ADVANCED
PRACTICAL
PHYSICS
FOR
STUDENTS
B.L.WORSNOP
AND
H.T. FLINT
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ADVANCED PRACTICAL PHYSICS
FOR STUDENTS
ADVANCED
PRACTICAL PHYSICS
FOR STUDENTS
BY
B. L. WORSNOP, B.Sc.
LECTURER IN PHYSICS, KING'S COLLEGE, LONDON
EXAMINER IN PHYSICS, UNIVERSITY OF LONDON
AND
H. T. FLINT, Ph.D., D.Sc.
READER IN PHYSICS, UNIVERSITY OF LONDON
EXAMINER IN PHYSICS, UNIVERSITY OF LONDON
WITH 404 DIAGRAMS AND ILLUSTRATIONS
SECOND EDITION, REVISED
METHUEN & CO. LTD.
36 ESSEX STREET W.G.
LONDON
First Published . . . April 1923
Second Edition, Revised J 9 3 7
PRINTED IN GREAT BRITAIN
PREFACE
THE course of Practical Physics described in this book
is based upon that followed in King's College, London,
by students who have completed their Intermediate
Course, and who are proceeding to a Pass or Honours Degree.
This has been extended, and it is hoped that the book will be
useful to a wider circle of students of Physics than those imme-
diately concerned with University Examinations.
A number of well-known Physicists have contributed to the
development of the King's College course, amongst whom we
may mention Professors H. A. Wilson, C. G. Barkla, H. S. Allen,
and W. Wilson, who formerly worked here in the Wheat stone
Laboratory, and Professor O. W. Richardson, the present occupant
of the chair.
The general aim has been to provide with each experiment a
short theoretical treatment which will enable the student to
perform the experiment without immediate reference to theoreti-
cal treatises. To aid this scheme an introductory chapter in the
Calculus has been included. This chapter is an innovation in a
book of this type, but it is hoped that the student will find here
a bridge over that period during which his Physics demands
more advanced mathematics than his systematic study of that
subject has yet given him.
We take this opportunity of expressing our gratitude to
Professor O. W. Richardson, who has allowed us to make use of
laboratory manuscripts and results of experiments. We are also
greatly indebted to our colleagues and to Mr. G. Williamson, who
have given us many suggestions, and to the Honours students of
vi ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the past session who have supplied us with numerical and graphical
results. We have been greatly helped by the ready assistance
on the part of The Cambridge and Paul Scientific Instrument Co.,
Messrs. Elliot Bros., Gambrell, Ltd., Adam Hilger, Ltd., W. G.
Pye & Co., and the Weston Electric Co., who supplied us with
the blocks for many of the illustrations.
B. L. W.
Wheatstone Laboratory, H. T. F.
University of London,
King's College.
March, ig2j
PREFACE TO SECOND EDITION
IN this revised edition we have removed the misprints and
errors which occurred in the first.
We have added to Chapter XXIV on Radioactivity,
and have included a chapter which contains a miscellaneous
collection of additional experiments.
We wish to thank those who have made suggestions for
additions and who have pointed out errors in the original text.
The adoption of all the suggestions would have made the book
unwieldy, but we hope that the additions made will increase its
value to the student.
To thank, individually, all students, colleagues, and others to
whom we are indebted would add greatly to the length of this
preface. But we feel that we owe our special thanks to Mr.
Brinkworth, Dr. D. Owen, and Prof. Rankine for suggestions ;
to Dr. K, G. Emeleus for suggestions and help with the proof ;
and to Prof, E. V. Appleton, who has added several experiments
with thermionic valves in the new chapter.
B. L. W.
H. T. F.
February 21, 192J
CONTENTS
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
XIII.
XIV.
XV.
XVI.
XVII.
XVIII.
XIX.
XX.
XXI.
XXII.
XXIII.
XXIV.
XXV
PACK
INTRODUCTION TO THE DIFFERENTIAL AND INTE-
GRAL CALCULUS - - - - I
MEASUREMENT OF LENGTH, AREA, VOLUME, AND
MASS - - - " " " 3 1
MOMENTS OF INERTIA AND DETERMINATION OF "g" 5 1
ELASTICITY - - * " - 80
SURFACE TENSION ----- 124
VISCOSITY ------ 149
THERMOMETRY AND THERMAL EXPANSION - - I78
CALORIMETRY - IQ^
VAPOUR DENSITY AND THERMAL CONDUCTIVITY - 217
MISCELLANEOUS EXPERIMENTS IN HEAT - - 243
REFLECTION - - - • -258
REFRACTION - - - " - 2J2
INTERFERENCE, DIFFRACTION, AND POLARISATION - 314
PHOTOMETRY _ _ - - - 380
SOUND - - . - - - - 39 2
MISCELLANEOUS MAGNETIC EXPERIMENTS - - 426
TERRESTRIAL MAGNETISM - 437
PERMEABILITY OF IRON AND STEEL - - 445
AMMETERS, VOLTMETERS, AND GALVANOMETERS - 456
RESISTANCE MEASUREMENTS - 4^9
RESISTANCE OF ELECTROLYTES - 526
MEASUREMENT OF POTENTIAL - 531
MEASUREMENT OF CAPACITY AND INDUCTANCE - 550
THE QUADRANT ELECTROMETER - 584
MISCELLANEOUS ELECTRICAL EXPERIMENTS - 605
ADDITIONAL EXPERIMENTS - &37
TABLES OF UNITS - 652
APPENDIX ------ 654
INDEX ------ 655
ADVANCED PRACTICAL
PHYSICS FOR STUDENTS
INTRODUCTION TO DIFFERENTIAL AND
INTEGRAL CALCULUS
The Differential Calculus
1. Any quantity % which may assume a series of values is called
a variable quantity or simply a variable, and if its value docs
not depend on that of any other quantity it is called an
independent variable.
