanical advantage of the screw is the ratio of 
the distance traversed by the effort in one turn to the pitch of 
the screw. 

In deducing the above relation the friction between the 
screw and the nut has been disregarded. This friction is 
always far from negligible, and in practice the mechanical 

advantage is considerably less than ? ^ -. 

8 

118. The Screw Gauge. The screw in the form of a screw 
gauge is used for measuring small dimensions. The object to 

be measured is placed be- 
tween the end of the screw 
O and the block B (Fig. 61). 
The nut is held on the other 
end of the strong curved 
arm. The head of the 
screw, or the cap D, is 
divided into some number 
of equal parts, say 50. The 
whole number of turns made by the screw is read on the 
scale .A, which is uncovered by the movement of the cap 
attached to the screw. The fractions of a turn are read on 
the scale on the edge of the cap. If, for example, the pitch 
of the screw is J mm., then for each turn the end of the screw 
moves |- mm. ; and if the scale on the cap reads 12 divisions 

12 1 

further, the screw has moved x = 0.12 mm. in addition. 

119. Sensibility of the Balance. The balance is an instru- 
ment for the comparison of equal masses. It is essentially 




Fig. 61 



MACHINES 



95 



a lever of the first kind with arms of equal length. It 
consists of a light, trussed beam, so as to have the requisite 
stiffness with the least weight. It is supported at its 
middle point by means of a knife-edge resting on agate 
planes. The scale pans are of equal weight and are sus- 
pended on knife-edges from the ends of the beam. 

A good balance must fulfill the following conditions : 
(1) It must be true ; that is, the beam must be horizontal 
with equal weights in the two scale pans. (2) It must be 
sensitive ; that is, a small difference between the two masses 
in the scale pans must produce an observable deviation of 
the beam from a horizontal position. (3) It must be stable ; 
that is, the beam must return to its horizontal position 
of equilibrium very precisely after displacement. (4) A 
fourth desideratum is that its period of oscillation about 
its position of stable equilibrium shall be as small as 
possible. 

Let the three points A, B, and O (Fig. 62) be in the 
same straight line. A and B are the knife-edges for the 
support of the pans, and A , 
is the knife-edge on which 
the beam rests. Let G- be . 
the center of gravity of the 
beam the weight of which is 
w. Let a weight P be placed 
in one scale pan and P+p 
in the other. If then the two 
arms of the balance are of 
equal length, P and P in the 
two pans have equal moments about O and counterbalance 
each other. They may then be omitted from the equation of 
equilibrium. Then the moment of the small excess weight 
p about is equal and opposite to the moment of the weight 
of the beam w about the same point. Therefore 



Fig. 62 



P+p 



p X B' 1= w x 



96 MECHANICS 

6r f is the position of the center of gravity of the beam 
when it is displaced by the excess weight p to the new 
position A'B' . Let OCr be denoted by r, and let I be the 
length of either arm of the balance. Then, if 6 is the 
angular displacement BOB 1 , 



Substituting in the preceding equation of equilibrium 

we have 

p x I cos 6 = w x r sin 6. 

Whence !HL^ tan0 = -^. (38) 

cos 6 wr 

The sensibility of the balance is measured by the angular 
displacement 6 of the beam with a given small difference of 
load p. If now 6 is small, it is equal to tan ( 23), and 
the sensibility for a given excess weight p is directly pro- 
portional to the length I of the balance arm, inversely 
proportional to the weight w of the beam, and inversely 
proportional to the vertical distance r of its center of gravity 
below the knife-edge. 

The conditions of sensibility do not agree in every particular with 
the requirements of a good balance. Especially is it true that a long 
beam I for high sensibility is directly antagonistic to a short period of 
oscillation ; for the longer the beam, the larger its moment of inertia and 
the longer its period. Then, too, a long beam means a larger weight w 
to secure sufficient rigidity ; and the sensibility is inversely as the weight 
of the beam. 

In the best modern balances the beam has been shortened, in order 
to secure lightness of the movable system and a short period of oscilla- 
tion. At the same time high sensibility has been secured by better 
workmanship on the knife-edges and bearing planes, and by the use 
of aluminum alloys to further reduce the weight of the beam. 

The bending of the beam under a load raises the point C with 
respect to the line through A and B. On this account an increase of 
the load generally produces a decrease in the sensibility. 



ELASTICITY 97 

VIII. ELASTICITY 

120. Strain and Stress. When a body is forced to change 
its size or shape, it is said to be strained, and the deformation 
it undergoes is called a strain. In general, a body resists a 
strain, and the internal restoring force, tending to cause the 
body to revert to its unstrained state, is called a stress. When 
the strained body is in equilibrium, the external deforming 
force is equal and opposite to the stress evoked, and the de- 
forming force may then be called a stress ( 51). The meas- 
ure of the stress is the force per unit area. In the c.g.s. 
system a stress is measured in dynes per square centimeter. 

121. Kinds of Stress. When the effect of a stress on a 
section of a body to which the stress is applied is to increase 
the dimensions of the body at right angles to the section, the 
stress is a tension; when the effect is to diminish this dimen- 
sion, the stress is a pressure. A stress which alters the form 
but not the size of a body is called a shearing stress. The 
deformation which a body undergoes under a shearing stress 
may be aptly illustrated by the aid of a pack of cards, lying 
on a table and forming a rectangular parallelepiped. Im- 
agine a horizontal force so applied as to cause each card to slip 
forward over the next one below it by the same amount 
(Fig. 63). Each card will then c D d 
move forward a distance propor- 
tional to its height above the 

table, and the pack has under- 
gone a shear. 

A stress is called hydrostatic 
pressure when the pressure at 

a point is the same in all directions. For measuring hydro- 
static and gas pressure, the column of mercury which it will 
support is often used. The unit of measure is then the 
pressure at a depth of one centimeter in mercury. A cubic 
centimeter of mercury weighs 13.596 gm. Hence the value 
of this unit of pressure is 13.596 x 980 = 13,324 dynes. 



98 MECHANICS 

The pressure of illuminating gas is measured in inches or 
centimeters of water. A column of water one centimeter 
high produces a pressure per square centimeter of one gram, 
or 980 dynes. 

122. Elasticity. Solid bodies react against a change either 
in shape or volume; fluid bodies react against a change in 
volume only. The property of a body by virtue of which 
it exerts such a reaction and tends to recover from a strain 
is called elasticity. 

A body which exhibits reaction under a change of volume 
has elasticity of volume; a body which reacts against change 
of shape, but without change of volume, has elasticity of form, 
or possesses rigidity. 

If a body suffers a strain, which does not change so long 
as the stress remains the same, and which completely disap- 
pears when the stress is removed, the body is said to be per- 
fectly elastic. Gases and liquids have perfect elasticity of 
volume; that is, they recover perfectly their initial volume 
when the initial pressure is restored. 

There is a limit to the elasticity of solids, called the elastic 
limit, beyond which they yield and are then incapable of re- 
gaining their original form and volume. The elastic limit 
for steel is high, and it breaks before there is much perma- 
nent distortion. On the other hand, lead scarcely recovers 
completely from any distortion, however slight. 

123. Elastic Fatigue. Even within the limits of elasticity, 
solids show distinct differences in their behavior. Some re- 
cover at once after the removal of the force of distortion. A 
fine thread of spun quartz recovers immediately from a twist 
after the torsional force is removed, while a steel wire may 
not recover completely for several hours, if it has been kept 
under torsion for some time. This delay in recovery from 
distortion is said to be due to elastic fatigue. It is very no- 
ticeable in the case of glass, and metals are never free from 
it. If a long glass fiber is kept twisted for some time, it will 



ELASTICITY 99 

largely untwist as soon as it is released from torsion, and it 
will then creep slowly back to its original condition. The 
larger the initial twist and the longer it lasts, the greater is 
the temporary set of the fiber. 

124. Hooke's Law. When the strain in an elastic body 
does not exceed the elastic limit, the reaction, due to the 
strain and tending to restore the body to its unstrained con- 
dition, is proportional to the distortion. This relation is 
known as Hooke's law. It has been verified for most mate- 
rials in common use. 

It follows from Hooke's law that if we know the stress 
corresponding to unit strain of any type, we can find the 
stress corresponding to a strain of any magnitude within the 
elastic limit and of the same type. According to this law, 
stress and strain are connected by the following relation: 

~ stress 

Stress = e x strain, or e = 

strain 

The quantity e is a proportionality factor called the modu- 
lus of elasticity. It is the quotient of stress by strain. 

125. Modulus of Volume Elasticity. If the strain is due 
to a change in the size of the body only, the measure of the 
strain is the diminution suffered by unit volume of the 
strained body; and e becomes the modulus or coefficient 
of volume elasticity. 

Let the body be subjected to a uniform normal pressure p 
over its entire surface. Let V be the original volume, and 
v the diminution in volume. Then v/ V is the compression, 
and this is the measure of the strain. The modulus of 
volume elasticity becomes 

TC= P +- = P V V. (39) 

This is the only modulus or coefficient of elasticity ap- 
plicable to liquids and gases. 



100 MECHANICS 

126. Young's Modulus. When a wire, for example, is 
stretched by a weight, the stress is the force per unit of cross 
section of the wire, the strain is the increase in unit length 
of the wire, and e is then called Young's modulus. 

Let L be the unstrained length of the wire,Z/ + I its length 
when stretched within the limit of elasticity. The strain is 
l/L. If F is the stretching force and A the sectional area 
of the wire, F/A is the stress per unit area. Therefore 
Young's modulus is 

HiH^f ' (40 > 

127. Energy stored in a Strained Body. The work required 
to stretch an elastic wire or slender rod within the limit of 
elasticity is stored in the strained body as potential energy. 
The expression for this energy may be derived from the case 
of a stretched wire. 

Let it be assumed that the load is added so gradually that 
it does not acquire appreciable velocity, so that none of the 
work done becomes kinetic energy. Since Hooke's law 
applies, the force for each increment of the elongation is 
proportional to the whole elongation corresponding. Then 
if L is the initial length of the wire, I the total elongation 
produced by the force -F, the mean working force is J.F, and 
the work done is W l Fl 

But from the definition of stress and strain, 

F = stress x A, and I = strain x L. 

The work done is therefore 

W '= | AL x stress x strain. 

But AL is the volume of the wire, and therefore the energy 
stored in unit volume of the stretched wire is 

W= stress X strain. 



ELASTICITY 10.1 

Problems 

1. What is the acceleration when a force of 40 dynes acts on a mass 
of 5 gm. ? How far will the mass move in 4 seconds ? 

2. A force of 60 dynes acts on a body for one minute and gives to it a 
velocity of 1200 cm. a second. What is the mass of the body? 

3. A mass of 500 gm. is whirled around at the end of a string 40 cm. 
long twice a second. What is the tension in the string in dynes, dis- 
regarding gravity ? 

4. An inelastic mass of 500 kgm., moving with a velocity of 30 m. a 
second, meets a similar mass of 300 kgm., moving with a velocity of 20 m. 
a second in the opposite direction. Find the velocity of the entire mass 
after impact. 

5. Compare the kinetic energy of a ball having a mass of 15 gm. and 
a velocity of 500 m. a second with that of a gun from which it was fired, 
if the mass of the gun is 10 kgm. 

6. A force of 1500 dynes acts continuously on a mass of 10 gm. for 30 
sec. Find the velocity acquired and the space traversed in the 30 sec. 

7. A shot weighing 20 Ib. is fired from a gun weighing 5 tons with 
an initial velocity of 1500 ft. a second. What is the initial velocity of 
recoil of the gun ? 

8. A rapid-firing gun fires 300 bullets of 30 gm. each per minute witli 
a velocity of 300 m. a second. Find the mean force of reaction on the 
gun. 

9. What angular velocity in a vertical plane must be given to an open 
vessel containing water so that no water may be spilled ? 

10. A man weighing 150 Ib. climbs to the top of the Eiffel Tower, 
height 984 ft. How many foot-pounds of work does he do? 

11. How many ergs of work are done in raising a mass of 1 kgm. 
vertically through a height of 5 m. ? 

12. An unbalanced force of 10 kgm. moves a mass of 100 kgm. through 
the distance of 100 m. How much work is done ? 

13. By means of a force of 1000 dynes, a mass is moved a distance of 
200 m. How many joules of work are done ? 

14. How much work in foot-pounds is done against gravity in haul- 
ing a load of 2000 Ib. to the top of a hill 200 ft. high ? If the hill is 2000 
ft. long, what force against gravity is necessary to pull the load up the hill ? 

15. At what rate is an engine working which raises 1000 tons of coal 
in 10 hr. from a mine 300 ft. deep ? 



102 ; MECHANICS 

16. A steam pnmp fills a tank with water in 4 hr. The capacity of 
the tank is 5000 gal. and the elevation is 40 ft. If a gallon of water 
weighs 8 lb., what is the horse power of the pump ? 

17. An electric motor raises an elevator cage whose unbalanced weight 
is 2000 kgm. through a height of 40 m. in 40 sec. What is the power of 
the motor in kilowatts? 

18. How many horse powers are transmitted by a rope passing over a 
pulley 16.5 ft. in circumference and making one revolution a second, the 
tension in -the rope being 200 lb. ? 

19. A dynamometer pulley is 32 cm. in diameter and the cord 1 cm. 
The weight at one end to produce tension is 1760 gm., and when the 
pulley makes 720 revolutions per minute, the tension of the cord shown 
by the spring S (Fig. 31) is 14 kgm. What is the power absorbed by the 
dynamometer in kilowatts and in horse powers ? 

20. Two men carry a weight of 150 kgm. slung on a pole 280 cm. long. 
If the weight be placed at the distance of 100 cm. from one end, what 
portion of the weight does each man carry ? 

21. Two cylinders of the same uniform material, each 30 cm. long, 
and of diameters 12 cm. and 8 cm. respectively, are joined end to end so 
that their axes are in the same straight line. Where is their common 
center of gravity ? 

22. A circle 20 cm. in diameter has cut out of it a smaller circle tan- 
gent to it and 12 cm. in diameter. Where is the center of gravity of the 
remainder ? 

23. A cylinder whose mass is 2000 gm. and radius 10 cm. rotates on its 
axis 300 times a minute. Find its kinetic energy of rotation. 

24. A pendulum beating seconds -at one place is carried to another 
station where it gains 10 sec. a day. Compare the accelerations of 
gravity at the two places ? 

25. A uniform slender rod 1 m. long is pivoted at one end so as to 
swing as a pendulum. Calculate its period of vibration at a place where 
g is 980 cm. per second per second. 



CHAPTER IV 

MECHANICS OF FLUIDS 
I. MOLECULAR PHENOMENA IN LIQUIDS 

128. Characteristics of a Fluid. A fluid has no shape of its 
own, but takes the shape of the containing vessel. It cannot 
resist a stress unless it is supported on all sides. A perfect 
fluid would offer no resistance to a shearing stress. The 
molecules of a fluid at rest are displaced by the application of 
the slightest force ; that is, a fluid yields to the continued 
application of a force tending to change its shape. Never- 
theless fluids exhibit wide differences in mobility, or readiness 
in yielding to a shearing stress. Alcohol, gasoline, and sul- 
phuric ether are examples of very mobile liquids ; glycerine 
is very much less mobile. In fact, liquids shade off gradually 
into solids, and there are intermediate bodies which exhibit to 
some degree the properties of liquids as well as of solids. A 
stick of sealing wax supported at its ends yields continuously, 
though very slowly, to its own weight. A cake of shoe- 
maker's wax on water, with bullets on it and corks under it, 
yields to both and is traversed by them in opposite directions. 
The wax will flow very slowly down a tortuous channel. It 
is therefore mobile, and its mobility increases with its tem- 
perature. At the same time both sealing wax and shoe- 
maker's wax when cold break readily under the blow of a 
hammer like a solid. 

129. Viscosity. The resistance of a fluid to flowing under 
stress is called viscosity. It is due to molecular fluid friction. 
The slowness of the descent of a fine precipitate in water is 

1G3 



104 MECHANICS 

due to the viscosity of the liquid ; and the slowness of the 
fall of fine raindrops, or a cloud, is due to the viscosity of the 
air. Viscosity varies through wide limits. It is less in gases 
than in liquids, and in general decreases as the temperature 
rises. Hot water is less viscous than cold water; hence the 
relative ease with which a hot solution filters. 

A body set oscillating in air has its vibrations damped by 
the molecular friction of the air, that is, by its viscosity ; if 
suspended in a liquid, the damping is much more pronounced. 
If the body is wetted by the liquid, the damping is due en- 
tirely to viscosity and is independent of the nature of the 
suspended body ; for there is no loss of velocity by friction 
between the solid arid the liquid. 

A perfect fluid would be entirely without rigidity and 
viscosity. 

130. Liquids and Gases. Fluids include both liquids and 
gases. The two may be distinguished by two characteristic 
properties : 

First. Liquids have a free surface, while gases cannot per- 
manently retain a free bounding surface, independent of the 
containing vessel. A gas introduced into an empty vessel 
completely fills it, whatever its volume. 

Second. Liquids are but slightly compressible, while gases 
are highly compressible, and tend to expand to an indefinitely 
large volume. A liquid offers great resistance to a stress 
tending to reduce its volume, while a gas offers relatively 
small resistance. Both have perfect elasticity of volume, but 
their coefficients of elasticity differ greatly. Water, for 
example, is reduced in volume only 0.00005 by a pressure of 
one atmosphere ( 171), while air is reduced to one half by 
the same increase of pressure above that of the atmosphere. 

131. Cohesion in Liquids. In liquids the molecules are 
within the sphere of one another's attraction. This attrac- 
tion accounts for the viscosity of even the most mobile liquids. 
A liquid is hindered in its flow by molecular friction. Molec- 



MOLECULAR PHENOMENA IN LIQUIDS 105 

ular attraction accounts for the fact that a small stream of 
liquid has a certain tenacity and does not break readily. 

If a clean glass rod be dipped in water and then withdrawn, a drop 
will adhere to the end of the rod until enough water has run down the 
rod to increase the weight of the drop to a point where it falls as a little 
sphere of water. Its spherical form is due to the attraction between its 
molecules, which gives to it uniform molecular pressure and a minimum 
surface. 

Cohesion in a liquid is due to the attraction existing among its mole- 
cules. It is to be noted that this attraction acts only at insensible dis- 
tances. It diminishes rapidly as the distance increases and vanishes at 
a range something like the twenty thousandth of a millimeter. 

132. Phenomena at the Surface of a Liquid. Bubbles of 
gas formed in the interior of a cold liquid rise to the surface 
and often show some difficulty in breaking through. A 
sewing needle carefully placed on water floats. The sur- 
face of the water around the needle is depressed and the 
needle rests in a little hollow. Let two bits of wood float 
on water a few millimeters apart. When a drop of alcohol 
is applied to the surface between them, they suddenly fly 
apart. A thin film of water may be spread over a chemically 
clean glass plate ; but if a drop of colored alcohol falls on 
this film, the film will break, the water retiring and leaving a 
dry area around the alcohol. 

The sewing needle indents the surface of the water as if 
this surface were a tense membrane or skin, and tough enough 
to support the needle. This membrane is weaker in alcohol 
than in water ; hence the moving apart of the bits of wood 
and the withdrawal of the water from the spot weakened by 
alcohol. 

133. Surface Tension. It is apparent that the surface of a 
liquid is physically different from the interior. The mole- 
cules composing the surface film are not under the same con- 
ditions of molecular equilibrium as those in the interior of 
the liquid. Let e be the range of molecular attraction. 
Then at any point in the interior of the liquid, at a distance 



. JEL ._}__ 




106 MECHANICS 

from the surface greater than e, each molecule is attracted 
equally in all directions. But at or very near the surface 
the attraction downward is not balanced by an equal attraction 
upward, and in consequence the molecules along the surface 
are crowded together so as to form what may be considered 
an elastic film. 

Let mn (Fig. 64) be the surface of the liquid, and let m'n' 
be a parallel plane at a distance e below the surface. For 

any point P above the 
plane m'n' the attrac- 
tion downward is 
greater than the attrac- 
tion upward. About 
P as a center describe 
a sphere with a radius 
e. Then the normal 

Fig. 64 

pressure on a plane 

through P perpendicular to mn is greater than on a plane 
parallel to mn. The upward attraction on P varies from a 
maximum at m'n' to zero at the surface. As P rises, the 
upper half of the sphere described about P contains a dimin- 
ishing number of molecules, but horizontally the attraction 
remains unchanged. From this inequality there arises a 
stress, causing the surface to contract. 

If we imagine the surface to be enlarged by forcing mole- 
cules out along the plane through P normal to the surface, 
then work must be done on them to transfer them from the 
interior and to spread them out against the force pressing the 
molecules together along the surface. It follows that an in- 
crease in the surface means an increase in potential energy. 
A surface film therefore possesses surface energy; and as 
potential energy always tends toward a minimum, the surface 
contracts to as small dimensions as possible. 

134. Shape of a Small Liquid Mass. A small mass of free 
liquid always tends to assume a spherical form, because the 



MOLECULAR PHENOMENA IN LIQUIDS 107 

volume inclosed by a sphere is a maximum relative to the area 
of its surface. The smaller the mass of the liquid, the larger 
its surface in proportion to its weight, and hence the smaller 
the influence of gravity in distorting it from the spherical 
form. Small globules of mercury on clean glass approach 
the nearer to a spherical form the smaller the globules. 
Drops of rain and dew are nearly spherical because of surface 
tension. 

When molten lead flows from a small orifice, the surface 
tension around the small stream throttles it and cuts it into 
segments ; again, surface tension molds these small detached 
masses into spherical form as soon as their oscillations have 
died out by the internal friction due to viscosity. If they 
rotate as they descend, they remain quite spherical and strike 
the water at the bottom of the shot tower as solid shot. 

An ingenious method of separating the perfect shot from the imper- 
fect ones consists in causing all to roll together down a smooth inclined 
plane. Near the bottom is a transverse slit. The perfect shot acquire 
enough motion to carry them safely across, while the imperfect ones 
hobble down and fall into the slot. 

135. Further Illustrations of Surface Tension. A mixture of 
alcohol and water may be made with the same density as that of olive oil. 
A small mass of the oil placed in this mixture will assume a globular 
form, since it is not distorted by its weight. If the attending conditions 
do not permit it to become spherical, 
it will in every case assume a form 
having the smallest surface under 
the given conditions. If, for ex- 
ample, a metallic ring be immersed 
in a large globule of olive oil sus- 
pended in the mixture and some of 
the oil be then removed by means 
of a pipette, the remainder will take 
the form of a double convex lens. 

Make a stout wire ring three or Fig 65 

four inches in diameter with a 

handle (Fig. 65). Tie to this a loop of thread so that the loop may hang 
near the middle of the ring. Dip the ring into a good soap solution con- 
taining glycerine, and obtain a plane film. The thread will float in it, 




108 



MECHANICS 




Fig. 66 



Break the film inside the loop with a warm pointed wire, and the loop 
will spring out into a circle. The tension of the film attached to the 
thread pulls it out equally in all directions. By tilting the ring from 
, side to side, the circle may be made to float about on 

the film. 

Interesting surfaces may be obtained by dipping 
skeleton frames made of stout wire into a soap solu- 
tion. The films in Figure 66 are all plane, and the 
angles where three surfaces meet along a line are 
necessarily 120 for equilibrium. 

A small bit of camphor gum placed on warm 
water, perfectly free from any oily film, will execute 
rapid gyrations across the surface. The camphor 
dissolves unequally at different points, and thus produces an unequal 
weakening of the surface tension in different directions. An interesting 
modification of this experiment is to make a miniature tin or wooden 
boat, with a notch in the stern to hold a bit of camphor gurn. The 
camphor weakens the tension astern, and the tension at the bow draws 
the boat forward. 

136. Surface Energy and Surface Tension. If we call the 
loss of potential energy, due to a diminution in the surface 
of one unit, the surface energy per unit area, it can be shown 
that this is numerically equal to the surface tension per 
unit width of the film. The following should be regarded as 
illustrating this relation rather than a model method of 
measurement. 

Let a soap film be stretched on a frame BOD (Fig. 67) 
with the light rod A movable. Denote the length of the 
rod between B and D by a, and let n 

the rod be drawn downward a dis- 
tance b. Then the increase in the 
surface for the two sides of the film 
is 2 ab ; and if E is the superficial 
energy per unit area, the increase 
in the potential energy of the sur- 
face is 2 Eab. 

The work done against the surface tension perpendicular 
to A may be placed equal to the increase in surface energy. 













B 








D 






*>! a 








- 


A 

Fig. 67 


- 





MOLECULAR PHENOMENA IN LIQUIDS 109 

Let T be the surface tension per unit width ; then the whole 
tension in the two sides of the film is 2 Ta. The work done 
against this force through the distance b is 2 Tab. Hence the 
equation 2 Tab = 2 Hab, 

or T= E. 

The surface tension per unit width is therefore equal to the 
surface energy per unit area. 

For pure water and air the surface tension is .75.8 dynes 
per centimeter width; for mercury it is 527.2 dynes, bo.th 
at C. 

137. Theory of the Spread of Oil on Water. Suppose a 
drop of oil to be placed on water (Fig. 68). There are then 
three fluids in contact : 
(a) air, (6) water, and 
(c) oil, and three surface 
tensions act on a particle at 
0; namely, T ab , between air ~ ^_&c . 

and water, T ac , between air 

and oil, and T bc , between water and oil, in the direction of 
the arrows respectively. Then if the three tensions are in 
equilibrium, they may be represented by the sides of a tri- 
angle taken in order ( 102), and the angles made by the 
three surfaces will depend only on the relative magnitudes 
of the three surface tensions. 

If now T ab > (T ac + T bc ), the two forces T ac and T bc cannot 
be in equilibrium with the third force T ab , and the particle 
at must move in the direction of the force T ab . 

For air,* water, and oil the tensions are as follows : 

^ = 75. 8 dynes; T ac = 36.88 dynes ; T bc = 20.56 dynes, 
all per centimeter width. 

Then 75.8 > (36.88 + 20.56), 

and the oil at moves until the whole surface of the water 
is covered with a thin film of oil. Although the oil is spread 
by the pull of the superior surface tension, it is customary to 




110 MECHANICS 

say that oil spreads itself on water. The so-called oiling of 
the sea takes place in this manner. 

If a small drop of turpentine be placed on clean water by 
means of a thin glass rod, and the surface be viewed by 
strong reflected light, the turpentine may be seen spreading 
over the water like a flash of light. 

138. Pressure within a Bubble. If a soap bubble is blown on a 
large thistle tube and the open end of the tube is held near a candle 
flame, the contraction of the bubble on account of surface tension will 
expel the air with sufficient force, it may be, to blow out the flame. The 
surface tension produces a normal pressure inward in excess of atmos- 
pheric pressure. Call this excess p per unit area. It may be expressed 
in terms of surface tension per unit width and the radius of the bubble 
as follows : 

Imagine the bubble to be divided into two halves by a plane through 
its center. Then the pressure on the two halves of the bubble normal 
to this plane, which is the sum of all the components of p normal to the 
plane on one of the halves, is the area of the section times the pressure 
p, or 7rR 2 p, R being the radius of the bubble. 

This pressure is equivalent to the surface tension all around the cir- 
cumference bounding the section, which for both inside and outside sur- 
faces equals 2 T x 2irR,or4:TrR T. Then 



and ,= . 

From this expression it will be seen that the pressure inside a 
bubble increases as the bubble gets smaller. If fog and cloud consisted 
of small vesicles of water, as some have supposed, they would still be 
heavier than the air displaced, both because of the weight of the water 
and because the air within is under greater pressure than the pressure of 
the atmosphere on the outside. 

139. Capillary Phenomena. If a fine glass tube, common- 
ly called a capillary tube, be partly immersed vertically in 
water, the water will rise higher in the tube than the level 
outside ; and the smaller the diameter of the tube, the higher 
will the water rise (Fig. 69). On the other hand, mercury 
in the tube is depressed below the level outside. 



MOLECULAR PHENOMENA IN LIQUIDS 



111 



Similarly, if two chemically clean glass plates, inclined at 

a very small angle, be supported with their lower edges in 

water, the height to which the water rises 

at different points is inversely as the dis- 
tance between the plates at the points, 

and the water line is a curve known as a 

rectangular hyperbola (Fig. 70). By 

coloring the water slightly this curve 

may be readily projected on a screen. 
It is easy to determine that the free 

surface of a liquid is not horizontal near 

the sides of the vessel containing it, but 

is noticeably curved. When the liquid 

wets the vessel, as water in glass, the surface is concave and 

the water rises along the glass; when the liquid does not 

adhere, as mercury in glass, 
the surface is convex and the 
mercury is depressed. 




Fig. 69 




Fig. 70 



The elevation or depression 
varies with the material of the 
tube and the nature of the liquid. 
Water in glass rises to a higher 
level than any other liquid. Capil- 
lary elevation for water is nearly 
three times as great as for sulphuric ether or bisulphide of carbon. A 
rise of temperature causes a decrease in the elevation and the depression 
respectively. 

Familiar examples of capillary action are numerous. Blotting paper 
absorbs ink in its fine pores, and oil rises in a wick, by capillary action. 
A sponge absorbs water for the same reason. The spread of water 
through a lump of sugar may be similarly explained. Small objects 
drift together on water or cling to the sides of the vessel because of 
capillary action; and for the same reason water rises around a fine wire 
and interferes with its free rotation. 

140. Capillary Elevation and Depression Due to Surface Ten- 
sion. The surface tension between air and a liquid acts 
around the inner circumference of the tube, downward in the 



112 



MECHANICS 



case of a convex meniscus, as the curved surface is called, 
and upward in the case of a concave meniscus. Let h (Fig. 

71) be the mean elevation 
of the liquid in the tube. 
Then if the liquid wets 
the wall of the tube, the 
angle is very nearly zero. 
The entire surface tension 
around the interior of the 
tube where the film adheres 
is %7rrT, r being the inner 
radius of the tube, and this 
force acts vertically. The 
force, lifting the liquid and depressing the tube, is in equilib- 
rium with the weight of the liquid column of the height h. 
Let 8 be the weight of unit volume of the liquid. The 
weight of the whole column is irr^hs. Consequently, 




Fig. 71 



2 T 
rs 



(41) 



or the elevation is inversely as the radius or the diameter of 
the tube. This expresses the law of capillary elevation and 
depression. 

141. Osmosis. If layers of chloroform and sulphuric ether 
are separated by a layer of water in a closed bottle, and are 
left undisturbed, in course of time the ether will pass into 
the chloroform. The ether dissolves to some extent in the 
water and is removed from the other side by the chloroform. 
The water does not permit the chloroform to pass to an 
appreciable extent into the ether. 

A thin sheet of India rubber between alcohol and water 
allows the alcohol to pass into it and to be removed from the 
other side by the water. This process is properly called 
diffusion through a membrane. 



MOLECULAR PHENOMENA IN LIQUIDS 



113 



When a porous partition is employed between the two 
substances, the passage of two liquids through it at different 
rates is called osmosis. Tie a piece of dampened parchment 
over the large end of a thistle tube so that it shall be water- 
tight. Fill to the bottom of the funnel part with a saturated 
solution of copper sulphate and immerse in water, so that 
the level of the liquids outside and in shall be the same. 
The level of the liquid within the tube will rise slowly, 
while the water outside will acquire a bluish tint ; thus, while 
some copper sulphate passes out through the membrane, 
more water passes in and raises the level inside. The 
process continues until the hydrostatic pressure in the long 
thistle tube produces an equilibrium of flow in the two direc- 
tions through the membrane. 

142. Osmotic Pressure. Pfeffer devised a porous parti- 
tion by depositing ferrocyanide of copper in the pores of an 
unglazed porcelain cylinder. Such a partition is called a 
semipermeable membrane, because it is permeable to water 
in one direction and only slightly to a solution of sugar in 
the other. The cylinder, filled with a solution of cane sugar, 
was submerged in water, and the pressure inside required 
to equalize the flow in the two directions was measured by 
Pfeffer by means of a column of mercury. The pressure is 
called osmotic pressure. Some of his results are the following : 



PERCENTAGE 
OF SUGAR IN 
SOLUTION 


OSMOTIC PRESSURE 
p IN CM. 
OF MERCURY 


MASS OF SUGAR 

PER CU. CM. OF 

THE SOLUTION 


VOLUME v CON- 
TAINING 1 GRAM- 
MOLECULE (342 
GM.) OF SUGAR 


PRODUCT 

pv. 


1 


53.5 


0.01004 


34,064 


1,822,400 


o 


101.6 


0.02016 


16,968 


. 1,723,600 


4 


208.2 


0.04063 


8,417 


1,752,500 


6 


307.5 


0.06119 


5,589 


1,718,600 



These measurements were made at the same temperature, 
and it will be observed that the product pv is a fairly con- 
stant quantity. 



114 MECHANICS 

Measurements of the change of osmotic pressure with tem- 
perature have shown that for a one per cent sugar solution 
the following relation holds: 

p = 49.62(1 + 0.003670- 

The constant 49.62 is the pressure p at 0. The coeffi- 
cient 0.00367 = 1/273 is the same as the coefficient of the 
change of pressure for air with change of temperature. 

A further discussion of these relations must be reserved 
for a later chapter in the subject of Heat. 

II. PRESSURE IN FLUIDS 

143. Laws of Fluid Pressure. The three fundamental char- 
acteristics of pressure in fluids at rest may be called the laws 
of fluid pressure. They are: 

I' Fluid pressure is normal to any surface on which it 
acts. 

II. Fluid pressure at a point in a fluid is of the same 
intensity in all directions. 

III. Pressure applied to a fluid is transmitted undimin- 
ished in all directions. 

Fluid pressure is measured in terms of the pressure per 
unit area. By pressure at a point is meant the pressure per 
unit area at the point. If the pressure over a surface is not 
uniform, then we may consider only a surface about the point 
so small that the pressure over it is practically constant, and 
the quotient of the force on it by this small area will be the 
pressure per unit area. 

144. Pascal's Principle. The first of the laws of fluid pres- 
sure is a consequence of the mobility of a fluid. If the pres- 
sure is not perpendicular to the surface, it can be resolved 
into a normal component and one parallel to the surface. 
This latter component would produce motion of the fluid 
parallel to the surface; but since the fluid is assumed to be 



PRESSURE IN FLUIDS 115 

at rest, no such motion can take place, and therefore the 
pressure exerted by a fluid on any surface is normal to that 
surface at every point. 

The other two laws are included in Pascal's principle enun- 
ciated in 1653. It is founded on the equal transmission of 
pressure in all directions. A solid transmits pressure only 
in the direction in which the force acts; but a fluid, either a 
liquid or a gas, transmits pressure in all directions. Hence 
Pascal's fundamental law of the mechanics of fluids: 

Pressure applied to an inclosed fluid is transmitted 
equally in all directions and without diminution to every 
part of the fluid and of the walls of the containing vessel. 

If a small cubical element of the fluid anywhere in the 
interior be imagined solidified without other change, this ele- 
ment will remain in equilibrium however it be turned about. 
But for equilibrium the forces acting on all the 
faces of this small cube balance, or the pressures 
on all the faces are the same. But the faces all 
have the same area ; therefore the intensity of 
the pressure on all the faces is the same. 

If a thin-walled bottle be filled with water and a close- 
grained cork be fitted to it, pressure applied to the cork 
(Fig. 72), with a lever if necessary, may cause the bottle 
to break, especially if it has flat sides. The whole burst- 
ing force is equal to the product of the area of the inner 
surface of the bottle and the pressure per unit area. 
Thus, if the inner surface be 40 square inches and the 
force applied to the cork 25 pounds per square inch, the p . ?2 

bursting pressure is 1000 pounds of force. 

When a balloon is inflated by pumping into it illuminating gas or 
hydrogen, the balloon swells out equally in all directions. This fact of 
equal pressures in all directions in the gas is evident in the inflation of 
a toy balloon or a soap bubble. 

145. The Hydraulic Press. Pascal's principle has an im- 
portant application in the hydraulic press invented by Joseph 
Bramah in 1795, and hence sometimes called a Bramah press. 




116 



MECHANICS 



It is employed for exerting great pressure, as in baling cot- 
ton, making lead pipe, and lifting heavy masses of metal in 
steel mills, locomotive works, and on warships. 

Figure 73 is a section showing the principle. A metal 
piston passes water-tight through the collar n of the large 
cylinder, while the piston p is worked up and down as a force 
pump, pumps water from a reservoir at the bottom and 
forces it through the pipe K into the cylinder B. When 
the plunger p is descending, the water transmits the applied 
pressure to the base of the large piston or ram, which is thus 




Fig. 73 

forced up with its load. If the cross-sectional area of the 
plunger p is a units and the force applied to it is P, the 
pressure per unit area on the water is P/a. This pressure is 
transmitted to B and acts on each unit of surface there. If 
the area of the base of the piston in B is A units, the force 
W acting on the piston is AP/a. Hence the mechanical 
advantage is . 2 



where D is the diameter of the ram and d that of the pump 
plunger. 

If friction be neglected, this machine conforms to the prin- 
ciple of work, for it is evident that the small piston travels 
as much farther than the large one as the force exerted on 



PRESSURE IN FLUIDS 



117 



the large piston is greater than the effort applied to the 
plunger of the pump. 

146. Fluids acted on by Gravity. The weight of each hori- 
zontal layer of a fluid at rest is transmitted to every layer at 
a lower level. The pressure in the lower layers is then greater 
than in the upper ones, since each layer supports the weight 
of all those above it. But the pressure throughout any hori- 
zontal layer is everywhere the same ; otherwise the fluid 
would flow from points of higher pressure in the horizontal 
plane to those of lower pressure, since 
no work would have to be done against 
gravity, so long as the motion is in the 
same horizontal plane. 

The pressure on any horizontal plane 
in a liquid due to the weight of the 
liquid itself is proportional to its depth 
below the surface ; for at every point 
of such a plane is supported a column 
of the liquid of the same height, and 
this height is the depth of the layer. 

Moreover, the pressure at a point is 
the same in every direction. If three 
glass tubes, bent as shown in Figure 
74, be filled to the same height with 
mercury, when they are immersed so 
that their inner open ends are at the 
same level, the bottom of the bends 
resting on the bottom of the tall jar, 
the difference in level of the mercury 
is the same in the three tubes ; so that at any point the pres- 
sure downwards, sideways, and upwards is the same. Also, 
the pressure measured in this way will be found to be pro- 
portional to the depth. 

It is immaterial whether the pressure on any horizontal 
plane is due to the weight of the liquid or is in part due to 




Fig. 74 



118 



MECHANICS 



an externally applied pressure. The equality of pressure in 
all directions is a consequence of the equal transmission of 
pressures in all directions. All the pressures in a liquid at 
rest must be balanced, since unbalanced pressure would pro- 
duce currents in the liquid. 

147. Pressure on the Bottom of a Vessel Independent of its 
Shape. Vessels known as Pascal's vases, made to screw into 
the same ring base with a removable bottom (Fig. 75), are 
used to demonstrate that the pressure on the detachable 

bottom is independent of the shape 
of the vessel. The bottom is sus- 
pended from one arm of a balance 
with an appropriate weight in the 
scale pan on the other side. The 
three vessels are successively screwed 
into the same base ; it is found that 
the bottom is detached and allows the 
water to escape when it reaches the 
same height, notwithstanding the 
difference in the weight of water 
required to fill the three vessels to 
the same level. 

.Therefore, the pressure on the 
F|S ' 75 bottom of a vessel is independent 

of the shape of the vessel. The apparent contradiction in- 
volved in the fact that unequal masses of water produce 
equal pressures has often been called the hydrostatic 
paradox. 

148. Total Pressure on an Immersed Surface. Liquid pres- 
sure depends on the depth and on the weight of a unit cube 
of the liquid. The pressure on any horizontal area may be 
calculated as follows : 

Let A denote the area pressed upon, j5"its depth below the 
surface, and w the weight of a unit of volume of the liquid. 
Then the pressure is equal to the weight of a cylindrical 




PRESSURE IN FLUIDS 119 

column of the liquid, the base of which has an area A, and its 
height H. It is therefore 

P=AHw. (42) 

In the metric system, w is 1 gm. per cubic centimeter ; in 
the English system, w is 62.4 Ib. per cubic foot, both for 
water. 

If the surface pressed upon is plane but not horizontal, 
then the average pressure over its surface will be the pressure 
on a unit area at the depth of its center of figure, or center 
of gravity. For the whole pressure on such a surface the 
expression is again P AH 

where His the depth of the center of gravity of the surface. 
Hence the general expression for the pressure on a surface 
immersed in any liquid is, the weight of a column of the liquid, 
the base of which is equal in area to the surface pressed upon, 
and its height, the depth of its center of gravity below the sur- 
face of the liquid. 

149. Surface of a Liquid at Rest. The free surface of a 
liquid at rest is horizontal. Consider a particle m at some 
point B of the free surface 
ABD (Fig. 76) which we may 
suppose is not horizontal. The 
vertical force on m is mg, repre- 
sented by BW. This force 
may be resolved into two com- 
ponents, one normal to the sur- 
face, and the other, B 0, parallel 
to the surface. Since the air 

pressure on the surface is everywhere the same, there is no 
hydrostatic pressure to resist this latter component of pres- 
sure; and as there is no friction of rest in a liquid, the 
particle m at B must move. When the surface is level, BO 
vanishes and there is no motion. 

The sea or any large expanse of water is a part of the 





120 MECHANICS 

spheroidal surface of the earth. When one looks with a field 
glass at a long straight stretch of the Suez Canal, the water 
and the retaining wall, as contrasting surfaces, may be seen 
to curve over in a vertical plane. 

III. BODIES IMMERSED IN LIQUIDS 

150. Buoyancy. An iron ball sinks in water and floats in 
mercury. An egg sinks in fresh water and floats in a satu- 
rated solution of common salt. A piece of oak floats in water, 
but the dense wood lignum-vitse sinks. When a swimmer 
wades up to his neck in sea water, he is nearly lifted off his 
feet by the heavy salt water which buoys him up. 

The resultant of the upward pressure of a liquid on a body 
immersed in it is a vertical force, and it counterbalances a 
part or the whole of the body's weight. This resultant 
upward pressure of a fluid is called buoyancy. 

Suspend a brass or iron weight from the hook of a spring balance and 
note the weight. Now bring a beaker of water up under the weight and 
submerge it. Its apparent weight will be diminished. If salt water is 
used, the apparent loss of weight will be greater ; if kerosene, it will be 
less. In the shortened form of popular language the body immersed is 
said to have suffered a loss of weight, though its real weight has not 
changed in the least. Another force has been brought to bear on it, 
namely, the excess upward pressure of the liquid, called buoyancy. 

151. The Principle of Archimedes. The law of buoyancy 
was discovered by Archimedes about 240 B.C. It is as 
follows : 

dL body immersed in a liquid is buoyed up by a force equal 
to the weight of the liquid displaced by it. 

Suppose a cube immersed in water (Fig. 77). The pres- 
sure on its vertical sides a and b are equal and in opposite 
directions. The same is true of the other pair of vertical 
faces. The resultant horizontal pressure is therefore zero. 
On d there is a downward pressure equal to the weight of 
the column of water having the face d as a base and a height 



BODIES IMMERSED IN LIQUIDS 



121 




Fig. 77 



dn. On the bottom c there is an upward pressure equal to 

the weight of a column of water having a base c and a 

height en. The upward pressure 

therefore exceeds the downward 

pressure by the weight of a prism 

of water, the base of which is the 

face c of the cube, and its height 

the difference between dn and en 

or cd\ and this is the weight of 

the volume of water displaced by 

the cube. 

In general, if we consider any 

portion of the mass of water solidified without other change, 
its equilibrium w r ould not be disturbed 
and its own weight may be considered as 
acting vertically downward through its 
center of gravity. The resultant liquid 
pressure 011 its surface must therefore be 
equal to its weight and must act verti- 
cally upward through its center of gravity 
for equilibrium. 

A metallic cylinder 5.1 cm. long and 2.5 cm. in 
diameter has a volume of almost exactly 25 cm. 8 
Suspend it by a fine thread from one arm of a 
balance (Fig. 78) and counterpoise. Then submerge 
it in water as in the figure. The equilibrium will 
be restored by placing 25 gm. in the pan above the cylinder. The 
cylinder displaces 25 cm. 3 of water weighing 25 gm. and its apparent 
loss of weight is also 25 gm. Strictly the temperature of the water 
should be 4 C. 

152. Equilibrium of Floating Bodies. A body cannot float 
partly immersed in a liquid unless it is specifically lighter 
than the liquid. If 110 other force acts on the body, it will 
sink until the weight of the displaced liquid exactly equals 
that of the body. 

In liquids the buoyancy is practically independent of the 




Fig. 



122 



MECHANICS 



depth so long as the body is completely immersed, but it 
will decrease as soon as the body begins to emerge from the 
liquid. Hence, 

When a body floats on a liquid, it sinks to such a depth that 
the weight of the liquid displaced equals its own weight. 

The weight of a body acts vertically downward, and the 
resultant pressure of the liquid acts vertically upward 
through the center of gravity of the displaced liquid, called 
its center of buoyancy, for equilibrium. These two forces 
must be equal and in the same vertical line. 




Fig. 79 

Let Figure 79 be the section of a floating body, such as a boat, G its 
center of gravity, and b the center of buoyancy. Then for equilibrium 
G and b must be in the same vertical line, as in A. If the body be dis- 
placed by tilting, as in B, the center of buoyancy will also be displaced 
to some point b r . The floating body is then acted on by a couple con- 
sisting of the upward pressure through b' and the weight of the body 
acting downward through G. This couple tends to restore the body to 
its position of equilibrium. The point M, where the new vertical 
through b r cuts the medial line Gb, is called the metacenter. For angular 
stability the metacenter must be above the center of gravity of the float- 
ing body, and the farther above the greater the angular stability. If 
the center of gravity of the floating body is so high and its shape such 
that its center of gravity is displaced further in tilting than the center 
of buoyancy, then the metacenter will be below, G the couple acting 
on the body increases the displacement, and the body is in unstable 
equilibrium. 

A floating body is stable so long as the metacenter remains above its 
center of gravity. If the metacenter coincides with the center of gravity, 
as in the case of a uniform sphere, the body is in neutral equilibrium ; 



DENSITY AND SPECIFIC GRAVITY 



123 



but if the metacenter is below the center of gravity, the equilibrium is 
unstable. The effect 6f ballast in a ship is to lower its center of gravity 
and so to increase its angular stability. 

153. The Cartesian Diver. Descartes illustrated the principle of 
Archimedes by means of a hydrostatic toy, called the Cartesian diver. It 
is made of glass, is hollow, and has a small opening 
near the bottom. The figure is partly filled with 
water so that it just floats in a jar of water (Fig. 80). 
When pressure is applied to the sheet rubber tied 
over the top of the jar, it is transmitted to the water, 
more water enters the floating figure, and the air in 
it is compressed. The figure then displaces less water 
and sinks. When the pressure is relieved, the air in 
the diver expands and forces water out again. The 
actual displacement of water is then increased and 
the figure rises to the surface. The water in the 
diver may be so nicely adjusted that the little figure ^ 
will sink in cold water, but will rise again when the 
water has reached the temperature of the room and 
the air in the figure has expanded. 

A good substitute for the diver is a small homoeopathic vial in a flat 
12-oz. prescription bottle filled with water and closed with a rubber 
stopper. By pressing on the flat sides of the bottle, the bottle yields, the 
air in the diver is compressed, and it sinks. 




IV. DENSITY AND SPECIFIC GRAVITY 

154. Density. The density of a body is the number of units 
of mass of it contained in a unit of volume. In the c. g. s. sys- 
tem it is the number of grams per cubic centimeter. If m 
denotes mass, v volume, and d density, then 



r m m i 7 

.= , v , and m vd. 
v d 



(43) 



155. Specific Gravity. The specific gravity of a body is the 
ratio of the mass of any volume of it to the mass of the same 
volume of pure water at 4 C. Specific gravity is, therefore, 
only the relative density as compared with water. The 
specific gravity of solids and liquids is numerically equal to 



124 MECHANICS 

their density when the latter is expressed in grams per cubic 
centimeter, since the density of water is tnen unity. 

Let m be the mass of a body and m' the mass of an equal 
volume of the standard, as water. Then the specific gravity 
8 = m/m r and m = m's. If the mass of the water is expressed 
in pounds and its volume in cubic feet, m r = v x 62.4, and 
m = v x 62.4 x s. 

Since the density of water at 4 C. in the c. g. s. system is 
sensibly unity, there is no occasion to use the term specific 
gravity. Whatever system of units is used to determine 
specific gravity, the result will be numerically equal to the 
density in the c. g. 8. system. Conversely, if the density is 
determined in the c. g. %. system, the numeral expressing the 
result is always the specific gravity. 

156. Density of Solids. A. Solids heavier than water. 
The density of a solid insoluble in water may be found by 
weighing the body first in air and then suspended in water. 
Its apparent loss of weight in water is, by the principle of 
Archimedes, equal to the weight of the water displaced, that 
is, the weight of a mass of water of the same volume as the 
immersed solid. Hence the quotient of the weight in air by 
the loss of weight in water is the specific gravity ; it is also 
the mass in grams per cubic centimeter of the solid, or its 
density. 

If the water is not at the temperature of maximum density, 
then the value found by the process just described should be 
multiplied by the density D of the water at the temperature 

of the 'observation, and 

d = sD. 

B. Solids lighter than water. Employ a sinker heavy 
enough to make the body sink in water. Counterbalance 
with the body in the scale pan and the sinker suspended from 
the same pan by a fine thread and immersed wholly in water. 
Transfer the body from the scale pan to the water and attach 
it to the sinker. The weight w', which must be added to the 



DENSITY AND SPECIFIC GRAVITY 



125 



scale pan to restore the equilibrium, is the weight of the water 
displaced by the body. It is not necessary to know the weight 
of the sinker. Then if w is the weight of the solid in air, 
the apparent density in grams per cubic centimeter is w/w'. 

If the solid is soluble in water, a liquid of known 
density, in which the body is not soluble, must be used 
instead of water. If D is the density of this liquid, then 
d = sD as above. 





Fig. 81 



157. Density of Liquids. A. By the specific gravity bottle. A spe- 
cific gravity bottle (Fig. 81) is usually made to hold a definite amount of 
distilled water at a specified temperature, for 
example, 25, 50, or 100 gm. To check this capacity, 
weigh the bottle empty and dry, and weigh again 
when filled with distilled water at the temperature 
marked on the bottle. The difference, corrected 
for the density of water at 
the given temperature, will 
give the volume in cubic 
centimeters. Then the bot- 
tle must be weighed again 
when filled with the liquid, 
the density of which is to 
be determined. The weight 
of the liquid divided by the 

capacity of the bottle in cubic centimeters will 
be the density of the liquid at the temperature 
marked on the bottle. The bottle must be filled 
at this temperature every time. 

B. By a glass sinker. "Weigh the glass sinker 
suspended by a fine platinum wire, first in air 
and then in water. The apparent loss of weight 
will be the weight of the water displaced by the 
sinker. Then weigh it again when suspended in 
the liquid. The loss of weight will now be the 
weight of the same volume as before, namely, 
that of the sinker. Divide the latter loss of 
weight by the former and the quotient will be 
the density of the liquid in grams per cubic centimeter. 

C. By the hydrometer. The common hydrometer is usually made of 
glass and consists of a cylindrical stem and a bulb weighted with mercury 



Fig 82 



126 



MECHANICS 



or fine shot to make it sink to the requisite level (Fig. 82). The stem is 
graduated so that the depth to which the instrument sinks can be read 
off, or at least a reading can be taken at the level of the water which 
may be made to give the density of the liquid in which the hydrometer 
floats. 

The stems of hydrometers are frequently graduated to give directly 
the density of the liquids in which they are immersed. In this case the 
length of the divisions on the stem decreases from the top downward. 

Hydrometers are sometimes provided with a thermometer in the stem 
to indicate the temperature of the liquid at the time of taking the density. 
Hydrometers of the type described are hydrometers of variable immersion 
as distinguished from those of constant immersion. The former are ex- 
tensively used in the arts for the approximate testing of the liquids used. 
They are generally graduated with reference to their specific use. Special 
names are then applied to them, such as lactometers for testing milk, 
alcoholmeters for determining the strength of spirits, acidimeters for 
testing the strength of acids, etc. 

158. Two Liquids in Communicating Tubes. If the heavier 
liquid, mercury for example, is first poured into a U-shaped tube 
(Fig. 83), and then the lighter one, water for ex- 
ample, is poured into one limb, the two liquids of 
unequal densities will rise to different levels above 
the surface of separation common to the two. If h 
and h' are the heights of their free surface above 
the common plane HH', then for equilibrium the 
pressures of the two columns h and h f on the com- 
mon plane are the same. 

Let d and d' be the densities of the two liquids. 
For the pressure on unit area the two columns have 
the volumes h and h' and masses hd and h'd'. Hence 
their weights are 




hdg = h'd'g, 



or 



Fig. 83 



The densities of the two liquids are therefore in 
the inverse ratio of their heights above the common plane of separation. 
This relation furnishes an additional method of comparing the densities 
of two liquids that do not react chemically. 



FLUIDS IN MOTION 127 

V. FLUIDS IN MOTION 

159. Velocity of Efflux. When a small opening is made 
in the side of a vessel containing a liquid, at a vertical dis- 
tance h below the surface, the liquid flows out with a definite 
velocity v. Torricelli's formula for the velocity of efflux is 

v*=2gh. (44) 

This is the same as the velocity acquired by a body falling 
through a height A in a vacuum. 

When a small mass m issues from the orifice, an equal 
mass m falls some distance a^ to take its place ; another equal 
mass must in consequence fall a distance a 2 , and so on through 
a series to the surface. 

The total loss of potential energy of the liquid is then 

mga^ + rnga^ -f mga^ -f- = mgJi, 

where h is the sum of the distances a v a 2 , a z , etc., or the 
height of the free surface of the liquid above the orifice. 

Then, neglecting viscosity, the loss of potential energy 
should equal the energy of motion acquired by the mass m. 

If so, we may write , n 9 

mgh = J mir. 

Solving for v 2 , we have 



which is Torricelli's formula. 

If the area of the orifice is a, the quantity of liquid dis- 
charged in time t should be avt. In point of fact the rate of 
discharge is less than this. If the opening be a simple orifice 
in the side of the vessel, the quantity of liquid discharged is 
about 62 per cent of the quantity avt. The difference is due 
chiefly to the convergence of the stream lines, which produces 
a contraction of the jet just outside the orifice. By the use 
of an orifice or projecting mouth-piece conforming to the 
conical shape of the contracted stream, a velocity but little 
short of ^/2gh may be attained. 



128 



MECHANICS 



160. Range of Jets. Let ED (Fig. 84) be the vertical side 
of a vessel of water, the surface being at E. EA, AB, BC, 
and CD are equal distances. Let v be the 
velocity of efflux from the orifice at J., the 
depth of which below the surface is h and its 
height above the horizontal plane 
through D is b. Then if t is the 
time of falling through the 
height 6, the horizontal 
range of the stream is 
a vt. (a) 

Also from equation (20), 




But if v 2 = 2gh by Torricelli's theorem, from 



Substitute this value of t 2 in (5), and 



4 

Whence a 2 = 4 bh and a = 2 VM1 

It follows that if b and h exchange values, the range will 
be the same, for their product bh is not changed. Hence the 
range from the orifice is the same as that from A. 

The greatest range is from the orifice B midway between 
the top and the bottom. This may be demonstrated as 
follows : 

On ED as a diameter describe a semicircle. The square 
of any half chord c is equal to the product of the two seg- 
ments into which the chord divides the diameter, or 



c 2 = bh and c = ^/bh. 

But e is a maximum at the point B. The range 2 
therefore also a maximum for the orifice B. 



s 



FLUIDS IN MOTION 129 

161. Flow of Liquids through Tubes. If a metallic tube 
be attached to an orifice, the velocity of efflux will be dimin- 
ished by reason of friction and viscosity. The layer of 
liquid in contact with the wall of the tube remains nearly at 
rest, especially if the tube be narrow and rough, and the 
velocity of flow increases toward the axis. The flow of an 
open stream of water is greatest near the middle of the 
stream and at the surface, where the friction is least. 

If an elastic rubber tube is attached instead of a rigid one, 
the efflux will be the same as by a rigid tube of the same 
length and diameter, so long as there are no sudden variations 
of pressure. If, however, the pressure be intermittent or 
pulsating, the stream from a rigid tube reproduces every 
variation of the pulsating pressure; while an elastic tube 
rapidly eliminates the inequalities of pressure, and if it be of 
sufficient length, the pulsations in the stream disappear. 
The mechanical effect is due to an alternate storage and 
restoration, or give and take, of energy by means of the elas- 
ticity of the tube. This action is analogous to that of a 
flywheel in producing a steady flow of energy from a recipro- 
cating steam engine. It has its analogue also in alternating 
currents of electricity. Attention will be drawn to this 
analogy in a later chapter ( 651). 

162. Flow of Liquids in Vertical Pipes. In a vertical pipe 
the rapidly descending liquid breaks into sections fitting the 
pipe more or less perfectly. They acquire increasing velocity 
in their descent through an unobstructed pipe, and act as 
liquid pistons. A partial vacuum is thus produced, and the 
outer pressure of the atmosphere forces the liquid into the 
pipe all the more rapidly. This partial vacuum in the long 
waste pipe from a washbasin or a bathtub accounts for the 
noisy flow of the water, and it may be sufficient even to with- 
draw the water from a siphon trap and leave the pipe open to 
the sewer. It is therefore essential that all such traps lead- 
ing to a sewer should have a separate vent to the outside air. 



130 



MECHANICS 



Fig. 85. 



163. Flow in Pipes of Variable Section. The flow of liquids 
through a pipe of variable cross section presents some inter- 
esting features. The variations of pressure in such a pipe 
may be shown by the height to which the liquid rises in 
vertical tubes attached to the pipe of variable diameter, as 
in Figure 85. When the flow is such as to keep the pipe 
full, the pressure is greatest in the widest parts and least in 

the narrowest. This ap- 
parent anomaly admits of 
ready explanation. 

It is evident that when 
the flow is steady, the 
velocity of the stream is 
greatest at the narrowest 
part of the pipe, since the rate of flow past every section 
must be the same, and the narrower the pipe the greater the 
velocity for the same rate of flow. Hence, when the liquid 
passes from a wider to a narrower part of the pipe, it is 
accelerated. With a pipe of uniform cross section and con- 
stant velocity, the loss of pressure is uniform, and is due to 
liquid friction against the pipe and to viscosity. But to 
produce an acceleration, additional pressure is required, or 
the pressure must fall from the wider section toward the 
narrower, in addition to the fall due to friction and viscosity. 
On the other hand, when the liquid flows from a narrow 
to a wide part, it loses velocity, or suffers a retardation. 
Hence the pressure ahead must be greater than behind, or 
the pressure increases from the narrower toward the wider 
section ahead. 

VI. PROPERTIES OF GASES 

164. Diffusion. The process by which two fluids mix in- 
dependently of external pressure is called diffusion. A gas 
diffuses very freely. Insert a rubber tube into the upper 
part of an inverted jar, about 30 cm. high, and allow illumi- 
nating gas to flow in. It will displace the heavier air and 



PROPERTIES OF GASES 131 

fill the tall jar. Now place the vertical jar filled with illumi- 
nating gas over the open end of another jar of the same 
size, and containing only air. The edges at the open end 
should be ground and smeared with grease to make an air- 
tight joint. 

Mixing of the two gases cannot take place by reason of 
gravity, for the lighter gas is above. Nevertheless, in the 
course of ten minutes there will be found in the lower jar an 
explosive mixture ; the lighter illuminating gas has diffused 
downward into the air, and the air has likewise diffused 
upward into the coal gas. 

The process of diffusion may be explained by the kinetic 
theory of gases, which supposes that the molecules of a gas 
are incessantly moving and colliding with one another. The 
mutual encounters between the gas molecules and the air 
molecules diminish the rapidity with which the two gases 
intermingle. Moreover, the rapidity of the process depends 
on the nature of the two diffusing gases, and it is especially 
rapid when one of them is hydrogen. 

165. Barton's Law of Diffusion. When the two gases have 
become uniformly mixed by diffusion, the pressure of the 
mixture remains the same as that under which the gases 
were before diffusion, assuming only diffusion and 110 chemi- 
cal reaction. Hence Dalton's law, according to which each 
gas in a mixture exerts the same pressure which it would 
exert if it were alone present ; and the pressure on the walls 
of the containing vessel is the sum of the partial pressures 
of the separate gases. If, for example, 21 parts by volume 
of oxygen and 79 parts of nitrogen are placed in a vessel and 
diffuse into each other under a pressure p, then after the 
completion of the diffusion, the pressure p l of the oxygen 
will be -fifo of the whole pressure, and that of the nitrogen 
p 2 will be -ffa, and pi + Pz = p . 

This is the mixture composing the major part of our 
atmosphere. 



132 



MECHANICS 



166 

pores 



Effusion. The passage of a gas through the fine 
of a partition is called effusion. A Florence flask 
nearly filled with water is surmounted 
with a jet tube and a funnel tube, to 
which is cemented air-tight a small 
battery jar or other porous pot (Fig. 
86). A stream of hydrogen is allowed 
to flow into the beaker inverted over 
the porous cup. If all the joints are 
tight, water will issue from the jet tube 
as a small fountain. The hydrogen 
passes through the fine pores of the 
unglazed cup by effusion. 

The rate of effusion of different 
gases is inversely proportional to the 
square root of their densities. Hydro- 
gen, for example, passes through a po- 
rous wall by diffusion four times as fast 
as oxygen, the density of which is six- 
teen times as great as that of hydrogen. 




Fig. 86 



167. Boyle's Law. The relation between the volume of a 
gas and the pressure to which it is subjected was discovered 
by Robert Boyle in 1662. It is therefore known as Boyle's 
law. In France it is called Mariotte's law, from Mariotte, 
who announced the law fourteen years later than Boyle. 
The law is as follows: 

At a constant temperature the volume of a given mass 
of gas varies inversely as the pressure to which it is subjected. 

If v and p are corresponding volume and pressure, then 
when the pressure is changed to p', the volume becomes v' ', 
and the relation between volumes and pressures is 



that is, pv = c a constant. 



PROPERTIES OF GASES 



133 



Boyle's experiments were made with a U-tube (Fig. 87) 
and they extended only from ^ of an atmosphere to 4 
atmospheres. The short leg A was closed at the 
top and mercury was introduced until it stood at 
the same level in both legs. The volume of air 
imprisoned in the short leg, which was under the 
same pressure as the atmosphere outside ( 171), 
was noted and more mercury was then poured 
into the tube. The difference in the level of the 

mercury in the two 
legs of the tube 
gave the excess 
pressure above that 
of the atmosphere 
in the open limb. 
The volume of air 



70 



50 



30 



20 



10 



-40 



-50 



-70 



-60 



.01 




was read each time 
from the shorter leg, which 
had previously been cali- 
brated. 

Equation (45) may be put 

in the form p = c , in which 

v 

c is a constant. This is the 
equation of a straight line. 
Hence, if the pressures in 
terms of the excess column 
of mercury be plotted as ordi- 
nates, and the reciprocals of 
the corresponding volumes as 
abscissas, the result should be 
a straight line (Fig. 88). 
Moreover, this line should in- 
tersect the axis of pressures at 
a point P below the zero equal to the pressure of the atmos- 
phere, for the pressure of the atmosphere is a part of the 



Fig. 88 



134 



MECHANICS 



pressure p under which the gas is. The data for Figure 87 
were obtained from a piece of apparatus essentially like a 
U-tube, but better adapted to secure reliable measurements. 

168. Researches of Regnault and Amagat. The application 
of Boyle's law over a wide range of pressures was not put 
to an experimental test until nearly 200 years after its dis- 




50 



250 



100 150 200 

Pressure in Meters of Mercury 

Fig. 89 

covery. Regnault carried the pressures as high as 20 meters 
of mercury. For the first 15 meters he showed that the 
air- and nitrogen are more compressible than they would be 
if they obeyed Boyle's law with precision, while hydrogen 
is less compressible. 



PRESSURE OF THE ATMOSPHERE 135 

In 1870 Amagat made use of a coal pit for his experiments 
on Boyle's law for the reason that it furnished a vertical shaft 
of nearly constant temperature, and a steel tube 330 m. 
long could be erected in it. This tube contained the mer- 
cury to measure the pressures. 

Amagat plotted the products pv as ordinates and the pres- 
sures in meters of mercury as abscissas. If Boyle's law were 
exactly true, the result would be a straight line parallel to 
the axis of pressures. The curves (Fig. 89) show that 
hydrogen is from the first less compressible, or more elastic, 
than the law of Boyle requires; while nitrogen is first more 
compressible up to a minimum of pv at a pressure of about 
40 m. of mercury, and from that pressure upwards is, 
like hydrogen, less compressible than Boyle's law requires. 

The diagram of carbon dioxide shows a more marked de- 
parture from a straight line ; at 100 C. it is an exaggeration 
of the nitrogen line, while at 35.1 C. carbon dioxide shows 
a distinctly marked minimum for pv at a pressure of 70 
m. of mercury. At a slightly lower temperature this 
minimum becomes the line of condensation of the gas into 
the liquid form. 

Gases which at low temperatures and pressures deviate 
from Boyle's law by being too compressible, at high tem- 
peratures and pressures resemble hydrogen and are less 
compressible, or more elastic, than they would be if they 
obeyed Boyle's law precisely. 



VII. PRESSURE OF THE ATMOSPHERE 

169. Air has Weight. Aristotle attempted to determine 
whether air has weight by weighing a bladder inflated with 
air and collapsed. But air, like other fluids, has buoyancy, 
and the change in buoyancy when the bladder collapsed was 
equal to the weight of the air removed. Hence Aristotle 
found no difference in the weight whether the bladder was 
inflated or not. Since the invention of the air pump it has 



136 



MECHANICS 



been determined that air and hydrogen have the following 
weights : 

1 liter of dry air at C. and under pressure of 76 cm. of 
mercury weighs 1.293 gm. 

1 liter of hydrogen under the same conditions weighs 
0.0895 gm. 

Therefore the density of air under the above standard 
conditions is 0.001293 gm. per cubic centimeter; the density 
of hydrogen, 0.0000895 gm. per cubic centimeter. The den- 
sity of oxygen is 0.0014279 gm. per cubic centimeter. 

170. Torricelli's Experiment. It was found by Galileo 
that water would not rise in the pumps of the Duke of Tus- 
cany to a height greater than about 32 feet. He suspected 
that the pressure of the atmosphere sustained a column of 
water of this height, but it remained for Torricelli, a pupil 

of Galileo, to demonstrate in 
1643 the truth of Galileo's sur- 
mise and to measure the pres- 
sure of the atmosphere. 

Torricelli filled with mercury 
a tube, about a meter long and 
closed at one end, inverted it, 
and dipped the open end below 
the surface of mercury in an 
open vessel (Fig. 90). The 
mercury settled down when 
the finger was removed from 
the bottom of the tube, and 
came to a rest at a height of 
about 76 cm. above the surface 
of the mercury in the vessel at 
A. Torricelli gave the true explanation to the effect that 
this column of mercury was supported by the pressure of the 
atmosphere on the free surface of the mercury in the open 
vessel. The pressure of 76 cm. of mercury at C. is equiva- 




Fig. 90 



PRESSURE OF THE ATMOSPHERE 137 

lent to the pressure of 33.9 feet of water at maximum density. 
Thus Torricelli confirmed the conjecture of Galileo that the 
water in the pumps of the Duke of Tuscany refused to rise 
above about 32 feet, not because " nature abhors a vacuum " 
as the ancients taught, but because the pressure of the 
atmosphere was insufficient to maintain a higher column. 

Torricelli's theory was confirmed by Pascal in Paris, who tried the 
experiment with tubes filled with oil, water, and wine, and found that 
the height of the column sustained was inversely proportional to the 
density of the liquid in the tube. It was Pascal also who suggested 
that the height of the column of mercury sustained by the pressure of 
the atmosphere should be less at the top of a mountain than at its base. 
He verified this prediction by carrying the inverted tube with mercury 
to the top of the Puy-de-D6me, about 1000 m. high. A fall in the height 
of the mercurial column, amounting to about 8 cm., was observed. Thus 
Torricelli's explanation was completely confirmed. 

171. Pressure of f the Atmosphere. The pressure of the 
atmosphere varies from hour to hour. It is also dependent 
on the altitude above the sea. It is therefore necessary in 
defining standard atmospheric pressure, which is measured 
by the weight of a column of mercury of unit cross-sectional 
area, to define the temperature and the value of the accelera- 
tion of gravity. The standard height chosen is 76 cm. of 
mercury, at a temperature of melting ice (0 C.), and at sea 
level in latitude 45. The density of mercury at C. is 
13.596, and the value of ' g at sea level in latitude 45 is 
980.6 cm. per second per second. Hence standard atmo- 
pheric pressure is 

76 x 13.596 x 980.6 = 1,013,250 dynes per sq. cm. 
This is a little over 10 6 dynes, or a megadyne. 

172. The Barometer. The barometer is an instrument 
based on the Torricelli experiment, and designed to measure 
the pressure of the atmosphere. It has become a highly 
important auxiliary in many branches of physics and in 
meteorology for weather predictions. 



138 



MECHANICS 



Barometers are either (1) liquid barometers, which meas- 
ure atmospheric pressure in terms of the height of a column 
of liquid; or (2) aneroid barometers, in which 
the pressure is measured by the deformation 
of the thin corrugated cover of an air-tight 
metal box. 

Mercury is practically the only liquid used in 
barometers of the first type, since it does not 
absorb moisture from the air as glycerine, for 
example, does, and its density is so great that 
the column sustained by atmospheric pressure is 
short enough to be manageable. 

The simplest form of mercurial barometer is a Torri- 
cellian tube in the form of an inverted siphon (Fig. 91). 
The short arm has a small hole near the top for the admis- 
sion of air. The height of 
the mercurial column is 
the difference between the 
readings on the two scales 
at the right ; for example, 
if the upper pointer stands 
at 78.4 cm. and the lower 
one at 4.2 cm., the ^ height 
of the barometer is 74.2 cm. 
Corrections must be made 
for temperature and eleva- 
tion above sea level. A 
good barometer must con- 
tain pure mercury and the 
mercury must be boiled in 
the glass tube to expel all 
air and moisture. 



Fig. 91 




173. Fortin's Barometer. A port- 
able barometer designed by Fortin 
is shown in Figure 92. The cistern Fig 92 

O of mercury is closed at the bot- 
tom by a leather bag B, the bottom of which may be lowered 
or raised by means of the screw 8. A glass cylinder Gr near 



PRESSURE OF THE ATMOSPHERE 139 

the top of the cistern permits the observer to see when the 
ivory point P and its image in the mercury touch each other. 
The point P is the zero of the barometer scale. The tube is 
attached to the cistern by means of a piece of chamois skin 
F, which prevents the escape of mercury but permits no dif- 
ference of pressure to exist between the outside and the 
inside air. The glass tube is inclosed in a metal case, which 
carries a scale and at the top a vernier for measuring frac- 
tions of a scale division. 

When the barometer is to be transported, the screw S is 
turned in till the cistern and the tube are entirely filled with 
mercury. The surging of mercury and the entrance of air 
are thus prevented. 

174. The Aneroid Barometer. The aneroid barometer consists es- 
sentially of a shallow cylindrical box B (Fig. 93), largely exhausted of 
air ; it has a thin cover corrugated in 

circular ridges to give it greater flexi- 
bility. The cover is prevented from col- 
lapsing under atmospheric pressure by 
a stiff spring attached to the center of 
the cover at M. This flexible cover 
rises and falls as the pressure of the 
atmosphere varies and its motion is 
transmitted to the pointer by means of 
levers and a chain. A scale graduated 
by comparison with a mercurial barom- Fj Q3 

eter is placed below the pointer. The 

advantages of an aneroid barometer are its portability and sensitiveness. 
It should be compared frequently with a mercurial barometer. 

175. Barometric Variations. Since the mercury in the ba- 
rometer tube is sustained by the pressure of the air on the 
mercury outside, changes in the barometric readings indicate 
corresponding fluctuations in atmospheric pressure. The 
greater changes follow no well-defined laws, but they herald 
important atmospheric movements associated with storms. 

Experience has shown that rapid barometric changes fore- 
tell changes in the weather. Thus a rapid fall of the barom- 




140 MECHANICS 

eter denotes the near approach of a storm, and a rising 
barometer is usually followed by fair weather. 

The words Rain, Fair, etc., often marked on aneroid barom- 
eters especially, are without significance in connection with 
the readings against which they are placed. Changes in the 
weather are indicated by rather rapid changes in the readings 
of the barometer. 

176. The Manometer. A manometer is an instrument de- 
signed to measure the pressure of a gas in a closed vessel. 
A bent tube partly filled with mercury may be used for 
the purpose. If the pressure to be measured does not 
greatly exceed ' one atmosphere, the outer end of the 
tube is left open, as at in Figure 94 A. The end D 
communicates with the closed vessel. The required 
pressure will then be the reading of 
the barometer increased by the dif- 
ference in level between M and L. 

A similar arrangement may serve 
for a vacuum gauge. For this pur- 
pose, the end communicates with 
the vacuum apparatus, and the other 
end is open to the air. The differ- 
ence in level of the mercury in the 
two legs of the tube will then be the vacuum. 

If the pressure to be measured is considerably greater 
than one atmosphere, use is made of the elasticity of the air. 
The outer end of the tube is then closed, as in Figure 94 B. 
The pressure is determined by applying Boyle's law and add- 
ing to the result the difference in level of the mercury in the 
two limbs of the manometer. In practice the pressure is 
read from a scale attached to the tube and graduated 
empirically. 

In the mechanic arts metallic pressure gauges, analogous to 
the aneroid barometer, are employed both for pressure and vac- 
uum. They are- provided with empirically graduated scales. 




INSTRUMENTS DEPENDING ON PRESSURE 141 



VIII. INSTRUMENTS DEPENDING ON PRESSURE OF THE AIR 

177. The Air Pump. The mechanical air pump for ex- 
hausting a closed vessel was invented by Otto von Guericke 
about 1650. The appearance 
of one of the best forms for 
general illustrative purposes 
is shown in Figure 95. The 
essential internal working 
parts appear in Figure 96. 
In the piston P is a valve S 
worked mechanically by the 
motion of the piston rod. 
The pump cylinder communi- 
cates with the outer air at its 
upper end by a valve V work- 
ing by air pressure, and with 
the~ pump plate by a me- 
chanically operated valve $', 
worked by a rod which passes 

rather snugly through the piston. It is lifted as the upstroke 

begins, but its ascent is 
arrested by a stop near 
the upper end of the rod. 
During the upstroke the 
air flows into the cylinder 
through S 1 from the re- 
ceiver on the pump plate. 
When the piston reaches 
the top of the cylinder, it 
hits the lever shown in 
Figure 96, and automati- 
cally closes the valve 8' 
before the descent begins. 
In the downstroke the 
Fig. 96 valve S opens automati- 





142 



MECHANICS 



cally, and the inclosed air passes through it into the upper 
part of the cylinder. The ascent of the piston again closes 
& and compresses the air above it. If its pressure at the top 
of the stroke exceeds that of the air outside, the valve V 
opens and air is expelled through the tube leading to the 
bottom of the cylinder. 

Each complete stroke of the pump removes a cylinder full 
of air ; but as the air becomes rarer with each stroke, the 
mass removed each time is less. On account of leakage, 
untraversed space, and absorption of air by the lubricating 
oil, the pressure in the vessel to be exhausted cannot be 
reduced much below one millimeter of mercury. 

178. The Fleuss Pump. The Fleuss pump (made under 
the commercial name of " Geryk " pump) is a type of air 
pump in which a non-volatile oil is 
used to cover both the piston and the 
outlet valve. The essential parts are 
shown in Figure 97. A piston N 
works in a cylinder M and around the 
piston rod is a valve G- opening out- 
wards. On its upstroke the piston, 
after passing the inlet from the 
surrounding chamber, compresses the 
air in the cylinder and insures 
the opening of the valve G- by means 
of the collar 0. The oil following 
leaves no untraversed space. On the 
downstroke the oil closes the valve Gr 
completely, a vacuum is produced 
above the piston, and when the piston 
reaches the bottom of the cylinder, air flows from any vessel 
connected with A into the pump cylinder. 

Two such pumps are often connected in series so that the second one 
pumps air from the barrel of the first. By such a pump the air pressure 
can be reduced to a small fraction of a millimeter of mercury; the 
vacuum produced is sufficient for incandescent lamps and Geissler tubes. 




Fig. 97 



INSTRUMENTS DEPENDING ON PRESSURE 143 




These pumps should always exhaust through a drying tube to pre- 
vent the admission of moisture, which is absorbed by the oil and 
given up again at low pressures. The vacuum will then be reduced 
to the vapor pressure of water instead of that of the heavy oil used 
to seal the valves. 

179. The Air Compressor. If the discharge pipe of an air 

pump were connected to a suitable vessel, while its inlet pipe 

were left open to the air, air would be 

forced into the vessel during the action 

of the pump. Such an arrangement 

would be an air compressor. For a 

pressure of several atmospheres, valves 

adapted to withstand high pressures 

must be employed ; and a pump de- 
signed for such pressures is suitable 

for compressing a gas (Fig. 98). The 

piston is solid and there are two metal valves at the bottom. 

Air or other gas is admitted through the left-hand tube when 
n the piston rises ; when it descends, it compresses 

the inclosed air, the pressure closes the left-hand 
valve and opens the other one, and the compressed 
air is discharged into the compression tank or 
other receptacle. 

A bicycle pump (Fig. 99) is an air compressor of a very 
simple type. In the cylinder A slides a piston B provided 
with a cup-shaped leather collar C. When the cylinder is 
pulled outwards, air is admitted past the leather collar, but 
when the stroke is reversed, a quick motion causes the com- 
pressed air to press the leather against the wall of the cyl- 
inder and closes it air-tight. The collar thus serves as a 
valve allowing the air to flow one way but not the other. 
The compressed air escapes through the tube forming the 
piston rod. The check valve to prevent the return of the 
compressed air is the valve in the bicycle tire inlet. 



180. Applications. Both the air pump and the com- 
pressor are used extensively in the arts. Sugar refiners employ the air 
pump to reduce the boiling point of the syrup ( 405) ; manufacturers of 
soda water use a compressor to charge the water with carbon dioxide ; in 



144 MECHANICS 

pneumatic dispatch tubes, now extensively employed for rapidly trans- 
porting small packages, both pumps are employed, the one to exhaust 
the air from the tube in front of the closely fitting carriage, and the 
other to force compressed air into the tube behind it, so as to propel it 
with great velocity. Compressed air is also used to facilitate the ventila- 
tion of buildings and mines, to operate pneumatic clocks, to control heat 
regulators, to work air brakes on cars, and to operate pneumatic ma- 
chinery especially in riveting, calking, and rock boring in tunneling. 

181. Buoyancy of the Air. The principle of Archimedes 
applies to gases as well as to liquids. The resultant pressure 
of the atmosphere on bodies is an upward force equal to the 
weight of the air displaced. A body weighs less in the air 
than in a vacuum if the volume of air displaced by it is 
greater than that displaced by the weights. 

A baroscope is a device to exhibit the upward pressure of 
the air. A thin hollow globe is slightly overbalanced by a 

brass weight on a short beam bal- 
ance (Fig. 100). When the baro- 
scope is placed under a large receiver 
and the air is exhausted, the hollow 
sphere sinks, showing that it is really 
heavier than the counterpoise. In 
the air it is buoyed up more because 
its volume is greater than that of 
the weight. It is perhaps needless to 
F 'g |0 say that the globe must be air-tight. 

182. Balloons. A soap bubble and a toy balloon filled with 
air sink because they are heavier than the air displaced; but 
a bubble filled with hydrogen rises in the air. Its buoyancy 
is greater than its weight including the hydrogen. The 
weight of the balloon with its car and contents must be less 
than that of the air displaced. It is not entirely filled with 
gas, but as it rises it expands as the pressure of the air out- 
side decreases. Its buoyancy then decreases but little when 
it rises into a rarer atmosphere. 

With hydrogen the upward force is about one kilogram 




INSTRUMENTS DEPENDING ON PRESSURE 145 



per cubic meter of gas ; with common illuminating gas it is 
about one half as much. The first ascent in a balloon filled 
with hydrogen was made by Charles, a professor of physics 
in Paris, in December, 1783. Gay-Lussac in 1804 ascended 
to a height of 23,000 feet. At this elevation the barometer 
sank to 12.6 inches. 

The most remarkable ascent ever made was by Messrs. 
Glaisher and Coxwell in England in 1861. At a height of 
29,000 feet, with the thermometer at - 16 C., Mr. Glaisher 
could no longer observe and fainted. According to an ap- 
proximate estimate, the two men reached an altitude of about 
36,000 feet, where the barometer stood at only 7 inches. 

In 1900 two long-distance balloon races were made from Paris in an 
easterly direction. One of the contestants, Count de la Vaulx, the win- 
ner in both races, reached Russian territory in both, having traveled the 
first time a distance of 766 miles in 21 hours and 31 minutes; and the 
second time, a distance of 1193 miles in 35 hours and 45 minutes. The 
greatest altitude reached was 18,700 feet. 

The aeronauts testify that when the sun shone on the balloon and 
heated it, the expansion of the gas increased the buoyancy, so that the 
balloon shot up to higher altitudes. It became necessary in consequence 
to let out some gas to cause the balloon to descend again. In the night, 
when the temperature fell, the buoyancy decreased. Ballast was then 
thrown out to lighten the balloon to prevent its descent. These alter- 
nate losses of gas and ballast at length exhausted 
the capacity of the balloon to keep afloat, and it 
descended to the ground. 

183. The Siphon. In its simplest form 
the siphon is a U-tube employed to con- 
vey liquids from one vessel to another at 
a lower level by means of atmospheric 
pressure. 

Let T (Fig. 101) be the height of the 
highest point of the siphon above the 
liquid in the vessel from which the dis- Fig - l01 

charge takes place ; and let x be the height of the same point 
of the siphon above its open end or the surface of the lower 




146 



MECHANICS 



liquid, if the longer arm dips below it. Let H be the height 
of the column of the liquid equal to atmospheric pressure. 
Then the pressure outward at the top of the siphon is 
(H y)d, and the pressure inward at the same point is 
(II ' x)d, d being the density of the liquid. The resultant 
pressure outward is the difference, 



- y)d - (H - x)d = (x - 



= dh. 



In the case of water d is unity, and the " head " producing 
the flow is h. If y exceeds J7, the liquid will not rise to the 
highest point of the siphon by atmospheric pressure and there 
will be no flow. 

A siphon made in the form shown in Figure 102 is called a " vacuum 
siphon. " The short arm ends in a jet tube inside a closed vessel. The 
pressure within the vessel is less than atmospheric 
pressure by the weight of a column of the liquid 
of unit cross section and of a length equal to the 
vertical distance of the outer end of the discharge 
arm below the surface of the upper liquid. 

The water in an S-trap may be siphoned off 
when .the discharge 
pipe is filled with 
water for a short dis- 
tance below the trap, 
unless the trap is 
ventilated. 



184. The Lift 
Pump. In the 
lift pump a piston 
<?, in which there is 
a valve v f , opening upward (Fig. 103), 
moves nearly air-tight in a cylinder. 
At the bottom of the cylinder is an 
opening provided with a valve t>, also 
opening upward. A pipe s leads 




Fig. 102 




Fig. 103 



from this opening down below the surface of the water. 
When the piston is drawn upward, a partial vacuum is 



INSTRUMENTS DEPENDING ON PRESSURE 147 



produced below it, and the pressure of the atmosphere on the 
water below forces water up the pipe s to a height to produce 
equilibrium. When the piston descends, the valve v 1 opens 
and v closes. Either air or water passes through the upper 
valve. Finally the upstroke lifts the water above the piston 
and the pressure of the air on the open water keeps the tube 
and the piston full. If the piston at the top of its course is 
less than about 33 feet above the water into which the pipe 
s dips, watel- will follow the piston to its highest point. 
The spout may be any reasonable distance above the upper 
valve ; the water above the piston is 
lifted mechanically and not by atmos- 
pheric pressure. 

185. The Force Pump. The piston of a 
force pump, like that of an air compressor, 
is solid and the valve through which 
the water is forced is below it. Other- 
wise a force pump is similar to a lifting 
pump. In Figure 104 d is the discharge 
-pipe, v' the discharge valve opening out- 
ward from the cylinder, and v the inlet 
valve closing the pipe s. 

In powerful pumps the pipe d is sur- 
mounted with an air chamber called the 
air dome. The air in it is compressed by water pressure, 
and it acts as an air spring to give steadiness to the flow 
of water from the delivery pipe. Fire engines and pumps 
operated by steam are fitted with an air dome. 

Problems 

1. The diameters of the cylinders of a hydraulic press are 6 in. and 
1 in. respectively. What is the force on the larger piston when a force 
of 200 lb. is applied to the smaller piston ? 

2. A hydraulic lift carries a weight of 3000 lb. If the piston sup- 
porting the lift is 8 in. in diameter, what pressure of water per square 
inch will be necessary ? 




Fig. 104 



148 MECHANICS 

3. A tank 5 ft. deep and 10 ft. square is filled with water. What is 
the pressure on the bottom ? What is it on one side ? 

4. A cubical block 10 cm. on each edge is submerged in water with 
its top face 100 cm. below the surface. Calculate the total pressure on 
the six faces. 

5. A vertical tube 2 cm. in diameter is filled with mercury (density 
13.6) to a depth of 2 m. What is the pressure per square centimeter at 
the bottom ? 

6. What is the pressure per square foot at a depth of 3 mi. in the 
ocean, sea water being 2| per cent heavier than fresh water? What is 
the buoyancy per cubic foot ? 

7. A body weighs 100 gm. in the air and 88 gm. in water at 20. 
Find its density. 

8. A glass stopper weighs 150 gm. in the air, 90 gm. in the water, 
and 42 gm. in sulphuric acid. Calculate the density of the acid. 

9. A piece of zinc weighs 70 gm. in air and 60 gm. in water. What 
will it weigh in alcohol of density 0.8 gm. per cubic centimeter? 

10. If the density of sea water is 1.025 gm. per cubic centimeter, and 
that of ice 0.9 grn. per cubic centimeter, what fraction of an iceberg 
floating in the sea is under water ? 

11. A hollow brass ball weighs 1 kgm. What must be its volume so 
that it will just float in water? 

12. A glass tube 72 cm. long and closed at one end is sunk in 
the ocean with its open end down. When drawn up it was found that 
the air in the tube had been compressed to within 6 crn.. of the top. 
Assuming a normal atmospheric pressure of 76 cm. of mercury, the 
density of mercury 13.6 and of sea water 1.025 gm. per cubic centimeter, 
to what depth did the tube descend ? 

13. Two bodies, having densities of 6 and 8 gm. per cubic centimeter, 
respectively, are of such relative volumes that their apparent weights 
in water are the same. Compare their weights in air. 

14. The mark to which a certain hydrometer weighing 90 gm. sinks 
in alcohol is noted. To make it sink to the same mark in water, it 
must be loaded with 22 gm. What is the density of the alcohol ? 

15. A block of wrought iron 10 cm. thick is floated on mercury. 
To what depth above the mercury must the vessel be filled with 
water so that the latter shall just reach the top of the block of iron ? 
Densities of iron and of mercury, 7.7 and 13.6 gm. per cubic centimeter, 
respectively. 



INSTRUMENTS DEPENDING ON PRESSURE 149 

16. A liter flask weighing 75 gm. is half filled with water and 
half with glycerine. The flask and liquids weigh 1205 gm. What is 
the density of the glycerine ? 

17. A body floats half submerged in alcohol of density 0.818 gm. per 
cubic centimeter. What part of its volume would be submerged in 
water ? 

18. If the surface tension of a soap solution is 25 dynes/cm., how 
much greater is the pressure of the gas inside a soap bubble 5 cm. in 
diameter than that of the air on the outside ? 

19. If a liter of air weighs 1.29 gm. when the barometer reading 
is 76 cm., calculate the buoyancy for a ball 20 cm. in diameter when 
the barometer stands at 70 cm., the temperature being the same. 

20. If an open vessel contains 250 gm. of air when the barometric 
pressure is 76 cm., how much will it contain at the same temperature 
when the barometric pressure is 70 cm. ? 

21. The volume of hydrogen gas in a graduated cylinder over mercury 
was 50 cm. 3 , the mercury standing 15 cm. high in the cylinder. and 
the barometer reading'75 cm. What would be the volume of the gas 
if it were under normal pressure ? 

22. When the barometer reading is 73 cm., what is the greatest 
possible length for the short arm of a siphon when used for sulphuric 
acid, density 1.84 gm. per cubic centimeter? 

23. A vertical cylinder is closed with a piston whose area is 60 sq. cm. 
The inclosed air column is 50 cm. high and is at the atmospheric pres- 
sure of 1000 gm. per sq. cm. If a weight of 100 kgm. be placed on 
the piston, how far will it descend, neglecting friction ? 

24. What is the total pressure on the vertical sides of a cylindrical 
tank 60 cm. in diameter and filled with water to the height of 2 m.? 

25. What is the vertical height of a column of water which will 
counterbalance a column of benzine 80 cm. high, density 0.9 gm. per 
cubic centimeter (Fig. 83) ? 



SOUND 

CHAPTER V 

WAVES 
I. WAVE MOTION 

186. Vibrations. Suspend a heavy ball by a long thread 
and set it swinging like a pendulum bob. The ball will 
return at regular intervals to the starting point. If it be set 
moving in a circle, the thread describing a conical surface, it 
will again return periodically to the point of departure. 

A vibrating or oscillating body is one which repeats its 
limited motion at regular intervals of time. The term oscil- 
lation is usually applied to motions of short period only. 
A complete or double vibration is the 

D' D n" 

motion between two successive pas- 
sages of the moving body through 
any point of its path in the same direc- 
tion, and its period is the interval of 
time taken to execute a complete 
vibration ( 92). 

Considered with respect to the vi- 
brating body, vibrations are transverse 
* 1 9F ~ when the motion is at right angles to 

ii the body. For example, clamp a thin 

steel strip in a vise (Fig. 105) ; draw 
the free end aside and release it. It 

Fig. 105 

will move from D f to D" and back 
again and its vibrations are transverse. 

An example of visible longitudinal vibration is the fol- 
lowing: Fasten the ends of a long spiral spring securely to 

150 




WAVE MOTION 151 

a fixed support with the spring slightly stretched. Crowd 
together a few turns of the spiral at one end and then release 
them. A vibratory move- 
ment will be transmitted 
from one end of the spiral 
to the other, and each 

turn will swing to and fro in the direction of the length of 
the spiral (Fig. 106). 

187. Wave Defined. The periodic motion of a single 
particle or rigid body has already been studied in Chapter 
II ; we have now to consider the related motions when the 
various particles of a medium are executing periodic vibra- 
tions simultaneously, while the phase of the motion ( 36) 
varies from particle to particle in a regular methodic way. 

To illustrate : Tie one end of a soft cotton rope to a fixed support. 
Grasp the other end and stretch the rope horizontally. Start a disturb- 
ance by an up-and-down motion of the hand. Each point of the rope 
will vibrate transversely with simple harmonic motion ( 36), while the 
disturbance will travel along the rope toward the fixed end as a wave. 
This progressive form, due to the periodic vibration of the particles of 
the medium through which it moves, is called a wave. The phase of 
each successive particle differs from that of the preceding particle by 
a definite fraction of a whole period. 

188. Transverse Waves. Suppose a series of particles, 
originally equidistant in a horizontal straight line, to have 
imparted to them transverse displacements, and that they 
vibrate with simple harmonic motion. Let the curve (Fig. 
107) represent the positions of these particles at some par- 
ticular instant. They will outline a transverse wave. At g 
the particle has reached its extreme displacement in the 
positive direction and is momentarily at rest ; the particle 
at s has reached its maximum negative displacement, and is 
also at rest. The particle at m is moving in the positive 
direction with maximum velocity, and the particle at y with 
maximum velocity in the negative direction. If the wave 
is traveling toward the right, then an instant later the dis- 



152 



SOUND 



placement of g will have diminished, and that of i will have 
increased to a maximum, the crest having moved forward 
from g to i in the short interval. The successive particles 
of the wave all differ in phase by the same amount. 







i 


i 








T 




j 










C T 




i 


T 








b } 




i 
i 


1 








a] 




1 


I t 






y 


1 






m , j 






rm 








1 ' 






i 








1 ! 






! ! ' 














i ' 



Fig. 107 

A transverse wave is one in which the vibration of the 
particles in the wave is at right angles to the direction in 
which the wave is traveling. 

189. Longitudinal Waves. Place a lighted candle at the 
conical end of a long tube (Fig. 108). Over the other end 
tie a piece of parchment or parchment paper. Tap the 
paper lightly with a cork or rubber mallet ; the transmitted 
impulse will cause the flame to duck, and it may easily be 
extinguished by a sharper blow. The air in the tube is 
agitated by the vibratory motion passing through it. A 




Fig. 108 

wave consisting of a compression followed by a rarefaction 
traverses the tube, and the dipping of the candle flame indi- 
cates the arrival of the compression. The first movement 
of the membrane compresses the air next to it ; the elasticity 
of the air forces these particles apart again, and this action 



WATER WAVES 153 

compresses the air further along in the tube. The process 
is continuous and carries the compression through the tube 
to the flame. Each particle vibrates longitudinally with 
approximately simple harmonic motion, the whole phenom- 
enon being quite similar to that of the vibrating spiral. 