On the other hand a quantity y, which bears a particular
relation to x, assumes values which depend on the values of x,
and for this reason is called a dependent variable. We may
have for example :
y = 2x - 3.
Here y takes values which depend in a quite definite manner
on those of x.
We may also have a dependence defined by the relations :
y = sin x, y = log x, and y m §•.
Such expressions as 2% — 3, sin x, log m, etc., are called functions
of x, and when we say that y is a function of % we mean that
y depends on the values that % assumes.
In case we do not specify definitely how y and % arc related
wc write
y = /(*)■
f(x) denotes any function of %.
It is often convenient in Physics to show by means of a diagram
the relation between two variables y and x. For example, a
record may be required of the atmospheric pressure at various
times. U Such a record is drawn automatically by a self-recording
barometer so that it can be seen how the pressure and time are
related. Here we have as independent variable the time and
the dependent variable is the barometric pressure.
2 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In fig. i, the curve represents the relation between x and y
and the shape depends on the way y and x are connected.
X
Fig. i
If y — 2X — 3, the curve becomes a straight line, and
if y = sin x, we have the familiar sine curve, fig. 2.
Fig. 2
A function is said to be a continuous function of a variable
when the graph representing it is a curve in which there is no
sudden change in value of ti\e ordinate at any point. In such a
curve, if we approach a point where x = a, from left to right,
we find a certain value for.y, and if we approach the point from
right to left we find the same value. In fig. 3 we have an example
of a function which is discontinuous at x = o. If we approach
the origin from left to right the value of y is very great and
DIFFERENTIAL AND INTEGRAL CALCULUS 3
negative in sign, while in approaching from the right y is very
large and positive. In nature we are chiefly concerned with
continuously varying quantities. If a train is at rest at a station
at a particular instant, and is observed to be moving with a
velocity of ten miles per hour ten minutes later, it must have
possessed every possible velocity between zero and ten miles per
hour during the interval.
The speed is continuous, and if it depends on the lapse of time
from the start it is said to be a continuous function of the time.
We do not contemplate the possibility that the train could
possess a speed of five miles per hour at one instant and at the
next without any interval whatever a velocity of six miles per
hour. If this were possible we should describe the speed as
discontinuous, because it had no value between five and six. If
this appeared to be the case we should consider that our powers
of observation were at fault, and we should describe the motion
as changing very rapidly between five and six miles per hour ;
so rapidly that we had failed to detect the lapse of time in which
the change took place.
Discontinuous functions are of frequent occurrence in Mathe-
matics. Consider as an example the case of y = -.
When x is a very small positive number, let us say — -;,
y is large and has the value 10 8 .
4 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
On the other hand if x = 6 , y is large in magnitude but
negative, it equals — io 6 .
As x passes through the value zero y suddenly leaps from an
enormously large negative value to a very great positive value,
and has no value between.
This is represented in the diagram, fig. 3. The curve has
two branches : they are the two parts of the rectangular
hyperbola
xy = 1.
We shall not be concerned with such functions so we dismiss
them briefly. It is to be borne in mind that our applications of
the Calculus are to continuous functions only. The results we
obtain must not be applied to discontinuous functions without
closer examination.
It is important to understand the meaning of the limit of a
function.
Suppose y depends on x, and that as x approaches the value a,
y approaches the value b.
b is called the limit of y as x approaches a, and we write :
Lim. y ► b
x > a
If reference be made to fig. 1, as x approaches the value OM t
y approaches the value M{9 X and M^ is actually the value
of y when x = a.
Cases occur in which the conception of a limit is not so simple.
If we examine the curve
1
J x
in the neighbourhood of the origin as x — > o, we obtain
a different value of y according as we begin on the right or left
of the origin.
On account of the discontinuity the limit of y as x approaches
zero is not definite.
Another case occurs in which x may continue to increase to
any extent while y continually approaches some particular value.
We may turn once more to the curve
y-l.
X
As x gets larger and larger, y gets smaller and smaller approaching
the value zero.
We may get as near zero as we please by making x larger.
For example, we may make y as small as one-millionth by
choosing x — 10 6 .
DIFFERENTIAL AND INTEGRAL CALCULUS 5
This is a very important point in denning a limit. It must
be possible to get as close to the limiting value as we please by
choosing x properly, although it may not actually be possible
to cause y to attain the limit. We have in our example a case
in point, y is never zero however large x may be, but it is
possible to make y nearer and nearer zero by increasing x.
The former definition of the limit of a function is not very
satisfactory. A limit is accurately defined as follows :
The limit of a function of x is some number, b, such that as x
approaches a particular value, a, the difference between b and the
function may be made as small as we please by taking x sufficiently
near a.
2. In describing natural phenomena by means of equations,
simplifications are often brought about by neglecting certain
terms in comparison with other more important terms.