Figure 109 illustrates the distribution of the air particles 
when disturbed by such a longitudinal wave consisting of 



A B CDEfOHIJ K L 

Fig. 109 

compressions and rarefactions. A, C, E, etc., are regions of 
rarefaction ; jB, Z>, F, etc., those of compression. 

A longitudinal wave is one in which the oscillations of the 
particles composing the wave are to and fro in the same 
direction as the wave is traveling. 

190. Wave Length. The length of a wave is the distance 
from any particle in the wave to the next one in the same 
vibration stage, that is, in the same phase. Such, for ex- 
ample, is the distance from a to y in Figure 107, or from A 
to (7, or B to D in Figure 109. In a longitudinal wave, a 
wave length comprises one compression and one rarefaction. 

Let X represent a wave length, n the frequency or number 
of vibrations a second, and v the velocity of the disturbance 
or wave. Each particle executes a complete vibration while 
the wave travels forward a wave length X. Hence there 
are n such waves in the distance v. Also, the vibration fre- 
quency n and the period T are the reciprocals of each other. 

Therefore. -x 

v = n\ = , or X = -. (46) 

T n 

II. WATER WAVES 

191. Gravitational Waves and Ripples. Waves on the surface 
of water or other liquid are of a distinct type. For small 
amplitudes water waves are similar to those arising from 



154 SOUND 

transverse vibrations; but water is so slightly compressible 
that these waves cannot be waves of compression and rare- 
faction in a vertical plane. The force by which long waves 
are produced on the surface of a liquid is the force of gravity. 
They are therefore commonly called gravitational waves. 

Suppose by some means the liquid is heaped up in the form 
BCD above the general level and scooped out into a trough 

DEF (Fig. 110). 

A .B^^^A^^j) F G Then the liquid 

=^=^^ ^v= = elevated above tke 

general level will 
tend to flow back 

Fig 110 

toward the level 

surface, and the upward hydrostatic pressure on the bottom 
of the trough will cause it to move upward. The simulta- 
neous downward flow of the portion BCD into the trough 
DEF, and the rise of the bottom of the trough by upward 
pressure, cause the forward movement of both crest and 
trough. Large waves on the surface of the sea are gravita- 
tional waves of this character. 

But gravity is not the only force tending to bring the dis- 
turbed surface of a liquid back to a position of stable equi- 
librium. The surface tension of a liquid ( 138), which acts 
like an elastic membrane stretched over the surface, also 
tends to remove all curvature in the surface, and thus acts as 
a force of restitution on the displaced liquid particles. The 
surface tension in a curved surface produces a normal pressure 
toward the concave side, and this normal pressure increases 
as the radius of curvature decreases ( 138). Since the mag- 
nitude of the surface tension is small, it comes into account 
in producing water waves only when the curvature is great, 
that is, when the waves are short. With short waves the 
weight of water displaced is small in comparison with the 
normal force due to surface tension. On the other hand, with 
long waves of slight curvature, surface tension is negligible 
in comparison with the gravitational effect. For waves 



WATER WAVES 155 

shorter than 3 mm. or about 0.12 inch, surface tension plays 
so important a part in the propagation of the wave that the 
gravity effect is negligible. Waves like these, in which the 
force of restitution is essentially due to surface tension, are 
called capillary waves or ripples. 

For waves longer than 10 cm. or 4 in., surface tension is 
negligible in comparison with the gravitational force of res- 
titution. The speed of ripples increases as the wave length 
\ diminishes ; the speed of gravitational waves increases as 
the wave length increases, provided always that the depth of 
the liquid is not less than the wave length. There must 
therefore be a certain wave length for which the speed is a 
minimum. For water the minimum speed is about 28 cm. or 
9 inches per second, and the corresponding wave length is 
1.72 cm. 

192. Oscillations in Water Waves. For waves on the surface 
of deep water the particles describe vertical circles, all in the 
same plane, containing the direction in which the disturbance 

traveling, as illustrated in Figure 111. The circles in the 




Fig. Ill 

diagram are divided into eight equal arcs, and the water par- 
ticles are supposed to describe these circles in the direction 
of watch hands, all at the same rate ; but in any two consecu- 
tive circles their phase of motion differs by one eighth of a 
period. When a has completed one revolution, b is one eighth 
of a revolution behind, c two eighths or one quarter, etc. A 
smooth curve drawn through the several simultaneous posi- 
tions is the outline or contour of a wave. 

When a particle is at the crest of a wave, it is moving in 
the same direction as the wave ; but when it is in the trough, 
its motion is opposite to that of the wave. 

As the depth increases, the particles there still move in 



156 SOUND 

circles, but the circles are smaller and smaller as the depth 
increases, till at the depth of a wave length the radius of the 
circles is only about -g-J-^ as great as at the surface. In 
shallow water these circular paths become ellipses with their 
major axes horizontal. The vertical axes decrease with the 
depth. 

The crests and troughs are not of the same size, and the 
larger the circles (or amplitude), the smaller are the crests 
in comparison with the troughs. Hence the tops of high 
waves tend to become sharp or looped, and the waves then 
break into foam or white caps. 

When a high wave comes into shallow water, its speed is 
diminished ; but since the frequency n remains the same, 
the wave length decreases. The effect of this shortening 
of the wave length is to increase the amplitude. Such a 
modification proceeds until an unstable condition is reached, 
and the wave breaks. 



III. PROPAGATION AND REFLECTION OF WAVES 

193. Wave Front and Rays. Suppose that a disturbance 
originates at a point P (Fig. 112) so that a wave is propagated 
outwards with the same speed in all directions. 
After a short interval of time, this disturbance 
will have spread so as to affect similarly a series 
of particles at a distance s = vt. If the wave 
spreads in two directions only, the particles at 
the distance vt will all lie on the circumference 
of a circle drawn about P with a radius vt. If it extends 
outwards in three dimensions, the similarly affected particles 
will lie on the surface of a sphere drawn about the same 
point as a center and with the same radius. The circum- 
ference of the circle in the one case and the surface of the 
sphere in the other everywhere pass through portions of the 
medium in the same phase of vibration, and the curve or 
the surface is called a wave front. 




PROPAGATION AND REFLECTION OF WAVES 157 



If the wave front is a sphere, the wave is a spherical wave ; 
if the wave front is either a straight line or a plane, the wave 
is called a plane wave. 

In a uniform medium the direction in which the disturb- 
ance is traveling at any point of the wave is normal to the 
wave front; in the case of a circular or a spherical wave 
front, this direction is radial. A line drawn to indicate the 
direction in which the disturbance is propagated, and along 
which the energy travels outward, is called a ray. In an 
isotropic medium, that is, a medium having the same physical 
properties in all directions, the wave front and the ray at any 
point are at right angles to each other. The rays are then 
straight lines. If the medium is not isotropic, the rays may 
be curved. 

A ray is quite as real as a wave front, and the geometrical methods of 
treatment founded on rays are as legitimate as the methods depending 
m wave fronts. In the elementary treatment of subjects involving wave 
lotion it is advisable to make use of either the geometrical or the wave 
lethod according as the one or the other may 
?st serve for the simple exposition of funda- m 
mental principles. 

194. Huyghens' Principle. Let a be 

the center of disturbance, and let it be 
surrounded by a surface or wave front 
(only part of which, men is shown in 
Fig. 113). Then it is clear that the 
lotion in the medium outside this sur- 
face must be fully determined by the 
motion existing for the moment in the 
wave front men. The single wave as it 
travels outward disturbs all the elements 
of the medium through which it passes. 
The disturbance of any element may then be considered as 
one cause of the subsequent disturbance of all the other 
elements. Hence Huyghens' principle that every point of 
the wave surface becomes a new center of disturbance, from 




Fig. 113 



158 SOUND 

which waves are propagated outward in the same manner as 
from the original center, and the aggregate effect at any 
point outside the surface mn is the resultant of all the com- 
bined disturbances due to the secondary waves from these 
new centers. Every particle on the wave surface mn has the 
same oscillatory motion, except in point of amplitude, as the 
first particle disturbed; it therefore stands in the same rela- 
tion to adjacent particles, and communicates motion to them in 
the same manner, or becomes itself a center of disturbance. 

The principle of Huyghens is the principle of super- 
position. The disturbance at any point is due to the super- 
position of all the disturbances reaching it at the same 
instant from the various points of the medium through which 
the wave front passed an instant earlier. 

Let the points of the surface, mn be centers from which 
waves proceed for a short distance cd. Then with these cen- 
ters and a radius cd describe circular waves. The number of 
such waves being indefinitely large, they ultimately coalesce 
to form the new surface m'n', which is the envelope of all the 
small secondary waves. The effective part of each secondary 
wave Huyghens supposed confined to that portion which 
touches the envelope. In fact Stokes showed that the sec- 
ondary waves mutually destroy one another, except at the 
surface enveloping them. 

The energy of mn is thus passed on to m f n r , and in the 
same form or manner from m'n' to m"n"^ etc. 

195. Reflection of a Plane Wave at a Plane Surface. Let 
CD (Fig. 114) be the reflecting surface and AB a portion of 
the plane advancing wave front. To find the relation be- 
tween the inclination of the incident wave front and that of 
ithe reflected wave front to the reflecting surface proceed as 
follows : 

When the point A reaches the reflecting surface, it be- 
comes by Huyghens' principle a new center of disturbance 
traveling back into the first medium. Then with A as a 



PROPAGATION AND REFLECTION OF WAVES 159 




Fig. 114 



center and with a radius A A' equal to BE 1 describe a circle. 
This circle serves to limit the distance traveled^, by the re- 
flected disturbance while the disturbance from B is traveling 
toB'. In the same 
time the disturb- 
ance from b would 
have reached b' if 
there had been no 
obstruction. But 
in point of fact it 
travels to E and 
is there reflected. 
Hence with E as a 
center and with a 
radius El' draw 
another circle. In 
the same manner draw any convenient number of. circles. 

Finally from B' draw a tangent to the first circle ; it will 
touch all the other circles and will be the reflected wave 
front. Draw A A' 1 to the point of tangency on the first circle. 
A A" is the path of the disturbance reflected from A and is there- 
fore a ray. The triangle AA" B' is symmetrical with A A B' 
and equal to it ; it is therefore equal to the triangle ABB' . 
Hence the angles BAB 1 and A" B' A are equal to each other. 
But the former is equal to the angle of incidence and the lat- 
ter to the angle of reflection, for they are equal respectively 
to BB'n and nB'r, and these are the angles made by the 
incident and reflected rays with the normal at the point of 
incidence. Hence, the angle of incidence is equal to the angle 
of reflection. The former is the angle between the incident 
wave front and the reflecting surface ; the latter is the angle 
between the wave front after reflection and the reflecting 
surface. 

196. Relation of Direct and Reflected Systems of Waves. Let 
(Fig. 115) be the origin of the incident spherical waves, 



160 



SOUND 




and let AB be the plane reflecting surface. Without re- 
flection these waves would take positions at equal successive 
intervals of time indicated by the dotted lines; but because 

they are reflected, 
their positions by the 
Huyghens' principle 
are the full line 
curves symmetrically 
situated in front of 
the reflecting surface. 
Let 01 be the path 
traveled by the dis- 
turbance from any 
point of the incident 
wave. Draw IMso that OJand IM make equal angles with 
the normal IN. Then IM is the path of the reflected dis- 
turbance : that is, it is normal to the reflected waves. 
Project IM backwards until it intersects at 0' the normal 
through to the reflecting surface. Then 0' is the center 
of the reflecting waves. The triangles 010 and 0' 10 are 
equal. Therefore 00 and O'O are equal, and the centers 
of the incident and reflected waves lie on a normal to the 
reflecting surface and are equidistant from it. 

197. Stationary Waves. When we are dealing with a train 
of waves, water waves for example, and not with a single 
wave, any point in the liquid may be affected at the same 
time by an incident wave and by one traveling in the op- 
posite direction after reflection. The result of two systems 
of equal waves traversing the same medium in. opposite di- 
rections is a system of stationary waves. 

Let A (Fig. 116) be a wave moving to the right and B a 
wave of the same frequency and amplitude moving to the 
left. At points 5, , c?, etc., midway between the points of 
zero displacement of the two waves, the displacement of the 
wave A is equal and of opposite sign to that of wave B. 



PROPAGATION AND REFLECTION OF WAVES 161 

Hence, 5, <?, e?, etc., are points of zero displacement of the 
medium and of the resultant wave. Also W and bb ff will 
increase and decrease together equally because the two waves 
are equal and travel with the same speed in opposite direc- 
tions. These points therefore remain at Test and are called 
nodes. Intermediate points vibrate from zero displacement 




to a maximum, first in one direction and then in the other. 
Points midway between the nodes and having the maximum 
amplitude of vibration are called antinodes. 

In II each wave has moved forward half a wave length as 
compared with I. The nodes remain fixed, but the stationary 
wave has its displacements at all points of opposite sign to 
those of corresponding points in I. There is no progressive 
movement of the crests and troughs of the resultant wave, but 
only a periodic increase and decrease of amplitude. Such 
waves are therefore properly called stationary waves. 



CHAPTER VI 

PRODUCTION AND TRANSMISSION OF SOUND 
I. NATURE AND MOTION OF SOUND 

198. Sound and Hearing. The word sound is commonly 
used in two distinct senses : first, to designate the sensation 
produced when a disturbance is conveyed to the brain by the 
auditory nerves ; and, second, the external cause of that sen- 
sation. These are the subjective and objective aspects re- 
spectively of the phenomena of sound. The external stimu- 
lus stands first in the series of energy changes leading to a 
sensation, but it is not like the sensation. All the external 
phenomena of sound may be present without the hearing ear. 
Sound should therefore be distinguished from hearing in the 
study of sound. 

Sound may be defined as that form of vibratory motion 
excited in an elastic body which affects the auditory nerves 
and produces the sensation of hearing. 

199. The Source of Sound a Vibrating Body. The source from 
which sound proceeds is always a body in a state of vibration. Sound 
and vibratory movement are so related that one is strong when the other 
is strong, and they diminish and cease together. 

If a mounted tuning fork is sounded and a light ball of pith or ivory, 
suspended by a thread, is brought in contact with one of the prongs at 
the back, it will be briskly thrown away by the energetic vibration of the 
fork. 

Partly fill a goblet with water and produce a musical note by drawing 
a bow across its edge. The tremors of the glass will be communicated to 
the water and will throw its surface into violent agitation in four sectors, 
with intermediate areas of relative repose. This agitation disappears as 
the sound ceases. 

162 



NATURE AND MOTION OF SOUND 163 

A stout glass tube, four or five feet long, may be made to emit a 
musical sound by grasping it by the middle and briskly rubbing one end 
with a cloth moistened with water. So energetic are the longitudinal 
vibrations excited that it is not difficult to break the tube near the hand, 
on the side opposite to the end rubbed, into many narrow rings. 

Any regular succession of taps produces a musical sound. The taps 
must be regular or periodic to make the sound musical ; otherwise it is 
only noise. The vibrating body producing sound may be solid, liquid, or 
gaseous. Only the first and last are employed in musical instruments, 
the first comprising all instruments with strings, reeds, or bars, and the 
last all wind instruments. 

200. Transmission of Sound to the Ear. Sound requires for 
its transmission to the ear an uninterrupted series of elastic 
bodies. They may be solid, liquid, or gaseous. If the 
vibrating source of sound be isolated so that there is an 
interruption in the elastic medium, the vibrations do not 
reach the ear and no sound is perceived. Hence the well- 
known experiment of suspending a bell by a thin string 
within the receiver of an air pump. The sound is greatly 
enfeebled by exhausting the air ; and if hydrogen be then 
admitted and the exhaustion repeated, the sound will cease 
altogether, though the hammer may still strike the bell. The 
bell has then no elastic medium in contact with it, to which 
it can give up its energy of vibration. 

The transmission of sound by a solid is illustrated by the acoustic or 
string telephone. It consists of a taut string connecting the centers of 
thin elastic membranes stretched over the bottom of two small conical 
boxes. When one speaks or even whispers into one of these boxes, a 
listener at the other can hear distinctly, even at a distance of several 
hundred feet. The membrane vibrating transversely sets up longitudinal 
vibrations in the stretched string. These are transmitted to the membrane 
of the receiving instrument, and its motions reproduce the sounds actuating 
the transmitter. 

Sound waves consist of a series of condensations and rare- 
factions, succeeding each other at regular intervals. Each 
particle of air vibrates in a short path in the direction of the 
sound transmission. Its vibrations are longitudinal as dis- 
tinguished from the transverse vibrations in water waves. 



164 SOUND 

201. Motion of the Particles and of the Wave. The motion 
of the particles of the medium conveying sound is quite dis- 
tinct from the motion of the sound wave itself. This 
distinction holds for all undulations transmitted through a 
medium of motion. A sound wave is composed of a conden- 
sation followed by a rarefaction. In the former the particles 
have a forward motion in the direction in which the sound is 
traveling ; in the latter they have a backward motion, while 
at the same time both condensation and rarefaction travel 
steadily forward with a speed nearly independent of that of 
the particles composing the wave. 

The independence of the two motions is aptly illustrated by a field of 
grain across which waves excited by the wind are coursing. Each stalk 
of grain is securely anchored to the ground while the w r ave sweeps onward. 
The heads of grain in front of the crest are found to be rising, while all 
those behind the crest and extending to the bottom of the trough are 
falling. They all sweep forward and backward, not simultaneously, butm 
succession, while the wave itself travels continuously forward. 

A series of particles along the line in which a wave is 
traveling are in successively different phases of motion ; and 
the distance from one particle to the next one in the same 
phase is a wave length. While any element of the medium 
merely oscillates about its position of stable equilibrium, there 
is a continuous handing on or flow of energy from point to 
point. 

202. Characteristics of Musical Sounds. Musical sounds 
differ from one another in three important particulars: 

1. Pitch. The pitch of a note depends on the frequency 
or number of vibrations per second. An acute note has a 
greater frequency than a grave one. Also, since with a given 
speed of transmission the wave length is inversely as the 
frequency, a note may be designated by its wave length as 
well as by its frequency. 

2. Loudness. The loudness of a sound depends on the 
energy of the vibrations transmitted to the ear. It involves 
also the pitch of the note. The energy of vibration is pro- 



NATURE AND MOTION OF SOUND 



165 



portional to the square of the amplitude ; but as it is obviously 
impracticable to express a sensation with mathematical pre- 
cision, it will suffice here to say that the loudness of a note 
increases with the amplitude of vibration. 

3. Quality. Two notes of the same pitch and loudness, 
such as those of a piano and a violin, are yet clearly distinguish- 
able by the ear. This distinction is expressed by the term 
quality or timbre. Helmholtz demonstrated that the quality 
of a note is determined by the presence of tones of higher 
pitch, whose frequencies are simple multiples of that of the 
fundamental or lowest tone. 

Pitch depends on the length of the 
sound wave, loudness on its ampli- 
tude, and quality on the form of the 
complex wave. 




203. The Siren. An instrument 
in which a note is produced by the 
escape of air at regular intervals 
through holes in a rotating disk is 

called a siren. Thus, if the perforated 
disk of Figure 117 be mounted on the 
shaft of a small electric motor and rotated 
at a constant speed, a musical note may 
be produced by blowing a current of air 
through either of the concentric series 
of equidistant holes. The pitch will be 
different for each circle of holes, and will 
rise and fall with increase and decrease 
of speed. 

The siren of Caignard de la Tour is 
shown in Figure 118. The rotating por- 
tion is mounted on a wind chest, which 
has a top perforated with a circle of equi- 
distant holes. The revolving plate D contains the same 
number of holes as the lid (7, and a puff of air issues simulta- 




166 SOUND 

neously from all the holes every time those in the disk coin- 
cide with those in the lid. For each rotation of the disk 
there are thus as many puffs as there are holes in the circle. 
The rotations of the disk are counted by means of a gear 
train similar to that of a gas or electric meter. 

The rotation of the disk and spindle of the siren of Figure 
118 is produced by air pressure. The holes in both lid and 
disk are drilled obliquely and slope in opposite directions, as 
shown at E. Hence, a certain pressure is exerted against 
one side of the holes sufficient to set the disk in rotation. 

The frequency of a given note may be determined by keep- 
ing the speed as nearly constant as possible, and throwing 
the counting train in gear with the spindle for an observed 
length of time. Then the product of the number of revolu- 
tions and the number of holes in the disk divided by the time 
in seconds gives the frequency of the note produced by the 
siren. 

II. VELOCITY OF SOUND 

204. Velocity of Sound in Air. Common experience shows 
that the velocity of sound in air is a very moderate quantity. 
Thus, when a man is seen at a distance striking a blow with 
a hammer, the observer sees the blow struck an appreciable 
interval of time before he hears the sound. The time taken 
by light to travel over the intervening distance is entirely 
inappreciable in comparison with that of sound. 

Again, one may hear a sound produced at a distance both 
through a telephone and through the air, the latter report 
arriving for a distance of 1100 feet about a second later than 
the former. 

When one utters a loud sound a few hundred feet in front 
of the plane surface of a large building, the sound returns as 
a reflected wave or echo after an interval of time required 
for the sound to travel to the reflecting surface and back 
again. This interval is the same as the sound wave would 
take to travel to the observer from the center of the reflected 



VELOCITY OF SOUND 167 

system of waves as far behind the reflecting surface as the 
observer is in front ( 196). 

Either of these three methods of observation may be made 
the basis of an experimental determination of the velocity of 
sound in free air. The problem, however, is beset with 
many difficulties. In the open air the disturbing effects of 
the wind and of local differences in temperature are beyond 
the control of the observer. The velocity of sound is 
affected also by the amount of moisture in the atmosphere. 
To these disturbances must be added the delay in sense per- 
ceptions after the arrival of the exciting cause of them. 
This would introduce no error if the delay or reaction 
periods were the same for sight and hearing ; but they are 
not the same, and neither is the same for different persons. 

The effect of the wind has been eliminated by firing a 
cannon from the two stations in succession, and taking the 
mean of the intervals between seeing the flash and hearing 
the report. 

Beginning with the determinations made by the French 
Academy of Sciences, the following are the most trustworthy 
results, all reduced to C. : 

1. Academy of Sciences, 1738 .... 332.00 m./sec. 

2. Bureau des Longitudes, 1822 .... 331.00 m./sec. 

3. Moll and Van Beck, 1823 332.25 m./sec. 

4. Bravais and Martins, 1844 332.37 m./sec. 

5. Stone, 1871 332.40 m./sec. 

In Stone's experiments in South Africa two observers were 
stationed at unequal distances from a time gun, and each 
gave a signal with an electric key on hearing the report. 
This method eliminated the eye observations entirely, and 
confined them to the ear. The observers were stationed 
respectively 641 feet and 15,449 feet from the gun. The 
signals were recorded on a chronograph at the Observatory 
in Cape Town. Reciprocal firing was not resorted to, but 
allowance was made for the wind. The results were cor- 



168 SOUND 

rected also for the difference in the personal equation of the 
observers, the time required to perceive and record an 
event. The difference in the distances of the two observers 
from the time gun divided by the interval recorded on the 
chronograph gave the velocity of sound. The final value 
obtained by Stone for C. was : 

F~ = 332.4 m./sec., or F = 1090.6 ft./sec. 

205. Eegnault's Experiments. In the years from 1862 to 
1866 Regnault made an extensive series of observations on the 
propagation of a sound pulse through water pipes in Paris, 
in lengths up to 4900 m., and of diameters from 10.8 cm. 
to 1.1 m. When a gun was used as a source of sound, the 
instant of firing was recorded electrically by the breaking of 
a wire stretched across the muzzle ; the interruption was 
recorded on the revolving drum of a chronograph. The 
receiver at a distant point was a thin rubber membrane 
stretched over the smaller end of a wide cone. The move- 
ment of this membrane or drum was made to close an electric 
circuit, and the arrival of the sound pulse was thus recorded 
on the cylinder of the chronograph. 

Since the time interval was recorded automatically, it 
might at first appear that the error due to the personal equa- 
tion was eliminated; but in fact such a membrane has as 
much a personal equation as an observer. Time is required 
by it to acquire sufficient energy from the wave to move the 
membrane, and there is thus an inevitable delay in respond- 
ing to the sound. This delay was the greater the feebler 
the sound. Regnault sought to allow for it, but it is prob- 
able that his results are slightly under the true value. 

Regnault found that the velocity of sound decreases with 
its intensity,- tending toward a lower limit for feeble sounds. 
Also, that it increases with the diameter of the tube, tending 
toward an upper limit in very wide tubes. His final result 
for a feeble sound in a wide tube at C. was: 

F = 330.6 m./sec., or F = 1084.6 ft./sec. 



VELOCITY OF SOUND 169 

206. The Krakatoa Eruption. At a temperature only a 
little above C. the velocity of sound is 1100 ft. /sec. 
Compare this with the speed of a railway train: 

88 ft. /sec. = 5280 ft./min = 1 mile/min. = 60 miles/hour. 
Add one quarter, and 

110 ft. /sec. = 6600 ft./min = 1.25 miles/mm. =75 miles/hour. 
Sound at 1100 ft. /sec. is then only ten times the speed 
of a very fast railway train, or 750 miles per hour. The 
air-line distance between New York and Chicago is about 
750 miles. Hence a very loud sound would travel from one 
city to the other in an hour. 

The Krakatoa eruption, August, 1883, furnished a very 
remarkable example of sound waves on a gigantic scale. A 
series of explosions culminated in a tremendous outburst, 
which blew a part of a mountain into the air. At a distance 
of 2000 miles these explosions sounded like the firing of 
heavy guns. The chief outburst produced a pulse or wave 
of such intensity that it left a noticeable trace on self-record- 
ing barometers as it traveled around the globe. A study 
of these records showed that the wave traveled with a 
speed of about 700 miles an hour. It culminated on the side 
of the earth opposite Krakatoa in 18 hours ; it then spread 
out again and culminated several times afterward, its trace 
not being entirely lost until after 127 hours. 

207. Theoretical Determination of the Velocity of Sound. 
The velocity of a longtitudinal wave may be calculated in 
terms of the coefficient of volume elasticity Tc and the density 
d of the medium.* The result 

F= x /| (47) 

ct 

was first deduced by Newton. This formula applies directly 
to transmission through solids and liquids. For gases it 
may be modified by substituting the pressure of the medium 

*Carhart's University Physics, Part I, p. 152. 



170 SOUND 

for the coefficient of elasticity ; for by Boyle's law, if P and 
$ are the original pressure and volume and P + p and S s 
the corresponding new values, then 



= PS + pS-Ps-ps. 

Neglecting the product ps of two very small quantities, we 
have o 

Ps = p$, or P=p- = k ( 125). 

8 

Hence, finally, substituting P for k, 

F-'-J? 

Vy (48) 

P = 1,013,250 dynes per cm. 2 and d = 0.001293. Therefore 



at C. F == 27,993 cm./sec., or 918.4 ft./sec. 



Direct observation gives 33,240 cm./sec. The discrepancy 
is about one sixth. 

208. Laplace's Correction. Newton attempted to explain 
the failure of his formula for the velocity of sound in air to 
give a result corresponding with the observed value; but 
his explanations were mere hypotheses, and it remained for 
Laplace to detect the error in 1816 and to point out the 
correction. 

The proof that the coefficient of elasticity of the air is 
numerically equal to the pressure depends on an application 
of Boyle's law ( 207); but Boyle's law is true only under 
the condition of a constant temperature. The corresponding 
coefficient is called the isothermal coefficient of elasticity of 
a' gas. But there is another coefficient called the adiabatic 
coefficient of elasticity. This is the coefficient when no heat 
leaves or enters the gas during the compression and expansion 



VELOCITY OF SOUND 171 

respectively. Now heat is generated in compressing a gas, 
and is absorbed when a gas expands. The compression and 
expansion of a gas are, therefore, not isothermal except 
when these changes take place very slowly so that equaliza- 
tion of temperature is possible. If they occur rapidly, there 
is no time for the temperature to come to equilibrium by 
conduction and radiation. 

In sound waves the compressions and rarefactions follow 
each other in such quick succession that there is no time for 
the compressed portion of a wave to share its excess of heat 
with the cooler rarefied portion. The result of this rise of 
temperature in compression is that the pressure for any given 
decrease in volume must be higher than when the tempera- 
ture is constant. So also the fall of temperature in rarefac- 
tion results in a greater decrease of pressure for a given 
increase in volume. The coefficient of elasticity is therefore 
greater than when these changes are isothermal. The rela- 
tion between pressure and volume is not then the relation 
expressing Boyle's law 

pv = a constant ( 167), 

but it is expressed by the formula of Poisson, 
pv y = a constant. 

The exppnent 7 is the ratio of the specific heat of a gas under 
constant pressure to its specific heat at constant volume 
(390). The corresponding adiabatic coefficient of elastic- 
ity is 7 times the isothermal coefficient, or <yP. 
For air 7 = 1.405. Hence 



VQ = -^ 1.405 = 33,181 cm./sec. 

This value agrees very closely with the best experimental 
determinations and appears to justify Laplace's supposition 
that the changes of volume are adiabatic. 



172 SOUND 

209. Correction for Temperature. Changes of pressure, 
such as a change in the barometric pressure, unaccompanied 
by changes of temperature, do not affect the velocity of 
sound in air; for P and d then vary in the same ratio, and 
their quotient Pfd remains unchanged. But this is not the 
case when a change of pressure is due to a change in tempera- 
ture. If we assume that the relation between pressure and 
temperature is expressed by the equation 



where a is the coefficient of expansion of a gas, ^rg- or 
0.00367, then the velocity at any temperature becomes 



or V= F V(l + 0.003670- 

For small values of t this expression becomes 
F=F (1 + J0 
= F (1 + 0.001830- 

The increase in the velocity of sound for 1 C. is therefore 
33,200 x 0.00183 = 60.7 cm. /sec., or 23.9 in. /sec. 

210. Velocity of Sound in Water. The compressibility 
of liquids is very slight and their isothermal and adiabatic 
coefficients of elasticity are substantially the same. There- 

ri 

fore the general formula for the velocity of sound V = %/-j 

is directly applicable to them. The compressibility of 
water at 4 C., that is, the decrease in volume of a unit vol- 
ume due to an increase of pressure of one atmosphere, is 
0.0000499. Hence 

k = * tre * 8 - 1*013,250 
~ strain ~ 0.0000499* 



VELOCITY OF SOUND 173 

The density of water at 4 C. is unity. Therefore the veloc- 
ity of sound in water at 4 C is 



= 142>50 cm - /sec " r l425 m -/ sec - 

In 1827 Colladon and Sturm measured with much care the 
velocity of sound in the water of Lake Geneva between boats 
anchored at a distance apart of 13,487 m. The mean time 
required for the transmission over this distance of the sound 
of a bell struck under water was 9.4 seconds. This gives 
for the observed velocity of sound in water 1435 m. at 8.1 C. 

A system of transmitting signals through water by means of sub- 
merged bells is in use by vessels at sea and for offshore stations. 
Special telephone receivers have been devised to operate under water 
and to respond to the sound signals. Indeed, the vessel itself acts as a 
sounding board arid as a very good receiver. 

211. Reflection of Sound. Whenever the medium trans- 
mitting sound changes suddenly in density, a part of the 
energy is transmitted and a part reflected. One system of 
waves then gives rise to two systems, and the intensity of 
the sound in either system is less than before reflection. 
The intensity of the reflected system is the greater the 
greater the difference in the densities of the two media. A 
dry sail reflects a part of sound and transmits a part ; but 
when wet it becomes a better reflector and almost impervious 
to sound. 

When an obstacle is interposed between the source of 
sound and the ear, if the dimensions of the obstacle are not 
much greater than the wave length of the sound, the waves 
close in around the edges of the obstacle and the intensity 
behind it is but little weakened. But for waves shorter 
than the dimensions of the obstacle, it casts a sound shadow, 
or the sound behind it is relatively faint. The dependence 
of the sound shadow on the dimensions of the obstacle as 
compared with the wave length may be strikingly shown 
in the following manner : Place the ear in line with a watch 



174 



SOUND 



and a clock, the watch much nearer the ear ; then interpose 
a quarto book or magazine, or even a folded newspaper, 
between the ear and the watch. The ticking of the watch 
will become almost or quite inaudible, while that of the clock 
will be scarcely changed. The wave lengths of the ticking 
of the clock are perhaps ten times as long as those of the 
watch. 

Reflection of sound may take place from the surface of 
a rarer medium as well as from that of a denser. Thus, 
it is easy to demonstrate that sound is reflected from a large 
flat gas flame, and that the angle of incidence is equal to 
the angle of reflection. In this case the phase of the 
reflected wave is changed by half a wave length, a condensa- 
tion being reflected as a rarefaction and conversely. 

When the reflection is from the denser surface, the motion 
of the air particles is reversed, and a condensation is reflected 
as a condensation ; but when the reflection is from the sur- 
face of the rarer medium, the motion of the air particles is 
not reversed, each particle moving forward beyond its normal 
position of equilibrium, and a condensation is reflected as a 
rarefaction. Mechanically the impact of two unequal elastic 
balls illustrates the same phenomena. When the smaller ball 
strikes the larger one, its motion is reversed ; but when the 
larger ball strikes the smaller, it moves forward without 
reversal, but with diminished velocity. 

212. Echoes. An echo is a sound reflected normally. A 
clear echo requires reflection from a vertical surface, the 
dimensions of which are large compared to the wave length 
of the sound. A cliff, a wooded hill, or the broad side of 
a building may serve as the reflecting surface. It must be 
smooth in the sense that its inequalities are small compared 
to the wave length of the incident sound. The same con- 
dition applied to the reflection of light requires highly 
polished surfaces for regular reflection as distinguished from 
diffusion. 



VELOCITY OF SOUND 175 

Parallel reflecting surfaces at suitable distances produce 
multiple echoes, as parallel mirrors produce multiple images. 
For short distances from the reflecting surface, the direct and 
reflected sounds are confused, as in the case of rooms with 
bad acoustic properties. A circular room, like the Baptistry 
at Pisa, may prolong a sound for ten seconds or more by 
successive reflections. The effect at Pisa is made more 
conspicuous by the good reflecting surface of polished marble. 

Echoes sometimes present peculiarities which may be re- 
ferred to the character of the reflecting surface. The re- 
flected sound, such as a shout, may be an octave higher than 
the incident sound. The octave is present in the shout itself, 
but is masked by the louder fundamental tone. The in- 
equalities of the reflecting surface are such that the waves 
constituting the fundamental tone are diffused, while the 
shorter waves of the octave are regularly reflected with much 
smaller diminution of intensity. 

Aerial echoes also are often observed. The reverberations 
of distant thunder are doubtless due in part to successive re- 
flection from clouds. Air almost perfectly transparent to 
light may have great acoustic opacity. When for any reason 
the atmosphere becomes unstable, vertical currents are formed 
and vertical sections or banks of different densities. The 
sound transmitted by one bank is in part reflected by the 
next, the successive reflections giving rise to a curious pro- 
longing of a short sound. Thus, the sound of a gun or a 
whistle is then heard apparently rolling away to a great dis- 
tance with decreasing loudness. 

213. Deflection of Sound Waves by the Wind. Sound is heard 
better with the wind than against it. Sound waves travel- 
ing in the same direction as the wind are deflected downward, 
while those going against the wind are deflected upward. 
Imagine plane vertical waves traveling horizontally. The 
upper layers of the air move faster than those next to the 
ground ; and if the wind is blowing in the same direction as 



176 SOUND 

the sound is traveling, the velocity of the wind must be added 
to that of the sound. Hence the upper parts of the sound 
waves travel faster than the lower, the wave fronts are tilted 
forward, and there is a condensation along the surface of the 
ground, with increased loudness. 

On the other hand, the velocity of sound against the wind 
is diminished by the wind, the upper parts of the waves are 
retarded more than the lower, the wave fronts are tilted 
backward, and the sound is deflected into the upper air with 
a consequent weakening near the ground. 

A similar deflection of sound waves occurs when there is a 
vertical temperature gradient. The velocity of sound in- 
creases about 0.6m. per degree C. If, therefore, there is a 
fall of temperature with increase in elevation, sound travels 
faster near the ground than higher up, the wave fronts are 
tilted backward and the sound may be deflected entirely over 
the listener. Such is the case on a hot summer day. 

If the temperature gradient is reversed, the wave fronts are 
tilted forward, and the sound is condensed along the ground. 
When the air is still after sunset, it cools near the ground 
more rapidly than above, and sounds are then heard at great 
distances. Similarly on frosty mornings when the air is still, 
it is cooled by contact with the ground, and sounds carry 
remarkably well. 

214. The Doppler Effect. If the source of sound is ap- 
proaching the ear with a velocity t>, then more waves reach 
the ear in a second than if the source were stationary, and 
the pitch of the note is correspondingly higher. 

The apparent wave length of the sound in the air is re- 
duced by the distance traveled by the source during the 
period T of the note. This distance is vT or v/n. Then the 
apparent wave length becomes V/n v/n = 1/n ( V t>), 
and the apparent frequency 



V v 



VELOCITY OF SOUND 177 

Similarly, if the source is moving away from the observer, 
the apparent frequency, that is, the frequency reaching the 

ear, isnl J. This phenomenon is known as the "Dop- 

pler effect." It may be readily detected if one notes the 
pitch of the whistle or of the bell of a locomotive when it 
approaches as compared with the pitch when it recedes in 
passing. 

A similar phenomenon is observable in light when the 
source is moving with respect to the observer with a velocity 
comparable with that of light ( 315). 

Problems 

1. How long will be required for sound to travel a distance of 2 mi. 
in air at 20? 

2. On a day when the temperature was 25, the interval between 
seeing a flash of lightning and hearing the thunder was 4 sec. How 
far away was the lightning? 

3. A gun is fired and after 4 sec. the echo from a distant hill is 
heard. How far distant is the hill, the temperature being 20? 

4. A shell fired at a target, half a mile distant, was heard to strike 
it 4.5 sec. after leaving the rifle. What was the average velocity of 
the shell, the temperature of the air being 20? 

5. If the velocity of sound in air at 20 is 1120 ft. per second, what 
would its velocity be in carbon dioxide at the same temperature, assum- 
ing carbon dioxide to be 1.44 times as heavy as air ? 

6. If sounds travel in air at the rate of 340 m. per second, and in- 
stantaneously through a telephone line, what would be the distance 
between the source and the observer if the interval between hearing the 
sounds by the two methods of transmission was 4 sec. ? 

7. If the velocity of sound in air is 340 m. per second, and in iron 
5130 m. per second, what would be the interval between hearing a sound 
through an iron bar 340m. long and through the air? 

8. A tuning fork gives a note of the same pitch as a siren, which 
rotates 270 times in 10 sec., the disk having 16 holes. What is the 
frequency of vibration of the fork? 



178 SOUND 

9. A locomotive is running at the rate of J- of a mile a minute. 
The difference between the apparent frequencies of vibration of the note 
given by the whistle when approaching the observer and after passing 
him is 37.5 vibrations per second. If the velocity of sound is 1120 ft. 
per second, what is the true frequency of the whistle? 

10. A stone is dropped into a well and is heard to strike the water 
after 4 seconds. What is the depth of the well if the velocity of the 
sound is 335 m. per second? 

11. If at night signals from a station are given simultaneously by 
means of a flash of light and a stroke on a submerged bell, what is the 
distance from the station of a ship on which the interval between seeing 
the flash and hearing the sound of the bell, by a telephone responding to 
the sound signal through the water, is 1| seconds, if the velocity of sound 
in the water is 1430 m. per second ? 

12. An observer is stationed at a distance of 6 km. from a gun, which 
is fired at noon ; if the wind is blowing at a speed of 80 km. an hour, and 
the observer is stationed directly to windward, at what time will he hear 
the report ? 



CHAPTER VII 

PHYSICAL BASIS OF MUSIC 

I. MUSICAL INTERVALS AND SCALES 

215. Musical Intervals The pitch of a musical note is 

the pitch of its gravest component, or fundamental tone. 
Pitch may be defined in two ways : 

1. Physically, as the number of vibrations per second in 
the fundamental tone of the note. 

2. Musically, by referring the note to its place in an arbi- 
trary scale of musical pitch. 

When two notes are sounded together, the ear recognizes 
a relationship between them, known as a musical interval. 
Its measure is the ratio of the frequencies of the two notes. 
Names have been given to many such intervals employed in 
music. When the ratio is 1, the interval is called unison; 

2, an octave; -, a fifth; -, & fourth; -, a major third; 

6 25 

, a minor third ; , a chromatic semitone. 

o 2i- 

Any three notes whose frequencies are as 4 : 5 : 6 form a 
major triad, and together with the octave of the lowest 
note, a major chord. Three notes whose frequencies are as 
10 : 12 : 15 form a minor triad, and together with the octave 
of the lowest note, a minor chord. 

A single tracing point may be set in motion by two or more systems 
of sound waves at the same time. If the surface on which the curve is 
to be inscribed is moved at right angles to the motion of the tracing 
point, the resultant curve will be due to the composition of the several 
motions in the same plane. The upper curve in Figure 119 is the result 

179 



180 SOUND 

of combining in this way three simple harmonic motions composing a 
major triad. The lower curve in the same way shows the composite 






4:5:6 




10 1 : 12:15 



X/\^VN/W\/\AA/ N/ V^^ 

Fig. 119 

motion resulting from a minor triad. It will be noticed that a 
complex wave recurs in both cases, but less frequently in the second 
than in the first. 

216. The Diatonic Scale. Continuity of tone from one 
note to the next, such as occurs in the whistling of the wind, 
or in the moaning of animals in pain, has always been avoided 
in musical compositions. The change of pitch in any melody 
takes place by steps or intervals, and not by continuous 
transition. A musical scale is the succession of notes by 
which musical composition ascends from one note, called the 
keynote, to its octave. The last note in one scale is regarded 
as the keynote of another series of eight notes with the same 
succession of intervals. In this way the series is extended 
until the limit of pitch established in music is reached. 

The common succession of eight notes in the major mode, 
called the diatonic scale, is one of many used at different 
times, and finally adopted by European peoples about three 
hundred and fifty years ago on account of its fitness to 
express the style of music cultivated by them. 

In the notation employed by Helmholtz the octave below 

bass c is written 

D E F a A B c. 

The octave above c is 

cdefgabc'. 

The octave above c' (middle (7) has one accent, and th< 
next higher two accents. 



MUSICAL INTERVALS AND SCALES 181 

In each octave there are three major triads for the key of O: 

c f : e' : g' \ 

g 1 :V :d"\ : : 4 : 5 : 6. 

fia' : c"\ 

It is the universal custom in Physics to take the frequency 
of c? as 256, because this number is a power of 2. This fre- 
quency is practically that of the middle O of the piano. If, 
then, c' is due to 256, or m, vibrations per second, the fre- 
quencies of the other notes of the diatonic scale may be found 
by simple proportion ; they are as follows : 

256 288 320 3411 384 426| 480 512 

c' d r e r f g' a' V c" 

Do Re Mi Fa Sol La Si Do 

m % m \m 4m f m 4 w - m 2 m 

o ~k o & o O 

If the fractions representing the vibration frequencies are 
reduced to a common denominator, the numerators may be 
taken to represent the relative frequencies of the eight notes 
of the scale. They are 

24 27 30 32 36 40 45 48 



The intervals from each note to the next higher are : 

Major Minor Half Major Minor Major Half 
Tone Tone Tone Tone Tone Tone Tone 

I tt I V I if 

81 

The interval between a major tone and a minor tone is ; 

80 

it is the smallest interval recognized in music and is called a 
comma. 

217. Transposition. In addition to the eight notes of the 
diatonic scale, use is made of additional notes derived from 
the eight by raising or lowering the pitch of each by a 

chromatic semitone, --. If the pitch is raised, the note is 



182 SOUND 

said to be sharp; if lowered, flat. Thus 6rj is read Gr sharp, 
and 6rt? is read 6r flat. 

Sharps and flats are introduced by changing scales. It 
is necessary to have a choice of scales with keynotes of 
different pitch to suit different voices. Also, to avoid 
monotony in the same composition, it is desirable to be able 
to change from one scale to another, whose keynote bears 
a simple relation in frequency to the keynote of the first 
scale. Hence arises a series of scales whose keynotes are 
the successive notes of the first one. Many notes are 
common to all these scales, and when negligible intervals 
are eliminated, only twenty notes remain to each octave. 
The process of changing from one keynote to another is 
called transposition. 

If, for example, the keynote is transferred from C to 6r, 
then the eight notes of the new scale must follow 6r with 
the same succession of intervals as in the key of O ( 216). 
The only notes in the key of Gr differing from those in the 
key of O are F and A. A numerical comparison of the two 
scales shows the precise difference. 

Key of O 

c' d' e' f g' a' b r c" d" e" f" g" 
256 288 320 341J 384 426| 480 512 576 640 682| 768 

Key of a 
256 288 320 360 384 432 480 512 576 640 720 768 

81 
The interval between the #'s of the two scales is only , 

80 

and may be neglected. The interval between the/'s is much 
larger and in the key of G- the / becomes /$. Instead of 
writing the sharp with this note every time it occurs, it is 
usually placed at the beginning of the staff and forms what 
is called the " signature " of the key. The signature of the 
key of 6r is therefore F sharp. 



MUSICAL INTERVALS AND SCALES 183 

218. Tempered Scales. Every transposition from one key 
to another increases the number of notes in an octave, and the 
number required for just intonation in all keys is in excess 
of what is mechanically practicable in instruments with fixed 
keys, such as the piano and organ. It is therefore necessary 
to adopt some system of accommodation by which the advan- 
tages of just intonation are sacrificed in order to reduce the 
number of notes. Such an adjustment or compromise is 
called temperament. 

By sacrificing the advantage of notes with frequencies in 
simple ratios to the keynote, in order to avoid the practical 
difficulties of musical execution, it has been found possible 
to make a sufficient variety of scales out of twelve notes to 
the octave. Since the interval from E to F and from B to 
O in the diatonic scale is already a semitone, no other note is 
interpolated at those points. The extra notes occur where 
there are whole tones, or in groups of twos and threes, repre- 
sented by the black keys on the piano, making thirteen notes 
to the octave and twelve intervals. 

The system of temperament most commonly applied to the piano and 
organ is the system of equal temperament introduced by Bach. This system 
ignores the difference between major and minor tones, and makes all the 
intervals from note to note the same. This ratio applied twelve times 
equals an octave, or 2. The half-tone interval in the system of equal 
temperament is therefore equal to v/2 = 1.05946. The results differ 
widely from pure intonation. The only accurately tuned interval in such 
a scale is the octave, all the others being more or less modified. 

The numbers following show the differences between the diatonic and 
equally tempered scales : 

c' d' e' f g' a' V c" 

Diatonic 256 288 320 341.3 384 426.7 480 512 

Tempered 256 287.3 322.5 341.7 383.6 430.5 483.3 512 

The frequencies of none of the corresponding notes differ as much 
as one per cent. In a melody where only single notes are sounded 
in succession, the difference between the two scales is hardly noticeable ; 
but when several notes are sounded in a chord, the contrast is more 
marked. 



184 SOUND 

219. Musical Pitch. The intervals between the notes of the 
scale are quite independent of the absolute frequencies, and 
depend on the ratios of these frequencies only. The actual 
pitch employed in music has varied widely in the last two 
centuries. In the time of Handel middle A (a') had a fre- 
quency of 424 vibrations per second, while for the organ in 
England in the middle of the eighteenth century a' was as 
low as 388 vibrations. In Paris in 1700 a' was 405, but in 
1857 it had gradually risen to 448. As late as 1857 a maker 
of pianos in New York tuned his pianos by an a' fork giving 
451.7 vibrations per second. Modern concert pitch has risen 
as high as 460 for a'. 

There are several possible reasons for this progressive rise 
of pitch ; the most obvious one is that instrument makers 
have intentionally raised concert pitch for the purpose of 
increasing the brilliancy of orchestral music. At the sanie 
time vocalists have been made to suffer. 

The frequency 435 for a' was chosen by the Paris Academy 
of Sciences, and at present probably this pitch receives the 
widest recognition. In 1834 the German Society at Stutt- 
gart recommended a' = 440 as the standard pitch. This 
makes c' = 264, or 11 times 24. The notes of the diatonic 
scale for the key of O corresponding to this standard may 
accordingly be found by multiplying the numbers 24, 27, 30, 
32, 36, 40, 45, 48, by 11. The corresponding frequencies for 
the diatonic and equally tempered scales are then the follow- 
ing, the a's being the same in both : 

c' d' e> f g' a' V c" 

Diatonic 264 297 330 352 396 440 495 528 

Tempered 261.6 293.7 329.6 349.2 392 440 493.9 523.3 

220. Limits of Pitch. In the modern piano of seven 
octaves, the bass A has a frequency of about 27.5, the high- 
est A, 3480. Allowing for slight variations from the stand- 
ard in tuning, the range of frequency for the piano is from 
27 to 3500 vibrations per second. 



Q 

MUSICAL INTERVALS AND SCALES 185 

The gravest note of the organ is the C of 16 vibrations 
per second, given by the 33-foot open pipe. Its wave length 

344 

in air at normal temperature is - = 21.5 m., or 70.5 ft. 

16 

The highest note is the same as the highest A of the piano, 
the third octave above a'. 

The cultivated voice of a singer has a range of about two 
octaves. The voice of women has about twice the frequency 
of that of men. The lowest note of the human voice, not 
including certain exceptional cases, is O of 65 vibrations. 
The entire range is included between this note and 
c" f = 1044, the two higher octaves belonging to women. 

The limits of hearing far exceed those of music. The 
range of audible sounds is about eleven octaves, or from 
<7 2 = 16 to c vlu == 32,768, though many persons of good hear- 
ing perceive nothing above c vu = 16,384. By means of a 
Galton's whistle of adjustable length, a series of short waves 
may be produced, gradually passing beyond the upper limit 
of hearing. 

221. Resultant Tones. When two loud notes, differing less 
than an octave, are sounded together, they give rise to a third 
tone, called a resultant tone, whose frequency is the difference 
of the vibration rates of the two. They were discovered by 
Sorge in 1740, and independently by Tartini in 1754 ; they 
are therefore often called Tartini's tones. 

Two notes c" and g", whose frequencies are 512 and 768, 
together give a distinct tone c 1 with a frequency of 768 512 
= 256. So also c" and e ff , with frequencies 512 and 640, 
give as a resultant tone 640 512 = 128, or c. 

Helmholtz called these tones difference tones to distinguish 
them from the summation tones, discovered by himself. The 
frequency of the latter is equal to the sum of the frequencies 
of the two notes producing them. 

Summation tones are fortunately much feebler than differ- 
ence tones, since they are mostly inharmonic. 



186 



I 
SOUND 



II. TRANSVERSE VIBRATION OF STRINGS 

222. The Sonometer. In sound a string signifies a cord or 
wire stretched between two fixed points with such a tension 
that when the string is slightly displaced and released, it vi- 
brates to and fro, and gives a musical note. For small dis- 
placements the increased force due to the flexure of the wire 
or cord is usually negligible in comparison with the restoring 
force due to the tension. 

The sonometer consists of a thin wooden box, across which 
is stretched a violin string or more commonly a thin piano 
wire (Fig. 120). The wire is fixed at one end and passes 




Fig 120 

over two bridges, A and J5, near the ends, and finally over 
the pulley N. It is desirable to have a comparison wire 
which can be tuned by means of a wrest pin, or by another 
pulley with weights. By means of a movable bridge B, 
sliding along a scale, the length of the wire may be short- 
ened at pleasure. 

223. Laws of Strings. The sonometer may be used to 
verify the following laws of strings: 

1 . The vibrations are isochronous for a given string at a 
given tension. If the string is plucked or bowed, the pitch 
of the note remains the same while its intensity dies away as 
the amplitude decreases. The pitch is therefore independent 
of the amplitude. 

2. The frequency of vibration is inversely as the length 
for a given tension. Tuning the two wires to unison, shorten 



TRANSVERSE VIBRATION OF STRINGS 187 

Q A q O 

one of them by moving the bridge B to -, ^, -, -, etc. The 
successive intervals between the notes given by the two wires 

will be ^, j, ~, ~, etc. Hence the notes given by the wire of 

o 4 o -^ 

variable length are those of the diatonic scale, and the fre- 
quencies are inversely as the lengths. 

3. When the length is constant, the frequency varies as 
the square root of tlxe tension. Starting with a given ten- 
sion and the strings in unison, increase the stretching load 
on AD four times ; it will then give the octave of the other. 
Increase it nine times, and AD will give the twelfth above 
the other with three times the frequency. These statements 
may be verified by dividing the comparison wire by a bridge 
into halves and thirds, so as to put it in unison with the 
other wire of variable tension. 

4. Tfo length and tension being constant, the frequency 
varies inversely as the square root of the mass per unit 
length. Stretch with equal tensions two strings differing in 
diameter and material, that is, in mass per unit length. 
Bring them to unison with the movable bridge. The ratio 
of the lengths will be the inverse of that of the square 
roots of the masses per unit length. If the strings are of 
the same material, their masses per unit length will be as the 
squares of their diameters. 

The last three laws are all expressed by the following 
equation, in which n is the frequency of vibration, I the 
length of the string, t the tension, and m the mass per unit 
length : . r- 

*-M (49) 

The figure 2 in the denominator comes from the fact that 
for one vibration the disturbance traverses the length of the 
string twice, that is, the wave length in the string is 2 1. 

224. Fundamental and Harmonics of a String. When a 
string is plucked or bowed at its middle point, it vibrates as 



188 



SOUND 



a whole and gives its lowest or fundamental tone. But if a 
string is lightly touched or damped at its middle point, and 
either half is bowed, it then vibrates in two segments and 
yields the octave above the fundamental. If it is damped at 
a point one third of its length from one end, and the shorter 
portion is bowed, it will give the twelfth above the funda- 
mental with three times its frequency. Damping in succession 
at points J, ^, J, etc., from one end, and bowing every time 
the shorter length, notes of 4, 5, 6, etc., times the frequency 
of the fundamental will be obtained. The series of notes 
obtained in this way are called harmonics of the fundamental. 

A simple addition to the above experiment serves to show that when 
a string is damped and bowed as described, it divides into vibrating 
segments, each equal in length to the one bowed, and separated by points 





Fig. 121 

relatively at rest. Cutting V-shaped slips of paper and placing them on 
the wire as riders (Fig. 121), it will be found that those at the middle of 
the segments are thrown off, while those at the points separating the 
segments remain seated. The two sets of riders are more easily distin- 
guished if they are of different colors. 

The intermediate points of least motion and the ends are called nodes ; 
the vibration sections, loops or segments; the middle points of the seg- 
ments, antinodes. The experiment illustrates stationary ivaves. The 
bowed segment sends waves along the wire which are reflected from the 
distant end, and the direct and reflected waves combine to produce 
stationary waves as described in 197. 

225. Melde's Experiment. This experiment illustrates in 
a most beautiful manner the division of a string into equal 



TRANSVERSE VIBRATION OF STRINGS 189 

vibrating segments. A light silk cord is stretched hori- 
zontally between a small fixed pulley and one prong of a 
vertical tuning fork. In Figure 122 A the plane of the 
two branches of the fork contains the cord, the end of 
which is moved horizontally in the direction of its length 
when the fork vibrates. The cord relaxes, falls to its 
lowest position with the forward movement of the fork, 
again rises to the horizontal, and then to its highest posi- 
tion when the fork is again in its most forward position. 



f 
f 



Fig. 122 



The longitudinal movement of the point of attachment 
thus gives rise to a transverse vibration of the cord, with 
a period double that of the fork. If W is the weight 
necessary to make the cord vibrate as a whole, IF/4 will 
make it vibrate in two segments, and IF/9 in three seg- 
ments. When the proper tension has been found, the cord 
spreads out in a pearl-white spindle, which appears to be 
fixed and stable. 

If now the fork is turned on its axis (Fig. 122 .5) so that 
it applies transverse impulses to the cord, the conditions 
then obviously require the fork and cord to vibrate in 
unison. Hence the cord vibrates in twice as many segments 
as in the first position; for if the frequency is doubled, the 
length of each segment must be halved with the same cord 
and stretching weight. 

This experiment demonstrates the law of lengths and the 



190 



SOUND 



law of tensions. Moreover, from such experiments we 
conclude that, 

A string may vibrate as a whole, or in any number of equal 
parts, the frequency being proportional to the number of parts. 

226. Stroboscopic Observation of a Vibrating String. The 

motion of a vibrating string is too rapid to permit it to be followed 
directly by the eye. A string may, however, be made to appear to 
execute its vibrations as slowly as one may desire by the " stroboscopic 
method " of illumination. 

If the string is illuminated by intermittent flashes of light, the period 
between flashes being equal to the period of vibration of the string, it 
will be visible only in one of the positions through which it passes in 
a complete vibration, while in all other positions it is not illuminated 
and is invisible. Hence it appears to be at rest. 

If the succession of flashes has a slightly different period from that 
of the string, they will find it in successive positions of its swing, and it 
will appear to execute a complete vibration in the period of time required 
for the intermittent illumination to gain or lose one flash as compared 
with the vibrations of the string. The more nearly the two periods 
coincide, the slower will be the apparent motion of the string. 

Figure 123 illustrates one method of arranging the apparatus for the 
stroboscopic method of illumination. Sunlight reflected from the plane 




Fig. 123 



mirror H passes through a converging lens L and is brought to a focus 
at 0. Just beyond O is placed a perforated disk B, mounted on an axis 
of rotation aa. The disk should be perforated with from two to four 
holes on a circle about aa as a center and at equal distances apart. 

Every time an opening comes to the position x, a flash of light passes 
through and illuminates the string S. The string is placed obliquely to 
the. diverging beam of light, so that its whole length is illuminated 
without any great difference in intensity. 

Instead of sunlight an electric lantern may be used ; and the string 
is preferably horizontal and may be kept in vibration by a fork, as in 



TRANSVERSE VIBRATION OF STRINGS 191 

Melde's experiment. The experiment is the more interesting and 
instructive if the string vibrates in two or three segments. 

The perforated disk may be mounted directly on the shaft of a email 
electric motor, and its rate of rotation may be adjusted by means of 
resistance in the electric circuit and by light controllable friction on the 
shaft. It is difficult to secure strictly uniform rotation of the disk, but 
its rotation may be kept constant long enough to show all the motions 
of the vibrating string. 

The same method of illumination gives most beautiful results when 
applied to a spiral spring vibrating in segments. The vibrations are 
maintained by attaching the spiral spring at one end to the prong of a 
fork which is kept vibrating by an electromagnet. The spring is 
placed next to the condenser of the lantern and is projected through the 
openings in the disk in the usual way. 

227. Coexistence of the Fundamental and Overtones. If the 

wire of a sonometer be bowed vigorously and then be lightly 
damped at its middle point, the fundamental will disappear 
and the octave with twice the frequency will be heard 
instead. If again the wire be bowed as before and then 
be damped at one third of its length, it will cease to give 
the fundamental, and the twelfth with three times the 
frequency will probably be heard by an ear capable of 
recognizing pitch. When a string vibrates, it emits a series 
of tones, the lowest of which is the fundamental. All these 
are initially present, though more or less masked by the 
fundamental tone. The particular members present depend 
on the manner in which the vibrations are excited. A string 
may execute simultaneously all the modes of motion which 
it is capable of adopting singly. Hence the production 
of harmonics or overtones together with the fundamental. 

The whole sound emitted by any source, such as a string, 
is termed a note. The trained musical ear assigns a pitch 
to this note, which is that of the lowest of the series ; each 
irresolvable component of a note is called a tone, the lowest 
the fundamental tone, and the others overtones or upper par- 
tial tones. Overtones whose frequencies are exact multiples 
of the fundamental are called harmonics. 



192 SOUND 

III. RESONANCE 

228. Free Vibrations. The vibrations which a body per- 
forms when it is disturbed and then left to itself, are called 
free vibrations. The period of such vibrations is the natural 
period of the system and is independent of the amplitude of 
vibration, provided only that the displacement is small. 
Thus, if a simple pendulum be drawn aside from its position 
of stable equilibrium and released, its vibrations are free. 
A ringing bell, a bowed tuning fork, or a plucked guitar 
string, left to itself executes free vibrations. 

Free vibrations gradually decrease in amplitude, because 
in nature all motion is checked more or less promptly, the 
moving body giving up its energy to other bodies. The 
amplitude therefore gradually decreases to zero, while the 
period remains constant. Such vibrations are said to be 
damped. 

229. Forced Vibrations. A body is often compelled to 
surrender its own free period and to vibrate with more or 
less accuracy in a manner imposed upon it by an external 
force. When a periodic force is applied to an oscillatory 
system, and the system ultimately vibrates with a period 
the same as that of the force, the vibrations are said to be 
forced. 

The two prongs of a tuning fork, with slightly different 
natural periods, mutually compel each other to adopt a com- 
mon frequency. Huyghens discovered that two similar 
clocks, adjusted to slightly different rates, kept time together 
when they stood on the same table. The more rapid clock 
was delayed and the slower one quickened by mutual influ- 
ence. These two cases are examples of mutual control, and 
the vibrations of both members of each pair are forced. 

If one prong of a fork be strongly pulled intermittently by an external 
force so predominant that its period cannot be altered by any resistance 
offered by the fork, the fork will ultimately be forced to vibrate at a rate 
determined by the external force. Such is the case of a fork controlled 



RESONANCE 193 

by an electromagnet operated by an intermittent current of a period 
slightly different from the natural period of the fork. The vibration at 
first is intermittent, because the vibrating system struggles to maintain 
its own period. The frequency of the intermittence is equal to the 
difference in frequency of the fork and the current. In time the inter- 
mittence dies away, and the fork adjusts itself in some way to the 
impulses of the current. The fork will maintain the imposed rate so 
long as it is compelled to do so, but it returns to its own normal fre- 
quency as soon as the current ceases. The nearer the rates of the forced 
and the free vibrations of a system agree, the wider is the amplitude. The 
amplitude reaches a maximum when the energy applied to the vibrating 
system exactly compensates for losses by friction and other causes. 

230. Resonance. The limiting case of forced vibrations, 
when the period of the external force is nearly or quite the 
same as the natural period of the oscillating system, is called 
resonance. The law of resonance is that vibrations will be 
taken up, and their energy absorbed, by any system capable 
of vibrating synchronously with them and exposed to their 
periodic impulses. 

Resonance depends upon the cumulative effect of small 
disturbances applied to a vibratory system in such a way as 
to synchronize with its own motions. Thus, one string takes 
up the vibrations of another which has the same vibration rate. 
If two tuning forks, mounted on resonant boxes, are adjusted 
very exactly to unison, the vibrations of one will excite the 
other, even at a distance of several meters. If one is bowed 
in the vicinity of the other and then silenced, the second will 
be sounding loudly. Then after a short interval, damping the 
second, the first will be found to be again sounding, and so on. 

Resonance may be mechanical without sound. A heavy weight sus- 
pended by a rope may be set swinging through a wide amplitude by tying 
to it a thread and pulling gently when the thread tends to slacken in the 
hand. Each effort then adds to the accumulated motion ; the series of small 
impulses at the right intervals unite to produce a large amplitude of 
oscillation. 

If two heavy pendulums, suspended side by side on knife edges on a 
slightly yielding stand, are carefully adjusted to swing in the same period, 
and one of them is set swinging, it will cause the other to swing. The one 



194 



SOUND 



initially at rest lags slightly in phase behind the other and absorbs its 
energy until the one first set in motion comes nearly to rest. The give- 
and-take process is then reversed. Many years ago a suspension bridge at 
Manchester in England was destroyed when a troop of cavalry was crossing 
it, the step of the troop keeping time with the natural swing of the bridge. 
Its swing reached an amplitude exceeding the limit of safety. It is now the 
custom to break step when bodies of either horse or foot cross a bridge. 

Many striking cases of resonance are observed in the domain of sound; 
resonance is also of frequent occurrence in the phenomena of alternating 
electric currents, and resonance in that branch of Physics is quite as 
striking as in sound. 

231. Air Resonators. The resonant body is frequently a 
partly inclosed mass of air. Any hollow vessel, such as a 

tall vase, has a natural note 
of its own, which may be 
found by blowing softly 
across the edge at the top. 
A sea shell is a very good 
resonator, and "the sound of 
the sea " heard when such a 
shell is held close to the ear 
is a case of resonance. The 
mass of air in the shell has 
its own vibration rate, and it 
responds to any faint sound 
of the same frequency. 

Hold an excited tuning 
fork over the mouth of a tall 
jar, the depth of which is 
greater than a quarter wave 

length of the note given by the fork (Fig. 124). Pour in 
water slowly; it will be found that at a certain level the 
note given by the fork is very greatly reenforced. Above 
or below that particular level, the intensity of the sound is 
not much affected by the presence of the column of air. 
Forks of different pitch require air columns of different 
length for marked reenforcement. 




Fig. 124 



ORGAN PIPES 195 

The box on which a tuning fork is mounted (Fig. 125) is 
a resonator tuned to the pitch of the fork and reenforcing 
its fundamental tone. 

The spherical resonator (Fig. 126) was designed by Helm- 
holtz for the purpose of picking out the overtones present in 





Fig. 125 Fig. 126 

a complex sound. The larger opening A is the mouth of the 
resonator; the smaller one B fits in the ear. When one of 
these resonators is held to the ear, it strongly reenforces a 
sound agreeing with it in pitch, but is silent to others. 

IV. ORGAN PIPES 

232. Air as a Source of Sound. In many musical instru- 
ments, classed as " wind instruments," the sonorous body is 
a column of air in a pipe. Columns of air are set vibrating 
in two ways : by a vibrating tongue, as in reed instruments; 
or by a stream of air striking against the sharp edge of a 
lateral opening in the tube, as in the whistle, flute, and organ 
pipe. 

Select several glass or metal tubes of different length and 
about 2 cm. in diameter ; close one end with the palm of the 
hand and blow sharply across the edge at the other. Each 
pipe will give its own note, and the longer the pipe the 
graver the fundamental tone. If four tubes have lengths 
32, 24, 20, 16 cm. respectively, the four notes will be <?', e', 



196 



SOUND 



g', c rf , forming a major chord. The lengths of the pipes are 
as 8, 6, 5, 4, and the frequencies of the notes as 4, 5, 6, 8 ; 
that is, frequencies are inversely as the lengths of the pipes. 

233. Length of Pipe and Wave Length of the Fundamental 
Tone. A comparison between the lengths of the pipes and 
the wave lengths of the notes given by them, as described in 
the last article, shows that a pipe closed at one end is ap- 
proximately one fourth the wave length of its note in air. 
Middle (V) of 264 vibrations per second has a wave 

O_l 1 9Q 

length at a temperature of 20 C. of -r- = 130.4 cm. The 




corresponding closed pipe has a length of ~T~~ 32.6 cm. 

When experiments are made with a fork over pipes of dif- 
ferent diameters, as in P^igure 124, it is found that the length 

of a pipe resounding to a given fork 
is not quite constant, the length 
diminishing slightly as the diam- 
eter increases. When the length of 
the pipe is several times its diameter, 
the correction to be added to its 
measured length is 0.6 times the 
radius ; that is, for a given fre- 
quency, length plus 0.6 radius is 
constant. Therefore the measured 
length of a pipe 2 cm. in diameter 
to give c' at 20 C. is 32.6- 0.6 
= 32 cm. 

When the prong at a (Fig. 127) 
moves to 5, it makes half a vibra- 
tion. It sends down the tube AB 



Fig. 127 

a condensation which is reflected at the surface B of the 

water and returns to the fork. If AB is -, the distance 

4 

down and back is , and the condensation returns to the 



ORGAN PIPES 197 

fork at the instant the prong reaches b. The prong next 
moves from b to a and sends a rarefaction into the tube 
following the condensation ; the rarefaction returns to the 
open end of the pipe at the instant the fork arrives at a 
and completes its vibration. The initial relations are then 
reestablished, and the air in the tall jar vibrates in synchro- 
nism with the fork. The disturbance has traversed the pipe 
four times while the fork has executed one complete vibra- 
tion ; hence a pipe closed at one end, called a stopped pipe, is 
one fourth the wave length of the fundamental tone emitted. 
For the fundamental tone there is a node at the closed end 
of the pipe and an antinode at the open end. The greatest 
motion and least variation of pressure occur at the open end, 
and the least motion and greatest variation of pressure at the 
closed end. An excellent illustration of the motion of the 
air in a closed pipe is furnished by a spiral spring firmly 
fastened to a support at its lower end and attached to one 
prong of a fork at the other, the fork and spring being so 
adjusted that they vibrate in unison, with a node in the 
spring at the lower end only. The prong of the fork must 
vibrate in the direction of the length of the spring. The 
whole of the spring then opens out during one half of the 
vibration and closes in during the other half. The great- 
est variation of tension and the least motion occur at the 
fixed end of the spring, and the greatest motion and the least 
variation of tension at the end attached to the fork. 

234. The Open Organ Pipe. The open pipe has an anti- 
node at each end, for there it is open to the air, and the air 
column at these points opens out most widely during vibra- 
tion and undergoes the least change in pressure. But two 
antinodes never succeed each other without an intervening 
node. We should therefore expect to find a node at the 
middle of an open pipe for the fundamental tone. 

The position of this node is readily found experimentally. 
Lower into an open pipe with one glass side a membrane 



198 SOUND 

covered with fine sand (Fig. 128). When the pipe gives its 
fundamental tone, the sand is agitated least near the middle 
of the pipe and most at the ends. 

Since the open pipe has a node at the middle,- 
it is equivalent to two stopped pipes with their 
closed ends together ; its length is therefore half 
the wave length of the fundamental tone emitted 
by it. 

To give the same note, the open pipe should 
be twice the length of the stopped pipe. If the 
correction 0.6 the radius be applied to each end 
of an open pipe and to the open end of the stopped 
pipe, which resounds to the same fork, the cor- 
rected lengths, within the range of experimental 
error, will be as two to one. 

235. Overtones of Stopped Pipes. The over- 
tones of a stopped pipe are due to a division of 
the air column into segments, as represented in 
Figure 129. Each complete segment may be re- 
garded as two pipes with their open ends turned 
toward each other, as represented in B and 0. 
The presence of an additional node in a stopped 
pipe for the first overtone requires an additional antinode 
also. The half vibrating segment is then one third the 
whole length of the pipe as in B. 

For the second overtone two nodes and two antinodes 
additional to those of the fundamental are required. The 
division of the air column is then as represented in (7, with 
nodes at J and ^ the length of the pipe from the open end. 

It is apparent that the overtones of the stopped pipe are 
the same as the fundamentals of pipes having lengths J, |> ^, 
etc., that of the given pipe These overtones are the odd har- 
monics with frequencies 3, 5, 7, etc., times the frequency of the 
fundamental tone. 

In adjacent half segments of an internodal space the mo- 



ORGAN PIPES 199 

tions are always in the same direction, or of the same sign. 
After a half period of the note produced, these motions are 
all again equal, but have changed sign. The motions of the 
air particles on the two sides of an inner node are always in 
opposite directions. 



J L 



Jl IL 



Fig. 129 

236. Overtones of Open Pipes. The overtones of open pipes 
are due to a division into segments of the vibrating column 
of air, as represented in Figure 130. For the fundamental the 
vibration in each half is like that in a pipe closed at one end. 
The motion of the air in the two halves of the pipe for a com- 
plete vibration is shown in .A, the upper arrows for the first 
half vibration, and the lower ones for the second half. The 
air at the node in the middle of the pipe remains at rest, but 
is subject to variations of pressure due to alternate com- 
pressions and rarefactions. 

The addition of another node and antinode gives four half 
segments as in B, instead of the two for the fundamental. 
The length of the segment for the first overtone is therefore 
half as great as for the fundamental, and the frequency is 
twice as great. 



200 SOUND 

In O is shown the division of the air column for the second 
overtone. There are now six half segments of equal length 
instead of two, and each has a frequency of vibration three 
times that for the fundamental. 

The overtones for an open pipe therefore form a complete har- 
monic series, with frequencies 2, 3, 4, etc., times the fundamental 
frequency. 



JL 



Fig. ISO 

237. Experimental Verification. The position of the antinodes for 
both open and stopped pipes may be verified by the simple device of 
piercing the side of the pipe with small holes at the points where the 
antinodes are indicated for the first, second, and third overtones. These 
openings in the side of a narrow wooden pipe may be covered by turning 
a small button (Fig. 131). 

The pressure at an antinode is always atmospheric. An opening made 
there will not then aifect the pitch ; but the uncovering of a hole at any 
other point in the pipe will be announced at once by a change of pitch. 
Suppose an open pipe is blown so as to give strongly the second overtone. 
The division of the air column is in segments with an antinode at one third 
the length of the pipe from either end (Fig. 130 C). If then a button be 
turned so as to open the pipe at either third of the length, there will be no 



ORGAN PIPES 



201 



change in the pitch of the overtone. Similarly for the first overtone a hole 
may be opened at the middle of the pipe without changing the pitch. 

For a stopped pipe, on the other hand, the opening for the first overtone 
must be made at one third the length of the pipe from the closed end, and 

for the second overtone one fifth or three fifths 

from the closed end (Fig. 129 B and C). 




Fig. 131 



Fig. 132 



238. Manometric Flames. Manometric flames, 
or flames showing variations of pressure, are very 
suitable for exhibiting to the eye the variations of 
pressure produced by the voice or by a large tuning 
fork, or the presence of stationary undulations in an organ pipe. 

A short cylinder, 3 or 4 cm. across, is divided into two chambers by a 
partition of goldbeater's skin or thin rubber (Fig. 132). Illuminating 
gas is conveyed into one of these chambers by one tube and out by another 
to a pin-hole burner, where it burns as a small flame. Any pure tone at 
the mouthpiece produces alternate compressions and rarefactions in the 
chamber on the left-hand side of the membrane, and these retard and aid 
the flow of gas to the burner. The flame changes shape and flickers, but 
its vibrations are too rapid to be seen directly. If now it is examined by 
reflection from the rotating mirrors attached to the faces of a cubical box, 
its image is a serrated band. 

Koenig fitted three of these manometric capsules to the side of an open 
pipe (Fig. 133), the membrane on one side of the gas chamber forming 



202 



SOUND 



part of the wall of the pipe. When the pipe is blown so as to sound its 
fundamental tone, the middle point is a node with greatest variations of 

pressure, and the flame 
at that point is more 
violently agitated than 
at the other two. When 
the air blast is increased 
so that the fundamental 
gives way to the first 
overtone, the middle 
point is an antinode 
with no variation of 
pressure and the middle 
flame does not flicker ; 
but the other two vi- 
brate, and the number 
of tongues of flame in 
the image is twice as 
great as for the funda- 
mental tone. 

The first band of 

Figure 134 shows the appearance of the image for the fundamental tone, 
the second for the first overtone ; the third band may be obtained by 
adjusting the blast of air so that the fundamental and the first overtone 
are produced at the same time. The same figure may be obtained by 
singing into the mouth-piece of Figure 132 the vowel sound o on the note 
5{7, showing that this vowel sound is composed of a fundamental com- 
bined with the first overtone. 

239. Kundt's Dust Tube. The division of a resonant pipe 
into segments is beautifully shown by means of a glass tube 





Fig. 135 

about 2 cm. in diameter and 40 cm. long. One end is closed 
and a common whistle is attached to the other (Fig. 135). 
Within the tube is placed a little sifted cork dust or amor- 
phous silica. When the whistle is blown, the powder is 
caught up by the moving air at the antinodes, and settles 
down in small circles at the nodes ; at the same time between 



ORGAN PIPES 203 

the nodes it is divided into thin, airy segments with vertical 
divisions, the agitation being sufficient to support the dust in 
opposition to gravity. The distribution of the light powder 
demonstrates the presence of stationary waves due to the 
superposed direct arid reflected systems. The subdivision 
changes when the blast of air is increased to give overtones. 
Kundt has given to the experiment an ingenious form, 
designed to compare the velocity of sound in air and other 
media. His apparatus consists of a glass tube about a meter 
long and 4 or 5 cm. in diameter, closed at one end a (Fig. 
136) by a membrane of thin sheet rubber, while at the other 
end is a cork piston b sliding freely in the tube. The inside 
of the tube, which must be very dry, is dusted with fin^ 






a 

Fig. 136 



sifted cork filings or amorphous silica. A rod of glass or 
metal is held by its middle point in a clamp #, and one end 
is furnished with a small disk of stiff paper, which rests 
against the rubber membrane at a. 

When the rod is made to vibrate longitudinally by friction, 
it vibrates like an open organ pipe, giving its fundamental 
tone. The paper disk at a communicates the vibrations of 
the rod to the air in the tube, and when the length of the 
air column has been properly adjusted by means of the piston 
ft, the cork dust is tossed about and gathers in small heaps, 
which indicate very neatly the exactness of the adjustment. 
The average distance I between nodes is then the half wave 
length of the note in the tube. The half wave length in the 
rod is its length L. These distances are traversed in the 
same time ; and, therefore, if v is the velocity of sound in air 
at the temperature of the experiment, and F'the velocity in 
the rod, we have the simple relation 

l_ = v_ 
L V 



204 SOUND 

V. QUALITY 

240. Definition of Quality. Two of the essential character- 
istics of musical sounds, namely, pitch and loudness, have 
already been considered. There is a third important differ- 
ence between musical sounds, known as quality. It is easily 
perceived that one musical note differs from another, not 
only in being more acute or grave, louder or softer, but also 
in respect to the character of the sound. We have no diffi- 
culty in distinguishing the notes of a piano from those of a 
violin, even though they are of the same apparent pitch and 
loudness. Similar differences enable us to distinguish one 
voice from another in speech or in song, even when some- 
what modified by transmission through the telephone, or 
when reproduced by the phonograph. Even the untrained 
ear recognizes characteristic differences in music produced 
by different instruments of the same class. 

All differences in musical notes, not assignable to pitch or 
loudness, are included under the term quality. Quality is 
the characteristic of a musical sound that enables one to 
refer it to its source. 

241. Quality due to Overtones. If an open and a stopped 
organ pipe, giving notes of the same pitch and loudness, are 
compared, a marked difference will be observed in their qual- 
ity. In the former the whole series of harmonics may be 
present, while in the latter only those whose frequencies are 
odd multiples of the fundamental are possible. The sound 
waves in each case are the result of compounding the funda- 
mental with the overtones present. . 

Pitch depends on the length of the sound wave, loudness 
on its amplitude, and quality on the only other physical 
difference between aerial waves, that is, their vibrational 
form. The form of a wave is the manner in which the dis- 
placements vary from point to point in the wave. It de- 
pends on the waves of higher frequency combined with the 
fundamental. Every change in the form of a complex sound 



QUALITY 205 

wave, not affecting pitch or loudness, is due to some change 
in the components of higher frequency than the fundamental, 
that is, in the overtones. Hence, quality is to be referred to 
the number, the order, and the relative intensity of the overtones 
associated with the fundamental. 

242. Quality of Musical Sounds. When the wave form is 
simple harmonic, it produces a pure tone, which even the 
musically trained ear is unable to analyze into components. 
When the wave form is complex and periodic, a well-trained 
ear can analyze the note into component pure tones, or a 
fundamental and harmonics. 

Tuning forks and wide-mouthed stopped pipes give nearly 
pure tones, which are not satisfactory as musical sounds be- 
cause of their dull or colorless quality. Musically, notes are 
described as full and rich when they consist of a fundamental 
accompanied by a series of moderately loud harmonics not 
higher than the fifth, with a frequency six times that of the 
fundamental. Such are the notes of open organ pipes, the 
French horn, and the softer tones of the human voice. 

The first harmonic, with a frequency twice that of the 
fundamental, forms the octave; the second, with a frequency 
of three, gives the octave plus a fifth; the third, with a 
frequency of four, gives the double octave; the fourth, with 
a frequency of five, gives two octaves plus a third; the fifth, 
with a frequency of six, gives two octaves plus a fifth. The 
sixth overtone, with a frequency of seven times the fun- 
damental, does not fall within the musical scale and is 
inharmonic. The same is true of the eighth overtone. 

When a large number of overtones or upper partials are 
present, the notes are said to be nasal. When overtones 
above the fifth are quite distinct, the quality is piercing and 
rough on account of the dissonances introduced by the higher 
overtones. Such notes are said to be metallic because of 
their resemblance to those given by a vibrating sheet of 
metal. 



206 SOUND 

VI. INTERFERENCE AND BEATS 

243. Principle of Interference. We have already seen that 
the result of combining direct and reflected sound waves in 
pipes is a system of stationary air waves with nodes and anti- 
nodes in fixed positions. These stationary waves are due to 
the superposition of the direct and reflected trains of waves. 
The principle of the superposition of two sets of similar 
waves traversing the same medium at the same time, while 
the resulting displacements at any point is the algebraic sum 
of the two displacements due to the two systems separately, 
is called interference. The name is singularly ill-chosen 
because each sound wave pursues its own way as if the other 
were not present. No observant person can fail to notice 
that several trains of waves may traverse the same surface of 
water at the same time in different directions, each train per- 
suing its way unimpeded by the others. Even the ripples 
ride freely over the crests of the long waves without hin- 
drance. "If this is interference, it is difficult to see what 
non-interference would be" (Lord Rayleigh). 

In the phenomena of interference the attention is directed 
to the occurrences at definite points in the medium rather 
than to the passage of waves. If two sound waves of equal 
length and amplitude agitate the same medium at any point, 
and if they are in opposite phase, so that the condensation 
of the one falls at the same place as the rarefaction of the 
other, then the medium at that point is at rest and the sound is 
extinguished there by interference. If at any point the two 
waves are in the same phase, the resulting displacement is 
double that of either taken separately, and the sound is 
greatly reenforced by interference. At the nodes in an organ 
pipe the two waves interfering are in opposite phase and annul 
each other; at the antinodes, they agree in phase and reen- 
force each other. 

244. Interference when the Frequencies are the Same. It is 
well known that the intensity of the sound of a vibrating 



INTERFERENCE AND BEATS 



207 




Fig. 137 



fork held freely in the hand near the ear and turned on its 

stem exhibits marked variations. In four positions the sound 

is nearly inaudible. 

Let A, B (Fig. 137) be the \ /' 

prongs of the fork. They vibrate \ . / 

with the same frequency, but in 

opposite directions, as indicated 

by the arrows. When the two 

branches approach each other, a 

condensation is produced between / 

them, and at the same time rare- / 

factions start from the backs at 

c and d. The condensations and 

rarefactions meet along the dotted lines of equilibrium, 

where partial extinction occurs. These lines lie nearly in 

planes passing through the axis of the fork and make angles 

of 45 with its face. 

It is easy to demon- 
strate that the weak- 
ening of the sound 
along the lines of mini- 
mum intensity is due 
to interference. Hold 
the vibrating fork over 
a cylindrical jar ad- 
justed as a resonator 
and turn it over till a 
position of minimum 
loudness is found. In 
this position cover one 
prong with a paste- 
board tube without 

touching (Fig. 138). The sound will be restored to nearly 

maximum intensity, because the paper cylinder cuts off the 

set of waves from the covered prong. 

When an organ pipe is sounded in a large room with good 




Fig. 138 




208 SOUND 

reflecting walls, regions of maximum and minimum loudness 
may easily be found. The variations in intensity are due to 
interference between the direct waves and those reflected 
from the walls. 

Interference between waves from the same source of sound may be 
demonstrated by means of the so-called " duplex phonograph." In 

this instrument the dia- 
phragm used to repro- 
duce the vibrations of 
sound recorded on the 
record plate has a 
" horn " connected with 
each side (Fig. 139). 
When this diaphragm 
vibrates, it produces 
simultaneously a con- 
densation on one side 
and a rarefaction on the 
other. Hence the sound 

waves reaching the medial line ef between the two horns a and b are 
always in opposite phase. As a fact of observation, when the ear of the 
listener is on the medial line near the horns, the intensity is noticeably 
less than at other points. 

The demonstration is made more complete by inserting rubber tubes 
in the small ends of the horns by means of tight-fitting corks, and bring- 
ing the two tubes of equal length together to a T -tube fitting the ear. 
The other ear should be closed. The two wave systems do not completely 
annul each other, but if the listener cuts off one system by pinching either 
tube, the intensity of the sound is increased to a surprising degree. 

245. Beats. When two sounds come from sources of 
slightly different period, interference gives rise to alternate 
swellings and subsidences in loudness, known by the term 
beats. When two tuning forks of the same pitch, mounted 
on resonant boxes, are sounded together, the sound is smooth 
as if only one fork were vibrating. Stick a small mass of wax 
to a prong of one of them ; this load increases the moment of 
inertia of the fork and so increases also its periodic time of 
vibration. If the two forks are now sounded together, the 
phenomenon of beats will be very pronounced. 



INTERFERENCE AND BEATS 



209 




Fig. 140 



Mount two organ pipes of the same pitch on a bellows, and 
sound together. If they are open pipes, a card gradually 
slipped over the open end of 
one of them will change its 
pitch enough to bring out 
strong beats. 

With glass tubes and small jets for 
burning coal gas or hydrogen set up 
the apparatus of Figure 140. One 
tube should be provided with a slider 
for the purpose of varying its length. 
When the gas jets are inserted a proper 
distance in the long tubes and the jets 
are turned down slowly, both tubes 
will give a loud sound. The agitated 
flame in a sounding tube is known as 
a " singing flame." By adjusting the 
height of the slider, the two tubes 
may be made to give notes of the same 
pitch ; the sound is then smooth and steady. If now the slider be moved 
either up or down, the sound will pulsate strongly with distinct beats. 

246. Number of Beats. Let two notes be produced by 
forks making, for example, one hundred and one hundred 
five vibrations per second respectively. Then in each sec- 
ond the fork of higher frequency gains five vibrations On 
the other; and five times during each second the two are 
vibrating in the same phase and five times in opposite phase. 
The same changing relation of phase occurs in the air 
transmitting the vibrations. Hence, subsidence of sound 
must occur five times a second, and there are five beats. 
Therefore, the number of beats per second is equal to the dif- 
ference between the vibration frequencies of the two notes. 

247. Beats due to Overtones. Beats are produced not only 
between two notes nearly in unison, but between notes whose 
interval is only approximately an octave, a major third, a 
fifth, and so on. These are attributed to the presence of 
overtones associated with the fundamentals. Thus, if two 



210 



SOUND 



notes have frequencies n and 2^ + 1, then the first overtone 
of the lower note will be due to 2 n vibrations per second, 
and this will produce one beat per second with the slightly 
higher note. So also if two notes are due to 2^ + 1 and 3 n 
vibrations per second respectively, then the second overtone 
of the first will correspond to 6 n + 3 vibrations, and the first 
overtone of the second note to 6 n vibrations per second, and 
these two overtones will give three beats per second, though 
the interval is otherwise indistinguishable from a fifth. 

Again, the interval between the fundamentals may be 
exact, but the frequencies of the overtones may not be exact 
multiples of the fundamental. Such may be the case with 
tuning forks, and beats are sometimes heard between their 
overtones of the same order. 

248. Lissajous's Figures. The optical combination of the 
simple harmonic motions of two tuning forks, vibrating in 
planes at right angles to each other, was first described by 
Lissajous ; the resulting curves are therefore known as Lis- 
sajous's figures. The method of obtaining them graphically 
has already been described in 41. Lissajous's figures fur- 
nish a method of observing beats optically as well as by the 
ear ; also of comparing the relative frequencies of two vibrat- 
ing bodies with great precision. 




Fig. Wl 



A beam of light falls in succession on the plane mirrors 
attached to the forks L and M (Fig. 141), and is finally 
reflected to a screen S. If now the fork L is set vibrating, 



INTERFERENCE AND BEATS 



211 



the spot of light on the screen will appear as a vertical band, 
because of the persistence of impressions on the retina of the 
eye. Similarly, if M only is vibrating, the spot will form a 
horizontal band. If both forks are vibrating, the spot will 
have both harmonic motions imparted to it, and will trace a 
characteristic Lissajous's figure. The forks should be of low 
pitch, and should be kept vibrating by means of electromag- 
nets. The same pair of forks may serve for several pairs of 
relative frequencies by tuning with sliding weights. 

















Fig. 142 

If the forks are in unison, the characteristic figure is an 
ellipse with oblique straight lines as limiting forms (Fig. 
142). If the tuning to unison is not exact but only approxi- 
mate, the figure takes the successive forms of the first line 
from left to right, and then the same ones recur in reverse 
order. One beat will be heard as often as the figure goes 
through the whole cycle of changes from the straight line 
at the left over to the right and back again. During this 
period one of the forks gains one vibration on the other. 



212 SOUND 

In Figure 142 are shown also the curves for frequencies 2 : 1 
and 3 : 2. The forms corresponding to no difference of phase 
are in the first column ; those differing by J of a period, in 
the second column, and so on. 

In all cases during the time the figure for any interval 
passes through a complete cycle of changes, one fork has 
executed one vibration in addition to the number required 
for the exact interval. 



VII. VIBRATION OF RODS, PLATES, AND BELLS 

249. Vibration of Rods. Rods of metal, of wood, and of 
glass may vibrate either transversely or longitudinally. A 
rod fixed at one end may be made to vibrate transversely by 
drawing the free end aside and releasing it, or by striking 
it with a suitable hammer. It may be set in longitudinal 
vibration by clamping at the middle and stroking lengthwise 
with a chamois skin dusted with powdered rosin. A moist 
cloth is better for glass. 

In the transverse vibration of rods, unlike that of strings, 
the force of restitution is the elasticity of flexure. The fre- 
quency of a rod with a rectangular section is independent of 
its width, but is directly proportional to its thickness. If 
two bars of the same material have the same length, while 
one is twice as thick as the other, the thick one will vibrate 
with twice the frequency of the other, whatever may be their 
relative width. 

Examples of the use of transversely vibrating rods in musical instru- 
ments are the reeds of an accordion or harmonium, the tongue of a jew's- 
harp or of a music box, the reeds of reed pipes in organs, the xylophone, 
and the tuning fork. 

The xylophone is a primitive instrument with bars free at both ends. 
It consists of a series of wood prisms of the proper length and thickness, 
supported by strings at the nodes, which are at points about one quarter 
of the length from each end. The prisms are adjusted to give the notes 
of the scale, and they are played by striking them in the middle with a 
little hammer having a soft, elastic face. 



VIBRATION OF RODS, PLATES, AND BELLS 213 



250. The Tuning Fork. The tuning fork is one of the 
most important applications of vibrating rods free at both 
ends. A straight elastic bar, when sounding its 
fundamental, has a node at a distance from each 

end of about one quarter of its length. As this 
bar is gradually bent into the form of a tuning 
fork (Fig. 143), the nodes approach each other; 
and when the fork has a stem, the nodes are near 
the bottom of each branch. The two branches 
then vibrate in unison, while the stem has a slight 
up and down motion, which is transmitted to the 
resonant box on which the fork is mounted. The 
overtones are of high pitch and feeble intensity, 
and soon vanish, leaving a pure tone. 

Fig. 143 

251 . Vibration of Plates. If a plate of elastic material, 
such as brass or glass, be clamped at its middle in a hori- 
zontal position, and fine sand be scattered evenly over it, the 

sand will gather 
along certain defi- 
nite nodal lines 
(Fig. 144) when 
the plate is 
thrown into vi- 
bration by draw- 
ing a bow across 
its edge. The 
sand figures in- 
crease in com- 
plexity as the 
number of seg- 
ments becomes 
larger with ris- 
ing pitch. These 
sound figures were first obtained by Chladni, and are known 
as Chladni's figures. The arrangement of nodal lines is 




Fig. 144 



214 SOUND 

determined by the position of the point bowed relative to 
those damped by the finger tips. 

The vibration of a glass plate may be beautifully shown by support- 
ing it close to the condensing lens of a vertical lantern, covering it with 
a thin film of water, and projecting the surface on the screen. The vibra- 
tion of the plate produces stationary waves in the film of water covering 
the middle portions of the vibrating segments between the nodes. The 
finer stationary waves correspond to the acuter modes of vibration. 

The fundamental tone of a round plate is produced by a division into 
four equal segments by two diameters ; the first higher tone comes from 
a division into six segments by three diameters, the second by eight seg- 
ments and four diameters. Adjacent segments, like those of strings, 
are always in opposite phases of motion. 

The vibration of plates is illustrated among musical instruments by 
the cymbal, the kettledrum, and the snare drum. They have small musi- 
cal value and are used solely to accentuate the rhythm. The best that 
can be accomplished by the tuning of a drum is to prevent its disturbing 
the harmony of other instruments. 

252. Vibrations of Bells. The modes of vibration of a bell 
are modifications of those of a cylinder. The motions of the 
rirn are both radial and tangential. 
Consider a thin circular shell (Fig. 
145) ; the forces resisting extension 
of this ring are very large compared 
with those resisting bending. We 
may therefore regard the perimeter 
as bending only and not changing in 
length. 

For the gravest tone the number 
of segments is four, with nodal meridians running down the 
bell. The motion is entirely radial at the middle points of 
the segments H, /, K, L. But while the nodes are at rest 
radially, they are not at rest tangentially ; for when the rim 
on one side of the node is inside its mean position, on the 
other side of the same node it is outside its mean position ; 
MH'Nis less than MHN, and ML' Q is greater than MLQ. 
If then there is no stretching, there must be tangential motion 





VIBRATION OF RODS, PLATES, AND BELLS 215 

of the nodes of radial vibration to allow for the variations in 
the length of the segments intercepted between adjacent 
nodes. The nodes for the radial motion are therefore anti- 
nodes for tangential motion. 