Suppose an equation is obtained which we may write :
A x + B x + C 2 + D 2 = E x + F 8 .
The suffix numbers denote the order of importance of the terms ;
that is to say, 1 denotes that the term is to be regarded as of
first importance, or it is of the first order of magnitude. The
2 and 3 denote that the terms are only of second and third
degrees of importance, they are of the second and third orders.
If we wish to include terms of the first and second orders we
omit F 3 , while if only terms of the first order are to be considered
the equation becomes :
h x + B x = E x .
Great care has to be exercised in thus drawing up a scale of
magnitude, and this leads to a short consideration of infinitesimals.
Suppose a quantity X is divided into 1000 equal parts, these
again subdivided in the same way, and so on. We then have a
series of values :
X, — 3— 6 ,etc,
10 3 10 6
which provides a scale of magnitude.
If circumstances do not permit of accurate observation of
quantities less than those of the same order as X we regard
— r» — / etc., as negligible.
io 3 io 6 ° °
Generally, if / is a small fraction, i.e. small compared with
unity :
/X,/*X,/ 3 X, etc.
are all small compared with X, and are said to be small quantities
6 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
of the first, second, third, etc., orders. If these small quantities
have zero limits they are called infinitesimals.
In equations between infinitesimals only the terms of the
lowest order are to be retained, i.e. the terms of greatest magni-
tude.
This is made clearer by an example which has important
Physical applications.
In fig. 4, AB represents the radius of an arc, BP, of a circle
which subtends an angle at A.
PN is normal to AB.
BT is also normal to AB cutting AP produced in T.
It may be regarded as an axiom that :
PN < arc PB < BT.
We shall examine the order of the differences between these
quantities if be regarded as of the first order of small quantities.
By expansion of sin and cos in powers of we have :
03 05
sin 6 = -, -1 , — , . (-i)
3l 51
02 04
cos = I r + — . — ( 2 /
214!
If only small quantities of the first order are retained :
sin = 0. -» .
cos = 1. / (3/
PN = a sin 0.
PB = a 0.
i 3 5 ^
.\ PB - PN ,= a(0 - sin 0) = a { - - - ( + .. . .1
This difference is of the third order.
Thus up to considerations of magnitude of the third order
PB = PN.
DIFFERENTIAL AND INTEGRAL CALCULUS 7
Again
BT =* a tan 0.
A 3 2
= a(0 + -~+ — 5 + - • • )»
as may be shown by division of the expressions for sin and cos0.
Thus BT - arc PB = a quantity of the third order of magni-
tude.
Fig. 5
BN — a — AN = a(i — cos 0) = a quantity of the second
order of magnitude.
Thus, if we regard as of the first order and retain only this
order in our equations we may write :
BN = o, PN = arc PB = BT.
and with the exception of BN = o this is true for the case when
second order quantities are retained.
Extensive use is made of these relations in Geometrical Optics
in the first study of reflection and refraction in mirrors and lenses.
In the case of a mirror, for example (see fig. 5), when the angle
is small, i.e. when the rays from an object, P, strike the mirror
at M not far from the pole, O, we establish certain formulas by
assuming that and N may be regarded as being coincident.
This is because we do not retain quantities of an order higher
than 0. Thus NO = o by the foregoing considerations.
Another important case is the calculation of the order of the
difference between the sum of two sides of a triangle and the
base when the base angles are of the first order of small quantities.
8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Referring to fig. 6, we have as for fig. 4,
PA — AN = a quantity of the second order in 0,
and similarly PA 1 — A X N is of the second order in 1
.'. PA + PA 1 - (AN + A X N) = a quantity of the second
order since and 1 are by the data small quantities of the first
order.
/. PA + PA 1 = AN + A*N = AA 1
to the first order.
This result is made use of in the establishment of Fermat's
Law of the extreme path which plays a fundamental part in
Optics. (See for example Houstoun's " Treatise on Light,"
p. 17.)
We consider as a final example the difference between a chord
and an arc both subtending the same small angle at the centre
of a circle.
Thus, again referring to fig. 4, we require the difference between
chord BP and arc BP.
a
arc BP — chord BP = ad — 2a sin — .
2
= a
e - 2( --- ^- + ...)
_ 1 0*
2 3! 8"
= a quantity of the third order.
We can thus regard the chord and arc as equal up to and including
quantities of the second order.
It should be noted that the successive orders are vanishingly
small with regard to the terms earlier in the scale, e.g. in com-
paring
aO, bd*, c0 s ,
hf}2 fia
if is of the first order, --- = — so that as > o 60 2 — > o
ad a
infinitely more rapidly than aO, and the same holds for any two
consecutive terms in the scale.
The ratio of two quantities of the same order will be a finite
quantity— not a vanishing or negligible quantity, but the ratio
of two quantities of differing order (higher order -^ lower order)
is vanishingly small.
We are concerned with small variations of this kind in the
differential Calculus.
3. The Differential Coefficient
Let y be a function of x, and suppose x varies by a small
quantity which we denote by dx. In consequence of this varia-
tion y will vary a small quantity, say dy.
DIFFERENTIAL AND INTEGRAL CALCULUS 9
The ultimate ratio 4^- when dx becomes very small is called
dx J
the differential coefficient of y with respect to x. It is denoted
by -j-y and written -j- and sometimes denoted simply by Dy.
In accordance with our notation we may write :
dy _ Lim. #y
dx ~ Sx > o dx'
In general the quantities dy and dx are of the same order of
magnitude, and -^- is a finite quantity.
In order to illustrate this, consider the relation :
y = 2.x — 3.
Let x become x + Sx, then the new value of y is
2{x + dx) - 3
i.e. y + <*y = 2 (* + **) — 3
= y + 2<S#.
.*. <5y = 2<5z.
• & - 2
Now no matter how small dx becomes, the ratio is always 2, for
dy is of the same order as dx, and their ratio is finite and equal
to 2.
We have a simpler case still in the differential coefficient of
a constant.
A constant is a number that does not depend on the variable.
Thus, if y = A it does not matter how x varies, y still remains
= A. Thus there is no change dy corresponding to a change dx.
Hence -j- = o if y is a constant.
It should be noted that -~ does not mean dy -^ dx. g~ is a short
. dy
notation for the operation of finding the ultimate ratio —
Nevertheless Physicists continually appear to use the coefficient
as if it meant dy -^ dx, and it is not a rare occurrence to find
an equation :
dx
written alternatively dy = x 2 dx.
This is, in fact, a very convenient mode of expressing the result,
io ADVANCED PRACTICAL PHYSICS FOR STUDENTS
and it means that dy and dx now no longer retain the same
significance. The second of these means :
dy = x 2 dx.
We have in the equation j- = x 2 an expression of the rate of
variation of y with respect to a; at a particular point on the
curve, which represents graphically the relation between y and x.
The alternative equation means that in the neighbourhood
of this point we can calculate a small change dy corresponding
to a small change dx. This point rarely causes difficulty in
practice, and it is obviously inconvenient to change to and fro
from d to 8, but to be strictly accurate we must bear the
distinction in mind.
The definition of the differential coefficient gives the clue to its
determination. We will not determine its value for more than
one or two cases but be content with reference to a table of
values of the important coefficients.
The Differential Coefficient for x n where n is any Number.
Write:
y = x*.
(Sx\ n
i+ — f
= x"
dx _j_ n(n — i) m (toy
I + "¥ + ^Tr^-i-»^+---
dy
fi(n — i)
= x n + nx*- 1 • dx + - L — — - - x n ~ 2 (dx} 2 +
X*2
= nx»-*dx + n ^~^ • x"- 2 (dx) 2 + . . .
1*2
ty n(n — i)
.'. v- = nx"* 1 -\ — • x*~ 2 dx + higher powers of dx.
dx is a quantity which we have called infinitesimal. In the
next step of finding the limit of ~- we shall suppose dx a quantity
of the first order of magnitude. It is therefore infinitesimally
small with regard to the finite quantity nx n ~K
We thus neglect all quantities of higher order than nx n ~ x and
have :
dy Lim. dy
dx dx >odx nx '
+
DIFFERENTIAL AND INTEGRAL CALCULUS 11
Differential Coefficient of sin x.
y -f. Sy = sin (x + dx) = sin x cos dx + cos x sin <5#.
We need retain only quantities of the first order on the right.
Thus : cos dx = I, sin dx — to, by equation (3).
•*• y + *y = sin x + cos x y - dx.
.*. dy — cos a; • dx.
dy
or -3^ = cos x.
dx
Similarly ■=- cos x = — sin #.
Differential Coefficient of log x.
y + <5y = log (X + <5*) = log a/ I + -^j
= log* + log(i + ^.
Retaining quantities of first order only :
dy = dx* —
"* x
; *1 = 1
dx x
d , t
or ~-\og X — _.
dx x
The same method of treatment can be applied to other cases.
In the case of a function f(x) we write :
dy^ _ Lim. f(x + dx) —f(x).
dx ~~ to > o fa
The Differential Coefficient of the Sum of two Functions.
If y = sin x + cos #
we have -r- = cos # — sin x.
dx
From the definition it follows that the differential coefficient
is the sum of the differential coefficients of sin x and cos x.
In the general case if :
y=yi+y 2
12 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
where y x and v 2 are any two functions of x :
dx dx dx
And similarly if y =y 1 — v 2
dy _ dyj _ dy 2 .
dx ~ dx dx
Differential Coefficient of a Product.
Let y = y^y 2
where y x and v 2 are any two functions :
e.g. we might have :
y — sinx x # n ,
sin # and #» are two functions of x.
Suppose that x becomes x + Sx and in consequence y increases
toy + dy,y 1 toy 1 + dy lt y 2 to v 2 + <*y 2 .
Then
y + «5y = (yx + «yi) (y« + <5y 2 )
Since jy — yiy2
•*• <?y =^i4y2 + ^1^2 + tyity*
dy 1 dy 2 is a term of the second order, and the other terms are
of the first order.
Thus we need not retain it.
Dividing throughout by dx.
&««,& + Si. y,.
dx yx dx 6x • ra
Hence in the limit :
In a product we differentiate one factor at a time, leaving the
others unchanged, and add all the resulting expressions together.
This is true for any number of factors, as may be shown in
the same way.
Thus, if
y =ytyzy*y±
dy dy, , dy 2 , dy 3 , dy*
& -at ' y w< +y > £** + ™* %* +y ^ y * it-
e S- y = sin x x % n
dy
-f- = sin x • nx n ~ x + cos x • *".
dx
DIFFERENTIAL AND INTEGRAL CALCULUS 13
The Differential Coefficient of a Quotient.