The tangential motion in the rim of a vibrating bell explains the 
method of sounding a wineglass by drawing the wet finger round the 
edge. The friction produces a tangential displacement 
which is accompanied by a radial displacement, just 
as a stroke on the bell produces a radial displacement 
which is accompanied by a tangential displacement. 

If there is lack of symmetry in the figure, or of homo- 
geneity in the rim, the nodes may slowly revolve around 
the rim, producing the beats often heard when a large 
bell is struck. 

The radial motion of a bell-shaped glass vessel may 
be illustrated by partly filling it with methyl alcohol 
before bowing the edge of the glass. The alcohol is Fig. 146 

thrown in drops from the sides of the glass toward the 
center ; these drops do not coalesce at once but form a cross (Fig. 146) 
for the fundamental, and a six-sided figure for the first higher tone. 

Problems 

1. If standard pitch is based on a! 435 vibrations per second, calcu- 
late the frequency for c', both for the diatonic and the equally tempered 
scale. 

2. Calculate the wave length of the note e', which is a third above 
c' of 256 vibrations per second, when the temperature is 10. 

3. A tuning fork is held over a long glass tube partly filled with ice- 
cold water. The most marked reinforcement of sound occurs when the 
column of air in the tube above the water is 32.5 cm. long. Find the 
frequency of the fork. 

4. A wire stretched with a weight of 8 kgm. gives the note C. What 
must be its tension for a note a fifth higher? 

5. A string 1 m. long has a frequency of 260 per second, when the 
tension is 28 kgm. What must be the tension so that one half the string 
shall have the same frequency ? 

6. A Kundt's tube, filled with air and thrown into vibration, divided 
into segments 2.5 cm. long, between successive heaps of dust. When 
the tube was filled with hydrogen, the segments for the same note were 
9.65 cm. long. Calculate the velocity of sound in hydrogen. 



216 SOUND 

7. Two telegraph sounders are in the same electric circuit which is 
closed five times every two seconds by the pendulum of a clock. If the 
listener is at one instrument, how far away must the other one be so that 
he may hear its nth click at the same instant that he hears the (n + l)th 
of the near one, the temperature being such that the velocity of sound 
is 340 m. per sec. ? 

8. What will be the lengths of two open organ pipes which give the 
notes c' and g' when the velocity of sound is 344 m. per second? 

9. A string, vibrated by a large fork, is stretched with a weight of 
270 gm. and divides into four segments. What must be the weight to 
cause it to divide into three segments with the same fork ? 

10. A cord vibrates synchronously with the attached fork by dividing 
into three segments. If it be replaced by a similar one of the same 
length and four times the sectional area, what relative weight will be 
required to cause it to divide into four segments ? 



LIGHT 

CHAPTER VIII 

NATURE AND PROPAGATION OF LIGHT 
I. GENERAL PHENOMENA 

253. Light and Radiation. The word light, as in the 
case of sound, has acquired two distinct meanings : first, the 
common and familiar one used to designate the sensation of 
vision; second, the external agency acting through the eye 
to excite visual impressions. Nearly all relating to the first 
aspect of the word light lies within the field of the psycholo- 
gist. Except in the study of the sensations of color, the 
word light in physics refers to the external cause of the sen- 
sation of luminosity, quite apart from the organ which re- 
veals it to us. In fact, the physicist now studies light as a 
phenomenon of the transference of energy by wave motion. 
The process is known as radiation. 

It has been found that the radiation affecting the normal 
eye also affects a photographic plate, a sensitive thermometer, 
and other appropriate detectors of radiant energy flowing 
from the sun, the electric light, and other similar sources. 
Further, the analysis of radiations from such sources reveals 
the fact that only a limited portion of them are capable of 
exciting vision. But the essential identity of these radia- 
tions with light is made evident by the fact that the various 
phenomena of optics may be reproduced by radiations which 
do not affect the eye. There is no fundamental difference 
between luminous and non-luminous radiations. The differ- 
ences are not qualitative but quantitative ; in other words, 
they are differences of wave length. 

217 



218 



LIGHT 



The limits of visible radiations are determined by the 
range of sensibility of the eye as a receiving instrument. 
It is sensitive to radiations lying within a certain rather 
definite range of frequency ; but radiations of higher and 
lower frequency are not distinguished in any way from those 
falling within the limits of perception, except by the physical 
difference of wave length. It is therefore convenient and 
almost essential to include under light the whole range of 
radiations, which are alike in their fundamental properties, 
but which were formerly classified as luminous, heat, and 
actinic radiations. 

254. Sources of Eadiation. The most important source of 
radiant energy for our planetary system is the sun. Other 
sources of light are solid bodies at a high temperature, such 
as lime at a white heat in the calcium light ; glowing carbon 
in the carbon arc and incandescent electric lights ; rare 
earths in the Nernst glower and the Welsbach mantle ; and 
luminous flames, such as those of a candle, an oil lamp, a gas 
jet, and certain metallic electric arcs. The light from ordi- 
nary flames is radiated by glowing carbon raised to incandes- 
cence by the burning 
vapors. Such flames are 
sooty because of the in- 
numerable solid carbon 
particles suspended in 
them, which get heated 
very hot and emit light. 

Gases do not appear to be- 
come luminous by high tem- 
perature alone, but by the 
passage of an electric spark or 
a current of electricity. The 
Fig | 47 luminosity of the faint nebulae 

in space still awaits an expla- 
nation. Many other subordinate but interesting sources of light are 
certain bodies at low temperature, which are said to give light by phos- 




GENERAL PHENOMENA 219 

phorescence. A common match, when rubbed gently, gives off fumes 
which are faintly luminous in a dark room. 

There are luminous bacteria which emit enough light to affect a sen- 
sitized photographic plate. Molisch of Prague has prepared a bacterial 
lamp (Fig. 147) by filling a glass jar with gelatine containing a colony 
of luminous bacteria. Its intensity is less than that of a candle, but 
bright enough to induce germinating peas to turn toward it as a source 
of radiant energy. 

255. Nature of Light. Light does not travel instantane- 
ously, but it has a finite velocity. Moreover, a continuous 
stream of energy flows from a luminous source like the sun, 
for it heats bodies on which it falls. Some means must exist 
for the transmission of this energy. The earth receives 
energy from the sun ; and as it takes over eight minutes for 
its transit across the intervening space, we are forced to seek 
for a vehicle by which it is conveyed. 

According to our experience, there are only two ways of 
transferring energy; either by the projection of material 
bodies through space like a cannon ball, or by transmission 
through an intervening medium by some form of wave 
motion. 

Newton chose the first method in his emission theory of 
light. He imagined the light-giving body projecting minute 
particles or corpuscles through space; and that they enter 
the eye and excite vision by impact on the retina. 

The other method requires a continuous medium, called 
the ether, and energy is handed on by it from point to point 
by undulations. In this way energy is conveyed in sound 
and by water waves. A luminous body is then the source 
of a disturbance in the universal ether, which transfers it by 
undulations through space. These waves travel with the 
velocity of light and excite visual impressions. 

Newton was aware of the wave theory, which his contem- 
porary Huyghens had formulated; but he found an insu- 
perable objection in the fact that water waves pass around an 
obstruction, and that sound shadows occur only under special 



220 LIGHT 

conditions. By analogy he concluded that if light is an un- 
dulation, it should also pass around bodies instead of casting 
a shadow. 

The answer is that short water waves or ripples do not 
pass behind obstacles, and sounds of high pitch and short 
wave length cast well-defined shadows. In light the phe- 
nomena of diffraction, as we shall see later, show that light 
waves also pass around obstructions into the shadow. 

Further, if light is transmitted by a wave motion, inter- 
ference ( 243) should apply to it as well as to sound. It 
was more than a century after Newton's time before Thomas 
Young showed by experiment the phenomena of interference 
in light. So great even then was the authority of Newton 
that Young's theories were derided by his English contem- 
poraries, and his writings were declared to be " destitute of 
every species of merit." It was not until Fresnel had taken 
up the theory in France and had greatly extended the 
experimental proofs that the undulatory theory finally 
prevailed. 

256. Light travels in Straight Lines. In a homogeneous 
medium light travels in straight lines. A ray of light is the 
direction in which the radiant energy is transmitted. Light 
passes by an obstacle in straight lines, so that the space be- 
hind it is screened from the radiation. The sharpness of a 
shadow is closely connected with the excessive shortness of 
light waves. But even when the source of light is so small 
as to be considered a point, the edge of the shadow cast by 
an opaque body is not perfectly sharp, for the reason that 
points at the edge of the obstacle become new centers of 
disturbance by Huyghens' principle ; and waves from these 
points overlap slightly the geometrical shadow. 

When the luminous source is a point, as L (Fig. 148), the 
shadow will be bounded by the cone of rays ALB, tangent 
to the object, and will consist wholly of the umbra, from 
which the light is entirely excluded. When the luminous 



GENERAL PHENOMENA 



221 



body has sensible dimensions, then outside of the total 
shadow or umbra, and surrounding it, is a region called the 




Fig. 148 



partial shadow or penumbra (Fig. 149). This region of 
partial shadow forms the transition from complete obscurity 
to the full light. Only a portion of the luminous body is 
visible to an eye situated within the penumbra. 




Fig 149 



Solar eclipses are produced by some portion of the earth's surface pass- 
ing through the shadow of the moon. The moon is smaller than the sun 
and its shadow is a limited cone. The apex of this shadow cone some- 
times reaches the earth, when new moon occurs near one of the lunar 
nodes; sometimes it falls short of it, the mean length of the lunar 
shadow being 6700 miles less than the mean distance of the moon from 
the earth. If the shadow cone reaches the earth, the eclipse is total for 
all points lying within the umbra ; within the penumbra the eclipse is 
partial ; if only the prolongation of the shadow cone encounters the 
earth, the eclipse is annular for all points touched successively by the axis 
of the shadow. 

257. Pinhole Images. The rectilinear propagation of light 
is illustrated by the images produced by a small orifice of 
any shape, such as a pinhole. If a white screen is placed op- 
posite a small hole in the shutter of a darkened room, an 
inverted picture of outside objects brightly illuminated by 
the sun will be formed on it in natural colors. The images 



222 



LIGHT 




Fig. 150 



will be sharper the smaller the hole, and distant objects will 
be more distinctly outlined than nearer ones. 

The principle is illustrated by a candle or an incandescent 
lamp (Fig. 150). Each point of the object, such as (7, is the 
A B vertex of a cone of rays 

passing through the aper- 
ture and forming an en- 
larged image of it on the 
screen, as at F. The images 
are symmetrically placed 
with reference to the points 
emitting the light, and together they form a figure having 
the same outlines as the luminous or illuminated object. 
Since a smaller number of these images are superposed near 
the edges of the picture, the edges are not so bright as the 
other portions. 

If a second pinhole is made near the first one, another pic- 
ture will be formed and the two will not quite coincide. 
Now a larger hole is the equivalent of a number of pinholes 
close together, and all the images overlap but do not exactly 
coincide. It is therefore easy to see why a very small hole is 
necessary for a sharp image. 

The image is not only inverted, but it is perverted. If it is viewed 
from the side of the screen toward the opening, and imagined turned 
around in its own plane so as to make it erect, it will be found that the 
right side of the image corresponds to the left side of the object ; that is, 
the image is perverted. If the screen is translucent and the image is 
viewed from behind, it will be inverted, but not perverted. An image in 
a plane mirror is perverted but not inverted. 

Pinhole pictures are said to have been first observed by the great Italian 
artist, Leonardo da Vinci. About a century later came the " camera 
obscura," a small darkened chamber or box, in which was an opening 
larger than a pinhole and containing a converging lens. This gave more 
light and a sharper image. The modern photographer's camera represents 
the latest form of this device. 

Landscape photographs and pictures of rather distant buildings of sur- 
prising softness and beauty may be made with a pinhole camera without 
a lens. 



VELOCITY OF LIGHT 



223 



258. Parallax. The apparent displacement of an object 
due to the real displacement of the observer is a well-known 
phenomenon called parallax. It depends on the rectilinear 
propagation of light. If, for example, an observer who is 
moving rapidly watches the moon hanging R 

low over a hill or forest at no great distance, 
it will appear to travel in the same direction 
as the observer, while intervening objects 
appear to move backward. 

When the observer is at (Fig. 151), an 
object at A appears to be at the angular dis- 
tance a to the left of the more distant object 
B\ but when the observer moves to A , the 
object at A appears at the angular distance 
/3 to the right of B. Hence the object A 
is apparently displaced with respect to B 
through an angle a -f in a direction oppo- 
site to the motion of the observer. 

If two objects are equally distant, their 
relative parallax vanishes. This fact serves a useful purpose 
in finding the position of an image formed by the objective 
relative to the cross thread of a telescope. When the motion 
of the eye right and left produces no displacement of the one 
with respect to the other, they coincide in position. 




Fig. 151 



II. VELOCITY OF LIGHT 

259. Homers Method. The eclipses of the inner satellite 
of Jupiter occur at average intervals of 48 hr. 28 min. 36 sec. 
These eclipses appear to take place quite suddenly, though 
each one is really a gradual phenomenon, and a single obser- 
vation is doubtful to half a minute. 

In 1676 Homer, a young Danish astronomer who was at 
the time an observer in the Paris observatory, announced 
that the observed eclipses of Jupiter's inner satellite differ 
systematically from the computed times. When the earth is 



224 



LIGHT 




Fig. 152 



receding from Jupiter, the interval between two successive 
eclipses is longer than the mean, and the more rapid the re- 
cession the greater the excess. The reverse is true as the 
earth approaches Jupiter. 

Let EE ! (Fig. 152) represents the earth's orbit and the 
larger circle JJ" the orbit of Jupiter. Then while the earth 
moves from .ZHhrough E' to E n ', or from opposition to con- 

junction, the eclipse intervals are 
longer than the mean, and the sum 
of all the excesses from opposition 
to conjunction is 16 min. 38 sec. x or 
998 Sec. From E" around to oppo- 
sition again the intervals between 
eclipses are shorter than the mean, 
and the sum of these deficiences is 
again 998 sec. 

Romer inferred that the speed 
of light is finite, so that the longer interval between two 
successive eclipses Vhen the earth is receding from Jupiter 
is due to the additional distance which light must travel to" 
reach the earth. This interval is greatest at E l ', where the 
earth is receding in a direct line from .the planet. The sum 
of all the excesses is the time taken by light to travel across 
\ /the earth's orbit. If .the diameter be 299,000,000 km., the 

speed of light will be 29 t 9 ?^ 000 = 99,600 'km. /sec. 

t7t7O _ - 

Romer's original suggestion was rejected by most astrono- 
mers for more than fifty years, and was not accepted until 
long after his death, when Bradley's discovery of the aber- 
ration of light confirmed the correctness of Romer's views. 

260. Bradley's Method. In 1727 Bradley, afterwards As- 
tronomer Royal of England, while attempting to measure 
the relative annual parallax of two stars, one of which was 
assumed to be much nearer than the other, discovered that 
every star appears to describe a small orbit about its mean 



<$\ s 



VELOCITY OF LIGHT 225 

position in the period of a year. This motion is equivalent 
to a negative parallax ; for instead of a displacement opposite 
to the motion of the earth, for which Bradley was looking, 
he discovered that a fixed star has an apparent displacement 
in the other direction. In the plane of the earth's orbit 
(the ecliptic) a star appears to move back and forth in a 
straight line, while a star near the pole of the ecliptic has an 
apparent motion in a circular orbit. 

A chance observation suggested to Bradley the explanation. 
He noticed that the direction of a wind vane on a boat sail- 
ing on the Thames was not that of the wind, but of the re- 
sultant of the wind and a virtual head wind due to the motion 
of the boat. In the same way, the apparent forward dis- 
placement of a star is the change in direction due to combin- 
ing the motion of the earth in its orbit with the motion of 
light. 

Suppose the wind blowing directly against the side of a 
vessel moving with a velocity u, the velocity of the wind 
being V. The motion of the vessel produces a head wind, 
and this combined with the real wind causes an apparent 
shifting forward of the point from which the wind comes by 
an angle 0, the tangent of which is u/ V. In the same way 
the velocity of light must be combined with one equal and 
opposite to that of the earth in its orbit in order to give the 
apparent direction from which the light comes, that is, to 
give the apparent position of a star. The phe- 
nomenon is known as the aberration of light. 

Let OA (Fig. 153) represent the velocity of 
light F", and A B the relative magnitude and direc- 
tion of the orbital velocity of the earth. Then 
CAD is the angle of aberration. This angle will 
be greatest when the motion of the earth is at 
right angles to the direction of the star. Then 
tan 6 = u/ V. The angle of aberration is known 
to be 20.445". Since the tangent of this angle is about 
follows that the velocity of light is about 10,000 




226 



LIGHT 



times the earth's orbital velocity; and as the latter is very 
nearly 30 kilometers a second, the velocity of light is in 
round numbers 300,000 km. /sec. = 186,400 mi. /sec. 

261. Fizeau's Method. It was more than a hundred years 
after Bradley's discovery of the aberration of light before 
Fizeau in 1849 made the earliest experimental determination 
of the velocity of light over a limited distance on the earth's 
surface. His method depends on the eclipse of a source of 
light by means of a rapidly rotating toothed wheel. 

A beam of light from a bright source S (Fig. 154) is re- 
flected from a piece of plate glass m and focused by a lens L 




Fig. 154 



on the teeth -of a wheel W. The light diverging from 1? 
emerges from the lens L^ as a parallel beam ; and after trav- 
ersing the intervening distance of 8633 m. to the distant 
lens Ly, it is focused on the mirror M> which reflects it back 
on its own path. The beam of light returns to the telescope 
and the inclined piece of plate glass ; a portion of it is reflected 
and a portion transmitted. The latter enters the eye of the 
observer at E, producing the appearance of a bright star 
at F. 

When the toothed wheel is rotated rapidly, a detached 
train of waves of light passes through a space between two 
teeth out toward M. If now the speed of light were infinite, 
the brightness of this artificial star would not be affected by 
the speed of rotation of the wheel; but if it is finite, a rate of 
rotation may be found such that a tooth will replace a space 
while the wave train travels to M and back ; the returning 



VELOCITY OF LIGHT 227 

/ 

light will then be intercepted by a tooth and the star will be 
eclipsed. 

What occurs in the experiment is the appearance at first 
of a bright star, which gradually diminishes in brightness as 
the speed of rotation increases, until at a definite speed it is 
entirely extinguished ; if the speed of rotation still increases, 
the star reappears, reaches its former maximum of bright- 
ness, again fades away, and is eclipsed a second time when 
the speed of rotation is three times as great as for the first 
eclipse. 

Fizeau found the first eclipse at a speed of 12.6 revolutions 
per second. The wheel contained 720 teeth, or 1440 equal 
divisions. Hence the time required for a tooth to take the 

place of a space was x = sec. The double 

12.6 1440 18144 

distance between the two stations was 17.266 km. The 
speed of light deduced from the experiment was, therefore, 
17.266 x 18,144 = 313,274 km. /sec. 

In 1874 Cornu repeated Fizeau's experiment with greatly 
improved apparatus. His final result was 300,330 km. in 
air. To obtain the speed of light in the ether of outer space, 
this result must be multiplied by the index of refraction of 
air ( 279) and this gives 300,400 km. /sec. 

262. Michelson's Modification of Foucault's Method. The 

original experiment of Foucault was intended to compare the 
speed of transmission of light through water and air as a 
crucial test between the emission and the undulatory theory 
of light ( 329). It was modified by Michelson in 1879 to 
make it capable of a very precise measurement of the velocity 
of light. 

An outline of Michelson's arrangement is shown in Figure 
155. S is a narrow slit, m a plane revolving mirror, L a 
lens of long focal length, and m' a fixed mirror. The illu- 
minated slit becomes .a source of light. It is incident on m y 
is reflected to m' whenever m is in a suitable position, and 



228 LIGHT 

forms an image of the slit at S'. When m is at rest, the 
light reflected from m r retraces its path to the slit S. 

If, however, m has rotated through an angle while the 
light is traveling from m to m' and back, the reflected pencil 
will be deflected through an angle 2 ( 270) and will form 
an image of the slit at S n '. 




Fig. 155 



If the deflection 88" is d, the distance between S and m 
is r, and that between m and m' is L\ and if n is the number 
of revolutions of the mirror m in a second and t the time 
required for light to pass from m to m' and back, then 
2 6 = d/r, t = 0/27rn, and 

v _ 2 L _ ^irnL _ &TrnLr 

"r s ~T~ ~d~ 

In Michelson's latest determination L = 605 m., r = 8.58 m., 
and n = 257. The displacement of the image of the slit 
was 113 mm. The final result, including all small cor- 
rections, for the velocity of light in a vacuum was 
299,853 50 km. /sec. 

By a similar method in 1882, and with a distance L between 
the Washington Monument and Fort Myer of 3721 m., the 
late Professor Newcomb obtained the value 299,860 30 
km./ sec. It is probable that the velocity of light in a 
vacuum does not differ from 300,000 km. /sec. by more than 
about one part in 3000. 




REFLECTION OF LIGHT 229 

^ III. REFLECTION OF LIGHT 

263. Regular Reflection. When a pencil of rays falls on 
a plane polished surface, the larger part of it is reflected 
in a definite direction 

(Fig. 156). The angle 
IBP between the inci- 
dent ray and the normal 
PB at the point of inci- 
dence is called the angle 
of incidence, and the 
plane containing the 
two lines is the plane of 

- 7 j.ii5 Fig - 156 

incidence ; the angle 

PBR between the reflected ray and the normal is the angle 
of reflection, and the plane containing these two lines is the 
plane of reflection. 

The law of regular reflection is : 

The angle of incidence is. equal to the angle of reflection, 
and the planes of incidence and reflection coincide. 

This law is an expression of experimental facts, and it 
coincides with the law of reflection deduced from the wave 
theory ( 195). 

264. Diffuse Reflection. When light falls on an unpolished 
surface, it is reflected in all directions. Light incident on 
finely divided matter in suspension, such as smoke in a glass 
jar, is similarly Reflected. The minute particles of floating 
smoke furnish a great many reflecting surfaces ; the light is 
reflected in as many directions, and the result is the diffusion 
of the beam. 

Diffuse reflection is usually selective; that is, though the 
incident light is white, the diffused light is colored. The 
rich red petal of a geranium may be illuminated by white 
light, but it reflects diffusely in all directions only the red. 

265. Visibility. Aeronauts have observed that at high 
altitudes the sky becomes black. If the atmosphere near 



230 



LIGHT 



the surface of the earth were free from minutely divided 
matter reflecting the shorter waves of light, the sky on a 
cloudless day would appear as black as at high altitudes. 
What we call the sky is the pale blue light coming to us by 
diffuse reflection from very minute particles suspended in the 
atmosphere. The heavens on a moonless night are black, 
except for the little light of the stars and planets, even though 
outside the earth's shadow cone the sun is flooding space with 
light. 

Objects are visible only when they are self-luminous or 
when they reflect irregularly and diffusely some of the light 
by which they are illuminated. Without diffuse reflection the 
eye would receive light from self-luminous bodies only. A 
beam of light admitted into a dark room is invisible if the 
air is dustless. It may pass through pure distilled water 
without illuminating it; but blow a puff of smoke across the 
beam, or add to the water a few drops of milk, and the light 

will flash out from the path of 
the beam. 

It is by diffuse reflection 
that objects become visible. 
Perfect reflectors would be in- 
visible. The trees, the ground, 
the grass, and particles float- 
ing in the air reflect light 
from the sun in all directions, 
and thus fill the space about us with light. 

266. Image of a Point in a Plane Mirror. - 
Let A be a luminous point in front of a plane 
mirror J/7V^(Fig. 157). The group of waves 
included between the limiting rays' AB and 
AC after reflection proceed as if from A', situated on the 
normal JJTand as far behind the reflecting surface as A is 
in front of it ( 196). An eye placed at DE receives these 
waves as if they came directly from a source A f . The point 




A 

Fig. 157 



REFLECTION OF LIGHT 



231 



A' is called the image of A in the mirror MN. It is known 
as a virtual image, because the light only appears to come 
from it. Therefore, the image of a point in a plane mirror 
is virtual, and is as far lack of the mirror as the point is 
in front. It may be found by drawing from the point a 
perpendicular to the mirror, and producing it till its length 
is doubled. 

267. Path of the Kays to the Eye. An image of an object 
is made up of the images of its points. Let AB (Fig. 158) 
represent an object in front of 
the plane mirror MN. Drop 
perpendiculars from points of 
the object to the mirror, and 
produce them till their length 
is doubled. In this manner 
the image of AB is found at 
A'B' ' . It is virtual, erect, of the 
same size as the object, and per- 
verted. 

It is important to observe that 
the image of any definite object 
is fixed in space, and is entirely 
independent of the position of the observer. The paths of 
the rays for the image for one observer are not the same as 
those for another. Let E and JE f be the position of the eye 
for two observers. To find the path of the rays entering the 
eye at E, draw lines from A' and B' to E. These lines are the 
directions in which the light enters the eye from A! and '. 
But no light comes from behind the mirror, and therefore 
the intersections of these lines with the mirror are the points 
of incidence of the rays from A and B which are reflected 
to E. In a similar manner the path of the rays may be traced 
for the position E 1 '. The full lines in front of the mirror 
represent the paths of the rays from A and B, which give the 
images at A' and B' . 




Fig. 158 



232 



LIGHT 




268. Multiple Reflection. Multiple images are produced 

by successive reflections from two reflecting surfaces. When 
these surfaces are parallel, the images all 
lie on the normal drawn from the object 
to the reflecting surfaces. The double 
image of a bright star or planet, and 
the several images of a gas jet in a 
thick mirror, are examples of multiple 
reflection. 

Let MM* and NN* (Fig. 159) be the 
two parallel surfaces of a thick mirror 
or piece of plate glass ; and let be the 
object. The first image 1 is found in 
the usual way. Part of the light enters 
the first surface, is reflected internally 
at the second surface, and returning to 
the first surface is in part internally 
reflected and in part transmitted, the 
transmitted portion forming the second 
image 2 , and so on. Geometrically the 

number of images is infinite ; but on account of the loss of light 

by successive reflections, only a limited number are visible. 
When the mirrors are inclined to 

each other, all the images lie on a 

circle, the radius of which is the 

distance between the object and the 

intersection of the planes of the two 

mirrors. The image in the first 

mirror becomes the object for the 

second, and this in turn is the object 

for a second image in the first mirror. 

In each case the light is incident on 

either mirror precisely as if it came 

from the next preceding virtual image in the other mirror. 

Let be a luminous point between the two inclined mir- 
rors AB and AC (Fig. 160). Then b is the first image in 




Fig. 159 




REFLECTION OF LIGHT 



233 



AB ; and since it is in front of A C, its image in the second 
mirror is at c' ; c' is in front of AB and has its image at b". 
But b ff is behind the plane of both mirrors and there is there- 
fore no image of it. 

In the same way the images e, b f , b", may be found by first 
finding the image of in AC. 

By equality of triangles it is easy to show that OA, bA, 
c'A, etc., are all equal, and that therefore the object and all 
its images lie on the circle, the center of which is A. 

269. Deviation by Successive Reflection from Two Mirrors. 
The deviation of a ray of light by two reflections from a pair 
of plane mirrors is 
twice the angle be- 
tween the mirrors. 

Let the ray be suc- 
cessively reflected from 
the two mirrors E and 
F inclined at an angle 6 
(Fig. 161). The devia- 
tion is the angle </>. 

We have 

<j> = 180 - 2 

= 90- 




N 



Fig 161 



Doubling the second equation and subtracting from the 
first, <-20 = 0, or = 20. 

270. Deviation due to the Rotation of a Plane Mirror. If a 

plane mirror on which a pencil of rays 
falls be turned through an angle about 
an axis perpendicular to the plane of 
incidence, the reflected pencil will be 
deflected through twice the angle. Let 
a ray AM be incident normally on the 
mirror (Fig. 162) ; it will retrace its 
path after reflection. If the mirror be 




234 LIGHT 

now rotated through the angle 0, the normal will be turned 
through the same angle, and the angles of incidence and re- 
flection will both be equal to 0. The deviation is then the 
angle AflfJS, or 2 (9. 

A plane mirror is extensively used to indicate, by the change in the 
direction of the reflected ray, the motion of the movable system of the 
instrument to which the mirror is attached. The reflecting galva- 
nometer takes its name from the mirror which is attached to the needle 
system, and which indicates the slightest rotational movement. 

271. Concave Spherical Mirrors. A spherical mirror is one 
whose reflecting surface is a portion of the surface of a sphere. 
If the inner surface is polished for reflection, the mirror is 
concave; if the outer surface, it is convex. Only a small 
portion of a spherical surface cut off by a plane is used as a 
mirror. The center of the mirror is the center of curvature 
of the sphere. The middle point of the reflecting surface is 
the pole or vertex of the mirror, and the straight line passing 
through the center of curvature and the pole of the mirror is 
its principal axis. 

Reflection from each element of the curved surface takes 
place in accordance with the fundamental law of reflection. 
A pencil of incident rays gives rise to a system of reflected 
rays, the direction of which can be geometrically determined. 
The position of the image in a spherical mirror can readily be 
determined by the application of Huyghens' principle, but 
the geometrical method is simpler and more generally useful. 
Let AB (Fig. 163) be a section of a concave spherical 
mirror through its principal axis A U. Uis a luminous point ; 

it is required to find 
the equation connect- 
ing its distance from 
the mirror with that 
of its image. If a ray 
from U meets the mirror at P, it will be reflected across 
the axis at V, so that the radius OP, which is the normal 
at the point of incidence, bisects the angle UP V. 




REFLECTION OF LIGHT 235 

Let the several angles be denoted by the letters indicated 
in the figure. Then from the triangles UPO and VPO, 

= z + A, O) 



Adding (V) and (T), 20 = A + < 00 

If now P is very near A, the angles 0, <, and A are very 
small, and their tangents may be put equal to the angles 
themselves ( 23). Let AU be denoted by p, AVby p r , 
and the radius by r. Also let y denote the distance PA. 
Then from (<?) 

??+#. 

r p p r 

Whence 1 + 1=^ (50) 

p p' r 

Since y does not appear in equation (50), it follows that 
for a given position of the radiant point U, the distance of 
V from the mirror is independent of the point of incidence 
P. The physical interpretation is that for small angles of 
incidence all the rays from CT, incident on the mirror, are 
reflected so as to pass through the common point V. F'is 
a real focus because the rays after reflection actually pass 
through this point. V and U are called conjugate foci, 
because if V were the radiant point, the focus after reflec- 
tion would be U. This relation follows from the symmetry 
with which p and p' enter the equation ; for the values 
of p and p' are interchangeable, or the object and image 
may exchange places. 

272, Focal Length. If the luminous point is at an infinite 
distance, that is, if the incident rays are parallel to the axis 
of the mirror, p is infinite, \/p is zero, and 

12 i r /. ^C-IN 

___,ory = -~/. (51) 



236 LIGHT 

The focus for rays parallel to the principal axis is the 
principal focus, and the length/, defined by equation (51), 
is the focal length of the mirror. Introducing it into equa- 
tion (50), 

t + h? (52) 

When p is greater than /, I/ 'p is less than I// and 1/p' 
is positive; that is, the image is in front of the mirror and 
is real. 

When JP is less than/, \jp is greater than I//, and 1/p 1 is 
negative; that is, p 1 is negative, or the image is back of the 
mirror and is virtual. 

273. Graphical Construction for Images. First. When the 
object is farther from the mirror than the principal focus. To 
find the image of -any point of an object, it is necessary to 
trace only two rays : one parallel to the principal axis, 
which after reflection passes through the principal focus; 
the other incident at the pole of the mirror, which makes 
the same angle with the principal axis after reflection as 
before. These rays are selected for convenience because 
they can be readily traced. They are of course not only 
rays, but radii of the spherical waves proceeding from the 
point of the object before reflection and converging toward 
the conjugate focal point or image after reflection. 

Let AB (Fig. 164) be the object. The ray AD after 
reflection passes through F, and the ray AP is reflected in a 

direction to make the 
angle r equal to the 
angle i. The two re- 
flected rays meet at a, 

B ^^ I which is therefore the 

Fig. 164 . * j on. .e 

image of A. The figure 

shows also the waves diverging from A and converging after 
reflection toward a. The point b may be found in a similar 
manner. The image ab is real and inverted. 





REFLECTION OF LIGHT 237 

Second. When the object is nearer the mirror than the 
principal focus. The ray AD (Fig. 165) parallel to the 
principal axis is a 

reflected through 
the principal fo- A 

cus F, halfway 
between the pole 
of the mirror and 

its center of cur- Fig |65 

vature. The ray 

AP makes equal angles with the principal axis before and 
after* reflection. The two reflected rays do not meet, but 
if their directions are prolonged backwards, the lines meet 
behind the mirror at a. This point is the virtual image of J., 
and the waves after reflection proceed as if from the center a. 
Similarly b is the conjugate focus of B and ab is the image 
of AB. It is erect, enlarged, and virtual. 

Since corresponding portions of the object and image 
subtend at P equal angles i and r respectively, the size of 
the object and the image are proportional to their respective 
distances from the mirror, or 



274. The Convex Mirror/ Since the center of curvature 
of a convex mirror is behind the reflecting surface, the 
radius r is negative. Then (50) becomes 

i+ii-4 

p p' r J- 

But the distance of the object p is necessarily always 
positive ; then p' must always be negative, and the formula 
for the convex mirror becomes 

-> 



238 



LIGHT 







Fig 166 



The image may be found by the construction applied 
to the concave mirror. The ray AD (Fig. 166) parallel to 

the axis is reflected in 
a direction which passes 
through the principal focus 
F\ the ray AP makes equal 
angles with the axis before 
and after reflection. These 
rays after reflection diverge 
as if from the point a, 
which is therefore the image of A. The group of rays in- 
cluded between AD and AP are the radii of waves 'diverging 
from A before reflection and from a after reflection. The 
image is erect, virtual, and smaller than the object. All 
possible images lie between F and the mirror. 

275. Spherical Aberration. A semicircular sheet of tin or 
polished brass is placed on a sheet of white paper with its 
concave surface toward a candle flame (Fig. 167). The light 
is focused on 
lines called caustic 
curves. Sunlight 
falling on a cup 
partly filled with 
milk, or on the 
inner surface of a 
plain gold ring 
lying on white 
paper, shows by 
reflection the 
same curves. 

Rays incident 
near the margin of 

Fig. 167 

the mirror, except 

when the radiant point is at the center of curvature, cross 

the axis at points nearer the mirror than the principal focus 




REFRACTION OF LIGHT 



239 



A 



F (Fig. 168). Waves reflected from a spherical mirror are 
not perfectly spherical, but they are normal at every point 
to the reflected rays. The surface envel- 
oping the group of intersecting reflected 
rays is called a caustic surface. The cusp 
F of this surface is the principal focus. 

Spherical aberration is the deviation from 
a spherical form of waves reflected from 
a spherical mirror. By decreasing the 

curvature of a mirror from the vertex ^_ 

outward, spherical aberration may be cor- 
rected. The reflecting surface then be- 
comes parabolic, a form used in lighthouses, in the head- 
lights of locomotives, and for reflecting telescopes. 



IV. REFRACTION OF LIGHT 

276. Refraction. When a pencil of light passes obliquely 
from one transparent medium into another, it undergoes a 
change in direction at the bounding surface of separation 
between the two media. The change in the course of light 
in passing from one transparent medium into another is 
called refraction. 

Thus, if a thin beam of bright light is admitted through a 
slit Am the cover of a glass jar (Fig. 169) so that it falls 

obliquely on the surface of 
water at B, its path may 
readily be seen and it changes 
direction sharply at B, the 
path BO making a smaller 
angle with the normal than 
that of the incident beam.^/ 
This sudden deviation in the v 
path of light in going from 
one medium to another was 
Fi. 69 known to the ancients. 




240 



LIGHT 




277. Laws of Refraction. Let BA denote a ray of light 
in air incident obliquely at A upon the surface MN (Fig. 

170) of another medium, 
as water. AQ is the re- 
fracted ray. The angle 
BAD between the incident 
ray and the normal to the 
surface at the point of inci- 
dence is the angle of inci- 
dence , i ; and the angle 
CAE between the refracted 
ray and the normal is the 
angle of refraction r. The 
angle OAH is the angle of 
deviation. The lines BF 
and OGr are drawn perpendicular to the normal DE. The 

ratio 

BF sin i ,rr^ 

xv = = //.. (OOI 

CO- sin r 

Snell, a Dutch mathematician, discovered, in 1621, that 
this ratio for the same two media is a constant, whatever 
the angles of incidence and refraction. The ratio /A is called 
the index of refraction. 

The following are the laws of single refraction : 

I. The planes of the angles of incidence and refraction 
coincide. 

II. The ratio of the sines of the angles of incidence and 
refraction is constant for the same two media. 

278. Velocity of Light in Different Media. A very sim- 
ple explanation of refraction was given by Huyghens by as- 
suming that the velocity of light changes in going from one 
medium to another. Newton attempted an explanation by 
assuming an attraction between the denser medium and the 
corpuscles of light. By the wave theory the velocity of 



REFRACTION OF LIGHT 241 

light in air is greater than in water ; by the emission theory 
it is less in air than in water. 

Up to the time of Foucault, while the emission theory of 
light had been practically abandoned in favor of the wave 
theory, there had been no crucial test by experiment between 
the two rival assumptions relating to the velocity of light in 
different media. The question was definitely settled in 
1850 by Foucault, who demonstrated that the velocity of light 
in water is less than in air, though he made no estimate of 
the ratio. In recent years Michelson measured the relative 
velocity of light in air and water, and found the ratio to be 
1.33 ; also that the ratio of the velocity in air to that in 
carbon bisulphide is 1.76. 

The retardation of light in transmission through trans- 
parent bodies is one of the fundamental facts of optics ; 
indeed, it is probable that there is no more important phe- 
nomenon in the whole domain of radiation. 

279. Explanation of Refraction by the Wave Theory. Let 
AB (Fig. 171) be a plane wave incident obliquely on the 
smooth plane surface AC, sepa- 
rating air and another trans- 
parent medium. Let the velocity 
in air be v, and in the second 
medium v'. Also let t be the 
time required for light to trav- 
erse the distance BC. The 
disturbance at A has just reached 
the second medium, and A be- 
comes a new center from which a spherical wave will travel 
in the second medium a distance AD equal to v't, while the 
disturbance in air travels from B to (7, a distance equal to vt. 

In the same way the disturbance from the point P will 
travel to Q, which then becomes a new center ; from this cen- 
ter the disturbance will spread into the second medium, a 
distance QT, such that QT/QT'=v'/v. All circles repre- 





242 



LIGHT 



senting traces of such spherical waves in the second medium 
ultimately coalesce along CD, drawn through C tangent to 
the circle described about A as a center. The envelope of 
all these spherical waves is, by Huyghens' principle, the new 
wave front in the second medium. 

The angles of incidence and refraction are equal to BAG 
and A CD respectively. They are the angles between the 
wave fronts in the two media and the refracting surface. 
Then 



= AC sin i=vt, 



Whence 



sn 
sin r 



vt 

v't 



(56) 



The index of refraction is therefore equal to the ratio of 
the velocity of light in the rarer medium to the velocity in 
the denser, and the constancy of the ratio of the sines has 
now a physical significance. The wave theory thus gives a 
satisfactory explanation of the laws of single refraction. 
The relative velocities of light in air and water, and in air 
and carbon bisulphide, found by Michelson, are very approx- 
imately the relative indices of refrac- 
tion for both pairs of transparent 
media. 

280. Object Viewed in a Direction 
Normal to the Boundary. Let NO 
(Fig. 172) be a normal to the plane 
bounding surface, and let the emer- 
gent group of waves between 10 and 
IB come from the luminous point 0. 
The center of these emergent waves 
is /, the image of 0. The angle AIB 
is i, and the angle A OB is r. Then 

AB = IB sin i = OB sin r. 




Whence 



sin ^ 
sinr 



OB 

IB' 



REFRACTION OF 'LIGHT 



243 



When OB is nearly normal, and the pencil of rays is only 
slightly divergent, OB is ultimately equal to OA, and IB to 
IA. IA is then equal to OA/fjL. There is then no lateral 
displacement, but only an apparent change of distance. 

The angle of the cone of rays entering the eye is limited 
by the size of the pupil, and is therefore small. Hence, 
when an object is viewed along a normal to the bounding 
surface, the distance of the object from the surface is p times 
that of the image. For air and water /*, is ^, and for air and 
glass about |. An object in water cannot appear more than 
three fourths of its real depth below the surface, and one in 
glass not more than two thirds its actual distance from the 
bounding surface. 

Viewed obliquely the depth of water appears less than 
three fourths of its actual depth. Hence the shoaling of 
still water when the bottom is visible. 

281. The Critical Angle for Internal Reflection, -r- When 
light passes from one medium into another in which the 
speed is higher, its path 
is the reverse of the one 
we have hitherto consid- 
ered, and a ray is deflected 
away from the normal in 
the second medium. We 
may still call the larger 
angle between the ray 

and the normal i and / / y^~-.~l^~-'' /' 

the smaller one r, even 
though the latter is in 
fact the angle of inci- 
dence. Then, since i is greater than f, there will always be 
a value of r for which i is 90, and the emergent ray will just 
graze the boundary surface. 

In Figure 173 the ray RO makes the angle r with the nor- 
mal in the optically denser medium, such as water, and the 




Water 



R 



N' 



Fig. 



244 LIGHT 

angle i in air after refraction. The angle i increases faster 

OB 

than r, and the former becomes 90 when equals the in- 

DJii 

dex of refraction for the two media. The angle CON' is 
called the critical angle. For values of r exceeding the criti- 
cal angle, such as ION 1 , the ray of light can no longer 
emerge into the second medium, but undergoes total internal 
reflection. For angles smaller than CON', part of the light 
is reflected and part refracted ; for larger values of r it is all 
reflected. 

To determine the critical angle for any medium whose 
relative index of refraction is //,, we have 

= sm9Q t Whence sin x = -, 
sin x fjL 

or the sine of the critical angle is the reciprocal of the index of 
refraction. 

For water the critical angle is 48 28' 

For crown glass about 41 10' 

For quartz . . 40 22' 

For diamond . 24 26' 

For chromate of lead 19 49' 

The smaller the critical angle for a jewel with regular facets, the 
larger is the proportion of the light incident on it which is internally 
reflected. This fact explains largely the bril- 
liancy of the diamond. 

The internal reflection of light is beautifully 
shown by focusing a beam of light by means of 
a lens L (Fig. 174) on the interior of a smooth 
jet of water at the point of issue from the 
side of a tall vessel. The angle of incidence 
on the inside of the jet exceeds the critical 
angle, and the light is reflected from side to 
side along the stream like sound in a speaking tube. The stream is visible 
on account of the light diffused from fine matter suspended in the water. 

282. Refraction through a Prism. A prism for optical pur- 
poses consists of a transparent medium bounded by two plane 





REFRACTION OF LIGHT 245 

faces inclosing an angle, which is less than twice the critical 
angle for the substance. This angle A (Fig. 175) is called 
the refracting angle of the prism, 
and the line along which the inclined 
faces meet is the refracting edge. 

Since light is refracted toward 
the normal when entering a denser Jrf"'^X ' -' '\\ 
medium, and away from it when ^ / ^ \ 
emerging into the rarer, the path B C 

of a ray of homogeneous light 

through a prism may be such as LIEO. The plane of inci- 
dence is perpendicular to both faces of the prism, and there- 
fore perpendicular to the refracting edge* 

The deviation at the first surface of the prism is i r ; at 
the second surface, i' r f . Therefore the total deviation is 

Z) = i-r + i'-r' = i + i f -(r + r'). 

The angle of the prism A equals the angle between the 
normals at P\ and since this angle is external to the tri- 
angle IPE, it equals the sum of the two interior opposite 
angles r and r'. Therefore A = r + r' . 

If the path of the ray through the prism is symmetrical 
with respect to the two faces, the ' deviation is a minimum ; 
then i = i' and r = r'. It follows that A = 2 r, and 

D= 2i-2r = 2i-A. 
Hence 




and = , 

sin r sin J A 

This is the formula in common use for measuring the in 
dex of refraction for any ray whose minimum deviation is D. 

When the angle of the prism is very small, an approxi- 
mate formula may be found -by making the angles A and D 



246 LIGHT 

equal to their sines. Then 



and D = A O - 1) 

This formula is applicable to thin prisms. 



(58) 




Fig. 176 



For crown glass the critical angle is about 
41 10'. Hence if a prism of crown glass has as 
a section a right-angled isosceles triangle BAC 
(Fig. 176), the refracting angle will be more than 
twice the critical angle, and a ray DE incident 
normally on either face inclosing the right angle 
will have an internal angle of incidence greater 
than the critical angle, and will be totally reflected 
at E. Such a prism is used frequently to change 
the direction of a beam of light by 90. 



V. LENSES 

283. Refraction at a Spherical Boundary. A lens is a trans- 
parent body bounded by two spherical surfaces, or by one 
plane and one spherical surface. Before proceeding, there- 
fore, to the subject of lenses, we must consider the case of 
refraction at a single spherical boundary. 




Fig. 177 



Let C (Fig. 177) be the center of curvature of the concave 
boundary, the point-source of light on the axis, OP the 
incident ray, and PE the ray after refraction ; produced 
backward the latter meets the axis at /. /is the image of 0. 



LENSES 247 

Then, from the law of refraction, 

LI sin r = sin i. 



Dividing both sides of this equation by sin 
sin r sin i 

Li = . 

sin sin 6 



oo 



Remembering that the sine of an angle 6 is equal to the 
sine of its supplement, and that the sides of a triangle are 
proportional to the sines of the angles opposite, we have, from 
the triangles OPJand QPO, 

i, and s4irti. 

sin u s 

If now the point P is near the vertex A, s f is very nearly 
equal to p' and s to p. Therefore equation (a) becomes 




Dividing through by r-^ and transposing, 

4_! = =-i. (59) 

p' p TI 

This equation may be considered as representing the gen- 
eral relation for light incident on a single boundary surface. 
For light reflected from the spherical surface, p = 1 and 
the formula becomes 119 

P P' " r i 

which agrees with equation (50) for reflection from a concave 
spherical mirror. If in addition r a is infinite, the reflecting 
surface is plane, 2/r x is zero, and p p', or the object and 
image are on opposite sides of the plane boundary and at 
equal distances from it ( 266). 



248 LIGHT 

284. Refraction at Two Successive Surfaces. If the ray PE 

. (Fig. 177) is incident on a second spherical surface near the 

r s first, having a radius of curvature r 2 , then, neglecting the 

jthickness of the denser medium (the lens), the apparent 

distance of the object for axial rays from this surface is p 1 ; 

and if q is the distance of the image, we have, from (59), the 

refraction being from the denser to the less dense, 




Multiply through by p, and 

i-i-t=Ji (60; 

q p' r 2 

Adding (59) and (60) 

(61) 



This is the approximate formula for a thin lens. The dis- 
tances p and q are called conjugate focal distances. 

285. Focal Length and Principal Focus, If the source of 
light is at an infinite distance, p = oo , and if we set f for 
the corresponding value of q, equation (61) becomes 



(62) 



The distance / is called the focal length of the lens; it is 
the distance from the lens to the focus of rays parallel to the 
principal axis, the line passing through the centers of curva- 
ture of the two surfaces. This focus is called the principal 
focus; its conjugate point is at an infinite distance on the 
principal axis. Rays from a point-source of light at an 
infinite distance on the principal axis are parallel, and after 



LENSES 



249 



refraction they either converge toward the principal focus, or 
diverge as if coming from it. If the radii of curvature are 
equal and the index of refraction is f , the focal length is 
equal to the radius of curvature, and the principal focus on! 
either side of the lens coincides with the center of curvature. 

286, Forms of Lenses. Lenses are in general portions of a 
refracting medium included between spherical surfaces. Of 
the six forms of Figure 178, the first three are converging 



Fig. 178 

lenses ; they are all thicker at the center than at the edge. 
The last three are diverging lenses ; they are thinner at the 
center than at the edge. 

Apply formula (62) to the double convex lens 1 of Figure 
178. Distances measured on the side of the lens from which 
the light comes are considered positive ; those on the other 
side of the lens are negative. Then, since r l is negative, 
(62) becomes 



The focal length f is therefore negative, or the principal 
focus for a double convex lens is on the side of the lens 
opposite to the source of light, and rays parallel to the prin- 
cipal axis are caused to converge toward the principal focus 
after traversing the lens. 

When formula (62) is applied to 2 and 3 of Figure 178, it 
will be found that in these cases also /is negative, and these 
forms are converging. 



250 LIGHT 

For the double concave lens 4 of Figure 178, r- is positive 
and r 2 negative. Therefore (62) becomes 



and /is necessarily positive, since p is always greater than 
unity. For the forms 5 and 6, / is also positive, and the 
principal focus lies on the same side of the lens as the source 
of light. The three forms, 4, 5, and 6, are therefore diverg- 
ing lenses. 

287. Effect of the Surrounding Medium on Focal Length. 

The focal length of a lens increases as the index of refraction 
decreases, or/ is inversely as JJL 1 from (62). What then 
will be the effect on the focal length of a lens by immersion 
in another medium than air, say water ? 

If the index of refraction for glass is f and for water |, 
then the relative index from water to glass is -|. Therefore 
for a lens with equal radii of curvature, the focal length in 
air is equal to its radius, but its focal length in water, obtained 

1 /9 \ 2 

from the relation == ( - - 1 ) , is/= 4 r v The effect of im- 
/ \o /r l 

mersion in water is therefore to increase ' the focal length 
fourfold. 




Fig. 179 



Figure 179 illustrates the change in focal length by im- 
mersion in water. F is the principal focus of the lens in 
air, as shown by the dotted lines, while the principal focus in 
water is at F f . 



LENSES 251 

If a converging lens is immersed in an oil in which the velocity of 
light is less than in glass, the converging lens becomes a diverging one. 
So also a hollow double convex lens filled with air and immersed in water 
is a diverging lens, for the speed of light in the medium inclosed in the 
lens is y greater than in water, and the plane wave front is converted into 
convex wave front by the air lens. A convex air lens in oil acts like a 
concave glass lens in air, and a concave air lens in oil like a convex glass 
lens in air. All a lens can do is to change the course of light waves by 
impressing new curvatures on the wave fronts, and the new curvatures 
depend on the relative .velocities of light in the lens and in the surround- 
ing medium. 

288. Universal Lens Formula. Comparing equations (61) 
and (62), we may write the general formula for refraction 
by a lens in the form 1 1 1 

i-i.L (63) 

p f 

For diverging lenses /is always positive, with the conven- 
tion here used respecting signs ( 286). Then, since J9, the 
distance of the point-source of light, or of the object, is 
necessarily positive, q must also be positive and smaller than 
p ; otherwise 1/q ~L/p would not be positive to conform to 
the formula (63). The image for diverging lenses is there- 
fore on the same side of the lens as the object, and is virtual. 

For converging lenses / is always negative. Then the 
general formula becomes 



q p 



/ 



When p >/, 1/p < I//, and q in the formula must be neg- 
ative, or object and image are on opposite sides of the lens, 
and the image is real. 

But when p </, l/p > I//, and hence q is positive. The 
object and image are then on the same side of the lens and 
the image is virtual. 

When p and q are numerically equal, the image is real, and 



252 LIGHT 

Object and image are then equidistant from the lens, and 
the distance between them is 4/. They cannot approach 
nearer than this for a real image. 

289. Object and Image at a Fixed Distance. When the ob- 
ject, such as an incandescent lamp filament, and the screen 
on which the real image is received are at a fixed distance, 
which must be greater than four times the focal length of the 
converging lens, there are two positions of the lens for a well- 
defined image. For the first the lens L (Fig. 180) is nearer 



1 I 
I / 
\l 

__ v _ [ --- 

~ Fig.lsO ~" 



the object and the image on the screen is enlarged ; for the 
second, the two focal distances are exchanged, the lens L' is 
nearer the image, which is then smaller than the object. 

Let I be the distance between the object and the screen, 
and a the distance between the two positions of the lens for 

distinct images. Then 

q -{- p = I, 

and q p = a. 

Adding these equations, we have 

1 + a 



Subtracting, p 

Therefore 



? = ~2- 
l-a 



2 

4? 



Whence / = !!_Z*L (65) 




LENSES 253 

This formula furnishes a very satisfactory method of 
measuring the focal length of a converging lens. 

290. Optical Center of a Lens. Let and C' (Fig. 181) 
be the centers of the two spherical surfaces of the lens. 
Draw any two parallel radii, 

as AC and BO'. Then the 
tangent planes at A and B 
are also parallel, and a ray 
incident at A and emerging 
at B passes through the lens 
as if through a plate with 
plane parallel sides, and there 

is no deviation ; for the angles of incidence and of refrac- 
tion on opposite sides of the lens are equal to each other. 

AB is the path of the ray through the lens. It cuts the 
axis at 0. Then, since AGO and BC'O are similar 
triangles, 00_ = OA 

Q'O C'B 

Since the radii are constant in value, CO and C'O are also 
constant, and is therefore a fixed point. It is called the 
optical center of the lens. Any ray passing through the lens 
without deviation crosses the axis at the optical center. 
Any straight line passing through the optical center, except 
the one joining the centers of curvature, is called a secondary 
axis of the lens. In the case of a thin lens, the optical cen- 
ter may usually be considered as coinciding with the center 
of the lens. 

291. Construction for the Optical Image of a Point in a 
Converging Lens. To find the image of a point, or of an object 
consisting of a collection of points, we have only to trace two 
rays and find their intersection after passing through the lens. 
The two rays traced are the following: 

1. Any ray parallel to the principal axis has after emer- 
gence a direction through the principal focus, and conversely. 



C* ' j 




254 



2. An incident ray along a secondary axis (through the 
optical center) emerges without deviation. 

To find the real image in a converging lens, with the ob- 
ject AB beyond the principal focus (Fig. 182), from the 




Fig. 182 

point A draw the ray AD parallel to the principal axis, and 
continue it after emergence through the principal focus F f . 
Also draw a ray from A through the optical center and 
find its intersection a with the first ray. Another ray 
through the principal focus F may be drawn ; it will emerge 
parallel to- the principal axis and will also pass through a. 
The point a is the optical image of A. Other points of the 
image ab may be found in a similar manner. 




Fig. 183 

For a virtual image the object AB (Fig. 183) is between 
the principal focus and the lens. Proceeding as before, the 
emerging rays now diverge as if they came from a, which is 
therefore the virtual image of A. 

The group of waves included between the two rays traced 
have their wave front changed from convex to concave in the 
first case for a real image; and in the second case, they have 
their curvature decreased for a virtual image. 



LENSES 



255 



292. Construction for an Optical Image in a Diverging 

Lens. The image formed by a diverging lens is always 
virtual and erect. It may be found by the same construction 
as for converging lenses. Thus, in Figure 184, AD, parallel 




Fig. 184 

to the principal axis, emerges in a direction passing through 
the principal focus JP, and AO does not suffer deviation. 
The two directions intersect at a, which is the virtual image. 
A ray from A in the direction through the principal focus F' 
emerges parallel to the principal axis. 

The group of incident waves have their center of curva- 
ture at A ; the emergent waves, at a. 

293. Spherical Aberration. The formula for conjugate 
focal distances has been derived for pencils of small aperture 
and for thin lenses, and it has been assumed that the emer- 
gent wave is spherical. But when the aperture is large, the 
focal length for marginal rays is less than for axial rays, or 
there is noticeable spherical aberration by refraction. The 
effect is to impair the distinctness of the images formed by 
the lens. In a simple lens this defect is greatly reduced by 
the use of a diaphragm to cut off the marginal rays, leaving 
only the central portion of the lens effective. In compound 
lenses spherical aberration is corrected by combining the 
spherical surfaces so that their respective aberrations mu- 
tually annul each other. In large objectives fqr telescopes, 
the curvature is made to diminish toward the edge, so that 
all rays parallel to the axis are brought to the same focus. 

If a plano-convex lens is turned with its convex surface 



256 LIGHT 

toward a source so distant that the incident rays are nearly 'or 
quite parallel, the path of the marginal rays through the lens 
is then nearly symmetrical with respect to the two surfaces, 
and the conditions are those for minimum deviation. The 
focal length for the marginal rays will then be approximately 
the same as for those near the axis. 



VI. DISPERSION 

294. Dispersion of White Light. When a horizontal beam 
of sunlight is admitted into a darkened room through a nar- 
row vertical slit, and is passed through a long-focus lens, a 

sharp image of the 
slit may be pro- 
jected upon a dis- 
tant white screen. 
If now a prism, 
with its refracting 
edge vertical, is 
placed near the fo- 
cus of the lens, the 
ribbon of white 
light not only un- 
dergoes deviation, 

F| |85 but it spreads out 

into a fan-shaped 

beam (Fig. 185) with its apex at the prism. This refracted 
beam is no longer white, but consists of a series of overlap- 
ping colored images of the slit, forming an apparently 
continuous band of brilliant colors, in which the hues vary 
continuously from red, falling nearest the white image of the 
slit, through all shades of orange, yellow, green, and blue to 
violet at the other edge of the diverging beam. 

This brilliant band of light, consisting of an indefinite 
number of colored images of the slit, is called a solar spectrum ; 
and the conversion of a pencil of parallel rays of white light 




DISPERSION 257 

into a divergent beam, in which the several colors diverge at 
different angles from the original direction, is called dispersion. 

This experiment, though not original with Sir Isaac New- 
ton, was first explained by him in 1666. He referred the 
colors to the complexity of white light, and concluded that 
the latter is to be regarded as a mixture of a series of tints, 
among which the " colors of the rainbow ," red, orange, 
yellow, green, blue, indigo, and violet, were selected by him 
as descriptive of the series. 

The deviation for red is less than for violet, and its index 
of refraction is therefore also less; and since the relative 
index is inversely as the velocity of light in the medium, it 
follows that red light is transmitted through the prism with 
greater velocity than violet. The other colors are trans- 
mitted with intermediate velocities. In fact Michelson has 
shown by direct measurement that the velocity of red light 
is 1.4 per cent greater in water and 2.5 per cent greater in 
carbon bisulphide than that of blue light. Dispersion is 
therefore due to the unequal retardation in the speed of 
transmission of the different colors through transparent 
media. Violet suffers a greater retardation, or travels more 
slowly, than red when it enters an optically denser medium. 
Measurements of wave length show that the undulations cor- 
responding to extreme violet are the shortest of all those 
lying within the visible spectrum. Physically the differ- 
ences in spectral colors are differences of wave length, and 
the short waves suffer greater diminution of velocity in a 
dense transparent body than long waves. In the ether of 
space waves of all lengths travel with the same velocity. 

295. Synthesis of White Light. Newton discovered that 
none of the component colors of solar light undergoes further 
resolution or change in kind by transmission through a 
second prism with its refracting edge turned in the same 
direction as that of the first. But when the second prism is 
exactly like the first and is reversed in position (Fig. 186), 



258 



LIGHT 




Fig. 186 



there is formed a colorless image on the screen. In this 
experiment there is actual synthesis of white light from the 
spectral colors. The incident beam S is resolved into its 

spectral colors by the first prism, 
and these are recombined into an 
emergent beam of white light E 
by the second prism. 

A beam of white light from any 
source, such as the electric arc, gives substantially the same 
succession of colors by dispersion as those obtained from 
sunlight. 



296. Fraunhofer Lines. If the slit used to obtain the solar 
spectrum be made narrower, the colored images of it will be 
narrower and the overlapping of the spectral images will be 
diminished. Wollaston was the first to use a narrow slit, in 
1802, and thus to secure an approximately pure spectrum, in 
which the colored spectral images do not much overlap. 
Wollaston 's method was to view a narrow slit at a distance 
of 10 or 12 feet through a prism held near the eye. He 
obtained in this way virtual images of the slit and was able 
to detect certain dark lines or spaces crossing the solar spec- 
trum, where the corresponding images were absent. Later 
Fraunhofer with better optical means counted about 750 of 
these dark lines, and marked the place of 354 of them on his 
map. They have since been called Fraunhofer lines. 

Fraunhofer designated the most conspicuous dark lines by 
the letters A, B, 0, AnEC D Eh p G H 
D, E, F, 




Red Orange I'ellow Or 



Fig. 187 



Ill 

187). The A lines 
are found in the ex- 
treme red, the D lines 
in the yellow, and the H lines at the limit of the violet end 
of the visible spectrum. 

Photography reveals the existence of lines like those of 
Fraunhofer in the ultra-violet portion of the solar spectrum 



DISPERSION 259 

beyond the ordinary visible limit (Fig. 188). All of them 
indicate the absence of certain colors, or radiations of definite 
wave length, in the light of the sun. They have been the 
means of making important discoveries relating to the con- 




pig. 188 

stitution and physical condition of the sun and of the stars. 
They serve also as a convenient means of reference for colors. 
Thus, when reference is made to any particular color, as D 
light, for example, light corresponding in wave length to the 
dark line D in the yellow of the solar spectrum is meant. 

297. Refractive Indices and Dispersive Power. In the fol- 
lowing table the indices of refraction of several substances 
are given for the most conspicuous Fraunhofer lines : 

AS C D E F G H 

Crown glass 1.5089 1.5109 1.5119 1.5146 1.5180 1.5210 1.5266 1.5314 

Flint glass . 1.6391 1.6429 1.6449 1.6504 1.6576 1.6642 1.6770 1.6886 

Water . . 1.3284 1.3300 1.3307 1.3324 1.3347 1.3366 1.3402 1.3431 
Carbon di- 

sulphide. 1.6142 1.6207 1.6240 1.6333 1.6465 1.6584 1.6836 1.7090 

The angle between the divergent rays for any two colors, 
produced by a prism from parallel rays of white light, is 
called the dispersion for these colors. These colors undergo 
not only dispersion, but deviation as well, and it is of great 
importance in optics to know whether the dispersion always 
bears the same relation to the mean deviation. The ratio of 
the dispersion for any two colors to the deviation of the 
mean between them is called the dispersive power of the sub- 
stance of which the prism is made. 

If /^ and /* 2 , D 1 and J> 2 , are the indices of refraction and 
the deviations respectively for any two colors, represented by 
the two Fraunhofer lines A and ^Tfor example, and JJL and D 



260 LIGHT 

are the same quantities for some intermediate color, such as 
the D line, then (58) 



-1). 

The dispersion, that is, the angular separation of these two 
extreme colors, is ,, -,, 



The following are the angular dispersions for the A and H 
lines in terms of the refracting angle A of the prism, taken 
from the table above : 

Crown glass ........ 0.0225 A 

Flint glass .... ..... 0.0495 A 

Water .......... 0.0147 J. 

Carbon disulphide ....... 0.0948 A 

Hence a hollow prism filled with carbon disulphide will 
give a spectrum 6.45 times as long as if it were filled with 
water. 

The dispersive power expresses the property of dispersion 
possessed by a substance irrespective of the refracting angle 
A of the prism, so long as this angle is small. It is the ratio 

.Z) 2 - Dj _ A (PI- jA t ) _ ^- ^ _ A/x 
D 



This ratio between the difference of deviations of two 
selected colors or lines of the spectrum and the mean devia- 
tion is constant for the same substance, so long as the refract- 
ing angle of the prism is small, but it is different for different 
substances. Thus for crown glass, and for the A, H, and D 
lines, the dispersive power is 0.0437, while for carbon disul- 
phide it is 0.1497. For the same mean deviation, therefore, a 
hollow prism filled with carbon disulphide will give a spec- 
trum 3.4 times as long as the one produced by a prism of 
crown glass. 



DISPERSION 261 

298. Chromatic Aberration. Since the homogeneous colors 
of white light have different indices of refraction, it follows 
from the formula . 




that a single lens has different focal lengths for different 

colors, and that / is less as n is greater. Hence violet light 

conies to a focus nearer the lens than red. Thus, in Figure 

189, v is the principal focus for violet 

rays, and r for red. The other colors 

have their respective foci between 

these two. If, therefore, a screen 

be placed at or near V, as at re, the 

image will be bordered with red ; if 

at y, near the focus /*, it will be fringed with violet. The 

image with least color will be obtained by placing the screen 

midway between the two foci v and r, where the refracted 

beam has the smallest cross section ; but it is impossible to 

get a colorless image with a single lens. This confusion 

of colored images is called chromatic aberration. 

299. Condition of Achromatism. It will be apparent from 297 
that by varying the refracting angles of two thin prisms of different 
materials, such as crown and flint glass, and by combining them with 
their sharp edges turned in opposite directions, it is possible to secure 
deviation without dispersion. 

The expression for dispersion is A A/A. Let the dispersion for a second 
prism be denoted by ^4 'A/A'. Then if the image of the slit is to be color- 
less, the dispersion of the colors to be reunited in one direction must 
equal their dispersion in the other direction ; that is, the angles of the 
prisms must be so chosen that the dispersion shall be the same for both. 
Then , 

A A/A = .4 'A//, or ^ = .. (66) 

. 1 iAyOt 

This equation expresses the condition for a colorless image, or achro- 
matism, for two prisms. Expressed in words it is, the refracting angles 
of the prisms must be inversely proportional to the differences between 
their indices of refraction for the pair of selected colors to be reunited. 
Strictly only the two colors selected are perfectly reunited, for the reason 



262 LIGHT 

that, while their angular separation is the same for the two prisms, the 
intermediate colors do not occupy precisely the same relative spaces in 
the two spectra. 

To unite B and H lines near the two ends of the spectrum, the ratio of 
the refracting angles for crown and flint glass may be found as follows : 

For soft crown glass and the B and H lines 

A/A = 1.5314 - 1.5109 = 0.0205. 

For dense flint glass and the same lines 

A = 1.6886 - 1.6429 = 0.0457. 




Whence = = 2 .23. 

A' 0.0205 

But since these two prisms have different dispersive powers and produce 
the same dispersion, the deviations cannot be the same ; the resultant 

^_- deviation is the difference in 
deviations and is in the direc- 
tion of that due to the prism 
of smaller dispersive power. 
In Figure 190, C is the crown 
glass prism and F the flint 

Fig . I90 S lass - The dis ~ 

persion due to C 

is neutralized by the dispersion of F in the opposite direc- 
tion, but there is an outstanding difference in the deviations ; 
that is, the emergent pencil is colorless and deviates from the 
direction of the incident pencil. 

In a similar way a converging lens of crown glass and a 
diverging lens of flint glass (Fig. 191) may be combined so as 
to give a nearly colorless image, and the pair may still have 
any desired focal length. For this purpose the focal lengths of the two 
lenses must be proportional to the dispersive powers of the two kinds of 
glass. The crown glass lens has the shorter focal length. Such a com- 
bination is called an achromatic lens. 

300. The Rainbow. The rainbow is the most conspicuous 
example in nature of dispersion on a large scale. It is due 
to sunlight refracted and internally reflected by spherical 
drops of rain. In general, the light thus refracted and 
reflected is dispersed in every direction, and not enough is 
received by the pupil of the eye to produce an impression of 




DISPERSION 263 

color except at a particular angle. This angle is the one at 
which an emergent pencil of parallel rays leaves the drop. 
For parallel rays the decrease in intensity with distance is 
small and is dependent on absorption in the intervening 
medium. Since the refrangibility is different for different 
wave lengths, the angle at which parallel rays emerge is not 
the same for red as for violet. Hence the dispersion and 
the spectral colors of the bow. 

Let (Fig. 192) be the center of a raindrop. It was 
demonstrated by Descartes that there is an angle of incidence 
on the drop which gives the 
least deviation from the 
original direction of the par- 
allel rays from the sun. For 
red this angle A OB is about 
59. If the angle of inci- 
dence is either less or greater 
than this, the deviation is 
greater. Now near a mini- 
mum (or a maximum) value 
the change is very slow. 
Hence a small parallel pencil 

of incident rays becomes at this angle a parallel pencil of 
emergent rays, and the intensity is sufficient to produce a 
visual impression. 

In the inner or primary bow, for red, the emergent pencil 
makes an angle of 42 with the line drawn through the sun 
arid the eye of the observer ; for violet, the index of refrac- 
tion of which is larger, the corresponding angle is 40. In 
the primary bow, therefore, the red is on the outside and 
the violet on the inside of the circle. Spherical drops at 
an angular distance of 42 from the axis through the eye 
and the sun, and in any plane through this axis, will there- 
fore send red light to the eye, and the bow is accordingly 
circular. 

The primary bow is the inner and brighter one; the 




264 



LIGHT 



secondary bow is much fainter because the light forming it 
suffers two internal reflections, neither of which* is total. 

Moreover, the order of 
colors, as compared with 
those of the primary, is 
reversed (Fig. 193). An 
observer at E, with his 
back to the sun, receives 
red light from drops at 
an angular distance of 
42 from the axis SO, and 
violet from those at an 
angular distance of 40. 
For the secondary bow, 
the angular distance is 51 
for red and 54 for violet. 




Fig 193 



Artificial rainbows may be made by covering the opening of a porte- 
lumiere with a large sheet of white cardboard, in which is a circular 
hole 3.75 cm. in diameter, and causing a horizontal beam of sunlight 
admitted through the hole to fall on a spherical glass bulb 4 pm. 
in diameter and filled with water. Two circular spectra, resembling 
rainbows, will be reflected back to the cardboard; the space between 
the inner and the outer bow will be quite dark. 



Problems 

1. An illuminated vertical object 6 ft. long is at a distance of 12 ft. 
from a shutter in which there is a minute hole. Inside is a vertical 
screen 4 ft. from the small aperture. How long is the image of the ex- 
ternal illuminated object? 

2. If a man is 5 ft. 11 in. tall, what is the length of the shortest plane 
mirror in which he can see his full-length image ? Does the vertical dis- 
tance of his eye below the level of the top of his head make any difference 
in the length of the mirror ? 

3. The ceiling of a room 20 x 30 ft. is 10 ft. 6 in. high. An observer 
stands with his eye in the center of the room. What is the least height 
of a plane mirror on one wall to enable him to see the image of the 
opposite wall from floor to ceiling? 



DISPERSION 265 

4. If a plane mirror is moved parallel to itself directly away from an 
object in front of it, how much faster does the image move than the 
mirror? 

5. The radius of curvature of a concave spherical mirror is 30 cm. 
If a pencil of light diverge from a point on its principal axis 90 cm. in 
front of it, at what point will it focus ? 

6. An object is 60 cm. in front of a concave spherical mirror and its 
image 20 cm. in front. What is the principal focal length of the mirror? 

7. An object 12 cm. long is placed symmetrically on the axis of a con- 
vex spherical mirror at a distance of 24 cm. from it; the image is 4 cm. 
long. What is the focal length of the mirror? 

8. A candle flame is placed at a distance of 30 cm. from a concave 
mirror made from a sphere of 30 cm. diameter. Find the position of 
the image. Is it erect or inverted ? 

9. If the image of an incandescent lamp is four times as far from 
a concave mirror as the lamp itself, what are the relative dimensions of 
object and image ? 

10. The diameter of the sun is T ^ of its distance from the earth. 
How many inches in diameter will be the sun's image in a concave mirror 
whose focal length is 48 ft. ? 

11. An incandescent lamp is moved from a position 25 cm. to one 
30 cm. in front of a concave mirror whose focal length is 20 cm. How 
far is its image shifted? 

12. Two plane mirrors face each other and are parallel. An object 
between them is distant 15 cm. from one and 20 cm. from the other. 
Find the positions of the first two images in each mirror. 

13. An object 5 cm. long in front of a converging lens has an image 
20 cm. long on a screen 100 cm. from the lens. What is the focal length 
of the lens ? 

14. At what distance from a converging lens of 30 cm. focal length 
must an object be placed so that the linear dimensions of the image will 
be four times those of the object? 

15. The focal length of a converging lens is 20 cm., and the distance 
between the object and the screen 100 cm. Where must the lens be 
placed to give a sharp image? 

16. The focal length of a glass lens in air is 20 cm. If the indices of 
refraction of glass and water are f and |, respectively, what is the focal 
length of the lens in water? 



266 LIGHT 

17. A converging lens is placed at a distance of 40 cm. from a lumi- 
nous object and forms an image of it on a screen. When the lens is moved 
50 cm. nearer the screen, another and smaller image is formed. What is 
the focal length of the lens? 

18. A glass beaker is filled with water to a depth of 5.2 cm. A cross 
scratched on the bottom of the beaker inside appears when viewed through 
the water with a microscope to be 1.29 cm. above the bottom. Calculate 
the index of refraction of water. 

19. Find the dispersive power of crown glass and flint glass for the 
lines A , H, and D. 

20. Two parallel walls are 16 ft. apart. Where must a converging 
lens of 3 ft. focal length be placed in order to project on one wall the 
image of an object on the opposite wall? What is the magnification? 

21. A copper cent is 19 mm. in diameter and a silver half dollar 
30.4 mm. At what distance from a converging lens, whose focal length 
is 10 cm., must a cent be placed so that its image shall be just the size 
of a silver half dollar? 

22. If the velocity of light in air is 300,000 km. per second, what is it in 
water, in flint glass, and in diamond (indices of refraction, 1.332, 1.65, 
and 2.47, respectively) ? 

23. Compute the critical angle for diamond, if the index of refraction 
is 2.47. 

24. If the apparent depth of a fish below the surface of still water is 
3 ft., what is its real distance? 

25. A plano-convex lens, radius of curvature 15 cm., index of refrac- 
tion 1.5, has at the center of the plane side a cross scratched. ' The thick- 
ness of the lens at the middle is 2 cm. If it lies with the flat side on 
white paper, how far down in the glass will the cross appear to an 
observer looking down along the axis ? 



CHAPTER IX 

LIGHT AS A WAVE MOTION 

I. INTERFERENCE AND DIFFRACTION 

301. Colors of Thin Films. Up to this point it has been 
assumed that light is a wave motion in a hypothetical 
medium called the ether. We are now to consider phe- 
nomena which do not appear to admit of explanation on 
any other theory. The phenomena of diffraction in light 
not only answer Newton's objection to the wave theory, but 
as developed by Young and Fresnel become the strongest 
evidence in its favor. 

If light consists of waves in an appropriate medium, it 
may be anticipated that interference phenomena will be 
observed similar to those in sound (243-246). In fact 
thin films of transparent substances, such as a soap bubble, 
a layer of oil on water, a coating of varnish on white card- 
board, a film of oxide on polished or molten metal, and a 
thin layer of air between good reflecting surfaces, all exhibit 
beautiful colors by reflection in white light; and in light of 
one color, alternate bright and dark bands or rings analogous 
to beats in sound. These iridescent colors are the residuals 
of white light after waves of 
definite length have been cut out 
by interference between the two A 
wave systems reflected from the B 

Fig 194 

parallel surfaces of the film. 

Let AA and BB (Fig. 194) be the two surfaces of the film. 
Light incident on the first surface is in part reflected and in 
part transmitted. The transmitted portion is in part reflected 

267 



268 LIGHT 

from the second surface, and it emerges from the first along 
with light externally reflected at the point of emergence. 

If now the additional path through the film traveled by 
the internally reflected light is a whole wave length, then 
the two systems will be in step unless a phase difference is 
introduced in some other way. If the difference of phase 
depends only on difference in path traveled, then when this 
latter difference vanishes with a film of infinitesimal thick- 
ness, the two pencils of light should mutually support each 
other, and the illumination should be a maximum. But the 
fact is that when a film is made as thin as possible, it becomes 
black. The light is extinguished by interierence. This 
happens for all wave lengths. 

The retardation of the one system with respect to the other 
by half a wave length occurs at the boundaries of the thin 
film. One of the two interfering systems loses half an un- 
dulation relative to the other because one is reflected in the 
rare medium next to the dense, and the other in the dense 
medium next to the rare. This change in phase is analogous 
to that of a sound wave reflected from the open end of an or- 
gan pipe. When a sound wave is reflected from the closed 
end of a pipe, there is a change in sign of the motion, but a 
condensation is reflected as a condensation ; when the reflec- 
tion is from the end of an open pipe, there is no change in 
sign of the motion, but the condensation changes sign, and 
is reflected as a rarefaction. So two pencils of light reflected 
under the corresponding opposite conditions have impressed 
upon them a difference of phase equal to half a period. The 
constraint of the ether in dense matter and its relative free- 
dom in space give rise to an effect similar to the phase dif- 
ference impressed upon two sound impulses when reflected 
respectively from the closed end of a pipe and from the free 
air at the open end. 

When the thickness of the film and the angle of incidence 
are such that one pencil falls behind the other in transmis- 
sion by a whole number of wave lengths, interference takes 



INTERFERENCE AND DIFFRACTION 



269 



place with extinction of light, since a phase difference of 
half a period must be added because the reflection at the two 
surfaces of the film is under opposite conditions. 

With white light the extinction of one spectral color by 
interference leaves colored fringes. Further, the thickness 
of film that impresses upon the internally reflected pencil 
a retardation of one wave length for violet produces a 
retardation of only about half a wave length for red ( 307). 
Therefore extinction of both colors cannot occur at the same 
part of the film. If the violet is cut out, the red remains. 
The case is similar with the intermediate colors. The 
reflected light is, therefore, fringed with color. 

302. Newton's Rings. Sir Isaac Newton ingeniously de- 
termined the relation between the colors given by a thin 
film of air and its thickness by means 
of a film between two pieces of 
glass, one flat and the other slightly 
curved. The curved surface is a 
convex lens of great radius of curva- 
ture. If this radius and the distance of any point from the 
point of contact (Fig. 195) are measured, the thickness 

of the film at the point is readily 
calculated. 

Between the lens and the plate 
there is a very thin film of air, in- 
creasing in thickness outward from 
the point of contact. At the contact 
there is a dark central spot, and 
around this are. concentric colored 
rings or "fringes" (Fig. 196). 
When the light is incident normally, 
the rings are circular; when the incident is oblique, the 
rings are elliptical. If the two pieces of glass are forcibly 
pressed together, the colored rings expand and the distortion 
of the glass is sufficient to distort the rings also. 





Fig 196 



270 



LIGHT 



When the illumination is by red light, the rings are dark 
and bright in succession, and their diameter is larger than if 
the light were blue, indicating that the waves of red light 
are longer than those of blue. A determination of the wave 
length of any spectral color may be made by measuring the 
diameter of a ring of that color and the curvature of the lens. 
This method is inferior to others now in common use. The 
wave length of yellow or D light is about 0.000059 cm. or 
1/43000 in. 

303. Transmission of Waves through Narrow Apertures. 
When a beam of sunlight is admitted into a darkened room 
through a very narrow slit, and is received on a white screen 
at some distance, there will be a central band of white light 
on the screen in the direct path of the beam, bordered with 
colored fringes. Through so narrow an opening light not 
only passes as a definite pencil, but it also diverges from all 
points of the opening as new centers of disturbance (Huy- 
g-hens). The colored bands are due to interference of the 
diverging secondary waves, and they are called diffraction 
fringes. 

Diffraction may be explained as a phenomenon of wave 
motion and applicable to both sound and light. Let ab 
(Fig. 197) be the width of a narrow aperture, through which 




Fig. 197 



come plane waves of one wave length X. When such a 
wave reaches #5, all points along the line ab are in the 
same phase of vibration. The lines ad and If are the bound- 
aries of the geometrical pencil or wave train. 

Now assume that the relation between the wave length and 



INTERFERENCE AND DIFFRACTION 271 

the width db is such that the point 2 , for which the distance 
bs 2 is exactly one wave length greater than as 2 , lies outside 
the geometrical beam df. Then the secondary waves from a 
and b will arrive at s 2 in the same phase, since the retarda- 
tion of the waves from b as compared with those from a is 
exactly one wave length. But the waves from a and c (the 
middle point of db) will arrive at s 2 in opposite phase, since 
cs 2 exceeds as 2 by half of X. They will therefore annul each 
other. 

Similarly for every point between a and c there is another 
between c and b so situated that the difference in their dis- 
tance from 8 2 is half a wave length. Hence all the disturb- 
ances coming from ac completely annul at # 2 the disturbances 
coming from the other half cb of the slit. 

Next consider a point s 4 , the distance of which from b 
exceeds that from a by two wave lengths. Then the dis- 
tance from c to 4 is one wave length greater than the dis- 
tance of a from s 4 , and the conditions for this half of the slit 
are those for complete extinction of the disturbance at 4 as 
just explained. The same is true for the other half cb. 
Therefore there is complete neutralization of all the disturb- 
ances of wave length \ arriving at s 4 from the slit db. The 
same is true for any point, the distances of which from the 
edges of the slit ab differ by any even number of times X/2. 

If the distances of a and b from the point s 3 differ by three 
halves X, then the slit may be divided into three equal parts, 
and the secondary waves from two adjacent ones of these will 
interfere at S B in the manner explained, while those from the 
third part alone are effective. Another point s & is so situ- 
ated that its distance from b exceeds its distance from a by 
five half wave lengths. Secondary waves from four of the 
five equal divisions of the slit then interfere in pairs at 6 , 
leaving only one fifth of the slit to send effective waves to 
6 . These equal divisions of the slit are called half period 
elements. 

In general if bs as on either side of 8 1 is an even number 



272 LIGHT 

of half wave lengths, there is an even number of half period 
elements in ab, the secondary waves from which interfere in 
pairs at s ; but if bs as is an odd number of half wave 
lengths, then ab may be divided into an odd number of half 
period elements, and the surviving secondary waves from only 
one of these arrive at s. With monochromatic light, bright 
and dark bands alternate on either side of s v The position 
of these bands is given by the equation 

bs - as = w|, (67) 

where n is even for the dark and odd for the bright bands. 
When the slit is illuminated by white light, in which X is 
different for the several spectral colors, extinction for these 
colors will occur at different distances from s v and therefore 
the fringes on the screen are colored. 

304. Conditions for Diffraction Phenomena. The resulting 
disturbance is a minimum at s 2 > 8 & etc., and a maximum at 
s 3 , a 5 , etc. It is obvious that the successive distances between 
maxima decrease outward ; moreover, these maxima them- 
selves decrease rapidly in intensity, for s 3 receives secondary 
waves from one third of the slit as a half period element, s 5 
from only one fifth, etc. Hence at no great distance on either 
side of Sj there is a region which no disturbance reaches. 
The width of the area within which diffraction phenomena 
may be observed is determined by the relation between the 
wave length of the disturbance and the width of the opening 
ab. It will readily be seen that if the width of the opening 
is many times the wave length, the points ,<? 3 , 6 , etc., for the 
chief maxima may all lie within the geometrical beam be- 
tween ad and If (Fig. 197). When, therefore, the wave 
length is very small in comparison with the width of the 
opening, the wave motion is propagated in straight lines 
through the opening and does not diverge appreciably into 
the geometrical shadow. This is the condition for sharp 
shadows for both sound and light. 



INTERFERENCE AND DIFFRACTION 273 

Again, if the opening through which the waves come is 
less than a wave length in width, there is no point such as 
S 2 , the distance of which from one edge of the aperture is a 
whole wave length greater than from the other edge. Under 
this condition there is no region of complete interference, and 
the geometrical outline of the shadow disappears. 

If these conclusions are applied to waves of light and of 
sound, it will at once be apparent why ordinarily sharp 
shadows are observed for light only and not for sound. Since 
the mean wave length for light is only about 0.00005 cm., 
the diffraction bands for the usual openings all lie so near 
the edge of the geometrical beam that they are indistinguish- 
able from it. They can be observed only when the opening 
is very narrow. 

On the other hand, sound waves of mean pitch have a 
length of about 125 cm., or 4 feet. Hence ordinary openings, 
such as a window, have a width comparable with the wave 
length, and the secondary waves diverge in all directions 
beyond the opening, or there is no well-defined sound shadow. 
The particular phenomena of diffraction in sound were not 
recognized until after their discovery in the case of light. 
(Consult 211.) 

305. The Diffraction Grating. The phenomena of diffrac- 
tion with a single slit are not readily observed. They are 
best shown by a diffraction grating, which was devised by 
Fraunhofer nearly a century ago. 

A diffraction grating consists of a surface containing a very 
large number of equidistant parallel lines for the transmission 
or reflection of light. A transmission grating is made by 
cutting the lines on a plate of glass with a diamond point by 
means of a dividing engine. The light passes through the 
transparent spaces between the ruled lines. A good grating 
has 1000 to 5000 lines to the centimeter. Reflection gratings 
are made by ruling the lines on a polished surface of specu- 
lum metal. 



274 



LIGHT 



Let AS (Fig. 198) denote the enlarged cross section of a 
transparent grating, a, &, c, etc., the transparent lines. Sup- 
pose a parallel beam of monochromatic light of wave length 
X incident normally on the grating. If the grating were 




Fig. 198 



absent, such a train of plane waves would be brought to a 
focus by the lens L at its principal focal point/. The inter- 
position of the grating has no other effect on this image than 
to reduce its intensity. It does serve, however, to introduce 
other images which are not present before it is interposed. 
The explanation is as follows : 

Each opening, a, , etc., becomes a new source of secondary 
waves, and a surface may be drawn through a and so inclined 
to AB that it will touch all these secondary waves at points 
having the same phase of vibration. Thus the distance of b 
from a is one wave length X ; of e?, 2 X ; of df, 3 X ; etc.; and 
all the waves from the parallel apertures of the grating 
arrive at aC in the same phase. Hence aO is another plane 
wave which the lens will bring to a focus at/j on a secondary 
axis. 

Similarly other lines may be drawn at such angles that be 
will equal 2 X, 3 X, etc. ; and for each position the line will 
be a wave front for all the secondary waves arriving from 
the apertures of the grating, and this resultant wave will be 
brought to a focus in the focal plane of the lens. 

For any inclination of a making be greater or less than an 
exact multiple of X, a O is not a wave front and the disturb- 



INTERFERENCE AND DIFFRACTION 275 

ances from the openings of the grating arrive at the corre- 
sponding focus out of phase and suffer interference. For 
example, suppose be to be less than \ by one per cent. Then 
the waves from the fifty-first opening arrive at a half a 
wave length ahead of those from the second ; those from the 
fifty-second, half a wave length ahead of those from the third; 
etc. Hence at the focus there is complete extinction of the 
light by interference. Between / and f v therefore, there is 
a black band; and between the successive colored images, 
called images of the first order, second order, and so on. are 
dark spaces. 

A Aeries of nearjyjjojiidistant images of a distant source of light may 
then be^oHamed by means of a grating and a lens. While such images 
are most perfectly produced by precise artificial means, still they are by 
no means lacking in nature. The colors of changeable silk, of the 
feathers of some birds, and of mother-of-pearl are imperfect replicas of 
those given by a reflection grating. The colored rays which may be seen 
when the sun near the horizon shines through the foliage of trees a mile 
away are diffraction fringes. When a small distant source of light, an 
open electric arc for example, is viewed through a semitransparent screen 
of regular structure, such as a linen handkerchief or a silk umbrella held 
close to the eye, two series of images at right angles to each other may 
be seen. There are two series because there are two sets of cross threads 
in the fabric. In fact the eyebrows or fine feathers may serve as im- 
perfect gratings to yield distinct images of a bright light at a distance. 

306. Measurement of Wave Lengths by a Grating. If Oi is the 

angle between the grating and the new wave front aC, which forms the 
image of the first order, that is, the angle between the direction of the 
incident light and that of the first image (Fig. 198), and if d is the dis- 
tance ab from center to center of the openings in the grating, then the 
triangle abe gives the relation 

be = \ = d sin r (68) 

For the second image 2 A. = d sin 0%. 

From equation (68) it is obvious that the longest waves are found at 
the greatest deviations. For the first image, for which O l is small, the 
wave lengths are nearly proportional to the deviations. 

The procedure for measuring wave lengths by a spectrometer fitted 
with a plane grating may be described by reference to Fig. 199. At the 



276 



LIGHT 



outer end of the collimating tele- 
scope L is a narrow adjustable slit 
n, which must be strongly illumi- 
nated by the D light of burning 
sodium, or the red light of burning 
lithium, for example. The lens at 
the other end of this telescope con- 
verts the divergent waves from the 
slit into the plane waves of a par- 
allel beam. This beam is incident 
normally on the grating at M, the 
lines of which are parallel to the slit. 
The view telescope jP is first set 
so that its cross wire coincides with 
the first colored image of the slit 
given by diffraction to the right of 
the line nm, and the reading on the 
circular, scale is noted. The tele- 
scope is then shifted until its cross 
wire coincides with the first image 
to the left of nm, and the reading 
is again taken. Half the angle be- 
tween the two sighted positions of 
the view telescope, equal to half the 
difference between the two readings 
on the circular scale, is the angle & r 
The grating space d may be ob- 
tained from the maker of the grat- 
ing, or it may be measured directly by a micrometer microscope. 
To illustrate by an example. 

Reading to the right 72 41' 

Reading to the left 55 28' 

Therefore O l = 8 36.5'. 

The distance d for the grating was 0.000446 cm. Substituting in for- 
mula (68), A = 0.000446 x 0.149679 = 0.0000667 cm. 

307. Dispersion by a Grating. When a grating is illumi- 
nated by white light instead of monochromatic light, the suc- 
cession of images of one color gives place to several colored 
bands on both sides of the central white image of the source. 
The white light undergoes dispersion, the colors in each 




Fig. 199 



INTERFERENCE AND DIFFRACTION 277 

band appearing in the order of their wave lengths, the violet 
nearest the central image and the red farthest away. Such 
a spectrum produced by a grating is called a normal spectrum ; 
the dispersion is due to diffraction and interference. 

It will be recalled that prismatic dispersion is due to the 
different velocities with which the several colors are trans- 
mitted through transparent media ( 294). Dispersion by a 
grating is due to diffraction and is dependent entirely on 
wave length. It is obvious from equation (68) that there is 
a different value of the deviation for each different wave 
length. Now since white light is composed of all the spectral 
colors, or the vibrations of all wave lengths which excite 
vision, it follows that when white light is incident on a 
grating, it will form a succession of colored images, one for 
each wave length, and these will shade off from one to the 
next in the order of the spectral colors. Violet is the psy- 
chological response to the shortest waves which excite the 
sensation of color, and red to the longest waves. 

The spectrum produced by a grating is said to be normal 
because, when 6 in the first spectral band is small it is pro- 
portional to \ (68), or the angular deviation of each color 
from the direction of the incident light is directly propor- 
tional to its wave length. 

A pure spectrum is one in which the colors do not over- 
lap. The spectrum of the first order is the only pure grat- 
ing spectrum. The wave length of extreme red is about 
0.00076 mm. ; of extreme violet, 0.00039 mm. Then 

For the red of the 2d order d sin = 2 x 0.00076 

For the violet of the 3d order d sin & = 3 x 0.00039 

Therefore d sin 6 > d sin 6', 

or the red of the second order overlaps the violet of the 
third. Grating spectra are limited to the first three orders, 
the overlapping in higher orders being sufficient to reproduce 
white light. 



278 



LIGHT 



The mean of the wave lengths for extreme red and extreme 
violet is 0.00054 mm. This is the wave length of the yellow 
between the Fraunhofer lines D and U. Yellow is therefore 
at the middle of all grating spectra, while in prismatic spectra 
it lies much nearer the red end. Figure 200 represents two. 




Fig. 200 

spectra of equal length, the upper one a grating spectrum 
and the lower a spectrum given by a flint glass prism. It 
will be noted that the D of the one corresponds very nearly 
with the F of the other ; also, that in the grating spectrum 
the red end is relatively more extended than in the prismatic 
spectrum, while the violet end is less. 

308. Wave Lengths and Frequencies of Vibration. The unit 

commonly employed in measuring wave lengths of light is the tenth- 
meter, of which 10 10 are required to make a meter. The following are 
the values for the principal Fraunhofer lines in air at 20 C. and 760 mm. 
pressure : 



A 7621.31 

B 6870.18 

C 6563.07 

D l 5896.18 

D 2 5890.22 



E l 5270.52 

E 2 5269.84 

F 4861.51 

G 4340.63 

#i 3968.62 



Taking the velocity of light as 300 million meters a second, or 300 x 10 18 
tem.h-meters, the frequencies of vibration corresponding to the above 
spectral lines may be found by dividing this velocity by the several wave 
lengths, since n = V/\. The following are the results : 



E, 


. . 569.2 x 10 12 


**l 


. . 569.3 x 10 12 


F 


. . 617.1 x 10 12 


G 


691 1 x 10 12 


H, 


756.0 x 10 12 



EMISSION AND ABSORPTION OF RADIATION 279 



A 393.6 x 10 12 

B 436.7 x 10 12 

C 457.1 x 10 12 

X 508.8 x 10 12 

D 9 509.3 x 10 12 



Thus the light entering the eye and producing the color of violet rep- 
resented by the line H l is due to 756 millions of millions of vibrations a 
second. A photograph of the sun has been taken with an exposure of 
only one twenty-thousandth of a second. During this short period a 
beam of light 15,000 meters (9.32 miles) in length enters the camera, 
and fully 375 x 10 8 or 37,500 millions of waves of violet light make their 
impression on the sensitized plate. 



II. EMISSION AND ABSORPTION OF RADIATION 

309. Types of Spectra. The methods of analyzing radiation 
by the dispersion produced either by a prism or a diffraction 
grating have already been described. The spectra from dif- 
ferent sources obtained by these methods fall into two general 
classes : 

A. Continuous Spectra. The radiation from a hot solid 
or liquid forms a continuous spectrum without any interrup- 
tions in the succession of colors and wave lengths from one 
extreme of visibility to the other. This is true however 
narrow the slit and pure the spectrum. Light emitted by a 
white-hot solid or liquid is composed of radiations of all pos- 
sible wave lengths between red and extreme violet. Such 
spectra are not characteristic of the substances producing 
them, but only of their temperatures. 

A continuous spectrum is given by the white-hot positive 
carbon of an electric arc lamp, by all gas and candle flames, 
and by molten metals. The light emitted by flames of burn- 
ing carbon compounds comes from the minutely divided solid 
carbon set free by the combustion of the gas or vapor and 
raised to incandescence by the intense heat of the flame. 

All bodies when heated begin to glow at about the same 
temperature in the neighborhood of 400. They become 
visible in the dark at first as a dull red. As the temperature 



280 LIGHT 

rises other colors are added to their spectrum in the order of 
wave lengths, violet appearing only at the highest tempera- 
ture. Conversely, when the continuous spectrum of the 
radiation from the positive carbon of an arc light is projected 
on a white screen, and the current is then cut off, as the car- 
bon cools, the violet, blue, green, yellow, and red disappear 
in succession. 

It has already been pointed out that not all self-luminous 
bodies are hot. Attention has been directed to the fact that 
there are many cases of faintly self-luminous bodies at low 
temperatures ( 254). Certain glowworms, fireflies, bacteria, 
and beetles emit a faint light, the wave lengths of which in 
general all fall within the limits of the visible spectrum. 
The luminous efficiency of this light is therefore very high as 
compared with artificial sources of illumination. 

B. Discontinuous Spectra. The spectra of incandescent 
gases and .vapors consist of a limited number of separate 
bright lines, each of which is an image in one color of the 
slit. These are called bright-line spectra to distinguish them 
from the discontinuous spectra crossed with dark Fraunhofer 
lines, such as those of the sun and fixed stars. 

Bright-line spectra are produced by volatilizing metallic 
salts in a Bunsen flame, by sending electric discharges through 
rarefied gases in glass tubes, or between metallic electrodes 
in the air. The characteristic spectrum of the luminous va- 
por of sodium is the yellow line D. With a spectroscope of 
sufficient resolving power this one line is resolved into two 
very near together, and each of these is found to be double. 
No two vapors give rise to the same series of bright-line 
images. The number varies greatly, ranging from about ten 
in the case of the alkali metals to several thousand in the 
spectra of iron and uranium. 

The lower spectrum in Figure 201 (from the Lick Observa- 
tory Bulletin, No. 62) is the bright-line spectrum of hydro- 
gen in the G-H region; the upper one is the bright-line 
spectrum of iron vapor in the same region. 



EMISSION AND ABSORPTION OF RADIATION 281 

The fact that all gases and vapors give discontinuous 
spectra was definitely established by Bunsen and Kirchhoff in 
1860. By means of this differentiation between the spectra 




of different metallic vapors, they discovered the rare metals 
rubidium and caesium. The same method of spectrum anal- 
ysis has been applied in recent times to the discovery of 
helium and some rare atmospheric gases. 

Bright spectral lines differ widely in width and intensity. 
This may be true even of lines due to the same element. 
Wide lines are not really monochromatic, nor are even the 
narrow ones strictly so. It has been shown by means of 
instruments of extreme resolving power that some apparently 
narrow lines are composed of a number of still narrower 
components, which ordinarily overlap. 

310. Absorption Spectra. A pure solar spectrum, crossed 
by the dark Fraunhofer lines, is a discontinuous spectrum, 
but it belongs to a subclass known as dark-line or absorption 
spectra. These are made discontinuous by losses due to 
absorption of radiation in the passage through transparent 
media. The absorption producing the dark lines of the 
solar spectrum takes place chiefly in the outer envelope of 
the sun's atmosphere. The incandescent photosphere of the 
sun, which would by itself present a continuous spectrum, is 
surrounded by a mass of gases and vapors through which the 
radiation from the photosphere must pass. Absorption takes 
place in this reversing layer ; and the dark lines, which are 
only relatively dark in comparison with the adjacent brighter 



282 LIGHT 

portions of the spectrum, are the inversion of those luminous 
radiations which form the emission spectra of the gases and 
vapors in the sun's outer envelope. 

The principle of absorption is the same as that of reso- 
nance or co-vibration in sound. Every gas or vapor when 
white-hot emits radiations of the same wave length as those 
which it absorbs from an independent source when at a lower 
temperature. Thus the D lines of the solar spectrum coin- 
cide exactly with the bright lines given by sodium vapor in 
a state of incandescence. Not only has the coincidence been 
established between the Fraunhofer D lines and the yellow 
lines of luminous sodium vapor, but the reversal of these lines 
by sodium vapor as the absorbing agent has been accom- 
plished. These results laid the foundation for the science of 
spectrum analysis, by which the approximate composition of 
self-luminous celestial bodies has been determined. 

The presence of hydrogen and iron in the sun's atmosphere 
is demonstrated by the correspondence between the bright 
lines composing their spectra and dark absorption lines in 
the middle spectrum of Figure 201, which is that of the sun. 

Kirchhoff established in 1860 the following law of spectrum 
analysis when the radiation is a pure temperature effect : 

The relation between the emissive power and the absorbing 
power, for any definite color or wave length, is the same for 
all bodies at the same temperature. If light from a lumi- 
nous vapor at a higher temperature traverses the same vapor at 
a lower temperature, the light absorbed in the latter is greater 
than the light emitted by it. The result is relatively dark 
lines, or a reversal. 

The reversal of the sodium line may readily be shown by 
means of a direct current arc lamp and a hollow prism filled 
with carbon disulphide. The positive carbon should be larger 
than the negative and should be placed below. In a cup- 
shaped cavity in its upper end is placed a small piece of 
metallic sodium. With a rather wide slit in front of the 
lantern condenser, the light of the burning sodium when the 



EMISSION AND ABSORPTION OF RADIATION 283 

current is turned on gives a broad yellow band on the 
screen. When the sodium vapor in the arc is most copious, 
the middle of the yellow band turns dark on account of 
absorption by the outer and cooler layer of sodium vapor 
surrounding the arc. This phenomenon is known as self- 
reversal by absorption. It is especially pronounced in 




Fig. 20& 

spectra given by the electric arc. Figure 202 shows the self- 
reversal in a portion of the arc spectrum of iron given by a 
grating. The fine dark lines in the middle of the bright 
bands are due to self-reversal. 

311. Infra-red and Ultra-violet Radiation. The radiation 
affecting the eye and forming the visible spectrum lies be- 
tween wave lengths of about 7500 and 3900 tenth-meters. 
The corresponding range of vibration frequencies is there- 
fore a little less than would be called an octave in sound. 

Besides these waves which give rise to sensations of color 
through the eye, there are many others which are entirely 
invisible, and yet possess all the physical characters of light 
waves. They may be reflected, refracted, diffracted, and 
polarized the same as waves exciting vision. Physically they 
differ only in wave length from those giving rise to optical 



284 LIGHT 

perception. The eye with its limited range of sensibility is 
not directly affected by them; but when other tests for 
detecting and registering waves are applied, there is no line of 
demarcation between the visible and the invisible radiations. 
The invisible portion of the solar spectrum is crossed by ab- 
sorption lines precisely as in the visible part. (See Fig. 188.) 

The radiation beyond the red end of the visible spectrum 
and of wave length greater than about 7500 is known as the 
infra-red; that beyond the extreme violet and of wave length 
shorter than about 3900, as ultra-violet. Both the infra-red 
and the ultra-violet radiations must be investigated by other 
physical effects than sight. 

To explore the infra-red end of the spectrum resort is had 
to the heating effect of radiation. Explorations by means of 
the thermopile (573) and by Professor Langley's bolometer 
show that the region of maximum heating effect is a little 
way into the infra-red region, and at wave lengths about 
9500 to 10,000 tenth-meters. Langley measured lunar 
radiation having a wave length of 170,000 tenth-meters 
(0.017 mm.), or more than twenty times as long as the long- 
est waves exciting vision. Professor Rubens has measured 
infra-red waves as long as 240,000 tenth-meters (0.024 mm.). 
These waves were thirty-two times as long as the longest 
ones affecting the eye as extreme red. They extend five 
optical octaves below the visible spectrum. 

Ultra-violet radiation is especially effective in producing 
photographic action, particularly with silver salts. Photog- 
raphy is therefore the most satisfactory method of exploring 
the ultra-violet end of the spectrum. It cannot be employed 
for the infra-red because it is difficult to make a film that is 
sensitive as far down even as the visible red extends. Row- 
land succeeded in photographing from wave length 7000 
tenth-meters in the red to 3000 in the ultra-violet. Ultra- 
violet radiation has been photographed to wave length 2020, 
or -to a frequency about twice as great as that of extreme 
violet light. By other means it has been detected to wave 



EMISSION AND ABSORPTION OF RADIATION 285 



length 1000 tenth-meters, or to two optical 
octaves above the visible spectrum. 

Figure 203 represents the entire spectrum from 
extreme infra-red up to wave length 3000 in the 
ultra-violet, the limit of Rowland's photographs. 
It is drawn to an even scale of frequencies instead 
of wave lengths. The extreme ultra-violet detected 
at wave length 1000 tenth-meters, or 100 micro- 
millimeters, with a frequency of 3000 billions of 
vibrations a second, lies twice the length of Figure 
203 beyond its short wave length end. A scale of 
even wave lengths would make a spectrum more 
than sixty times the length of the visible spectrum. 

Only a few of the principal absorption 
lines are represented in Figure 203. The 
last two dotted lines in the infra-red mark 
the limit reached by Langley and Rubens, 
respectively. 

312. Reemission of Absorbed Radiation. 

In general it may be said that a part of 
the radiation absorbed by a body is again 
emitted, but always with a change of wave 
length. The reemitted radiation is not 
due to high temperature and does not 
follow Kirchhoff s law. The reemission as 
waves too long to affect the retina by a 
black body absorbing light and transform- 
ing it into heat is a familiar case in point. 
In many cases the transformed radiation 
falls within the range of visual perception. 
The phenomenon is then called lumines- 
cence, a name applied by Wiedemann to 
Cover all cases of visible radiation not 
directly due to high temperature. 

Wiedemann recognized eight causes of 
luminescence, including radiation, chemi- 





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286 LIGHT 

cal action, heat, friction, and X-rays. Attention is directed 
here chiefly to the first of these. 

Many substances may be stimulated into giving out light 
by exposing them to ultra-violet radiation. In all such cases 
the wave length emitted is longer than that of the stimulat- 
ing radiation. Photo-luminescence, or luminescence stimu- 
lated by light, is subdivided into two groups, fluorescence 
and phosphorescence, the essential difference between the two 
depending upon the duration of the luminescence after the 
stimulus is withdrawn. 

313. Fluorescence. The emission of light by a body stim- 
ulated by radiation of a different wave length, in so far as 
the emission continues only during the period of stimula- 
tion, is called fluorescence. Fluorspar was the first substance 
in which the phenomenon was noticed; hence the name 
fluorescence. 

Many substances exhibit this property. Uranium glass, 
solutions of the sulphate of quinine, eosin (a product of coal- 
tar distillation), and petroleum possess the property of fluo- 
rescing in varying degrees. When a beam of sunlight passes 
through a green solution of chlorophyll, its path is marked 
by a bright red streak. Uranium glass is yellow, but it has 
a beautiful green surface tint. Petroleum when strongly 
illuminated and viewed obliquely shows a splendid blue 
color. 

Most of these substances have the property of absorbing 
ultra-violet radiation and transforming it into longer waves 
visible to the eye. When a beam of light from an electric 
lantern is passed through a dark violet glass, it loses nearly 
all visible light; but if it is then directed upon a block of 
uranium glass, the invisible waves are transmuted into a bril- 
liant green, and the glass stands out vividly in the darkness. 
A vessel of kerosene oil in the ultra-violet beam appears 
azure blue. These substances shine most brilliantly in ultra- 
violet light just beyond the violet end of the visible spectrum. 



EMISSION AND ABSORPTION OF RADIATION 287 

Quinine in solution, on the other hand, possesses the property 
of converting long infra-red waves into shorter ones visible 
to the eye. This and a few other aqueous solutions are excep- 
tions to the general law that reemission increases the wave 
length of the radiation. 

It is possible by means of fluorescence to photograph invis- 
ible objects. Thus, an inscription written on white drawing 
paper with colorless sulphate of quinine dissolved in a solution 
of citric acid is invisible in white light. This substance 
fluoresces and so transmutes the highly active ultra-violet 
radiation into longer waves, 
which do not act so vig- 
orously on a photographic 
plate. When therefore such 
a sheet is illuminated by 
means of the arc lamp and 
is photographed, the parts 
to which the sulphate of 
quinine has been applied 
come out darker than the 
untouched surface of the 

paper (Fig. 204). With one or two sheets of blue glass in 
front of the lantern, the inscription written with the sulphate 
of quinine solution stands out in nearly white letters on a 
blue ground. 

314. Phosphorescence. In fluorescence the emission ceases 
as soon as the stimulating radiation ceases to fall on the sub- 
stance. But many substances, notably the sulphides of cal- 
cium, of barium, and of strontium, continue to emit light for 
appreciable periods after stimulation is withdrawn. Bodies 
which thus store radiant energy and continue to give it out 
as light afterwards are said to be phosphorescent. The name 
is derived from the similar appearance of phosphorus during 
slow oxidation. Some diamonds have this property, but 
sulphide of calcium, especially when mixed with small quan- 




288 LIGHT 

titles of other substances, exhibits this property in a most 
extraordinary degree. Balmain's paint is a preparation 
mainly of sulphide of calcium and a trace of bismuth. 
The excitation due to sunlight during the day enables 
it to give out light all night without losing its whole 
store of luminous energy. Even after having been kept 
in darkness for many months, a sheet of luminous paint 
may give off enough invisible radiation to fog a photo- 
graphic plate. 

The warming of a sheet of luminous paint causes it to shine 
more brightly. Conversely, chilling it dims its light. On 
the other hand Professor Dewar has discovered that many 
substances acquire the property of phosphorescence only 
when cooled in liquid air to a temperature of about 200 C. 
below zero. Such are gelatine, horn, paper, ivory, and egg 
shell. At this low temperature they absorb radiant energy 
and store it for emission as light when warmed. 

A curious instance of phosphorescence is afforded by the termite or 
white ant hills in the region of the Amazon. These hills are from five 
to ten feet high, are made of hard clay, and are bare of vegetation. In 
the night'they glow and scintillate with a shifting phosphorescent light, 
which is visible at a distance when the night is dark. The weight of 
opinion appears to be that the phosphorescence is in the hills themselves 
and not in the insects. 

315. Applications of Doppler's Principle. The Doppler 
effect in light ( 214) is the apparent change in wave length 
produced by the relative motion of the observer and the 
source in the line of sight. When the observer and the 
source of light are approaching each other, the apparent 
wave length is shortened, and vice versa. This apparent 
shortening or lengthening of the waves gives rise in discon- 
tinuous spectra to the shifting of the lines toward the violet 
end or the red end of the spectrum, respectively. Since the 
velocity of light is so very great, a large velocity of approach 
or- recession is necessary in order to produce a measurable 
shift in the position of spectral lines. 









EMISSION AND ABSORPTION OF RADIATION 289 

Referring to 214, the apparent wave length is 



n 



Whence 



v 



n(\-\') = - 

A. 



where A\ is the apparent change in wave length, and v the 
relative velocity of approach or recession. It is easy to cal- 
culate what this velocity must be to produce any assumed 
change in apparent wave length, 0.1 of a unit in tenth- 
meters, for example. Take V, the velocity of light, as 
300,000 km. a second, and \ as the wave length of the 
F line of hydrogen, 4861 tenth-meters. Then 



v = 



300000000 



x 0.1 x 10- 10 = 6170 m. = 6 + km. 



4861 
per second. 

The linear velocity of rotation of a point on the sun's 
equator is 2 km. a second. The difference in velocity of 
\ points at the two ends of the solar equator with respect to 
{ an observer is then 4 km. a second. Vogel, in 1871, suc- 
ceeded in detecting the displacement of Fraunhofer lines 
by comparing the spectra from the two limbs of the sun, 
v and Young determined the sun's period of rotation by the 
same method. 

Some of the dark lines in the spectra of sun spots are not 
infrequently broken and greatly distorted. This distortion 
is due to the swift motion 
of matter, especially hydro- 
gen, in the line of sight. 
The broken lines of Figure 
205 are three views of a 
hydrogen line taken at in- 
tervals of about four min- 
utes. In this particular case the gas was projected outward 
toward the observer with a velocity of about 480 km. (300 
miles) a second. 



Fig. 205 



290 LIGHT 

By the refined measurement of the displacement of Fraun- 
hofer lines in the photographs of stellar spectra, the veloc- 
ities of stars in the line of sight may be determined with an 
error not exceeding one tenth km. per second. The investi- 
gation of a great many stars shows that most of those in one 
direction in the heavens have an apparent motion toward 
the sun, and those in the opposite direction an apparent 
motion away from the sun. The inference is that the solar 
system is moving through space in the other direction. 

316. Spectroscopic Binary Stars. One of the most interest- 
ing applications of the Doppler principle is the investigation 
of very close double stars, which are entirely unresolvable by 
the most powerful telescopes. These are known as spectro- 
scopic binary stars. 

When light comes from a double star, both members of 
which are self-luminous, the Fraunhofer lines in the stellar 
spectrum become double, separate to a maximum distance, 
and then come together again. The complete cycle of these 
changes takes place in the period of revolution of the com- 
ponents of the double star about their common center of 
mass. 

The lines in the spectrum of such double stars appear sin- 
gle when one member is behind the other in the line of sight ; 
but when the two components of the binary system are side 
by side, one is approaching the observer and the other re- 
ceding from him. Any given Fraunhofer line coming from 
the two sources is then shifted in opposite directions and 
appears double. 

A more interesting type still is that of spectroscopic bi- 
naries in which one member is dark. These are usually 
variable stars, such as Algol. By means of the displacement 
of lines in its spectrum it was found that Algol varies in its 
velocity in the line of sight, positive and negative motions 
alternating with each other, the period of these motions 
agreeing with the period of the variable luminosity. It has 



EMISSION AND ABSORPTION OF RADIATION 291 

thus been shown that Algol is a binary system composed of 
one bright and one dark member, both rotating around their 
common center of mass. 































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Fig. 206 



Other stars of this type have been investigated. When 
the observations are plotted with times as abscissas and 
radial velocities (velocities in the line of sight) as ordinates, 
the resulting curve is a sine curve. Figure 206 was plotted 





Fig. 207 



from data derived from some of the observations on the star i 
Peyasi made at the Lick Observatory. The curve shows 



292 LIGHT 

that the orbit of this star projected on a plane containing the 
line of sight is almost perfectly circular, and that the star is 
the luminous member of a binary, the other member of which 
is dark. The dotted line denotes the center of mass of the 
system. The period of revolution in the orbit is 10.213 
days, and the diameter of the circular orbit is 13,480,000 km. 
The displacement of stellar lines in the spectrum of this 
spectroscopic binary is shown in Figure 207. The middle 
spectrum is that of the star, and the comparison spectra are 
the bright lines given by titanium. In the upper group 
the star is receding from the observer, and in the lower one 
it is approaching. The stellar lines are displaced in opposite 
directions in the two groups. 

Problems 

1. If the grating space d = 589 x 10~* mm., and 2 = \ 1 32' for D light, 
what is its wave length [equation (68) ] ? 

2. If D light falls normally on the grating (Fig. 198), for which d =' 
589 x 10~ 5 mm., and the screen is 5 m. from the lens L, find the distance 
ff\ for the first bright line image. 

3. If the angle 2 for the blue line of cadmium (A 48 x 10~ 5 mm.) 
is 12, calculate the grating space d (68) . 

4. The velocities of the source in the line of sight were 36 and 43 km. 
per second respectively for the two spectra of Figure 207. What were 
the corresponding displacements of the line X = 4383.7 x 1Q- 10 m., in 
terms of the unit of wave length, 10" 10 m. ? 

5. The maximum computed velocities of i Pegasi in the line of sight 
were 43.7 and - 52.1 km. per second ( 316). Calculate the diameter 
of the orbit of the visible member of this binary on the assumption that 
its orbit is a circle with the line of sight in its plane. 



CHAPTER X 
SENSATIONS OF COLOR 

317. Modes of Producing Color Color itself has no objec- 
tive existence. It is the response^f^e^atipn to the stimulus 
of light. The only physical difference corresponding to 
different simple colors is a difference of wave length. The 
extreme red of the spectrum is a sensation excited by the 
longest ether waves affecting the eye ; extreme violet, by 
the shortest. The production of spectral colors from white 
light involves some process of separating the waves of dif- 
ferent length, so as to get their individual effect. Such a 
separation we know is brought about by dispersion and inter- 
ference. Other modes of producing colors from white light 
are reflection and selective absorption. Then there are other 
sensations of color depending on over-stimulus or fatigue of 
the retina. 

Although each light wave of different length produces a 
definite simple color sensation, it is not true that every mixed 
color sensation is produced only by one set of waves of defi- 
nite frequencies. The combined effect of all the visible 
light waves from a white-hot solid is the sensation of white ; 
but it is also possible to produce the sensation of white light 
by combining only two kinds of waves. For example, the 
superposition of light from the red end of the spectrum and 
that from a region between the green and the blue also pro- 
duces the sensation of white. A similar sensation is excited 
by the combined effect of violet and yellowish green. Any 
two colors, the combined effect of which is the sensation of 
white, are said to be complementary* 

293 



294 LIGHT 

318. Selective Absorption. Substances which absorb some 
radiations and transmit or reflect others are said to exercise 
selective absorption. The absorption of radiation is usually 
selective. Familiar examples occur within the limits of the 
visible spectrum. Thus, red glass colored with the suboxide 
of copper transmits red and absorbs all other visible radiation. 
For this reason " ruby glass " is used in photographic rooms ; 
it does not transmit the radiations which fog a sensitized 
plate. 

Many colored liquids exhibit the same peculiarity of selec- 
tive absorption. Thus amyl alcohol, dyed with aniline red, 
cuts off by absorption every tint except red ; cupric chloride 
dissolved in hydrochloric acid cuts off all except green and 
some blue. 

Many substances are quite transparent within the limits of the visible 
spectrum, but show selective absorption in the infra-red and ultra-violet 
regions. Glass, for example, is opaque to waves shorter than 3500 tenth- 
meters, and longer than about 30,000 tenth-meters. Quartz is trans- 
parent to radiations between wave lengths 1800 and 70,000; rock salt 
between 1800 and 180,000 ; while fluorite transmits the shortest known 
ultra-violet waves, and down to wave length 95,000 in the infra-red. 

A body which shows selective absorption in the infra-red or ultra- 
violet only appears colorless, but in a physical sense it is similar to a 
colored body. If we obtained visual sensations from ultra-violet radia- 
tions, glass, which absorbs ultra-violet strongly, would not appear trans- 
parent and colorless. The insensibility of the eye to ultra-violet light 
appears to be due to the fact that these radiations are absorbed by the 
media of the eye before they reach the retina. 

319. Color of Opaque Bodies. All bodies, except those 
with highly polished surfaces, reflect light by irregular reflec- 
tion, and in most cases the light penetrates more or less into 
the medium before it is diffused. If all the constituents of 
white light are reflected in the same proportion, the body 
appears white or gray. Such is the case with a sheet of 
white paper or a white screen on which the solar spectrum is 
projected. White bodies reflect diffused light in all direc- 
tions, and without preference for light of particular wave 



SENSATIONS OF COLOR 295 

lengths. Hence all the colors of the spectrum on such a 
screen appear the same as when they are received directly 
into the eye placed in the path of the beam diverging from 
the prism. 

If, however, a body exhibits selective absorption for some 
frequencies lying within the visible spectrum, then the light 
reflected from it consists of a mixture of the color components 
of white light which the body reflects. The colors of opaque 
bodies are therefore chiefly the residuals left after absorption. 

The illumination of opaque bodies by colored light is very instructive. 
Project a spectrum of the sun or of the electric arc, with a rather wide 
slit, on a white screen in a dark room, by means of a carbon disulphide 
prism. Select a flower with rich red petals in large masses, such as the 
tulip or certain geraniums, and pass it through the different colors of the 
spectrum on the screen. In the red the flower will shine with its usual 
bright red color; but as it is passed along into the green, it becomes 
black, and shows no power of reflection for the remaining colors of the 
spectrum. All the colors except red are almost completely absorbed. 
The red, on the contrary, is reflected, and gives color to the body. A 
piece of ordinary red flannel is brilliantly red in the less refrangible end 
of the spectrum, but suddenly turns to a dirty brown, and then a dead 
black, when moved away from the red toward the violet. 

It is obvious that the color of a body does not inhere in the body itself, 
for it can exhibit no color not already present in the light which illumi- 
nates it. With homogeneous illumination differences of color are no 
longer possible. This fact is strikingly illustrated by viewing objects of 
various colors in a room lighted only with burning sodium. The most 
healthful face is of an ashen hue, and brilliant flowers are reduced to a 
faded yellow. It requires the white light of the sun, in which innumer- 
able colors are blended, to disclose to our eyes the variegated tints of 
nature. 

320. Color of Transparent Bodies. A body transparent 
to certain radiations affecting the eye, and not to others, 

1 appears colored by transmitted light, and the color is due to 
the mixed impression excited by the transmitted radiations. 
The colors of transparent bodies are due to their power of 
selective absorption. If a piece of blue cobalt glass be inter- 
posed in the solar beam, the spectrum will consist of a small 



296 



LIGHT 



amount of extreme red and all of the indigo-violet (Fig. 
208, 2). A piece of glass, colored red with the suboxide 
of copper, allows only the red and orange-red rays as far as 
the D line to pass through (Fig. 208, 1). All the rest of 
the spectrum is completely stopped by this glass. If now the 
light be passed in succession through the copper red and the 
cobalt blue glasses, the only spectral color surviving the pro- 
cess of double absorption will be the extreme dark red below 
the line jB, which is transmitted by both. 

A solution of potassium bichromate transmits the less refran- 
gible part of the spectrum only up to the Fraunhofer line I. 



if 




Fig. 208 

A solution of the ammoniated oxide of copper transmits only 
the more refrangible part of the spectrum from the b line on 
(Fig. 208, 3 and 4). These two colors, therefore, contain all 
the spectral tints, and are complementary to each other. But 
if the two solutions in flat glass cells be placed in the path of 
a beam of sunlight, one behind the other, the combination 
scarcely permits the passage of any light at all. The light 
that struggles through the one solution is stopped by the 
other. 

If the blue ammoniated copper oxide solution be placed in 
front of a yellow solution of normal potassium chromate of 
the proper density, the light transmitted by the two will be 



SENSATIONS OF COLOR 297 

green. If their separate spectra be examined, it will be 
found that both solutions transmit green. Green is the only 
color common to the two, and the only one not stopped by 
absorption in the one solution or the other. 

321. Surface Color. Some thin metallic films and solid aniline 
dyes reflect one color and transmit another. Thus gold reflects char- 
acteristic yellow light, but a very thin sheet of gold leaf is green by 
transmitted light. The reflected and transmitted colors are com- 
plementary. 

Fuchsine has a superficial metallic green color ; an alcoholic solution 
of it forms a brilliant red dye and transmits no green. In such cases the 
surface color appears to be due to selective reflection at the surface, the 
other colors being transmitted through a thin layer or absorbed by a 
thick one. 

Silver exhibits similar properties if we include the ultra-violet. The 
surface color of silver is gray. A thin film of silver deposited on glass 
transmits nearly white light, and the ultra-violet is transmitted through 
a film so thick that no visible light gets through. Hence, with a 
quartz lens heavily silvered so as to be opaque to spectral colors, photo- 
graphs may be taken by means of the transmitted ultra-violet rays. In 
such a photograph any surface which absorbs ultra-vio]et light, such as a 
surface covered with a fluorescent substance, appears dark or nearly 
black. A polished silver object shows dark because it absorbs the ultra- 
violet rays by which the photograph is taken. 

322. Mixing Colored Lights. For the perception of the 
mixed effect of two or more colored lights, it is necessary 
that they reach the retina either simultaneously or in quick 
succession. Visual impressions persist for a small fraction 
of a second; and, if one remains until after the arrival of 
another, both impressions are present at the same time. 

If two partially overlapping disks of light be projected on 
a screen, and transparent colored bodies be placed in the path 
of the two beams, the light reflected to the eye from the over- 
lapping area will consist of a real mixture of the two colored 
lights. Thus, if the ammoniated copper oxide solution be 
placed in the path of one beam and the potassium chromate 
in the other, the area common to the two disks will be white 
or gray, with the proper density of the two solutions. When 



298 



LIGHT 




Fig. 209 



these colored lights are added, they cannot in any way be 
made to produce green. So the cobalt-blue and the copper- 
red glasses will give beams of light 
which by addition produce white. 

If a disk of cardboard, colored in sections, 
be rapidly rotated (Fig. 209), the result is a 
mixture or a superposition of visual impres- 
sions. But very different mixtures may pro- 
duce the same visual impression. The eye 
has no power of analyzing light into its con- 
stituent colors. It can tell nothing about 
the composition of colored light. Colored 
light must be analyzed by means of a spec- 
troscope armed with a prism or a diffraction 
grating. 

323. Pigment Colors. The effect of 
mixing pigments is not at all the 
same as that of mixing colored lights. 
In the case of pigments, the light reaching the eye is white 
light which has lost some of its components by absorption 
within the pigment which it slightly penetrates. A pigment 
color is the residual left after this absorption, and the mixture 
of two pigments gives rise to a double absorption instead of 
the addition of two colored lights. 

The result of mixing pigments is the same as the passage 
of the light in succession through two colored glasses or 
colored liquids. Blue paint absorbs nearly every color except 
the blue and some green ; yellow paint absorbs all but the 
red, yellow, and green. Therefore with white light incident 
on a mixture of fine yellow and blue pigments, the only color 
that escapes absorption is the green. If the ammoniated 
copper oxide solution and the potassium chromate solution be 
mixed, the transmitted light will be green, the same as when 
the light is transmitted through the two solutions in succession. 

324. The Three Primary Color Sensations. In the theory of 
color vision originated by Thomas Young and later elaborated 




SENSATIONS OF COLOR 299 

by Helmholtz, all color sensations are referred to three simple 
primary sensations of color, which have their counterpart in 
the nerve terminals of the retina. These are the sensation of 
red, the sensation of yellowish green, and the sensation of 
Mue-violet. Red light stimulates but one of these sensations 
in the nerves of the eye. Yellow light stimulates two, the 
red and the green. The recognition of different colors is due 
to the excitation of the three pri- 
mary sensations in varying degrees. 

With three colored circles of red, 
green, and blue- violet light partially 
overlapping on the screen (Fig. 
210), the resulting mixtures are 
readily made out. The overlapping 
of^the red and green gives yellow ; 
of the green and the blue-violet, 
peacock blue ; of the blue- violet 

and the red, purple ; while the overlapping of the three 
colored lights gives white. The tint produced by the over- 
lapping of two of the primary color sensations is comple- 
mentary to the third. 

The theory of the three primary sensations of color is the 
foundation of all the methods of color photography. 

325. Color Blindness. The Young-Helmholtz theory of color sen- 
sation affords a simple explanation of the facts of defective color vision 
known as color blindness. 

A red-blind person is one who is entirely deficient in the nerve fibers 
by which the primary sensation of red is perceived. Only two primary 
color sensations remain for such an abnormal eye. In such cases red 
produces only a moderate stimulation of the nerve terminals giving green, 
and little or no stimulus of the blue-violet. Red therefore appears to such 
a person as a green of low luminosity. To such an eye there is no dis- 
tinction between the green leaves of a cherry tree and the red ripe fruit. 
To a red-blind person the luminosity of the red end of the spectrum is 
much below normal. 

Green blindness and blue blindness also occur, though rarely. To a 
green-blind person equal stimulation of the red and the blue-violet corre- 
sponds to the sensation produced by white. When such a color-blind 



300 LIGHT 

person looks at a spectrum of white light, he sees red at one end, violet at 
the other, and a white band between them. 

By far the commonest form of defective color vision is red blindness. 
It occurs chiefly in the male sex, and affects three or four per cent of the 
community. 

326. Subjective Colors. The theory of primary color sen- 
sation furnishes a ready explanation of colors due to fatigue 
of the retina. It is well known that objects are quite invisi- 
ble to one entering a faintly lighted room after exposure of 
the eyes for some time to bright light. After subjection to 
a strong stimulus the eye loses its sensitiveness to a weak 
one. When one enters from outdoors a room in which sen- 
sitized photographic films are cut and rolled, nothing is at 
first visible except a few dim red lights; but after a short 
time the eye recovers its sensibility; one then sees a room 
filled with machines and the operatives attending to all 
details of the work without difficulty. 

This liability to fatigue is characteristic not only of the 
retina as a whole, but of any portion of it giving one of the 
primary color sensations. Fatigue of the retina causes it to 
lose the power of responding to the stimulus of any color 
long observed; and if the eye is then directed toward a 
dimly illuminated surface, a tint appears complementary to 
the one which has- produced the fatigue. This is one form 
of subjective color. If the eye is fixed for half a minute on 
a colored picture, red for example, in a strong light, and is 
then directed toward a dimly lighted white wall, an image 
of the picture will be seen in green, enlarged if the wall is 
farther from the eye than the picture itself. The eye, 
fatigued for red, still retains its sensitiveness for green. 
Hence, the relatively faint white light is sufficient to stimu- 
late for green, while it is an insufficient stimulus for red. 
The state of the retina serves to suppress the sensation due 
to a portion of the radiation, leaving the remainder. 

'It has been shown that when the eye is fatigued by white 
light it recovers its sensibility for different colors successively 



SENSATIONS OF COLOR 301 

after different intervals of time. If one looks out of a small 
window, such as a porthole, on a strongly illuminated sky, 
and then closes one's eyes, the after image of the window 
will appear in dissolving colors of brilliant hues. 

Simultaneous color contrasts are another form of subjective 
colors. These are well displayed by laying thin tissue paper 
over black letters printed on green cardboard. The letters 
in a strong light are pink by contrast. The tissue paper fur- 
nishes a faint illumination compared to the strong green, and 
the unfatigued nerve terminals giving red cause the letters 
to appear pink in contrast with the complementary green. 
In this way certain colors are heightened by contrast, partic- 
ularly if they are complementary. 



CHAPTER XI 

POLARIZED LIGHT 

I. DOUBLE REFRACTION 

327. Phenomena of Double Refraction. In the study of sin- 
gle refraction in Chapter VIII it was tacitly assumed that 
when light enters a dense transparent medium it has but one 
velocity of transmission, and that there is consequently only 
one refracted ray for each incident ray. But all crystal- 
line substances, except those whose fundamental form is the 
cube, possess the property of dividing each incident ray into 
two refracted ones. This phenomenon is called double refrac- 
tion. It belongs also to many animal and vegetable sub- 
stances and to homogeneous media, like glass, which are 
unequally strained in different, directions. On account of 
this unequal strain the physical properties are not the same 
in all directions. 

When a thin pencil of light traverses a double refracting 
body, it emerges as two pencils. One of these obeys the 
laws of single refraction ( 277), and is called the ordinary 
ray. The other in general does not obey these laws, and is 
called the extraordinary ray. 

In every double refracting crystal there is at least one di- 
rection in which light may pass without bifurcation. This 
direction is called the optic axis. The refracted rays diverge 
most widely when the incident pencil is perpendicular to the 
optic axis. Both rays then follow the ordinary laws of single 
refraction. 

It should be noted that the angular separation of the ordi- 
nary and the extraordinary ray in many cases is so slight 

302 



DOUBLE REFRACTION 



303 



that they are nearly superposed and not distinguishable as 
separate rays. The double refraction in such cases is clearly 
demonstrated by polarized light in a manner soon to be 
explained. 

328. Double Refraction in Iceland Spar. Iceland spar ex- 
hibits double refraction to a remarkable degree. It is crys- 
tallized carbonate of calcium, and occurs in rhombohedra, 
whose faces form angles with each 
other of 105 5' or 74 55'. The 
angles of the crystal are the same 
in all specimens, but the lengths of 
the three edges may have any ratios 
whatever. Two of the solid angles 
at the extremities of the diagonal ab 
(Fig. 211) are contained between 
three obtuse plane angles. A line 
equally inclined to the three edges 
bounding one of these solid angles 
is the axis of the crystal. This axis is a definite direction 
rather than a definite line. Iceland spar has but one such 
axis, and hence is said to be uniaxial. 

Any plane containing the axis is a principal plane. If the 
two solid angles bounded by the three obtuse angles are cut 

away by planes perpen- 
dicular to the axis of 
the crystal, a parallel 
pencil of light, incident 
normally on either of 
these polished planes, 
will pass through with- 
out bifurcation. 

If a crystal of Ice- 
land spar be laid on a 

printed page, the letters will in general appear double 
(Fig. 212) ; or if a pencil of sunlight be admitted through a 





304 LIGHT 

small round opening in a shutter, and a crystal of Iceland 
spar be placed in the path of the pencil, so that it is normal 
to a face of the crystal, two equally illuminated disks will ap- 
pear on the screen. 

When the plane of incidence is perpendicular to the optic 
axis, both rays follow the laws of refraction ; the index of 
refraction for the ordinary ray and the D line is 1.658 ; for 
the extraordinary ray it is 1.486. It follows that the extraor- 
dinary ray is transmitted through Iceland spar with greater 
velocity than the ordinary ray. 

Crystals like Iceland spar, in which the index of refraction 
for the extraordinary ray is smaller than for the ordinary, 
are called negative uniaxial crystals. Quartz is also double 
refracting; but in quartz the angular separation of the two rays 
is smaller than in Iceland spar, the indices of refraction being 
1.544 and 1.553 for the ordinary and the extraordinary rays 
respectively with sodium light. Such crystals, in which the 
index of refraction for the ordinary ray is the smaller of the 
two, are called positive uniaxial crystals. 

329. Uniaxial Prisms. A prism may be cut from a uniaxial 
substance either so that its optic axis makes equal angles 
with the faces which unite at the refracting edge, or so that 

it is parallel to the refracting edge. 
In the former case a pencil of light 
traversing the prism in the direction 
for minimum deviation will not di- 
vide into two rays, for its path 
through the prism is in the direc- 
tion of the optic axis ; but in the 
B second case (Fig. 213) the plane of 

Fig. 213 t t ^ i.ii 

incidence is perpendicular to the 

optic axis, and the divergence of the ordinary and the ex- 
traordinary rays is a maximum. Two spectra will then be 
formed, the ordinary having the greater mean deviation for 
an Iceland spar prism, and the smaller for a quartz prism. 




DOUBLE REFRACTION 305 

Such a prism may be used to measure the refractive 
indices by setting it so that first the ordinary and then the 
extraordinary ray is at minimum deviation, and employing 
formula (57), 282. The following are a few values for 
sodium light : 

NEGATIVE CRYSTALS 

Ho f^e 

Iceland spar 1.658 1.486 

Sodium nitrate . . . ."'.,,. . 1.587 1.536 

POSITIVE CRYSTALS 

Ho He 

Quartz . , . . 1.544 1.553 

Ice . : . '" r f 1.309 1.310 

330. Wave Surfaces in Uniaxial Crystals. If there is a 
radiant point within a transparent isotropic body, that is, 
one whose physical properties are the same in all directions, 
such as water or well-annealed glass, the wave surfaces 
about the radiant point are spheres, for the velocity of light 
is the same in all directions. If, however, the body is not 
isotropic, the velocity varies with the direction of trans- 
mission, and the wave surfaces are no longer spherical. 

Now, in double refracting crystals there are two rays, and 
two indices of refraction ; hence, there are two different wave 
surfaces ; and, since the refractive index for the extraor- 
dinary ray is not the same in every direction, the velocity 
of this ray varies with the direction between limits. It 
follows that the wave surfaces for this ray are not spherical. 

Huyghens demonstrated that in uniaxial crystals the wave 
surface for the extraordinary ray is a spheroid or ellipsoid of 
revolution, that is, the surface generated by rotating an 
ellipse about one of its axes coinciding with the optic axis 
of the crystal. If, further, there is only one ray in the direc- 
tion of the optic axis, there is in general only one velocity 
in this direction and the sphere and spheroid touch each 
other on this axis. 

There are, however, two cases corresponding to positive 



306 



LIGHT 



and negative crystals. In the former the velocity of the 
ordinary ray is the greater of the two, and in the latter 
it is the less, except along the optic axis. In the former a 
section of the wave surfaces through the radiant point and 
containing the optic axis is like Figure 214 a ; in the latter, 





it is like Figure 214 5. The two rays have the same velocity 
along A A', but unequal velocities in all other directions, 
the difference being the greatest along BB\ perpendicular 
to the optic axis. Also, the direction of propagation of the 
extraordinary ray is not in general normal to its wave 
surface, while that of the ordinary ray is normal. 

In certain uniaxial crystals, quartz for example, the 
velocity of the two rays is not the same even in the direction 
of the optic axis, the spectral lines in the spectroscope 
appearing double, indicating slight double refraction. In 
such crystals the two wave surfaces are not tangent to each 
other, but either the one lies wholly within the other, or the 
two Intersect at four points symmetrically situated with 
respect to the optic axis. 

331. Construction for the Refracted Rays in Iceland Spar. 
The path of the two refracted rays in Iceland spar may be 
readily constructed from the two wave surfaces, especially in 
the selected cases in which both rays lie in the plane of 
incidence. 



DOUBLE REFRACTION 



307 




Fig. 215 



Let db (Fig. 215) be the direction of the incident light and 
Id its plane wave front. MN is the surface of the spar, ; 
Also let the optic axis bi 
lie in the plane of the 
paper and therefore in 
the plane of incidence. 
The path of the ordinary 
refracted ray is found 
by the construction al- 
ready employed in 279. 
About b as a center, and 
with a radius which 
bears to df the ratio of 

1 to 1.658 (v f to v~), describe a circle ; from/ draw a tangent 
to this circle, and connect b and the point of tangency o. 
Then bo is the ordinary ray. 

Since the optic axis coincides with bi, the intersection of 
the plane of the paper with the wave surface of the extraor- 
dinary ray is the ellipse tangent to the circle at i. To find 
the direction of the extraordinary ray, draw from / the line 
fe tangent to the ellipse, and connect frand the point of tan- 
gency e. Then be is the extraordinary ray. The new wave 

fronts are/o and/e. 

As a second example, 
let the optic axis be per- 
pendicular to the plane of 
the figure, and therefore 
to the plane of incidence. 
The intersections of the 
plane of the diagram with 
the two wave surfaces are 
then both circles (Fig. 
216). From / draw tan- 
gents to both circles, and the lines bo and be, drawn from b to 
the points of tangency, are the directions of the ordinary 
and the extraordinary ray respectively. The new wave 





Fig. 216 



308 



LIGHT 



fronts are fo and fe. This is the construction for the 
greatest divergence of the two refracted rays. 



II. POLARIZATION BY REFLECTION 

332. Plane Polarized Light. A ray of common light has 
on all sides the same peculiarities. When a pencil of com- 
mon light is received on a plane mirror, it is reflected, in 
whatever direction the plane of incidence may pass through 
the pencil. But there is a peculiar modification of light 
which has not the same properties on all sides ; this asym- 
metry about the line of propagation is known as polarization, 
and light having this modification of asymmetry MS called 
polarized light. 

The term polarization was introduced by Malus in 1811. 
Mains discovered that light reflected at a particular angle 

from plane glass does not 
possess symmetrical properties 
of reflection with respect to its 
direction of transmission. The 
important discovery of Malus 
made it possible to explain the 
earlier observations of asym- 
metry in connection with 
double refraction, and to de- 
termine that the vibrations 
in light are transverse to the 
line of propagation, rather 
than longitudinal, as in the 
case of sound. 

When a ray of common light 
6 (Fig. 217) falls on a plate 
of plane glass at an angle of 
incidence of about 57, it is re- 
flected in the direction be, in accordance with the laws of ordi- 
nary reflection. The ray be is mostly polarized by reflection. 




Fig. 217 







LARIZATION BY REFLECTION 309 

/ 

If this polarized ray is incident on a second plate of glass 
parallel to the first one, the plane of incidence for the upper 
mirror coincides with that of the lower, and the ray is re- 
flected like ordinary light. If, however, one turns the upper 
mirror around be as an axis, keeping the angle of incidence on 
< the ujDpeji mirror at 57, the two mirrors will no longer be 
parallel, arid the plane of reflection of the upper mirror will 
revolve with the mirror. When the upper mirror is thus 
turned out of parallelism with the lower, the intensity of the 
light reflected from the second mirror decreases continuously, 
until the angle of rotation is 90, when the ray be ceases to be 
longer reflected. Turning the upper mirror still further^ be 
is again reflected and increases in intensity up to 180, when 
the two planes of incidence again coincide. By further ro- 
tation of the upper mirror, the reflected light again becomes 
fainter and fails at 270. The light reflected from the lower 
mirror at an angle of about 57 is polarized, for be shows a 
two-sidedness with respect to reflection. 

333. Vibrations in Light Transverse. If the vibrations in 
light were longitudinal, as in sound, it is not conceivable 
that a ray of light could be modified in such a way that it 
would not be equally susceptible of reflection in all direc- 
tions. On the other hand, if the vibrations are transverse, 
they may then be reduced to a single plane containing the 
ray, and will thus be asymmetrical about its direction of 
transmission. Now reflec- 
tion from glass is one of the 
means of reducing ether vi- 
brations to a single plane, 
and light thus modified is 
called plane polarized light. 

In light polarized by re- 
flection the vibrations are 
parallel to the reflecting surface. When, for example, a ray 
of light ab (Fig. 218) is incident on a plate of glass at an 




310 LIGHT 

angle of 57, the reflected ray be is polarized and the ether 
vibrations are parallel tofd and at right angles to the plane 
of incidence containing ab and be. 

When such a plane polarized pencil of light cb is incident 
on a glass mirror RS, whose plane is parallel to the ether vi- 
brations, a portion of the pencil is transmitted through the 
glass without change in its plane of vibration, and another 
portion is always reflected in the direction ba with its vibra- 
tions unchanged and parallel to fd. 

334. The Polarizing Angle. The degree of polarization of 
a reflected pencil of light varies with the angle of incidence. 
The angle of incidence for which the polarization is greatest 
is called the polarizing angle. It is different for different 
substances and indeed for different colors. The following 
simple law relating to the polarizing angle was discovered by 
Brewster : 

The polarizing angle is the angle of incidence for whichr 
the reflected and the refracted rays are at right angles. 

Thus, in Figure 219, if be, the reflected ray, makes a right 
angle with bd, the refracted ray, the polarization is most 
nearly complete, and the angle i is the polarizing angle. 
a The plane of reflection is called the 
plane of polarization. The' ether 
vibrations are at right angles to it. 
Since the angle dbc equals 90, 
so also i + r = 90 ; consequently 
sin r = cos i. Therefore 

sin i sin i 

= = tan i = /JL. 

sin r cos i 

Hence the polarizing angle is 
the angle of incidence whose trigo- 
nometrical tangent is equal to the index of refraction of 
the reflecting substance. Since the different colors have 




POLARIZATION BY DOUBLE REFRACTION 311 

not the same index of refraction, their polarizing angles for 
the same substance also differ slightly. 

It must not be inferred that for every substance there is an angle of 
complete polarization. The polarization always increases with the angle 
of incidence up to a maximum, and then decreases again, after passing 
the angle of greatest polarization. This maximum is the polarizing 
angle of the substance. Only a few substances, with a refractive index 
of about 1.46, polarize light completely by reflection. If the substance 
is transparent, the refracted ray is also polarized, and in a plane perpen- 
dicular to that of the reflected ray. 

in. POLARIZATION BY DOUBLE REFRACTION 

335. Polarization by Tourmaline. If a slice of brown or 
green tourmaline, cut parallel to its optic axis, be held so 
that a beam of light falls on it normally, the transmitted 
light will differ in no respect to the eye from the incident 
beam, except that it is slightly colored by selective absorp- 
tion in the tourmaline. 

But if the transmitted light be examined by means of a 
piece of plate glass, it will be found to have undergone 
a remarkable change. While in one direction it is reflected 
in the same manner as common light, yet when the glass is 
turned around the beam as an axis, the light varies in inten- 
sity ; and in one position the reflected light vanishes entirely. 
The light__ transmitted by the plate of tourmaline may he 
reflected in the plane passing through the axis at all angles 
of incidence ; but in a plane at right angles to this, it is im- 
perfectly reflected, and at an angle of incidence of about 57 
it is not reflected at 
all. 

"further, if a beam |f . ^fltM^ 

transmitted through 
one plate of tour- 
maline be examined 

, , . ., Fig. 220 

by a second similar 

plate, it will be found that in one Relative position jjJLthe 

two the light is freely transmitted (Fig. 220, AB), while it 






312 LIGHT 

becomes feebler and feebler if either plate be turned around 
in its own plane, as in A'B 1 ; and when the two parallel 
plates have their longer dimensions at right angles, as in 
A" B" ', no light whatever passes through them. The light 
is completely extinguished by crossing two transparent and 
nearly colorless crystals. . 

The light transmitted by the tourmaline is plane polarized. 
Tourmaline is a double refracting substance, and it has the 
property of absorbing the ordinary ray so rapidly that this 
ray does not emerge at all if the tourmaline plate is from one 
to two millimeters thick. The first plate of tourmaline is 
called the polarizer ; and the plate glass reflector, or the 
second tourmaline, the analyzer. 

336. Common Light Compared with Plane Polarized Light. 
It is important to observe that the proportion of the beam 
coming from the polarizer which the analyzer transmits de- 
pends on the orientation of the second plate with respect to 
the first; and that the beam transmitted by the first plate is 
completely extinguished by the second when the two are 
crossed. Now^Ji-the vibrations of the ether were longitu- 
dinal, it is inconceivable that the crossing of the plates 
should stop the light, since the rotation of the second plate _ 
could not modify longitudinal vibrations. This phenome- 
non of extinction is therefore held to demonstrate that the., 
vibrations of the ether in light, and in radiation in general, 
are transverse. 

In common light the vibrations are transverse to the direc- 
tion of propagation and in all planes containing the ray. In 
other words, the ray is perfectly symmetrical with respect to 
its direction of transmission. Hence it is that the beam of 
light incident on the first plate of tourmaline is transmitted 
equally well in every position of the plate. The rotation of 
the plate about the ray as an axis produces no change in the 
intensity of the transmitted light. -The action of the tour- 
maline is to resolve the transverse vibrations of the common 



POLARIZATION BY DOUBLE REFRACTION 313 



D 



a 



Fig. 221 



light into simple harmonic components in two directions, one 
parallel to the optic axis and the other at right angles to it. 
The former constitute the extraordinary ray, which is trans- 
mitted; the latter, the ordinary ray, which is absorbed. 

Thus when the ray of common light 10 is incident on the 
slice of tourmaline with its faces containing the optic axis AB 
(Fig. 221), the vibrationsjrf thejtrans_- 
mitted ray are confined to planes par- 
allel to AB, as indicated by the short 
arrows. When the axis of the second 
parallel tourmaline plate is inclined to 
that of the first, its effect is to resolve 
the incident vibrations parallel to the 
plane ABOD into two components, one 
in the direction of its axis and con- 
stituting the extraordinary ray, which is transmitted, and 
the other at right angles to its axis and composing the ordi- 
nary ray, which is absorbed. When the axes of the two 
plates are crossed, the vibrations from the polarizer enter 
the analyzer without resolution, but they form the ordinary 
ray, which is absorbed. There is then no extraordinary ray, 
and hence no light gets through the crossed plates. 

Again, when the axes of the two plates are parallel, the 
vibrations from the polarizer are not resolved by the 
analyzer, but in this case they form the extraordinary ray, 
which passes through without change. 

Common light shows no such evidences of polarization. It 
is accordingly necessary to assume that, while the motions 
constituting common light are at right angles to the direction 
in which the light is traveling, the character of this trans- 
verse vibrational form is continually changing, so that in a 
very short period the vibrations are performed in all azimuths 
with respect to the ray. On account of this shifting of the 
plane of vibration, it is impossible to produce interference be- 
tween two trains of waves from different sources of light, 01 
even from different parts of the same source. 



314 LIGHT 

337. Polarization by Iceland Spar. The-two beams of light 
into which a single one is divided by Iceland spar ( 328) 
are both plane polarized. Their polarization may be readily 
shown by examining them by reflection from plane glass, or by a 
plate of tourmaline. If the tourmaline is placed in the path 
of the two rays, with its plane at right angles to them, and if 
it is slowly turned about the rays as an axis, a position is readily 
found in which only one image appears on the screen. If then 
the tourmaline plate is turned further in its own plane, the 
second image will become visible. It will increase its bright- 
ness, while the first one will become fainter and will disappear 
when the angle of rotation is 90, the second image only re- 
maining visible. 

The positions of the tourmaline when there is only one 
image show that the ether vibrations of the light forming 
the ordinary image are perpendicular to a principal plane, 
while those of the extraordinary image are always in a prin- 
cipal plane. The two rays are therefore polarized in planes 
at right angles to each other. 

338. The Nicol Prism. A beam of plane polarized light 
may be obtained by reflection from glass, or by transmission 
through a plate of tourmaline. By the former method the 
polarized light is usually mixed with a small amount of com- 
mon light, and by the latter it is colored by the tourmaline. 
Any device for getting rid of one of the rays given by Ice- 
land spar leaves a colorless beam of plane polarized light. 

The best device for this purpose was invented by Nicol in 
1828, and is called the Nicol prism. It is made of Iceland 
spar in such a way that the ordinary ray is stopped by total 
internal reflection. 

A long rhomb of the spar has its end faces cut so that the 
natural angle of cleavage is reduced to 68 (Fig. 222, ACS). 
The plane of the figure is a principal plane. The rhomb 
is .cut through along a section perpendicular to a prin- 
cipal plane. The trace of the section is AB, which makes 




POLARIZATION BY DOUBLE REFRACTION 315 

the angle BAG 90. The two faces of the section are polished 
and then cemented together by a thin layer of Canada balsam, 
which has a mean 
refractive index 
(1.54) intermediate 
between those of the 
ordinary and the 
extraordinary ray. 

When therefore a ray enters the Nicol prism at 5, the ordi- 
nary raj bo meets the balsam at an angle somewhat greater 
than the critical angle from the spar to the balsam (68 15'). 
It thus suffers total internal reflection and fails to pass 
through. The extraordinary ray is not totally reflected at 
the first surface of the balsam because it goes from a medium 
of lower refractive index for it to one of higher ; and it is 
not reflected at the second surface of the balsam because its 
angle of incidence is less than the critical angle for the extraor- 
dinary ray in the two media. Moreover, since the cemented 
section is at right angles to a principal plane of the crystal, 
the vibrations of the ordinary ray are parallel to this section 
and the ordinary ray is therefore readily reflected by it. 
Thus the extraordinary ray alone passes through. 

The direction of vibration for the ordinary ray is the longer 
diagonal of the end of the prism ; for the transmitted ray it 
is the shorter diagonal EF. The light emerging from the 
prism is polarized at right angles to the principal plane. 

339. Extinction of Light by Crossed Nicol Prisms. When 
the light which has passed through one Nicol prism falls on 
a second, the amount transmitted depends on the relation of 
the principal planes of the two. If the shorter diagonals are 
parallel, the plane polarized light from the polarizer composes 
the extraordinary ray in the analyzer, and passes on through. 
But if the analyzer be turned about the beam of light as an 
axis, the transmitted beam will decrease in brightness, and 
will disappear entirely when the rotation reaches 90. The 



316 LIGHT 

Nicol prisms are then said to be crossed ; the light from the polar, 
izer now forms the ordinary ray for the analyzer, and is lost 
by internal reflection. In intermediate positions the recti- 
linear vibrations of the plane polarized light coming through 
the polarizer are resolved into two rectangular components 
by the analyzer and in directions corresponding to its two 
planes of vibration. This resolution takes place in accord- 
ance with the usual mechanical law for the resolution of a 
motion into two components at right angles ( 32). 

340. Effect of interposing Double Refracting Substances Let 

the Nicol prisms be crossed, the direction of vibration for the 
polarizer being vertical and that for the analyzer horizontal. 
The field of view is then dark. The introduction between 
the Nicols of a thin sheet of mica or selenite, with its plane 
at right angles to the beam of light, in general restores the 
light on the screen, or to the field of view for direct vision. 
But if the thin plate is rotated in its own plane, four positions 
will be found in every revolution in which the interposed 
plate does not restore the light. These positions are 90 
apart. At points 45 from them the transmitted light reaches 
its greatest intensity. 

Mica and selenite are double refracting substances, and the 
vibrations of the ordinary and of the extraordinary ray are 
in planes at right angles to each other. Now when the plate 
is in such a position that its two directions of vibration co- 
incide with those of the polarizer and the analyzer, then the 
extraordinary ray from the polarizer passes through the 
crystalline plate without resolution into two components, and 
it is stopped by the analyzer as the ordinary ray. But for 
any other position of the mica or selenite plate the case is 
different. 

Assume the mica plate rotated 45 from any position of 
extinction. Let the full line in Figure 223 a be the vibration 
in the plane polarized light from the polarizer and incident 
on the mica plate. As soon as this light enters the crystal- 



POLARIZATION BY DOUBLE REFRACTION 317 

line plate, the single vibration is resolved into two at right 
angles to each other, o and e, as represented in b. These two 
trains of waves are transmitted through the mica with 

/K 





Fig. 223 

unequal velocities ; and since their frequencies are the same, 
the one with the slower velocity will gain in phase over the 
other. Assume the thickness of the plate such that the 
difference in phase between the two wave trains at the point 
of emergence is half a period for sodium light, as represented 
in c by the reversal of the arrow o. Moreover, since the 
plate is thin and the beam comparatively broad, the two rays 
are not visibly separated, and the same portions of the ether 
are agitated by the two wave trains after emergence from 
the mica plate. The two vibrations of c therefore combine 
into a single vibration, as represented in d. 

The light emerging from the mica, compared with the 
incident beam, is still plane polarized, but its direction of 
vibration has been rotated through a right angle by the 
phase difference of half a period introduced by the crystal. 
Since the analyzer is supposed to be set so as to transmit 
vibrations in a horizontal plane, it will transmit the light 
emerging from the mica. In order to extinguish the light 
now, the analyzer must be turned into a position parallel 
with the polarizer. 

341. Elliptical Polarization. If the reader will recall 38 
and 41 on the composition of simple harmonic motions, it 
will at once be apparent that the ellipse must be the general 
form of vibration resulting from the resolution of the simple 
harmonic vibrations of plane piolarized light into two compo- 
nents at right angles to each other by transmission through a 
double refracting body, and the introduction of a phase 



318 LIGHT 

difference between the two components by the retardation of 
the one ray with respect to the other on account of the differ- 
ence in their velocities of transmission. The ellipse becomes 
a straight line as one of its limiting forms when the phase 
difference is a whole number of half wave lengths. 

Assuming the relations of the two Nicol prisms and the 
interposed mica plate to be the same as in the last article, 
then a of Figure 224 is the vibration of the plane polarized 

o e o 

V' 
* 



< a 

\ 



, 

Fig. 224 





light incident normally on the mica, b the two rectangular 
components of this vibration immediately after entering the 
mica, and c the two as the light leaves the mica, its thickness 
being such that the one ray o has fallen behind the other one e 
by one eighth of a wave length, as indicated by the head of 
the arrow in o. These two simple harmonic vibrations com- 
pound into an ellipse d, as may easily be shown by the 
graphical method of 41. Hence the light emerging from 
the mica plate is elliptically polarized. The vibrational 
form is an ellipse with its plane perpendicular to the direction 
of transmission. 

When this elliptically polarized light enters the analyzing 
Nicol, the elliptical vibration is resolved into two simple 
harmonic components, forming the ordinary and extraordi- 
nary rays. With a phase difference of one eighth of a period, 
as represented in Fig. 224, the analyzer will transmit a maxi- 
mum amount of light in a position parallel with the polarizer, 
and a minimum when the two Nicols are crossed. 

342. Circular Polarization. The reader who has made himself 
familiar with the composition of simple harmonic motions at right angles 
win readily anticipate the possibility of circular polarization of light. 
Two simple harmonic motions of the same period and at right angles 



POLARIZATION BY DOUBLE REFRACTION 



319 



compound into uniform circular motion when their amplitudes are equal 
and their phase difference one quarter of a period ( 41). Now when the 
mica plate is set so that its directions of vibration are at an angular dis- 
tance of 45 from that of the incident plane polarized light, then the two 
components into which the plane polarized light is resolved are of equal 
amplitude, and they have the same period. If, further, the thickness of 
the prepared mica plate is just enough to introduce a phase difference 



V 

\ 






Fig. 225 



of one quarter of a period for some selected color, such as yellow sodium 
light, the emergent beam is circularly polarized (Fig. 225), and the mica 
is called a quarter wave plate. 

In a train of circularly polarized waves the successive ether particles 
at any instant all lie on the thread of a screw, the pitch of which is the 
wave length. In right-handed circular polariza- 
tion (Fig. 226 72) the ether particles rotate 
counter-clockwise; in left-handed polarization 
they rotate clockwise (Fig. 226 L). In R the 
particles b, c, d are moving upward simultaneously 
and /, ff, ~h, downward ; in L b, c, d are mov- 
ing downward and /, g, h, upward, while in both 
the waves are advancing in the direction of the 
arrow A. 

When circularly polarized light is transmitted 
through a Nicol prism, there is always an extraor- 
dinary ray in the Nicol, with an amplitude of 
vibration equal to the radius of the circle ; there- 
fore no change in the intensity of the transmitted 
light accompanies the rotation of the prism. If, 

however, a second quarter wave plate be interposed in the path of 
circularly polarized light, it will introduce a relative retardation of 
another quarter wave for the same color, and the emergent light will 
be plane polarized in an azimuth which may be found by rotating 
the analyzing Nicol. Two quarter wave plates may thus serve simply 
to rotate the plane of polarization. 

343. Rotation of the Plane of Polarization. If the two Nicol 
prisms are set for extinction, and a plate of quartz, cut 




320 LIGHT 

perpendicular to its optic axis, is introduced between them, 
the light will be to some extent restored. It can be extin- 
guished again by rotating the analyzer through a small angle. 
Quartz rotates the plane of polarization during the pas- 
sage of light though it. Some quartz crystals rotate the 
plane of polarization to the right, looking in the direction 
of transmission, and these are called right-handed; others 
rotate the plane of polarization to the left, and are called 
left-handed. 

The rotatory power of quartz is intimately related to its crystal- 
line structure. Fused quartz is not double refracting and does not produce 
rotation. Further, in some cases when a right-handed and a left- 
handed quartz crystal are placed side by side, they are found to be 
symmetrical with respect to a plane between them, like an object and 
its image in a plane mirror. In other words, they are right-handed 
or left-handed crystals, corresponding to their power of rotating the 
plane of polarization to the right or to the left. In many quartz crys- 
tals the secondary planes or facets, which distinguish them as right- 
handed or left-handed, are lacking; but even then their contrasting 
crystalline structure may be demonstrated by etching. 

344. Rotation of the Plane of Polarization Explained. In 
quartz the double refraction does not vanish even along the 
optic axis. The two wave surfaces are therefore not tangent 
( 330), but the spheroid lies wholly within the sphere with- 
out touching. Further, both rays in the direction of the 
axis are circularly polarized, the rotations of the two com- 
ponents being in opposite directions around equal circles. 
When these circularly polarized rays emerge from the crys- 
tal, they recombine into plane polarized light, but with ro- 
tation of the plane of polarization. 

Let the two uniform circular motions o and e (Fig. 227 
A) be resolved into two linear displacements at right angles 
to each other. They are symmetrical with respect to the line 
P; the components b and b r are equal and opposite, while a 
and a' are equal and in the same direction. Hence, if the two 
uniform circular motions are simultaneously impressed on 



POLARIZATION BY DOUBLE REFRACTION 321 




Fig 227 



the same particle, its velocity parallel to the line P at any 

instant will be double that of o or e, and perpendicular to it 

will be zero. The particle 

will therefore describe simple 

harmonic motion along the 

line P with an amplitude 

twice the radius of the circle. 

Conversely, the simple har- 
monic motion P is resolv- 
able into two equal uniform 
circular motions of the same 
period as that of P and of 
equal amplitudes. ', 

Plane polarized light incident normally on a plate of 
quartz, cut at right angles to its optic axis, is thus resolved 
into two circularly polarized rays ; and if the velocity of 
transmission of the two through the quartz were the same, 
the plane of polarization of the light emerging into the air 
would be parallel to that of the incident ray. But the cir- 
cular component 0, constituting the ordinary ray, travels 
faster through the quartz than the component 0, while the 
two still have the same frequency. Hence the plane of vi- 
bration of the emergent light is rotated in the direction of 
the circular motion of the component which rotates through 
the greater angle in traversing the quartz plate. This is the 
component 6, which has the smaller velocity of transmission. 
Its wave length is shorter than the other, and it rotates 
through a larger angle in traversing the same thickness of 
the crystal. 

Thus, in Fig. 227 B, o and e are the relative angular posi- 
tions of the two circular motions as they emerge from the 
quartz. The component e has gained in phase as compared 
with 0, and the two are symmetrical with respect to the line 
P\ the perpendicular displacements, b and o 7 , annulling each 
other, while the parallel displacements, a and a 1 , are addi- 
tive. The line P' is then the direction of vibration of the 



322 LIGHT 

emergent light. The plane of polarization has been rotated 
through the angle 8, which is half the angular retardation of 
the one ray with respect to the other. 

The rotation at 20 C. for different wave lengths produced by a plate 
of quartz one millimeter thick is as follows : 

A B C D E F G 

12.67 15.75 17.32 21.70 27.54 32.77 42.60 

The angular rotation is nearly inversely as the square of the wave 
length. This is known as Biot's law. 

345. Rotatory Power of Liquids. It was discovered by 
Biot in 1815 that some liquids also possess the same power 
of rotation as quartz, but to a much smaller degree. Of 
these, solutions of cane sugar have received the most atten- 
tion, for the reason that the commercial test for the percent- 
age of sugar present is the rotation produced by a column of 
sugar solution of fixed length and at a definite temperature. 
This method of quantitative estimation applied to sugar is 
commonly known as saccharimetry '. 

When heated in solution with dilute acids, cane sugar 
takes up water and splits into two molecules. The two have 
the same chemical composition, but one rotates the plane of 
polarization to the right, and is therefore called dextrose; 
the other rotates it to the left, and is called levulose. The 
levulose rotates the plane of polarization more strongly than 
the dextrose, and the mixture therefore rotates to the left, or 
the rotatory action of the cane sugar has been inverted. 
Whence the name "invert sugar." 

IV. COLORS BY POLARIZED LIGHT 

346. Colors produced by an Interposed Film. When the 
polarizer and the analyzer are set so that the latter quenches 
the light transmitted by the former ( 340), a lamina of 
mica or selenite, held obliquely in the path of the polarized 
white light between the two Nicol prisms, not only brings 



COLORS BY POLARIZED LIGHT 323 

light into the dark field, but shows brilliant colors. The 
restoration of the light has already been explained; it re- 
mains now to account for the colors. 

Let us suppose that the selenite film (which gives more 
brilliant colors than mica) is just thick enough to introduce 
a difference of phase between the two component vibrations 
in the selenite of half a period for the longest waves of red. 
Then, since the short waves of extreme violet are about one 
half the length of those of extreme red, the selenite lamina 
will produce a phase difference of a whole period for the 
violet. The two components for the latter will thus emerge 
from the selenite without any phase difference, and will re- 
combine into plane vibrations in the same direction as those 
of the polarized light incident on the lamina. The red, how- 
ever, will emerge from the selenite with one of its component 
vibrations half a period out of phase with the other, and the 
two will combine into plane vibrations at right angles to 
those of the incident beam, as represented in Fig. 223 d. 
The plane of vibration for red has been rotated in this man- 
ner through a quarter turn, and it will form the extraordi- 
nary ray for the analyzer and will pass through. The 
violet, on the other hand, is polarized in a plane at right 
angles to that of the red and will suffer extinction by the 
analyzer. If then only red and violet light were mingled in 
the plane polarized beam incident on the selenite, the ana- 
lyzing Nicol would cut out all the violet and transmit all the 
red ; but if the analyzer were rotated through a quarter 
turn, it would cut out all the red and transmit all the violet, 
and the continued rotation of the analyzer would allow the 
red and the violet to be transmitted alternately. 

If, now, the incident light is white, all wave lengths be- 
tween those for red and for violet are present, and the vibra- 
tions for these after passing the selenite are ellipses of various 
forms. The vibrational forms for the orange and yellow 
near the red are ellipses approaching straight lines, and their 
components parallel with those for red are much larger than 



324 LIGHT 

those parallel with the vibrations for violet. The orange 
and the yellow will therefore be largely transmitted along 
with the red ; while those nearest violet, namely, the blues 
and shorter greens, will be largely transmitted with the 
violet. It follows that the light transmitted by the analyzer 
when crossed with the polarizer will be some shade of red ; 
when parallel with the polarizer, some shade of blue. The 
colors transmitted in the two positions make up the whole of 
the white light incident on the selenite, and they are there- 
fore complementary to each other. 

A double refracting lamina thick enough to produce a phase difference 
of many periods for any color will also produce a phase difference of 
many periods for a number of other colors distributed throughout the 
spectrum. These colors may then be cut out by the analyzer, and the 
remaining transmitted colors will also be so widely distributed through- 
out the entire range of the spectrum that they will together reproduce the 
effect of white light. Hence, colors are produced in polarized light only 
when the double refracting lamina gives rise to a phase difference of only 
a few periods. 

347. Colors Due to a Plate cut at Right Angles to the Optic 
Axis. Extremely beautiful effects are produced by plates of 
uniaxial crystals cut perpendicular to the optic axis, such as a 
section of Iceland spar, in a beam of converging plane polarized 
light. The plate is placed between the two Nicol prisms in 
a converging beam so that the central ray of the converging 
cone of light is normal to the plate. This central portion of 
the cone of light passes along the optic axis of the section 
without undergoing double refraction, but all the other rays 
traverse the crystal section obliquely and are doubly refracted. 
The further any ray is from the axis of the cone, the greater 
is the obliquity of its path through the plate and the greater 
the thickness traversed. Also, the more oblique the ray, the 
greater the difference in the velocity of transmission of the 
two component disturbances. Since at the same distance 
from the optic axis each of these two causes of phase differ- 
ence has a constant value, it follows that there must be the 




COLORS BY POLARIZED LIGHT 325 

. 

same phase difference for any one color at all points of a 
circle conceived as drawn on the screen around the axial ray 
of the cone of light as a center. Hence, a system of concen- 
tric rings appears on the screen in iridescent colors like those 
of Newton's rings ( 302). 

When the polarizer and analyzer are crossed, the colored 
rings are traversed by a black cross (Fig. 228). This cross 
is explained as follows : Since the 
optic axis of the plate of spar is 
perpendicular to its surface, every 
diameter of the system of rings is 
the trace of a principal plane. 
The vibrations of the ordinary ray 
are normal to a principal plane and 
therefore tangential to all the con- 
centric circles ; those of the ex- 
traordinary ray are in a principal Fig - 228 
plane, or radially in the circles. Hence, along the two 
diameters representing the planes of vibration of the polar- 
izer and the analyzer, the directions of vibration for the inter- 
posed plate are the same as those of the polarizer and the 
analyzer. In these two directions, therefore, the thin plate 
does not resolve a ray into two components, and since the 
Nicols are crossed, the field is dark. Along all other diam- 
eters, the tangential and radial directions of vibration for 
the plate are inclined to those of the polarizer and the 
analyzer ; therefore double refraction takes place with colors 
as already explained. 

If the analyzer is turned so as to be parallel with the 
polarizer, a white cross takes the place of the black one. 
The colored rings are then projected on a bright field as 
a background. 

Similar dark and bright crosses are obtained from artificial crystals of 
saliciu on glass. The crystallization starts from many centers, and 
around them grow crystals with a radial structure and circular in outline. 
The maximum resolution of the plane polarized light occurs in each 



326 



LIGHT 




Fig. 229 



circle midway between the planes of vibration of the polarizer and the 
analyzer, while in these latter directions there is no resolution of the 
polarized light. Along these lines therefore 
the field remains dark, giving the black 
crosses (Fig. 229). When the analyzer is 
turned through 90, the black crosses change 
into white ones, and all the little crystals 
appear to revolve with the analyzer. 

348. Colors Produced by Stresses in 
Glass. Well-annealed glass does not 
show double refraction; but it may 
be made to acquire this property by 
unequal stresses in different directions. It is then no longer 
an isotropic body. Such stresses are evoked in glass by 
sudden chilling after heating. When glass so treated is 
traversed by plane polarized 
light, chromatic phenomena 
are obtained analogous to 
those due to double refract- 
ing crystals. A consider- 
able variety may be given 
to the colored curves by 
varying the form of the 
glass plates, making them circular, square, rectangular, or 
triangular in pattern. 

The introduction of one of these plates between 
the crossed Nicol prisms allows light to pass 
through, and colors spring forth of surprising 
brilliancy. Figure 230 A represents the black 
cross appearing on the dark field with a square of 
unannealed glass ? and Figure 230 j5, the figure 
when the plate is turned around in its own plane. 

:^j>l II Similar effects are produced by the mechanical compression 

of annealed glass. When a square piece of thick plate glass 
is compressed by turning a screw with the thumb and finger 
(Fig. 231), the light appears in the field where the pressure is applied 
at opposite points of the block, and it extends farther into the block 





Fig. 230 




Fig. 231 



COLORS BY POLARIZED LIGHT 



327 





Fig. 232 



as the pressure is increased, showing distortion through the whole 
body. 

The double refraction produced by a bending stress applied to a long 
polished strip of thick glass by means of a small hand press (Fig. 232) is 
very instructive. The polarized light 
passes through at right angles to the 
bending stress. The glass restores the 
light on both edges, but leaves a dark 
band through the middle from end to 
end. The concave surface of the bent 
strip is compressed, and the convex 
surface is stretched. The middle is 
the neutral axis, which is neither com- 
pressed nor elongated. 

Another very instructive experiment 
in double refraction in glass under 
stress was devised by Biot as long ago as 1820. It demonstrates the 
alternate compression and extension of a strip of glass at a node when 
vibrating longitudinally ( 199 and 249). A glass tube serves as well 
as a strip of glass and is more convenient to handle. It should be placed 
obliquely across the field of polarized light between the two Nicols and 
should be held firmly by its middle point. This point is a node, and the 
light should pass through the glass as near the node as practicable. If 
the tube is inclined to the planes of vibration of the crossed Nicol prisms, 
the field remains dark until the tube is thrown into vigorous vibration 
by rubbing it lengthwise with a moist cloth. Then every time a note is 
produced a band of light flashes across the field of view. The distortion 
of the tube due to the passage along it of Compressional waves renders it 
double refracting. 

Since the tube is alternately compressed and stretched, it. is obvious 
that twice during every complete vibration it is in its original isotropic 
state; therefore the illumination of the field must be interrupted by 
short periods of darkness. The change occurs too rapidly to be observed 
directly ; but Kundt verified this conclusion by observing the light in a 
rotating mirror. The elongated band of light seen in the mirror was 
crossed by dark spaces. Moreover, Kundt determined that the glass is 
optically positive like quartz while it is extended, and optically negative 
like Iceland spar while it is compressed. 



CHAPTER XII 




OPTICAL INSTRUMENTS 
I. THE EYE 

349. The Photographic Camera. The essential parts of a 
photographic camera are a dark chamber, or "camera 
obscura," and an achromatic converging lens provided with 
a screen on which to form a real image. In Figure 238, BO 
is the dark chamber, which is blackened inside and usually 

adjustable in length, 
LL ! is the converg- 
ing lens combina- 
tion, and E the 
screen or sensitized 
plate. 

In front of the 
^^^^ lens is a shutter and, 
if the lens is single, a diaphragm with small openings. This 
cuts off the marginal rays and reduces spherical aberration. 
It also gives greater depth of focus for landscape photogra- 
phy, so that objects at different distances may be approxi- 
mately in focus at the same time, analogous to the pinhole 
camera ( 257), in which the focus is nearly independent of 
the distance- of the object. 

The requirements of a good photographic lens are quite different from 
those of instruments used for visual impressions. As the photographic 
films in common use are most sensitive for the shortest waves, the lens 
must be achromatically adjusted around the blue-violet as a mean 
instead of the yellow-green, which is brightest to the eye. The photo- 
graphic camera is expected to form an image covering a wide area 
which must be flat and free from distortion, and all points of it must 

328 



Fig. 233 



THE EYE 



329 



be focused together. Distortions cannot be entirely eliminated by a 
diaphragm and a single lens. With two lenses or sets of lenses and 
a diaphragm between them, the distortions correct each other. Such a 
combination is called a rectilinear doublet. 

350. The Eye. The eye resembles a small camera, into 
which light enters only through a lens. In the camera the 
lens forms an inverted image on the sensitized plate, where 
the light initiates a chemical change in the sensitizing silver 
salts ; the lens of the eye forms an inverted image on the 
retina and agitates its sensitive nerve terminals. 

Figure 234 is a vertical section through the axis of the 
eye. The sclerotic coat If is a tough, fibrous substance, 
known as " the white 
of the eye." It is 
opaque except in 
front, where it be- 
comes a transparent 
coat A, called the 
cornea. Behind the 
cornea is the iris D, 
a colored curtain per- 
forated by a circular 
aperture called the 
pupil. The pupil ad- 
justs itself in size automatically with reference to the inten- 
sity of the light. Between the iris and the crystalline lens E 
is a transparent fluid called the aqueous humor. The chamber 
behind the lens is filled with a thin albuminous fluid called 
the vitreous humor. The choroid coat, filled with a black pig- 
ment to prevent internal reflection, lines the walls of this 
chamber ; on it is spread the retina, which is traversed by a 
network of nerves branching from the optic nerve M. 

The lens is built up of transparent horny layers, which 
increase in refractive index toward the center. This increase 
in the index of refraction toward the axis serves for the 
partial correction of spherical aberration ( 293), which is 




Fig. 234 



330 LIGHT 

further diminished by the iris diaphragm, as in the photo- 
graphic camera. 

In the camera the distance between the lens and the plate 
is adjustable for objects at different distances. In the eye 
the corresponding distance is fixed. When the eye is at 
rest, or meditatively fixed on space, it is adjusted for vision 
for infinitely distant objects. To look at nearer objects 
requires an effort of accommodation . Muscles attached to the 
periphery of the lens change its curvature by contraction or 
relaxation, and thus enable it to focus on the retina either 
very distant or very near objects. The range of accommo- 
dation is the same as if the eye were provided with a series 
of lenses of all focal lengths for objects ranging in distance 
from infinity to about twenty centimeters. This power of 
accommodation wanes with advancing age, so that the image 
for near objects falls behind the retina. Hence the necessity 
for converging reading glasses. 

351. The Blind Spot. The point in the retina where the 
optic nerve enters it is not sensitive to light; it is accord- 
ingly called the blind spot. When the image falls on this 
spot, there is no visual impression produced. This can 
easily be demonstrated by the aid of Figure 235. Hold the 



Fig. 235 

book with the circle opposite the right eye. Then close 
the left eye and turn the right to look at the cross. Move 
the book toward the eye from a distance of a little more 
than 30 cm. (1 foot), and a position may readily be found 
where the black circle will disappear. Its image then falls 
on the blind spot. It may be brought into view again by 
moving the book either nearer the eye or farther away. 

352. Defects of the Eye. A normal eye in its passive con- 
dition focuses parallel rays on the retina; but many eyes are 
not normal and have defects of several sorts. Those of most 



THE EYE 331 

frequent occurrence are near-sightedness, far-sightedness, 
and astigmatism. 

If the eye when relaxed focuses parallel rays in front of 
the retina, it is near-sighted. The length of the eyeball 
from front to back is then too great for the focal length of 
the crystalline lens. The correction consists in placing in 
front of the eye a concave lens that makes with the lens of 
the eye a less convergent system than the crystalline lens 
itself. 

Let d be the greatest distance for distinct vision of a near- 
sighted person. Then if the focal length of the concave lens 
is d, and if it is held close to the eye, parallel rays from a 
distant object will enter the eye as if they came from the 
principal focal point of the lens, which is at the greatest dis- 
tance for distinct vision d (Fig. 236). Provided with such 




a lens and possessed of the power of accommodation, the 
near-sighted eye may then have distinct vision for objects at 
all distances. .". * 

If the eye when relaxed focuses parallel rays from a 
distant object behind the retina, it is far-sighted. The 
length of the eyeball is then too small to correspond with 
the focal length of the crystalline lens. The correction for 
far-sightedness consists in placing in front of the eye a con- 
verging lens, making with the lens of the eye a more 
converging system than the crystalline lens alone. 

In this case let d be the least distance for distinct vision, 
and D that of normal distinct vision for small objects like 
the letters on this page. It is customary to take D as 25 cm. 



332 LIGHT 

(10 in.). Then the focal length of the correcting lens may 
be found by substitution in formula (64). Since d and D 
are both positive, \/D > \/f, or D is less than /. The con- 
verging lens then gives a virtual image, and light from an 
object at a distance D enters the eye as if it came from a 
distance d (Fig. 237). 

Sometimes the front of the cornea has different curvatures 
in different planes through the axis; that is, it has a some- 
what cylindrical form. Horizontal and vertical objects at 




Fig. 237 

the same distance are not then in focus at the same time. 
This defect is known as astigmatism. It is corrected by the 
use of a lens one surface of which at least is not spherical, 
but differs from it in the opposite sense to that of the de- 
fective eye. The astigmatism of the two eyes is not usually 
the same. 

353. Binocular Vision For distances in excess of a few 

hundred feet there is little difference between vision with 
one eye and with both if the illumination is good. But for 
near objects binocular vision introduces a new element. It 
is easy to demonstrate that when both eyes are directed to 
the nearer side of an object, the images of a point farther 
away do not fall on corresponding retinal points, and hence 
there is double vision. Hold two lead pencils vertically in 
front of the eyes with an interval of about six inches between 
them. When the eyes are fixed on the nearer pencil, the 
'farther one is seen double. Closing the right eye causes the 
right image to disappear, and closing the left eye, the left 



THE EYE 333 

image. If the gaze is fixed on the more distant pencil, the 
nearer one is seen double, but now the right image is given 
by the left eye and the left image by the right eye. 

The perception of depth in binocular vision is attained by 
perfect fusion of the images of a point at one distance from 
the eyes, while there is imperfect fusion of the images of 
points more remote and less remote. One eye occupies a 
position sensibly different from that of the other, and the 
retinal images are therefore not identical. On the whole, 
the impression received is that of a single object; but the 
right eye sees more of the right side of this object, and the 
left eye more 'of the left. Binocular vision thus comprehends 
more than vision with one eye. The two images are uncon- 
sciously combined into a single impression embracing the 
features of both retinal pictures. The effect of this fusion is 
the perception of depth, and the observer is as unconscious 
of double images as he is of the inversion of all retinal 
images. 

Pictures to be viewed with binocular vision in a stereo- 
scope are taken with a camera provided with two lenses, 
giving dissimilar images like those of the two eyes. A print 
from one of these negatives is viewed with the right eye and 
one from the other with the left eye only. The fusion of 
the two impressions gives the effect of relief in the picture. 

354, Irradiation. The increase in the apparent size of an 
object as it becomes more highly luminous is known as 
irradiation. Thus, the filament of an incandescent lamp ap- 
pears to become thicker as it passes from a red-hot to a 
white-hot temperature ; the crescent of the new moon appears 
to belong to a larger circle than the remainder of the disk, 
which is only faintly illuminated by light reflected from the 
earth ; a candle or gas flame appears to be continuous, though 
the incandescent particles of carbon are not in contact with 
one another. Strongly illuminated white objects, or those of 
a very bright color, appear larger against a dark background 



334 LIGHT 

than they really are. Irradiation probably arises from the 
fact that the impression produced on the retina extends 
beyond the outlines of the geometrical image. 

355. Persistence of Visual Impressions. When a piece of 
ignited charcoal is rapidly whirled about in a circle, the ap- 
pearance produced is a circle of fire. The spokes of a rapidly 
rotating wheel cannot be seen as separate images, but they 
blend into one another and give the impression of a translu- 
cent disk. Drops of falling rain have the appearance of 
liquid threads. The roughened part of a small vertical 
stream of water looks continuous below the smooth portion, 
while in reality it consists of distinct drops. An electric 
arc lamp, fed with alternating current, appears to give a con- 
tinuous light if the frequency is above about twenty-five 
cycles a second; but when the arc is photographed on a 
rapidly falling plate, it is found to be extinguished with 
every reversal of the current, and is really discontinuous. 
If it is viewed at night with a sudden turn of the head and 
eyes, the discontinuity may be seen by catching several 
stages of the illumination on different portions of the retina. 

The apparent continuity in all such cases is due to the fact that visual 
impressions persist after the external stimulus ceases. The explanation 
is physiological rather than physical, but the persistence of impressions 
made on the eye is involved in so many physical phenomena, such as 
manometric flames ( 238) and Lissajous's figures ( 248), that it cannot 
be ignored in physics. The duration of a visual impression depends on 
the sensitiveness of the retina and the intensity of the light. Plateau 
found an average duration of half a second. 

II. MICROSCOPES AND TELESCOPES 

356. The Simple Microscope. A converging lens of rather 
short focal length may be used as a simple microscope or mag- 
nifying glass. The object is usually placed just within the 
principal focal distance ; the image is then virtual, erect, and 
enlarged. The image ab (Fig. 238) subtends at the center 
of the lens the same angle as the object AB ; and since the 



MICROSCOPES AND TELESCOPES 



335 




Fig. 238 



eye is placed close to the lens, the angles subtended by the 
image at the center of the lens and at the eye are nearly 
the same. The 
image is usually 
brought to the 
distance of nor- 
mal distinct vision 
for the eye, say 
25 cm. The mag- 
nification is then 
approximately 25//, where /is the focal length of the lens in cm. 
The value of a simple magnifier consists in the ability it 
offers of bringing the object much nearer the eye than is 
possible without it, and thus in effect increasing the size of 
the retinal image. The lens may be considered as merely 
extending the power of accommodation of the eye. 

357. The Compound Microscope. The linear magnification 
of a simple magnifier cannot be extended much beyond one 
hundred diameters. To obtain still higher magnifying 
powers a combination of converging lenses, known as a com- 
pound microscope, is used. In its simplest form it consists 
of an object lens or train of lenses 0.(Fig. 239), which 

forms a nfagnified real image ab, 
and an eyepiece O used as a 
simple magnifier with ab as the 
object. The eyepiece gives a 
virtual image A'B' . Magnifi- 
cation is produced both by the 
objective and the eyepiece. The 
former is of short focal length 
and the object is placed just beyond its principal focus, so 
that a real and inverted image is formed very near the focus 
of the eyepiece. The eyepiece as a simple microscope may 
give a virtual image at a distance from the eye of normal 
distinct vision ; but since the normal eye at rest is adjusted 




Fig. 239 



336 



LIGHT 



for objects at a great distance, it is easier for the eye ii 
an optical instrument is so arranged that the rays entering 
the eye are nearly parallel. This can easily be done with 
the compound microscope by placing the eyepiece so that the 
image ab coincides with its principal focus. The focus- 
ing is then done by the instrument instead of using the 
accommodation of the eye. 

The above is only an outline of the principle of the microscope. The 
reader is referred to special treatises for the complete theory of the modern 
microscope. In it both the objective and the eyepiece consist of a com- 
bination of lenses, the former sometimes containing as many as ten 
separate ones to correct for spherical and chromatic aberration and to give 
a flat focal surface. 

358. The Astronomical Telescope. In the astronomical tele- 
scope, which gives an inverted image, the objective forms a 
real image which is viewed with the eyepiece or ocular, as in 
the compound microscope (Fig. 240). The employment of 




a large objective is for the purpose of collecting enough light 
to permit of large magnification without too much loss in 
brightness. 

Since the opposite angles between the two secondary axes 
through are equal, the angular magnitudes of the object 
and its real image ab, seen from 0, are the same ; but if the 
eye approaches ab to the least distance of distinct vision, the 
image will increase In magnitude in the ratio of the distances 
of the objective and the eye from the image. If F is the 
focal length of the objective in centimeters, the magnification 

TT 

would be . But the magnification is increased by the 



MICROSCOPES AND TELESCOPES 337 

25 

eyepiece by the factor , where / is the focal length of 

J 

the eyepiece (356). The total magnification then becomes 
TT 05 w 

~ x = , or the focal length of the objective divided by 
2o / / 
that of the eyepiece. 

If the telescope is turned toward a bright sky, the eyepiece 
may be focused on the illuminated objective, and an image of 
it, called the ocular circle, may be received on a piece of white 
paper very near the eyepiece. Now, the size of the objective 
and that of its image are directly proportional to their 
respective distances from the eyepiece, F +/ and a (a the 
distance of the real image from the eyepiece); also equation 

(64) may be written -- \- - = - . Multiplying both sides 

a J 



of this equation by F+f and subtracting unity, we have 

TT XT 

= . But is the expression for the magnification. 

/ / 

It follows that the magnification is equal to the ratio of the 
diameter of the objective to that of the ocular circle. 

The ocular circle is the smallest area traversed by the light after it 
leaves the telescope, and it marks the best position for the eye of the ob- 
server. To facilitate the placing of the eye in this position, a brass 
diaphragm, with a hole in its center, is screwed into the eye end of the 
telescope ; the proper place for the eye is close to this hole. If the pupil 
is smaller than the ocular circle, there is loss of light. Now the smaller 
the ocular circle, the larger the magnifying power of the telescope ; hence, 
the lowest power that can be used to advantage is one that makes the 
ocular circle no larger than the pupil of the eye. This latter in feeble 
light may be assumed to be from two fifths to one half a centimeter in 
diameter. It follows that the lowest power which can be used to advantage 
is from two to two and a half times the diameter of the objective in 
centimeters. 

359. Galileo's Telescope. G-alileo's telescope is the simplest 
of all telescopes giving an erect image. To the astronomical 
telescope an inverting system must be added if the image is 
to be erect. If an ocular system, similar to a compound 



338 



LIGHT 




Fig. 241 



microscope, be substituted for the ordinary eyepiece of an 
astronomical telescope, the image will be erect. But this 
substitution means more surfaces and more loss of light. 

Galileo's telescope consists of only two lenses, the objective 
MN, and the diverging system RS (Fig. 241). The distance 

between the two 
lenses is approx- 
imately the dif- 
ference of their 
focal lengths ; in 
the astronomi- 
cal telescope it 
is their sum. It 
was invented by Galileo in 1609, and about six months later 
he discovered with it four satellites of Jupiter, and soon 
thereafter the mountains on the moon, spots on the sun, and 
the variable phases of the planet Venus. 

In consequence of the divergence of the rays emerging from 
the eyepiece, Galileo's telescope has only a small field of view, 
and the greater magnification, the smaller the field. For this 
reason the use of this type of telescope is restricted to cases 
where only moderate magnification is required. The bin- 
ocular opera glass consists of a pair of Galilean telescopes 
combined with their axes parallel. It produces an image in 
each eye and thus secures greater brightness. The magnifi- 
cation in opera glasses is rarely more than three diameters. 

A peculiarity of this telescope, which precludes its use for 
astronomical measurements, is that no cross wires can be used 
in it. To be of service a cross wire must coincide with the 
real image given by the objective; but no such image is 
formed in Galileo's telescope. Another peculiarity is the 
absence of the ocular circle. There can be none because the 
image of the object glass formed by the eyepiece is virtual. 



MICROSCOPES AND TELESCOPES 339 

Problems 

1. The frequency n for the spectral line H l is 756 x 10 12 . What ve- 
locity of a star in the line of sight is derived from a displacement of H^ 
equal to 0.15 of a unit toward the violet end of the spectrum? 

2. Since light travels in straight lines, the intensity of illumination is 
inversely as the square of the distance. If an incandescent light and an arc 
light of 16 and 400 candle power respectively are placed 180 cm. apart, 
where' must a white screen be placed to receive equal illumination from 
the two sources ? 

3. If a crystal of Iceland spar has its faces cut parallel to its optic 
axis and 4 cm. apart, how far below the upper surface are the ordinary and 
extraordinary images of a mark on its lower face when viewed normally ? 

4. A narrow aperture 0.05 mm. wide is placed parallel to a screen 3 m. 
distant. If the aperture be illuminated with a beam of parallel rays of 
sodium light ( X = 5890 x 10- 10 m. ), what will be the distance between the 
central image and the first dark band of the diffraction image on the screen ? 

5. A glass grating is ruled with 4245 lines to the centimeter. When 
plane waves of sodium light are incident on this grating normally, the 
image of the second order is at an angular deviation of 30 (0 2 ) Find 
the wave length. 

6. If the greatest distance for distinct vision for a near-sighted person 
is 10 cm., what should be the focal length of glasses to read at a distance 
of 25 cm. ? 

7. If the nearest distance for distinct vision for a far-sighted person is 
80 cm., what should be the focal length of converging lenses to read at a 
distance of 25 cm. ? 