We use the same notation as before and apply the same
principles.
y* + *y* y* v (i + ^)
(retaining only terms of first order).
*■"
y* 2
fy y * dx yx dx
6x ^ y 2 2
jV — tan x —
dy __ cos x cos * — sin x( — sin #)
dx ~~ cos 2 #
, sin#
e S> y — tan # =
J COS #
= sec z #.
Differential Coefficient of a Function of a Function.
The expression :
y — a sin 9 # -f- b sin x -f c
in which a, 6, c are constant quantities, is a function of sin #,
sin x is itself a function of x.
Thus, y is a function of a function of x.
We proceed to determine the differential coefficient -j~ in this
complex case.
Before attacking the general problem we will consider a
special case.
Let y = log sin x.
and write z = sin x.
14 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Then y — log z
and as above :
dy = — dz,
J z
In this step we have regarded y as a function of z.
But z = sin x.
.*. dz = COS X'dX.
<5y = - • cos x ' dx.
dy
z
COS X
"
dx
~ sin x
Note that from
dy
i
~ z
dz we have :
dy i dz
dx ~ z dx
Hence
dy i dz
dx ~~ z dx'
Now
.
dz z
dy dy ^ dz
dx ~~ dz dx'
The rule is :
Differentiate the whole function first
as if the
inner function
(in this case sin x)
were the independent '
variable,
and thus obtain
~, then multiply by the differential coefficient of the inner
function. .
This rule, which has been established in a special case, can
readily be proved generally.
Let y = F(z) where z =/(*).
fy = Q .*:
'dx ~ dz ' dx
Then
In the limit : ~ = -j- • -j-.
dx dz dx
e.g. y = sin 2 * == (sin x) 2 .
Put z = sin x.
then y =z\
. & =
dz
dy _
dy dz
~ = 2Z, -5- = cos x.
dz dx
, — 2z cos x — 2 sm X cos X.
DIFFERENTIAL AND INTEGRAL CALCULUS 15
It will be convenient at first to introduce z in this way, but
with practice this intermediate step may be omitted.
5. The Second Differential Coefficient
We have seen that when y = sin x t
dy
-£ = cos x.
ax
Thus-f- is itself a function of x and will have a differential
ax
coefficient.
Let -f- be denoted by z.
dx J
z = cos x.
. dz
. . -j- = — sm x.
dx
Thus the differential coefficient of -f- is — sin x in this case.
dx
This is called the second differential coefficient of v, and is written
Similarly we define higher coefficients -=-^, etc., but we shall
not be concerned with higher orders than the second.
As another example consider y =x n .
~ — nx n ~ x .
dx
d 2 v
j+=n(n- i) x»-*, etc.
6. Applications in Dynamics
When a particle describes a path under the action of a
force or set of forces, its position will vary with the time. Thus
if a stone is thrown vertically upward the position at any instant
will depend on the time that has elapsed since the moment
of projection.
If this distance is measured by y,
Suppose the time at a particular instant is measured by t and
the corresponding value of the distance is y. At a small time
16 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
dt later y will become y -\- dy. We may say the average velocity
. <5y
is -jf- in this short interval. The smaller we make the interval
ot
the more accurately will this ratio represent the velocity near
the position denoted by y. Thus proceeding to the limit in
which dt ultimately vanishes we have :
, ., Lim. dy dy
velocity v — CJ ~ = -£
J dt >0 dt dt
It is often convenient to measure the position of a point by its
position measured from some point O along the arc it is describing
(fig. 7).
Fig. 7.
ds
In this case -j? is the velocity at P along the arc or the velocity
in the direction of the tangent at P, where s measures the arc OP.
dv
In the same way -.— measures the acceleration of a particle
moving with velocity v.,
dv d 2 s . ds
But W = df> "^ V = dt
Thus the acceleration along the arc is measured by the second
differential coefficient.
7. A Geometrical Application
Referring to fig. 1, let Pj denote the point (x- y) and P,
the point {x •+■ ** • y + <5 3')-
Then P 2 N = dy and PjN = M^ = 6x.
The ratio ^ = |^ = tan V^P^ = tan v .
ox r x JN
As the limit is approached the line P 2 P t becomes closer and
closer to the tangent at P x and finally actually coincides with it.
T , Lim. dy _ dy
Thus . ^ *■ — r-=tanv
dx ► o dx dx
DIFFERENTIAL AND INTEGRAL CALCULUS 17
where y now denotes the inclination of the tangent at Pj to the
axis of x.
If a curve in the course of its extent is of the character shown
in fig. 8, the turning points BjBa are called maximum and B 2 B 4
minimum values of the ordinate.
Bx is a point such that it possesses the greatest value of y
for points in its immediate neighbourhood. It is not necessarily
the greatest ordinate of the curve.
Fig. 8
A similar remark applies to the minimum values. At such
extreme points the tangent is necessarily parallel to the axis of x.
Thus y = o and tan y also vanishes.
Hence for maxima and minima :
£-0.
ax
This relation is also of great use in Physical problems, as in
the case of the minimum deviation of a prism or in the theory
of the formation of the rainbow.