HEAT 

CHAPTER XIII 

NATURE AND EFFECTS OF HEAT 
I. HEAT AND TEMPERATURE 

360. Sensations of Heat and Cold. If one grasps an iron 
bar, heated in a blacksmith's forge, it is said to feel hot ; on 
the other hand, if one holds in one's hands a block of ice, it is 
said to feel cold. The words hot and cold belong to primi- 
tive language and express sensations with which every one 
is familiar. They are not given by sight or hearing, but by 
a specialized sense which is often confused with that of touch. 
The iron bar feels hot because it imparts heat to the hand ; 
the ice feels cold because the hand loses heat to it. 

Experience has thus made us acquainted with a physical 
state of bodies which is quite independent of their mass 
motions or the forces acting on them. This physical condi- 
tion is made known to us in common experience by the sense 
of heat, and the experience is often coupled with painful 
consequences. The sensation and its consequences are asso- 
ciated with fire and they are ascribed to heat as the cause. 
It is with heat as a physical quantity that we have now 
to deal. 

361. Heat a Form of Energy. Up to the beginning of the 
last century heat was thought to be a very tenuous form of 
matter, or subtle fluid, that could penetrate liquid and solid 
bodies with ease. This imaginary fluid was called caloric ; 
hence the caloric theory of heat. 

340 



HEAT AND TEMPERATURE 341 

About the beginning of the nineteenth century, the classi- 
cal experiments of Count Rumford and Sir Humphry Davy 
demonstrated that the caloric theory was no longer tenable. 
The former observed that in the boring of brass cannon with 
a blunt drill in the arsenal at Munich much heat was gener- 
ated, while only a small quantity of metal was abraded. 
This abraded material showed no change in its capacity for 
heat, and at the same time no stage was reached where the 
heat of the metal showed any signs of exhaustion. All that 
was required for the continued generation of heat was the 
further expenditure of work in turning the drill against fric- 
tion. Rumford concluded that the heat which a system of 
bodies can thus continue to furnish indefinitely cannot be a 
material substance. 

About the same time Davy, then a young man of only 
twenty-one, showed by experiment that enough heat could 
be developed by the friction of one block of ice against an- 
other to melt the ice. At the same time the water, instead 
of having its heat capacity diminished by the process, as was 
claimed by the calorists in the case of the brass abraded in 
boring the cannon, has actually more than twice the capacity 
for heat that the ice has from which it comes. 

Near the middle of the last century Joule of Manchester 
in England demonstrated by extensive experiments that a 
definite amount of mechanical work is the equivalent of a 
definite amount of heat. It was thus finally settled that 
there is no such substance as caloric, but that heat is a form 
of energy. 

If the work done on a system of bodies is not directed by 
the environment or the mechanism into some other form of 
energy, it always appears as an equivalent of heat. Thus, 
an expert blacksmith may heat a small bar of iron red-hot 
by rapid dextrous hammering; the kinetic energy of a 
swiftly moving cannon ball is converted into heat when it 
strikes the target, as evinced by the flash of light ; the fric- 
tion of a piece of steel on an emery wheel converts the 



342 HEAT 

energy expended in turning it into sufficient heat to raise 
the temperature of the abraded particles to the point of igni- 
tion ; the kinetic energy of a heavy moving railway train is 
converted into heat chiefly by the friction of the brakes. Glow- 
ing bits of metal from the wheels often give evidence of the 
high temperature of the abraded iron. The heating of a 
compression pump, when used for the inflation of a bicycle 
or an automobile tire, bears testimony to the conversion of 
mechanical work into heat by the compression of air. The 
converse expenditure of heat in doing mechanical work 
by means of a heat engine is now so common an operation 
that comment relating to its bearing on the question of 
the nature of heat as a form of energy is superfluous. In 
the case of high-speed machinery, it is difficult to prevent 
the reversion of this energy into wasteful heat at the 
bearings. 

We conclude therefore that heat is a form of energy ; and, 
if so, it can be measured in ergs, joules, kilogram meters, or 
foot pounds. 

362. The Molecular Theory of Heat. The kinetic theory of 
matter, from which most of the properties of a gas have been 
deduced, assumes that the molecules of a body are in a state 
of incessant agitation ( 164). The innumerable collisions 
between molecules, and their mutual jostling, after disturb- 
ance from without, must speedily produce a uniform state, in 
which the molecular motions are in all possible directions. 

After this steady state has been established, the molecular 
energy is uniformly distributed throughout the body ; it is 
largely kinetic, but may be in part potential. Heat then is 
supposed to be the energy due to the irregular motion of the 
molecules of a body. The caloric theory demanded the con- 
servation of heat as a necessary corollary. It was supposed 
to be invariable in amount, but might become hidden or 
"latent." In the modern theory heat is convertible into 
other forms of energy and is incessantly varying in amount. 



HEAT AND TEMPERATURE 343 

The caloric theory is inconsistent with the doctrine of the 
conservation of energy. Some of its terminology still re- 
mains as relics of an obsolete theory of heat. 

363. Temperature. Common observation teaches us that 
when a hot body is placed in contact with a cold one, the 
latter becomes warmer and the former cooler. This process, 
tending toward thermal equilibrium, is assumed to be the 
passage of heat from the hot body to the cold one, and the 
assumption is entirely independent of any theory of heat. 
The two bodies between which there is a passage of heat are 
said to be at different temperatures. If A is the hot body 
and B the cold one, the temperature of A is higher than 
that of B, and the heat flows from the body of higher tem- 
perature to the one of lower. , 

Temperature may be defined as the thermal state of a body 
which determines the transfer of heat between it and other bodies. 
In the unaided or spontaneous transfer of heat, the body 
losing heat during the process of equalization is said to be 
at the higher temperature. 

Temperature should not be confused with quantity of 
heat. A cupful of boiling water is at a higher temperature 
than a pailful of tepid water, but the latter gives out the 
greater quantity of heat in cooling down to the freezing 
point because of the greater quantity of water. 

364. Sensation not a Reliable Measure of Relative Tempera- 
ture. Incorrect estimates of relative temperatures are often 
drawn from the sensation of touch. Our judgment of the 
temperature of the atmosphere is influenced by the wind and 
by moisture in the air. If the left hand is held in hot water 
and the right in ice water for a few seconds, and immediately 
thereafter, if both are thrust into tepid water, the latter will 
feel cold to the left hand and hot to the right. The sensa- 
tions are dependent in no small degree on immediately 
preceding experience. Even scientific instruments for meas- 
uring temperature are not entirely independent of their pre- 



344 HEAT 

vious history, but they are much more so than the sensations 
of the hand. 

It is evident that some method independent of physical 
sensation must be employed for the reliable measurement of 
the relative temperatures of bodies. The method most com- 
monly used is the increase in the volume of a body attending 
a rise in its temperature. 

II. THERMOMETRY 

365. The Thermometer. An instrument designed to meas- 
ure temperature, as a physical quantity, is called a thermome- 
ter ; if the temperature to be measured is high, it is called a 
pyrometer. 

The choice of the thermometric substance depends largely 
on the purposes for which the thermometer is to be used. 
For the construction of a standard scale of comparison, hy- 
drogen gas is the substance most commonly employed. It 
may be inclosed in a constant volume thermometer, in which 
the increase of pressure is measured for a given increment 
of temperature when the gas is inclosed in a vessel of con- 
stant volume ; or in a constant pressure instrument, in which 
the increase of volume under constant pressure for a given 
increment of temperature is measured. In the former it is 
assumed that the change of pressure is proportional to the 
change of temperature producing it. In the latter form the 
same assumption is made relative to the volume. Neither 
assumption can be demonstrated experimentally with preci- 
sion, for to do so implies the previous possession of an accu- 
rate instrument possessing this same property. But there 
are theoretical reasons for supposing the law to be correct, 
at least for gases of sufficient tenuity, which follow Boyle's 
law most closely. 

For domestic and commercial purposes, the thermometric 
substance in common use is mercury in glass. For tempera- 
tures below the freezing point of mercury ( 370), alcohol 
is often employed, but on account of its irregularities, 



THERMOMETRY 



345 



toluene is to be preferred. Pentane remains liquid at a 
much lower temperature than alcohol, and can be used for 
extremely low temperatures. All such thermometers must 
be carefully compared with a standard hydrogen scale. 

366. The Mercurial Thermometer. The mercurial thermom- 
eter consists of a capillary glass tube of uniform bore, at one 
end of which is blown a bulb, either spherical or cylindrical 
(Fig. 242). The bulb and a portion of the tube are filled 
with mercury. Small changes in the volume of 
the mercury in the bulb due to changes of tem- 
perature show themselves by an appreciable 
motion of the end of the thread of mercury in 
the capillary tube. 

The observed dilatation of the mercury is its 
excess above that of the glass envelope. The 
assumption is that a change of temperature is 
proportional to the apparent change in volume 
of the mercury. But thermometers made of 
different kinds of glass do not quite agree 
among themselves, and none of them agree 
precisely with the normal hydrogen scale. 

Mercury is chosen as the common thermometric sub- 
stance for several reasons. Among them is the fact that 
it can be obtained quite easily in a pure state by distilla- Fig 242 

tion in a vacuum ; also, its dilatation is relatively large 
in comparison with that of glass, so that it indicates small changes in 
temperature ; in a pure state it does not adhere to the walls of the glass 
tube, and the end of the thread of mercury is therefore well defined ; its 
capacity for heat is small, and there is thus little transfer of heat between 
it and the body whose temperature is to be measured in coming to thermal 
equilibrium. 

On the other hand, the glass is always more or less affected by its pre- 
vious history. The bulb contracts slowly for a long period after it has been 
blown in acquiring molecular equilibrium, and this slow contraction 
raises the readings for all temperatures of the scale. Moreover, the 
apparent dilatation of the mercury is affected by the irregular dilatation 
of the glass; besides, on account of the great density of mercury, its hydro- 



346 HEAT 

static pressure enlarges the bulb, and the readings of a sensitive vertical 
instrument, in which the bulb is large and the stem fine, are not in exact 
agreement with those in a horizontal position. " A sensitive thermometer 
should, therefore, always be compared with a standard thermometer in 
the position, horizontal or vertical, in which the former is to be used. 

367. Two Fixed Points on the Scale. If the expansion of 
mercury by heat is to be utilized in measuring temperature, 
riot only must a scale be chosen, but there must be certain 
fixed points of reference before the scale can be applied. No 
two bulbs and their tubes agree in size, and each scale must 
be graduated separately. This may be done by comparison 
with a standard, but the graduation is commonly applied 
first and the correction afterwards. 

The fixed points of temperature, which may be verified at 
any subsequent time, are the melting point of pure ice and 
the temperature of steam from water boiling under standard 
atmospheric pressure. They are called the freezing point 
and the boiling point. The first is determined by marking on 
the stem the position of the end of the thread of mercury 
when the thermometer is surrounded with pounded ice in a 
room above freezing temperature. It is of prime importance 
that the ice should be free from contamination such as salt; 
otherwise the freezing point will be too low. 

The determination of the boiling point is much more diffi- 
cult, for the temperature of steam in contact with boiling 
water varies with the pressure under which it is formed. 
The thermometer is placed in the steam and not in the water 
because under the same pressure the steam is always at the 
same temperature, while that of the boiling water depends 
slightly on the vessel and on the presence of foreign sub- 
stances in it. Also, if the steam escapes freely into the open 
air, the pressure of the steam is the same as that of the at- 
mosphere, and this may be measured by the barometer. If 
the reduced height of the barometer is not 760 mm., a cor- 
rection must be applied to the observed boiling point ( 405). 
That two fixed points on the scale of a mercurial thermom- 



THERMOMETRY 347 

eter are necessary in order to make the readings of one 
thermometer comparable with those of another appears to 
have been first recognized by Newton in 1701. 

368. Thermometric Scales. Fahrenheit appears to have 
made his first thermometers about 1714, but the earliest pub- 
lished description of them he contributed to the Philosophical 
Transactions in 1724. He had already attained some celeb- 
rity, for in that year he was elected to the Royal Society of 
London. 

In 1724 the scale of Fahrenheit's thermometer for meteor- 
ological purposes began at and ended at 95. He de- 
scribes his scale as depending on the determination of three 
points : the lowest was the and was found by a mixture of 
ice, water, and sea salt ; the next was the 32 point and was 
found by dipping the thermometer in a mixture of ice and 
water without the salt ; the third was the 96 point to which 
alcohol expanded " if the thermometer be held in the mouth 
or armpit of a healthy person." The divisions were called 
degrees. When this scale was extended, the boiling point 
was found to be 212. It has since been determined that 
the normal temperature of the human body is not 96, but 
98.4, on Fahrenheit's scale. 

It has also been supposed that Fahrenheit's division of the 
scale from freezing to boiling into 180 was in imitation of 
the division of a semicircle into 180 degrees of arc. The 
division of a circle into 360 is a survival of the sexagesimal 
system, and is convenient because 360 has a large number of 
divisors. 

The Centigrade scale was introduced by Celsius, professor 
of astronomy in the university of Upsala, about 1742. He 
divided the interval between the freezing and the boiling 
point into 100 equal parts, but he placed the at the boiling 
point and the 100 at the freezing point. The inversion of 
this scale, making the the freezing point, was due to 
Stromer, a colleague of Celsius, eight years later. The sim- 



348 HEAT 

plicity of Celsius's mode of division of the distance between 
the two points of reference has led to its general adoption 
in all countries for scientific purposes. 

The scale of Reaumur, on which the freezing point is 
marked and the boiling point 80, has nothing to recom- 
mend it except that he avoided the misplaced zero of 
Fahrenheit. It is said that "he found that spirit of wine, 
mixed with one fifth water, expanded between the freezing 
and the boiling temperatures of water from 1000 to 1080 
volumes ; so he divided the intervening distance on the stem 
into 80 parts." This scale is still used for domestic purposes 
in Germany, but usually alongside the Centigrade scale. 

The number of thermometric scales used in the eighteenth 
century was at least nineteen. Fortunately all but three of 
them have passed into ancient history, and the sooner the 
Centigrade becomes the sole survivor the better. 

Each of the three scales is extended beyond the fixed 
points as far as desired. The divisions below are read as 
minus and are marked with the negative sign. The initial 
letters F. , C. , and R. indicate the Fahrenheit, the Centigrade, 
and the Reaumur scales respectively. In this book Centigrade 
degrees will be understood unless another scale is indicated. 

369. Conversion of Readings from One Scale to Another. AB 
in Figure 243 is a thermometer with the three scales attached, 

P is the head of the 

A 7> T> 

so thread of mercury, 



Fahrenheit 



and F, C, and R are 

, . , the readings on the 

Reaumut ' ' ' 





three respective scales. 

On the Fahrenheit 

scale AB = 180 and AP = F - 32, since the zero is 32 divi- 
sions below A ; on the Centigrade AB = 100 and AP (7; 
on the Reaumur AB = 80 and AP = R. Then the ratio of 

TT _ OO SY ID 

AP to AB is = JQQ = Q % substituting in this 



THERMOMETRY 349 

equation the reading on any scale, the equivalent on either of 
the others may easily be found. For Fahrenheit readings 32 
must be subtracted algebraically to find the number of 
degrees between the freezing point and the reading. Thus 
50 F. is 50 - 32 = 18 above freezing; and - 10 F. is - 10 
_ 32 = - 42, or 42 below freezing. 

As an example of conversion from one scale to another, if 
it is desired to express 68 F. in degrees on the Centigrade 

seale, then ^_=_2 = _ and C== 2 0. 



370. Limits of the Mercurial Thermometer. Since mercury 
freezes at 38.8, the mercurial thermometer cannot be 
used for temperatures below that point. The scale may be ex- 
tended downward by alcohol to about 110, and by pentane 
to 200. But the dilatation of these liquids is not 
uniform ; and thermometers filled by them must be *^ 
calibrated by comparison with a standard. 

Mercury boils under atmospheric pressure at about 
350. For higher temperatures up to 500, mercu- 
rial thermometers may be used if the space above the 
mercury is filled with nitrogen under pressure to 
prevent boiling (405). This is the upper limit for 
mercury in glass because the glass softens at higher 
temperatures. If the tube is made of fused quartz, the 
temperature of the mercury under sufficient pressure 
may be carried as high as 700. The portion of the 
scale above 100 must be graduated by comparison 
with the normal hydrogen scale. 

371. The Clinical Thermometer. The clinical thermometer 
in universal use among physicians and surgeons is a sensitive 
instrument of short range for indicating maximum temperatures. 
It is usually graduated from 95 to 110 F., or from about 35 to 

45 C. There is a constriction in the tube just above the bulb Fig 2 44 
(Fig. 244), which causes the thread of mercury to break at that 
point when the temperature begins to fall, leaving the top of the 
disengaged thread to mark the highest temperature registered. The 



350 



HEAT 



mercury can be forced down past the constriction by utilizing its inertia 
in tapping or jarring the thermometer. 

372. Beckmann's Thermometer. To increase the length of 
a division for a Centigrade degree, so that it maybe divided into 
hundredths on the scale, the bulb must be large and the capillary 
tube very narrow. The range of such an instrument is then quite 
limited, seldom exceeding ten degrees. To extend its availability 
Beckmann's thermometer has a reversed bulb at the top as a 
reservoir for excess mercury (Fig. 245). The bulb is 10 or 12 mm. 
in diameter and the stem is graduated to hundredths of a degree. 
The range is usually six degrees. When it is desired to use this 
thermometer for temperatures above 6, it is raised to a tempera- 
ture a little above the one to be measured, the excess mercury 
flowing into the upper bulb. Then the thread is detached with a 
shake, and the mercury remaining in the bulb and tube is used 
in the ordinary way. Such a thermometer is commonly employed 
for measuring small differences of temperature very accurately ; 
the temperature corresponding to any point on the scale may be 
determined by comparison with a standard instrument. 

373. Galileo's Air Thermometer. The first air ther- 
mometer was invented by Galileo and it served in his 
time for the detection of fever. In its 

early form it consisted of a glass bulb 
at the end of a narrow tube, which 
was supported vertically in front of a 
scale and dipped into a vessel of 
colored liquid (Fig. 246). A small 
portion of the air in the bulb is ex- 
pelled by warming, so that the colored 
alcohol will rise in the tube when 
the bulb again cools. If now the 
temperature rises, the liquid column 
falls ; if the temperature falls, the 
liquid column rises. The instrument 
is remarkable for its sensitiveness, but 
until it has been greatly modified, it is 
Rg. 245 ou iy a thermoscope, because its readings 
change with barometric pressure as well as with temperature. 




EXPANSION 



351 



The practical methods of using a gas as a thermometric substance are 
described in memoirs and large treatises. A description of a simple form 
of gas thermometer and the definition of the " absolute zero " are reserved 
for a later section. 

III. EXPANSION 

374. Expansion of Solids. It is a familiar fact that solids in gen- 
eral expand when heated and contract again when their temperature 
falls. Gravesande's ring (Fig. 247) illus- 
trates the expansion of a metal ball 
which just passes through the ring when 
both are at room temperature; but if 
the ball is heated in boiling water, it 
will no longer pass through the ring 
until it has again cooled. 

If a strip of sheet iron and one of 
copper are riveted together (Fig. 248) 
and supported at the ends, heating with 
a spirit lamp or a Bun sen burner will 
cause the compound bar to bend into an arc of a circle with the copper 
on the convex side, because the expansion of 
the copper is greater than that of the iron. 

A metal wire increases in length when 
heated. In Figure 249 an iron wire about a 
meter long is attached to a wooden support at 
one end A and at the other end to a screw 

eye in a long wooden pointer BC. The pointer is free to turn around a 
pin at B very near the screw eye, and its weight keeps the wire stretched. 
When the wire is heated by passing through it a suitable electric cur- 




Fig. 247 




Fig. 249 

rent, its expansion is indicated by a long sweep of the pointer. The wire 
cools quickly after the current is cut off and the pointer returns to its 
former position. 

Wagon tires are shrunk on wooden wheels by putting them on while red- 
hot and suddenly cooling. Steel rims are shrunk on street car wheels in 
the same manner. Large guns at the Washington Navy Yard are made 



352 HEAT 

by slipping a red-hot outer cylinder over a cold inner one in a vertical 
position; when the hot cylinder cools, it contracts powerfully on the 
cold one. 

375. Coefficient of Expansion. The question as to how much 
a body changes in dimensions with a given change of tem- 
perature can always be answered if its coefficients of expan- 
sion are known. 

Let 6? be the numerical measure of any dimension of a 
body at 0, d its value at , and Ad the change in this dimen- 
sion per degree. Then the relation between the values of 
the dimension at the two temperatures may be expressed 
with sufficient accuracy for most purposes by the linear 
equation 



If now be substituted for the ratio Ad/c? , the equation 
becomes (69) 



The coefficient of expansion may be denned as the ratio 
of the increase in the dimension per degree to the dimension 
at zero. 

376. Coefficients of Expansion of a Solid. A solid has three 
coefficients of expansion, according as d is regarded as a 
length, an area, or a volume. Equation (69) has three cor- 
responding forms, 



(70) 

v = 

The constants X, o-, a are the coefficients of linear, superfi- 
cial, and cubical expansion respectively. 

The relation between these three coefficients is a simple 
one in the case of isotropic bodies which expand equally in 
all directions, that is, have the same linear coefficient in three 
directions at right angles. Then the coefficient of expansion 
in volume is three times that in length. The volume of a 



EXPANSION 353 



cube whose edge is Z at is 7 3 (1 4- 

v (l + at). Equating these values, and remembering that 

V Q = Z 3 , we have 

1 + at = (1 + \e) 8 = 1 + 3 \t + 3 X 2 2 + X 3 * 3 . 

But since X is a number of the order of 1/100,000, its 
higher powers may be neglected in comparison with the 

first, or 1 + = 1 + 3 M, and a = 3 X nearly. 

In a similar manner it may be shown that cr = 2 X. 

In general crystals have three axes of expansion at right 
angles, and the three linear coefficients in these directions 
are not identical. The coefficient of volume is then equal to 
the sum of the three linear coefficients. In such cases a 
crystalline sphere at one temperature ceases to be spherical 
at other temperatures ; also a crystalline body in the form 
of a cube at one temperature does not remain cubical when 
the temperature changes, unless the crystal belongs to the 
cubic system. 

When quartz crystals are strongly heated, their unequal 
expansion in different directions causes them to burst into 
small pieces. Large diamonds have been known to break 
into fragments by the heat of the hand when they are first 
taken from the earth. 

LINEAR COEFFICIENTS OF SOLIDS 

(Centigrade Scale) 

Glass. ...... 0.0000088 Mercury (cubical) . . 0.000180 

Platinum ..... 0.0000089 Quartz || axis .... 0.000007 

Copper ...... 0.000017 Quartz _L axis . . . 0.000014 

Gold ....... 0.000015 Quartz fused .... 0.0000004 

Iron ....... 0.000012 Ebonite ...... 0.00008 

Lead ....... 0.000029 Porcelain ..... 0.0000027 

Silver ...... 0.000019 Nickel steel (36% Ni) 0.0000009 

Zinc ....... 0.000030 

377. Applications of Expansion. In addition to the instances of 
expansion already mentioned, the following more or less familiar facts 



354 HEAT 

may be accounted for by the expansion taking place with changes of 
temperature : 

When hot water is poured into a thick glass vessel, especially if it is 
not well annealed and so under stress, as shown by polarized light 
( 348), it will probably break, because the temperature difference be- 
tween the outer and inner portions produces stresses greater than the 
glass can sustain. The low temperature of a mixture of solid carbon 
dioxide and sulphuric ether is equally destructive. 

After quartz has been fused it loses its crystalline structure and its 
linear coefficients of expansion become equal and very small. It is for 
this reason that vessels made of fused quartz may be raised to a 
red heat and plunged into cold water without breaking. 

The inequality in the coefficients of expansion of different 
substances is utilized in thermometers, in the compensated clock 
pendulum, and in the compensated balance wheel of a watch. 

Graham's mercurial pendulum bob consists of one or more 
glass jars, nearly filled with mercury, and attached to the lower 
end of the pendulum rod (Fig. 250). A rise of temperature 
lengthens the rod, and lowers the center of oscillation of the 
pendulum ; but the mercury expands upwards and compensates 
by raising the center of oscillation. The adjustment of the 
compensation is made by raising or lowering the cylinder of 
mercury by means of a screw. 

The rate of a watch or a chronometer depends largely on the 
balance wheel. Unless this is compensated, it expands when 
the temperature rises and the watch loses time, the larger wheel 
oscillating more slowly under the force supplied by the elas- 
Fig 250 ticity f the hairspring. Compensation is effected by making 
the rim of the wheel in two or more sections, each being made 
of two materials (like Fig. 248) and supported by one end on a separate 
arm (Fig. 251). The more expansible* metal is on the outside. Then 
when the temperature rises and the wheel as a 
whole expands, the loaded ends a, a of the sections 
move inward, thus compensating for the increase 
in the length of the radial arms. The small 
screws on the rim are for the adjustment of the 
moment of inertia of the wheel. In practice the 
compensation is seldom perfect, and the rate of 
the chronometer must be checked by stellar ob- 
servations, that is, by means of the uniform rota- 
tion of the earth on its axis. 

'It has been found possible to make a nearly perfect compensation for 
expansion by a particular percentage composition of an alloy of steel and 




EXPANSION 



355 



nickel, named "invar" by its inventor, Guillaume of Paris. It contains 
36 per cent of nickel and its coefficient of linear expansion is only one 
tenth that of glass and less than one part in a million. This property 
commends it for use in the construction of standards of length, of pendu- 
lums, and for other instruments in which invariability is highly desirable. 

378. Expansion of Liquids. In the case of liquids and 
gases expansion in volume only comes into consideration. 
With the exception of water and aqueous solutions, the 
expansion of fluids is positive and larger than that of solids. 
Two values of this coefficient for fluids must be carefully 
distinguished, the absolute and the apparent. The former is 
defined by the third equation of (70) ; the latter is the 
observed coefficient when its value is affected by the expansion 
of the containing vessel. The absolute coefficient is the sum 
of the^ apparent coefficient and the coefficient of the vessel. 

The expansion of liquids in general increases with the 
temperature and becomes large at high temperatures. The 
volume of liquids cannot be expressed with one coefficient 
as a linear function of the temperature, but it requires two 
or more constants in the equation connecting volume and 
temperature. The equation has the form v = v c 
+ 7^ 3 + ) The coefficients above the 
first are usually relatively small, but they 
are not negligible. 

379. Expansion of Water, Water exhibits 
the remarkable property of contracting when 
heated at the freezing point. This contrac- 
tion continues up to 4 ; at this temperature 
expansion sets in, so that the greatest density 
of water is at a temperature of 4, and its 
density at 6 is nearly the same as at 2. 

The peculiar behavior of water is illus- 
trated by Hope's apparatus (Fig. 252). It consists of a 
glass jar with thermometers inserted near the top and bottom. 
Around the middle is an annular reservoir. If the vessel is 




Fig. 252 



356 



HEAT 



filled with water at about 10, the upper thermometer will 
show at first a slightly higher temperature than the lower 
one. If now the trough at the middle be filled with a freez- 
ing mixture, the first effect will be the gradual fall of the 
lower thermometer to 4 without much change of the upper 
one. After the lower thermometer becomes stationary, the 
upper one falls rapidly until its reading is reduced to zero 
and ice forms at the surface. The water at 4 sinks to the 
bottom, while that below 4 is lighter and rises to the top, 

where the freezing begins. For this 
reason ice forms at the surface of a 
body of cold water, which freezes 
from the surface downward, instead 
of from the bottom upward. 

The relation between the volume 
and the temperature of water near 
the freezing point may be determined 
by means of a large thermometer 
filled with distilled water. If the 
apparent volumes of the water in 
glass are plotted as ordinates and 
the corresponding temperatures as 
abscissas, the curve is approximately 
a parabola abc (Fig. 253). The ver- 
tex is somewhat above 4 and this is 
the temperature of the least apparent 
volume. The observations for this 
curve include the dilatation of both 
the glass and the water. The true volume-temperature curve 
for water may be found by adding to the ordinates of this 
curve the expansion of the glass. For this purpose, it is only 
necessary to draw a line OD, making with the axis of tem- 
peratures an angle whose tangent, expressed in terms of the 
vertical and horizontal scales, is the dilatation of glass for 
one degree. If then the vertical distance between OX and 
OD are added to the corresponding ordinates of abc, the result 



140 


















I 










1 










1 i 








1 


/ 




a 




/ 


/ 




V 


tl 


/ 




20 


^ 


.^x^ 


' 





1 X 



Fig. 253 



EXPANSION 357 

is the curve adf. The point of least volume and greatest 
density corresponds to the shortest vertical distance between 
OD and the curve abc. This may be found by drawing a 
tangent to the curve parallel to OD. The tangent touches 
the curve very near the ordinate passing through 4, and this 
is the temperature of least volume. 

380. Expansion of Gases Law of Charles. The law of 

expansion of gases, discovered by Charles in 1787 and con- 
firmed by Gay-Lussac in 1802, is the following: The volume 
of a given mass of any gas, under constant pressure, increases 
from the freezing to the boiling point by a constant fraction 
of its volume at zero. This law is known as the law of 
Charles or of Gay-Lussac. For the Centigrade scale the 
constant fraction is 0.3665 for dry air. This is equivalent 
to 0.003665 for one degree C. A near approximation is 
1/273. Therefore 30 cm. 3 of a gas at become about 41 
cm. 3 at 100. 

It follows from the law of Charles that the expression for 
the dilatation of a solid may also be applied to that of a gas 
under constant pressure, or 

v = V Q (1 + at) = v (l + 0.0036650- (71) 

The investigations of Regnault and others have shown 
that this law, like that of Boyle, is not exact, but is a close 
approximation only in the case of gases which most nearly 
obey Boyle's law. 

A gas which would obey Boyle's law exactly is known as 
an ideal or perfect gas. For such a gas, the product pv of 
the pressure and the volume, for a constant temperature, is a 
constant. It follows that this product is some function of 
the temperature, or 



It is obvious from this expression that the changes pro- 
duced by the application of heat to a gas may be investigated 
either by observing the changes of volume under constant 



358 



HEAT 



pressure, or the changes of pressure at constant volume. 
The two methods have been found to give nearly identical 
results. 

COEFFICIENTS OF EXPANSION AND PRESSURE 





CONSTANT PRESSURE 


CONSTANT VOLUME 


Hydrogen 


003660 


003663 




3663 


3664 


Air 


3671 


3665 




3671 


3668 




3710 


3686 


Cyanogen 


3877 


3682 


Sulphur dioxide .... 


3903 


3670 









The easily liquefiable gases at the bottom of the list have a larger co- 
efficient of dilatation than those which are liquefied with great difficulty. 
It was found that the coefficient of the gases investigated approach 
equality as the initial pressure decreases ; and the more highly rarefied 
they are, the more nearly do they approach the ideal limit of exact 
obedience to Boyle's law. 

381. The Absolute Scale of Temperature. The law of 
Charles leads to another scale of temperature known as the 
absolute scale. By this law the volumes of any mass of gas, 
under constant pressure at 0, and at any other temperature 
, are connected by the relation 

v t = v + the expansion = vi 1 H ). 

/ +i \ 

Also v tf =v + the expansion = v Q [ 1 H ). 

\ 273y 



t 

273 



Then 



Suppose now a new scale be chosen, whose zero is 273 de- 
grees Centigrade below the freezing point of water, and let 



EXPANSION 359 

temperatures on this new scale be denoted by T. Then 
273 + t will be represented by T, and 273 + t' by T' on the 

new scale, and 

v t = 273 + t = T 

v t , ~ 273 + 1' T'" 

or, the volumes of the same mass of gas, under constant pressure, 
are proportional to the temperatures on this new scale. The 
point 273 below is called the absolute zero, and the tem- 
peratures on this scale, absolute temperatures. Up to the 
present time it has not been found possible to cool a body to 
the absolute zero; but by evaporating liquid hydrogen under 
low pressure, Sir James Dewar reached a temperature which 
he estimated to be within 9 of the absolute zero. In phys- 
ical theory the absolute zero is more important, and, for 
practical purposes, it is more convenient, than the arbitrary 
zero of the Centigrade scale. 

382. The Laws of Boyle and Charles Combined. The laws of 
Boyle and Charles may be combined into one expression, 
which is known as the gas law, though it has greater gener- 
ality than its method of derivation from gases would imply. 

Let VQ, p Q , T Q IOQ the volume, pressure, and absolute tem- 
perature of a gas under standard conditions of tempera- 
ture and 76 cm. of mercury pressure. 

Also let v, p, T be the corresponding quantities at temper- 
ature T. 

Then, applying Boyle's law to increase the pressure to the 
value p, the temperature remaining constant, we have 



By changing the pressure from p Q to p, the volume has 
become v'. 

Next apply the law of Charles, keeping the pressure con- 
stant at the value p, and starting with the value v f . Then 

v' : v :: T Q : T. 



360 HEAT 

These changes have taken place by two successive steps. 
From the first proportion ^ = $- ; and from the second ~ = Q. 
Multiplying the two equations together member by member, 

= = a constant. 



This constant is usually denoted by R. We may there- 
fore write pv = RT (72) 

Since R in this equation is a constant, it follows that in a 
perfect gas obeying the laws of Boyle and Charles, both the 
pressure at constant volume and the volume under constant 
pressure are directly proportional to the absolute temperature. 

383. Numerical Value of the Constant R. If v in equation 
(72) is the volume containing the unit mass of the gas under 
standard conditions of temperature and pressure, the con- 
stant R will be inversely as the density of the gas; for, 
with p and T constant, R is proportional to v, and v is 
inversely as the density d. 

If, however, the same volume v be taken for different 
gases, then R will have the same value for all. But what 
shall this volume be ? The hypothesis of Avogadro is that 
equal volumes of all gases at the same temperature and pres- 
sure contain the same number of molecules. The masses of 
equal volumes are therefore proportional to the molecular 
weights. The particular volume chosen for the gas equation 
is the volume containing the gram molecule, that is, a number 
of grams equal to the molecular weight. 

The constant value of R may be found from the following 
data: 

The standard pressure p of one atmosphere is 1,013,250 
dynes per square centimeter ( 171). The density of oxy- 
gen at and under a pressure of one atmosphere is 0.0014279 
gm. per cubic centimeter, and its molecular weight is 32. 



EXPANSION 



361 



Hence the volume v containing 32 gm. is 32/0.0014279 = 
22,410 cm. 3 . Therefore 



= JZ 



x 



83,175,000. 



Since T is a number and the product pv is work, 
R = 8.317 x 107 ergs = 8.317 joules/degree. 

384. The Constant Volume Gas Thermometer. In the gas 
thermometer the increase in temperature is assumed to be 
proportional to the increase in pressure of the gas when its 
volume is kept constant. 

The dry gas is inclosed in a suitable bulb a (Fig. 254), 
which is connected by means of a capillary tube to one of 
larger cross section at 1$. The tubes 
CE and BD are joined by means of 
stout rubber tubing, which permits CE 
to be raised or lowered so as to keep 
the mercury surface at B. In the larger 
tube at -B is a hook pointing downward ; 
the surface of the mercury is always 
brought just in contact with this hook 
before a reading is taken. The volume 
of the gas is then a constant, subject to 
a small correction for the expansion of 
the glass. The difference in the level 
of the mercury at B and E, added to 
the height of the barometer, both cor- 
rected for temperature, gives the pres- 
sure under which the gas is in the bulb. 

Since hydrogen at high temperatures 
diffuses through the walls of a contain- 
ing vessel, nitrogen is sometimes used 
instead. For temperatures between 
and 100 the difference between the readings on the normal 
hydrogen scale and those with nitrogen are at most only a 




Fig. 254 



362 HEAT 

few hundredths of a degree ; at both lower and higher tem- 
peratures they become appreciable. 

If p Q is the pressure at and p the pressure at some 
higher temperature , then since the absolute zero is 273 
below the zero of the Centigrade scale, we may write 



. Whence * = 273-l (74) 

p 9 V A ) 

The pressure at zero is determined by surrounding the bulb 
of the thermometer with melting ice and taking readings. 
Any other temperature is then measured by observing the 
pressure necessary to keep the surface of the mercury at the 
fiducial point near B. 

TV. MEASUREMENT OF HEAT 

385. Unit Quantity of Heat. For the measurement of heat 
as a physical quantity no knowledge of the ultimate nature 
of heat is required ; the methods of measurement are based 
on some property or effect attributed to heat. The meas- 
urement of heat is called calorimetry. 

Heat, like other physical quantities, must be expressed in 
terms of a convenient unit. The unit quantity of heat is the 
heat required to raise the temperature of unit mass of water 
one degree. If the unit mass is the gram and the unit of 
temperature the degree Centigrade, the unit of heat is called 
the calorie. In engineering practice in England and America 
the British thermal unit (B. T. U.) is commonly employed. 
It is the heat required to raise the temperature of one pound 
of water one degree Fahrenheit. 

Since the quantity of heat which will warm a gram of 
water one degree at different temperatures between and 100 
is not rigorously the same, the temperature on the scale 
should be stated for an exact definition of the calorie. No 
agreement with respect to this point has been reached by 
physicists. If the interval is chosen from 15 to 16, the 
calorie will then be the one hundredth part of the heat re- 



MEASUREMENT OF HEAT 363 

quired to raise the temperature of one gram of water from 
to 100. 

386. Thermal Capacity. The thermal capacity of a lody 
is the number of heat units required to raise its tempera- 
ture one degree. The thermal capacity in calories of any 
mass of water is numerically equal to that mass in grams. 

The thermal capacities of equal masses of different sub- 
stances differ widely. Thus, if 100 gm. of mercury at 80 
be mixed with 100 gm. of water at 20, the temperature of 
the whole mass will be about 22. The heat given up by 
the mercury in cooling 58 heats the same mass of water 
only about 2. The thermal capacity of the water is about 
thirty times as great as that of the mercury. So also if 
100 gm. of copper at 100 be cooled in 100 gm. of water at 
0, the final temperature will be only 9.1. The heat lost 
by the copper in cooling through 90.9 is sufficient to raise 
the temperature of the same mass of water only 9.1. 

387. Specific Heat. The specific heat of a substance is the 
ratio between the heat capacities of equal masses of the sub- 
stance and of water. This amounts to saying that the thermal 
capacity of unit mass of a substance is its specific heat. 

For example, the heat capacity of one pound of iron is 
0.112 B.T.U. Also, the heat capacity of one gm. of iron is 
0.112 calories. Its specific heat is therefore 0.112. Specific 
heat is a ratio and is independent of the unit of heat meas- 
urement. The thermal capacity of a body is the product of 
its specific heat and its mass. The numerical value of ther- 
mal capacity depends on the units employed. 

388. Specific Heat by the Method of Mixtures. The method 
most commonly applied to determine specific heats is the 
method of mixtures. It is based on the experimental fact 
that when an exchange of heat takes place between bodies in 
thermal contact, the quantity of heat lost by one part of the 
system is the same as that gained by the other. In apply- 



364 HEAT 

ing this principle of equal heat exchanges, it is necessary to 
guard against its application in cases where there is either 
absorption or generation of heat, due to some reaction taking 
place between the parts of the mixture. For example, it 
cannot be applied to the determination of the specific heat 
of a salt solution by mixing it with water, for in general 
there is a change in temperature, due to the heat of dilution, 
when a salt solution is mixed with water, both at the same 
temperature. 

Let two bodies A l and A 2 have masses m 1 and m 2 , temper- 
atures ^ and 2 , and specific heats 8 1 and s 2 . If they are 
placed in thermal contact, they will reach some intermediate 
temperature , which is higher than ^ and lower than t 2 . 
Then the quantity of heat lost by A 2 will be w 2 s 2 ( 2 ), 
and the heat gained by A 1 will be m 1 s 1 ( f x ). Assuming 
now no generation or absorption of heat and that the only 
exchange is between A 1 and A 2 , the heat lost by one is 
gained by the other, or 

- = s (* ~ *) and ^ = 





If A 1 is water and its specific heat is unity, the last 
equation becomes s 9 = m ^ ~ l \ 



In practice it is necessary to take into account the ther- 
mal capacity of the calorimeter itself, since both the vessel 
and its contents change temperature equally. The number 
of heat units required to raise the temperature of the calo- 
rimeter one degree is called its water equivalent, because it is 
equal to the mass of water having the same thermal capacity 
as the calorimeter. 

Let the water equivalent be m. Then the heat gained by 
the calorimeter and its contents is (m -f- mS)(t ^), and the 
specific heat s 2 becomes T|i 






MEASUREMENT OF HEAT 365 

For the radiation correction for the exchange of heat 
between the calorimeter and its surroundings the reader is 
referred to laboratory manuals. 

SPECIFIC HEATS AT 50 

Antimony 0.051 Mercury 0.033 

Bismuth 0.031 Platinum 0.032 

Cadmium 0.059 Silver 0.056 

Copper 0.095 Tin 0.054 

Glass (Jena 16 1 ") .... 0.199 Zinc 0.093 

Iron 0.112 Ice (-30 0) 0.505 

Lead 0.031 Alcohol (10 15) .... 0.602 

Magnesium ...... 0.245 Turpentine 0.468 

389. Specific Heat of Water. Water has a higher thermal 
capacity than any other substance except hydrogen. The 
specific heat of water is nearly twice as great as that of ice 
(0.505), and more than twice as great as that of steam under 
constant pressure (0.477). 

The distribution of large quantities of heat in buildings 
by means of hot water is practicable because of its large 
thermal capacity. As it is, the radiating surface must be 
larger for heating by hot water than by steam. 

The beneficent influence of the water of the ocean in 
equalizing climatic differences between summer and winter 
at once suggests itself. The ocean stores the heat of summer 
and gives it out gradually in winter. Hence the absence of 
extremes in an island climate. 

The specific heat of water is not a constant. The exhaustive 
experiments of Rowland on the dynamical equivalent of heat 
were the first to demonstrate that the specific heat of water 
decreases from to about 30 and then increases again. The 
precise point of this minimum value is difficult to determine, 
since the change in the specific heat near the minimum is 
very small. 

390. Two Specific Heats of a Gas. The specific heat of a gas 
is an indeterminate quantity unless the conditions are defined. 



366 HEAT 

It may be measured in two ways : under the condition of a 
constant pressure, or at a constant volume. The former is 
the specific heat under constant pressure, and the latter the 
specific "heat at constant volume. If the pressure is maintained 
constant, the volume must increase, and in consequence the 
gas does work ( 66). The heat applied to a gas expanding 
under pressure not only raises the temperature and increases 
the kinetic energy of the molecules, but it also does the work 
of expanding under pressure. The specific heat under con- 
stant pressure is therefore greater than the specific heat at 
constant volume. The ratio between them is involved in the 
calculation of the velocity of sound in gases ( 208). 

The following table gives the value of the specific heat under constant 
pressure for several gases : 

Air 0.2374 Chlorine. ;,'.. ^. .<:. . 0.1240 

Bromine 0.0555 Hydrogen . . -, ; i. u , . 3.4000 

Carbon dioxide .... 0.2169 Nitrogen , ) ...,,.. . . . 0.2438 

Carbon monoxide . . . 0.2405 Oxygen 0.2175 

391, Thermochemical Equations. A chemical reaction is nearly 
always attended by a thermal change, which is generally an evolution 
of heat. In equations denoting thermochemical reactions the symbols 
serve to indicate the substances taking part in the reaction as in ordi- 
nary chemical equations; they denote also definite quantities of these 
substances in grams equal to their atomic or molecular weight. Further, 
they are energy equations and are incomplete unless the energy involved 
in the reaction is also included in them. Thus 

Na + Cl = NaCl + 20,400 cal. 

means that the union of 23.05 gm. of sodium and 35.45 gm. of chlorine 
produces 58.5 gm. of sodium chloride (one gram molecule), together with 
the generation of 20,400 calories of heat or the equivalent energy in some 
other form, such as that of an electric current. The 20,400 calories rep- 
resent the difference between the energy associated with these definite 
quantities of sodium and chlorine on the one side, and of the sodium 
chloride on the other. 

Thermochemical processes are called exothermic when heat is evolved, 
and endothermic when it is absorbed. In the former the energy in calories 
(or in joules) has the positive sign ; in the latter, the negative. 



CHANGE OF STATE 367 

The heat of reaction has different names according to the process pro- 
ducing it. Thus we have heat of formation, heat of hydration, heat of 
solution, heat of dilution, etc. The following examples illustrate them : 

(1) Heat of formation : 

Pb + 2 I = PbI 2 + 39,800 cal. (exothermic). 
C + 2 S = CS 2 - 28,700 cal. (endothermic). 

(2) Heat of hydration : 

ZnSO 4 + 7 H 2 O = ZnSO 4 7 H 2 O + 22,690 cal. 
CuSO 4 + 5 H 2 O = CuSO 4 5 H 2 O + 18,550 cal. 

(3) Heat of solution and dilution : 

ZnSO 4 . 7 H 2 O + 393 H 2 O = ZnSO 4 400 H 2 O - 4260 cal. 
CdSO 4 . | H a O + 397 H 2 O = CdSO 4 400 H 2 O + 2660 cal. 



(4) Heat of dilution : 

ZnSO 4 20 fr 2 O -f 380 H 2 O = ZnSO 4 - 400 H 2 O + 400 cal. 
ZnCl 2 . 5 H 2 O + 395 H 2 O = ZnCl 2 400 H 2 O + 8020 cal. 

(5) Heat of combustion : 

2 H + O = H 2 O + 69,400 cal. 
C + 2 O = CO 2 + 103,000 cal. 

V. CHANGE OF STATE 

392. Changes of State Produced by Heat. When a crystal- 
line solid is heated its temperature rises until it begins to 
pass into the liquid state. So long as the supply of heat is 
moderate, the temperature remains constant until the entire 
mass has fused or melted, after which the temperature rises 
again to the point at which boiling sets in. Again the 
temperature remains constant until the whole of the liquid 
has changed to vapor. The temperature will then again rise. 
At either stationary temperature and under a constant pres- 
sure the two states of the substance are in equilibrium with 
each other, and a mixture of the two will remain unchanged 
if no heat is supplied or lost. 



368 HEAT 

Most substances are -capable of existing in more than on6 
state or modification. Thus, water may exist as ice, liquid 
water, or water vapor; and sulphur as a vapor, a liquid, or as 
either of two solids crystallizing in different systems. Such 
modifications, when they exist together, and may be separated 
from each other, as ice from water, are known as phases. 
The physical conditions affecting equilibrium of these phases 
are temperature and pressure. 

At and under a pressure of one atmosphere ice and water 
have the same vapor pressure and are in equilibrium. They 
may exist together at this temperature and pressure in all 
relative proportions, and these proportions will remain un- 
changed so long as the mixture neither gains nor loses heat. 
So also at 100 and under atmospheric pressure, water and 
water vapor exist together, and the two phases are in 
equilibrium. 

393. The Melting Point. The melting or fusing point of a 
substance is the temperature at which the solid and liquid 
states or phases are in equilibrium under some defined pres- 
sure, usually atmospheric. Above this temperature the sub- 
stance is a liquid; below it, normally a solid. Solidification 
or freezing is the converse of fusion, and the temperature of 
solidification of any substance is normally the same as the 
melting point. 

The melting point of ice is sharply marked, and there is 
no appreciable difference of temperature between the melt- 
ing ice and the liquid phase into which it passes. This is 
generally true of crystalline substances, but the case is very 
different with amorphous solids, like wax, glass, and wrought 
iron, which cannot be said to have a definite melting point. 
Such substances soften and become plastic before reaching a 
more or less viscous liquid state. It is on account of this 
property that glass can be bent, moulded, drawn out into 
rods and tubes, or blown into various forms. Similarly the 
softening of wrought iron at a temperature much below the 






CHANGE OF STATE 369 

melting point permits the metal to be rolled, forged, and 
welded. In the fusion of wax the outer portions are softer 
than the interior and are at a slightly higher temperature. 
The experiments of Person go to show that ice begins to 
increase in specific heat between 2 and 0, and that there 
is a very small range of temperature within which it softens 
and melts. The difference between it and wax from this 
point of view is one of degree. 

In general crystalline bodies have a definite fusing point, 
or a temperature at which they may exist either as a solid or 
a liquid ; amorphous bodies, on the other hand, pass gradually 
from the solid to the liquid phase. A liquid which passes 
abruptly from the solid to the liquid state may be carefully 
cooled several degrees below the normal freezing point with- 
.out solidifying. Thus water, protected from vibrations or 
covered with oil, will remain liquid at 10, or in capillary 
tubes at 20. This phenomenon is called undercooling. 

Undercooled liquids are in unstable equilibrium ; for if 
they be jarred, or if a solid fragment of the same substance 
be dropped into the liquid, solidification takes place rapidly 
with disengagement of heat, the temperature rising to the 
normal freezing point. 

As examples, water in a small bulb containing also a thermometer 
may be cooled oy a freezing mixture to 8 or 10. When it finally 
freezes the temperature rises rapidly to 0. 

Hyposulphite of sodium may be melted at 50 and then cooled without 
solidifying to room temperature. The addition of a small crystal of the 
salt destroys the equilibrium, solidification sets in around the crystal, 
and the temperature rises to the melting point of 47.9. So also sulphate 
of sodium, NaSO 4 10 H 2 O, which has a constant melting point of 32.38, 
may be undercooled, and when it solidifies its temperature rises to the 
normal melting point, for this is the only temperature at which the solid 
and liquid can exist permanently in contact. 

394. Freezing Point of Solutions. The liquid solution of 
any substance has a lower freezing point than that of the 
pure solvent. An aqueous solution of any salt must be cooled 



370 



HEAT 



below the freezing point of water before ice separates out ; foi 
small concentrations the lowering of the freezing point is 
proportional to the amount of the substance in solution. The 
freezing point of a solution is the temperature at which the 
solution is in equilibrium with the solvent in the solid form. 

Practical applications of the lowering of the freezing point by dis- 
solving substances in water are familiar. Snow or ice is melted by salt, 
and alcohol or glycerine added to water lowers the freezing point. A 
so-called non-freezing mixture for the radiator of a motor car is made by 
adding alcohol to water. Twenty-five per cent of alcohol lowers the 
freezing point to 13. 

* A dilute solution of common salt is in equilibrium with ice at a tem- 
perature a little below 0. If now the temperature be reduced below 
this particular one for equilibrium, ice will separate, the concentration 
of the remainder of the solution will be increased, and the freezing 
point will be lower than that of the original solution. This process 
does not continue indefinitely because there is a limit to the solubility of 
salt in water, and this finite solubility sets a limit to the lowering of the 
freezing point. Beyond that limit, both ice and salt separate out and 
leave the concentration unchanged ( 399). 

395. Change of Volume in Fusion. Most bodies occupy a 
larger volume in the liquid than in the solid state. Water, 

bismuth, and cast iron 
are exceptions. These 
expand when they 
solidify. The increase 
in volume when water 
freezes is very marked. 
Hence the bursting of 
water pipes, and the 
disruption of rocks by 
the freezing of water 
in cracks and crevices. 
The expansion of cast iron and type metal when they solidify 
contributes to an exact reproduction of the mould in which 
they are cast. 

Figure 255 shows graphically the changes in the volume of 



1.10 



1.08 



1.06 



1.04 



1.02 



Fig. 255 



CHANGE OF STATE 371 

a gram of water from 20 to 50. When water is under- 
cooled, instead of the abrupt expansion at accompanying 
freezing, the undercooled water expands gradually along the 
dotted curve, which has its vertex at 4. 

396. Influence of Pressure on the Melting Point. In 1849 
Professor James Thomson showed from the dynamical theory 
of heat that a substance which expands on solidifying, like 
water, has its melting point lowered by pressure; but if the 
body contracts when it solidifies, like paraffin, then pressure 
raises its melting point. He calculated that in the case 
of water the freezing point is lowered 0.0075 by an in- 
crease of pressure of one atmosphere. Later Sir James 
Dewar found experimentally a reduction of 0.0072 per 
atmosphere up to 700 atmospheres. Hence, under a pressure 
of 1000 atmospheres water would not freeze above 7.2. 
In other words, if water is confined so that it cannot freeze 
at 7.2, it must be under a pressure of 1000 atmospheres. 

A rough numerical statement is that a pressure of 145 
kgm. per square centimeter (one ton per square inch) lowers 
the freezing point of water to 1. 

On the other hand, Bunsen found that the melting point 
of paraffin was raised from 46.3 to 49.9 by a pressure of 
100 atmospheres. 

397. Regulation. Intimately connected with the lowering 
of the freezing point of ice by pressure is the phenomenon 
of re-freezing, or regelation, when the pressure is relieved. 
Regelation was first observed by Faraday. 

Familiar examples are the hardening of snowballs under 
the pressure of the hands, and the formation of compact ice 
in a roadway, where it is compressed by vehicles and the 
hoofs of horses. Frozen foot-forms may often be seen to 
persist in compact ice after the loose snow has melted around 
them. The descending mass of snow from high altitudes 
becomes ice after melting under the pressure of its own 
weight and relief at lower levels. Such solidification occurs 



372 HEAT 

only when the snow is soft and near its melting point. In 
cold weather snow will not pack. If two pieces of ice are 
firmly pressed together, even under warm water, they will 
adhere by regelation when the pressure is removed. 

A glacier makes its way down its course by very irregular 
movements. Ice is probably plastic to some extent, but 
without doubt regelation plays an important role in glacial 
motion. The ice melts where it is subjected to enormous 
pressure by the descending masses above it. The melting 
permits the ice to accommodate itself to abrupt changes in 
the rocky channel, and a slow ice-flow results. As soon as 
the pressure at any surface is relieved, the water again 
freezes. The motion thus takes place by alternate melting 
and freezing. The middle of the flow moves faster than the 
borders of the stream, because the pressure there is greater 
and the consequent melting is more extensive. 

Bottomley's experiment to illustrate regelation is very instructive. A 
stout block of ice is supported horizontally by wooden supports at its two 
ends. On it is hung a weight by means of a wire passed over the ice at 
the middle. The pressure melts the ice under the wire, and the water, 
passing around it and relieved of the stress, again freezes. In this way 
the wire cuts its way down through the ice, but the block of ice remains 
intact, though the track of the wire through it remains visible. The ice 
is as liable to break at any other place as along the track of the wire. 

398. Heat of Fusion. Since the temperature of a crystalline 
solid remains constant while heat is applied to melt it, it is obvi- 
ous that heat disappears during the process. The heat that 
fuses a crystalline solid without raising its temperature be- 
comes potential energy in doing the work of changing its state. 
When the liquid solidifies, this work is restored as heat. 

The heat of fusion is defined as the number of heat units 
required to convert unit mass of the substance from the solid 
to the liquid state, without change of temperature. The 
whole quantity of heat H required to melt a, mass m of any 
substance is proportional to its mass, or 

ff=lm. (76) 



CHANGE OF STATE 373 

The proportionality factor I is the heat of fusion. It is 
numerically equal to the number of units of heat necessary 
to melt unit mass of the substance. The unit in the e.g. 8. 
system is the calorie per gram. 

The heat of fusion may be determined by the method of 
mixtures. Let m^ be the mass, and t the temperature of the 
water and the calorimeter ; also let m be the water equiva- 
lent of the calorimeter and m 2 the mass of the ice, the heat 
of fusion of which is to be found. If the temperature of the 
mixture after the ice is melted is , then the heat lost by the 
calorimeter and its contents may be equated to the heat of 
fusion of the ice and its gain in heat in rising from to , 



or (m -f Wj) 

Whence I = ( + i)(*i-0 _ ^ ( 77) 

2 

The most probable value for the heat of fusion of ice is 80 
calories per gram, on the basis of a minimum specific heat 
of water at about 30. 

When the heat of fusion of ice is known, the ice calorimeter 
furnishes one of the standard methods for the determination 
of the specific heat of any substance which does not dissolve 
in water. The body of mass m 2 is placed in a cavity made 
in a block of ice of temperature 0, and the mass m^ of the 
ice that melts, while the body cools from temperature t to 0, 
is measured. Then sm t = Im 

The specific heat s is given by this equation. 

399. Heat lost in Solution. Heat is absorbed when a body 
passes from the solid to the liquid state, although no heat is 
applied. If the liquefaction is brought about by solution in a 
proper solvent without chemical action, heat is still required to 
give mobility to the molecules, and the temperature of the solu- 
tion falls. A thermometer will show a sensible fall of tem- 
perature when some finely divided ammonium nitrate is added 



374 



HEAT 



to a little water in a test glass. If a delicate thermoscope is 
used, such as a thermopile ( 573), and a galvanometer, the 
heat absorbed by the solution of sugar in water may be de- 
tected. A still larger effect is produced by dissolving com- 
mon salt, while quite a notable reduction of temperature is 
produced by dissolving nitrate of sodium in water. 

A converse experiment, designed to show that heat is evolved when a 
substance becomes a solid by crystallization from a solution, is easily 
arranged by making a saturated solution of sodium hyposulphite at 30 
in a small flask, and slowly cooling to about 20. The solution is then 
undercooled. As soon as a very small crystal of the salt is dropped in, 
rapid crystallization sets in and extends through the whole solution. A 
thermometer in the solution shows that the temperature rises at the 
same time to about 30. 

Freezing mixtures are based on the absorption of heat nec- 
essary to give fluidity. Salt water freezes at a lower temper- 
ature than fresh water. When salt and snow or pounded 
ice are mixed together, both become fluid and absorb heat in 
the transition from one state to the other. By this mixture 
a temperature of 22 may be obtained. 

Figure 256 shows graphically why the temperature of a 
mixture of ice and salt water cannot be lower than 22. 

The curve A.B has freezing 
points as abscissas and concen- 
trations as ordinates. The 
curve CD is the solubility curve 
of common salt (sodium chlo- 
ride). The concentration of 
the salt solution decreases with 
falling temperature. 

From the point B to the in- 
tersection of the two curves, or 
from to 22, ice separates, 
leaving the solution increas- 
ingly concentrated with a continuous lowering of the freez- 
ing point. At P the solution has become saturated by the 




CHANGE OF STATE 375 

removal of water as pure ice. After this point has been 
reached, ice and salt separate together in a constant pro- 
portion of about 25 per cent of salt to 75 of ice and at a con- 
stant temperature. The solid mixture is called a cryohydrate. 
After the solidification is complete, with continued cool- 
ing, the temperature falls along the line PQ; but then 
there is no longer a solution, but a mixture of ice and salt. 
Hence, 22 is the lowest temperature at which a liquid 
salt solution can exist in equilibrium with ice. A freezing 
mixture should be composed of about one quarter by weight 
of salt and three quarters ice. 

400. Vaporization. Vaporization is the transition of a sub- 
stance from the solid or the liquid state into that of a gas. 
There are four distinct forms of vaporization, depending on 
the conditions under which the transition occurs : 

1. Evaporation, in which a liquid is converted into a vapor 
at its free surface at a relatively low temperature, and with- 
out the formation of bubbles. 

2. Ebullition, or boiling, a rapid evaporation at a higher 
thermal equilibrium, when the liquid is visibly agitated by 
internal evaporation. 

3. Spheroidal state, in which quiet evaporation, at a lower 
rate than boiling, goes on with the vapor acting as a cush- 
ion between the liquid and a surface at a relatively high 
temperature. 

4. Sublimation, in which a solid passes directly into the 
gaseous form without passing through the intermediate liquid 
state. 

Whether the gaseous condition is reached by one of these 
processes or another, heat is always absorbed, although the 
vapor formed is at the same temperature as the solid or the 
liquid from which it comes. The heat absorbed in the tran- 
sition into the form of a gas is called the heat of vaporization. 

401. Evaporation in a Closed Space. The dynamical theory 
of heat assumes that the molecules of a liquid are in a state 



376 HEAT 

of incessant agitation. The free path of the molecular 
motion in liquids is at least very limited; the migratory 
track of any individual molecule depends on its innumerable 
encounters with other molecules. While the average molec- 
ular velocity is determined by the temperature, the velocity 
of individual molecules may greatly exceed this average ; and 
whenever a molecule at the free surface of a liquid has a 
normal component of velocity sufficient to carry it through 
the surface film, it escapes into the free space above the 
liquid. The concurrent escape of many molecules in this 
manner constitutes evaporation. 

When evaporation takes place in a limited closed space, the 
free molecules may again return into the liquid. This return 
is called condensation. When the number of molecules mak- 
ing their escape equals the number returning through the 
surface film, the vapor in contact with the liquid 
is said to be saturated. It is in equilibrium with 
its liquid, and its pressure is the greatest it can 
have at the existing temperature. This maximum 
pressure is the vapor pressure at that temperature. 
The saturated vapor pressure is independent of 
the volume and depends only on the substance 
and its temperature. The value of this pressure 
and its independence of the volume are easily 
shown by means of Torricellian tubes. If the 
three tubes of Figure 257 are filled with mercury 
free from air, and are then inverted in a vessel 
of mercury, the mercurial columns in the long 
tubes will settle down to the height of the 
barometer at the time. If now by means of a 
small pipette, with its tip bent upward, a little 
sulphuric ether is allowed to enter two of the 
tubes, it will quickly evaporate and the mercury 
in them will fall to the same level through s. 
The column of mercury cs measures the vapor pressure of 
the ether. The mercury surfaces in the two tubes containing 




CHANGE OF STATE 377 

the ether will stand at the same level, provided the two are 
at the same temperature with only a small quantity of liquid 
ether above the mercury. A rise in the temperature of the 
ether increases the' vapor pressure and causes a further de- 
pression of the column of mercury. 

If the volume of the saturated vapor is diminished without 
change of temperature, some of the vapor will condense to a 
liquid ; if the volume is increased, more of the liquid will 
evaporate so as to maintain the same vapor pressure. 

402. Ebullition and the Boiling Point. Evaporation absorbs 
heat. If only a moderate amount of heat is supplied to 
a liquid, the evaporation is confined to its surface ; with 
an increased supply the evaporation increases until the rate 
at which heat is supplied equals the rate of loss by evapora- 
tion. At a still higher temperature, this equilibrium of 
quiet evaporation can no longer be maintained, and bubbles 
of vapor form in the interior and at points of contact with 
the walls of the containing vessel. If the vapor pressure is 
insufficient to support these bubbles as they rise into the 
cooler liquid, they collapse with the familiar sound of " sim- 
mering." At a slightly higher temperature evaporation 
takes place into the bubbles themselves, and they rise with 
sufficient buoyancy to break through the surface film and 
escape. This process of rapid evaporation from the interior, 
as well as at the surface, is called ebullition or boiling. 

The temperature at which this new equilibrium is estab- 
lished is called the boiling point of the liquid. It is constant 
for the same pressure. The normal boiling point assumes 
the standard atmospheric pressure of 76 cm. of mercury. 

The boiling point of a liquid is the temperature at which 
it gives off vapor at a pressure equal to that sustained by the 
surface of the liquid. The vapor pressure in boiling equals 
the pressure of the surrounding atmosphere. 

403. Superheating. When the air has all been boiled 
out of a liquid and the containing vessel is clean, the tern- 



378 HEAT 

perature may rise several degrees above the normal boiling 
point before ebullition begins. It then proceeds with almost 
explosive violence, and continues at this high rate until the 
temperature of the liquid falls to the point corresponding to 
the existing pressure. Air-free water may thus have an 
abnormally high initial boiling point. In clean glass ves- 
sels this condition gives rise to "bumping." It may be 
prevented by placing in the vessel insoluble bodies with 
rough surfaces, or by introducing some fresh liquid con- 
taining air. 

While the temperature of the liquid may thus be above 
the normal boiling point, that of the saturated vapor is nor- 
mal. Determinations of the boiling point are therefore made 
with the thermometer bulb enveloped by the saturated vapor 
just above the liquid. 

A liquid may be superheated by keeping it out of contact 
with its gaseous phase, just as it may be undercooled by 
excluding the solid phase ( 393). Drops of water 10 mm. 
in diameter have been kept liquid at 120 when suspended in 
a mixture of linseed and clove oils, and drops 1 mm. in 
diameter remained liquid up to 178. They exploded when 
touched with a glass rod or by the side of the vessel. 

404. Boiling Point of Solutions. The vapor pressure of a 
salt solution is lower and the boiling point higher than 
those of the pure solvent. From solutions which do not 
contain a volatile constituent, such as alcohol, only the sol- 
vent evaporates. When the solution becomes saturated, the 
excess of salt crystallizes out. 

The temperature of an aqueous solution of common salt 
may be as high as 110; the boiling point of a saturated 
solution of calcium chloride is 180, but the steam from the 
boiling solution falls quickly to 100 under normal atmos- 
pheric pressure. In other words, the temperature of the 
saturated vapor is the boiling point of the solvent and not 
that of the solution. 



CHANGE OF STATE 



379 



240 

200 

160 
Cm. 
120 

80 
40 



Water 



Va 



60 
Fig. 258 



80 100 120 140' 



405. Influence of Pressure on the Boiling Point. The vapor 
pressure and the boiling point of a liquid rise together, the 
pressure rising more rapidly 
than the boiling point. 
The vapor pressure curve 
for water is plotted in Fig- 
ure 258 with pressures in 
centimeters of mercury as 
ordinates and temperatures 
as abscissas. This curve 
divides the diagram into 
two regions: below it the 
conditions permit the exist- 
ence of water as an unsaturated vapor only ; above it, as a 
liquid only; along the curve the liquid and the saturated 
vapor exist together in equilibrium. At 9.2 cm. pressure 
the boiling point falls to 50 and at a 
pressure of 4.6 mm. it is the same as 
the freezing point. Near 100 the 
change in the boiling point is 0.1 for 
a change of pressure of 2.71 mm. of 
mercury. Hence at high altitudes 
water boils below 100. At Quito the 
boiling point is about 90. 

Boiling under reduced pressure is 
resorted to in concentrating sugar 
solutions. The ' same principle is 
utilized in removing moisture from 
coils of insulated wire by heating 
them by means of steam pipes in an exhausted chamber. 

The boiling of water at a reduced temperature may be 
conveniently shown by first boiling it in a round-bottomed 
flask, then corking tightly, inverting, and supporting in a 
ring stand (Fig. 259). The boiling may be renewed by 
applying cold water, which condenses the water vapor and 
reduces the pressure within the flask. 




Fig. 259 



380 HEAT 

406. The Spheroidal State. Wlien a drop of water is care* 
fully placed on a clean hot stove, it will often take a flattened 
globular form and roll about with rapid but silent evaporation. 
It is then in the spheroidal state. So liquid oxygen assumes 
the spheroidal state on water. It boils at 180, and the 
water is at a high temperature relative to it. Spheroidal 
sulphur dioxide has a temperature low enough to freeze a 
drop of water placed in it. This may happen in a red-hot 
crucible because the sulphur dioxide in the spheroidal state 
is below its boiling point, and this is below the freezing point 
of water. The globular form of a spheroidal drop is ac 
counted for by surface tension. 

A liquid in a spheroidal state is not in contact with the hot 
surface, but rests on a cushion of its own vapor. If the drop 
is not too large, light may be projected through between it 
and a hot surface of platinum. The temperature of spheroidal 
water is from 90 to 98. Under the receiver of an air-pump 
it may be as low as 80. 

407. Sublimation. A body sublimes when it passes directly 
from the solid to the gaseous form without going through the 
liquid state. Ice and snow below freezing temperature grad- 
ually waste away by evaporation. Solid carbon dioxide dis- 
appears by sublimation without melting. It evaporates only 
as it gets heat to convert it into the gaseous form* 

Camphor and ammonium carbonate sublime at room tem- 
perature. Iodine, ammonium chloride, and arsenic sublime 
when heated gently at atmospheric pressure. Arsenic may 
be fused if the pressure is increased. 

Below a certain critical pressure for each, ice, mercuric 
chloride, and camphor cannot be melted, but they pass 
directly into the gaseous form. Whenever the boiling point 
of a substance may be lowered by reduced pressure to its 
freezing point, then at or below that pressure the solid will 
sublime. At a pressure less than 4.6 mm. of mercury ice is 
converted into water vapor by heat without melting. 



CHANGE OF STATE 



381 



408. Heat of Vaporization. The heat of vaporization of a 
substance is the quantity of heat required to vaporize unit 
mass of it without change of temperature. The heat of vapori- 
zation of a liquid is usually understood to refer to its evapo- 
ration at its normal boiling point. In the e.g. s. system it 
is expressed in calories per gram. 

The heat of vaporization varies with the temperature at 
which the vaporization takes place. The following expression 
is derived from the investigations of Regnault and Griffiths, 
and it applies to the evaporation of water between and 
100 0. : 

L = 596.73 0.601 1 cal./gm. (calories at 15). 

L is the number of calories required to evaporate 1 gm. of 
water at t. At 100, therefore, the heat of vaporization is 
536.6 cal./gm. To convert 1 gm. of water at 100 into steam at 
100 requires 53L6 calories ; conversely, when 1 gm. of steam 
at 100 condenses to water at the same temperature, 536.6 
calories of heat are produced. The energy carried by steam 
is large because of the high heat of vaporization of water. 

409. Cooling by Evaporation. If the heat required to 
vaporize a liquid is not supplied from an external source, the 
evaporation will be accompanied by a lowering of its tem- 
perature. A little ether, alcohol, or gaso- 
line on the hand feels cool because its 

evaporation takes away heat. 

Wollaston's cryophorus (Fig. 260) was 
designed to freeze water by its own evapo- 
ration. It contains no air, but only water 
and water vapor. The water is collected 
in bulb A, while B is surrounded with a 
freezing mixture. The vapor condenses 
rapidly in B, and the lowering of the vapor 
pressure causes equally rapid evaporation of the water in A. 
Heat is thus carried by the vapor from A to the freezing 
mixture until the water in A freezes. 




Fig. 260 




382 HEAT 

Water may easily be made to boil until it freezes. A thin 
flat dish is supported over a broad shallow glass vessel con- 
taining strong sulphuric acid (Fig. 261), and the whole is 
inclosed in a low receiver on an air pump. 
The success of the experiment depends 
on removing the water vapor so rapidly 
that the boiling point is reduced to the 
freezing point. A good air pump is 
needed and the volume of the receiver 
should be no greater than is needed to 
cover the two dishes. When the air is 
exhausted, the pressure is reduced until 
the boiling point falls to the temperature of the water. The 
water then boils violently ; if the vapor is removed rapidly 
enough by the pump and by absorption by the acid, the pres- 
sure may be reduced below 4.6 mm. of mercury, and the 
boiling will then continue until the water freezes. The heat 
to do the internal work of evaporation is drawn from 'the thin 
dish and the water itself. 

Very low temperatures may be produced by the vaporiza- 
tion of more volatile substances. Thus, by opening a small 
orifice in a strong cylinder containing liquid carbon dioxide 
under great pressure, the spray escaping into a wooden box 
lined with asbestos cools so rapidly by evaporation that some 
of it freezes in the form of fine white snow at a temperature 
of - 79. 

The property of the absorption of heat by liquids evaporating at a 
low temperature is applied in ice machines. Ammonia, for example, is 
first condensed by pressure and cooling to a liquid with about one-tenth 
of its weight of water. It is then evaporated under reduced pressure 
obtained by vacuum pumps, and its temperature falls low enough to 
freeze water in vessels about it or submerged in it. The process is made 
continuous by returning the gaseous ammonia to a condensing chamber 
cooled with water. It thus passes repeatedly through the same cycle of 
changes. 

410. Aqueous Vapor in the Atmosphere. The atmosphere 
always contains aqueous vapor, and if it is sufficiently cooled 



CHANGE OF STATE 383 

a temperature will be reached at which the pressure of the 
aqueous vapor equals the saturation pressure. The vapor 
then condenses in the air as rain or snow and on the sur- 
face of bodies as dew or hoar frost. This temperature is 
known as the dew point. 

Condensation commences about nuclei, and in the absence 
of these the air may be supersaturated with moisture. Each 
minute mote of dust floating in the air serves as such a 
nucleus. The more numerous the nuclei, the smaller the 
drops, and small drops sink through the air very slowly on 
account of surface friction. A fog is composed of much 
smaller drops than those falling as rain. The prevalence of 
fogs in large cities in a moist climate is accounted for by the 
presence in the air of innumerable particles of dust and soot. 
A rain clears the air by dragging down these particles 
weighted with moisture. 

It is now known that air ionized by ultra-violet light, by 
Roentgen rays, or by radioactive substances, also furnishes 
nuclei for condensation ( 675). 

Condensation of aqueous vapor may readily be shown by passing a 
beam of strong light through a large glass receiver on an air pump in a 
darkened room. With only moderately moist air, a single stroke of the 
pump produces a thick cloud of condensed vapor, showing splendid 
iridescent diffraction effects. The expansion of the air under pressure 
cools it below the dew point. The condensed water is in a state of fine 
division. 

411. Relative Humidity. Air is said to be damp when it is 
nearly saturated with water vapor. Since the saturation pres- 
sure rises rapidly with the temperature (Fig. 258), the heat- 
ing of the air, while the quantity of aqueous vapor in it 
remains unchanged, removes it farther from the saturation 
point and diminishes its dampness. A cubic meter of air 
saturated at contains 4.87 gm. aqueous vapor ; at 15 it 
contains 12.76 gm. ; and at 30, 30.15 gm. Therefore if air 
which is saturated with moisture at 15 is heated to 30, its 
humidity will be only about two fifths the amount required 



384 



HEAT 



for saturation. Thus, when damp air from outdoors passes 
through a hot-air furnace, it becomes dry, not because it has 
lost any aqueous vapor, but because its capacity to take up 
this vapor has been greatly increased by the rise of tempera- 
ture. In winter the humidity is usually greater than in 
summer because the temperature is lower, and the amount of 
aqueous vapor necessary to saturate the air is less. 

Relative humidity is the ratio of the pressure of the aqueous 
vapor present at a given temperature to the saturation pressure 
at the same temperature. It is measured by determining the 
actual pressure of the aqueous vapor in the air and compar- 
ing it with the maximum pressure at the same temperature 
obtained from tables. This is the method applied by all dew 
point instruments, which are called hygrometers. 

412. Regnault's Hygrometer. The superior form of hy- 
grometer devised by Regnault consists of two thin polished 
silver thimbles, A and B (Fig. 262), into 
which are fitted glass tubes open at both 
ends. The tube A is half filled with sul- 
phuric ether and is closed with a stopper 
through which pass a thermometer t f and 
a bent tube extending down nearly to 
the bottom of the silver thimble. The 
other tube contains only a thermometer 
t. The exhaust tube DE leads to an 
aspirator. 

To make an observation, the air is 
drawn through C by the aspirator. It 
bubbles up through the ether and 
causes it to evaporate rapidly. 
The temperature of A is thus 
lowered ; and when the dew point 
is reached, it is indicated by a dim- 
ming of the silver tube A as compared with .Z?, which remains 
at atmospheric temperature. The thermometer t' is read as 




LIQUEFACTION OF GASES 385 

soon as the dimming is apparent. The aspirator is stopped 
at the same time, and the temperature is again read at the 
instant when the dew disappears. 

The temperature given by t ! is then the dew point. The 
saturation pressures corresponding to both temperatures are 
taken from the table, and their ratio is the relative humidity. 
For example, if the dew point is 7 and the temperature of the 
air 20, the corresponding saturation pressures are 7.49 and 
17.39 mm. respectively. The actual pressure of the aqueous 
vapor is therefore 7.49 mm., and the pressure for saturation 
at 20 is 17.39 mm. Hence the relative humidity is 
7.49/17.39 = 0.431, or 43.1 per cent. 

VI. LIQUEFACTION OF GASES 

413. Conditions for Liquefaction. Under atmospheric pres- 
sure a number of substances are known to us in both the 
liquid and the gaseous states. Water is liquid below 100 
and a vapor at higher temperatures ; alcohol is liquid below 
78 and a vapor above ; sulphuric ether is liquid below 35 
and a vapor above. If we had no means of reducing the tem- 
perature below freezing, sulphur dioxide at atmospheric pres- 
sure would be known to us as a gas only, since it boils at 8. 

When the temperature of a substance in the gaseous state 
is lowered artificially, and its boiling point is at the same 
time raised by pressure, the two 
temperatures approach each other ; 
and if these processes are carried 
far enough to bring the two tem- 
peratures together, liquefaction 
ensues. 

Faraday was one of the first to 
liquefy chlorine, carbon dioxide, 
and ammonia, by combined cooling and pressure. His appa- 
ratus was of the simplest character, consisting merely of a 
stout bent tube (Fig. 263). The materials to liberate the 




386 HEAT 

gas by heat were placed in this tube, and it was then sealed 
off. The limb b was surrounded by a freezing mixture, the 
limb a was heated, and the pressure was produced by the 
gas itself. When a was filled with sodium carbonate and 
was heated, the released carbon dioxide condensed to a 
liquid in b. 

By means of a small inclosed pressure manometer it was 
found that the pressure in every case increased to the point 
where condensation set in ; after that it remained constant 
so long as the temperature of the condensed liquid was kept 
the same. This pressure was that of the saturated vapor at 
the given temperature. 

414. The Critical Temperature. The renowned experiments 
of Dr. Andrews, published in 1869, established the existence 
for gases. of a critical temperature, above which no amount 
of pressure produces liquefaction. The phenomena discov- 
ered by Andrews may be shown by means of a stout glass 
tube half full of liquid carbon dioxide and sealed before the 
blowpipe. Careful heating of the tube causes the liquid to 
expand rapidly ; at the same time the pressure increases, 
until the surface of separation between the liquid and the 
vapor becomes less clearly marked, and finally the two phases 
merge into each other at a temperature of 81. The entire 
tube is then filled with a homogeneous fluid. When this 
fluid has cooled a little, a thick cloud suddenly makes its 
appearance on the top, and the surface of separation between 
the liquid and the vapor is again visible. The temperature 
at which the liquid surface disappears or reappears is the 
critical temperature. Above its critical temperature a gas 
cannot be liquefied by any pressure, however great. Above 
the critical temperature of 31 the distinctions between liquid 
and gaseous carbon dioxide cease to exist. 

If the pressure on the gas above 31 be increased to 150 
atmospheres, the volume will diminish, but there will be no 
sudden decrease at any point. If now the temperature be 



LIQUEFACTION OF GASES 



387 



--85 



gradually lowered to about 20, the fluid is then clearly a 
liquid. The substance has passed from the gaseous to the 
liquid form by a continuous process and without any sudden 
evolution of heat. A gas and a liquid are then only widely 
separated forms of the same condition of matter, and the 
passage from one to the other may be made without breach 
of continuity. 

The phenomena attending the condensation of carbon 
dioxide may be shown by plotting pressures and volumes 
as coordinates. Each 
curve (Fig. 264) corre- 
sponds to a single tem- 
perature and is therefore 
called an isothermal line. 
The coordinates of the 
point P denote the vol- 
ume and pressure at a 
temperature of 21.5. 
The volume decreases to 
a point A with increas- 
ing pressure. At A 
condensation sets in and 
continues along the line 
AB with constant pres- 
sure until the whole 
mass is liquefied. A 
further increase of pressure produces but a slight change in 
volume along the line BC. 

The region below the dotted line, which passes through all 
such points as A and B, is one in which the liquid and the 
vapor exist together in equilibrium. Above this line the 
substance is either wholly a liquid or a gas. The top of 
the dotted curve corresponds to the critical temperature, and 
above it the isothermals show only a point of inflection. The 
isotherm for 31 touches the dotted line where the former is 
flattened. 




Fig. 264 



388 HEAT 

The following are the critical temperatures for several 
substances : 

Hydrogen -220 Chlorine 130 

Nitrogen 146 Sulphur dioxide .... 155 

Air -141 Ether 197 

Oxygen - 118 Acetone 246 

Fluorine - 120 Alcohol 239 

Carbon dioxide .... 31 Carbon bisulphide . . . 273 

Ammonia 130 Water 365 

415. Distinction between a Gas and a Vapor. The discov- 
eries of Andrews permit of drawing a distinction between 
a gas and a vapor. By a vapor is meant a substance in the 
aeriform state at a temperature below the critical point. A 
gas is a substance in the aeriform state above its critical tem- 
perature. A vapor can be converted into a liquid by pres- 
sure alone; and can exist in contact with its own liquid. A 
gas cannot be liquefied by pressure alone, but only by first 
reducing its temperature below the critical temperature and 
thus converting it into a vapor. Below 31 carbon dioxide 
may exist as a vapor ; above 31 it is a gas and cannot be 
liquefied. At the critical temperature the heat of vaporization 
becomes zero. 

416. Gases with Low Critical Temperatures. Oxygen, nitro- 
gen, and hydrogen resisted liquefaction for several years after 
the discovery of a critical temperature. The critical tem- 
perature of these gases is below the temperature of solid 
carbon dioxide. 

In 1877 Cailletet and Pictet liquefied oxygen and nitrogen 
by resorting to a two-cycle process of cooling by evaporation 
under reduced pressure, and then by the cooling effect of the 
sudden expansion of a gas under great pressure. Oxygen 
under a pressure of 500 atmospheres was surrounded by 
liquid carbon dioxide, and this in turn by liquid sulphur 
dioxide. Both of these cooling liquids were rapidly evapo- 
rated under low pressure by exhaust pumps. The sulphur 



LIQUEFACTION OF GASES 389 

dioxide cooled by evaporation withdrew heat from the liquid 
carbon dioxide, and the rapid evaporation of the latter finally 
reduced its temperature and that of the oxygen to 130. 
At this stage the pressure of the oxygen fell to 320 atmos- 
pheres, indicating some liquefaction. It was then allowed to 
issue from a stopcock with great violence, and its sudden ex- 
pansion reduced its temperature to such an extent that some 
of it was collected in the liquid state. 

Later Sir James Dewar employed two circuits of liquid 
nitrous oxide and ethylene in successive cycles for cooling 
oxygen and nitrogen under great pressure. The sudden 
expansion of oxygen as it rushed out of a small orifice cooled 
it finally to the liquefying point. It was mixed with some 
solid carbon dioxide from the air, from which it was freed by 
filtration through an ordinary filter paper. It has a delicate 
sky-blue color, and its temperature when evaporating under 
atmospheric pressure is 182. Nitrogen was liquefied by 
Dewar in the same manner, though its critical temperature 
is lower than that of oxygen. 

417. The Regenerative Process, The process of Linde and of 
Hampson, introduced in 1895, differs from the older method 
in leading the gas cooled by sudden expansion back through 
spiral tubes around the pipe containing the gas flowing out 
under great pressure. This method is known as the regen- 
erative process. The gas to be liquefied is put under a 
pressure of 150 to 200 atmospheres by a compressor, cooled 
by some means to a low temperature, and then allowed to 
escape into a chamber in which the pressure is kept down to 
a few atmospheres. The sudden expansion of the gas cools 
it, and the cooled gas passes back through spiral tubes 
around the compressed gas flowing toward the expansion 
orifice. The gas is thus further cooled before the expansion, 
and this reduction continues until the temperature falls below 
the critical point and liquefaction begins in the low-pressure 
chamber. After this stage is reached, the process is continuous. 



390 HEAT 

In 1898 Dewar liquefied hydrogen by cooling it under a pressure of 180 
atmospheres to 205 by means of a bath of liquid oxygen boiling under 
reduced pressure, and then allowing it to escape into a triple-walled 
silvered vacuum flask at atmospheric pressure. The internal silvering 
and the vacuum are for the purpose of heat insulation. The boiling point 
of hydrogen is 21 absolute scale. By rapid evaporation under a pres- 
sure of 25 mm., the liquid was reduced to a frothy solid, the density of 
which was 0.086. The density of liquid hydrogen is 0.07. Glass tubes 
containing air, when immersed in boiling hydrogen, have the pressure 
in them reduced to one millionth of an atmosphere by the liquefaction 
of the air in them. 

Problems 

1. Express the following temperatures in Fahrenheit degrees: the 
boiling point of nitrogen, 195.5; melting point of hydrogen, 257; 
alcohol flame, 1705. 

2. At what temperature will the reading on the Fahrenheit scale be 
the same as that on the Centigrade scale? 

3. At what temperature will the reading on the Fahrenheit scale be 
double that on the Centigrade? 

4. At what temperature will the reading on the Centigrade scale be 
double that on the Fahrenheit? 

5. If a thermometer scale were marked 10 at the freezing point 
and 60 at the boiling point, what would 35 on this scale mean in Centi- 
grade degrees? 

6. The testing of a Centigrade thermometer shows that the freezing 
point is 0.6 and the boiling point 101. What is the meaning of 50 
on this scale if the tube is of uniform bore? 

7. A glass flask holds 200 cm. 3 of water at O 3 . What is its internal 
capacity at 100? The coefficient of linear expansion for glass is 
0.0000083. 

8. The density of a piece of silver at is 10.5 gm. per cubic 
centimeter. What is its density at 100 if the coefficient of cubical 
expansion is 0.0000583? 

9. A brass pendulum keeps correct time at 15, but at 35 it loses 16 
seconds a day. Find the linear coefficient of expansion of brass. 

10. A solid displaces 500 cm. 8 when immersed in water at ; but 
in water at 30 it displaces 503 cm. 3 . Find its coefficient of cubical 
expansion. 

11. If 3 kgm. of iron (specific heat, 0.11) at 95 are put into 3 1. of 
water at 10, what will be the rise in temperature of the water? 



LIQUEFACTION OF GASES 391 

12. A mass of 500 gm. of copper at 98 put into 500 grn. of water at 
0, contained in a copper vessel weighing 150 gm., raises the temperature 
of the water to 8.3. Find the specific heat of copper. 

13. Into a mass of water at are introduced 100 gm. of ice at 12; 
7.5 gm. of ice are frozen and the temperature of all the ice is raised to 0. 
If the heat of fusion is 80, find the specific heat of ice. 

14. If 1 kgm. of copper at 100 (specific heat, 0.095) be placed in a 
cavity in a block of ice at 0, and if 119 gm. of ice are melted, find the 
heat of fusion of ice. 

15. What will be the volume of air measuring 500 cm. 3 at if the 
temperature be raised to 273 and the pressure be doubled ? 

16. A liter glass flask of air at is heated to 30. How many cubic 
centimeters of air escape at 30, neglecting the expansion of the glass? 

17. How much gas must be collected at a temperature of 20 and 
74 cm. barometric pressure to give 100 cm. 3 at and 76 cm. pressure? 

18. A vessel filled with air at is heated to 67, when it is found that 
20 cm. 8 of air at the latter temperature have escaped. What was the 
capacity of the vessel at 0? 

19. A mass of hydrogen at 20 occupies a volume of 500 cm. 8 . Find 
its volume at 100, the pressure remaining the same. 

20. The area of Lake Erie is 9900 sq. mi. If the density of ice is 
62.3 Ib. per cubic foot, and the heat of combustion of anthracite coal is 
14,000 B. T. U., how many tons of coal would be required to furnish by 
combustion enough heat to melt the ice ft. thick over the entire surface 
of the lake? 



CHAPTER XIV 

TRANSMISSION AND RADIATION OF HEAT 
I. CONDUCTION 

418. Three Modes of Distribution of Heat. The distribution 
of heat between the different parts of a body or of a system, 
differing in temperature, takes place by three distinct modes. 
These are : 

1. Conduction, in which heat is transmitted from one por- 
tion of matter to another in contact with it and at a lower 
temperature. This is a slow process, depending on differ- 
ences of temperature and on the nature of the conducting 
substance. 

2. Convection, in which heat is carried from one place to 
another by sensible masses of matter. In this manner build- 
ings are heated by the circulation of hot water, and heat is 
conveyed by hot air and by steam. 

3. Radiation, in which the energy is transferred from one 
body to another by the same physical process as the one by 
which light is transmitted, and without heating the interven- 
ing medium. By the first two modes heat is distributed 
through the agency of matter ; in tjie third the ether is the 
medium of transmitting it as radiant energy. 

419. Conduction in Solids. When the molecules of a solid 
are agitated by the motion of heat, their oscillations impart 
similar motions to adjacent molecules. The slow trans- 
mission of this motion of heat from particle to particle is 
conduction. 

Conduction tends to establish equilibrium of temperature. 
If one end of an iron rod be placed in the fire of a forge, the 

392 




CONDUCTION 393 

other will in time become hot. In this mode of distribution 
heat is handed on from the hotter to the colder parts of a 
body by a slow process. 

Different substances possess the capacity of conducting 
heat in very different degrees. Metals are the best conduc- 
tors, while glass, wood, fire clay, 
Wool, and feathers are poor con- 
ductors. If a sheet of writing 
paper is tightly wrapped around 
a cylinder of uniform diameter, 
half brass and half wood (Fig. 265), 
a Bunsen flame may be applied for 
a short time without charring the 
paper in contact with the brass, 
while around the wood it will be 

scorched. The metal conducts away the heat so rapidly 
that it keeps the temperature of the paper in contact with 
it below the point of ignition. 

A Norwegian cookstove (or fireless cooker) is a box heavily lined with 
felt or other poor conducting material. In it fits a metallic dish with a 
cover. The food to be cooked in water is first boiled 
in the dish in the usual way and is then transferred 
to the box and is inclosed in it. The conductivity of 
the felt and the imprisoned air is so poor that the heat 
is retained, and in three hours the temperature falls 
not more than 10 or 15 and the cooking is completed 
without further heating. 
. The good conductivity of metal gauze was utilized 

jllpgApi^iigj by Davy in his safety lamp for miners. The flame is 
completely inclosed in metal and fine wire gauze. By 
conducting away heat the gauze keeps any fire damp 
outside the lamp below the point of ignition and pre- 
vents explosions. The action of the gauze is easily 
illustrated by holding it over the flame of a Bunsen 
Fig 266 burner (Fig. 266). The flame does not pass through 

unless the gauze is heated to redness. If the gas is 
first allowed to stream through the gauze, it may be lighted on top with- 
out being ignited below. 




394 



HEAT 




420. Coefficient of Thermal Conductivity. Let AB and CD 
(Fig. 267) be two parallel surfaces of a homogeneous body, 
the thickness of which is I, one of the surfaces being main- 
tained at a temperature t and the other t 1 . The length AB 

may represent the temperature t, and the 
length CD, the temperature t'. Then the 
temperature gradient (tt f )/l is repre- 
sented by the slope of the line BD, that 
is, by tan BDCr. The quantity of heat 
transmitted in time T through any cross- 
section S of the plane EF and at right 
angles to the two surfaces is proportional 
to the area S, to the temperature gradient, 

and to the time, or 

f ff 

TT TfQ >Tf /' I 7Q\ 

J2 _1O J. { O ) 

Fig. 267 / 

The proportionality factor K, which depends on the nature 
of the substance, is the coefficient of thermal conductivity. It 
is the time rate with which heat is transmitted through unit area 
when the temperature gradient is unity. 

If the temperature is measured in Centigrade degrees, the 
dimensions in centimeters, and the time in seconds, the 
quantity of heat is in calories. 

Practical methods of measuring thermal conductivity are not applied to 
such a body with parallel sides, but to the flow of heat along a bar, one 
end of which is maintained at a constant temperature, while the other 
is at the temperature of the room. The temperature gradients are then 
represented by the tangents to a curve obtained by measuring the tem- 
peratures at equal distances along the bar. The heat flowing past any 
cross section of the bar is all dissipated from the surface beyond the 
section. The relative conductivities of two bars can be determined by 
getting their temperature gradients; but to measure the coefficient K, 
another experiment is necessary in order to find the rate of cooling. 
Then the total quantity of heat traversing any section of the bar may 
be calculated. 

421. Conductivity in Wood and Crystals The investiga- 
tion of many kinds of wood shows that heat is conducted 




CONDUCTION 395 

better along the fibers than across them. Further, the con- 
ductivity perpendicular to the fibers and to the ligneous 
rings is greater than in a direction tangential to them. The 
conductivity in the first of these three rectangular directions 
is from two to four times as great as in the last. 

A similar difference of conductivity has been found in 
the case of laminated rocks, , the conductivity being better 
along the planes of cleavage than 
across them. The same statement 
may be made with respect to bismuth. 

If two plates be cut from quartz 
crystals, one perpendicular to the 
optic axis and the other parallel to it, 
and if a minute hole be made through 
each plate for the passage of a fine 
wire which may be heated by an elec- 
tric current, then a film of wax on the crystal will be melted 
in the form of a circle when the section is at right angles to 
the axis, and in the form of an ellipse when the section is 
parallel to the axis (Fig. 268). Quartz and calc-spar con- 
duct heat best along the axis and equally well in all direc- 
tions perpendicular to it ; tourmaline conducts best at right 
angles to its axis. 

422. Conductivity of Liquids and Gases. Liquids have low 
coefficients of conductivity as compared with solids. Gases 
are still poorer conductors of heat. If liquids or gases are 
heated at the bottom, the heat distributed by convection 
masks any distribution by conduction. This difficulty is 
overcome in part by heating at the top, but the results are 
still complicated with diffusion and with conduction by the 
containing vessel. 

All liquids except molten metals are poor conductors. 
The upper strata of water in a test tube may be boiled for 
some time without melting a lump of ice confined at the 
bottom of the tube. Support a simple air thermometer by 



396 



HEAT 




Fig. 269 



a ring stand (Fig. 269), and cover the bulb to a depth of a 
centimeter with water; pour a spoonful of ether on the 

water and set it on fire. The index 
of the thermometer will show that 
little if any heat is conducted to the 
bulb. So feeble is the flow of heat 
through liquids .that the results of 
measurement are always open to the 
suspicion that the transport has been 
brought about by diffusion and con- 
vection. 

The difficulties encountered in 
measuring the conductivity of liquids 
are exaggerated in the case of gases, 
so that they become almost insuper- 
able. Many facts, however, go to 
show that heat is conveyed very im- 
perfectly by gases, except under con- 
ditions favorable to convection. The interstices filled with 
air in bodies made of wool, hair, feathers, or fur, make them 
poorer conductors than they are after the air spaces have 
been diminished by compression. Some solids which con- 
duct fairly well are very poor conductors when reduced to a 
powder. The solids are made discontinuous by the intro- 
duction of air. 

II. CONVECTION 

423. Convection in Liquids. The transmission of heat by 
convection is accomplished by the translatory motion of 
heated matter. Convection of heat by currents of warm 
water may be illustrated by heating a large beaker of water 
containing some bits of cochineal. A stream of warm water 
ascends along the axis above the burner, and currents of 
cooler water descend along the sides. In Faraday's appa- 
ratus to illustrate convection currents (Fig. 270), the flask 
and connecting tubes are completely filled with water up to 



CONVECTION 



397 




Fig. 270 



a point above the open end of the vertical tube C. The 
water begins to circulate as soon as heat is applied to the 
flask by means of a Bunsen burner. To make 
the circulation visible, the liquid in the flask 
may be colored red with some aniline dye, and 
that in the reservoir at the top blue. 

This experiment illustrates the method of heating 
buildings by hot water. A pipe rises from the top of the 
boiler to an expansion tank in the upper part of the 
building. From this tank the hot water is distributed 
through the several radiators, and finally enters the boiler 
again at the bottom. The water loses heat in the radi- 
ators of large surface, and becomes denser. The heat of 
the boiler and the loss by radiation and convection at the 
radiators produce unequal hydrostatic pressures, which 
give rise to continuous currents as long as the heat is 
applied. 

The Gulf Stream is a convection current on a gigantic scale, and it 
transports enormous quantities of heat from the equatorial regions, and 
distributes it over the British Islands and the western part of the conti- 
nent of Europe, thus contributing largely to their mild climate. A 
counter current of cold water flows south from Greenland and washes the 
Atlantic coast of America. Hence the contrast between the climate 
along the Hudson and the Tiber in about the same latitude. 

424. Convection in Gases. Convection currents are more 
easily set up in gases than in liquids. The heated air over 
a flame rises rapidly, and its place is taken by a lateral in- 
rush of cold air. The current of hot air over a gas flame or 
a bar of hot iron may easily be shown by placing them in the 
path of a beam of light from a projection lantern. The 
wavering outlines of the rising stream of air can be dis- 
tinctly seen because the refraction for the hot air is less than 
for the cold. When the atmosphere is in unstable thermal 
equilibrium, especially over heated sand, a telescope focused 
on a vertical straight object gives a wiggling line as an 
image. 

Convection air currents on a large scale are present near 
the seacoast. The wind is a sea breeze during the day, 



398 



HEAT 



because the air moves from the cooler ocean to take the place 
of the air rising over the heated land. As soon as the sun 
sets, the ground cools rapidly by uncompensated radiation, 
and the air over it is cooler than over the sea. Hence the 
reversal in the direction of the wind, which is now a land 
breeze. 

The trade winds are similar currents on a still larger 
scale. The earth and the air are highly heated by the ver- 
tical rays of a tropical sun ; the air expands, rises, and over- 
flows northward and southward. The denser air from both 
hemispheres flows in to take the place of the 
ascending mass. The air currents toward the 
equatorial belt lag behind the west-east rotat- 
ing earth, and blow as northeast and southeast 
trade winds. 

425. Convection by Hydrogen. The rapid con- 
vection of heat by hydrogen gas formed the 
subject of a celebrated experiment by Dr. 
Andrews. A thin platinum wire, which could 
be heated by an electric current, was stretched 
along the axis of a glass tube. Inlet and 
exhaust tubes were provided for filling and 
exhausting (Fig. 271). When the tube was 
exhausted of air, the current was adjusted so 
as to heat the wire to vivid brightness without 
fusing it. The introduction of air sensibly 
diminished the brightness of the wire ; but 
when the tube was filled with hydrogen, the 
wire was scarcely red hot. In an atmosphere of 
hydrogen the light and rapidly moving mole- 
cules carry frequent charges of heat from the 
wire to the cooler walls of the tube. 

The incandescent lamp is made with the filament in a high 
vacuum to avoid the loss of heat by convection. For this 
reason even an inert gas, like nitrogen, cannot be used to fill 



CONVECTION 



399 



the bulb, because the energy of the heated filament is then 
rapidly conveyed from it to the glass bulb, and heats it at 
the expense of the brightness of the filament. The bulb 
heats to a still higher temperature when the filament is in- 
closed in an atmosphere of hydrogen. Very hot bulbs indi- 
cate lamps with too low a vacuum. 

426. Ventilation. The office of a chimney is to produce a 
convection current of air. The heated air in a chimney rises 
because it is lighter than the air without. The external 
pressure is therefore only partly counterbalanced by that of 
the air in the flue. If the chimney happens to be colder 
than the external air, there is a down draft, or the chimney 
smokes. 

The office of a lamp chimney is to increase the supply of 
oxygen to the flame. The air within it is heated by the 
flame and rises ; at the same time 
cold air flows in at the bottom to 
restore the equilibrium. 

Place a lighted candle at the 
bottom of a tall lamp chimney. 
Ingress of air may be prevented 
by pouring a little water into the 
outer dish (Fig. 272). The flame 
soon goes out for lack of oxygen. 
If a T-shaped partition be now 
placed in the chimney and the 
candle relighted, it will continue 
to burn. If a piece of smoldering 
brown paper be held over the tube, 
the smoke will descend on one side of the partition and 
ascend on the other. This is a true convection current, 
supplying oxygen to the candle and carrying off the products 
of combustion. 




Fig. 272 



400 HEAT 

III. RADIATION AND ABSORPTION 

427. Cooling- by Radiation. A hot body imparts its 
energy of heat to other bodies, not only by conduction and 
by convection currents, but also by radiation without the 
agency of any intervening material. We feel the heat 
radiated from the sun or from a hot stove and warm our- 
selves before an open grate fire. In the best vacuum of 
an incandescent lamp bulb, the hot filament cools rapidly 
by radiation unless the energy is supplied as fast as it is 
dissipated. 

The transmission of the energy given out by a hot body 
through a medium without heating it is a physical operation 
identical with the transmission of light. The experimental 
evidence of the physical identity of " radiant heat " and light 
is overwhelming. It is reflected, refracted, polarized, and is 
subject to interference the same as light. This energy dur- 
ing transmission is not heat, but it is transformed again into 
heat by absorption in bodies which do not transmit it. Light 
waves, heat waves, and electric waves are essentially identi- 
cal except in the length of the waves, and they all travel 
with the stupendous velocity of 300 million meters a second. 
The light effects, the heat effects, and the chemical effects 
of radiation are only different aspects of the same physical 
phenomena. 

428. Law of Cooling. Newton's law of cooling is that 
the rate is proportional to the difference of temperature be- 
tween that of the hot body and the inclosing chamber. This 
law represents the facts fairly well when the difference of 
temperature does not exceed a few degrees, but it fails nota- 
bly when the difference is as much as 50. 

The law of Stefan, discovered in 1879, is that the rate of 
emission is proportional to the fourth power of the absolute 
temperature. The heat actually lost by a body is the differ- 
ence between what it radiates and what it receives from sur- 
rounding bodies. Then by Stefan's law the loss of heat R 



RADIATION AND ABSORPTION 



401 



per square centimeter per second may be expressed as 



(79) 






where T l is the absolute temperature of the hot body and T 2 
that of the inclosure. The constant c for a perfectly black 
body has been found to be about 1.28 x 10~~ 12 calories. 

When T T 2 is only a few degrees, Stefan's law is equiv- 
alent to Newton's law of cooling. 

429. The Radiometer. Among the instruments which have 
been used to measure the intensity of radiation is a modifi- 
cation of Crookes's radiometer due to Professor E. F. Nichols. 
Sir William Crookes invented the radiometer in 1873 while 
investigating the properties of highly attenuated gases. It 
consists of a glass bulb exhausted to a pressure not exceed- 
ing 7 mm. of mercury (Fig. 273). Within the bulb is a light 
cross of aluminum wire carrying small 
vanes of mica, one face of each being 
coated with lampblack ; the whole is 
mounted so as to revolve lightly on a ^M 

vertical pivot. When the radiometer 
is placed in sunlight, or receives the 
radiation from a hot body, the cross 
revolves with the blackened faces of 
the vanes retreating from the source 
of radiation. 

The explanation of this interesting 
instrument is found in the kinetic 
theory of gases that the mean free 
path of the molecules between col- 
lisions with other molecules, at this 
low pressure, becomes equal to the 
distance between the vanes and the 
wall of the bulb. The infrequent collisions among the mole- 
cules in such a vacuum prevents the equalization of pressure 
throughout the tube. The blackened side of the vanes 




Fig. 273 



402 



HEAT 



absorbs more heat than the other, and the gas molecules 
rebound from the warmer surfaces with greater energy than 
from the opposite ones, and thus give the vanes an impulse 
as a reaction the other way. This impulse is the equivalent 
of a pressure, but the residual gas has lost the power of rapid 
adjustment of pressure throughout the bulb. When the 
vacuum is not good, the increased energy of the molecules 
projected from the blackened vanes is so rapidly distributed 
that the differential pressure on the two sides of a vane 
becomes evanescent. 

In the Nichols instrument the rotating system is delicately suspended 
by a quartz fiber, and the radiation is admitted through a window of 
fluorite, which is remarkably transparent to radiations of all wave 
lengths. The suspended system carries a very light mirror, and the de- 
flections are read by means of a telescope and scale. 

430. Reflection of Invisible Radiation. Aside from the 
simple observation (as shown by a fire screen) that radiant 
heat travels in straight lines like light, the most obvious 
analogy between the two is found in their common obedience 
to the law of reflection. 

Two large concave mirrors are placed several meters apart, 
as in Figure 274. The two may be adjusted in position by 





Fig 274 

means of a candle placed at the focus of one of them. The 
position of its image may be marked at the focus of the other 
mirror. Then if the candle be replaced by a heated iron 
ball, and if the blackened bulb of a thermometer, or the black- 
ened face of a thermopile ( 573), be placed at the marked 
focus, either instrument will show that the radiant heat is 
reflected to the same focus as the light of the candle. The 



RADIATION AND ABSORPTION 



403 



reflection of the non-luminous radiation from the two mirrors 
takes place in the same manner as that of the luminous radi- 
ation, for both converge to the same point. If the ball be 
heated to a dull red, the convergence of the heat at the focus 
may be felt by the hand. The thermopile will detect it 
when the ball has cooled to such an extent that it may be 
held in the fingers. 

If the ball be replaced by a piece of ice, a delicate ther- 
mopile and galvanometer will show that the thermopile is 
cooled. In this case the thermopile radiates more heat to 
the ice than the ice radiates to it. It therefore cools. 

431. The Law of Inverse Squares. Melloni was the first to per- 
form an ingenious experiment to demonstrate that the invisible heat 
radiation received 

from any small area 

varies inversely as 

the square of its 

distance from the 

source. BC (Fig. 

275) is a tank filled 

with hot water and 

coated on its front 

side with lampblack. 

Let the thermopile 

with a converging 

cone be placed -in 

the position A, and let the deflection of the galvanometer be noted. 

Then let the thermopile be moved to double the distance from the tank 

at A'. The galvanometer will indicate the same current as before. The 

radiating surfaces in the two cases are the bases of the dotted cones. 

Their linear dimensions are as one to two and their areas as one to four. 

Since the radiation from a fourfold area produces the same effect at twice 

the distance, the intensity of the radiation received from any small area 

must vary inversely as the square of the distance. In the same way 

a uniformly red-hot surface, viewed through a ^ tube, appears equally 

bright at all distances, so long as the surface fills the field of view 

through the tube. 

432. Refraction of Heat Radiation. Herschel observed dark 
heat radiation in the solar spectrum beyond the red end, or 




Fig. 275 



404 HEAT 

the invisible infra-red rays. They are refracted and dis- 
persed along with radiation of shorter wave length from a 
source of high temperature like the sun. 

Melloni discovered that rock salt transmits all wave 
lengths of non-luminous radiation with nearly equal facility, 
while almost every other substance absorbs them with 
avidity. Clear glass is as opaque to radiation from a non- 
luminous source as black glass is to visual radiation. By 
employing rock salt prisms and lenses Melloni demonstrated 
that the radiation from a body at a temperature as low as 
100 may be refracted and converged to a focus the same as 
light. 

It has also been shown that the refrangibility decreases 
with the temperature of the source, and that obscure heat 
rays are of lower refrangibility, or longer wave length, than 
visual rays. . 

433. Absorption of Radiation. When luminous radiations 
are incident on a body, in general, one portion is reflected, 
another is transmitted, and the remainder is absorbed. Thus, 
a piece of red glass reflects a part of the incident beam, 
transmits only wave lengths near the red end of the visible 
spectrum, and absorbs the rest, converting its- energy into 
heat. If the transmitted portion is reduced to zero, the body 
is opaque ; if the surface is covered with lampblack, the re- 
flected light is sensibly zero and the entire incident beam is 
absorbed. The absorption that rejects the red only is called 
selective absorption, while that of lampblack is general. 

This division of the incident radiation, either by general 
or selective absorption, is not peculiar to those radiations 
that affect the eye. Bodies which transmit radiant heat are 
said to be diathermanous, while those that absorb it are called 
athermanous. A body transparent to light is not on this 
account transparent to non-luminous radiation. Clear glass 
is transparent to radiation somewhat beyond the violet of the 
spectrum, but it is very athermanous to long heat waves. 



RADIATION AND ABSORPTION 405 

If a sheet of glass be held between the heated ball and the 
mirror in the experiment of Figure 274, little or no heat will 
be detected at the focus of the distant mirror. All glass ex- 
hibits selective absorption, and colored glass has its range of 
absorption extended to some portions of the visible spectrum. 
In a physical sense clear glass itself is not colorless. 

Hard rubber in thin sheets is opaque to light, but trans- 
parent to long heat waves. Carbon disulphide transmits in 
almost equal degree the luminous and the non-luminous rays ; 
but if iodine be dissolved in it, more and more light will be 
cut off as iodine is added, until at length the solution be- 
comes opaque. But the solution is still diathermanous. Tyn- 
dall inclosed it in a hollow lens with rock salt faces and 
showed that it transmits enough heat from an electric arc 
light to raise platinum to incandescence at the focus. 

Such facts as these lead to the conclusion that selective 
absorption is not peculiar to the visible spectrum. There 
are Fraunhofer lines, or gaps in the continuity due to absorp- 
tion, in both the infra-red and the ultra-violet, as well as in 
the visible spectrum. In the case of liquids the absorption 
bands in the spectrum are fairly broad; with gases and 
vapors as the absorbing media, the bands are sharply denned 
and consist of a number of narrow lines in different parts of 
the spectrum. 

434. Theory of Exchanges. If a warm body, such as a ther- 
mometer, be hung in an inclosure cooler than itself, it will 
lose heat, and even in a vacuum thermal equilibrium will at 
length be reached by radiation alone. Does all radiation 
cease when the body and the inclosure are at the same tem- 
perature, and does the body radiate no heat when sur- 
rounded by an inclosure warmer than itself ? A cold body 
introduced into the inclosure would at once receive heat by 
radiation ; but its presence can have no effect on the radia- 
tion of other bodies within the same inclosure. 

Prevost came to the conclusion many years ago that radia- 



406 HEAT 

tion continues all the time, and that its rate has no relation 
to the temperature of other bodies in the inclosure, but is a 
function of the nature of its surface and of its temperature. 

In the experiment of Figure 274 with the mirrors, it would 
be unscientific and unnecessary to suppose that cold is radiated 
by the ice. The thermopile radiates toward the ice exactly 
as it does toward the hot ball, but it receives from the ice 
less heat than it loses by radiation, and its temperature 
therefore falls. 

The two processes of radiation and absorption go on 
together, and there is a continuous interchange of energy 
between bodies by radiation. Heating and cooling depend 
on the differential effect of the radiation emitted and ab- 
sorbed. A stationary temperature is maintained only so long 
as the emission and the absorption balance each other. 

435. Absorptive and Emissive Power. A body that absorbs 
all radiation incident on it is known as a " perfectly black " 
body. The incident energy is all transformed by such a body 
into heat. Lampblack is an approximately " black " body. 

The emissive power, or emissivity, of any surface is defined 
as the ratio of the energy emitted by it to that emitted by a 
black body of equal area at the same temperature and in the 
same time. 

The absorptive power of a surface is the ratio of the energy 
absorbed by it to that absorbed by a black body of equal area 
in the same time. 

Since a black body absorbs all radiation incident on it, the 
absorptive power of a body may be more simply defined as 
the fraction of the whole incident energy which it absorbs. 

Emissive and absorptive powers are connected by the 
simple relation, that for any given surface at a given tem- 
perature, the two are equal to each other. 

436. Equilibrium of Radiation. The equality of the emissive 
and the absorptive powers, which flows from the principle of 
Prevost, applies even to such specific differences as wave 



RADIATION AND ABSORPTION 407 

length and polarization of radiation, whether luminous or 
non-luminous. 

Sodium vapor is a remarkable illustration. When heated 
it emits radiations of two wave lengths differing but little 
from each other. Now, if light from a white-hot solid is 
transmitted through relatively cool sodium vapor, the spec- 
troscope reveals two dark absorption lines identical in position 
with the two bright lines which self-luminous sodium vapor 
emits. The energy absorbed by the cooler sodium vapor is 
greater than the energy emitted ; but if its temperature be 
raised to that of the source of the radiation, the emissivity 
will equal the absorptive power and the absorption lines will 
disappear. 

If the absorptive and emissive powers are equal, then good 
reflectors are poor radiators. A pot of red-hot lead examined 
in the dark is more luminous where the surface is covered 
with dross than where it is clear. 

If a piece of platinum foil, with a figure drawn on it in 
ink, be heated in a Bunsen flame held under it, the blackened 
portion will be more highly luminous than the rest when viewed 
from the tarnished side. If it be viewed from the reverse 
side, the figure in ink will be darker than the adjacent surface. 
Since the tarnished surface radiates more than the clean sur- 
face, it is cooler and appears dark by contrast on the reverse 
side. 

In a striking experiment of Stewart a piece of white stone- 
ware with a black pattern on it was heated to white heat. 
In the dark, the black surface shone much more brightly 
than the white, so that the white and black in the pattern 
were curiously exchanged. 

A transparent piece of tourmaline, cut parallel to its axis, 
absorbs nearly all the light polarized in a plane parallel to 
the axis. If Prevost's principle extends to polarization, such 
a plate when heated red-hot should emit light polarized in 
the same plane as the light which it absorbs. This proved to 
be true. 



408 HEAT 

Again, any incombustible colored body in a bright coal 
fire does not alter the color of the light emitted after it has 
attained the temperature of the hot coals. A piece of red 
glass, for example, transmits red from the glowing coals and 
radiates the greenish light which it absorbs when cold. The 
light which it radiates exactly compensates for the light 
which it absorbs. 

Problems 

1. What will be the resulting temperature if 5 kgm. of water at 90 
are mixed with 5 kgm. of ice at 0? 

2. 4 kgm. of ice at are put into 6 kgm. of water at 40. Deter- 
mine the result. 

3. liow much steam at 100 will be required to melt 310 gm. of 
ice at and to raise the temperature of the water to 16 ? 

4. 10 gm. of steam at 100 are blown into a mixture of ice and 49.7 
gm. of water at 0. The final temperature of the water is 5. Find the 
quantity of- ice. 

5. What is the heat of vaporization of water derived from the follow- 
ing data : 10 gm. of steam at 100 condensed in 610 gm. of water at 15 
raised its temperature to 25? 

6. When heat was supplied at a constant rate to a certain block of tin 
it was found that the temperature rose 2 a second. After the melting 
point was reached, the temperature remained constant for 130 seconds, 
when all the tin was melted. If the specific heat of tin is 0.054, find its 
heat of fusion. 

7. How many degrees would the air in a room 6x4x3 m. be warmed 
by the condensation in the radiator of 1 kgm. of steam at 100, if one 
liter of air weighs 1.29 gm. ? 

8. If the coefficient of thermal conductivity of iron is 0.164, how 
many calories will be conducted through a plate of iron 1 m. square and 
0.3 cm. thick if the two sides are kept at and 60 for one hour? 

9. A brass plate 1 cm. thick and 100 sq. cm. in area was kept in 
contact with steam at 100 on one side and with melting ice on the other. 
If 22.9 kgm. of ice were melted in 10 minutes, what is the coefficient of 
thermal conductivity of the brass? 

10. A plate of glass 2 cm. thick and 3x4 m. in area separates two 
rooms which remain at and 25 respectively. If the coefficient of 
thermal conductivity of glass is 0.015, how much heat is given off by the 
glass per minute? 



CHAPTER XV 

THERMODYNAMICS 
I. MECHANICAL EQUIVALENT OF HEAT 

437. First Law of Thermodynamics. Evidence has been 
accumulating that heat is a form of energy. It is a familiar 
fact that mechanical energy is readily converted into heat; 
also that in purely mechanical processes the total energy 
remains constant. It is also true that in a system of bodies 
between which only thermal processes take place, the total 
quantity of heat remains constant. The inquiry now relates 
to the constancy of energy during the reciprocal conversion 
of heat and mechanical work. 

The first law of thermodynamics relates to this reciprocal 
conversion. It is the principle of the Conservation of 
Energy applied specifically to heat. Maxwell expressed it 
as follows: 

When work is transformed into heat, or heat into work, 
the quantity of work is mechanically equal to the quantity 
ofheat. 

In symbols this relation may be written 

W=JH, (80) 

where TFis the work measured in ergs, .ffthe quantity of 
heat in calories, and 7the constant ratio between W and H 
known as the "mechanical equivalent of heat." (Obviously 
H and IF" may be measured in any suitable units.) 

438. Joule's Determinations. The investigations of Joule in 
support of the first law of thermodynamics and to find the 
mechanical equivalent of heat extended from 1842 to 1878. 

409 



410 



HEAT 



They demonstrated that whenever mechanical energy is con- 
verted into heat, a definite ratio always exists between the 
numerical expressions for the two, whatever may be the 
method by which the one is converted into the other. Joule's 
experiments left little for subsequent investigators except 
refinement of details and an increase of the scale on which 
the experiments were conducted. 

In the last experiment of Joule in 1878 water was churned 
with paddles, and the ratio was measured between the work 
expended in turning the paddles and the number of heat units 
generated. The work was measured by the following devices : 
A calorimeter h (Fig. 276), containing the water, was 
supported on a hollow cylindrical vessel w, which floated in 

water in v. It was thus free to 
turn around a vertical axis and 
the pressure was taken off the 
bearings. The paddles in the 
calorimeter were carried on a 
vertical axis &, about which 
the calorimeter could also turn. 
There was a horizontal flywheel 
at /, and the paddles were 
turned by the handwheels d 
and e. 

To prevent the calorimeter 
turning by the friction between 
the paddles and the water, two 
thin silk strings were wound 
in a groove around it, and car- 
ried weights k, k, beyond the pulleys /, j. These weights 
were adjusted until they remained stationary, while the shaft 
and paddles revolved at a suitable uniform speed, which was 
recorded by the counter g. The weights then gave the 
torque necessary to keep the calorimeter at rest, that is, 
a torque equal to the moment of the force exerted by the 
paddles on the water. To measure the work done, it was 




Fig. 276 



MECHANICAL EQUIVALENT OF HEAT 411 

only necessary to multiply this moment by the angulai 
velocity of the shaft. The arrangement was in fact equiva- 
lent to a friction dynamometer ( 81). 

Let m be the mass of each weight, r the radius of the groove 
in the calorimeter, and n the number of rotations a second. 
Then the moment of the two weights is 2 mgr, and the angular 
velocity is 2 irn. The former is the torque T and the latter 
Co. Hence in n turns the work done is 

Tco = 2 mgr x 2 Trn == 4 nrnrmg ergs. 

If M is the mass of the water and M f the water equivalent 
of the calorimeter and the paddles, and if t is the rise in 
temperature, the heat generated is (M+ M')t calories. 
Then the ratio of the two, or Joule's equivalent, is 



Joule expressed his results in terms of the temperature 
given by a mercury-in-glass thermometer. They were later 
reduced by Rowland to the hydrogen scale, with the result 

J= 4.182 x 10 7 ergs per calorie at 15, 
or 1 calorie = 4.182 x 10 7 ergs = 4.182 joules. 

439. Rowland's Determination of /. In 1879 Rowland re- 
peated the work of Joule in a series of very elaborate and 
precise experiments. Rowland's plan was the same in prin- 
ciple as Joule's, the chief differences being that the paddles 
were turned by power from a steam engine, and the revolu- 
tions were recorded on a chronograph. On the same 
chronograph were recorded the transits of the mercury over the 
divisions of the thermometer. The rate at which heat was 
generated in Rowland's apparatus was about 50 times as great 
as in Joule's. His series of determinations at different tem- 
peratures showed that the specific heat of water is a minimum 
at about 30. 



412 HEAT 

When reduced to the hydrogen scale, the final value for J 
deduced from Rowland's experiments is 

J= 4.189 x 10 7 ergs/calorie, at 14.6, 
or 1 calorie = 4.189 x 10 7 ergs = 4.189 joules. 

It is apparent that the numerical value of the mechanical 
equivalent depends on the units employed. Rowland ex- 
pressed his results also in gram meters in the latitude of 
Baltimore. Expressed in this way 

1 calorie = 427.52 gram meters. 

The interpretation is that if the work done in lifting 
427.52 grams one meter high is all converted into heat, it 
will raise the temperature of one gram of water one degree 
centigrade at 14.6. 

In terms of the English system, the equivalent is 

1 B. T. U. = 778.1 foot pounds at Greenwich. 

This is equivalent to 1400.6 foot pounds per degree centi- 
grade ; that is, if the work done in lifting 1400.6 pounds 
one foot high in the latitude of Greenwich is converted into 
heat, it will raise the temperature of one pound of water one 
degree centigrade. Obviously 778.1 and 1400.6 stand to 
each other in the relation of 5 to 9. 

440. Mechanical Equivalent from Specific Heats of a Gas. 

The mechanical equivalent of heat was first calculated by Robert Mayer, 
a German physician, in 1842, on the assumption that the internal work 
done during the expansion of a gas is zero. It has since been shown by 
Joule and Thomson that, while this internal work is not zero, it is very 
small in the case of gases not easily -liquefied. 

Let v be the volume of unit mass of the gas at absolute temperature T. 
Since the volume is proportional to the temperature on the absolute 
scale ( 381), v/T is the change in volume per degree, or. the expansion, 
and pv/T, the product of the pressure and the small change of volume, 
is the work done by the gas during the expansion under constant pressure 
while the temperature changes one degree ( 66). 

The specific heat .? is the number of calories required to raise the tem- 
perature of unit mass of a gas one degree when the volume is kept con- 



CARNOT'S CYCLE 



413 



stant ; the specific heat s p is the number of calories required to raise the 
temperature one degree when the pressure is kept constant and the gas 
expands. If there is no internal work done, the latter will exceed the 
former by the thermal equivalent of the work done by the gas in expand- 
ing under pressure. 

Reduced to mechanical units, this difference may be expressed in the 
form of the equation, 



or 



T r" 

J ~Y' 



Under standard conditions of temperature and pressure, T = 273, 
p = 1,013,250, v = l/d = 1/0.001293 gm. per cm. 3 , s p = 0.2375, s, = 0.1684. 
Substituting in the last equation, 



1013250 



273 x 0.001293 x 0.0691 



= 4.16 x 10 7 ergs/calorie. 



II. CARNOT'S CYCLE 

441. Isothermal Processes. An isothermal process is one 
in which the temperature remains constant. The simplest 
example of such a process is the isothermal expansion or 
compression of a gas obeying Boyle's law. If the state of 
the gas be represented by a pressure-volume Qp, v) diagram, 
each curve drawn for one tem- 
perature, T v T 2 , etc., and called 
an isothermal line, is a rectan- 
gular hyperbola (Fig. 277), for 
the equation is 

pv = RT=& constant ( 382), 

and this is the criterion of a 
rectangular hyperbola. 

If the gas of volume v 1 and 
pressure p l expands isother- 
mally at a temperature T^ to the 
state (v 2 , jt? 2 ), it performs external work and absorbs heat 
from the outside to maintain constant temperature. The 







414 HEAT 

work done during the process is measured on the scale of the 
diagram by the shaded area p^p^Vv included between 
the curve, the axis of volumes, and the two ordinates, p 1 
and p 2 ( 66). 

Conversely, if the gas is compressed isothermally between 
the same limits, the same amount of work is done on the gas, 
and an equivalent quantity of heat is given out by it. 

442. Adiabatic Processes. An adiabatic process is one in 
which the working substance neither receives heat from 
other bodies nor loses heat to them. For a perfect gas the 
equation for an adiabatic process is 

pvy = a constant. 

The corresponding pressure-volume curve is an adiabatic 
line. Since work is done during an adiabatic expansion 
without the application of heat, the gas cools and the 
pressure falls below the pressure for isothermal expan- 
sion. The slope of an adiabatic line through any point 
is therefore steeper than that of an isothermal line through 
the same point. 

The work done in adiabatic expansion is again measured 
by the area included between the adiabatic curve, the axis 
of volumes, and the two limiting ordinates. 

The coefficient of elasticity for the isothermal expansion 
of a gas is numerically equal to the pressure ( 207). But 
for any increment of volume in adiabatic expansion, the 
decrease of pressure is greater than for the same isothermal 
expansion. Hence the adiabatic coefficient of elasticity of a 
gas is greater than the isothermal coefficient. 

An instance of an adiabatic process occurs in the transmis- 
sion of sound through air, where the periodic compressions 
and rarefactions are too rapid to permit of an equalization 
of temperature. It is, therefore, the adiabatic coefficient of 
elasticity that enters into the calculation of the velocity 
of sound in gases ( 208). 



CARNOT'S CYCLE 



415 



443. Carnot's Cycle. To Carnot belongs the credit of in- 
troducing a cycle of operations, consisting of isothermal and 
adiabatic processes, by which external work is done and the 
working substance is returned to its initial state. The advan- 
tage gained by carrying the working substance, such as a gas, 
through a complete cycle of operations is that it neither gains 
nor loses internal energy. Whatever work has been done must 
therefore be credited to energy derived from external sources. 

Carnot's cycle consists of four processes, two isothermal 
and two adiabatic: 

1. An isothermal expansion from the state A to the state 
B (Fig. 278). The work done equals the area ABv^v^ and 
heat IT*! is absorbed from 

without at the tempera- 
ture T r 

2. An adiabatic expan- 
sion from the state B to 
the state 0. The work 
done equals the area 
BCvJvJ. No heat is ab- 
sorbed, but the tempera- 
ture falls from T^ to T v 

An isothermal compres- 
sion from the state O to 
the state D. Work is 
done on the gas equal to 
the area CDv^v^, and a quantity of heat ff 2 is generated and 
given out at the lower temperature T z . 

4. An adiabatic compression from the state D to the initial 
state A. The work done on the gas equals the area DAv^Vp 
and the temperature rises from T 2 to T r 

The net results of such a cycle of operations, by which the 
gas is returned to its initial volume, pressure, and tempera- 
ture, are the following : 

a. The absorption of a quantity of heat ff l at the higher 
temperature T v 




Fig. 278 



416 



HEAT 



b. The evolution of a quantity of heat H 2 at the lower tem- 
perature T z . 

c. The performance of an amount of mechanical work 
represented by the shaded area ABCD inclosed by the two 
isothermal and the two adiabatic lines, for 



ABv 1 'v 1 4- 

This work is obtained from the heat ff l Jf 2 in accordance 
with the first law of thermodynamics. The gas at the close 
of the operations is in the same state in every respect as at 
the beginning and has neither gained nor lost energy. It is 
only the agency by means of which heat is converted into 
work ; it is therefore called the working substance. 

444. Carnot's Engine. Carnot's engine is an ideal one 
designed to show that his cycle may be used for the continued 
production of work from heat. Suppose the working sub- 
stance D (Fig. 279) inclosed in a cylinder impervious to heat 
except through its bottom, which is assumed to be a perfect 






Fig. 279 

conductor. A and B are two stands which are maintained 
at the temperature T and T 2 respectively. is a third 
stand, the top of which is assumed to be perfectly non- 
conducting. The four operations of Carnot's cycle may then 
be realized in the following manner : 

1. The cylinder at the temperature T^ is placed on stand 
A and the substance D expands isothermally. Heat flows in 



CARNOT'S CYCLE 417 

through the bottom of the cylinder to maintain a constant 
temperature. 

2. The cylinder is transferred to the stand O and the 
expansion is continued adiabatically until the temperature 
falls to T 2 . 

3. The cylinder is placed on the cooler stand B and the 
substance D is compressed isothermally. Heat flows out 
through the bottom of the cylinder and the temperature 
remains constant. 

4. The cylinder is finally returned to the stand C and the 
substance D is further compressed adiabatically until the 
temperature rises to T r 

The physical results of this cycle of operations are the 
same as those described in the last article. With a suitable 
heater A and refrigerator B, this cycle may be repeated 
indefinitely with continued conversion of heat into work. 

These four operations may be performed in the reverse 
order. The working substance is carried around the Carnot 
cycle in the reverse direction, and the physical and mechani- 
cal operations are reversed. Starting with the substance in 
the state A (Fig. 278), the cylinder is placed on the stand C 
and the first expansion is adiabatic, the temperature falling 
from T to T 2 . This operation is followed by isothermal 
expansion with the cylinder on the stand A. Heat _T 2 is 
absorbed at the temperature T z . The third operation is an 
adiabatic compression with the cylinder on the stand 0. The 
temperature rises to T during the compression. The cylin- 
der is finally placed on B and the substance is compressed 
isothermally at the temperature T until it returns to its 
initial state at A (Fig. 278). During this compression heat 
H-i is returned to the heater at the higher temperature. 

The net result of this reverse order is that heat has been 
transferred from the refrigerator to the heater, but only at 
the expense of mechanical work equal to the area ABCD. 
More work has been done on the working substance than by 
it, and the excess has been converted into heat. Because all 



418 HEAT 

the operations may be carried out in the reverse order, Car- 
not's engine is said to be reversible. 

445. Efficiency of Car-net's Engine. If temperatures are 
measured by a gas thermometer with a perfect gas as the 
thermometric substance, the quantities of heat absorbed and 
given out by the working substance are proportional to its 
absolute temperatures during the two operations of compres- 
sion and expansion, or 



Efficiency of conversion is the ratio between the heat trans- 
formed into work and the heat absorbed by the working sub- 
stance at the higher temperature, or 

Efficiency = Heat utUized ~ H ~ 



Heat absorbed H 1 T 

446. Carnot's Principle -- The important principle of 
Carnot, derived from his reversible engine, is as follows : 

" If a given reversible engine, working between the upper 
temperature T^ and the lower temperature T y and receiving 
a quantity H^ of heat at the upper temperature, produces a 
quantity w of mechanical work, then no other engine, what- 
ever be its construction, can produce a greater quantity of 
work when supplied by the same amount of heat and working 
between the same temperatures." 

Suppose an engine M to have a higher efficiency than a 
reversible engine N. Let it be coupled to N working in the 
reverse order. Then, since M converts a larger portion of 
the heat H^ into mechanical work than N requires to restore 
the same quantity of heat ff l to the source, the two engines 
constitute an automatic device by which M, while drawing 
heat Jff 1 from the heater, supplies to N sufficient energy 
to enable it to transfer from the refrigerator to the heater 
more heat than M withdraws. In other words, the two 



CARNOT'S CYCLE 419 

coupled engines would run perpetually, transferring heat 
all the time from colder bodies to hotter ones. Heat might 
thus be collected automatically from the earth or the air and 
be stored in a reservoir, from which it could be drawn for 
the continuous performance of mechanical work. But such 
an operation is denied by universal experience and is inad- 
missible. It follows that no engine can be more efficient 
than the ideal reversible one of Carnot. As a practical fact 
no heat engine has as high an efficiency as the theoretical 
one of an ideal reversible engine. 

Since no engine can have a higher efficiency than a rever- 
sible engine, it follows that no one reversible engine can have 
a higher efficiency than another. This means that the effi- 
ciency is independent of the working substance and depends 
alone on the temperatures between which the engine works. 

447. The Second Law of Thermodynamics. The second law 
of thermodynamics embodies the Carnot principle, and was 
stated by Clausius as follows : 

"It is impossible for a self-acting machine, unaided by 
any external agency, to convey heat from one body to another 
at a higher temperature." 

Lord Kelvin expressed it in a slightly different form : 

" It is impossible, by means of inanimate material agency, 
to derive mechanical effect from any portion of matter by 
cooling it below the temperature of the coldest of the sur- 
rounding objects." 

These statements apply only to the performance of devices 
working in cycles. Without this limitation, it is evident 
that the heat of a body, that of compressed gas, for example, 
may be converted into work by cooling the body below the 
temperature of surrounding objects ; but before the operation 
can be repeated, the working substance must be restored to 
its initial condition, and this can be done only by applying 
energy from without. 



420 HEAT 

448. Thermodynamic Scale of Temperature. A new scale of 
absolute temperatures was early suggested by Lord Kelvin. 
Since the ratio H^/H^ of the quantities of heat taken in and 
rejected between the temperatures t 1 and t 2 depends on these 
temperatures alone, new numbers T^ and T^ having the same 
ratio as ff l and H^ might be taken to denote these tempera- 
tures in centigrade degrees, thus forming a thermodynamic 
scale independent of the expansion coefficient of a gas. 

Such a scale leads to the conception of an absolute zero; 
for if the temperature T 2 is conceived to be lowered to the 
point where no heat is ejected by the engine, the efficiency 
will be unity. No lower temperature than this can exist; 
for if it did, by selecting this lower one as the temperature 
of the refrigerator, more work could be obtained from the 
heat absorbed than its mechanical equivalent. But such a 
result violates the first law of thermodynamics. Therefore 
there can be no negative temperature below T% = 0, which is 
thus the absolute zero. Kelvin's thermodynamic scale is 
independent of the properties of the working substance, and 
it has been shown to be identical with that of a perfect gas 
thermometer. In fact, the difference between it and the 
hydrogen scale between 50 and 150 is less than 0.001. 
Even at 1000 the reading on the hydrogen scale is only 
0.044 lower than on the thermodynamic scale. 

449. The Steam Engine. The most important devices for 
converting heat into mechanical work are the steam engine 
and the gas engine. In the reciprocating steam engine a 
piston is moved alternately in opposite directions by the 
pressure of steam applied first to one of its faces and then 
to the other. This reciprocating motion is converted into a 
rotatory motion by the device of a connecting rod, a crank, 
and a flywheel. 

The working parts, in longitudinal section, of the cylinder 
and slide valve of a single expansion engine are shown in 
Figure 280. The piston B is moved in the cylinder A by the 



CARNOT'S CYCLE 



421 



R 




pressure of the steam admitted through the inlet pipe a. 
The slide valve d works in the steam chest cc and admits 
steam alternately to the 
two ends of the cylinder 
through the steam ports 
at either end. 

When the valve is 
in the position shown, 
steam passes into the 
right-hand end of the 
cylinder and drives 
the piston toward the 
left. At the same time, 
the other end is in 

connection with the ex- 
Fig. 280 

haust pipe ee, through 

which the expanded steam escapes, either into the air, as 

in a high-pressure non-condensing engine, or into a large 

condensing chamber, as in a low-pressure condensing 

engine. In the condenser the steam is condensed to water, 

thus reducing the back pressure on the exhaust side of the 

piston. 

The slide valve d is moved through the valve rod R by 
means of an eccentric, which is a round disk mounted eccen- 
trically on the engine shaft and has the effect of a crank. 
The flywheel, also mounted on the shaft, has a heavy rim 
with a large moment of inertia. It has, therefore, the ca- 
pacity of storing enough energy during its acceleration to 
carry the shaft over the dead points when the piston is at 
either end of the cylinder. There is in the flywheel a give- 
and-take of energy twice every revolution, the result of 
which is a fairly steady rotation of the shaft. 

450. The Indicator Diagram. A steam indicator is a de- 
vice for the automatic tracing of a diagram representing the 
relation between the volume and the pressure of the steam 



422 HEAT 

in the cylinder during one stroke. This diagram is techni- 
cally known as an "indicator card " (Fig. 281). 

The interpretation of the successive operations, which com- 
pose the cycle and are recorded on the diagram, is the follow- 
ing : From a to 6 the inlet port is open 
and the pressure on the piston is the full 
pressure of the steam ; at b the inlet port 
v ^^^ is closed and the steam 

expands in a mixed adia- 
batic and isothermal man- 




Volunie 

Fig 28| ner from b to <?, when the 

exhaust port opens ; at d 

the pressure is reduced to its lowest value and it remains 
sensibly constant during the return movement of the piston 
until e is reached, when the exhaust port closes and the 
remaining steam is compressed adiabatically from e to /. At 
/ the inlet port opens and the pressure rises abruptly to the 
initial maximum, thus completing the cycle. 

The diagram bears some resemblance to the Carriot cycle, 
the portions ab and de corresponding to isothermal lines, and 
be and ea to adiabatic lines. The work done during the 
stroke is represented, as in the Carnot cycle, by the inclosed 
area abcdef. It is readily calculated if the length of stroke 
and the scale of pressures are known. 

451. The Gas Engine. The gas engine is a type of internal 
combustion engine, which includes prime movers consuming 
illuminating gas, blast furnace gas, producer gas, gasolene, 
kerosene, or alcohol as fuel. The fuel is introduced into the 
cylinders of the engine either as a gas or as a vapor, mixed 
with the proper proportion of air to produce a good ex- 
plosive mixture. The mixture is usually exploded at the 
right instant by means of an electric spark. The force of 
the explosion drives the piston forward in the cylinder. 

In the four-cycle type of gas engine, the explosive mixture 
is drawn in and exploded every other revolution of the engine, 



CARNOT'S CYCLE 



423 



while in the two-cycle type an explosion occurs every revo- 
lution. The former type is used in nearly all motor car 
engines. 

The operation of the four-cycle engine may be understood 
from a description of the four steps in the complete cycle, as 
illustrated in 1, 2, 3, and 4 of Fig- 
ure 282. The inlet valve a and 
the exhaust valve b are operated 
by the cams c and d. Both 
valves are kept normally closed 
by springs surrounding the valve 
stems. The small shafts to which 
the two cams are fixed are driven 
by the small spur wheel e on the 
shaft of the engine. This wheel 
engages with the two larger spur 
wheels on the cam shafts, each 
having twice as many teeth as e 
and forming with it a two-to-one 
gear, so that c and d rotate once 
in every two revolutions of the 
crank shaft. The piston m is supplied with packing rings ; 
h is the connecting rod, k the crank shaft, and I the spark 
plug. 

The cycle is completed in four strokes, or two revolutions 
of the crank shaft. If the engine is running, the flywheel 
carries the piston down and draws in the charge through the 
open valve a, as represented in 1. In 2 both valves are 
closed and the piston compresses the explosive charge. 
Shortly after the piston reaches its highest point, the charge 
is ignited by a spark at the spark plug, and the working 
stroke then takes place, as in 3, both valves remaining closed. 
In 4 the exhaust valve b is opened by cam d, while a remains 
closed, and the products of combustion escape through the 
muffler, or directly into the open air. 

A single engine is commonly constructed with two, four, or 




Fig. 282 



424 HEAT 

six combined cylinders, giving one, two, or three impulses 
for every revolution of the shaft. The increase in the num- 
ber of cylinders contributes to the steady running of the 
engine. 

Problems 

1. Victoria Falls in South Africa are 340 feet high. How much is 
the water heated by the fall if no heat is lost by evaporation? 

2. A mass of 100 gm. moving with a velocity of 50 m. per second is 
suddenly stopped. If all its kinetic energy were converted into heat, how 
many calories would be generated? 

3. An iron bullet (specific heat, 0.112) weighing 50 gm. strikes a 
target with a velocity of 400 m. per second. Assuming that 20 per cent 
of the energy of the moving bullet remains in it as heat, how many 
degrees will its temperature be raised? 

4. How much heat is generated in stopping a train of 100,000 kgm. 
mass, running at 36 km. an hour? 

5. How much work would be done if all the heat of combustion 
(7850 cal./gm.) of 1 kgm. of anthracite coal could be converted into work? 
It would be equivalent to how many horse power hours? 

6. How much work is done against atmospheric pressure when 1 kgm. 
of water is converted into steam at 100? What would be the heat of 
vaporization of water if this energy were not included ? (At 100 the 
volume of the steam is 1582 times that of the water.) 

7. If the heat of combustion of pure anthracite is 7844 calories per 
gm., find the thermal value of 1 Ib. anthracite in B. T. U. Find also the 
maximum horse power hours obtainable from 1 Ib. of anthracite if the 
combined efficiency of the boiler and steam engine is 25 per cent. 

8. One calorie is the equivalent of 4.189 x 10 7 ergs. Find the num- 
ber of foot pounds of energy required to raise the temperature of 1 Ib. of 
water one degree Fahrenheit at New York, where g equals 980 cm. per 
second per second. 

9. If a perfect reversible engine takes steam from a boiler at 153, at 
what temperature must it exhaust into a condenser to have a theoretical 
efficiency of 40 per cent? 

10. From the following data compute the indicated horse power of 
a steam engine, the indicator diagrams being the same on both sides of 
the piston : mean effective pressure, 56 Ib. per square inch ; diameter 
of piston, 10 in. ; length of stroke, 12 in. ; revolutions per minute, 300. 



MAGNETISM AND ELECTRICITY 

CHAPTER XVI 

MAGNETS AND MAGNETIC FIELDS 
I. PROPERTIES OF MAGNETS 

452. Fimdamsntal Facts. Black oxide of iron, commonly 
called magnetite, is widely distributed and is sometimes 
found to possess the property of attracting iron. This 
property has been recognized from ancient times; it was 
exhibited in a marked degree by iron ores from Magnesia in 
Asia Minor, and they were therefore called magnetic stones 
and later magnets. They are now known as natural magnets, 
and the properties peculiar to them are called magnetic 
properties. 

If a piece of natural magnet be suspended by an un- 
twisted thread (Fig. 283), its longer dimension will point 
nearly in a north-and-south direction. 
This property of orientation, which led 
to the invention of the compass, has been 
known since the beginning of the thir- 
teenth century ; and because of this direc- 
tive property the magnet in early times 
acquired the name of lodestone, or leading 

Fig. 283 

stone. 

Little more was known of the fundamental properties of 
natural magnets until Gilbert published his book entitled 
De Magnete in 1600. He named the centers of attraction 
near the ends of a magnet the poles, and the straight line join- 
ing the poles he called the magnetic axis. 

425 




426 



MAGNETISM AND ELECTRICITY 




Fig. 284 




Fig. 285 



453. Artificial Magnets. If a slender piece of hardened 
steel be stroked lengthwise with a lodestone, it will acquire 

magnetic properties ; fine iron 
filings will cling to it in tufts 
near the ends (Fig. 284) ; and 
if it be suspended, or floated 
on a piece of cork in water (Fig. 285), it will come to rest 
in a north-and-south line with the same end always point- 
ing north. This end 
is called the north- 
seeking pole, and 
the other the south- 
seeking pole. No 
iron filings cling to 
the middle of the 
bar. This middle region is called the equator. 

A long thin rod, magnetized in the direction of its length, 
has its centers of attraction, or poles, very near the ends. 

The remainder of the magnet is 
apparently devoid of magnetic 
properties, but only apparently 
so. In short, thick magnets the 
poles are much less sharply 
defined. 

Magnetized bars of hardened 
steel are called artificial magnets 
to distinguish them from natural magnets or lodestones. 
The most common forms are the bar and the horseshoe (Fig. 
286), so called from their shape. 

454. Magnetic Substances A magnetic substance is one 

capable of being affected by a magnet. Faraday showed 
that most substances are influenced by magnetism, but only 
a few show magnetic properties in a marked degree. At- 
traction takes place between a piece of soft iron and either 
pole of a magnet, but the soft iron does not retain the prop- 




Fig. 286 



PROPERTIES OF MAGNETS 



427 



erty of attracting other pieces of iron, and it has no direc- 
tive force when freely suspended horizontally. Neither has 
it fixed poles and an equatorial region. 

Other substances attracted by a magnet are nickel, cobalt 
manganese, and chromium. Only nickel and cobalt show 
decided magnetic properties comparable with iron. Some 
gases are feebly magnetic ; liquid oxygen exhibits conspicu- 
ous magnetic properties. 

Another class of substances are apparently repelled by a 
magnet. These are called diamagnetic, to distinguish them 
from paramagnetic bodies like iron and nickel. Among them 
are bismuth, antimony, tin, copper, and some others still less 
strongly diamagnetic. 

455. First Law of Magnetic Force A thin pointed bar of 
magnetized steel, having at the middle a cap with an inset 
of agate, so that the bar 
may turn freely on a 
sharp steel point around 
a vertical axis, is called 
a magnetic needle (Fig. 
287). If the S-seeking 
pole of a bar magnet be 
presented to the N-seek- 
ing pole of a magnetic 
needle, they will mutu- 
ally attract each other; but if the N-seeking pole of the 
magnet be brought near the same pole of the needle, there 
will be repulsion. The law of attraction and repulsion is 
accordingly formulated as follows : 

Like magnetic poles repel and unlike magnetic poles at- 
tract each other. 

j 456. Magnetic Induction. When a magnet attracts a 

piece of soft iron, the iron first becomes a temporary magnet 

by induction. This piece may in turn act inductively on a, 




Fig. 287 




428 MAGNETISM AND ELECTRICITY 

second one, and so on in a series of temporary magnets of 
decreasing strength (Fig. 288). But if the first piece be 
detached from the magnet and be slowly 
withdrawn, all of them will fall apart, 
and they will not again attract one an- 
other until they are once more brought 
under the inductive influence of a per- 
manent magnet. A bar of iron near a 
magnet is attracted because it becomes 
a temporary magnet by induction, 'with 
Flg ' 28 the pole nearest to the pole of the in- 

ducing magnet of the opposite sign or name (Fig. 289). 
Induction thus precedes attraction. 

457. Permanent and Temporary Magnets. Permanent and 
temporary magnets differ only in the degree with which 
they retain their magnetism. The 

softest iron retains a small amount 
of magnetism after it has been 
brought under the influence of a 
magnetizing force, while hardened 
steel retains a large proportion of Fig 289 

it. The latter loses some of its 

magnetism when the magnetizing force is withdrawn, while 
the former loses nearly all of it. A much larger magnetizing 
force is required to magnetize hard steel than soft iron to the 
same magnetic strength. The ratio between the part lost 
and the part retained depends on the quality and hardness 
of the iron and on its after treatment. Cast iron retains an 
appreciable fraction of the magnetism induced in it, and this 
property is utilized in starting the excitation of dynamo ma- 
chines. The property of resisting magnetization or demag- 
netization is called retentivity. The retentivity of hardened 
steel is much greater than that of soft iron. 

458. Effect of Heat on Magnetism. If a permanent mag- 
net be heated to a bright red heat, all signs of magnetism 



PROPERTIES OF MAGNETS 



429 



disappear. Up to 680 iron shows but a slight change in its 
magnetic properties ; above this temperature a rapid decrease 
in magnetic susceptibility takes place, and at about 750 it 
ceases entirely to be magnetic and is quite indifferent toward 
a magnet. Nickel loses its magnetic properties at about 
350. Chromium ceases to be magnetic at about 500. Man- 
ganese is magnetic at temperatures near only. According 
to Dewar, when iron is chilled to 200 in liquid oxygen, 
its susceptibility is twice as great as at 0. 

The loss of magnetization by heat in the case of nickel is beautifully 
shown by the simple apparatus of Figure 290, designed by Bidwell. A thin 
tongue of nickel is soldered to a copper 
disk and the whole is blackened and sus- 
pended by silk threads. A permanent 
magnet is held in such a position that it 
retains the nickel tongue just over the 
flame of an alcohol lamp. When the nickel 
reaches the temperature of about 350, the 
magnet releases it, and the nickel-copper 
bob swings as a pendulum. It loses enough 
heat in one or two vibrations to recover its 
magnetic properties, and it is then again 
attracted and held by the magnet. The 
operation is repeated as soon as the nickel 
is again heated by the lamp. 

459. Unit Magnetic Pole. A unit 
magnetic pole, or a magnetic pole of 
unit strength, exerts on an equal pole 
at a distance of one centimeter in 
air a force of one dyne. A pole of 
strength m exerts a force of m 
dynes on a unit pole in air at a 
distance of one centimeter. If therefore m and m' are the 
pole strengths of two magnets, the mutual force between 
the two poles in air at a distance of one centimeter is mm- 
dynes. If the poles are of opposite sign, the product mm 1 is 
negative, or the negative sign means attraction. 




Fig. 290 



430 



MAGNETISM AND ELECTRICITY 



460. Second Law of Magnetic Force. Toward the end of 
the eighteenth century Coulomb investigated the law of at- 
traction and repulsion between magnet poles with the follow- 
ing quantitative result : 

The force between two magnetic poles in air is propor- 
tional to the product of their strengths and inversely pro- 
portional to the square of the distance between them. 

The magnets must be relatively long and the distance so 
great that the poles may be regarded as mere points. Com- 
bining this law with the definition of unit pole, we may write 



mm' 



(81) 



461. Magnetic Moment. The moment of a magnet is the 
product of the strength of its poles and the distance between 

them, or 

I 1 



IV 
\m 



m 



: ml. (82) 

Let the dotted lines of Figure 291 
be the direction of the magnetic force 
in a region where this force is uniform 
and of unit value, that is, where the 
force on a unit pole is one dyne ; and 
let ns be a magnet with pole strength 
m. Then the force on either pole is 
m and the two forces form a couple. 
The moment of this couple, when the 
magnetic axis of ns is perpendicular to the direction of the 
magnetic force acting on the magnet, is ml ; ml is the mag- 
netic moment of the magnet. 



Fig 291 



II. THE MAGNETIC FIELD 

462. Lines of Magnetic Force Since a magnet acts on a 
magnetic needle anywhere in its neighborhood, the space 
around the magnet is distinguished from other regions by its 



THE MAGNETIC FIELD 



431 



magnetic properties. This distinction is expressed by saying 
that a magnet produces around it a field of force called a 
magnetic field. 

If a sheet of cardboard or of thin glass be placed over a 
bar magnet, and if soft iron filings be evenly sifted over the 
upper surface, the filings will arrange themselves in curved 
lines when the paper or glass is gently tapped to set them 




Fig. 292 

free. They cling together in lines which diverge from one 
pole of the magnet and converge again toward the other. 
These lines are called lines of magnetic force, or of magnetic 
induction. The direction of a line of force at any point is 
the direction of the re- 
sultant magnetic force 
at the point. 

Figure 292 is reproduced 
from a photograph made by 
sifting iron filings on the 
sensitized side of a photo- 
graphic plate, with a piece 
of magnetized watch spring 
under it. 

Figure 293 shows the 
magnetic induction or field 
of force between the unlike poles of two similar magnets. The lines 
from the north pole of the one stretch across to the south pole of 
the other. Lines of magnetic force are under tension or tend to 
shorten. They act like stretched cords mutually repelling one another. 




Fig. 293 



432 



MAGNETISM AND ELECTRICITY 




Fig. 294 



The two poles of opposite sign are drawn together by the tension along 
these lines, or this figure is a picture of magnetic attraction. 

Figure 294 was made 
from two like poles. No 
lines extend across be- 
tween them. The resili- 
ency of these lines r.nder 
distortion is such as to 
force the magnets apart, in 
order to permit the lines to 
recover their normal dis- 
tribution about the poles. 
This figure is therefore a 
picture of magnetic re- 
pulsion. 

463. Direction of Lines of Force. The direction of a line 
of force at any point is that of a line drawn tangent to the 
curve at the point ; the direction along it is the same as that 
in which a north pole is urged. The north pole of a mag- 
netic needle is repelled from the north pole of a bar magnet. 
Hence, if an observer stands with his back to the north pole 
of a magnet, he is looking in the direction of .the lines of force 
coming from that pole. 

464. Intensity of a Magnetic Field. The intensity of a 
magnetic field at any point is the force exerted on unit pole 
placed at the point. Intensity of field is conventionally 
denoted by the number of lines of force passing through one 
square centimeter at right angles to the direction of the field. 
It is designated by the letter 8?8. The magnetic flux through 
any area s is equal to the product of this area and the strength 
of field, or s&6. 

465. Magnetic Flux from Unit Pole. Imagine a sphere 
of one centimeter radius described about a unit magnetic pole 
as a center. The intensity of the field at every point on the 
surface of this sphere is unity, or one line passes through 
every square centimeter. Then, since the surface of the 
sphere is 4-Tr square centimeters, the number of lines belong- 



THEORY OF MAGNETISM 



433 



Fig. 295 



ing to unit pole and passing through the surface of the sphere, 
that is, the total magnetic flux, is 4?r; and for a pole of 
strength ra, the flux is 4 irm lines. 

466. Consequent Poles. A bar of steel or other magnetic 
body may be magnetized in such a manner that it will have 
a succession of poles alternating in sign. Thus, in Figure 295 
there are north 

poles at N N, and ^N .N s\^ 

south poles at S S. 
The two poles not 
at the ends of the bar are consequent poles. A consequent 
pole belongs to two or more magnetic fields. The lines 
emerging from a consequent N pole go to two or more S poles. 

Consequent poles are commonly 
used in the field magnets of 
dynamo-electric machines. 

A ring may be magnetized so 
as to have consequent poles, or 
else in such a way that it will not 
exhibit external magnetic effects. 
In the latter mode (Fig. 296) there 
are no poles, that is, no points at 
which lines of force pass into the 
air. Such a ring constitutes a 
closed magnetic circuit, that is, one in which the magnetic 
flux is entirely in the iron. It has no external magnetic influ- 
ence, so long as there is no change in its magnetism, because 
there are no external lines of force. Closed magnetic circuits 
are more retentive of magnetism than open ones. 

III. THEORY OF MAGNETISM 

467. Magnetization and Mechanical Stress. Joule discov- 
ered that an iron rod increases in length when it is magnet- 
ized longitudinally. He concluded that if it were magnetized 
circularly, so that the lines of magnetization are circles 




Fig. 296 



434 MAGNETISM AND ELECTRICITY 

around the axis of the rod, it should shorten. This conclu- 
sion he verified by experiment. 

Bidwell extended Joule's discoveries by showing that at 
a certain degree of magnetization the elongation reaches a 
maximum, and that for magnetizing forces beyond that point 
the elongation is less and less, until finally the dimensions of 
the iron are unchanged ; any increase of the magnetizing 
force beyond this latter point causes the rod to shorten. 
Effects of the same kind occur in rings forming closed mag- 
netic circuits ; the diameter is increased by small magnetizing 
forces and is decreased with larger ones. 

A circularly magnetized iron wire, when twisted, becomes 
magnetized longitudinally ; and, conversely, torsion in weak 
fields diminishes longitudinal magnetization and produces 
circular magnetization. It is conversely probable that the 
superposition of circular and longitudinal magnetizations will 
result in torsional strain. Wiedemann demonstrated that 
this is true in the case of iron. With small magnetizing 
forces the twist is in one direction, but when the magnetizing 
forces are large there is a reversal in the direction of the 
twist. 

468. Magnetism Molecular. A great many facts point to 
the conclusion that magnetism is a property belonging to the 
smallest particles composing a body. If a piece of magnetized 
watch spring be broken in halves, each half will be a magnet 
with its poles pointing in the same direction as in the original 
magnet. Smaller subdivision of the spring simply increases 
the number of poles without destroying the magnetism. The 
inference is that the ultimate particles or molecules of iron 
and steel are magnets, and that they are naturally and per- 
manently such. 

If a glass tube be nearly filled with fine iron filings, it may be magnet- 
ized by stroking it from one end to the other with one pole of a strong 
m.agnet. If it then be shaken so as to rearrange the particles, all signs 
of magnetism disappear. The demagnetization of an iron bar by strong 
vibration is a similar phenomenon. 



TERRESTRIAL MAGNETISM 435 

Beetz deposited iron electrolytically in a thin line on silver parallel to 
the lines of induction in a strong magnetic field. The iron was found to 
be so highly magnetized that no more permanent magnetism could be 
induced in it. 

469. Ewing's Theory of Magnetism. According to Weber 
the molecules of iron and other paramagnetic bodies are 
natural magnets ; but in the unmagnetized state of the mass, 
their magnetic axes lie in all directions crisscross. Ewing 
has shown that the more probable arrangement of the particles 
is in closed magnetic circuits, or perhaps stable configurations, 
under the action of their mutual magnetic forces. A group 
of such molecules arrange themselves so as to satisfy their 
relative attractions and repulsions. To illustrate his theory 
Ewing constructed a model, consisting of lozenge-shaped 
magnets pivoted on points and arranged at equal distances in 
a horizontal plane. Any small number of these may group 
themselves into several stable configurations. After agitation 
they settle down into groups of equilibrium. With a small 
external magnetizing force these needles turn through a 
small angle only ; when the force reaches a larger value, some 
of the needles suddenly turn around and new groupings re- 
sult, with most of the needles pointing in the direction of the 
magnetizing force. Any further increase of the magnetizing 
force produces but little additional effect. These three 
stages correspond to three similar ones often observed in 
the magnetization of iron by electric currents ( 602). 

IV. TERRESTRIAL MAGNETISM 

470. The Earth a Magnet. The behavior of a magnetic 
needle in pointing generally not far from north suggested to 
Dr. William Gilbert, the leading English man of science in 
the reign of Queen Elizabeth, that the earth itself is a great 
magnet. Since then overwhelming evidence has accumulated 
to confirm his suggestion. 

Navigators have located the magnetic pole of the northern 



436 MAGNETISM AND ELECTRICITY 

hemisphere in extreme North America within the Arctic 
Circle. In 1831 Sir John Ross found it in latitude 70 N. 
and longitude 96 46' W. In 1907 Amundsen placed it in 
latitude 75 5' N. and longitude 96 47' W. It is not sta- 
tionary, but is subject to a slow cyclic movement. The mag- 
netic pole of the southern hemisphere has never been reached, 
but it is probably near latitude 73 S. and longitude 150 E. 
Since the N-seeking pole of a magnetic needle points 
toward the north, it is obvious that the lines of magnetic 
force about the earth run from the magnetic pole in the 
southern hemisphere toward the corresponding magnetic pole 
in the northern hemisphere; in other words, the northern 
hemisphere of the earth has the polarity of the S-seeking 
pole of a magnet. 

471. Terrestrial Magnetic Induction. Select a piece of gas 
pipe 3 or 4 cm. in diameter and about a meter long and care- 
fully free it from magnetism. If it is held horizontally east 
and west, either end of it will attract both the N-seeking and 
and the S-seeking poles of a magnetic needle. Gradually tilt 
it into a vertical position; its lower end will become an N 
pole and will repel the N pole of the needle. Reverse it and 
the lower end will again be an N pole and the upper end an S 
pole. Hold it vertically, or better in the meridian and in- 
clined about 75 below the horizontal toward the north, and 
strike it a sharp blow on the upper end with a hammer. It 
has now acquired permanent magnetism with the N pole at 
the lower end. By reversing it and striking it on the other 
end the polarity may be reversed, and by graduating the 
strength of the blow the pipe may be nearly or quite demag- 
netized. 

The earth as a magnet acts inductively on the pipe, as any 
other magnet does on a piece of iron, putting it under mag- 
netic stress. The vibration due to the blow gives a certain 
freedom of motion to the molecules, and they arrange them- 
selves to some slight extent under the influence of the earth's 



TERRESTRIAL MAGNETISM 437 

magnetic force. When the molecules are thus arranged, the 
pipe is a magnet. 

Bars of iron or steel in a vertical or in a horizontal north- 
and-south position acquire magnetism by induction from the 
earth. This is particularly true if they are subjected to fre- 
quent jarring or vibration. Drills, railway iron, beams, and 
posts are usually found to be magnetized. 

472. Magnetic Declination. The magnetic meridian is the 
vertical plane coinciding in direction with the earth's mag- 
netic field and containing, therefore, the axis of a suspended 
magnetic needle. This meridian does not in general coincide 
with the geographical meridian. The angle between the two 
is called the magnetic declination. The declination is east or 
west according as the N pole of the needle points to the east 
or to the west of the geographical meridian. The existence 
of magnetic declination was not known in Europe until the 
thirteenth century and was first distinctly shown on a map in 
1436. To Columbus belongs the undisputed discovery that 
the declination is different at different points on the earth's 
surface. In 1492 he discovered a place of no declination in 
the Atlantic Ocean north of the Azores. 

Lines connecting points of equal declination are called 
isogonic lines ; a line of no declination, separating regions in 
which the declination is westerly from those in which it 
is easterly, is called an agonic line. The magnetic needle 
everywhere on this line points due north. Such a line in 
1900 ran from the north magnetic pole across the eastern end 
of Lake Superior, through Michigan, Ohio, West Virginia, 
and South Carolina, and left the mainland near Charleston on 
its way to the magnetic pole in the southern hemisphere. 
The magnetic declination at Augusta, Maine, is about 16 Wo 
and at Tacoma, Washington, 23 E. 

473. Magnetic Inclination or Dip. If a magnetic needle 
be carefully balanced on a horizontal axis through its center 
of gravity before it is magnetized, its N-seeking pole after 



438 



MAGNETISM AND ELECTRICITY 





Fig. 297 



magnetization will incline below the horizontal in the north- 
ern hemisphere by an angle ranging from to 90. This 

angle is called the inclina- 
tion or dip. Norman, a 
London instrument maker 
who first measured the 
dip in 1576, made a dip- 
ping needle, which is a 
magnetic needle free to 
turn about a horizontal 
I axis in a vertical plane 

^ ^5^ > ^|d|BBr and provided with a grad- 
jJP uated vertical circle (Fig. 

297). The magnetic poles 
of the earth are points 
where the dip is 90; the 
dip at the magnetic equator is 0. The dip in London for 
1900 was 67 9' and in Washington 70 18'. It reached its 
maximum value in London in 1820 and has 
fcince been decreasing. Lines of equal in- 
clination on the earth's surface are called 
isoclinic lines. The magnetic equator is a line 
of no inclination in the vicinity of the geo- 
graphic equator. 

474. Magnetic Intensity. The intensity or 
strength of the earth's magnetic field may be 
measured in terms of the force in dynes act- 
ing on unit magnetic pole. The actual meas- 
urements made are those of the angle of dip and the hori- 
zontal component of the intensity. If & is the angle of dip 
(Fig. 298), then obviously the following relations hold : 




Fig. 298 



~ 

= cos 8, 



= sin S, and = tan 



The value of % at Washington is about 0.6 dyne, and that 
of $> is about 0.2 dyne per unit pole. 



TERRESTRIAL MAGNETISM 439 

475. Magnetic Variations. Several distinct changes ot 
variations in the magnetic elements of the earth are con- 
stantly taking place. Among them are the following : 

1. Diurnal Variation. The declination of the magnetic 
needle has a daily period. During the morning hours the 
swing of the N -seeking end is toward the east and it reaches 
its most easterly elongation in general before 9 A.M. It then 
pauses, turns westward, crosses its mean position, and by 
3 P.M. reaches its most westerly elongation. Some time be- 
fore midnight it again crosses its mean position on the east- 
ward swing, and then repeats the cycle of the day before. 
The amplitude of its oscillation is about 14' in the northern 
portion of the United States and about 4' in the most south- 
ern. These limits vary with the season, being greater in 
summer and less in winter. 

2. Secular Variation. In adding to the daily oscillations 
of the needle, the mean position about which it oscillates is 
subject to a slow change of long period. This secular varia- 
tion is different for different localities and for different 
periods of time. The change in the value of the declination 
has been recorded in London for more than 300 years. In 
1660 it was zero, and it attained its maximum westerly value 
of 24 in 1810. In 1900 it had decreased again to 15. If 
this secular change is periodic, it has a period of about 470 
years. The annual change on the Pacific coast is about 4', 
and in New England about 3'. 

3. Extraordinary Variations. In addition to the regular 
diurnal and slow secular variations, magnetic recording in- 
struments show sudden changes in the magnetic elements, 
due to some violent terrestrial or cosmic events. For ex- 
ample, such instruments usually furnish a record of earth- 
quake shocks corresponding very nearly in time with the 
record obtained from the seismograph. While the connection 
between earthquake shocks and magnetic disturbances has 
been clearly established, the reason for it has not yet been 
made out. 



440 MAGNETISM AND ELECTRICITY 

476. Terrestrial Magnetism and Sun Spots. The occurrence 
of large sun spots and solar outbursts has often been fol- 
lowed by marked magnetic disturbances, as shown by the 
automatic records of the magnetograph. Recently Professor 
Hale has obtained evidence at the Mount Wilson Solar Ob- 
servatory that a sun spot is a magnetic field, or has polarity. 
The evidence cannot be reviewed here, but it is conclusive 
in respect to polarity, sometimes N-seeking and sometimes 
S-seeking; for the effect on spectroscopic lines is qualita- 
tively the same as one obtains in the laboratory with the 
source of light in a magnetic field ( 661). The sudden pro- 
duction of these solar magnetic poles during the initial stages 
of a solar vortex or whirlpool does not appear to affect the 
earth's magnetism by the reaction between the solar and 
terrestrial magnetic fields. 



CHAPTER XVII 

ELECTROSTATICS 
I. ELECTRIC CHARGES 

477. Electricity and Electrification. The simple fact that a 
piece of amber (a fossil gum), rubbed with a flannel cloth, 
acquires the property of attracting bits of paper, pith, or 
other light bodies, has been known since about 600 B.C. 
But it appears not to have been known for the following 
2200 years that any bodies except amber and jet were capable 
of this kind of excitation. About the year 1600 Dr. Gilbert 
discovered that a large number of substances possess the 
same property. These lie styled electrics (from the Greek 
word for amber, electron), but the word electricity to desig- 
nate the invisible agent to which the phenomenon should be 
referred appears to have been introduced by Boyle in 1675. 

A body excited in this manner is said to be electrified; 
electrification is the result of the work done in electrifying, 
or charging with electricity. Electrification, or electricity 
under pressure, is a form of potential energy, just as air 
under pressure or water at an elevation above the earth 
represents potential energy. Electricity, air, and water are 
not themselves energy, but only the agents which store or 
carry energy. 

Electricity appears to be as indestructible as matter and 
energy. Its distribution is subject to control. It may rep- 
resent energy of stress or energy of motion ; but when its 
energy has been spent in producing physical effects, its 
quantity remains unchanged. None can be created or gen- 
erated and none destroyed. 

441 



442 



MAGNETISM AND ELECTRICITY 



478. Electrical Repulsion. If several pith balls are sus- 
pended by fine linen threads from a glass rod, and an electri- 
fied glass tube is brought near 
them, they are first attracted to 
the tube but soon fly away from 
it and from one another (Fig. 
299). Their mutual repulsion 
continues after the tube is with- 
drawn ; and, if the hand is brought 
near them, they move toward it as 
if attracted, showing that the balls 
are electrified. 

It thus appears that bodies are 
electrified by coming in contact 
with other electrified bodies, and that electrification is shown 
by repulsion as well as by attraction. 

479. Attraction Mutual. Boyle discovered that the at- 
traction between an electrified and an unelectrified body is 
mutual. Excite a glass tube by rubbing it with silk, and 
lay it in a light stirrup suspended by a silk thread (Fig. 300). 
If the hand is presented to it, it will 

swing around by the attraction. 

Force, whatever its origin, is of the 

nature of a stress, and action and 
reaction are equal. 




Fig. 299 




480. Two Kinds of Electrification. 

Not all electrified bodies repel one 

another. If a second excited glass 

tube be presented to the one hung 

in the stirrup (Fig. 300), there 

will be mutual repulsion between 

them. On the other hand, a rod of gutta-percha or a stick 

of sealing wax excited by rubbing with flannel will attract 

the electrified glass tube ; also, if a pith ball be charged by 

contact with the rubbed sealing wax, it will be repelled 



Fig. 300 



ELECTRIC CHARGES 443 

by the sealing wax, but attracted by the glass tube rubbed 
with silk. 

From such facts as these, Du Fay drew the inference that 
there are two kinds of electrification. Electricity manifests 
itself under two aspects analogous to the two poles of a mag- 
net. Du Fay called the two kinds vitreous and resinous 
respectively ; but since the result of the friction depends on 
the rubber as well as on the material rubbed, the terms posi- 
tive and negative introduced by Franklin have been adopted 
everywhere as preferable. The kind of electrification which 
makes its appearance on glass when rubbed with silk is 
called positive, while the kind excited by rubbing sealing 
wax with flannel is called negative. 

These terms are purely arbitrary, and if their choice had 
been postponed until the present, it is highly probable that 
many phenomena of recent discovery would have dictated 
the exchange of the words positive and negative as applied 
to electrification. 

481. First Law of Electrostatics. All the phenomena simi- 
lar to those described in the last article are included under 
the first law of electrostatics, namely : 

Like kinds of electrification produce mutual repulsion 
between the bodies charged; unlike electrifications produce 
mutual attraction. 

482. Conductors and Nonconductors. To all substances 
which do not show electrification by friction Gilbert gave 
the name " nonelectrics " ; but in 1729 Stephen Gray dis- 
covered that these substances convey away the "electric 
virtue " as fast as it is excited. If a metal rod be held by 
a dry glass handle, it can be electrified by rubbing with silk. 
Gray conveyed electric charges to a distance of 700 feet by 
means of a hempen thread supported by silken loops. 

Ever since Gray's discovery, substances have been classi- 
fied as conductors and nonconductors or insulators. These 



444 MAGNETISM AND ELECTRICITY 

are only relative terms, for all substances may be arranged 
in a graded series with the best conductors at one end and 
the poorest at the other. None conduct perfectly and none 
insulate perfectly. The difference in relative conductivity, 
however, is enormous. Thus, silver conducts more than 60 
times as well as mercury, and 2500 times as well as gas car- 
bon, while the conductivity of mercury is 100,000 times 
greater than that of dilute sulphuric acid. 

% 

483. The Electroscope and the Electrometer. The electro- 
scope is an instrument for detecting electric charges. The 
most common form is the gold leaf electroscope. Through 
the top of a glass jar passes a brass rod, terminating in a ball, 
or horizontal plate above, and in two strips of gold leaf on 
the inside hanging parallel and close together. The metal 
rod must be insulated from the glass either by a heavy coat 
of melted shellac or by a stopper of sulphur. 

If the knob of an electroscope be touched with an excited 
glass tube, the gold leaves will repel each other with positive 
charges. The approach of any other charged body will cause 
them to diverge more widely if the body is charged posi- 
tively, and to approach each other if it is charged negatively. 

The study of ionization and of radioactivity in recent years has led 
to a modification of the electroscope with the object of making it a 
measuring instrument. Of the many forms proposed, 
that of Wilson is typical. The indicating system con- 
sists of a rigid piece of flat brass B (Fig. 301), to which 
is attached a narrow strip of gold leaf G. This system 
is supported by a block of sulphur /, which in turn is 
suspended by a rod fitting tightly in a block of ebonite 
E. A charging wire W passes through the ebonite and 
is bent at right angles at the bottom. By rotating the 
upper bent end of W, the arm at the bottom may be 
brought in contact with the brass strip ; it may be dis- 
connected as soon as the system is charged. In some 
instruments the gold leaf is viewed with a microscope of low magni- 
fying power. 

For measuring small electrostatic potential differences ( 500) Lord 




ELECTRIC CHARGES 



445 




Kelvin devised the quadrant electrometer. The essential parts are the 
cage, or quadrants, and the needle. The four insulated quadrants 
together form a short hollow cylinder 
with parallel ends (Fig. 302). The 
needle, a long, thin piece of aluminum 
with broad rounded ends, is suspended 
by a fine wire or fiber so as to turn in a 
horizontal plane inside the four quad- 
rants. It is connected at the bottom 
with the jar B by a fine platinum wire 
dipping into sulphuric acid. Opposite 
quadrants are connected electrically. 

If all the quadrants are equally 
charged, the needle will take a position 
depending only on the torsion of the 
suspending fiber ; but if two of the diag- 
onally opposite quadrants are charged 
positively and the other two negatively, 
the needle being positive, the latter will 
rotate in one direction or the other, 
according to the connections, until the 
couple due to the electrostatic attrac- 
tions and repulsions is just balanced by the torsion of the suspension. 

484. Positive and Negative Electrifications Equal. When 
a body is electrified by friction, the body rubbed and 
the rubber are equally electrified, but with charges of oppo- 
site sign. The equality consists in the ability of the one 
charge to exactly neutralize the other. 

If a stick of sealing wax, provided with 
a flannel cap with a silk cord attached 
(Fig. 303), be excited by turning it 
around a few times inside the cap, it will 
show no signs of electrification if pre- 
sented to the knob of an electroscope 
without removing the cap ; but if the 
cap be withdrawn by the cord, the sealing wax will cause 
the gold leaves to diverge less and the flannel cap will cause 
them to diverge more, provided the electroscope is charged 
positively. 




Fig. 303 



446 MAGNETISM AND ELECTRICITY 

The inferences are : first, that one kind of electrification is 
not produced without the other ; second, that the two kinds are 
produced in equal quantities. The electrification of a body 
consists in the separation of two equal charges of opposite sign 
against their mutual attraction. It follows that the medium 
between them is strained by the operation, and work is done. 
The slightest positive charge at one point always means an 
equal charge of the opposite sign as near it as the conduct- 
ance of the insulators separating the two charges permits. 

485. Charge External. When a conductor is electrified 
by friction or by a charge conveyed to it from some external 

source, the charge always resides 
on the outside. A simple demon- 
stration of this principle may be 
made by means of a hollow metal 
sphere with a hole at the top and 
insulated on a glass stem as a 
support (Fig. 304). It may be 
tested by means of a proof plane 
(Fig. 305), which consists of a 
small metal disk cemented to one 
end of an ebonite or shellac handle. 
If the proof plane be applied to 
the outside of the charged sphere, 
a small charge may be removed 
and tested by an electroscope. If 
the proof plane be passed through 
the hole in the sphere and touched 
to the inner surface, it will not 
show any trace of electrifica- 
tion. If in fact it be charged 
from the outside of the sphere, 
and then be made* to touch the 
interior, it will lose all its charge and will show none on 
withdrawal. 





i 



ELECTRIC CHARGES 447 

486. Distribution of the Charge. The quantity of electric- 
ity on a square centimeter of the surface of a conductor, 
or the ratio of the quantity on any small area to the area 
itself, is called the surface density. The distribution of an 
electric charge on an insulated conductor is not such as to 
give uniform surface density over it, except in the case of 
a sphere remote from other conductors and electrified bodies. 
The surface density on different parts of a conductor may 
be examined by means of a proof plane and an electroscope. 
After the proof plane has been brought in contact with 
the rounded end of a charged cylindrical conductor, it will 
produce a much greater divergence of the leaves of an elec- 
troscope than after it has touched the side of the cylinder. 

On a cylinder with rounded ends, the surface density is 
greatest at the ends. 

On a flat disk the density is much greater at the edges 
than on the flat surface ; the distribution over the latter is 
fairly uniform except near the edges. 

The surface density is greatest on those parts of a con- 
ductor which project most and have the greatest curvature 
outwards. On sharp points, such as that of a needle, the 
density is very great, and the charge escapes rapidly from 
them into the air. It is for this reason that the edges of 
conductors are rounded and made smooth. 

487. Second Law of Electrostatics. The law of the force 
between two electric charges was first investigated by 
Coulomb. He demonstrated that the stress between two 
electric charges is directly proportional to the product of the 
two quantities, and inversely proportional to the square of 
the distance between them. 

The law of distance does not hold unless the charged con- 
ductors are very small in comparison with the distance be- 
tween them. The distribution of the charges on the two 
bodies is then not appreciably affected by their mutual 
action. 



448 MAGNETISM AND ELECTRICITY 

The second law of electrostatics may be succinctly ex- 
pressed in terms of algebraic symbols as follows : Let q and 
q f denote the electric charges of the two small bodies, d the 
distance between them in air, and O a proportionality factor. 
Then Coulomb's results are expressed by the equation 



. (83) 

488. Unit Charge. The definition of unit charge or unit 
quantity is so chosen that the constant O in the foregoing 
equation becomes unity. The electrostatic unit charge is that 
quantity which exerts on an equal quantity one centimeter dis- 
tant in air a force qf one dyne. 

It is necessary to say " in air " because, as will be seen 
later, the force between two charged bodies depends on the 
nature of the medium between them. 

489. Nature of Electrification. It was suggested by Fara- 
day, and a great many facts tend to confirm his view, that 
the electrification of a body produces a strained condition 
of the ether around it. Conductors differ from insula- 
tors in this respect : in the former the molecular mobility 
is such that the state of strain is continually giving way ; 
while in the latter considerable distortion is possible before 
the molecular structure yields to the stress. The phenomena 
of attraction and repulsion exhibited by electrified bodies are 
due to the effort of the strained ether in and about the 
bodies to return to its normal condition. In producing 
electrification, work is done in distorting the medium ; hence 
electrification is a form of potential energy. 

490. Electric Field. Any region within which the medium 
is under stress is a field of force; and an electric field is one 
in which the stress is due to an electric charge. The inten- 
sity of an electric field at any point is defined as the force 
sustained by a unit charge placed at the point. A unit elec- 
tric field is one in which unit charge is acted on with a force 



ELECTRIC CHARGES 



449 



of one dyne. Electric intensity is measured in the same 
fundamental units as mechanical forces. 

491. Lines of Electric Force. An electric field, like a 
magnetic field, is conveniently represented by lines of force. 
At any point in the field a line of force takes the direction 
of the electric force at that point. It is conceived to start 





Fig. 306 



Fig. 307 



on a positive charge and to end on a negative one. The 
positive direction is therefore the direction in which a small 
positively charged body tends to move along the line of 
force. 

For a positive charge on a very small body, the lines of 
force are radial lines emanating from the charge. For two 
equal charges of opposite sign on very small 
bodies, the lines of force run as in Figure 306. 
For like charges, they take the form shown in 
Figure 307. No two lines cross, for if they did, a 
charge placed at the intersecting point would tend 
to move in two directions at the same time. 

The lines of force in an electric field tend to 
shorten like stretched cords. When one electrified 
body attracts another, they are drawn together by 
these taut lines extending from one to the other. 
If two parallel plates face each other and are 
charged oppositely (Fig. 308), the lines of force 
stretch across from the positive to the negative, and the 
tension in the medium pulls the plates together. When the 



Fig. 308 



450 MAGNETISM AND ELECTRICITY 

plates are close together, the field is nearly uniform and 
the lines of force are parallel except near the edges. 

II. ELECTROSTATIC INDUCTION 

492. Induction Phenomena. A charged conductor exerts 
'influence, or acts inductively, on neighboring bodies. If it 
be charged positively, lines of electric force spring from it 
and proceed to an equal negative quantity on adjacent 
bodies; the influence or induction is exerted along these 
force lines. 

Let an insulated sphere A (Fig. 309), charged positively, 
be placed near an insulated cylindrical conductor B. HA 

and both ends of B 
be now examined by 
means of a proof 
planQ and an elec- 
troscope, it will be 
found that the 

charge on A has been redistributed, so that the surface 
density on the side toward B is greater than on the remote 
side ; also, that the end a of the cylinder has negative elec- 
trification, the central portion is neutral, and the end b has 
positive electrification. The surface density at b is less than 
at #, and the neutral line is somewhat nearer a than b. 

As soon as A is removed or is discharged by connecting 
with the earth, all signs of electrification on B disappear. 
The electrification of B is due to the charge on A, but is not 
at the expense of that charge. The separation of the posi- 
tive and negative charges on B through the influence of the 
charge on A. is called electrostatic induction, or electrification 
by influence. 

493. Charging by Induction. The next step is to give to 
B a charge of one sign by induction. While it is still under 
the influence of the charge on A, let it be connected for an 
instant with the earth. The effect is to enlarge indefinitely 




ELECTROSTATIC INDUCTION 451 

the dimensions of B, and the positive electrification goes to 
the remote part of this enlarged conductor, that is, to the 
earth. If now A be removed, while B remains insulated, 
the negative charge on the latter will distribute itself over 
the entire conductor, and B is said to have been charged by 
induction. This induced charge is of the opposite sign to 
the inducing charge on A. 

Charging by induction may also be referred to lines of 
force in the enveloping medium. When B (Fig. 309) is 
brought into the field of A, some of the lines of force spring- 
ing from A terminate on the nearer end of B\ an equal 
number leave B at the distant end; that is, the end a is 
charged negatively and b positively. When B is connected 
to the earth by a conductor, the stress in the medium, repre- 
sented by the lines running from B, is removed, and only 
that between A and B remains. Hence, B has then a 
negative charge only. 

494. ElectrificatioV with Like Charges by Induction. It is 

quite possible to charge by induction so that the induced 
charge shall be of the same 
sign as the inducing charge. 
Imagine the conductor B pro- 
vided with sharp points at 
the end a (Fig. 310), and a 
circular glass plate revolving 
with its edge between A and 
B. The negative charge on /^ ^X 
a will then acquire so great f -j- A J 
a density on the points that X^^^^/ 
they will discharge it on the p . 3|Q 

revolving plate. If another 

row of points <?, connected with the earth, be placed opposite 
the same side of the glass plate, but away from the inductive 
action of A, the revolving plate will give up to c the negative 
charge acquired at a, and c will convey it to the earth. In 



a B 6+ 



452 MAGNETISM AND ELECTRICITY 

this way B is left with a positive charge. The electrification 
still represents energy, for work is done in turning the glass 
plate against the attraction of the unlike charges on it and A. 

495. Attraction due to Induction. The simple facts of 
electrostatic induction furnish an explanation of the attrac- 
tion between electrified and unelectrified bodies. The in- 
duced charge of opposite sign always accumulates on the 
part of the conductor nearest the inducing charge, while the 
repelled charge goes to the most distant parts of the con- 
ductor, or to the earth if a conducting path is furnished. 

If an excited glass rod (Fig. 311) be presented to an 
uncharged pith ball suspended by a silk thread, a negative 
charge will be induced on the ball at a and positive at b. 
\ Since the former is nearer O than the latter, the attrac- 
\ tion will be greater than the repulsion, and 

the pith ball on the whole will be attracted. 
If it be touched while under induction, the 
repelled positive charge will disappear and 
the attraction will be increased. 

496. The Inducing and Induced Charges Equal. The charge 
induced on a conductor can never exceed the inducing 
charge. If all the force lines from the inducing charge 
go to the induced charge on the conductor, 

the two charges are equal. If, for example, a 
charged ball be nearly surrounded by a hollow 
conductor (Fig. 312), all the lines of force 
from the ball A end in the induced charge on 
the inclosing conductor. No sensible number 
escape through the small opening. A nega- 
tive charge then spreads over the interior of B Fjg 3I2 
equal in quantity to the positive charge on A. 

In this case the charge on B is on the inside instead of the 
outside. If B is insulated while under the inductive influence 
of A, and A is then removed without making contact with 
-B, the negative charge on B will all go to the outside. 




ELECTROSTATIC INDUCTION 



453 




Fig. 313 



497. Faraday's Ice Pail Experiment. Faraday employed 
a pewter ice pail as a convenient hollow conductor to test 
the question of equality be- 
tween the induced and the 

inducing charges. A is a 
section of a well-insulated pail 
(Fig. 313). The outside is 
connected to a gold leaf 
electroscope E. A positively 
charged ball O is let down 
into the pail by means of a 
silk thread. As soon as it 
enters the pail the gold leaves 
begin to diverge, and the 
divergence increases until the 
ball reaches a certain depth, depending on the relative 
dimensions of the pail. Beyond this depth the divergence 
remains constant. The divergence increases up to the point 
where all the lines of force from O terminate in the nega- 
tive charge on the inside of the pail. 

If now the ball be allowed to touch the pail, not the slightest 
change in the divergence of the gold leaves can be detected. The 
inference is that the positive charge on the outside of the pail, 
when the ball is acting inductively on it, is exactly equal to the 
charge communicated to the pail when the ball makes contact. 

The experiment was varied by touching the pail while 
under the influence of the charge on the ball. The gold 
leaves collapsed. But when the ball was withdrawn, they 
again diverged to the same extent as before, but with a nega- 
tive charge. If next the charged ball was replaced and 
made to touch the pail, all signs of electrification disap- 
peared, indicating again an equality between the induced 
charge and the positive conveyed by the ball. 

498. The Electrophoms. The simplest induction machine 
is the electrophorus invented by Volta. By means of it an 



454 



MAGNETISM AND ELECTRICITY 



indefinite number of small charges may be obtained by 
induction from a single charge due to friction. It con- 
sists of a plate of hard rubber or resin 
and a metal cover provided with an in- 
sulating handle (Fig. 314). The hard 
rubber plate usually rests on a metal 
plate or sole. 

To use the electrophorus, the hard 
rubber or resin is electrified by beating 
it with a catskin, and the metal cover is 
placed on it. Since the cover touches 
the nonconducting disk in a few points 
only, it does not remove the negative 
charge due to the friction of the catskin, 
but it is acted on inductively by the 
negative charge on the disk. The cover 
is then touched for an instant with the 
finger; the repelled negative charge passes off, leaving the 
cover with an induced positive charge on its lower surface. 
The cover may then be lifted by the insulating handle ; the 
bound charge becomes free, and is available for charging other 
bodies. 

Evidently this process may be repeated an indefinite num- 
ber of times without removing any appreciable part of the 
charge from the hard rubber plate. It is, however, slowly 
dissipated in damp air or if the hard rubber is not dry. 




Fig. 314 



III. ELECTRIC POTENTIAL 

499. Definition of Electric Potential. The term potential^ 
introduced by Green in England in 1828, was originally a 
mathematical function, but it has now the greatest practical 
significance in the science of electricity. 

First. Mutual potential energy. Consider two unlike 
electric charges ; the mutual potential energy of such a system 
in any given position is the work which must be done 



ELECTRIC POTENTIAL 455 

against their mutual attraction in separating them to an 
infinite distance, or in conveying one to the boundary of 
the field produced by the other. It should be carefully 
noted that the force worked against is purely electrical ; 
all others are excluded. 

Second. Potential. The potential at any point, due to a 
given charge, is the mutual potential energy of this charge 
and unit quantity of electricity placed at the point. In other 
words, it is the work which must be done on a unit charge 
in carrying it from the point to an infinite distance, or to the 
boundary line of the field of force due to the given charge. 

For convenience the potential of the earth is usually taken 
to be the arbitrary zero of potential, just as the sea level is 
an arbitrary level from which altitudes are measured. 

500. Difference of Potential. Consider two points A and 
B, and let the potentials at these points be denoted by V 1 
and V 2 respectively. Since work equal to F" x ergs is required 
to convey unit charge from A to infinite distance, and work 
F 2 ergs from B to an infinite distance, it is obvious that the 
work done against the electric forces in displacing a positive, 
unit charge from one point to the other is Fj V%. This 
work is independent of the path followed in going from A 
to B ; otherwise it would be possible, by making a quantity 
of electricity circulate between A and B by suitable paths, 
to gain energy without a corresponding expenditure of 
work. If the work expended in conveying the charge from 
A to B by one path should be less than the energy recovered 
by allowing it to return by electric forces from B to A, 
energy would be accumulated in every cycle, and this is 
contrary to the law of conservation of energy. The same 
principle applies to work done against gravity between two 
points at different levels. The work done is independent of 
the path traversed between the two points ; otherwise, " per- 
petual motion " and continuous work without a corresponding 
expenditure of energy would be possible. 



456 MAGNETISM AND ELECTRICITY 

501. Equipotential Surfaces. An equipotential surface is 
the analogue of a level surface. The potential at all points 
of an equipotential surface is the same. There is then no 
difference of potential between points on an equipotential 
surface, and no work is done in carrying a charge of elec- 
tricity between any two points on the same equipotential 
surface. 

The surface of a charged conductor is an equipotential 
surface, for its charge is in equilibrium and there is no force 
along the surface of the conductor and therefore no potential 
difference. But a charged conductor produces a field of 
force, and the only direction in which a charge can be 
moved without work is at right angles to the lines of force. 
It follows that the lines of force are everywhere normal to 
the equipotential surface. They terminate at the charge on 
the exterior, for there is no force on the inside of a charged 
conductor. 

502. Work between Equipotential Surfaces. Consider two 
equipotential surfaces, the potentials of which are V l and F^. 
The work done in displacing unit charge from the one 

* surface to the other is the difference of potentials, V\ V 2 . 
It is independent of the path traversed and of the position 
of the point of departure and the point of arrival on the 
two surfaces. If a quantity q units is conveyed from one 
surface to the other, the work done is q times as much as for 
one unit, or q (Fi F^). The numerical 
measure of the electrical work is thus a 
product of two factors, one of them a 
potential difference and the other a quan- 
tity of electricity. If the potentials of the 
two surfaces differ by unity, then one erg 
of work must be done to convey unit 
Fig. sis charge from the one surface to the other. 