8. Integration
The process of Differentiation is to derive from a function
its differential coefficient or rate of variation with respect to
the independent variable.
We have the inverse problem in Integration, where from the
differential coefficient we have to derive the function., This
operation is more difficult and cannot be accomplished in every
case. .
There are many standard cases which can be readily recognized,
*e.g. we have seen that for_y = #**
Y-= nx"- 1
ax
i8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
so that when we are given that
ax
we remember that y has the value x n .
But if y has the value x n + A, where A is some constant,
and hence has a zero differential coefficient we still have :
dx
nx
•»— 1
Thus, there is an uncertainty about the value of y to the extent
of an unknown constant. Such a constant must always be taken
Fig. 9
into account in integration and is called the integration constant.
In Physical problems this constant usually has some special
value determined by the conditions of the problem.
We shall approach the Integral Calculus by a Geometrical
consideration.
In fig. 9, NM represents a curve, y =/{x). OA and OB are
two fixed values of x and the area between the ordinates AN
and BM, the curve NM and the axis of x is divided by drawing
ordinates P^, P 2 Q 2 , P 3 Q 8 , etc., between A and B, so that
AP X = dx x , P X P 2 = dx 2> P a P 8 = dx 3J etc.
DIFFERENTIAL AND INTEGRAL CALCULUS 19
All these elements are to be reduced and ultimately vanish.
Let x v x 2 , # 3 , etc., denote the values^ of the abscissae for points
lying in the strips AP 1} P^, P 2 P 3 , etc., respectively.
Of these the ordinate at X 3 , which denotes the point with
abscissa x z , is a typical example :
Let the ordinates corresponding to these abscissae be denoted
by y x , y 2 , y 3 , etc
We shall suppose that between all the points such as NQ X ,
QiQ2> Q2Q3. etc., * ne curve is continuously increasing or decreas-
ing. At every point where there is a maximum or minimum,
as at C, we shall draw an ordinate. The curve is drawn with
only one such turning point, and this is sufficient for our purpose
since the argument may be extended to deal with cases where
several such points occur.
From the points N, Q lf Q., etc., draw perpendiculars NR 1?
Q X R 2 , Q 2 R 3 , etc., to the ordinates ¥ x Q lf P 2 Q 2 , P 3 Q 3 , etc., respec-
tively, and from Q lf Q 2 , Q 3 , etc., draw perpendiculars Q^, Q 2 S 2 ,
Q 3 S 3 , etc., to the ordinates AN, P^, P 2 Q 2 , etc., respectively,
as shown in fig. 9.
We have thus two step-like figures which we will call the
outer and inner stepped figures.
Now consider the expression y z 8x 3 .
This lies between Q 3 P 3 x P 2 P 3 and Q 2 P 2 x P 2 P 3 .
It thus represents an area intermediate between that of the
rectangle P 2 P 3 Q 3 S 3 and P 2 P 3 R 3 Q 2 .
In the same way y x fa x lies between APiQ^j and AP^N.
Thus, if we consider the sum :
y 1 6x 1 -\-y i 6x 2 + y z 8x 3 + . . . +y n fi%n = tydx,
where y n represents an ordinate in the last strip which terminates
at B and has a width 8x n , we know that it lies between the
outer and inner stepped figures.
We have to inquire into the value of the sum when we make
the number of divisions very large or, what is the same thing,
the values d$ lt 6x 2 , 8x 3 , etc., very small.
In performing this summation we meet with the difficulty
that although each term becomes small, the total number in-
creases, and the question arises as to whether the sum remains
finite under these circumstances. It seems almost axiomatic that
the sum will in the limit prove to be the area between the
curve, the extreme ordinates and the #-axis.
However small the values, 8x, become the sum concerned
always lies between the outer and inner stepped figures. The
difference between these areas is the sum of the rectangles :
SjQ.RiNi, S.QtR.Qp etc.
20 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
There will always be one strip which has a width greater than
that of any of the others, or at any rate its width will not be
less than that of any other, though of course it is possible to
have another strip of equal width. Let this be the strip lying
between P r and P r+1 , and draw the two rectangles as before,
viz. P r P r+1 Q r+1 S r+1 and P r P r+1 R r+1 Q r l
In our figure the greatest ordinate is CD. Continue P r +1 Q r , ,
to T 1 and P r S,+i to T so that P r+1 T* = P r T = CD.
Now all the small rectangles S^, S 2 R 2 , S 3 R 3 , etc., lying
between AN and the maximum ordinate at C could be removed
and placed within the rectangle TP r+1 . Thus the difference
between the two stepped figures lying on the left of C cannot
be greater than the rectangle TP r+1 .
As the widths 6x lt 8x 2 , dx z , etc., are diminished the area
TP r+1 diminishes and vanishes in the limit.
Thus in the limit when the widths of the strips become infinitely
small, the difference between the two stepped areas also diminishes
indefinitely.
In other words, the two areas become equal in the limit, and
each is then equal to the area between the curve, the ordinate,
AN, the maximum ordinate and the #-axis.
The same process may be applied to the area between the
maximum ordinate, BM, the curve and #-axis. Thus adding
the two parts together we find that the total area is equal to
the sum :
Sv<5#
in which 8x tends to zero.
If there occur any finite number of maxima or minima in the
curve, it may be divided at each and the same process applied
to each partial area.
When the several areas are summed the result applies to the
whole curve.
ydx and it is called the integral
of y with respect to x.
a and b denote the values OA and OB, and since these are definite
abscissae the integral is called a definite integral.
In some cases when we do not fix the limits we write simply
J ydx and the integral is called indefinite. The lower limit
is then tacitly assumed to be some convenient starting point,
while the upper limit is any variable point which we denote
by the variable x.
DIFFERENTIAL AND INTEGRAL CALCULUS 21
9. Connection between Differentiation and Integration
If we consider the area included between the limits A and X
where X is any point with co-ordinate X, the magnitude of the
area depends on the positions of the points A and X, and the
form of the curve. The form of the curve is fixed by its equation,
and when X is changed the area will vary by an amount depending
on the change in X.
The area thus depends on the value of X at which the summa-
tion is stopped. If the student feels any difficulty about this
point he should draw a semicircle and choose the diameter as
x— axis. Choose one end of the diameter for the point, O,
and determine the area up to an ordinate drawn at a point, X,
taken on the diameter. If OX is denoted by x it will be found
that the area can be expressed as a function of x.
Thus we may write :
fydx = a function of X, say, /(X)
Write :
= jydx.
J a
We are about to show what is the relation between A and v.
Proceed a step farther with the integration, up to the point,
Xi. Denote XX 1 by <5X.
The area will alter by the amount, XX^Y, where YY 1
means the arc of the curve between Y and Y 1 .
Write this change = dA.
Then the new area is
A + dA
= jydx.
= area up to XY + XX^Y.
But A = area up to XY.
/. 8A = XX^Y.
= Y x <5X in the limit,
where Y is an ordinate in the strip, XX 1 , and in the limit we
shall suppose the strip to possess a vanishingly small width.
_. T . dA dA
We have used X and Y to denote the variables for convenience,
but we have not specified any particular values for them ; they
22 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
denote any abscissa and corresponding ordinate so that we may
just as well write :
_dh
y ~ dx
Thus y is the differential coefficient of the integral which has
been denoted by A.
The problem thus resolves itself into determining A if we
know its differential coefficient.
The occurrence of the arbitrary constant is seen to be connected
with the choice of the starting point.
We may summarize this by stating that if :
= lydx
then y =^
J dx
There is no general rule for passing from y to A, i.e. from a
function to its integral.
We can only perform the operation by recognizing a standard
form, and many devices have to be studied for the reduction
of forms not directly recognizable, to more familiar ones.
For these the student is referred to textbooks on the Integral
Calculus.
Reference should be made to tables in Mathematical textbooks
for the important standard forms.
10. Evaluation of a Definite Integral
Suppose that A has been determined in this way, and that it is
now expressed as a function of x.
Then A = lydx = f(x) + constant.
= /(*)+B(say)
B is the constant which it is necessary to add for the reason
explained above.
B depends on the arbitrary starting point so that there is
some connexion between B and a.
Write x = b, i.e. perform the integration up to a fixed point b.
Then fydx = /(&) + B.
If we write x = awe have lydx.
This must vanish, for it means the calculation of an integral
DIFFERENTIAL AND INTEGRAL CALCULUS 23
of no extent, or referring to our illustration it indicates an area
of no width.
.-. fydx=f(a) +B = o.
/. B = -/(a).
This is the relation between B and a.
Hence : fydx = f(b) - f{a).
The rule is therefore : Find the function of which y is the
differential coefficient and express it as a function of x. Substitute
in it the values of the upper and lower limits and subtract the results
so obtained.
/•io
xHx.
# 2 is the differential coefficient of
e.g.
-3
3"'
10 IQ 3 j%
x*dx= ^- = 333-
A
I.
* sin xdx.
sin x is the differential coefficient of — cos x.
IT
.-. P sin xdx = [" — cos x\ = - cos ^ - ( — cos o) = 1.
This measures the area of the part of the sine curve from the
origin up to its maximum ordinate at -, lying above the x axis.
11. An important application of the Integral Calculus is to
the calculation of moments of inertia. (Chapter II.)
A body is considered as made up of a large number of small
masses 8m. Let r denote the shortest distance of dm from an
axis. Then the product r 2 8m is called the moment of inertia
of the particle 8m about the axis. If all the particles of the
body are taken into consideration and we make the summation
Er 2 <5m for the whole body, we obtain the moment of inertia of
the body. Thus Sr 2 <5w or IrHm is the moment of inertia of
the body about the axis.
24 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If we have an area A divided into small elements, da, each
of which has a corresponding distance r from some axis, we require
sometimes the value fr 2 da, and this may be called the
moment of inertia (shortly M. of I.) of the area about the axis.
Such an expression occurs in the treatment of bending of beams,
and will be employed in the calculation of Young's Modulus
by bending.
As an example of the calculation of a M. of I., consider the
case of a thin rod of total mass M and length zl. Let its mass
per unit length be m, so that M = ilm.
Assume that the axis is through the e.g. and perpendicular to
the rod (fig. 10).
Z
Ac
- — x— \
G
1 *
Fig. io
The form of section is of no consequence, but the ends at A
and B are perpendicular to the length.
The element of length dx at distance x from GZ will have a
mass mdx and its moment of inertia about GZ is mix • x z .
Thus the total M. of I. = Jmx 2 dx the limits indicating
between what limits x extends.