503. Force in Terms of Potential. Let S and S' be two 
equipotential surfaces very near together (Fig. 315), and 




ELECTRIC POTENTIAL 457 

let J^and V 1 be their potentials. Let F be the electric 
force along a normal between P and P r . If n is the 
distance PP 1 ', the work done by the force F in conveying 
a unit quantity from one of the surfaces to the other is 
F x n. Then 

Fn = V- V, and .F= V ~ V ' . (84) 

^ 

The electric intensity along a line of force is therefore equal 
to the decrease of the potential per unit length along the 
line. In general the intensity in any direction equals the 
rate of the diminution of potential in that direction. It 
follows that the intensity between two equipotential surfaces 
is greatest in the direction in which the distance n is least. 

504. Equilibrium of a Conductor. When a charge of 
electricity is given to a conductor, it at once distributes 
itself over the surface and comes to equilibrium. Moreover, 
since there is no force inside a charged conductor, there is 
no difference of potential throughout its entire volume, 
since force equals the rate of variation of potential. All parts 
of a charged conductor have therefore the same potential. 

The surface of an insulated conductor under the influence 
of a charged one is also an equipotential surface, because the 
charges on it are in equilibrium and there is no electric flow 
along it. For example, the potential at a (Fig. 316), due to 
the positive charge + 

ative charge near a 
lowers the potential 

of the nearer half of the cylinder, and the positive near b 
raises the more distant half to the same potential level as at a. 
If now the cylinder be connected to earth, its potential will 
be reduced to that of the earth, that is, to zero. The cylin- 
der will remain charged negatively, but its potential will be 



458 MAGNETISM AND ELECTRICITY 

zero. The positive potential due to the charge on A and 
the negative due to its own charge everywhere equal each 
other, and the resultant is zero. It is obvious then that 
surface density and potential are in no sense the equivalents 
of each other. 



505. Potential equals r' ^ ne P ro ^ em i g to find the 
expression for the potential at A, at a distance r from an ele- 
ment q of the charge on the conductor (Fig. 317). Let B 



o (q 



r' 

Fig. 317 

in the line OA be at a distance r' from 0. Let the distance 
AB be supposed divided into n very small elements, so that 
the points of division are distant r v r 2 , r 3 , etc., from 0. 

Then, since the electric intensity is the force on unit 
charge, the intensity at the distance / is q/r\ at the distance 
rj it is q/r-f, etc. Since r and r^ are very nearly equal, we may 
put the equivalent intensity between the two adjacent points r 
and r^ equal to q/rr r Similar expressions obtain for the 
other elements of the distance AB. 

Hence the work in carrying unit charge 

from r to r, =-^-(r^ j*) = gl ]; 

rr l Vr rj 

from /*! to r 2 = 



from r^ to r' = _L(r' - r n _J= q(- - I\ 

r n l r V-l r / 

The whole work done in carrying the unit charge from A 
to .B is the sum of all these small quantities ; but in sum- 



ELECTRIC POTENTIAL 459 

ming up it will be observed that all the intermediate terms 
cancel out ; hence the work between A and B is 



This expression is the difference of potential between A and 
B due to the element of charge q at 0. 

If we suppose the point B removed to an infinite distance, 
1/r' becomes zero, and 

q 
work from A to an infinite distance is -. 

But by definition this is the potential at A due to the charge 
q. The expressions for other elements of the charge are simi- 
lar ; and since potential is a scalar quantity, the resulting 
potential at A is the algebraic sum of the potentials due to 
the several elements of the charge, or 



506. Potential of a Sphere. If the sphere has a charge 
Q, every element q of this charge is at a distance r from the 
center of the sphere, and the potential at the center due to an 
element q is q/r, r being the radius of the sphere. The poten- 
tial at the center due to the entire charge is then 



But as all points of a conductor in equilibrium have the 
same potential, the potential of every point of the sphere 
due to a charge Q is Q/r. 

A charge uniformly distributed over a sphere acts at exter- 
nal points as if it were collected at its center ; hence the po- 
tential at any point external to the sphere and distant d u^its 
from its center is Q/d. 



460 MAGNETISM AND ELECTRICITY 

Problems 

1. What would be the potential difference between A and B (Fig. 317) 
if were charged with 100 units of electricity, the distance r being 10 cm. 
and r 1 20 cm. ? 

2. Positive charges, 150, 424, and 300 units, are placed at the three 
corners A, B, C, of a square 40 cm. on a side. Calculate the potential at 
the fourth corner D. 

3. Positive charges of 50 units each are placed at the three corners of 
an equilateral triangle whose sides are 100 cm. Find the potential at 
the center of the circumscribing circle. 

4. Find the potential at the center of the square in problem 2, and 
the work required to carry a unit charge from D to the center. 

5. Two charges, 100 positive and 70 negative, are placed 30 cm. apart. 
Find the potential at a point on the line joining them 70 cm. from the 
negative charge and 100 cm. from the positive ; also, calculate the force 
on a charge of 40 positive units at the same point. 

IV. ELECTRIC CAPACITY AND CONDENSERS 

507. Definition of Capacity. The electric capacity of a 
conductor is defined as the ratio of its charge to its potential 
when all other conductors within its field are at zero potential. 
This definition is equivalent to saying that the numerical 
value of its capacity is the number of units of electricity that 
will raise its potential from zero to unity. In symbols, if 
denotes capacity, n 

0= & (88) 

Also <?=<7Fand F=^. 

C 

508. Capacity of an Insulated Sphere. The capacity of a 
sphere in air remote from other conductors is numerically 
equal to its radius in centimeters. For the potential of such 
a sphere is Q/r. Hence 

C==Q+9 =r . (89) 

The radius is in centimeters because the centimeter is the 
unit of length employed in defining the unit of charge. 



ELECTRIC CAPACITY AND CONDENSERS 461 

509. Condensers. Two conductors placed near each other 
with an insulator, called the dielectric, between them form 
with the dielectric a condenser. The effect of the additional 
conductor and the dielectric is to increase the charge without 
any increase of potential. In other words, the capacity of the 
one conductor is greatly increased by the presence of the 
other. The ratio of the charge on either conductor to 
the potential difference between two is the capacity of the 
condenser. 

510. Capacity of Two Concentric Spheres. It is not difficult 
to calculate the capacity of a condenser when the plates have 
certain simple geometric forms. Let r be 

the radius of the inner sphere and r' that of 
the outer one (Fig. 318), and assume that 
the outer surface is connected to the earth, 
so that its potential is zero. Then all the 
lines of force from the insulated charged 
sphere A run to the outer sphere j5, and their 
charges are equal and of opposite sign, + Q 

i /-i Fig- 318 

and Q. 

The potential at 0, the common center of the two spheres, 




This is the potential of the entire inner sphere because the 
potential inside a charged conductor is the same as at any 
point on its surface. From the last equation 



When r 1 r = t is very small, that is, when the two spheri- 
cal surfaces are very near each other, the capacity becomes 
very large. The expression for the capacity is then 



rr r _r(rt) 

~-~~~ 



462 



MAGNETISM AND ELECTRICITY 



where t is the thickness of the dielectric, which is here assumed 
to be air. When t is very small compared with r, the expres- 
sion for the capacity becomes 



(90) 



where A is the surface area of the inner sphere. 




511. The Leyden Jar. The Leyden jar was the earliest 
known form of electric condenser. It derives its name from 
the fact of its accidental discovery in the 
city of Leyden. As now made, it consists 
of a wide-mouthed jar of thin flint glass, 
coated inside and out with tin foil to about 
three fourths its height (Fig. 319). The 
metal knob is connected to the inner coat- 
ing by a rod terminating in a piece of 
chain. 

The jar may be charged by holding it in the 

hand, touching the knob to one electrode of an 

influence machine, and 

bringing the outer coating 

so near the other electrode 

that a series of sparks pass across. If charged 
to too high a potential, it will discharge along 
the glass over the top. It may be safely dis- 
charged by a discharger (Fig. 320) held by 
the glass handles, one ball being brought into 
contact with the outer coating and the other 
with the knob. 

Fig. 320 

If A is the area of the tin foil and t 

the thickness of the glass, with air as the dielectric the 
capacity is approximately the same as with concentric spheres 
of the same conducting area A, that is, 

c= A 

4-7rt" 



Fig. 319 




ELECTRIC CAPACITY AND CONDENSERS 



463 



The effect of interposing glass instead of air between the 
two coatings is to increase the capacity by a factor K, so that 



a constant of the dielectric depending on the kind of 
glass, and varying in value from about 3 to 7 for different 
specimens. 

512. Residual Charge. If a Leyden jar be left standing 
for a few minutes after it has been discharged, the two coat- 
ings gradually acquire a small potential difference and a 
second small discharge may be obtained from it, due to the 
residual charge. Several of them, of decreasing intensities, 
may sometimes be observed. The potential of the residual 
charge depends on the potential difference to which the jar 
has been charged, the length of time it is left charged, and 
the kind of glass forming the dielectric. 

513. Seat of the Charge. The Leyden jar with remova- 
ble coatings (Fig. 321) is due to Franklin. By means of 
it he demonstrated that the charge is 

apparently on the surface of the glass. 
The real significance of the experiment is 
somewhat different. 

The inner and outer coatings are 
metallic and fit the glass jar. If the 
jar be charged in the usual manner and 
be placed on an insulating stand, the 
inner metal vessel may be lifted out by 
means of the curved rod; the glass jar 
may then be withdrawn from the outer 
vessel. The two metallic coatings are 
now completely discharged ; but after the parts are all 
replaced, the jar may be discharged with a bright flash. 

The electrification of the jar is to some extent a phenom- 
enon of the glass. During the charging of the jar the glass 




Fig. 321 



464 MAGNETISM AND ELECTRICITY 

is strained ; the conductors carry the charge and facilitate 
the release from strain. Thin glass jars may be crushed or 
perforated by overcharging. The residual charge indicates 
that the glass acts as if it were distorted, like a twisted glass 
fiber, and that it does not return at once to its normal 
unstrained state when the jar is discharged. 

514. Energy of a Charged Conductor. The energy expended 
in carrying Q units of electricity through a potential difference 
V V is Q(V V^). If the charge Q were transferred 
from the earth, whose potential is zero, to a conductor 
whose potential remained V units, the work done would 
be QV. But in charging a condenser, or any other con- 
ductor, the potential is zero at the beginning of the charg- 
ing process and Fat the end. Since the potential at every 
instant is proportional to the charge, the mean potential to 
which the charge is raised is ^ F", and the work done in charg- 
ing the condenser, which equals the energy of the charge, is 



Since Q = (7F", other equivalent expressions for the energy 
are Q2 

(92) 



Illustrations of similar expressions for mechanical work done are 
readily found. Thus, the work done against gravity in building a tower 
of uniform cross section (corresponding to capacity) is equal to the prod- 
uct of the mass of the material and half the height, or \ MH. To build 
the tower twice as high requires four times as much work, because double 
the mass is lifted to twice the mean height. 

So also the work done in compressing gas into a cylinder of fixed vol- 
ume is proportional to half the product of the quantity and the final 
pressure, or \ QP. To fill it to a pressure twice as great requires forcing 
in twice the mass of gas against twice the mean pressure, or the work 
done is proportional to the square of the pressure. Height in the one 
case and pressure in the other are the analogues of potential. 

515. Energy Lost in dividing a Charge. Let O l and <7 2 be 

the capacities of two condensers, and let the first be charged 






ELECTRIC CAPACITY AND CONDENSERS 465 

with a quantity Q. If the potential is F", the energy of the 
charge is 



After the charge has been divided by connecting the two 
condensers in parallel, the capacity has been increased to 
C\ 4- O y and the energy is therefore reduced to 




The stored potential energy after the division of the charge 
is less than before it so long as 6 Y 2 has any value in compari- 
son with O r If the two capacities are equal, the energy 
stored after the division is half as great as before it. The 
other half is represented by the energy of the spark at the 
moment of the division. Energy is always lost by the divi- 
sion, whatever be the relative values of the two capacities. 

516. Energy of Similar Condensers in Parallel. If n con- 
densers of the same capacity are charged in parallel, n 
Leyden jars for example, with all 

their outside coatings connected 
together, also their inside coatings 
(Fig. 322), the capacity of the 
whole is n times the capacity of a 
single jar ; the effect is simply to 
increase the size of the coatings. 

Since the energy of discharge of 

a single condenser is J<7F 2 , for n condensers of the same 
capacity it is Tr= x j (7F , = j.crp,. 

The energy of the charge is proportional to the number of 
similar condensers. 

517. Energy of Condensers in Series. If several Leyden 
jars are insulated and the outside of one is joined to the in- 





466 MAGNETISM AND ELECTRICITY 

side of the next (Fig. 323), they are said to be connected in 
series or in " cascade." Let the series be charged to a poten- 
tial difference of V units between 
the inside coating of the first and 
the outside of the last. Then the 
potential difference between the 
coatings of each jar is V/n, and 
the energy of its charge is | CV 2 /^?. 
Fig. 323 The energy of the n charged jars is 

w=^. 

2 n 

The energy of the charge in the n similar jars is 1/w of the 
energy of one of the jars charged to the same available poten- 
tial difference V between its two coatings. 

518. Electric Strain. The phenomenon of the residual 
charge may be best explained by considering the dielectric as 
the medium through which the induction takes place. The 
charging of a Leyden jar is accompanied by the straining of 
the glass. If the potential difference is raised to a sufficiently 
high value, the glass may be strained beyond its elastic limit 
and may give way with a disruptive discharge. The glass is 
shattered at the point through which the discharge takes 
place. In the case of air or other fluid dielectrics, such as 
insulating oils, the dielectric may be broken down by a dis- 
ruptive discharge, but the damage is automatically repaired 
by the inflow of the insulating fluid. 

By subjecting plate glass to powerful electrostatic stress and passing 
plane polarized light through it at right angles to the lines of force, Kerr 
discovered that the glass becomes double refracting, and is strained as if 
it were compressed along the lines of force. Quartz behaves in the same 
way. Kohlrausch and others have pointed out the analogy between the 
Kerr effect and the elastic fatigue of solids after they have been subjected 
to a twisting stress. A fiber of glass does not immediately regain its 
initial form when released from stress, but a slight set remains from which 
it slowly recovers. Its after recovery from distortion due to an electric 
charge sets free energy which is represented by the residual charge. 



ELECTRIC CAPACITY AND CONDENSERS 



467 



Hopkinson has shown that it is possible to superpose several residual 
charges of opposite signs. In the same way a glass fiber may be twisted 
first in one direction and then in the other, and the .residual twists will 
appear in reverse order. No residual charge can be obtained from air 
condensers. 

519. Electric Displacement. Electricity exhibits some of 
the properties of an incompressible fluid. Electric charges 
show themselves only at the boundary between conductors 
and dielectrics. All cases of electrification are examples of 
the transfer of electricity. Hence Maxwell proposed his 
theory of electric displacement. It supposes that when an 
electromotive force acts on a dielectric, as in induction, elec- 
tricity is displaced along the lines of induction. The electro- 
motive force transfers electricity by distorting the dielectric. 
The strained dielectric by its elastic reaction produces a back 
electromotive force, and the discharge is a reverse electric 
transfer to restore the equilibrium. 

520. The Dielectric Constant The density of the charges 
on the surfaces of the plates of a condenser, with a given 
potential difference between them, depends not only on their 
distance apart, but also on the 

facility with which the dielec- 
tric permits electric displace- 
ment. 

Let A, B, O (Fig. 324), 
be three insulated conducting 
plates. From the back of A 
and C are suspended pith balls. 
Let B receive a positive charge 
and let A and O be charged j^ 
negatively by induction. If 
they are touched with the 
finger, the pith balls collapse and remain in contact with the 
plates. If now A, for example, be moved nearer to B, both 
pith balls will diverge, the one on A with a positive charge 
and the one on O with a negative one. 




Fig. 324 



468 MAGNETISM AND ELECTRICITY 

Now replace A in the first position, with B charged as 
before and the pith balls not diverging. Interpose between 
A and B a thick plate of glass or sulphur. Both pith balls 
will again diverge as if A had been moved nearer to B. 
The effect is the same as the reduction of the thickness of 
air between the plates ; the capacity of a condenser depends 
on the nature of the insulating medium between the plates. 

The dielectric constant of a substance is the ratio of the 
capacity of a condenser with the substance as the dielectric 
to its capacity when the dielectric is air. To this constant 
of a dielectric Faraday gave the name specific inductive 
capacity. 

The following are approximate values of jfiTfor some com- 
mon dielectrics : 

Glass 7.4 Ebonite ..... >r, , .< . . .3.1 

Mica . 6.6 Petroleum '. . 3.0 

Sulphur 4.0 Paraffin . ' , '".- J . V . . .2.3 

Shellac 3.2 Vaseline 2.0 

521. Oscillatory Discharge. It appears to have been first 
observed by Savary in 1824 that when a Ley den jar is dis- 
charged through a coil of insulated wire, consisting of a few 
turns and inclosing a sewing needle lying along its axis, 
the needle is not always magnetized in the expected direc- 
tion. This curious fact was first explained by Joseph Henry 
in 1842. He discovered that the discharge of a Leyden jar 
consists of electric surges, first in one direction and then in 
the other, and that the energy of the discharge becomes 
smaller with each oscillation until it is all expended in heat. 
The last discharge strong enough to reverse the magnetism 
of the needle (which is confined to a superficial shell of the 
steel) determines its polarity. 

In 1853 Lord Kelvin gave the mathematical theory of 
electric oscillations, and in 1858 Fedderson analyzed the 
spark of a small discharge into a string of images by a re- 
volving mirror. These observations have since been con- 



ELECTRIC CAPACITY AND CONDENSERS 



469 



firmed by many observers. When the discharge is through 
a low resistance, the spark is a periodic phenomenon. 

522. Electrical Resonance. If two Ley den jars of the 
same capacity are attached to two similar discharge circuits, 
they should have the same period of oscillation and should 
therefore exhibit the phenomenon of resonance. Sir Oliver 
Lodge has found this to be true. 

The two jars are connected to discharge circuits of the 
same size (Fig. 325) ; but while that of A is interrupted by 
a spark gap, that of B is closed 
and is adjustable by means of the 
slider S. If now the coatings of 
A are connected to the two elec- 
trodes of an influence machine, 
this jar discharges across the gap, 
and the oscillations at every dis- 
charge disturb the circuit of B, 
exciting in it feebler oscillations 
of the same period. If the two 
circuits are tuned to unison by 
moving the slider, the oscillations 
in B become sufficiently violent to 
make it overflow through the tin 
foil strip <?, which comes over from 
the inner coating and nearly touches the outer one. The 
strip furnishes an easy overflow path, so that when the jars 
are near together and the two discharge circuits are parallel, 
every discharge of A is accompanied by a bright spark at the 
air gap c. 

Problems 

1. Two Leyden jars are charged with quantities as 1 to 4. The tin 
foil surface of the second jar is twice as large as that of the first and the 
glass is half as thick. Find the relative energy of the two charges. 

2. An insulated conducting sphere whose radius is 10 cm. is charged 
to a potential of 50 units. Find the number of units of charge. 




Fig. 325 



470 MAGNETISM AND ELECTRICITY 

3. If one of two conducting spheres, 25 cm. in diameter, be charged 
to a potential of 200 c. g. s. units ( 500), and then be connected to the 
other sphere by means of a long thin wire, find the energy of the dis- 
charge between them. 

4. A conducting sphere of 30 cm. radius is charged with 900 c. g. s. 
units. If it divides its charge with another insulated sphere of 10 cm. 
radius, what will be the charges on the two spheres ? 

5. Find the capacity of a spherical condenser, the radii of the opposed 
surfaces being 9 and 10 cm., and the dielectric constant of the paraffin 
dielectric 2.3. 

6. An insulated metal ball of 20 cm. radius and removed from all 
other conductors is charged with 200 c. g. s. units of electricity. What 
will be its potential if it is then surrounded by a smooth conducting shell 
of 20J cm. radius connected to earth ? 

7. Two concentric spheres of radii 10 cm. and 10 cm. are separated 
by air and are charged to a potential difference of 300 units. Find the 
charge. 

8. Find the capacity of a Leyden jar with coatings of 900 cm. 2 , the 
glass 2 mm. thick, and the dielectric constant 7. 

9. What is the energy of the charge if the jar of the last problem is 
charged to a potential of 1500 c.g.s. units? 

10. What is the intensity of field between two parallel condenser 
plates which are \ cm. apart and differ in potential by 250 c. g. s. units? 

V. ATMOSPHERIC ELECTRICITY 

523. Lightning an Electrical Discharge. Some of the early 
philosophers surmised that the lightning flash is an electrical 
discharge, but this view 'obtained little currency until after 
Franklin's suggestion to apply his discovery of the discharg- 
ing power of points had been carried into effect. In 1752 
D'Alibard, acting on Franklin's suggestion, erected an insu- 
lated iron rod 40 feet high and drew sparks from passing 
clouds. About the same time Franklin sent up his famous 
kite by means of a linen thread, during a passing storm, and 
held it by means of a silk ribbon between his hand and a key 
attached to the thread. After the thread had been wetted 
by the rain, sparks were drawn from the key and a Leyden 



ATMOSPHERIC ELECTRICITY 471 

jar was charged. A year later Richmarm of St. Petersburg 
was killed by lightning while experimenting with a rod simi- 
lar to that of D'Alibard. 

524. High Potential of Thunder Clouds. The source of 
the electrification of the atmosphere and of clouds remains 
unsettled. But given ever so slight electrification of aqueous 
vapor, it is not difficult to account for the high potential of 
thunder clouds. Assume that each particle of water vapor 
has an initial charge, and that the larger globules are formed 
by the coalescence of smaller ones. If, for example, 1000 such 
particles unite to form a single large one, the diameter of the 
large drop will be only 10 times that of the component parti- 
cles; but while the charge of the larger sphere is 1000 times 
that of the smaller ones, its capacity is increased only tenfold, 
since the capacity of a sphere is numerically equal to its radius. 
Its potential, however, has risen a hundred fold (V Q/0). 
This rise of potential increases the inductive action between 
drops and the tendency to discharge from drop to drop in 
a cloudy atmosphere. 

525. Effect of Electrification on Condensation. A small 
ascending jet of water is resolved into drops, which describe 
different paths. By reason of the different velocities and 
directions of motion of the individual drops, they come into 
frequent collision and rebound. The influence of electrifi- 
cation on the recoil after collision is marked and interesting. 
The subject was investigated by Lord Rayleigh. 

If the ascending jet is strongly electrified, the repulsion 
between the drops scatters them and prevents collision ; but 
with feebler electrification the drops coalesce on impact and 
the stream is much smoother. The coalescence was shown 
to be due to slightly different degrees of electrification of 
the impinging particles. Their attraction and union appear 
to be due to induction, the resulting force of which is always 
an attraction. 

The bearing of these results on the precipitation of aque- 



472 MAGNETISM AND ELECTRICITY 

ous vapor is obvious. Innumerable minute globules of water^ 
feebly charged to different potentials, collide and coalesce 
into drops which descend by gravity. A slight degree of 
electrification of the atmosphere is therefore favorable to 
aqueous precipitation. 

It is an observed fact of frequent occurrence that a vivid flash of 
lightning is quickly followed by a sudden downpour of rain. It is 
clearly impossible to tell which is antecedent to the other, the discharge 
or the condensation ; for, while the flash reaches the observer first, light 
travels from the place of condensation in negligible time, and the dis- 
charge may therefore be subsequent to the sudden condensation. If the 
condensation occurs before the discharge, it is accompanied by a sudden 
rise of potential in the enlarged drops, leading to an electric discharge. 

526. Lightning Flashes. Lightning flashes are discharges 
between two oppositely charged conductors. They may oc- 
cur between clouds, or between clouds and the earth. The 
rise of potential of a cloud causes an accumulation by induc- 
tion in the earth underneath ; and unless this accumulation 
is carried off by the silent action of points, when the electric 
stress in the air reaches a certain limiting value, the air as 
a dielectric breaks down with a sudden subsidence to equi- 
librium. Sir J. J. Thomson estimates the dielectric strength 
of the air under the usual conditions of pressure and temper- 
ature to be about 400 dynes per square centimeter. When 
the electric tension along lines of force becomes greater than 
this, a disruptive discharge takes place. 

527. Discharge with Steady Strain. If the stress in the 
dielectric is increased gradually, and the medium is finally 
strained to the point of rupture, a discharge takes place. 
Under these conditions a point will effect a silent discharge 
and afford protection. This condition Sir Oliver Lodge has 
named the " steady strain," and has illustrated it as follows : 

A and B are the terminals of an influence machine, L is a 
Leyden jar, T and T two metal plates connected with the 
two coatings of the jar as shown in Figure 326. On the lower 
plate are three conducting terminals of different heights. 



ATMOSPHERIC ELECTRICITY 



473 




Fig. 326 



When the machine is turned and the jar is charged, the 

stress increases gradually ; but the pointed conductor, even 

when it is low compared 

with the others, prevents 

a discharge altogether. 

If the point be covered 

or removed, and the 

knobs be positive, long 

flashes may be obtained, 

but generally to the 

smaller ball unless it is 

much farther from the 

upper plate than the large one. The air breaks down at 

the weakest point, or where the stress is greatest, and this 

is at the surface of greatest curvature. 

528. Discharge with Impulsive Rush. In the arrangement 
to illustrate the condition of " steady strain " the potential 
difference between the parallel plates increases gradually un- 
til the limit of the dielectric strength of the air is reached. 
Lodge has arranged a different experiment to illustrate the 

very sudden production 

A f a high potential differ- 

ence and a discharge with 
an "impulsive rush." 

The two Leyden jars in 
series (Fig. 327) stand on 
a wooden table or, better, 
have their outer coatings 
connected by a moist 
string. They charge 
slowly, the outer surfaces through the string, and finally 
discharge at A. The discharge between the inner coatings 
releases the charges on the outer ones, a violent rush takes 
place, producing a very sudden stress in the air between the 
plates, and one of the conductors is struck. The small ball 



. I Q 




Fig. 327 



474 MAGNETISM AND ELECTRICITY 

no longer protects the large one, nor does the point afford 
any certain protective influence. All three terminals are 
equally liable to be struck, if of the same height. In fact 
all three may be struck at once. 

In this case the electric pressure is developed with such 
impulsive suddenness that the dielectric appears as liable 
to break down at one point as another. Such sudden rushes 
may occur when two charged clouds discharge into each 
other, and one overflows into the earth. The highest and 
best conducting objects may then be struck irrespective of 
points and terminals. The conditions of the discharge in 
the case of an impulsive rush are entirely different from 
those of the steady strain, and points are incompetent to 
afford protection by preventing the discharge. 

529. Potential of the Air. In clear weather the potential of the air 
is sometimes nearly as high as during a storm, but it shows smaller fluc- 
tuations. The value of the potential gradient found by McAdie at the 
Blue Hill Observatory was 0.013 electrostatic units per meter of eleva- 
tion. On certain clear days the variation of potential with the elevation 
reached twice this value. During thunder storms the potential gradient 
may amount to 0.12 electrostatic units per meter. 

By means of kites McAdie has shown that the potential difference in 
clear weather increases as the kite rises; and, further, that it is pos- 
sible to obtain sparks from a perfectly cloudless sky, and generally at an 
elevation not exceeding 500 meters. 

530. The Aurora. The aurora, or polar light, is due to 
silent discharges in the upper regions of the atmosphere. 
Within the arctic circle it occurs almost nightly, and some- 
times with indescribable splendor. Lemstrorn has shown 
that the illumination of the aurora is due to currents of posi- 
tive electricity passing from the higher regions of the atmos- 
phere to the earth. In our latitude these silent streamers in 
the atmosphere are infrequent. When they do occur, they are 
accompanied by great disturbances of the earth's magnetism 
and by earth currents. Such magnetic disturbances occur 
at the same time in widely separated parts of the earth. 



CHAPTER XVIII 

ELECTRIC CURRENTS 

I. VOLTAIC CELLS 

531. An Electric Current. When a condenser is discharged 
through a wire, there is produced in and around the wire a 
state called an electric current. Electrification is a condition 
of strain in the dielectric ; an electric displacement through 
the discharging conductor rapidly relieves this strain. If 
the state of strain is reproduced by some agency as fast as 
it is relieved by the conductor, the result is a continuous 
electric current. The expression, "current of electricity," 
was introduced when electricity was regarded as a fluid 
flowing from higher to lower potential through a conductor, 
just as water flows through a pipe from a higher to a lower 
level. 

To produce a continuous electric current through a con- 
ductor, a continuous potential difference must be maintained 
between its terminals. This may be accomplished by means 
of chemical energy, as in a voltaic cell ; by the application 
of heat, as in a thermal couple ; or through the agency of 
mechanical energy, as in the dynamo-electric machine. In 
all these cases the energy applied is converted, wholly or in 
part, into the energy associated with the transport of elec- 
tricity under the electric pressure, which it is the function 
of the device or machine to establish and maintain. The 
energy of an electrostatic charge is potential energy ; that 
of an electric current is kinetic energy. 

The magnitude or strength of a current is the quantity of 
electricity passing any section of the conductor per second. 

475 



476 MAGNETISM AND ELECTRICITY 

Quantity of electricity includes the sum of the positive trans- 
fer in one direction through the conductor and the negative 
in the other ; for the flow of negative electricity in one 
direction has the same effect as that of an equal quantity of 
positive in the other. It is not necessary to distinguish be- 
tween the two fluxes in opposite directions except through 
fluid conductors. 

532. Simple Voltaic Cell. The era of electric currents 
dates from Galvani's discovery in 1786 that violent muscular 
contractions are produced at the instant when a bimetallic 
arc of iron and copper connects the lumbar nerve and the 
crural muscle of a freshly killed frog. In the hands of his 
contemporary Volta this observation resulted in the discov- 
ery that a potential difference exists between two different 
metals, such as zinc and copper, when they are separated by 
moist cloth, damp muscle, or a conducting liquid. Hence 
the simple voltaic cell invented by Volta in the year 1800. 

If a strip of zinc, amalgamated with mercury, be placed 
in sulphuric acid diluted with about twenty times its volume 
of water, bubbles of hydrogen will collect on the zinc, 
but the chemical action will soon apparently cease. No 
change is produced by placing a strip of clean copper in the 
same solution unless the two metals are connected directly 
or by means of some good conductor 
(Fig. 328). The acid then attacks 
the zinc, hydrogen is freely liberated 
at the surface of the copper plate, and a 
dense solution of zinc sulphate streams 
down from the zinc. The liquid prod- 
uct of the electrochemical reaction 
appears at the zinc plate and the 
Fig 328 gaseous product at the copper. As 

soon as the electrical connection be- 
tween the two metals is interrupted, chemical action ceases 
and hydrogen is no longer disengaged. 




VOLTAIC CELLS 477 

Such a combination of two conductors, immersed in a 
liquid, called an electrolyte, which is capable of reacting 
chemically with one of them, is called a voltaic cell. 

If an electroscope, terminating in a plate instead of a ball, 
be supplied with an extra loose plate insulated with shellac 
varnish, it becomes a condensing electroscope. If now the 
zinc strip of a voltaic cell be connected momentarily with 
the top plate or cover and the copper strip with the lower 
plate of the electroscope, when the cover is lifted with its 
insulating handle, the gold leaves will diverge slightly. 
Tested by means of an excited glass tube, they will be found 
to have a positive charge. If the zinc strip were connected 
with the lower plate, the gold leaves would diverge with a 
negative charge. Hence, the copper is positive and the zinc 
negative. The copper and zinc strips are called electrodes, 
the copper the positive electrode and the zinc the negative 
electrode. 

When the two plates of a voltaic cell are joined by a con- 
ducting wire a number of new phenomena appear, which are 
characteristic of an electric current flowing through the con- 
ductor from the copper to the zinc, and through the liquid 
from the zinc to the copper. 

533. The Circuit. The circuit of a voltaic cell comprises 
the entire path traversed by the current, including the elec- 
trodes, and the liquid as well as the external conductor. 
Closing the circuit means completing it by joining the two 
electrodes by a conductor ; breaking or opening the circuit is 
disconnecting them. When the circuit is closed, the zinc 
wastes away, and the energy of its union with the acid is in 
part given out by degrees as the energy of the electric cur- 
rent, which may be made to do work or to generate heat. 

534. Electrochemical Action in a Voltaic Cell. The theory 
of dissociation furnishes an explanation of the manner in 
which an electric current is conducted through a liquid. It 
is briefly as follows : When a salt or an acid, such as hydro- 



478 MAGNETISM AND ELECTRICITY 



chloric acid (5"07), is dissolved in water, some of the mole- 

+ 
cules split into two parts (H and Cl, for example), one part 

having a positive electrical charge and the other a negative 
one. The two parts of the dissociated substance with their 
electrical charges are called ions. (The term is from a Greek 
word meaning to go; ions are travelers or carriers.) An 
electrolyte is a compound capable of such dissociation into 
ions. It conducts electricity only by means of the migration 
of the ions resulting from the splitting in two of the mole- 
cules. The separate ions convey their charges with a slow 
and measurable velocity through the liquid. Electroposi- 
tive ions, such as zinc and hydrogen, carry positive charges 
in one direction ; electronegative ions, such as chlorine and 
"sulphion" ($0 4 ), carry negative charges in the opposite 
direction ; the sum of the two kinds of charges carried 
through the liquid per second is the measure of the current. 
The active components in a voltaic cell, set up with hydro- 
chloric acid, may be represented as follows : 

+ + + + + 
H H H H H 

Zn Ou 

Cl Cl 01 01 01 

Immediately after the circuit has been closed this becomes 

++ + + + 

Zn H H H 

Zn 2#, Ou 

01 01 01 01 01 

Zinc goes into solution as zinc chloride (ZnCl^), and hydro- 
gen appears as free hydrogen gas at the copper plate. Zinc 
ions crowd out hydrogen ions, while the positive and negative 
charges brought to the copper and zinc plate respectively 
reunite as a current through the external conductor. 



VOLTAIC CELLS 479 

535. Electromotive Force. A voltaic cell is an electric 
generator. It is analogous to a rotary pump which produces 
a difference of pressure between its inlet and its outlet. Such 
a pump may cause water to circulate through a system of 
horizontal pipes against friction. In any portion of the sys- 
tem the force producing the flow is the difference of water 
pressure between the two ends of this portion. The force is 
all applied at the pump, and it produces a pressure throughout 
the whole circuit. 

A voltaic cell generates electric pressure called electromo- 
tive force. It produces electric pressure to set electricity 
flowing. The seat of the electromotive force (E. M. F.) in a 
voltaic cell is at the contact of the dissimilar substances in the 
cell, and chiefly at the contact surfaces between the elec- 
trodes and the electrolyte. The E. M. F. of any form of vol- 
taic cell depends on the materials employed, and it is entirely 
independent of the shape and size of the electrodes ; it is 
modified by oxidation and by the concentration of the 
solutions. 

The E. M. F. of a cell is the measure of the work required to 
transport unit quantity of electricity around its entire circuit. 
Work is required to effect this transfer, because all con- 
ductors offer resistance to the passage of a current, and back 
or opposing E. M. F.'s are sometimes present. The energy 
expended against resistance goes to heat the conductor. 

536. Difference of Potential. The difference of potential 
between two points on the external conducting circuit is the 
work done in carrying unit quantity of electricity from one 
point to the other. If E denotes this potential difference 
and Q the quantity conveyed, then the work done is the 
product EQ. But the quantity conveyed by a conductor per 
second is the strength of current /. The energy trans- 
formed, therefore, when a current / flows through a con- 
ductor, under electric pressure or potential difference of E 
units between its ends, is JET" ergs per second. 



480 MAGNETISM AND ELECTRICITY 

537. Polarization. When the circuit of a simple voltaic cell 
is closed, the current falls off rapidly in intensity, and at 
length almost ceases to flow. The hydrogen covering the 
copper plate as a film produces a state known as the polariza- 
tion of the cell. The accumulation of ions on the electrodes 
changes the potential difference between them and the elec- 
trolyte. The hydrogen on the positive electrode not only 
introduces more resistance to the flow of the current, but it 
diminishes the electromotive force to which that flow is due, 
by setting up an opposing difference of potential. 

538. Remedies for Polarization. Any device that will prevent 
the liberation of hydrogen and its deposit on the positive electrode is a 
remedy for polarization. These remedies are mostly chemical, as illus- 
trated by the following experiment, which was devised by Mr. D. H. 
Fitch about 1879 : 

Set up a cell by placing in a small glass jar enough clean mercury to 
cover the bottom, and filling it with a saturated solution of common salt. 
Hang a plate of zinc in the liquid, and thrust into the mercury the ex- 
posed end of a platinum wire, sealed into a glass tube, to connect with 
the mercury as the positive electrode. Close the circuit through some 
simple current indicator, such as a common telegraph sounder of low re- 
sistance. The armature will be drawn down strongly at first ; but in the 
course of a minute or two the magnet will release it, showing that the 
current is very greatly weakened by polarization. The sodium released 
on the surface of the mercury attacks the water, producing sodium 
hydroxide and hydrogen. 

Keeping the circuit closed, drop into the cell a very small piece of 
mercuric chloride (HgClz) no larger than the head of a pin. The arma- 
ture of the sounder will be drawn down suddenly, showing recovery of 
the cell from polarization. The mercuric chloride furnishes chlorine ions 
which unite with the hydrogen ions on the surface- of the mercury elec- 
trode and reduce the polarization. The chlorine will be exhausted in a 
few minutes, and polarization will again ensue. 

539. The Daniell Cell. The Daniell cell illustrates the 
chemical method of avoiding polarization by replacing the 
hydrogen ions by others, such as copper or mercury, which 
do' not produce polarization when they are deposited on the 
positive electrode. In the Daniell cell the copper plate is 



VOLTAIC CELLS 



481 



surrounded with a saturated solution of copper sulphate 
(CuSO^), so that copper instead of hydrogen is deposited on 
the copper electrode. 

A zinc bar Z (Fig. 329) is immersed in acidulated water, 
or in a dilute solution of zinc sulphate (ZnSO^), in an un- 
glazed earthenware cup; the copper 
plate O is a cylinder of sheet copper 
surrounded with a saturated solution 
of copper sulphate. Spare crystals of 
this salt are added to keep the solu- 
tion saturated during the action of the 
cell. The porous cup allows the ions 
to pass through its pores, but prevents 
a rapid admixture of the two electro- 
lytes. 

Both electrolytes undergo partial 
dissociation into ions; and when the 
circuit is closed the electropositive 
zinc and copper ions both travel 
toward the copper electrode. The zinc ions do not reach it, 
because zinc in copper sulphate solution replaces the copper, 
forming zinc sulphate. The relation of the several ions in 
the Daniell cell may be represented graphically as follows : 




Fig. 329 



+ + ++ + + + + + + 
Zn Zn Cu Cu Cu 



Zn 



As soon as the circuit is closed, ZnSO is formed at the 
zinc electrode and copper is deposited on the copper electrode. 

The so-called gravity cell is a Daniell cell, and in it advantage is taken 
of the difference in density of the two sulphate solutions to keep them in 
a measure separate. The copper electrode is placed in the bottom of the 



482 



MAGNETISM AND ELECTRICITY 




Fig. 330 



jar with the CuSOt solution, and the lighter dilute ZnSO solution floats 
on top with the zinc suspended in it (Fig. 330). The cell must be kept 

at work to prevent the diffusion of the 
CuS04 upward as far as the zinc. 

540. The Leclanche CeU. The 

Leclanche cell belongs to a class 
containing a depolarizer, the office 
of which is to supply oxygen to 
unite with the hydrogen and form 
water. In this cell the depolarizer 
is solid manganese dioxide (MnO^). 
It is a zinc-carbon couple with a 
saturated solution of ammonium 
chloride (NHQl) as the electrolyte. 
The carbon electrode is packed in a 
porous cup with the manganese dioxide in granules mixed 
with broken carbon to increase the conductance. The zinc 
is a rolled rod about one centimeter in 
diameter. Figure 331 shows a com- 
plete cell. The porous cup in this par- 
ticular form* has a flange resting on the 
top of the glass jar. This closes the 
jar and prevents evaporation. 

If the circuit be kept closed for 
several minutes, the accumulation of 
hydrogen on the carbon plate produces 
some polarization ; but when the circuit 
is opened again, the depolarizer slowly 
removes the Irydrogen and the cell 
recovers its normal E.M.F. No serious local chemical action 
takes place on open circuit; the cell will stand without 
material waste for months. It is this characteristic that 
makes it suitable for many domestic purposes, or for inter- 
mittent service. 

The initial relation of the ions in the Leclanche cell may 
be represented as follows: 




Fig 331 



VOLTAIC CELLS 483 



Zn O 

Cl Cl Cl 01 

After the circuit is closed the first step is 

Zn NH NHt 

Zn 2NIT a , 2ff, 

Cl Cl Cl Cl 



One atom of zinc unites with two atoms of chlorine, or 
zinc has a valence of two and carries two ionic charges as 
compared with hydrogen or chlorine. Ammonia is released 
and the free hydrogen unites with oxygen from the manga- 
nese dioxide in accordance with the chemical reaction 

2 H+ 2Mn0 2 = H 2 + Mn z O s . 

The so-called "dry cells" in common use are substantially 
Leclanche cells, and the equations above describe the electro- 
chemical action going on in them. 

541. Standard Cells. For many methods of precise elec- 
trical measurement and for purposes of standardizing instru- 
ments, a definitely known electromotive force is necessary. 
This requirement is met by a standard cell, which is never 
employed to furnish a current, but only a known potential 
difference to be very precisely balanced against some other 
potential difference to be measured. 

The standard cell chiefly in use for many years was 
invented by Latimer Clark and known as the Clark cell. 
The negative electrode is a 10 per cent amalgam of zinc in 



484 



MAGNETISM AND ELECTRICITY 



a neutral saturated solution of zinc sulphate ; the positive 
electrode is pure mercury covered with a paste of mercurous 

sulphate and zinc 
sulphate. The cell 
(Fig. 332) is nearly 
filled with zinc sul- 
phate crystals and 
the saturated solu- 
tion of this salt. 
Both legs of the cell 



Cadmium sulphate 
crystals and solution 



Cadmium 




Platinum wire 
Meiicuty 



Fig. 332 



are hermetically 
sealed. 

The Clark cell has now practically been replaced by the 
" Weston Normal Cell " because of its great advantage over 
the Clark i'n that its change of E. M. F. with temperature is 
only one thirtieth as great. It is set up in the same manner 
as the Clark, except that cadmium and cadmium sulphate 
take the place of zinc and zinc sulphate. The negative 
electrode is an amalgam containing from 12 to 12.5 per cent 
of cadmium. 

The E. M. F. of the Weston cell is given by the equation 

E= 1.0184 - 0.0000406 (* - 20) - 0.00000095 (t - 20) 2 . 

(The E. M. F., E, is in volts equal to 10 8 c. g. s. electromag- 
netic units as explained in 584.) 

542. Effects of Heat on Voltaic Cells. Two different effects 
are produced by heating a voltaic cell. The resistance of 
the liquid to the passage of the current is reduced, and the 
E. M. F. suffers a small change, either an increase or a 
decrease. 

Professor Daniell found that a larger current flowed from 
his cell when he heated it to 100. This increase was due 
to the fact that the relative decrease in the internal resist- 
ance of the cell was larger than the relative decrease in its 
E. M. F. The curve in Figure 333 shows the relation between 



ELECTROLYSIS 



485 



the internal resistance and the temperature of a Daniell cell 
between 13 and 68. The resistance at the upper tempera- 
ture is reduced to less than one half its initial value. 

The temperature coefficient of a Daniell cell is only about 
O.OOT percent; that is, the E. M. F. falls 0.007 volt for a 
rise of temperature of 100 degrees. 



DEGREES C 
Fig. 333 



7.5 



II. ELECTROLYSIS 

543 . Electrolytes . When an electric current passes through 
metals or carbon (often called conductors of the first class), 
heat is generated, but the conductor undergoes no change in 
chemical composition or physical state, except the change 
caused by the rise in temperature. Many liquids also con- 
duct electricity, notably a large number of chemical com- 
pounds, either fused or in solution (often called conductors 
of the second class) ; the passage of electricity through these 
is always accompanied by chemical decomposition. For this 
reason the process is called electrolytic conduction or electroly- 
sis, and the substance decomposed, an electrolyte. 

The conductors of the first class by which the current 
enters and leaves the electrolyte are called electrodes, as in 
the case of voltaic cells. The current enters the electrolyte 
by the anode and leaves it by the cathode. The electrolyte is 



486 MAGNETISM AND ELECTRICITY 

split in two, the two parts migrating in opposite directions ; 
the part migrating toward the anode is called the anion, and 
the other part, migrating toward the cathode, is the cation. 
All these terms are due to Faraday. 

The initial products of the electrolysis are not always set 
free at the electrodes, because secondary chemical reactions 
sometimes take place between them and the electrodes or the 
solvent. Such is the case in a voltaic cell when the hydrogen 
is oxidized by the depolarizer. These secondary reactions 
are sometimes termed "secondary electrolysis." 

544. Electrolysis of Copper Sulphate. Copper sulphate pre- 
sents one of the simplest examples of electrolysis. If the 
electrodes are copper, the passage of an electric current 
simply transfers copper from the anode to the cathode. 
The copper deposited on the cathode is very pure, and the 
process is now employed on a colossal scale for refining 
copper. 

When copper sulphate is dissolved in water, it under- 
goes dissociation to some extent. If electric pressure is 

applied to the solution through the electrodes, the electro- 

+ + 
positive ions ((7w) are set in motion from higher to lower 

potential, and the electronegative ions ($0 4 ) in the opposite 
direction. The Ou ions are driven against the cathode, and, 
giving up their charges, become metallic copper. The S0 
ions go to the anode, and, giving up their charges, unite with 
the copper of the anode, forming copper sulphate. Copper 
is thus removed from the anode as fast as it is deposited on 
the cathode. 

The relation of the ions to the current may be represented 
graphically as follows : 

+ + ++ ++ +i 
Ou Cu Ou Ou 

Anode Ou Ou Cathode 



ELECTROLYSIS 



487 



The passage of a current through an electrolyte is accom- 
plished in the same manner, whether it is in a voltaic cell or 
in an electrolytic cell. Molecules not dissociated are elec- 
trically neutral and take no part in the transfer of electricity. 

Since metallic copper is deposited from the solution on 
the cathode, and S0 migrates from the cathode toward the 
anode, the concentration of the solution at the cathode is 
diminished by electrolysis. At the anode, on the other 
hand, the concentration is increased. The change in con- 
centration at the cathode may be shown by arranging the 
anode at the bottom of a large glass tube about 20 .cm. long, 
filling with saturated CuSO solution, and electrolyzing for 
some time with large current density. The liquid near the 
cathode will become nearly or quite colorless. 

545. Electrolysis of Water. Water appears to have been 
the first substance decomposed by an electric current ; but it 
was a mooted question for about three 
quarters of a century whether the de- 
composition of the water was a result of 
a primary electrolysis or only that of a 
secondary chemical reaction. It is now 
known that pure water does not conduct 
an appreciable current of electricity; 
but if it is acidulated with a small 
quantity of sulphuric acid, it is decom- 
posed as a secondary action. 

In Hofmann's apparatus (Fig. 334) 
the acidulated water is poured into the 
bulb at the top, and the air escapes by 
the glass taps until the tubes are filled. 
The taps are then closed, and if connec- 
tion is made with a battery of three or 
more cells in series, bubbles of gas will 
be liberated on the pieces of platinum foil at the bottom. 
The gases collecting in the tubes may be examined by allow- 




Fig. 334 



488 MAGNETISM AND ELECTRICITY 

ing them to escape through the taps. Oxygen will be found 
at the electrode by which the current enters the apparatus, 
and hydrogen at the other ; that is, oxygen collects at the 
anode and hydrogen at the cathode. 

The electrochemical action may be represented as follows : 



Anode Pt Pt Cathode 



Hydrogen is set free at the cathode, while at the anode 
the secondary electrolysis is 



The primary electrolysis is that of the dissociated sul- 
phuric acid ; the water is decomposed at the anode by the 
S0. As often as one atom of oxygen is set free at the anode, 
two of hydrogen are liberated at the cathode. The volume 
of the hydrogen is not exactly twice that of the oxygen, be- 
cause the latter is more soluble in water than the former, and 
about one per cent of it is evolved in the denser form of 
ozone ; on the other hand, more hydrogen than oxygen is 
absorbed or occluded by the platinum electrodes. 

546. Electrolysis of Sodium Sulphate. When a salt of one of 
the alkali metals is electrolyzed between platinum electrodes, 
secondary reactions take place at both electrodes. If the 
electrolyte is sodium sulphate, free sodium cannot exist at 
the cathode, but it unites with water, forming sodium 
hydroxide ; at the anode the S0 decomposes water and 
liberates oxygen. 

Fill a V-shaped tube (Fig. 335) two thirds full with a solution of 
sodium sulphate, colored with the extract of purple cabbage or purple 
violets. Close the ends with corks and thrust through them platinum 



ELECTROLYSIS 489 

wires terminating within the tube in strips of platinum foil. When the 

current is passed, the liquid turns red at the anode and green at the cathode, 

showing the presence of an acid at the 

former and an alkali at the latter. Stop 

the flow of current and .mix the liquids 

at the two electrodes ; both the red and 

green color will disappear with the 

restoration of the faint purple, demon- 

strating that the acid and the alkali 

are produced in chemically equivalent 

... . Fig. 335 

quantities. 

The final result is the liberation of oxygen at the anode and hydrogen 
at the cathode. 



2Na 2Na 2 Na 2 Na 
Anode Pt Pt Cathode 




H >2 + S0 4 = H 2 S0 4 + O. 2Na + 2 H 2 = 2 NaOH + 2 H. 

If mercury is used as the cathode in an aqueous solution of sodium 
sulphate or sodium chloride, the separated metallic sodium amalga- 
mates with the mercury, and only a little sodium hydroxide is formed. 
When fused sodium chloride (common salt) is electrolyzed between a 
carbon anode and molten lead as the cathode, the metallic sodium alloys 
with the lead. These two facts form the basis of two methods of manu- 
facturing caustic soda by electrolysis. 

547. Electrolysis of Lead Acetate. Place the solution, which 
may be made clear by the addition of a little acetic acid, 
in a flat glass tank and electrolyze between two lead wires 
as electrodes. The lead separated from the clear solution 
will be deposited on the cathode in the form of shining 
crystals, which grow rapidly, giving rise to the "lead tree." 
If the process is not conducted too rapidly, these crystals 
assume very beautiful forms. The lead goes into solution 
at one electrode and comes out of solution at the other. 

After a few minutes reverse the current; the first crystal- 
line deposit will gradually disappear, and another one will 



490 



MAGNETISM AND ELECTRICITY 




Fig. 336 



form on the other lead wire. In this way the disappearance 
of the lead at the anode and its appearance out of a clear 
solution at the cathode may be observed at the same time. 
The reaction is exactly the same as that of copper sulphate 
between copper electrodes. 

548. Faraday's Laws of Electrolysis. When several elec- 
trolytic cells are joined in series, in the manner shown in 

Figure 336, the same cur- 
rent passes through all 
of them. When the sev- 
eral cathodes are weighed, 
it is found that the masses 
of the different elements 
deposited are directly as 
their atomic weights and 
inversely as their valences. Thus, silver has a valence of 
one and an atomic weight of 107.94, while copper has a 
valence of two and an atomic weight of 63.4. The masses 
of the two deposited by the same quantity of electricity are 

ft() A 

therefore as 107.94 : . The ratio of the atomic weight 
2 

of an element to its valence is known as its "chemical 
equivalent." 

From an extensive study of electrolytic phenomena Fara- 
day deduced the following laws : 

I. The mass of an electrolyte decomposed by an electric 
current is directly proportional to the quantity of electricity 
conveyed through it. 

II. When the same quantity of electricity is conveyed 
through different electrolytes, the masses of the different ions 
set free at the electrodes are proportional to their chemical 
equivalents. 

Thus, if the same current be passed through a series of 
electrolytic cells, in which it liberates hydrogen, chlorine, 
copper, and silver, for every gram of hydrogen set free, 



ELECTROLYSIS 491 



35.46 gm. of chlorine, 31.7 of copper, and 107.94 of silver 
will be separated. 

The electrochemical equivalent of an element is the number 
of grains of it deposited by the passage of unit quantity of 
electricity. When a current has unit strength, unit quan- 
tity flows through any cross section of the conductor per 
second. Faraday's laws may then be combined in the one 
statement that the number of grams deposited by the passage 
of a constant current through an electrolyte is equal to the 
continued product of the strength of the current (in am- 
peres), the time in seconds during which it flows, and the 
electrochemical equivalent of the element. If z denotes the 
electrochemical equivalent and I the current strength, then 
in symbols 



The coulomb is the unit of quantity in the electromagnetic 
system of measurement to be described later. The current 
strength is an ampere when the quantity transmitted is at 
the rate of a coulomb per second. The electrochemical 
equivalent of silver has been determined by absolute meas- 
urement of the current in amperes and observing the time. 
The accepted value is 0.001118 grams per coulomb. 

549. Charge conveyed by the Gram Equivalent. The gram 
equivalent of an element is the number of grams equal 
to its chemical equivalent. Thus, the gram equivalent of 
silver is 107.94; of copper, 31.8; of zinc, 32.7. Since one 
coulomb of electricity separates 0.001118 gm. of silver, it 
will require 107.94/0.001118 = 96,540 coulombs to separate 
one gram equivalent. An equivalent statement of Faraday's 
second law is that the gram equivalent of any element trans- 
ports the same quantity of electricity or carries the same 
charge ; this quantity is 96,540 coulombs. 

Since the gram equivalents of univalent ions are proportional to their 
atomic weights, it follows that the charge carried by each ion is the same 
for all univalent atoms. If e is this charge in coulombs and m the mass 



492 



MAGNETISM AND ELECTRICITY 



in grams of an atom of hydrogen (the gram equivalent of which is one), 
the " . = 96,540m, 

if the atom of hydrogen is the mass of its ion. The charge carried by a 
divalent ion is 2 e; by a trivalent ion, 3 e, etc. 

The c. g. s. electromagnetic unit of quantity is ten times the coulomb. 

If, then, e is in c.g.s. units, fi - A 

6 = ",oo4 m 

The ratio e/m in the case of hydrogen in liquid electrolysis is thus 
approximately 10 4 . 

550. Polarization of an Electrolytic Cell. If the two plati- 
num electrodes of Hofmann's apparatus be connected to a 
sensitive galvanometer ( 588) immediately after they have 
been used for the electrolysis of sul- 
phuric acid, it will be found that some 
energy has been stored, for the cell will 
furnish a current. The chemical and 
electrical functions are now reversed ; 
the hydrogen and oxygen in contact 
with the electrodes unite to form water, 
and a reverse current flows through the 
cell. The apparatus may be set up as 
in Figure 337. B is the battery for 
the electrolysis of the sulphuric acid. 
Hydrogen accumulates in the tube H 
and oxygen in the tube 0. Let the two- 
point switch & be now turned so as to 
cut 'off the battery and to join the elec- 
trolytic cell to the galvanometer Gr. The 

needle will be sharply deflected by the current from the 
Hofmann's apparatus. To determine its direction, a thermal 
couple ( 571), consisting of a copper and an iron wire 
soldered together and placed in the circuit of the galva- 
nometer at T, is convenient. When such a couple is slightly 
heated, a current passes across from Cu to Fe. It may be 
tried before charging the electrolytic cell, and the direction 
of the deflection of the galvanometer should be noted. It 




Fig. 337 



ELECTROLYSIS 



493 



will be found that the current due to the polarization of the 
electrolytic cell flows out from A and in at (7, or in the 
reverse direction to the current which separates the gases 
oxygen and hydrogen. The E. M. F. of polarization is there- 
fore a back or resisting E. M. F. 

551. The Storage Cell. If the platinum electrodes of the 
sulphuric acid electrolytic cell be replaced by lead, we have 
the Plante storage cell, which is the basis of all modern stor- 
age batteries. 

Attach two lead plates, to which are soldered copper wires, to opposite 
sides of a block of dry wood, and immerse them in a twenty per cent 
solution of sulphuric acid (Fig. 338). Pass 
a current through the cell for a few min- 
utes. The oxygen liberated at the anode 
will oxidize the lead, forming a dark brown 
or chocolate-colored coating of peroxide of 
lead. An ordinary electric house bell may 
be connected to the cell by a switch, as in 
Figure 337. When the switch is turned, 
cutting off the charging battery and con- 
necting the lead electrolytic cell with the 
bell, the latter will ring vigorously for a 
few seconds. The operation may be re- 
peated, showing that energy is stored in 
the cell by the process of electrolysis. 
Plante subjected his cells to repeated 
charging in opposite directions, so that both plates should be modified 
to an appreciable depth by alternate oxidation and reduction. 

In most modern storage cells the lead plates, cast or rolled 
in the form of grids, are filled with lead oxides called the 
active material. These oxides are changed into peroxide at 
the anode, and reduced by hydrogen to spongy lead at the 
cathode, during the process of charging. The chemical reac- 
tions of the storage cell are complex and to some extent unde- 
termined. Sulphuric acid is formed during the charging 
of the cell and disappears during the discharge. Some sul- 
phate of lead is also formed during the discharge, and may be 




Fig. 338 



494 



MAGNETISM AND ELECTRICITY 



reduced by hydrogen with slow charging. The electrode 
which is the anode when charging and the cathode when dis- 
charging, is called the positive plate. 
Figure 339 shows a complete storage 
cell containing one positive plate be- 
tween two negatives. 

Many voltaic cells are ' reversible ; 
that is, when a reverse current is sent 
through them, the electrochemical reac- 
tions in them are all reversed as com- 
pared with those taking place when the 
current is in the normal direction, and 
energy is stored by means of elec- 
trolysis. A storage cell is a reversible 
one specially designed to store the ap- 
plied electrical energy, so that it can 
be recovered at a subsequent time in the form of a current. 




Fig. 339 



Problems 

1. How much silver will a current of one ampere deposit in an hour? 

2. The silver deposited on the cathode in the electrolysis of silver 
nitrate was 2.8095 gm. in 45 minutes. What was the average current? 

3. How much copper will a current of 1000 amperes deposit per hour? 

4. What current will deposit 0.5 gm. of copper in an hour? 

5. If the current in the last problem is furnished by a Daniell battery, 
how much zinc will go into solution and how much copper will be de- 
posited in each cell during the hour? 

6. If doubling the current through a given circuit requires doubling 
the number of similar cells in series, how much more zinc will be con- 
sumed per second in the entire enlarged battery ? 

III. OHM'S LAW AND ITS APPLICATIONS 
552. Ohm's Law. When an electromotive force produces 
a current flow in any given circuit, the strength of the current 
is always proportional to the value of the electromotive force. 
This law was discovered by Dr. Ohm of Berlin in 1827, and 
it has since been known as Ohm's Law. 



OHM'S LAW AND ITS APPLICATIONS 495 

If E be the potential difference between two points of a 
conductor and / the numerical value of the current flowing 
through it, then if suitable units be chosen, 

E=RI, (94) 

where R is a proportionality factor called the resistance of 
the conductor. The resistance is independent of the value 
and direction of the current flowing, and depends only on the 
material of the conductor, its length and sectional area, and 
its temperature. 

Equation (94) is an expression of Ohm's law. It is usu- 
ally written in the equivalent form 

' ' ' /-f (95) 

If the practical units now adopted by international agree- 
ment be employed, Ohm's law may be expressed without 
symbols as follows: The number of amperes flowing between 
two points of a circuit is equal to the number of volts of 
potential difference divided by the number of ohms of resist- 
ance between the same points. 

When Ohm's law is applied to the entire circuit, which 
may contain several sources of E. M. F. of different signs, 
and both metallic and electrolytic resistances, attention must 
be paid to the signs of the electromotive forces. If, for 
example, there are several voltaic cells in the circuit, some of 
them may be connected in the wrong direction so that they 
oppose the flow of current; or the circuit may include elec- 
trolytic or storage cells or motors, which offer a resistance 
to the flow of the current in the form of a counter 
E. M. F. All such electromotive forces must be regarded 
as negative. 

There may be also a number of consecutive resistances in 
series, but resistance is not a directed quantity, for it restricts 
the flow of the current, whether it be in one direction or the 
other. 



496 MAGNETISM AND ELECTRICITY 

Ohm's law may then be written 

T- El + El 

~ 



where each E. M. F. must be taken with its proper sign. 

553. Resistance. Resistance has already been denned as 
a proportionality factor in the equation expressing Ohm's law. 
It is moreover the property of a conductor by virtue of which 
the energy of a current is converted into heat. It is inde- 
pendent of the direction of the current, and the conversion 
of electrical energy into heat occasioned by it is an irrevers- 
ible one ; that is, there is no tendency for the heat energy to 
revert to the energy of an electric current. 

The practical unit of resistance is the ohm. It is repre- 
sented by the resistance offered to an unvarying current by a 
thread of mercury at the temperature of melting ice, 14.4521 
gm. in mass, of uniform, cross-sectional area, and of a length 
0/106.3 cm. This definition is equivalent to saying that the 
cross section of the thread of mercury is one square milli- 
meter, but it avoids any assumption respecting the density 
of mercury. 

554. Resistivity. The resistances of diverse conductors 
are found to conform to the following laws : 

I. The resistance of a uniform conductor is directly pro- 
portional to its length. 

II. The resistance of a uniform conductor is inversely 
proportional to its cross-sectional area. 

III. The resistance of a uniform conductor of given 
length and cross section depends upon the material of 
which it is made. 

This property of a conductor, which determines its resist- 
ance and depends on the nature of its material, is called 



OHM'S LAW AND ITS APPLICATIONS 497 

'tr 

In symbols the three laws of resistance may be expressed 
by the equation , 

R*K, (97) 

a 

in which p is the measure of the resistivity. Obviously 
p R when both I and a are unity ; or, if I is in centi- 
meters and a in square centimeters, p is numerically equal to 
the resistance of a conductor 1 cm. long and 1 cm. 2 cross- 
sectional area. Resistivity is expressed in c. g. 8. units ; the 
ohm is equal to 10 9 c. g. s. units of resistance. 

TABLE 

Resistivities in c. g. s. Units at 

Lead ....,';.".. . 20,380 Iron 9,065 

Thallium . . . ' ''. '. . 17,633 Zinc 5,751 

Tin .... . . . . 13,048 Magnesium 4,355 

Nickel ..'; ,'-.,. . 12,323 Aluminum 99% .... 2,563 

Platinum, v,* :* -.* --... . . 10,917 Gold 2,197 

Palladium . . v .- . . 10,219 Copper 1,561 

Cadmium 10,023 Silver 1,468 

555. Conductance. The inverse of a resistance is a con- 
ductance. A conductor whose resistance is r ohms has a 
conductance equal to 1/r. When a number of conductors are 
joined in parallel, the conductance of the whole is the sum of 




the conductances of the several branches. If two conductors 
of resistances R and R 2 are connected in parallel between 
the points A and B (Fig. 340), and if V\ and V^ are the 
potentials of A and B respectively, putting V l V^ = E, we 
have by Ohm's law, ^ E E 



498 MAGNETISM AND ELECTRICITY 

The first member of this equation is the whole current flow- 
ing, and this is equal to the sum of the currents through the 
two branches. R is the combined resistance of the two con- 
ductors in parallel. Hence 

! = JL + !. 

R R l R 2 

From the last equation", 

R= iH^- < 98 > 

ri^ -f- ^ 2 

556. Relation between Resistivity and Temperature. The 

resistivity of all pure metals and of most alloys increases with 
rise of temperature. The resistivity of carbon and of elec- 
trolytic conductors decreases when the temperature rises. 
Thus, the resistance of a carbon incandescent lamp filament 
is only about half as great at normal incandescence as when 
cold. Solutions of zinc sulphate and of copper sulphate have 
a temperature coefficient somewhat over 0.02, or 2 per cent 
per degree C. 

The curves of Figure 341 show the variation of resistivity 
with temperature for a number of pure metals. All these 
curves tend toward a point of convergence of zero resistivity 
at a temperature near the absolute zero of 273 on the 
centigrade scale. Over comparatively short ranges of tem- 
perature these curves are approximately straight lines, or the 
change in resistivity is nearly proportional to the change in 
temperature. It follows that the resistivity at any tempera- 
ture t may be expressed by the relation 

Pi =Po (1 + 0i (") 

in which p t is the resistivity at any temperature , p that 
at 0, and a is the temperature coefficient. The same expres- 
sion holds for the resistance of any particular conductor. It 
will be seen from the diagram that the temperature coefficient 
of platinum is very nearly a constant over a wide range of 



OHM'S LAW AND ITS APPLICATIONS 



499 



temperature. It is the same as the coefficient of expansion 
of a perfect gas, 0.00367 ( 380). In fact, the relation be- 
tween the resistivity of platinum and its temperature might 
be used to define the absolute zero of temperature just as well 
as the expansion of a perfect gas. The temperature coefficient 
of the other pure metals is a quantity of the same order of 
magnitude as that of platinum, but in general a trifle larger. 



20,000 



10,000 



77 



/ 



LA. 




-300 -200 -100 +100 -t-200 

Fig. 341 

The temperature coefficient of alloys is smaller than that of 
pure metals. German silver has a coefficient only about one 
tenth as great as that of copper. Manganin, an alloy of 
copper, manganese, and nickel, has a temperature coefficient 
nearly equal to zero ; at certain temperatures manganin has 
a small negative coefficient ; that is, its resistivity diminishes 
slightly as the temperature rises. 

557. Loss of Potential Proportional to Resistance. If V l and 
V^ are the potentials of two points on a conductor carrying a 
current J, by Ohm's law (94), 

Fi - F 2 = R J. 



500 MAGNETISM AND ELECTRICITY 

It is obvious from this relation that the potential difference 
between any two points of a conductor through which a con- 
stant current is flowing is proportional to the resistance be- 
tween them, provided the conductor is not the seat of an 
E. M. F. Even when electromotive forces are 
encountered, the loss of potential, when a 
given current flows through a conductor, is 
proportional to the resistance of 
the conductor. 

Let distances measured along 
Or denote resistances (Fig. 342), 
and those along Ov potential dif- 
ferences. Lay off AP equal to 
V\ and BQ to V^ ; also the dis- 

tance PQ stands for the resistance R between the points A 
and B on the conductor. Then AC equals V l V 2 t and the 
slope of the line AB represents the rate at which the potential 
drops per unit length along the resistance R. Moreover, 
since 




it is evident that the tangent of the angle of slope equals the 
strength of the current. 

The principle that the loss of potential is equal to the 
resistance passed over, when the current is constant, is one of 
very frequent application in electrical measurements. 

558. Wheatstone's Bridge. The combination of resistances 
which is more commonly used than any other for a com- 
parison of two of them is known as a Wheatstone's net- 
work ; and when it is embodied in a piece of apparatus for 
measuring resistances, it is called a Wheatstone's bridge. 

A Wheatstone's network consists of six conductors con- 
necting four points ; in one of the conductors is a source of 
E. M. F., and in another branch is a galvanometer or other 
sensitive detector of current. 



OHM'S LAW AND ITS APPLICATIONS 



501 




In the diagram, Figure 343, A and J) are maintained at 
a fixed potential by the battery B' . Then the fall of poten- 
tial from A to D is the same 
whether one considers the path 
ABD or the path ACD -, and 
if a point, B, is chosen on the 
one path, another one, (7, may 
always be found on the other 
such that the fall of potential 
from A to B is the same 
as from A to (7; in other 
words, B and O have the same 
potential, and if these points 

are connected through a galvanometer, no current will flow 
through it. If O has the same potential as B, by the prin- 
ciple of the last article the ratio of the fall of potential in 
AB to that in BD is equal to R^/R^ and the ratio of the fall 
in AC to that in CD is equal to R^/R^ But these ratios are 
equal to each other ; hence 

I? = j?- (1) 

In practice three of these resistances are fixed and the ad- 
justment for a balance is made by varying the fourth. It is 

necessary to know only the ratio 2 , for example ; the equa- 
tion (100) gives the relation between J2 X and M 8 . 

559. Cells in Series. If n similar voltaic cells, with an 
electromotive force E and internal resistance r for each cell, 
are joined in series by connecting the negative of the first 
with the positive of the second, and so on through the series ; 
and if R is the resistance external to the battery, by the ap- 
plication of Ohm's law the current is 



7 = 



nE 
R + nr 



(101) 



502 MAGNETISM AND ELECTRICITY 

If R is small in comparison with r, / = E/r nearly, or the 
current is no greater than could be obtained from a single 
cell ; but if R is large in comparison with r, then the current 
with n cells in series is nearly n times as large as one cell 
would give with the same external resistance. The internal 
resistance of a storage cell is very low, and the short-circuit- 
ing of a single cell gives as large a current as the short- 
circuiting of a large number of cells in series. 

560. Cells in Parallel. A battery is said to be connected 
in parallel, or in multiple, when all the positive electrodes 
are joined together for the positive terminal, and all the 
negative electrodes for the negative. The object aimed at 
is the reduction of the internal resistance, except that 
storage cells are joined in parallel when it is desired to 
draw from them a larger current than the normal dis- 
charge current for a single cell. With several cells in 
parallel, the current in the external circuit is divided 
among them. 

When n similar cells are connected in parallel, the E. M. F. 
is the same as for a single cell, but there are n internal paths 
of equal resistance, and the internal resistance is r/n. Hence 

(102) 



Problems 

1. Three Daniell cells are joined in series ; the E. M. F. of each cell 
is 1.1 volts and the internal resistance 2 ohms. If the external resist- 
ance is 5 ohms, find the current. 

2. Two Leclanche cells are joined in parallel ; each has an E. M. F. of 
1.5 volts and an internal resistance of 4 ohms. If the external resist- 
ance consists of two parallel conductors of 2 'and 3 ohms respectively, 
find the current through each branch. 

3. A cell whose E. M. F. is 2 volts gave a current of } ampere through 
an external resistance of 3 ohms. What was the internal resistance of 
the cell? 



THERMAL EFFECTS OF A CURRENT 503 

4. Four similar cells, each having an E. M. F. of 1.5 volts, are joined 
in series through a resistance and give a current of one ampere; and 
when joined in parallel through the same resistance, the current is one 
third less. What is the internal resistance of each cell? 

5. Three wires are connected in parallel ; their resistances are 20, 30. 
and 60 ohms. Find their combined resistance in parallel. 

6. The resistance between two points A and B of a circuit is 25 ohms ; 
when another wire is connected in parallel between A and B, the resistance 
becomes 20 ohms. Find the resistance of the second wire. 

7. A current of \ ampere flows through an incandescent lamp under 
a pressure of 220 volts. What is the resistance of the lamp? 

8. What is the resistance of a set of coils carrying a current of 10 
amperes when the fall of potential between the terminals of the coils is 
450 volts ? 

9. If the ratio of two of the resistances forming a Wheatstone's 
bridge is 10, and the resistance of the third branch is 20 ohms, what is 
the resistance of the fourth branch for a balance ? 

10. Two voltaic cells are joined in series with a given resistance and 
produce a current of 3 amperes ; one of the cells is then reversed and the 
current falls to 1 ampere. Find the ratio of the electromotive forces of 
the two cells. 

IV. THERMAL EFFECTS OF A CURRENT 

561. Conversion of Electric Energy into Heat. It is now a 

familiar fact that electric energy is readily converted into 
other forms. If an electric current encounters a back E.M.F. 
anywhere in the circuit, work is done by the passage of the 
current against this opposing E.M.F. Such is the case in 
most examples of electrolysis and in charging a storage 
battery. All the energy of an electric current not so con- 
verted, or stored in some form of stress as potential energy, is 
dissipated as heat. Heat is generated wherever the circuit 
offers resistance to the current. In a simple circuit containing 
no devices for transforming and storing energy, all of it is 
converted into heat. 

The heating of a carbon or a tungsten filament to incan- 
descence by the passage of an electric current is a fact of 



504 



MAGNETISM AND ELECTRICITY 



everyday observation. Electric cars are heated by currents 
passing through resistance coils. 

The heat evolved by dissolving 33 gm. of zinc in sulphuric acid Favre 
found to be 18,682 calories. When the same weight of zinc was consumed 
in a voltaic cell, the heat evolved in the entire circuit was 18,674 calories. 
The operations were conducted in both cases in a large calorimeter. 
The two quantities are nearly identical, or the heat 
generated is the same whether the solution of the 
I* zinc is associated with an electric current or not. 

When a definite amount of chemical action takes 
place in a battery and no work is done, the dis- 
tribution of the heat evolved is altered, but not its 
amount. 




562. Laws of the Generation of Heat. 

The laws of the generation of heat in an 
electric circuit were discovered experi- 
mentally by Joule and Lenz. The latter 
experimented with a simple calorimeter 
like that of Figure 344. A thin platinum 
wire, joined to two stout conductors, was 
inclosed in a wide-mouthed bottle containing alcohol. A 
thermometer t passed through a hole in the stopper. The 
resistance of the wire was known, and the observations con- 
sisted in measuring the current and noting the rise of tem- 
perature. Joule found that the number of units of heat 
generated in a conductor is proportional 

a. To its resistance; 

b. To the square of the current strength ; 

c. To the length of time the current flows. 

563. The Heat Equivalent of a Current. When Q units of 
electricity are transferred through a potential difference of 
V units, both in e.g. s. measure, the work done is QV ergs, 
since the potential difference between two points is defined 
as the work required to transport one unit of electricity from 
the one point to the other. 



THERMAL EFFECTS OF A CURRENT 505 

If/' is the current in c.g.s. units, Q = I't. Hence the 
W0rk ' - ^ 



But W = JH ( 437) . Substituting for W in the last equa- 
tion and solving for H, 

rr rVt I'Vt 

-- =Cal0 



If, however, the current is measured in amperes I and the 
potential difference in volts E, then since /' = Jx 10" 1 and 
V=Ex 10 8 ( 584), it follows that 

PVt = IEtx 107 ergs, 

lEt X 10 7 lEt , 

4.186 x!0r = 4386 Cal0neS - (1 8) 

But by Ohm's law E = RI. Then 

= 0. 239 x I*Rt calories. (104) 



Obviously equation (103) may be written 

HJ= lEt x 10 7 ergs = lEt joules. 

This relation has been used by Griffiths and by Barnes as 
the basis of a very accurate electrical method of determining 
J". By passing a current through a coil of platinum wire in 
a water calorimeter and measuring /, E, and H for an ob- 
served time of t seconds, all the quantities in the equation 
are known except 7, which may then be calculated. 

564. Counter E. M. F. in a Circuit. The total activity, or 
rate at which a generator is supplying energy to a circuit, 
is represented in part by the heat evolved in accordance with 
Joule's law, and in part by work done, such as chemical 
decomposition and storage in electrolysis, the mechanical 



506 MAGNETISM AND ELECTRICITY 

work of a motor, etc. In every case of doing work the 
energy absorbed by it is proportional to the current strength 
instead of its square. But the whole energy expended per 
second by the generator is the product of the current strength 
and the applied E. M. F. We may therefore write for the 
whole energy transformed in t seconds, 

lEt = PRt + Alt. 

The first term of the second member of this equation is the 
waste in heat ; the second, the work done ; A is a constant 
or proportionality factor. Dividing through by It and solv- 

ingf0rl> T _E-A 

^r 

It is obvious from the form of this equation that the con- 
stant A is .of the nature of an E. M. F. Since it has the 
negative sign, it is a counter E. M. F. The effective E. M. F. 
producing the current in accordance with Ohm's law is the 
applied E. M. F. less the counter E. M. F. This counter 
E. M. F. is necessarily present in every case in which work 
is done by an electric current. 

565. Expression for the Work Done. If the counter E. M. F. 
be represented by E' , the equation for the current by Ohm's 
law is 

J=: "' (105) 



Therefore the heat waste in watts becomes 
PR = IE- E' = IE -IE 1 . 



Now IE is the total activity in the portion of the circuit con- 
sidered, that is, the whole energy applied to it per second. 
The heat generated per second in this same portion of the 
circuit of resistance R is less than the energy applied by 
IE 1 watts. Hence, the energy spent per second in doing 



THERMAL EFFECTS OF A CURRENT 



507 



work is the product of the current strength and the counter 
E. M. F. 

The ratio of the work done to the heat waste is 

IE' E' 



E-E 1 

With a given value of E, the value of this ratio increases 
with E' ; that is, the efficiency with which electric energy is 
converted into work increases with the counter E. M. F. 

566. Applications of Electric Heating. Of the many appli- 
cations of heating by an electric current the following are some of the 
more important: 

1. Electric cautery. A thin platinum wire heated to incandescence 
is employed in surgery instead of a knife. Platinum is used because 
it is infusible except at a high temperature, and it is not corrosive. 

2. Safety fuses. Advantage is taken of the low 
temperature of fusion of some alloys, in which 
lead is a large constituent, in making safety fuses 
to open a circuit automatically whenever the cur- 
rent becomes excessive. Safety fuses should be 
mounted on noncombustible bases or inclosed in 
a protecting tube. 

3. Electric heating. Electric street cars are 
sometimes heated by a current through suitable 
iron wire resistances embedded in cement, asbes- 
tos, or enamel. Similar devices for cooking have 
now become articles of commerce. Small fur- 
naces for fusing, vulcanizing, and enameling in 
the operations of dentistry are in common use. 
Large furnaces for melting iron and the reduc- 
tion of iron ores by electric heat are in use to 
some extent. They are also employed in many 
chemical operations requiring high temperatures. 

4. Electric welding. If the abutting ends of 

two rods or bars are pressed together while a large current passes through 
them, enough heat is generated at the junction, where the resistance is 
greatest, to soften and weld them together. The rod becomes uniform 
when the weld is complete and the heat is no longer localized. This 
method has been perfected by Elihu Thomson. Figure 345 shows three 
small welded joints. 




Fig. 345 



508 MAGNETISM AND ELECTRICITY 

V. ELECTRIC LIGHTING 

567. The Carbon Arc. In 1800 Sir Humphry Davy dis- 
covered that when two pieces of charcoal, connected by 
suitable conductors to a powerful voltaic battery, were 
brought into contact and then separated a slight distance, 
brilliant sparks passed between them ; but no mention was 
made of the electric arc until 1808. In 1810 Davy exhibited 
the arc light at the Royal Institution in London. 

With a battery of 2000 simple elements, when the car- 
bons in a horizontal position were drawn apart to a distance 
of several inches, the carbon was apparently volatilized, and 
the current was conducted across in the form of a curved 
flame or arc. Hence the name electric arc was given to this 
form of electric lighting. 

The white-hot charcoal electrodes of Davy burned away 
very rapidly unless they were inclosed in a vacuum. It 
was not till 1844 that Foucault surmounted the difficulty by 
the use of dense carbon from a gas retort instead of wood 
charcoal. 

When the carbon points are separated, the heat due to the 
current volatilizes some of the carbon and ionizes it, and 
this ionized carbon vapor conducts the current across. The 
dazzling light is emitted chiefly by the vividly hot carbon 
electrodes, and especially by the positive one. In it is 
formed a small cavity by the transport of carbon across to 
the negative. Violle has estimated the temperature of this 
depression or crater at 3500 C. The positive carbon rod 
wastes away about twice as fast as the negative one. 

Duddell has found a thermal electromotive force between hot carbon 
and carbon vapor, and directed from the latter to the former. At the 
positive electrode there is therefore a back E. M. F. against which the 
current flows; the current thus gives up energy there, and the only form 
it can take is heat. The reverse is true at the negative electrode. The 
principal resistance to the passage of the current is offered by the layer 
of carbon vapor near the electrodes. The counter E. M. F. at the posi- 
tive accounts for the higher initial temperature on this side; a denser 



ELECTRIC LIGHTING 509 

layer of carbon vapor is thus formed there, which in turn increases the 
resistance with the generation of more heat than at the negative. 

The efficiency of an electric lamp is denned as the ratio of 
the electrical power expended to the candle power emitted. 
The efficiency of the carbon arc is about 1 watt per candle 
power. 

568. The Inclosed Arc, When the electric arc is main- 
tained between rods of hard retort carbon in the open air, 
the carbon burns away quite rapidly. The potential differ- 
ence between the carbons is then from 45 to 55 volts for a 
10-ampere current. 

In the " inclosed arcs " the lower carbon and a portion 
of the upper one are inclosed in a small globe, which 
is air-tight at the bottom, but allows the upper carbon to 
slip through a check- valve at the top. Soon after the arc is 
formed, the oxygen is absorbed and the arc is thereafter 
inclosed in an atmosphere of nitrogen and carbon monoxide. 
The inclosed arc is longer than the open arc, and the poten- 
tial difference required to maintain it is about 80 volts, but 
the current for the same consumption of energy is smaller 
than the open arc requires. The carbons for the inclosed 
arc last about ten times as long as in the open air. 

569. Other Arc Lights. Other arc lamps are now in com- 
mercial use in which the light comes chiefly from the incan- 
descent stream between the electrodes. They have a higher 
efficiency than the carbon arc. In the " metallic arc " pow- 
dered magnetite in an iron tube is used for one electrode 
and a block of copper for the other. The arc flame is very 
white and brilliant ; the light comes from the luminous iron 
vapor. 

" Flaming arcs " are made by the use of a positive elec- 
trode impregnated with salts of calcium. The light from 
the flaming arc is yellow and is adapted to outdoor illumina- 
tion only. 



510 MAGNETISM AND ELECTRICITY 

570. The Incandescent Lamp. In the incandescent lamp 
the heat is due to the simple resistance of a thin con- 
ducting filament inclosed in an exhausted glass bulb. The 
terminals of the filament are connected through the glass 
by means of two short pieces of platinum wire. Platinum 
is used for this purpose because its coefficient of expansion 
is about the same as that of glass ; and so, when the lamp 
becomes hot in use, it neither leaks air around the wires nor 
cracks. 

The carbon filament is made usually from cellulose 
obtained from cotton. After preliminary treatment it is 
carbonized by raising to a cherry-red heat out of contact 
with the air. It is then siirrounded by an atmosphere of 
rarefied hydrocarbon vapor, and is heated white hot by an 
electric current. The heat decomposes the vapor, and the 
carbon residue is deposited in a dense form on the filament. 
By this treatment the filament acquires a hard, steel-gray 
surface and greater uniformity. 

The temperature to which a carbon filament can be raised 
is limited by the tendency of the carbon to disintegrate at 
high temperatures. This disintegration rapidly reduces the 
thickness of the filament and blackens the glass bulb. 

In recent years filaments have been made of the rare 
metals osmium, tantalum, tungsten, and some other mate- 
rials. The tungsten lamp is rapidly displacing the carbon 
lamp because of its higher efficiency, in spite of the fact that 
it is much more fragile. Its efficiency is as high as from 1 
to 1.25 watts per candle power. 

The ordinary commercial unit for the carbon filament is 
the 16-candle-power incandescent lamp. On a 110-volt 
circuit it takes about 0.5 ampere. Since the power in watts 
consumed is J2I, this lamp consumes about 55 watts, or 3.5 
watts per candle power. The tungsten 25-watt lamp gives 
20 candle power, and the 40-watt lamp 32 candle power. 



THERMOELECTROMOTIVE FORCE 511 

VI. THERMOELECTROMOTIVE FORCE 

571. Thermoelectric Junction. If a circuit be formed by 
two wires of different metals joined together at their ends, 
no current will flow through it so long as both junctions 
are at the same temperature. Any electromotive forces at 
the two junctions are then equal and in opposite directions 
around the circuit. Seebeck found about 1822 that if one 
junction is at a higher temperature than the other, there 
is in general an electromotive force in the circuit and a 
current will flow. 

If a copper wire and an iron wire be twisted together 
and their free ends be connected to a sensitive galvanometer, 
and if the twisted junction be warmed to a higher tempera- 
ture than the rest of the circuit, a current will flow across 
the warmed junction from copper to iron. Such a current 
is called a thermoelectric current, and the electromotive force 
at the junction of the wires, thermoelectromotive force. 

The dissimilar substances composing a thermoelectric junc- 
tion or pair may be two metals, a metal and a liquid, two 
liquids, or even two pieces of the same metal at different 
temperatures or in different physical states. 

572. The Neutral Temperature, Assume two copper wires 
connected to a suitable galvanometer and their free ends 
joined by an iron wire. If now one junction of the copper 
and iron be kept at a constant temperature, while that of 
the other is gradually raised, the current in the circuit will 
increase up to a certain temperature of the hot junction. 
This temperature is called the neutral temperature or neutral 
point. 

If the hot junction be heated still more, the current will 
decrease and, finally, it will become zero when the tempera- 
ture of the hot junction is as much above the neutral point 
as that of the cold one is below it. If the temperature of 
the hot junction be raised still higher, there will again be a 
current, but its direction will be reversed. 



512 



MAGNETISM AND ELECTRICITY 




Fig. 346 



573. The Thermopile. The E. M. F. of a single thermal 
couple is very small ; to get a larger E. M. F. a number of 

similar couples may be joined in 
series. With n couples in series the 
potential difference between the ex- 
treme terminals is n times that of a 
single couple. Figure 346 shows the 
manner of connecting in series. If 
the bars are A antimony and B bis- 
muth, then heating the junctions c, c, c 
will cause a current to flow through the circuit in the direc- 
tion of the arrow ; but if these junctions are cooled, or the 
alternate ones, c?, c?, heated, the current will 
flow in the other direction. 

When a number of bars of antimony and 
bismuth are soldered together in this manner, 
and are closely packed in the form of a cube, 
with insulating material between adjacent 
bars, so that opposite faces of the cube form 
alternate junctions, the instrument is called a 
thermopile (Fig. 347). When the face of such a pile is black- 
ened with lampblack and is provided with a reflecting cone, the 
instrument becomes a sensitive detector of radiation ( 427). 

574. The Peltier Effect. In 1834 Peltier discovered the 
phenomenon which bears his name ; it is the converse of See- 
beck's discovery. If a bismuth-antimony junction be heated, 




Fig. 347 



Fig. 348 

the current flows from the former to the latter. Peltier dis- 
covered that if a current from an external source be sent 
through such a compound bar from bismuth B to antimony 
A (Fig. 348), the junction will be cooled ; but if it be sent 
the other way, the junction will be heated. 



THERMOELECTROMOTIVE FORCE 513 

The long arrow shows the direction of the current sent 
through the bar; the small arrows at a and b indicate the 
direction of the thermoelectromotive forces at the junctions. 
At a the thermal E. M. F. is in the direction in which the 
current is flowing. Hence at this junction work is done on 
the current, and the heat of the metals is converted into the 
energy of the current. At b the thermal E. M. F. opposes 
the current, which therefore does work on the junction and 
heats it. 

The thermal effect of a current at a junction of dissimilar 
substances differs greatly from the thermal effect due to 
simple ohmic resistance. The Peltier effect is reversible, the 
current heating or cooling the junction according to its 
direction, and the quantity of heat evolved or absorbed varies 
simply as the current strength ; the heat due to resistance is 
independent of the direction of the current, and is propor- 
tional to the square of the current strength. 

575. Experiment to demonstrate the Peltier Effect. Connect a 
Leclanche cell, B', with a thermopile and a sensitive galvanometer, as in 
Figure 349. S is a two-point switch ; when 
it is turned in the direction of the full line, 
the circuit through the cell and the thermo- 
pile is closed and the galvanometer circuit 
is open. When it stands in the direction 
of the dotted line, the cell is cut off and 
the thermopile is connected to the galva- 
nometer. To show that the current from the 
thermopile P is opposite in direction to the 
current sent through it by the voltaic cell, 
insert in the circuit of the galvanometer at 
T a copper-iron couple. With the switch at 
b, the current produced by heating this junc- 
tion flows from Cu to Fe, and the direction of the galvanometer deflection 
may be noted. Turn the switch for a moment to a and then back again 
to b. The galvanometer will show a current coming from the thermo- 
pile, and the direction of the deflection will be the same as when the 
junction T was warmed. Hence B must be the positive and A the nega- 
tive electrode of the thermopile as a generator. But the current from 




514 MAGNETISM AND ELECTRICITY 

the cell enters the pile at B and leaves it at A. The thermal effect pro- 
duced by the current through the pile is such as to generate a counter 
E.M.F. 

576. Thermoelectromotive Force between Metals and Liquids. 
The therrnoelectromotive forces having their seat at 
metal-liquid contacts have special interest because of their 
relation to the temperature coefficient of voltaic cells. These 
electromotive forces are larger than most of those between 
metals. Thus, the therrnoelectromotive force of Zn-ZnSO 
is 0.00076 volt per degree for a mean temperature of 18.5; 
that of Cu-CuSO is 0.00069 for about the same mean tem- 
perature. In microvolts (millionths of a volt) these are 760 
and 690 respectively. Since the metal is in both cases posi- 
tive to the solution, and there is no appreciable E. M. F. at 
the contact of the two liquids, the temperature change per 
degree in the E. M. F. of a Daniell cell is the difference be- 
tween the therrnoelectromotive forces at the zinc and copper 
electrodes, or 0.00007 volt per degree. It is, moreover, neg- 
ative because the thermal E. M. F. on the zinc side is greater 
than on the copper side. A rise of temperature of one degree 
on the zinc side of the cell lowers the E. M. F. 0.00076 volt; 
the same rise of temperature on the copper side of the cell 
raises the E. M. F. 0.00069 volt. The same rise of tempera- 
ture of the whole cell therefore lowers the E. M. F. by the 
difference of the two thermoelectromotive forces, or by 
0.00007 volt. This conclusion has been fully verified by 
experiment. 

This method of analysis of the temperature coefficient has 
been applied to other cells, such as the Clark standard cell 
and the Weston normal cell ; the results in every case show 
that the temperature coefficient is determined by the thermo- 
electromotive forces at t the contacts of the dissimilar sub- 
stances in the cell, whenever it is not complicated by the 
solution and recrystallization of salts. 

The Peltier phenomenon applies to junctions between 
solids and liquids. When a Daniell cell furnishes a current, 



THERMOELECTROMOTIVE FORCE 515 

heat is absorbed at the positive or copper electrode and is 
generated at the zinc electrode, because the thermoelectro- 
motive force on both sides is directed from the liquid to the 
metal. Research has shown that the observed difference of 
temperature between the two sides of a suitably designed 
Daniell cell, due to the flow of a known quantity of electric- 
ity through it, conforms to the calculated value. 

With metal-liquid junctions there may be an E. M. F. in 
the circuit without any differences of temperature, because 
the thermoelectromotive force is a function of the concen- 
tration of the solution as well as of the temperature. 

Problems 

1. The electrodes of a voltaic cell are joined by two wires alike in 
every respect, except that one is twice as long as the other. What are 
the relative quantities of heat generated in the two? 

2. The E. M. F. of a battery is 20 volts and its internal resistance 2 
ohms. The potential difference between its poles when connected by a 
wire A is 16 volts ; it falls to 14 volts when A is replaced by another 
wire -B. Calculate the number of calories of heat generated in the 
external circuit in 3 min. in the two cases. 

3. A current of 10 amperes flows through a resistance of 2 ohms for 
14 sec. Find the number of calories of heat generated. 

4. What current would have to flow for an hour through a resistance 
of 20 ohms to produce enough heat to raise the temperature of a kgm. of 
water from the freezing point to the boiling point ? 

5. 500 incandescent lamps in parallel are supplied with one half am- 
pere each at a potential difference of 110 volts between lamp terminals. 
The drop of potential in the line is 2.2 volts. What is the resistance of 
the line and how much power is lost in it? 

6. A battery has an E. M. F. of 8.5 volts; the total resistance in the 
circuit is 20 ohms, including an electrolytic cell. The heat generated 
per second in a 5.12 ohm coil included in the circuit is 0.12 calorie. 
What is the counter E. M. F. of the electrolytic cell? 



CHAPTER XIX 



ELECTROMAGNETISM 
I. MAGNETIC RELATIONS OF A CURRENT 

577. Oersted's Discovery. The discovery by Oersted at 
Copenhagen in 1819 was one of prime importance, for he 
was the first to find any connection between electricity and 
magnetism. He observed that when a magnetic needle is 
brought near a long straight wire conveying a current, the 
needle tends to set itself at right angles to the length of the 
wire ; also that the direction in which the needle turns de- 
pends on the direction of the current through the conductor. 
The experiment of Oersted shows that the region around a 
wire conveying an electric current has magnetic properties, 
that is, it is a magnetic field. At this point the analogy 
between an electric current and a stream of water flowing 
through a pipe fails, for such a stream produces no effect in 
the region surrounding the pipe. 

Assume a current flowing through the conductor above the 

needle NS from 
north to south, 
as indicated in 
Figure 350. The 
north pole will 
turn toward the 
east. If the cur- 
rent be reversed, 
or if it be placed 
under the needle while still flowing in the same direction, 
the north pole will turn toward the west. 

516 




Fig. 350 




MAGNETIC RELATIONS OF A CURRENT 517 

If the wire be carried around the needle in a rectangular 
loop (Fig. 351), both branches of it will contribute to the 
force of deflection, and the north-seeking pole at the left 
will turn toward the east. : 

578. Direction of Deflection 
with Respect to the Current. 
All the movements of a mag- 
netic needle under the influence 
of a current may be summed up 

in one rule : F . g . 351 

Stretch out the right hand in the direction of the wire, 
with the palm turned toward the magnetic needle, and with 
the current flowing in the direction of the extended fin- 
gers ; the outstretclwd thumb will point in the direction in 
which the north pole is deflected. 

579. Magnetic Field about a Wire. A little consideration 
will show that if the current flows in the direction of the 
long arrow (Fig. 352), the resulting magnetic force is in 

the direction of the small arrows 
around the circle ; conversely, if the 
current flows around the circle clock- 
^ wise, the positive direction of the 

^^^LL^ magnetic force, or the direction in 
which a north pole is urged, is along 
the axis of the circle away from the 
observer. If the fingers of the closed 
Flg ' 352 right hand represent the circle with 

the current flowing around in the direction of the finger 
tips, the outstretched thumb points in the direction of the 
lines of force, and conversely ( 580). The lines of force 
due to a current are, therefore, concentric circles about the 
conductor as a center. 

Such circular lines of force may be shown by a fairly strong current 
through a vertical wire passing through a hole in a horizontal glass plate, 




518 



MAGNETISM AND ELECTRICITY 



on which are evenly sifted fine iron filings. When the plate is gently 
tapped, the filings are left free to arrange themselves in circles, indicat- 
ing the lines of magnetic force around 
the wire. Figure 353 is reproduced 
from a photograph of such circular 
lines. They show that the ether about 
the current is under stress, and there- 
fore possesses potential energy. It is 
rather more important to direct the 
attention to this magnetic stress in the 
ether than to what goes on within 
the conductor itself. 




Fig. 353 



f/ 



580. Magnetic Properties of a 
Circular Conductor. Bend a 
copper wire into the form shown 
in Figure 354, and suspend it by a long untwisted thread so 
that the ends dip into the mercury cups shown in section at 
the bottom. When a current is sent 
through the suspended wire, a magnet 
pole near the circular conductor will 
cause the latter to turn around a ver- 
tical axis and take up a position with 
its plane at right angles to the axis of 
the magnet. 

This experiment shows that a cur- 
rent in the form of a loop acts like a 
magnetic shell or disk. The lines of 
force about the circular conductor pass 
through it and come around from one 
face to the other through the air out- 
side the loop. An electric circuit is 
in every case equivalent to a magnetic 
shell whose contour coincides with the 
circuit. The clbsed circuit and the 
magnetic shell have in their vicinity similar magnetic fields. 

The north-seeking side of the loop is the one from which 
the magnetic lines issue ; to an observer looking toward this 




Fig. 354 




MAGNETIC RELATIONS OF A CURRENT 519 

side, the current flows around the loop counter-clockwise 
(Fig. 355). The direction of the current and that of the 
lines of force are related to each other 
as the direction of rotation and of trans- 
lation in a right-handed screw. 

581. Magnetic Intensity at the Center of 
a Circular Coil. The intensity of the 
magnetic field at any point is the force 
in dynes acting on a unit magnetic pole 
( 459) placed at the point. Faraday 
showed that the magnetic intensity pro- 
duced by a current is proportional to the current strength; 
Biot and Savart demonstrated experimentally that for a cur- 
rent of indefinite extent it is inversely proportional to the 
distance between the conductor and the point ; Laplace 
proved that this latter result follows from the law of in- 
verse squares as applied to the mutual action between an 
element of the conductor and unit pole at the point. Hence 
the intensity due to the current in an element I of the con- 
ductor, at a point P on a perpendicular from the element, is 



where r is the distance between the current element and the 
point P, k is the constant or proportionality factor, and 1 
the strength of the current. 

If the point P is at the center of a circle of radius r and 
a current / is flowing around the circle, then the magnetic 
intensity at the center due to the current in the entire cir- 
cumference is 



If now the unit current is so defined as to make Jc equal 
to unity, then o _. T 

J aE..S*S, (106) 

r 



520 MAGNETISM AND ELECTRICITY 

582. The Electromagnetic Unit of Current. The electromag- 
netic system of electrical units in common use is based on 
the magnetic effects of a current. The starting point is 
the magnetic intensity due to a conductor conveying a 
current. 

If an element of a conductor one centimeter long be bent 
into an arc of one centimeter radius, the current through it 
will have unit strength when it exerts a force of one dyne on 
a unit pole at the center of the arc. 

This definition is equivalent to making the constant k in 
the last article equal to unity. If the field due to unit cur- 
rent in unit length of the conductor is unity, the field due 
to the same current through the whole circumference will be 
2 TT ; and if the current is I units, the field will be 2 irl. If, 
further, the radius is not unity, but r, the circumference will 
be 2 TIT, and then 



The ampere is one tenth of this c. g. s. electromagnetic unit 
of current. The unit of quantity in the electromagnetic 
system is the quantity which passes any cross section of the 
conductor in one second when the current through it is one 
e.g. 8. unit. The practical unit of quantity is the coulomb; 
it corresponds with the ampere, and is one tenth of the c. g. s. 
unit of quantity. 

583. Electromagnetic Units. It will be convenient for reference to 
bring together the several electrical units expressed in electromagnetic 
measure in the c. g. s. system. 

Unit Strength of Current. A current has unit strength when a length 
of one centimeter of its circuit, bent into an arc of one centimeter radius, 
exerts a force of one dyne on a unit magnetic pole at its center. 

Unit Quantity. Unit quantity is the quantity conveyed by unit cur- 
rent in one second. 

. Unit Potential Difference. Unit potential difference exists between 
two points when the transfer of unit quantity from one point to the 
other requires the expenditure of one erg of work. 



MAGNETIC RELATIONS OF A CURRENT 521 

Unit Resistance. A conductor offers unit resistance when unit poten- 
tial difference between its ends causes unit current to flow through it. 

Unit Capacity. A conductor has unit capacity when unit quantity 
charges it to unit potential. 

584. Practical Electromagnetic Units Several of the c.g.s. elec- 
tromagnetic units are inconveniently small and others are inconveniently 
large for practical purposes. Hence the following multiples arid sub- 
multiples of them have been universally adopted as the practical units : 

Current. The ampere, equal to 10" 1 c.g.s. unit; it is represented by 
the current which will deposit silver from silver nitrate solution at the 
rate of 0.0011182 gm. per second. 

Quantity. The coulomb, equal to 10" 1 c.g.s. unit of quantity; it is 
the quantity conve} T ed by a current of one ampere in one second. 

Electromotive Force. The volt, equal to 10 8 c.g.s. units; it is 
10,000/10,1 of the E.M.F. of a Weston normal cell at 20. 

Resistance. The ohm, equal to 10 9 c.g.s. units; a volt produces an 
ampere through a resistance of one ohm ; practically the ohm is repre- 
sented by the resistance at of a uniform thread of mercury 106.3 cm. 
in length and 14.4521 gm. mass. 

Capacity. The farad, equal to 10~ 9 c.g.s. unit; it is the capacity of 
a condenser which is charged to a potential difference of one volt by one 
coulomb. The microfarad, chiefly used in practice, is one millionth of 
a farad, or 10~ 15 c.g.s. unit. 

Work. The joule, equal to 10 7 ergs ; it is represented by the energy 
expended per second by one ampere under a pressure of one volt. 

Power. The watt, equal to 10 7 ergs per second ; it is equivalent to 
the power of a current of one ampere flowing under a pressure of one 
volt, or to one joule per second ; it is very approximately ^ of a horse 
power. 

Induction. The henry, equal to 10 9 c.g.s. units; it is the induction in 
a circuit when the electromotive force induced in this circuit is one volt, 
while the inducing current varies at the rate of one ampere per second 
(623). 

The prefixes kilo- and milli- combined with any of the preceding units 
signify a thousand and a thousandth respectively. Thus, a kilowatt is 
a thousand watts, and a millivolt is a thousandth of a volt. The pre- 
fixes mega- and micro- signify a million and a millionth respectively. 
Thus, a megohm is a million ohms, and a microfarad is a millionth of a 
farad. 



522 MAGNETISM AND ELECTRICITY 

II. GALVANOMETERS 

585. Types of Galvanometers. When currents are com- 
pared by means of their magnetic effects, the instrument 
used for the purpose is called a galvanometer. 

The three types of galvanometers most in common use are 
as follows : (1) those in which the current through a fixed 
coil of wire causes a deflection of a suspended magnetic 
needle, usually at the center of the coil ; (2) those in which 
the coil itself is movable around a vertical axis between the 
poles of a fixed magnet ; (3} these two types are applicable to 
direct currents only ; for both direct and alternating currents 
another kind is employed, in which both the fixed and the mov- 
able parts are coils ; these are known as electrodynamometers. 

586. Nobili's Astatic Pair. Galvanometers of type (1) 
are commonly used to detect small currents. The first req- 
uisite in such instruments is sensitiveness, that is, a very 
small current must produce an observable deflection of the 
needle. For this purpose the controlling couple of the 
earth's field on the movable magnetic system must be 
reduced. This may be done by means of a weak compen- 
sating magnet, placed either above or below the movable 

magnetic needle, with its north-seeking pole turned 
toward the north. The field produced by it is then 
opposed to the earth's field. 

Nobili's astatic pair illustrates the other method in 
common use. It consists of a pair of needles, or two 
groups of needles, mounted in the same vertical plane, 
but with their similar poles turned in opposite direc- 
tions (Fig. 356). If the two needles have equal mag- 
netic moments, the resultant action of the earth's field 
on the system is zero. Since such a system has no directive 
tendency, it is called astatic, because it will remain at rest in 
any azimuth. In practice the astatic system is never exactly 
balanced magnetically, and the earth's field always has some 
directive influence. 



GALVANOMETERS 



523 



If both needles are surrounded with coils so connected 
that the current flows around them in opposite directions, 
the two forces of deflection turn the system in the same 
direction, while the opposing controlling force is reduced to 
a small value. 

587. The Astatic Mirror Galvanometer. In Figure 357 the 
front coils are swung open to expose to view the astatic 
system. It consists of minute 
pieces of magnetized watch- 
spring at the centers of the coils 
above and below. In very sen- 
sitive instruments like this one 
a small mirror is attached to the 
movable system ; the instrument 
is then called a mirror galva- 
nometer. In the figure the mir- 
ror is midway between the two 
sets of small magnets. Some- 
times a beam of light from a 
lamp is reflected 
from the small mir- 
ror back to a scale, 
and sometimes the 
light from a scale 
is reflected back 
to a telescope, by 
means of which the 
deflections are read. 
In either case the 
beam of light becomes a long pointer without weight. 




Fig. 357 



588. The D'Arsonval Galvanometer. It is immaterial from 
a magnetic point of view whether the magnet or' the coil 
of a galvanometer is made movable, since the action be- 
tween them is reciprocal. In the D'Arsonval galvanometer 
(Fig. 358) a coil, suspended by a fine wire, swings between 



524 



MAGNETISM AND ELECTRICITY 



the poles of a strong permanent magnet. The current is 
led in by the suspending wire and out by the wire or spiral 

spring connecting the coil to 
the bottom. The great advan- 
tage of this type of galvanom- 
eter is that it has a strong 
field of its own, which is only 
slightly affected by the earth's 
magnetism or by iron or other 
magnetic material in its neigh- 
borhood. A small mirror for 
reflecting a beam of light is 
attached to the coil. Inside 
the coil is a soft iron tube sup- 
ported from the back. This 
has the effect of strengthening 
the narrow magnetic field in 
which the coil swings. 




Fig. 358 



589. Potential Galvanometers. Galvanometers designed to 
determine the potential difference between two points of a 
circuit must be of high resistance. If they are graduated to 
read in volts, they are called voltmeters. Any sensitive gal- 
vanometer may be used as a voltmeter by adding a sufficiently 
large resistance in series with it. Unless the resistance of 
the voltmeter is high, the application 
of its terminals to two points of a cir- 
cuit A and B (Fig. 359), so as to put 
it in parallel with a resistance s through 
which a current is flowing, will dimin- 
ish the potential difference which it is 
desired to measure. 

For direct currents the most con- 
venient portable voltmeter is made on the principle of the 
D'Arsonval galvanometer. The inside of the instrument is 
shown in Figure 360. A portion of the magnet and of one 





ELECTRODYNAMICS 525 

pole piece is cut away to show the coil and the springs. 

The current is led in by one spiral and out by the other. 

Attached to the coil is a 

light aluminum pointer, 

which moves over a scale on 

which the voltage is read 

directly. 

A similar instrument, 
called an ammeter, is designed 
to measure currents in am- 
peres. The resistance of a 
voltmeter should be so high 
that it will take the smallest 
operating current ; the re- Fi 36Q 

sistance of an ammeter 

should be as small as possible, so that it will not increase 
the resistance of the circuit in which it is placed. 

III. ELECTRODYNAMICS 

590. Magnetic Field about Parallel Currents. The term 
electrodynamics is applied to that part of the science of elec- 
tricity which is concerned with the force exerted by one cur- 
rent on another. The reciprocal action between conductors 
conveying currents was discovered by Ampere in 1821, 
shortly after Oersted's discovery of the reciprocal action 
between a current and a magnet. 

Every conductor through which a current is flowing is sur- 
rounded by a magnetic field, and the magnetic fields of two such 
conductors react on each other. The reciprocal action between 
conductors carrying currents is purely magnetic, and may be 
accounted for by the stresses set up in the surrounding medium. 

The magnetic field about a single conductor is composed, 
as we^have seen, of concentric circles ; but when the fields of 
two conductors are in part superposed, the composite mag- 
netic figures will be those due to the resultant of the two 
sets of forces in the field. 



526 



MAGNETISM AND ELECTRICITY 










Figure 361 is the field shown by iron filings about two 
parallel wires passing through the two holes and with the 
_________ currents flowing in 

the same direction. 
In addition to the 
small displaced circles 
immediately around 
the conductors, there 
are continuous curves 
inclosing both cir- 
cuits. These are due 
to the coalescence of 
a number of circles 
belonging to the two 
currents. The two conductors are drawn together by the 
tension along these inclosing lines. 

Figure 362 is the field about two parallel conductors with 
the currents flowing in opposite directions. Midway between 
the two wires the 
lines of force have 
the same direction 
in space, and pro- 
duce a uniform field 
over a small area. 
The circles about 
the two wires are 
all eccentric, but 
there are no lines 
common to the two ^ 
conductors ; the re- 
siliency of these lines, or their tendency to recover from dis- 
placement, forces the conductors apart. 

591. Motion of a Circuit in a Magnetic Field. The law 

applying to the electrodynamic action between conductors 
conveying currents is that their relative motion is always 




V>' 









ELECTRODYNAMICS 



527 




Fig. 363 



such as to make the flux* of magnetic lines around them a 
maximum. Hence, two circuits tend to move toward coin- 
cidence. Each is urged to a 
position that makes the lines 
of force common to the two as 
numerous as possible. 

Similar statements hold with 
respect to a magnet and a cir- 
cuit. When a bar magnet and 
a helix (Fig. 363) come into 
the relative position where the 
middle point of the former 
coincides with the mean plane of the latter, the lines of force 
of the two are identical in direction through the helix, and 
the position is one of stable magnetic eq,ui- 
CK^ 6 ^ librium. Hence the law of parallel currents : 

Parallel conductors conveying currents in 
the same direction attract each other ; if the 
currents are in opposite directions, they repel 
each other. 

592. Electromagnetic Rotations. A large num- 
ber of different devices have been designed for the 
purpose of producing continuous rotation by the 
action between 
a magnet and 
a circuit. In 
Figure 364 a 
copper wire is 
hung by a hook 
Fig. 364 (better by a lig- 

ament of fine 

copper wires), and the lower end 

dips into mercury M surrounding 

the pole of a magnet. If the current flows down through the wire and 

the upper end of the magnet is a north pole, the bottom of the wire will 

rotate around the pole clockwise. 

Barlow's wheel (Fig. 365) is one of the oldest devices to secure con- 





Fig. 365 



528 



MAGNETISM AND ELECTRICITY 




tinuous rotation by the action between a current and a magnet. It is 
therefore a direct current motor without a commutor ( 635). Contact 
is made by means of mercury in the trough C, and the action of the 
magnetic field between the poles of the horseshoe magnet is on the radial 
current from the mercury to the axis A of the copper wheel. 

593. Electrodynamometers. The electrodynamometer is an 
instrument designed originally by the German physicist 
Weber to measure the strength of a current by the electro- 
dynamic action between two coils of 
wire, one fixed and the other movable 
about a vertical axis through its own 
plane. The two coils are set with their 
magnetic axes at right angles (Fig. 
366), and the free coil moves in a 
direction to make their axes coincide. 
AB is a single turn of the fixed 
coil, and CD one of the suspended 
coil. The ends a and b of the latter 
dip into mercury cups not shown, and 
the two coils are in series. The mov- 
able coil is suspended by silk threads 
(or on a point resting in a jewel), and 
a helix is rigidly connected with it 
at the top and with a torsion head T. 
The movable conductor is subjected 
to a system of forces, that is, to a 
torque, tending to turn it in the direc- 
tion indicated. 

When the coil CD is deflected by sending a current through 
the two coils in series, the torsion head is turned by hand so 
as to bring the movable coil back to its initial position. 
The couple due to the magnetic action between the two coils is 
then in equilibrium with the couple of torsion of the twisted 
helix. The couple of torsion is proportional to the angle of 
torsion by Hooke's law. The electrodynamic action between 
the coils is proportional to the square of the current, since 




Fig. 366 



ELECTRODYNAMICS 



529 



doubling the current doubles it through both coils, doubles 
the magnetic field of both, and therefore quadruples the 
force. The square of the current is therefore proportional 
to the angle through which the helix at the top is twisted to 
restore the suspended coil to its zero position, or 



Whence 



(107) 



A is a constant of the instrument depending on the windings 
and the helix. 

Figure 367 is one form of the complete instrument, show- 
ing the coils, the helix, and the scale at the top with the 
pointers, one moving with 
the suspended coil and the 
other with the helix. The 
fixed coil may be consid- 
ered as furnishing a mag- 
netic field corresponding 
to that of the permanent 
magnet in the D'Arsonval 
galvanometer ; but in this 
instrument the field re- 
verses with the reversal of 
the current, and therefore 
the deflection is in the 
same direction whether 
the current goes through 
the instrument in one direc- 
tion or the other. It may 
thus be used with alter- 
nating or reversing currents as well as with direct ones. 




Fig. 367 



594. Convection Currents. Two parallel currents in the 
same direction attract and two like electric charges repel each 
other. According to Maxwell, the electrodynamic attraction 
should exactly equal the electrostatic repulsion when the 



530 MAGNETISM AND ELECTRICITY 

electric charges move with the velocity of light. Faraday 
assumed that a stream of particles (or ions) carrying electric 
charges of the same sign has a magnetic effect like a current 
of electricity. Rowland demonstrated the truth of this as- 
sumption in 1876; he found that a charged disk, when 
rapidly rotated, had a feeble field equivalent to a circular 
current. Conversely, such convection currents are acted on 
by a magnet. The electric arc behaves like a flexible con- 
ductor. It may even be ruptured by the deflecting influence 
of a powerful magnet. Elihu Thomson has utilized this 
effect to extinguish an arc started by lightning on an electric 
lighting circuit. 

The titanic whirlpool constituting a sunspot has a mag- 
netic field along its axis. The hypothesis is that the ions 
composing the whirl are charged, and their rapid rotation in 
circles is the equivalent of circular currents, with a magnetic 
axis coinciding with the axis of the whirlpool. 



IV. ELECTROMAGNETS 

595. Solenoids. Since a circular current is the equiva- 
lent of a plane magnetic shell ( 580), a helix composed of 

equal circular currents, all on the 
same axis, and with their similar 
faces turned in the same direction, 
is the equivalent of a cylindrical 

magnet. Such a system of circular currents constitutes a 
solenoid (Fig. 368). Each turn of the helix 
may be resolved into a plane circular current 
AB(7(Fig. 369), and a linear current AC per- 
pendicular to the plane of the circle. An 
entire helix of n turns is therefore equivalent 
to n circular currents and a linear current along 
the. axis of the helix. If the conductor returns 
along the axis, as in the figure, the external field is due to 
the circular elements only. 




ELECTROMAGNETS 531 

If such a solenoid is suspended so as to turn freely, it will 
set its axis in the magnetic meridian when a current is passed 
through it. It is therefore equivalent to a magnet, and its 
poles may be determined by the method described in 579. 
They will be attracted or repelled by a permanent magnet 
like those of a magnetic needle. 

596. Effect of Introducing Iron. When an iron bar is 
introduced into a solenoid through which a current is pass- 
ing, the iron will be magnetized by the magnetizing force 
along the axis of the helix. If the iron core is absent, many 




Fig. 370 



of the lines of induction leak out at the sides of the helix 
(Fig. 370). The core not only diminishes the leakage, but 
adds many more lines to those previously running through 
the solenoid. Hence the magnetic strength of a helix is 
indefinitely increased by the iron core. 

If the bar be of soft iron, it will exhibit notable magnetism 
only so long as the current flows through the magnetizing 
coil. The loss of magnetism is not complete when the cur- 
rent is interrupted; the small amount remaining is known 
as residual magnetism. Temporary magnets produced by 
magnetic induction within a helix are called electromagnets. 
The north pole of an electromagnet is the one about which 
the current in the helix appears to circulate counter-clockwise 
to one looking toward the pole. The circular currents in 
the helix and the lines of induction in the core are linked 
together by the right-handed screw relation. In fact, no 



532 



MAGNETISM AND ELECTRICITY 




exception has ever been discovered to the general fact that with 
every electric current there are always linked lines of induction 

in this same relation. 



Iron filings arranged in circles 
about a linear conductor may be 
regarded as flexible magnetized 
iron winding itself in a helix 
around the current ; conversely, a 
free flexible conductor carrying a 
current winds itself around a 
straight bar magnet. The flex- 
ible conductor of Figure 371 may 
be made of tinsel cord or braid. 
Directly the circuit is closed, the 
conductor winds slowly around 
the vertical magnet ; if the cur- 
rent is reversed, the conductor 
unwinds and winds up again in 
the reverse direction. 



597. The Horseshoe Mag- 
net. --The most common 
form of electromagnet is 

the U -shape or horseshoe type (Fig. 372). The windings on 
the two iron cylinders or cores must be in a direction to 
make the two poles of opposite signs. It is the same as if 
the two cores were straightened out and the bar wound con- 
tinuously from end to end. The armature (not shown in the 
figure) consists of a flat bar like the 
yoke at the other end, and extending 
across from pole to pole. As a rule, the 
cores, the yoke, and the armature should 
form a closed magnetic circuit, that is, 
one in which the lines of induction are 
entirely in the iron. 

If a ring be wound continuously with a right-handed 
helix so as to form a closed circuit, and if connection with 
the winding be made at two points diametrically opposite 



Fig. 371 




MAGNETIC INDUCTION 



533 



each other (Fig. 373), and a divided current be sent through, 
there will be a consequent south pole where the current en- 
ters and a consequent north pole 
where it leaves the ring. The poles 
are consequent because they belong 
to two magnetic circuits, or to a 
divided circuit through the iron. 
Either half of the ring may be re- 
garded as the armature of the other 
half. 

V. MAGNETIC INDUCTION 




Fig. 373 



598. Magnetic Permeability. The 
effect of placing iron in a magnetic 
field is to increase greatly the num- 
ber of lines of induction running through the space occupied 
by the iron. When these lines of induction traverse the iron, 
it is magnetized. The increase in the number of lines due 
to the iron may amount to several thousand per square 
centimeter. 

Let 6B stand for the magnetic induction, or number of 
lines per square centimeter through the iron. Then the 
ratio between 6B and &6> the magnetizing force, is called the 
permeability of the iron, or 



where /z stands for the permeability. It expresses the fact 
that iron transmits the inductive effect better than air, or is 
more permeable. Magnetic induction is //. times the mag- 
netizing force. 

599. Magnetic Susceptibility. The intensity of magnetiza- 
tion is the pole strength per unit area of the polar surface. 
Magnetic susceptibility is the ratio between the intensity 
of magnetization 8 and the strength of the field, or, in sym- 

bols ' 



534 MAGNETISM AND ELECTRICITY 

The concept involved in permeability rather than the one in 
susceptibility is the modern one derived from Faraday. 

600. Paramagnetic and Diamagnetic Substances. A clear 
distinction between paramagnetic and diamagnetic substances 
may be drawn by means of their relative permeability as 
compared with that of air. Paramagnetic substances are 
those whose permeability is greater than unity; and since 
the permeability of air is practically unity, paramagnetic 
substances are more permeable than air. On the other 
hand, diamagnetic substances have a permeability less than 
unity, or they are less permeable than air. 

Paramagnetic substances concentrate the magnetic flux 
and diamagnetic substances diffuse it. If iron be placed in 
a magnetic field, it will cause more lines of induction to pass 

through than through air; 
if bismuth be placed there 
instead of iron, fewer lines 
will pass through it than 
through the air previous to 
its introduction. 

If an iron sphere be 

Fig- 374 . r 

placed in a uniform mag- 
netic field, the effort of the lines of induction will be to run 
as far as possible through the sphere (Fig. 374). This 
action proceeds on the prin- 
ciple that the potential 
energy of a system always 
tends to as small a value as 
possible ; for when the same 
magnetic flux passes through 
iron as through air, there is 

l i -, Hg. 375 

less energy per unit volume 

in the iron, for it requires a smaller magnetizing current to 

send the flux through. 

If the sphere in Figure 375 is bismuth, the effort of the 




MAGNETIC INDUCTION 535 

lines of induction will be to avoid it because it is less per- 
meable than air. For the same flux density, the energy per 
unit volume is greater in bismuth than in air. 

When lines of magnetic induction pass from air into a 
paramagnetic substance, as in Figure 374, they are bent away 
from the normal to the surface in the substance; when they 
pass from air into a diamagnetic substance, they are bent 
toward the normal. 

601. Movement of Paramagnetic and Diamagnetic Bodies in a 
Magnetic Field. Faraday examined the magnetic behavior 
of a large number of bodies in the intense field between- 
the pointed poles of a powerful 
electromagnet. A small bar of iron 
suspended between the poles (Fig. 
376) turns in the axial direction ab, 
while a bar of bismuth sets its longer 
axis in the equatorial direction cd 
across the field. If the bismuth is 
in the form of a cube or a sphere, 

it is repelled to one side. Iron moves into the stronger parts 
of the field ; bismuth into the weaker. They are examples of 
the two classes into which bodies are divided with respect td 
the action of magnetism on them. 

These movements may be satisfactorily explained by the 
relative permeability of the body and the medium in which 
it is suspended. Feebly magnetic bodies behave as if they 
were diamagnetic when surrounded by a more highly per- 
meable medium. A small glass tube containing a weak 
solution of ferric chloride is paramagnetic in air ; but when 
it is suspended in a denser solution of ferric chloride, it 
takes a cross position like a diamagnetic body. When any 
body assumes the equatorial position, the only inference 
which can justly be drawn from this behavior is that its 
permeability is less than that of the air or other medium 
surrounding it. 




536 



MAGNETISM AND ELECTRICITY 




8,000 



4,000 



2,000 



I* JC 



In general liquids are diamagnetic ; liquid oxygen and 
solutions of salts of the paramagnetic metals are exceptions. 

602. Curves of Magnetization. The curves of Figure 377 
show the relation between the magnetizing force and the 

masfnetic induction for 

12,000 1 1 1 1 i 1 1 1 1 

0& 



three samples of iron ; a is 
the curve for mild steel, b 
for wrought iron, and c for 
cast iron. The magnetiz- 
ing force &6 is plotted 
horizontally, and the in- 
duction 68 vertically ; the 
resulting curves represent 
the successive stages of 

Fig. 377 ,. . 

magnetization. 

If the ratio of 68 to &6 were constant, the curve of mag- 
netization would be a straight line. For small magnetizing 
forces the curve is nearly straight; after this it bends 
sharply upward, and then becomes gradually flatter and 
flatter, so that for large values of the magnetizing force it is 
again a straight line. In the figure the scale for 68 is much 
smaller than for 8f6* If the same scale were used for the two, 
the curves for large values of BS would be a straight line 
inclined to either axis at the angle of 45. Under these 
conditions the iron is said to be saturated, and the permea- 
bility /A is a constant. 

Since /*, is the ratio of 68 to $g it is evident that the 
permeability is a maximum for that point of the curve for 
which a straight line joining the origin and the point, as P, 
no longer cuts the curve, but is tangent to it. For smaller 
as well as for larger values of the coordinates the permea- 
bility is less. 



603. Hysteresis. If the magnetization of a ring of iron 
is carried through a complete cycle by increasing the mag- 
netizing force by successive steps from zero to some definite 



MAGNETIC INDUCTION 



537 



value, decreasing it from that value by small steps through 
zero to an equal value in the other direction, and then again 
reducing it to zero and complet- 
ing the cycle, the curve connect- 
ing 6B and BS will not be the 
same with decreasing values of 
BS as with increasing ones (Fig. 
378). The induction M lags 
behind the magnetizing force. 
Thus, when BS is reduced to 
zero from its maximum positive 
value, 68 has the value 05, and 
BS must be given a negative 
value equal to oc before the 
induction becomes zero. So 
when BS returns from its maxi- 
mum value in the other direc- 
tion to zero, the induction 
decreases only to the value oe. 
This phenomenon of the lag of 
the induction behind the magnetizing force Ewing has called 
magnetic hysteresis. The result of plotting the correspond- 
ing values of 68 and BS through a complete cycle is a curve 
inclosing an area, and this area represents the heat lost per 
cubic centimeter in the iron in carrying it through a single cycle. 

604. Remanence and Coercive Force. A cyclic magnetiza- 
tion curve serves among other things to give definiteness 
to the terms remanence or retentivity and coercive force. 
The residual value of 68 when BS is reduced to zero is ob 
(Fig. 378). This value is the remanence. It depends on 
the quality of the iron, the limit to which the magnetization 
has been pushed, and whether the magnetic circuit of the 
iron is open or closed. The figure applies to a magnetic 
circuit consisting of a ring. The value of BS required to 
reduce the residual induction to zero, namely, oc, is the 




Fig. 378 



538 MAGNETISM AND ELECTRICITY 

measure of the coercive force. Mechanical vibration due to 
external forces has the effect of diminishing residual mag- 
netism, coercive force, and hysteresis. If the iron in thin 
plates be carried rapidly through successive cycles of mag- 
netization by alternating currents, some vibration will be set 
up in the plates unless they are rigidly clamped together. 
Any vibration sustained by means of the current absorbs 
energy, and increases the area of the hysteresis curve. 



VI. THE MAGNETIC CIRCUIT 

605. Law of the Magnetic Circuit. The idea of a mag- 
netic circuit in a vague form is more primitive than that of 
an electric circuit, for it appears to go back to the mathema- 
tician Euler in 1761. Later Joule asserted that the resist- 
ance to magnetic induction is proportional to the length of a 
closed magnetic circuit. Faraday made the very apt com- 
parison of an electromagnet with open magnetic circuit to a 
voltaic cell immersed in an electrolyte of poor conductivity. 
The low permeability of the air corresponds to the low con- 
ductivity of the electrolyte. Maxwell said, "In isotropic 
media the magnetic induction depends on the magnetic force 
in a manner which corresponds with that in which the elec- 
tric current depends on the electromotive force." 

The first definite expression of the law of the magnetic 
circuit in the form of an equation, like the equation express- 
ing Ohm's law, was given by Rowland in 1873; he says 
expressly that it "is similar to the law of Ohm." 

In 1883 Bosanquet introduced the term "magnetomotive 
force," corresponding to electromotive force in the electric 
circuit. We may then write 

Magnetic flux = Magnetomotive force, 
Magnetic reluctance 

Before attempting to write a more detailed equation for 
the magnetic circuit, it is necessary to introduce certain 




THE MAGNETIC CIRCUIT 539 

general propositions which furnish an expression for the 
magnetomotive force. 

606. Rotation of a Closed Circuit in a Magnetic Field. 
Conceive a current of I c.g.s. electromagnetic units flow- 
ing through the half 

circle abed (Fig. 379), 

and let there be a unit 

magnetic pole at the 

center P. Then the field 

produced at P by the 

current urges the pole in a direction normal to the plane of 

the ring ( 580). The circuit is urged by an equal force in 

the opposite direction by Newton's third law. 

Let be be an element of the curve. Then by the recipro- 
cal action between a magnet and a current, which has been 
experimentally demonstrated, the force / on the unit pole at 
P, due to the current / in 1 cm. of the circular conductor, 
is equal to I/r 2 . Hence, the work done in rotating any 
short arc be against this force through 360 about the axis 
ad is / x be x 2 7rr f . But this expression is / times the area 
of that portion of the spherical surface generated by be during 
the rotation. Therefore, the entire work done against the 
magnetic reaction between the current in the whole semi- 
circumference and the unit pole at the center, for one revo- 
lution, is the product of / and the numerical value of the 
surface of the sphere whose radius is r, or 

IF = /x47rr 2 = ^x47rr 2 =47r/. (108) 

Since 4 IT lines of force radiate from unit pole ( 465), 
and all of these are cut by the semicircle during one rota- 
tion around the axis ad, it follows that the work done is equal 
to the product of the whole number of lines cut by the con- 
ductor and the strength of the current flowing through it. 

607. The Electromotive Force Generated. Assume the rota- 
tion described in the last article to take place in a period 



540 MAGNETISM AND ELECTRICITY 

of t seconds, that the resistance of the conductor between 
the points a and d is R, and that E is the' applied potential 
difference between the same points. Then from the law of 
the conservation of energy, the whole work done is the sum 
of the energy spent in heating the conductor and the work 
done in rotating it in the magnetic field, or 

Elt = I*Rt + 4 TT/ 

as the energy equation. 

Therefore JE=IE+< 

I/ 

and /= j?-47rA (1Q9) 

R 

This is an expression for the current in the form of Ohm's 
law. It shows that there is generated by the rotation an 
E. M. F. equal to 4 ir/t. But this fraction is the rate at 
which the 4 TT lines of force from the unit pole are cut by 
t*he rotating conductor. The E. M. F. generated by a con- 
ductor cutting across lines of magnetic force is, therefore, equal 
to the rate at which they are cut. 

In estimating the number of lines cut attention must 
be given to the direction in which they are cut, and the 
algebraic sum must be taken in all cases. Use will be 
made of this principle in the next chapter on induced 
electromotive forces. 



608. Force at a Point due to a Current of Indefinite 
Length. Let ab (Fig. 380) be a portion of the straight 
conductor carrying a current of strength Ic.g.s. 
units, and let P be the point at a distance r from it. 
Then if a unit pole be at P and if the conductor be 
P 380 carried around it at the distance r, or the pole around 
the conductor at the same distance, all the lines of 
force from the pole will be cut once. The work done is 
4 7rJ. Also, if the field produced by the current at the point 



THE MAGNETIC CIRCUIT 541 

P is &6, the work done is the product of the magnetic force 
and the distance 2 TT/-, or 2 irrSS. Hence, 

2 irr&6 = 4 TT 7, 
or, 96=2I/r. 

If the current Jis in amperes, the force in dynes at the point 



is 



88 = 21/10 r. 




609. Force within a Helix. Let AB (Fig. 381) represent 
a section through the axis of a long helix, and let unit 
pole be at the point P. Let 
there be n turns of wire per rx^SOC^5XXTTTTTT~) 

centimeter parallel to the axis of A ^^yp'Vp' B 

the helix, each turn carrying a 

current of lo. g. s. units. Then 

if the unit pole be carried along Fig 38| 

the axis from P to P', a distance 

of one centimeter, each of the 4 TT lines from the pole will be 

cut by n turns of wire ; the whole number of lines cut will 

be 4 ?m, and the work done 4 TTH!. Since the distance moved 

is one centimeter, the work is numerically equal to the force, 

'(110) 



If the current I is in amperes, 

(Ill) 



This is the value of the field of force at points distant 
from the ends of the helix. If the helix or solenoid forms a 
closed curve, so that there are no ends to the helix, the field 
along the magnetic axis is everywhere the same. 

610. Magnetomotive Force. The reader will recall that the 
electromotive force in a circuit is equal to the work required 
to carry unit quantity of electricity entirely around the cir- 
cuit ( 535). So magnetomotive force is equal to the work 



542 



MAGNETISM AND ELECTRICITY 



done in carrying a unit pole once around the magnetic circuit. 
If L is the length of the solenoid, the work done is L times 
the strength of the field, or 4 TrnIL, if the field is uniform. 
Now nL is the entire number of turns of wire in the sole- 
noid. Denote this by N. Then the magnetomotive force 

(112) 



if the current is expressed in amperes. The quantity Nlis 
called the ampere turns. The magnetomotive force in a long 
solenoid is therefore 1.257 times the ampere turns. 

Similarly, the difference of magnetic potential between 
two points of a solenoid is 1.257 times the number of ampere 
turns between the same points. 

611. Reluctance. The magnetic reluctance of a bar of iron 
may be calculated from its length, its sectional area, and 
its permeability, just as the electrical resistance of a con- 
ductor may be calculated from its length, its cross sec- 
tion, and its conductivity. Let the length of the bar be I 
centimeters, its section S square centimeters, and its permea- 

bility fji. Then its reluctance is 

(113) 




For example, the reluctance of the 
magnetic circuit of the electromagnet of 
Figure 382 is made up of two parts, that 
of the core and that of the armature. 
Let the lengths, sections, and permea- 
bilities be denoted by ^ and Z 2 , ^ and 
$ 2 , and /^ and /* 2 respectively. Then 
the reluctance of the whole circuit is 









Fig. 382 



612. Law of the Magnetic Circuit Applied. When the mag- 
netic circuit is not closed, the lines of induction must be 



THE MAGNETIC CIRCUIT 543 

forced across the air gap between the faces of the iron 
parts of the circuit. Suppose the armature removed a short 
distance ? 3 from the poles (Fig. 383). Then the length 
of the circuit is thereby increased 21 8 cm., and additional 
reluctance is introduced equal to 2 Zg/jtfg, 
where S B is the cross section of the air 
traversed by the induction. The per- 
meability of the air is unity, and does 
not appear in the expression. 

We may therefore write for the flux 
of magnetic induction 




, T . , . Fig. 383 

where /is expressed in amperes. 

While this expression is simple in theory it is rendered 
difficult of application because /*, unlike conductivity, is 
not a constant, but is a function of the magnetization or 
induction in the iron. In applying the formula to any par- 
ticular magnetic circuit it is necessary to know the curve of 
magnetization or the quality of iron used, and to ascertain 
from it or from tables the values of p corresponding to the 
degree of saturation which it is desired to use. When this 
has been determined, the formula gives the number of 
ampere turns of excitation required. For open magnetic 
circuits an allowance must be made for leakage of lines of 
force through the air between parts of the magnet. This 
leakage requires excitation, but contributes nothing to the 
purpose for which the magnet is designed. The allowance 
for it must be estimated from experience with the particular 
form of magnet employed. The electromagnets of dynamos 
are designed by a process similar to this. 

613. Superficial Magnetism by Electric Discharges. Thin 
steel rods and sewing needles may be magnetized by passing 



544 



MAGNETISM AND ELECTRICITY 



an electric discharge around them, or even across them at 
right angles to their length. It has long been known that 
hard steel is sometimes magnetized by lightning. 

If a Leyden jar be discharged through a strip of tin foil 
across which lies a sewing needle, the needle will be mag- 
netized by the discharge. Better results will be obtained 
by surrounding the needle with an open helix of rubber- 
covered wire and discharging through it. It was with 
simple means like these that Joseph Henry discovered the 
oscillatory character of the Leyden jar discharge. 

Anomalous results have sometimes been observed in the 
relation of the poles to the direction of the discharge around 
the needles or rods, the poles being turned in the direction 
opposite to what the rule would lead one to expect. This 
result is due to the oscillatory discharge combined with the 
superficial character of the magnetism imparted. If small 
steel rods, magnetized by electric discharges, be examined 
by removing the external portions with acid, it will be found 
that the magnetized part is confined to a thin shell, the 
underlying parts remaining unmagnetized. If a second 

discharge succeeds the first in 
the opposite direction, it will 
reduce the external magnetism 
to zero if the magnetism of 
half the shell is reversed. Two 
shells of equal magnetic moment 
will then be superposed in oppo- 
site senses. If therefore the 
reverse discharge have more 
than half the magnetizing effect 
of the first, the resultant mag- 
netism will be apparently " anom- 
alous"; but it is accounted for 
by the direct and reverse discharges, and does not constitute 
an exception to the law of magnetization. 

Figure 384 contains the curves obtained from two glass- 





























/ 


\^ 




















V 






\ 


















/s, 






\ 




















\ 






\ 




















V 




\ 


B 


















\ 
























\ 
























A 


\ 




\ 




















\ 






\ 
























\ 




















V 






\ 


















\ 






\ 




















\ 






\ 




















* 












24 20 10 1$ 

Fig. 384 



THE MAGNETIC CIRCUIT 545 

hard steel rods, 6 cm. long and 1.8 mm. in diameter, mag- 
netized by ten successive discharges of a small Leyden jar, 
all in the same direction. The relation of the two magnet- 
izing coils was such that the first reverse oscillation was 
more powerful with B than with A. The data for these 
curves were obtained by removing successive portions of the 
outside with acid and measuring the magnetic moments 
after each removal. Moments are plotted as ordinates, and 
decreasing weights as abscissas. The moment of B at first 
increases to a maximum, and then decreases parallel to the 
A curve. B had a thin external shell magnetized in a sense 
opposite to that of the underlying portions. When this 
had all been removed, the magnetic moment was a maxi- 
mum. 

Problems 

1. An iron bar 50 cm. long and 3 cm. in diameter was magnetized 
to 15,780 lines per square centimeter, when //, equaled 800. Find the 
reluctance and the total induction through the bar. 

2. A ring of soft iron 20 cm. in diameter and 3 cm. 2 sectional area 
is wound uniformly with a magnetizing helix. Find the number of 
ampere turns required to magnetize to 13,640 lines per square centimeter, 
with p equal to 2200. What will be the total induction ? 

3. A straight wire carries a current of 10 amperes. Find the force in 
dynes on a pole of strength 20 at a distance of 5 cm. from the wire. 

4. Find the strength of the magnetic field 8 cm. from the center of 
a coil of one turn in the line of its axis if the coil is 12 cm. in diameter 
and carries 0.5 ampere. 

5. Two straight insulated conductors, indefinitely long, cross each 
other at right angles ; one carries a current of 50 amperes and the other 
100. Find the force on unit pole in their plane, at a distance of 6 cm. 
from the former and 8 cm. from the latter. 

6. What current in amperes through a straight wire of indefinite length 
will produce at a distance of 10 cm. a field equal in dynes to the weight 
of 10 mg. 



CHAPTER XX 

ELECTROMAGNETIC INDUCTION 

I. FARADAY'S EXPERIMENTS 

614. Faraday's Discovery. The discovery made by Oer- 
sted in 1819 led speedily to the discovery of magnetization 
by electric currents, and to the mechanical action between 
conductors conveying them. Faraday completed this cor- 
related group of electromagnetic phenomena by discover- 
ing, in 1831, the laws of electromagnetic induction, that 
is, the laws of the production of induced currents by means 
of other currents or by magnets. These discoveries are 
of very great interest and importance, for all modern 
methods of generating large electromotive forces by dy- 
namo machines, arid all induction coils and alternate current 
transformers, are based on the principles of electromagnetic 
induction. 

Induced electromotive forces and currents are those pro- 
duced by the action of magnets and other currents. Strictly 
only electromotive forces are induced ; induced currents flow 
as a consequence when the circuit in which the electromotive 
force is generated is closed. But electromotive forces or 
potential differences may be induced just the same when the 
circuit is open. 

615. Induction by a Magnet. Assume a coil of many turns 
of insulated wire connected in circuit with a sensitive gal- 
vanometer (Fig. 385), and that the north pole of a bar 
magnet is quickly thrust into the coil. The galvanometer 
will indicate a transient current, which flows only during 

546 



FARADAY'S EXPERIMENTS 



547 




Fig. 385 



the motion of the magnet. If the magnet be suddenly with- 
drawn from the coil, a transient current will flow in the 
reverse direction. If the 
south pole of the magnet be 
thrust into the coil and then 
withdrawn, the currents in 
both cases will be the reverse 
of those with the north pole. 
The momentary electromo- 
tive forces generated in the 
helix are known as induced 
electromotive forces, and the 
currents as induced currents. 
The magnet carries with it 
into the coil its lines of in- 
duction ; and when the rela- 
tive position of a magnet 
and a coil are so altered 
that a variation is produced in the induction linked with 
the coil, an induced electromotive force is generated in it. 

If a coil of fine wire be wound around the armature of 
a permanent magnet (Fig. 386), when the armature is in 
contact with the poles the flux of induction through the coil 
is a maximum. When it is pulled away, the magnetic flux 
through the armature and the coil decreases rapidly, and an 

E. M. F. is generated. This 
experiment illustrates Fara- 
day's method of producing 
electric currents by the aid 
of magnetism. 

When the armature is in 
contact with the magnet, and one or more fine iron wires 
are dropped so as to join the poles, the galvanometer, if 
sufficiently sensitive, will show an induced E. M. F 1 . The 
iron wires divert magnetic flux from the armature by form- 
ing a divided magnetic circuit, and the sudden decrease of 




Fig. 386 



548 



MAGNETISM AND ELECTRICITY 



induction through the armature generates an E. M. F. in 
the coil surrounding it. 

616. Direction and Value of an Induced Electromotive Force. 
The direction of the induced E. M. F. Faraday determined 
by experiment, but it can be deduced from considerations 
with which we are already familiar. 

Suppose a magnet NS thrust into a helix (Fig. 387). Since 
an E. M. F. is generated and a current circulates through the 

coil, the energy of the 
current must be derived 
from the work done 
in moving the magnet. 
-" There must therefore be 
resistance opposing the 
"* movement ; this resist- 
ance is magnetic. It is 
due to the helix consid- 
ered as a magnetic shell, 
and the current must 
flow around it in a direc- 
tion to make a north pole of the side which the north pole of 
the bar magnet enters. Its direction is therefore against 
watch hands, as indicated by the arrows. If the observer 
looks along the positive direction of the lines of induction, a 
current flowing clockwise is said to be direct; if counter- 
clockwise, it is indirect. We have therefore the following 
law relating to the direction of the induced electromotive 
force : 

An increase' in the number of lines of induction threading 
through a helix produces an indirect electromotive force ; a 
decrease in the number of lines produces a direct electromo- 
tive force. 

The numerical value of an induced electromotive force in 
e.g. s. units may be expressed as follows : 




Fig. 387 



FARADAY'S EXPERIMENTS 



549 



The electromotive force induced is equal to the rate of 
change in the number of lines of induction threading 
through the circuit. 

If d3> is the change in the magnetic flux through the 
circuit taking place in the short time dt, the induced 
E. M. F. is 



The minus sign indicates that a direct electromotive force 
corresponds to a decrease in the flux of induction. 

617. Induction by Currents. Since a current through a 
solenoid produces a magnetic field equivalent to that of 
a magnet, the same induction 
effects will be produced by in- 
serting a helix conveying a cur- 
rent into the long coil (Fig. 
388) as by thrusting in the 
equivalent magnet. Let the 
circuit P include a battery and 
a key, and the circuit S a galva- 
nometer. The former is called 
the primary and the latter the 
secondary. 

If the current through P is 
kept constant while the coil is 
moved about, when P approaches 
S, an E. M. F. is generated in S 
tending to send a current in 
the opposite direction to that 
around P ; while if P is moved 
away from $, the E. M. F. induced in S is in the direction of 
the current around P. These electromotive forces in S act 
only so long as P is moving. If P is kept fixed while 8 is 
moved, the results are the same. 

Next, let P be in a fixed position near S with the key 
open. Then on closing the key in P the galvanometer 




Fig. 388 



550 MAGNETISM AND ELECTRICITY 

needle will be deflected. This deflection is not a permanent 
one, but the needle oscillates and finally returns to its initial 
position of rest, indicating the passage of a sudden discharge 
through the galvanometer. The direction of this momen- 
tary current is opposite to that through P. On opening 
the key another similar momentary current passes through 
S, but in the same direction as through P. Thus the start- 
ing or stopping of a current in P is accompanied by the 
induction of another current in a neighboring circuit S. 

The sudden increase of the current in P produces an 
opposite current in S, and the sudden decrease of the cur- 
rent in P produces a current through S^XL the same direction 
as through P. 

If when P remains inside of $, or coaxial with it, a bar of 
soft iron is placed within it, there is an increase of magnetic 
flux through both P and S, and the E. M. F. generated in S 
is in the same direction as that produced by closing the key 
in P, moving P toward $, or increasing the current through 
P. The withdrawal of the iron produces the opposite effects 
to its insertion in the coil. 

The law of the direction and magnitude of the E. M. F. 
generated inductively by another current is the same as that 
given in the last article. When the magnetic flux changes, 
an E. M. F. is produced equal to the rate of change in the 
magnetic flux passing through the circuit. The positive 
direction of the E. M. F. and of the flux through the circuit 
are related to each other as are the rotation and the translation 
of a right-handed screw. 

618. Faraday's Ring. Faraday wound upon an iron ring 
two coils of wire P and (Fig. 389). When a battery 
and a key were included in the circuit P and a galvanometer 
in S, whenever the circuit of P was closed or opened a 
momentary current was produced in the closed circuit S. 
In this experiment the iron is the medium through which 
the induction between P and S takes place. The current 




FARADAY'S EXPERIMENTS 551 

through P magnetizes the iron ring as a closed magnetic 
circuit. The starting of the current in the circuit P sends 
magnetic lines through S and 
produces in it an inverse cur- 
rent ; the stopping of the primary 
current withdraws lines and pro- 
duces a direct current through 
the secondary. A larger deflec- 
tion of the galvanometer will be Fj 389 
produced by the first closing of 

the primary circuit than by opening it, or by closing it a 
second time unless the current is reversed. The reason is 
that the ring forms a closed magnetic circuit, and its reten- 
tivity or remanence is so great that only a small part of the 
lines of induction drop out when the magnetizing current 
ceases to flow. But if the current through the primary be 
reversed, all the lines will be taken out and will be put in 
again the other way round. Hence, a large induction will 
take place in S. A closed magnetic circuit is not well 
adapted, therefore, to produce induction effects by merely 
opening and closing the primary circuit. 

The relation between P and S is a mutual one. If S is made 
the primary, induced electromotive force will be generated in 
P as the secondary. The Faraday ring with its two coils is 
the type of the modern transformer for alternating currents. 

619. An Inductive System a Conservative System. It is in- 
structive to consider a system of two circuits, or of a cir- 
cuit and a magnet, as a conservative system. The action 
between the parts of the system always tends to maintain 
unchanged the number of lines of induction threading 
through the circuits. Thus, in Figure 387 the approach of 
the magnet to the coil increases the magnetic flux through 
the coil, and the induced current is in a direction to send a 
counter flux through it so as to keep the flux linked with 
the coil a constant. In Figure 389 the primary current pro- 



552 MAGNETISM AND ELECTRICITY 

duces magnetic flux in the ring, and the current induced in 
the secondary produces magnetic flux in the other direction 
around the ring ; that is, the induced current opposes any 
change in the flux. After the primary current has produced 
a steady magnetic flux through the iron, the opening of the 
primary circuit induces a secondary current in the same 
direction around the ring as the primary, and this tends to 
maintain the flux of induction in the ring unchanged. 

The same principle may be applied to two coils without 
iron. There is no exception to the law that induced currents 
are always in a direction to conserve the status of the magnetic 
flux through the circuit in which the induction takes place. 
This law means that the magnetic flux through a circuit 
does not change abruptly a property of magnetic induction 
analogous to inertia in matter. 

620. Lenz's Law. When induced currents are produced 
by the motion of a conductor in a magnetic field, the circuit 
is acted on by a mechanical force. Lenz's law is that the 
direction of this force is such that the force opposes the mo- 
tion which induces the current. Lenz's law is a particular 
case of the principle of the conservation of magnetic flux. 
Every action on an electromagnetic system which involves a 
transformation of energy sets up reactions tending to preserve 
unchanged the state of the system. 

An example of Lenz's law is afforded by a coil revolving 
in a magnetic field. The mechanical action of the field on 
the current induced in the coil produces a couple tending to 
stop the rotation. The oscillations of the coil of a D'Arsonval 
galvanometer subside quickly when the coil is short-circuited. 
The galvanometer is then a magneto-electric machine, and 
the currents induced in its closed coil bring it to rest. 

Broadly stated, Lenz's law is as follows: 

The direction of an induced current is always such that it 
produces a magnetic field opposing the motion or change 
which induces the current. 




FARADAY'S EXPERIMENTS 553 

621. Arago's Rotations. When a magnet is suspended 
horizontally over a copper disk and the disk is rotated, 
induced currents are produced in it. These give rise to a 
force opposing the rotation. Since the force between the 
magnet and the disk is a mutual one, a couple acts on the 
magnet and turns it, if it is free to move, in the same direc- 
tion as the disk. Or if the magnet is 
spun around a vertical axis and the 
disk is movable, it is dragged after 
the magnet. These motions are called 
Arago's rotations ; they were discovered 
by Arago, but were first explained by 
Faraday. Induced currents flow in 
closed circuits through the disk, and 
the action between them and the mag- 
net tends to stop the disk; or if the magnet oscillates, the 
induced currents damp its motion. Thus in Figure 390, if 
the needle ab oscillates over the disk, as it moves in the 
direction of the arrows, a current is induced on the M side 
which repels the needle, and one on the N side 
attracting it ; or the current under it flows from 
the center toward the circumference if a is an 
N-seeking pole. * 

622. Other Examples of Lenz's Law. Let a copper 
cube or cylinder be suspended between the pointed poles of a 
powerful electromagnet (Fig. 391). The cube may be set 
rotating by twisting the thread and releasing it. When the 
electromagnet is excited the cube is instantly brought to 
rest ; it begins to spin as soon as the magnetizing current is 
cut off, and is again arrested when the circuit is closed. This 
resistance to motion in a magnetic field is sometimes called 
Fjg 39| magnetic friction. 

In another experiment a disk of copper is made to rotate 
rapidly between the poles of an electromagnet (Fig. 392). When the 
magnet is excited, the disk appears to meet with a sudden resistance. 
Foucault found that if it is forced to rotate, it is heated by the induced 
currents flowing in it. These induced currents in masses of metal are 




554 MAGNETISM AND ELECTRICITY 

often called Foucault currents. There is a pair of eddy currents in the 
part of the disk passing the poles ; and these currents, as in Arago's 

rotation, hold the disk back. 

The drag due to eddy currents is propor- 
tional to the speed and to the square of the 
magnetic field ; for the force is proportional 
to the field and the current, and the current 
is proportional to the field and the speed. 
When the field is constant, the force is 
therefore proportional to the speed of 
rotation. 

The principle is employed to produce 
Fj 392 damping in rotatory meters. A copper disk, 

attached to the shaft to which is connected 

a dial train, rotates between the poles of fixed magnets. The drag on 
the copper disk keeps the speed proportional to the torque. 

II. SELF-INDUCTION 

623. Self-induction. Joseph Henry discovered that a 
current through a helix with parallel turns acts inductively 
on its own circuit, producing what has long been known as 
the extra current, and a bright spark across the gap when the 
circuit is opened. The effects are not very marked unless 
the helix contains an iron core. 

Even a single circuit is a conservative system as regards 
the magnetic flux through it. When the current magnetizes 
the core, the effect is the same as if a magnet had been 
plunged into the helix; that is, the induced E. M. F. is a 
counter E. M. F. tending to prevent the flux of magnetic 
induction through the circuit. The result is that the current 
in such a circuit does not reach its maximum value abruptly, 
but only after a short interval depending on the value of the 
coefficient of self-induction, or simply the inductance. On the 
other hand, when the circuit is opened the induced E. M. F. 
is direct and tends to prolong the current, or to resist the 
diminution in the magnetic flux. 

The unit of inductance, the henry, is the inductance in the 
circuit when the E. M. F. induced in this circuit is one volt, 



SELF-INDUCTION 555 

while the inducing current varies at the rate of one ampere 
per second. 

624. Growth of Current in Inductive Circuits. When a con- 
stant electromotive force is impressed on a circuit having 
self-inductance, the current does not attain its permanent 
value instantly. During the variable stage its value is not 
given by the simple application of Ohm's law; the inductance 
is another property of the circuit, in addition to its resistance, 
which determines the instantaneous value of the current. 
When the circuit is closed the self-induced electromotive 
force opposes the applied electromotive force and retards the 
growth of the current to its full value. When the circuit is 
opened, the self-induced electromotive force prolongs the 
current, or acts to retard its decrease to zero. 

Figure 393 is reproduced from a graph made by the current itself by 
means of an instrument called an oscillograph.* The oscillograph is in 
principle like a D'Arsonval galva- 
nometer but with an extremely 
short period of oscillation, so that 
the moving system is able to follow 
all the changes of the current. A 



reflected beam of sunlight is re- Fig 393 

ceived on a falling photographic 

plate and thus records the motion of the mirror. The vertical line is 
the trace of the beam immediately after closing a n on inductive circuit ; 
the other line is the trace obtained with a parallel inductive circuit. The 
opposing electromotive force of self-induction is greatest at the instant 
the circuit is closed ; as it dies away, the effective electromotive force 
increases, and the current rises to the value given by Ohm's law. 

625. Energy stored in a Magnetic Field. The inductance 
is the property of a circuit by virtue of which the passage of 
a current is accompanied by the absorption of energy in the 
form of a magnetic field. If no other work is done, part of 
the energy flowing from the source is converted into heat, 

*My thanks are due to Professor Benjamin F. Thomas for his beautiful oscillo- 
grams from which this illustration and a few subsequent ones were made. 

H.S.C. 



556 



MAGNETISM AND ELECTRICITY 



and the rest is stored in the ether as the potential energy of 
the field. This storage of energy goes on while the current 
is rising from nothing to its steady value. The work repre- 
sented by this energy is done by the current against the 
electromotive force of self-induction. 

The storage of energy may be strikingly illustrated as follows : 
M (Fig. 394) is a large electromagnet, B a storage battery, L an incan- 
descent lamp of a normal voltage equal to 
that of the battery, and K a circuit breaker. 
The circuit is divided between the electro- 
magnet and the lamp ; and since the former 
is of low resistance, when the current reaches 
its steady state most of it will go through the 
coils of the magnet, leaving the lamp at only 
a red glow. At the instant when the circuit 
is closed, the self-induction of the magnet 
acts against the current, like a large resist- 
ance, and sends most of it around through 
the lamp. It accordingly lights up at first, 
but quickly grows dim as the current rises to 
its steady value through M. 

On breaking the circuit and cutting off 
the battery, the lamp flashes up brightly. 
The lamp and the electromagnet are then together on a closed circuit. 
The energy stored in the magnetic field, as a strain in he ether about 
the magnet, is converted into electric energy, and a reverse current 
through the lamp lights it momentarily. 









M 






1 

























4 


1 


1 


B 
1 


1 1 



K 

J 



Fig. 394 



III. THE INDUCTION COIL 

626. The Induction Coil. An induction coil is commonly 
employed to obtain transient flashes of high electromotive 
force in rapid succession. In modern terms it is a step-up 
transformer with open magnetic circuit. About an iron 
core, consisting of a bundle of fine iron wires to avoid in- 
duced or eddy currents in the metal of the core, is wound 
a primary coil of comparatively few turns of stout wire ; 
outside of this, and carefully insulated from it, is the second- 
ary of a very large number of turns of fine wire. In Spottis- 



THE INDUCTION COIL 



557 



wood's great coil, which gave 42J-inch sparks, the secondary 
contained 280 miles of wire wound in 340,000 turns. 

The primary must be provided with a circuit breaker 
(Fig. 395), if the .coil is to be used with direct interrupted 
currents. It is com- 
monly made automatic 
by a vibrating device 
actuated by the core and 
similar to that of a vi- 
brating electric bell. 

In large coils the sec- 
ondary is wound in flat 
spirals, and these are 
slipped on over the primary and separated from one another 
by insulating rings. The difference of potential between 
adjacent turns of wire is then not so large as when the entire 
coil is wound in layers from end to end, and it is easier to 
maintain the insulation. The ratio of the transformation of 
the electromotive force is nearly the same as the ratio between 
the number of turns of wire on the primary and the secondary. 




Fig. 395 




Fig. 396 



627. Action of the Coil. The essential parts of an induc- 
tion coil are shown in Figure 396. The current from the 
battery passes through the heavy primary wire PP, thence 
through the spring A, which carries the soft iron block F, 



558 MAGNETISM AND ELECTRICITY 

then across to the platinum-tipped screw 6, and so back to 
the negative pole of the battery. The attraction of F by 
the magnetized core breaks the primary circuit at 5; the 
core is then demagnetized, and the release of F again closes 
the circuit. Electromotive forces are induced in the second- 
ary coil SS, both at the make and break of the primary. 

The self-induction of the primary has an important bearing 
on the action of the coil. At the instant the circuit is closed, 
the counter E. M. F. opposes the battery current, and pro- 
longs the time of reaching its greatest strength. Conse- 
quently the E. M. F. of the secondary is diminished by the 
self-induction of the primary. The E.M.F. of self-induction 
at the break of the primary is direct, and this added to the 
E. M. F. of the battery produces a spark at the break points. 

628. The Condenser. The addition of a condenser in- 
creases the E. M. F. of the secondary coil in two ways : 
First. It gives such an increase of capacity to the primary 
coil that at the moment of breaking the circuit the potential 
difference between the contact points at the break does not 
rise high enough to cause a spark across the opening. The 
interruption of the primary is therefore more abrupt, and 
the E. M. F. of the secondary is increased. Second. After 
the break, the condenser C, which has been charged by the 
E. M. F. of self-induction, discharges back through the pri- 
mary coil and the battery. The condenser causes an electric 
recoil in the current, and returns the stored charge as a 
current in the reverse direction through the primary, thus 
demagnetizing the core, increasing the rate of change of 
magnetic flux, and increasing the induced E. M. F. in the 
secondary. The condenser momentarily stores the energy 
represented by the spark which occurs without the condenser, 
and then returns it to the primary and by mutual inductance 
to the secondary, as indicated by the longer spark or the 
greater current. When the secondary terminals are sepa- 
rated, the discharge is all in one direction and occurs when 



THE TELEPHONE 



559 



the primary current is interrupted. With a suitable con- 
denser the conditions are those described by the word 
resonance. The current through the primary is rendered 
oscillatory by means of the condenser. 

629. The Tesla Induction Coil. Tesla's device to produce 
electric discharges of very high frequency employs induc- 
tive processes in two stages ; in the second stage the oscil- 
latory discharge of a Leyden jar ( 521) serves as the 
interrupter. The terminals of the secondary of an induction 
coil Q (Fig. 397) are connected to the inner and the outer 
coating respectively of a 
Leyden jar J. The dis- 
charge circuit of the jar 
is through the primary 
winding of the Tesla coil 
and the discharge balls 1$^. 
The primary A of the 
Tesla coil consists of a 
few turns of heavy wire 
without a magnetic core. 
The secondary B has a 
much larger number of 
turns and it is separated 

from the primary by air or oil insulation. The oscillations 
of the Leyden jar discharges at S 1 may have a frequency of 
several millions per second. The discharges at $ 2 from the 
secondary of the Tesla coil are not only of very high fre- 
quency, but they are of such high E. M. F. that they produce 
auroral displays of astonishing brilliancy and marked induc- 
tion effects at a distance of several feet. 




Fig. 397 



IV. THE TELEPHONE 

630. The Telephone. The telephone was invented by Gra- 
ham Bell and Elisha Gray in 1876. It consists of a per- 
manent magnet (Fig. 398), one end of which is surrounded 



560 



MAGNETISM AND ELECTRICITY 




Fig. 398 



by a coil of many turns of fine insulated copper wire with 
its ends connected to the binding posts t and t. At a dis- 
tance of a millimeter or less from the pole of 
the magnet on which the coil is wound is an 
elastic iron diaphragm or disk a; it is kept 
in place by the conical mouthpiece d. 

Sounds may be transmitted to a distance by 
the use of two telephones, using one as a 
transmitter and the other as a receiver. The 
two wire coils are connected in series by the 
line wire between the two stations. 

The diaphragm is magnetized by induction, 
and this induced magnetism reacts on the per- 
manent magnet 0, the amount of the reaction 
depending on the distance between the diaphragm and the 
surface of the pole. When words are spoken into the mouth- 
piece, the vibrations of the air cause the diaphragm to vibrate 
in unison ; and the to-and-fro motion of the disk causes the 
induction between it and the magnet to vary in unison with 
the vibrations of the air. The variation in the number of 
lines of induction through the coil b produces a correspond- 
ing series of induced currents through the circuit in which 
the coil is connected. 

When these varying induced currents, alternating in direc- 
tion, traverse the coil of the receiving instrument, the mag- 
netic field due to the coil, combined with that due to the 
magnet, alter intermittently the force between the magnet 
pole and the disk ; the disk is therefore set in vibration in 
such a way as to reproduce the vibrations in the diaphragm 
of the transmitting instrument. The vibrations of the dia- 
phragm of the receiving instrument are communicated to the 
air in contact with it, and thus the sounds made near the 
transmitter are reproduced by the diaphragm of the receiver. 

631. The Microphone. The currents produced by induction 
in a telephone as a transmitter are very feeble and serve 



THE TELEPHONE 



561 



to transmit the voice over short distances only. The vari- 
able resistance transmitter, discovered by Berliner in 1877 
and independently by Hughes in 1878, vastly improved the 
efficiency of transmission by causing the vibrations of the 
diaphragm to vary the resistance of a battery circuit by 
means of one or more contacts, the resistance of which varies 
with the pressure. The first recorded and definite transmis- 
sion of speech was in fact accomplished by the variable re- 
sistance transmitter of Elisha Gray as early as 1876. It 
employed the simple device of a fine platinum wire soldered to 
the center of a diaphragm and dipping a little way into an 
acid solution. The motion of the diaphragm varied the 
depth of immersion of the wire and so the resistance of trans- 
mission by the electrolyte. 

One form of the microphone, invented by the English elec- 
trician Hughes, consists of a rod of gas carbon A (Fig. 399), 
pointed at both ends and 
resting lightly in conical 
hollows made in blocks 
CO of the same material 
attached to a sounding 
board. These blocks are 
placed in circuit with a 
battery and a telephone 
receiver by means of the 
wires X and T. The least 
disturbance of the sounding 
board, such as that caused by the ticking of a watch lying 
on the base D, disturbs the rod and the pressure with which 
it rests against the upper carbon block. Since the resistance 
of a carbon contact varies with the pressure, the disturbance 
of A causes variations in the resistance of the circuit, and 
therefore a continuously varying current. The receiver 
responds to the varying current ; and so great is the sensi- 
tiveness of the instrument that a fly may be heard walking 
on the sounding board. 




Fig. 399 



562 



MAGNETISM AND ELECTRICITY 



632. The Solid Back Transmitter. Instead of the loose carbon con- 
tact of the early microphone, carbon in granules between carbon plates 
is now extensively used. The form of transmitter employed for long 
distance transmission is the "solid back" transmitter (Fig. 400). The 

figure shows only the 
essential parts in sec- 
tion, with minor de- 
tails omitted. M is the 
mouthpiece, and F and 
C the front and back 
parts of the metal case. 
The aluminum dia- 
phragm D is held 
around its circumfer- 
ence by a soft rubber 
ring. The block W 
has a recess in front 
to receive the carbon 
electrodes A and B. 
Between them are the 
carbon granules. The 
block E is attached to 
the diaphragm and is 
insulated from W ex- 
F, g . 400 cept through the car- 

. bon granules. The 

transmitter is placed in circuit by wires connected to W and E. 

Provision is made for an elastic motion of the diaphragm and block 
E. Sound waves striking the diaphragm cause a varying pressure be- 
tween the plates and the carbon granules; this varying pressure varies 
the resistance offered by the granules and so varies the current. The 
transmitter is in circuit in the line with the primary of a small induction 
coil, the secondary being in a local circuit with the telephone receiver. 
The induced currents in the secondary have all the modifications of the 
primary current ; and when they pass through a receiver, it responds and 
reproduces sound waves similar to those which disturb the diaphragm of 
the transmitter. 




CHAPTER XXI 



DYNAMO-ELECTRIC MACHINES 
I. DIRECT CURRENT MACHINES 

633. Ideal Simple Dynamo. A dynamo is a machine for 
converting mechanical energy into the energy of an electric 
current* It is a generator of electromotive force, and is 
based on the principles of electromagnetic induction discov- 
ered by Faraday. It consists of a system of conductors, 
called an armature, revolving in a magnetic field in such a 
manner as to vary continuously the magnetic flux through 
them. 

Suppose a single loop of wire revolving in a uniform mag- 
netic field between the poles N S of a magnet (Fig. 401) 
around a horizontal axis 

.^ *. 

in the direction of the 
arrow. The loop of wire 
in the position shown in 
the figure incloses the max- 
imum magnetic flux <l>. 
When it has revolved 
through an angle 0, the 
flux through it will be reduced to <3> cos 6 ; for the projec- 
tion of the loop on a plane perpendicular to the field varies 
as the cosine of the angle of displacement from that plane. 
After a quarter turn the loop does not inclose any lines 
of induction ; as it revolves further they thread through in 
the opposite direction, and this is equivalent to a continued 
diminution of the magnetic flux through the loop. During 

663 



5 


E^E 


K 

Jv-A 




N 

s 


rf 


P 


N 






w^ 




Vv 





Fig. 401 



564 



MAGNETISM AND ELECTRICITY 



the second half of the revolution the opposite changes take 
place ; when the loop has revolved through 360 it returns 
to its initial relation to the magnetic field. 

The magnetic flux through the loop varies therefore as the 
cosine of the angle of displacement 0. During the first half 
revolution a current flows around the loop in the direction 
of the arrows; during the second half it is in the reverse 
direction. Thus the E. M. F. changes sign twice every 
revolution. Such a loop, or coil composed of a number of 
parallel turns, generates an alternating electromotive force. 

634. Law of the Electromotive Force. The induced electro- 
motive force is equal to the rate of change of the mag- 
netic flux through the circuit ( 616). But the flux inclosed 
by the loop varies as the cosine of the angle defining the 
position of the loop; and when the flux is a maximum, its 
rate of change is a minimum and conversely. Hence when 6 
is zero or 180, the E. M. F. generated is zero ; while for the 
positions 90 and 270 the E. M. F. is a maximum. The 
trigonometrical function that is related in this way to the 
cosine is the sine. Hence the law of variation of the electro- 
motive force generated by the revolution of the loop in a 
uniform magnetic field is the same as the variation in the 
value of the sine of the angle of position. 

If, therefore, we plot uniform distances along a straight 
line to represent equal increments of 0, and erect perpendicu- 



II 





Fig. 402 N 

lars to denote the values of the corresponding sines of 0, the 
curve connecting the extremities of these ordinates will be a 
sine curve. In Figure 402 the heavy line I is the cosine curve, 




DIRECT CURRENT MACHINES 565 

representing the law of change in the magnetic flux through 
the loop ; the light line II is the sine curve, whose ordinates 
denote the rate of change of the flux, or the E. M. F. Their 
maximum values differ in phase by 90, or a quarter of a 
period. When the magnetic flux decreases through its zero 
value at J5, its rate of change is greatest and the electromo- 
tive force is a maximum and positive. 

635. The Commutator. The current in a single loop, or 
coil composed of several parallel turns of wire, has alternately 
equal values in opposite directions 

through the loop. To make it unidirec- 
tional in the external circuit a two-part 
commutator must be used. The two 
halves of a split tube (Fig. 403), in- 
sulated from each other and mounted 
on the shaft of the armature, are con- 
nected to the two terminals of the rotat- 
ing coil. The brushes leading to the external circuit are so 
placed that they exchange contact with the two commutator 
segments in passing through the positions where the current 
changes its direction in the coil. The pulses are then all in 
one direction in the external circuit. Alternate loops of the 
sine curve are thus reversed, so that all of them lie on the 
same side of the zero line. One of the brushes is then always 
positive and the other negative. 

When the armature consists of two loops or coils placed at 
right angles to each other, the commutator has four segments, 
and the frequency of the pulses in the external circuit is 
doubled. 

636. The Gramme Bing. The use of a commutator with 
more than two segments is conveniently illustrated in con- 
nection with the Crramme Ring (Fig. 404). The armature 
has a core made either of iron wire or of thin disks at right 
angles to the axis of rotation. The iron is divided for the 
purpose of preventing eddy currents in it, which heat the 




566 MAGNETISM AND ELECTRICITY 

machine and waste energy. A number of coils are wound 
in one direction and are all joined in series in a closed circuit. 

Each junction between coils is 
joined to a commutator bar. 
When a coil is in the highest 
position in the figure, the maxi- 
mum flux passes through it; 
as the ring rotates, the flux 
through the coil decreases, and 
after a quarter of a revolution 

Fig. 404 

there is no flux through it. 

The current through each coil reverses twice during each 
revolution, exactly as in the case of a single loop, when the 
magnetic field has only two poles. No current flows entirely 
around the armature, because the E. M. F. generated in one 
coil at any instant is exactly counterbalanced by the E. M. F. 
generated in the coil opposite. 

When the external circuit connecting the brushes is closed 
a current flows in opposite directions through the two halves 
of the armature. The current through the armature has, in 
other words, two paths, and one brush is constantly positive 
and the other negative. 

By increasing the number of coils, the potential difference 
between the brashes never drops to zero, as it does with a 
single coil twice every revolution, but it is kept nearly con- 
stant and at the highest value given by half the coils in 
series. The brushes must bear on the commutator near the 
part of the field where the E. M. F. in any coil passes through 
zero value and reverses. 

637. The Field Magnet. In direct current machines the 
magnetic field in which the armature rotates is produced by a 
large electromagnet excited by a current from the armature. 
After this field magnet has once been magnetized, the resid- 
ual magnetism is sufficient to start the induced current, which 
is carried wholly or in part around the field magnet coils. 



DIRECT CURRENT MACHINES 



567 



When the entire current passes through the coils of the field 
magnet, the dynamo is said to be series^wound. When the 
field magnet is excited by coils of many turns of wire con- 
nected as a shunt to the external circuit, the dynamo is said 
to be shunt-wound. The two methods of connecting the field 
magnet with the armature are shown in Figure 405. The 





Fig. 405 

series-wound machine is adapted to furnish a constant current 
with varying potential only; the shunt- wound machine is 
designed for a constant potential with varying current. In 
the latter when the current changes as a result of a change 
in the external resistance, the excitation of the field magnet 
remains nearly the same and the E. M. F. generated is nearly 
constant. 

A compound-wound dynamo consists of a combination of 
shunt and series coils on the field magnet. It is designed to 
maintain the potential difference between the brushes more 
nearly constant than is possible with a shunt machine. The 
larger the current through the armature, the larger the loss 
of potential in the armature itself, the smaller the potential 
difference between the brushes, and the smaller the excitation 
of the field magnet. By carrying the whole current around 
the field magnet in a series coil of a few turns, the increased 
excitation thus produced compensates for the loss of potential 



568 



MAGNETISM AND ELECTRICITY 



in the armature and maintains a constant potential difference 
between the brushes. If the armature were without resist- 
ance, compounding would not be necessary except for the 
demagnetizing effect of the armature considered as an electro- 
magnet. 

638. The Drum Armature. The modern drum armature 
for direct currents is an evolution from a single coil shuttle 
armature. It is shown at A (Fig. 406), which represents the 
parts of a four-pole machine. 

In the drum armature the iron core, which is made up of 
laminated disks, contains a series of grooves, parallel to the 




Fig. 406 

shaft, and coils wound in them at equal angular distances 
around the circumference. These sections of the armature 
may all be joined in series, and the junctions between them 
are then connected with the commutator bars (7, as in the 
Gramme ring. The machine of Figure 406 has four poles, 
four parallel circuits through the armature, and four brushes. 
Two of the brushes are positive and two negative ; the two 
positives are connected in parallel as the positive terminal, 
and the two negatives as the negative terminal. With such 



DIRECT CURRENT MACHINES 



569 




a winding, divided into numerous sections with corresponding 
commutator bars, the current in the external circuit is almost 
perfectly continuous and free from small fluctuations. 

639. Reactions in the Field of a Generator and a Motor. An 
electric motor for direct currents is constructed in the 
same manner as a generator. The study of a magnetic field 
through which a cur- 
rent is passing throws 
much light on the in- 
teractions between the 
field and the armature 
current. Figure 407 is 
a reproduction of a 
photograph of the field 
between unlike poles 
distorted by a current 
through a loop of wire 

i i ,1 i Fig. 407 

which came up through 

one hole and went down through the other. The lines of 
induction, outlined by means of fine iron filings, are so dis- 
torted that some of them thread through the loop. 

If we conceive this loop to rotate clockwise around an 
axis perpendicular to the plane of the paper, it is clear that 
mechanical force must be applied to keep up the motion, 
because the tension along the lines of induction holds the 
loop back. The armature then turns against the magnetic 
forces or torque of the field acting on it, and operates as a 
generator, and the field of the machine is distorted in the 
direction of the rotation. 

If, on the contrary, this loop of wire be conceived to rotate 
as an armature in the direction of the magnetic stress on it, 
then with the current in the same direction the field remains 
the same, and the armature reverses its direction of rotation. 

When the machine is used as a generator mechanical power 
is converted into electrical energy, because the rotation of 



570 MAGNETISM AND ELECTRICITY 

the armature is maintained against the internal magnetic 
reactions in the field. Work is then done on the machine 
as a generator. On the other hand, when it is used as a 
motor, electrical energy is converted into mechanical work, 
because the rotation takes place in the direction of the mag- 
netic effort or torque between the field and the armature. 
Work is then done by the machine as a motor. 

640. Direction of Rotation as a Motor. A series machine when 
used as a motor runs in the opposite direction to its motion as a gen- 
erator. It will rotate in the same direction whether the current goes 
through it in one direction or the other, since it is reversed through the 
armature when it is reversed through the field. 

A shunt machine runs in the same direction as a motor and as a 
generator. If in the shunt machine of Figure 405 the current from an 
external source enters the lower brush, as in the figure, its direction 
through the armature remains unchanged ; through the field coils, how- 
ever, it goes in the opposite direction to the arrows, for the armature and 
the field are now in parallel with reference to the external circuit or 
source of current. When used as a generator, the external circuit and 
the field are in parallel with respect to the armature as the source of the 
electric pressure. The field is therefore reversed. But as a motor the 
machine runs with the magnetic torque, and as a generator against it ; so 
that running with the torque when the field only is reversed is the same 
as running against it before the field is reversed. For this reason the 
shunt-wound machine runs in the same direction whether used as a 
motor or a generator. 

641. Counter Electromotive Force in a Motor. The armature 
of an electric motor revolves in a magnetic field and gen- 
erates an E. M. F. A little consideration will show that 
this E. M. F. must be an opposing one tending to reduce 
the current through it. In Figure 408 a generator and a 
motor are shown connected together. The direction of 
rotation of the two machines is the same. The direction 
of the electromotive forces generated in the two armatures 
is shown by the arrows. They are toward the lower brush 
in both, because both armatures revolve in the same direc- 
tion in similar fields. But in the generator the current is in 



DIRECT CURRENT MACHINES 



571 



the same direction as the E. M.F. generated in the armature, 
while in the motor it is against this generated E. M. F. The 
E. M.F. of the latter therefore opposes the current. 





Fig. 408 



If the motor is provided with a fly wheel to keep up its motion when the 
generator current is cut off, a voltmeter placed across its terminals, as V 
in the figure, will show only a slightly diminished E. M. F. immediately 
after the circuit is broken, if there is no load on the motor to produce a 
quick slackening of speed. The voltmeter indicates no reversal of cur- 
rent when the generator is cut off. This fact shows that the positive 
brush of the generator is connected to the positive of the motor, or that 
the E. M.F. of the motor is a back E. M. F. The voltmeter may be 
replaced by an incandescent lamp ; it will glow nearly as brightly for 
a few seconds directly after the main circuit is opened as before. 

642. Relation of Speed to Field Strength. It has already 
been demonstrated that the work done against an opposing 
E. M.F. per second is measured electrically by the product 
of this counter E. M. F. and the current, or E'I( 565). 

The two factors of the power, measured mechanically, are 
the torque and the speed (that is, torque and angular veloc- 
ity, or Tai). The torque is the moment of the couple pro- 
ducing the rotation ; it is proportional both to the strength 
of the field and the current in the armature. If the field is 
kept constant, the torque is proportional to the current ; the 
back E. M. F. is proportional to the speed. Hence we may 



E'I=AnT, 



(115) 



572 



MAGNETISM AND ELECTRICITY 



where T is the torque, n the number of rotations per second, 
and A a constant or proportionality factor (A = 2 TT). 

When the motor is working under a fixed load, an increase 
in the field strength increases the torque and decreases the 
speed n ; weakening the field, on the other hand, diminishes 
the torque and increases the speed. Both these conclusions 
follow from the constancy of the product nT under the 
assumed condition of the fixed load. In both cases the 
speed changes until the counter E. M. F. acquires the same 
value that it had before the change was made in the field 
strength. 

II. ALTERNATING CURRENT MACHINES 

643. Alternators. If the brushes of a dynamo bear on two 
continuous rings on the shaft, instead of a commutator, 
the current, in the external circuit will alternate or reverse 
every time the armature turns through the angular distance 
from one field pole to the next. A complete series of changes 
in the current and the E. M. F. takes place while the arma- 
ture is turning the angular distance of one pole to the next 

one of the same sign. This 
series of changes is called a 
cycle. The frequency, or 
number of cycles per 
second, is the product of 
the number of pairs of 
poles of the field magnet 
and the number of rota- 
tions per second. Frequen- 
cies are now restricted to 
the limits of about twenty- 
five and sixty cycles per 
second. Multipolar ma- 
chines are used to avoid excessive speed of rotation. 

Figure 409 is a diagrammatic sketch of an alternator with 
a stationary field and a rotating armature. In large ma- 




Fig. 409 



ALTERNATING CURRENT MACHINES 



573 



chines the armature is usually the stationary member and 
the field the revolving one. The field is excited by a direct 
current machine. The armature coils are reversed in wind- 
ing from pole to pole; they are all joined in series and the 
terminals are brought out to the two slip rings at B. The 
brushes bearing on these rings lead to the external circuit. 

644. Lag of the Current behind the Electromotive Force. 
When an alternating electromotive force is applied to a 
circuit having inductance, one of the novel and essential 
facts is that the current reaches its maximum value later 
than the electromotive force; and, as a consequence, Ohm's 
law is no longer adequate to express its value. The effect 
of self-inductance is not only to introduce an additional 
electromotive force, but to produce a lag of the current in 
phase behind the electromotive force impressed on the cir- 
cuit by the generator. 

The effort required to get an understanding of the phase 
lag of the current is well worth while. Let an alternating 
current, following the simple harmonic law, be represented 
by the heavy sine curve I in Figure 410. The induced electro- 






Fig. 410 



motive force is proportional to the rate of change of mag- 
netic flux, and hence to the rate of change of the current 
producing it when there is no iron in or about the circuit. 
The induced electromotive force curve is represented by the 
light line II. This is also a sine curve, for the rate of vari- 
ation of a sine function is itself a sine function. The curve 
of induced E. M. F. reaches its maximum value a quarter 



574 MAGNETISM AND ELECTRICITY 

of a period later than the current curve; for, when the 
current is a maximum at A, its rate of change is zero and 
the induced E. M. F. is zero; when it decreases through its 
zero value at B, its rate of change is a maximum, and the 
induced E. M. F. is a maximum in the positive direction. 
Hence the induced E. M. F. is a quarter of a period behind 
the current, or the current and induced electromotive force 
are said to be in quadrature. 

The electromotive force producing the current by Ohm's 
law corresponds in phase with the current itself. The 
electromotive forces consumed by resistance and self-induc- 
tance may therefore be represented by the two adjacent sides 
of a right triangle (Fig. 411), where ab and be are the 

c electromotive forces con- 
sumed by resistance and 
inductance respectively. If 
these electromotive forces 
are added geometrically as 
if they were vector quan- 
tities, their resultant is the 




hypotenuse ac. which is 

Fig. 411 / r 

therefore the maximum im- 
pressed E. M. F. But the current agrees in phase with ab; 
it therefore lags behind the impressed electromotive force by 
the angle <. In the absence of capacity in the circuit, this 
angle of lag becomes zero only when the inductance is zero. 
One component of the impressed E. M. F. goes to neutral- 
ize the E. M. F. of self-induction; the other is lost in pro- 
ducing the current against resistance by Ohm's law. 

Figure 412 A was reproduced from one of Professor Thomas's oscillo- 
grams and shows the current lagging behind the E. M. F. nearly a quarter 
of a period in a circuit of large self-inductance. Self-inductance operates 
to check sudden changes in the impressed E. M. F. Hence the small 
inequalities of the E. M. F. curve, due to the presence of higher harmon- 
ics or multiples of the fundamental frequency, nearly or quite disappear 
in the current curve. 



ALTERNATING CURRENT MACHINES 575 

The effects of capacity in a circuit are exactly the opposite of those of 
self-induction. Capacity causes the current to lead the impressed E. M. F. 






Fig. 412 A Fig. 412 B 

and it exaggerates the higher harmonics instead of reducing or suppress- 

ing them in the current (Fig. 4125). Figure 412 C is an oscillogram 

from the same machine, but in this 

case the circuit was non-inductive 

and without capacity. The current 

and E. M. F. are in the same phase 

and the current at the maximum 

reproduces all the small irregulari- 

ties of the E. M. F. due to the pres- 

ence of odd harmonics. 

Fig. 412 C 

645. Value of an Alternating 

Current. The maximum value of the E. M. F. of self-induc- 
tion is Lcol, where L is the coefficient of self-induction and &> 
the angular velocity, or 2 TTH. The impressed E. M. F. repre- 
sented by the hypotenuse of the triangle is 7(jR 2 + IPaPy- 
Hence 

E = J(^ 2 + 2 o>2)*, and 1= E/(IP + JEW)* (116) 

The expression (.72 2 + .Z/ 2 a> 2 )^ is called the impedance of the 
circuit. It shows that the effect of inductance 'on the value 
of the current is equivalent to an additional resistance. 

646. Effective Volts and Amperes. The heating effect of 
an alternating current, as in the filament of an incandes- 
cent lamp, is proportional to its mean square value, and all 
instruments for measuring alternating currents and pressures 
indicate u square root of mean square " values and not the 



576 MAGNETISM AND ELECTRICITY 

arithmetical mean. Thus the electrodynamometer ( 593), 
whether used to measure current or voltage,. integrates the 
forces operating it, and these are proportional either to the 
squares of the current or of the electric pressure. If the 
current and the electromotive force follow the sine law, 
the mean given by the measuring instruments is 0.707 of 
the maximum value. When a voltmeter on an alternating 
circuit reads 70.7, the voltage alternately rises to + 100 and 
sinks to 100 as positive and negative maxima. The values 
given by measuring instruments for alternating current cir- 
cuits are effective volts and effective amperes. 

647. Transformers. A transformer is an induction coil with 
a primary of many turns, a secondary of a smaller number, 
and a closed magnetic circuit. It is employed with alter- 
nating currents as a " step-down " instrument for the purpose 
of reducing the high electromotive force on the transmission 
line to a low electromotive force for lighting and power. It 
is entirely reversible and can be used equally well for the 
" step-up " process with alternating currents. 

The primary and the secondary coil are both wound around 
the same iron core, but are insulated from each other as per- 
fectly as possible. The iron (Fig. 413) serves as a path for 

the flux of magnetic in- 
duction. The relation of 
the flux and the current 
s is a reciprocal one, so that 
they may always exchange 
places. In the " shell 
type" of transformer the 
iron surrounds the wire, 
while in the " core type " 
the wire surrounds the 

iron. With either relative arrangement of the iron and the 
coils, nearly all the lines of induction produced by the pri- 
mary pass also through the secondary, and vice versa. 




ALTERNATING CURRENT MACHINES 



577 



When the secondary is open the primary acts simply as a 
" choke coil," and the only current passing is the very small 
one required to magnetize the iron for the generation of a 
counter E.M.F. of self-induction nearly equal to the im- 
pressed E.M.F., and to furnish the small amount of energy 
lost in hysteresis. The electromotive forces generated in 
the two coils are directly as the ratio of the number of turns 
on the two ; while the currents in them when the secondary 
is closed are very nearly in the inverse ratio of the number 
of turns. The energy in the secondary circuit is nearly the 
same as that expended on the primary. The small loss is 
chargeable to loss in the copper of the primary and to losses in 
heating the iron on account of hysteresis and eddy currents. 

The secondary current is nearly opposite in phase to the 
primary, and causes a diminution in the apparent self-induc- 
tance of the primary coil ; so that the larger the secondary 
current, the larger the primary. A transformer is thus 
nearly self-governing. 

648. Polyphase Currents. It has long been known that 
two or more alternating currents of the same frequency, but 
differing in phase, may be obtained from one generator. If, 
instead of a commutator, four insulated rings on the shaft 
be connected to four 
equidistant points on 
either a drum arma- 
ture or a Gramme 
ring, the currents in 
the externally sepa- 
rate circuits will differ 
in phase by a quarter 
of a period. Each 

1 . Fig. 414 

pair ot rings con- 
nected to points on the armature 180 apart act as terminals 
for an alternating circuit. It is obvious that one current 
passes through its maximum at the instant that the other 




578 



MAGNETISM AND ELECTRICITY 




passes through its zero value (Fig. 414). In a similar way, 
three-phase currents may be taken from rings as terminals if 

they are connected to 
conductors in the arma- 
ture 120 apart. If 
there are only three con- 
ductors, each one serves 
as a return for the other 
two, since the algebraic 
sum of the two currents 
having the same sign at 
any instant is always 
equal and opposite to the third (Fig. 415). 

649. The Rotatory Field. When an alternating current is 
transmitted through a coil without iron, it produces an alternat- 
ing magnetic field along the axis of the coil. If the current fol- 
lows the sine law, the magnetic flux will follow the sine law also. 

Let two such coils be set with their axes at right angles, 
and let equal two-phase alternating currents pass through 
them. Two simple harmonic motions of equal amplitude, 
at right angles, and differing in 
phase by a quarter of a period, com- 
bine to produce uniform circular 
motion ( 38). In a similar way 
the two coils A A and BB (Fig. 416) 
produce a rotating magnetic field 
near their common center. In 1888 
Ferraris in Italy mounted within 
them a hollow copper cylinder on 
pivots at top and bottom. When 
two-phase currents are passed 
through the two circuits of the Ferraris apparatus, the 
copper cylinder is set rotating in the direction of the rotating 
field. The rotation of the field produces currents in the 
copper, as in Arago's rotations. By Lenz's law the motion 




ALTERNATING CURRENT MACHINES 579 

of the cylinder is in the direction to check the induction in 
it ; hence the cylinder is dragged around in the same direc- 
tion as the rotation of the field ; for, if the speed of the 
cylinder were the same as that of the field, no current would 
be induced in the copper. 

If one current is reversed with respect to the other, that 
is, if its phase is changed by 180, the direction of rotation 
of both the field and the cylinder is reversed. The cylinder 
tends to run up to synchronism with the field, but never 
quite reaches it ; the difference in their speeds, called the 
slip, is just sufficient to induce currents to supply the requi- 
site torque. If the rotation of the field generates a direct 
electromotive force, the rotation of the cylinder, which is 
equivalent to the rotation of the field in the other direction, 
generates a counter electromotive force, and the latter is 
always smaller than the former. 

650. The Induction Motor. Rotation of the magnetic 
field may also be secured by winding the coils for the two 
circuits on an iron ring (Fig. 417). The coils A and A! 
are wound so as to make consequent poles at B and B' , 
while the coils B and B 1 are wound to produce consequent 
poles at A and A'. 

At the instant when the current through coils A and A 1 
in phase A has reached its maximum value at A (Fig. 414), 
the current in phase B is zero and the poles in the ring are 
concentrated at B and B f . An eighth of a period, or 45, 
later the currents in the two phases are positive and equal. 
The poles in the ring then extend from A to B and from A r 
to B 1 . An eighth of a period later, or at B in phase B 
(Fig. 414), the current in phase A is zero while the current 
in phase B is a maximum. The poles are now at A and A 1 , 
and they have moved forward counter clockwise a quarter 
of the way around the ring. 

The rotation of the field within the ring of Figure 417 may readily be 
shown by placing a glass vessel in the horizontal ring, filling with water 



580 



MAGNETISM AND ELECTRICITY 



to the level of the middle of the ring, and sifting powdered iron on the 
surface of the water. With small two-phase currents traversing the 

coils, the individual iron particles rotate 
rapidly ; with larger currents, the whole 
surface rotates in the direction of the 
rotation of the field. 

The A connections for a three- 
phase winding of the stationary 
member, or stator, of an induction 
motor are shown in Figure 418. 
Instead of three coils, P, Q, and R, 
there may be any multiple of three. 
Inside the two-phase or the 
three-phase ring is mounted a 
rotor, consisting of a laminated 
iron cylinder with heavy conductors embedded in the periph- 
ery parallel to its axis of rotation. They are connected 
together at the ends of the h 

cylinder so as to form a 
" squirrel cage " of copper. 
The induced currents in this 
cage produce a torque which 
turns the cylinder in the direc- 
tion of the rotating field. 




Fig. 417 




651. Power in an Alternating 
Current Circuit. It has been 
shown that for a direct con- 
stant current the energy per 
second is the product El of the 
E. M. F. and current ( 536). 
Similarly the power of an alter- 
nating current is the product 
ei of corresponding instantaneous values of the E. M. F. and 
current. But this product is incessantly changing; it is 
instructive to resort to the graphical method to illustrate the 
changes taking place during a few cycles. 



Fig. 418 



ALTERNATING CURRENT MACHINES 



581 



In .Figure 419 t