Now CmxHx = [y^] =| ml* = *m(2t)l 2 = —.... (4)
The application of integration to the calculation of moments
of inertia is extended in Chapter II.
12. Oscillatory Motion
.Many experiments make use of the fact that in a number
of cases bodies slightly displaced from their position of equilibrium
perform periodic isochronous vibrations about that position.
Examples of this occur in the case of the simple pendulum, in
cases of torsional oscillations, and movements of galvanometer
needles.
DIFFERENTIAL AND INTEGRAL CALCULUS 25
We usually require the complete period of the oscillation in
making calculations.
Such motion is best and most simply treated by means of the
Differential Calculus.
A simple harmonic vibration is by definition one in which
the body moves so thajt it is under the action of a force tending
to restore it to the position of equilibrium, the magnitude of
the force being proportional to the measure of its displacement.
Thus, if a point is moving to and fro in a straight line
about a position O, and is performing S.H.M. it is always under
a force directed towards O and proportional to thd distance
OP ( = x).
d 2 x
But the acceleration is -j^' and if we write,:
force = — kx
where k is constant and the negative sign denotes the direction
of the force, we have by Newton's Second Law of Motion
d 2 x .
m M = -**"
y It is usual to write this in the form :
dH
dt*
where * 2 = — ■•
r m
This is an example of a differential equation.
We do not here consider any series of arguments leading
logically to the solution of this equation, we merely state the
solution and verify the truth of the statement.
This equation and another slightly more complicated are so
important in Physics at an early stage that the solutions should
be remembered and the student prepared to apply them with
ease.
Consider x = A sin (j>t + a) (6)
A and a are arbitrary constants not occurring in the equation.
They occur for the same reason that B occurred in § 10 ; in fact,
we are proceeding from a differential coefficient to the function
from which it is derived and so are performing an integration
when solving the above equation. It should be remembered
also that in the complete solution of a differential equation
there must occur the same number of arbitrary constants as the
number of the order of the highest differential coefficient in the
equation.
In the present case the highest order is 2, and so we have A
and a.
■[£•+#* -o. (5)
26 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
dx
d 2 x
57 i - = -^ 2 Asin(^ + a)
Now ~£ = pA cos (pt+«), ^
= —p 2 x.
Hence the value chosen for x satisfies the equation.
Since x = A sin {pt + a) we shall have the same value of x
occumng T sees, later when a complete period has elapsed.
/. x = A sin [p(t + T) + a] = A sin (pt + a).
.\ sin (pt + a + pT) = sin (£* + a).
This is true if pT = 2*, 47c, etc.
Thus the first recurrence of the value of x is after an interval
T = 2*/p, and this is the period of the S.H.M. described bv
our equation.
We note that T = 2tu -f- square root of the coefficient of x in
the reduced equation / 7 \
O ^9
Fig. 11
Thus in considering any problem in which T is required we
have only to write down the equation of motion and but it in
this form ; we can then write down the value of T immediately
We consider the case of the simple pendulum (fig. 11).
The displacement is measured by the angle 6.
The force along the tangent to the circle described by P is
?fx Sm - e ? T ™ 8 ° if 6 is smaU - Note that the m °tion is only
b.H M. if 6 is small, since the force is only then proportional
to the displacement.
The acceleration in this direction is ~
dt 2
where s = arc OP.
or since s = Id, it is /
dH
dt 2
DIFFERENTIAL AtiD INTEGRAL CALCULUS 27
The more complicated case occurs when the effect of friction
has to be included. The force of friction depends on the sum
of a number of terms proportional to the powers of the velocities.
In slow motions, such as those occurring in the movements of
a galvanometer needle, the term of paramount importance is
that containing the first power of the velocity. As this discussion
is for the purpose of considering oscillations such as occur in
galvanometers we consider this as a typical example.
The position of the needle or coil is defined by its angular
displacement from a normal position.
For example, let OH denote the direction of a magnetic field
in which the needle normally sets.
mH
A 1
"^J 3
A
B - — H
Fig. 12
In the new position, A 1 !* 1 , let measure the angle BOB 1 .
If is small, the restoring couple due to the field H is 2mUl sin
= 2mMd, where / = half the length of the magnet. If the
magnetic moment be M this couple may be written MHO.
m denotes the pole strength. If I denotes the M. of L of the
needle, neglecting friction we have :
!»?--»=• <9)
and the motion is simple harmonic.
We assume in accordance with what has been stated above
that there is a frictional force proportional to the angular velocity
4i and write the force = c -& where c is constant.
at at
The equation then becomes :
T d 2 d dd __„ , v
and may be simplified by dividing throughout by I and changing
the notation of the constants^
28 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
It will now be written :
dH , „ dd , , a . .
— + K Jt+ n*6 = o. .. (ii)
The derivation of this result has been made by reference to a
magnetic needle oscillating in a magnetic field, let us say, supported
by a thread of negligible torsion.
The same result is obtained for such oscillations as occur in
the case of bars vibrating at the end of stretched strings under
torsion or for oscillations in the electrical discharge of a condenser.
We shall denote ~rr by 0" and ^ by 0*.
Again, we do not attempt to deduce the solution of the equation,
we merely verify a stated result.
The solution of such equations can be obtained by writing
6 = Ae mt
for on using this value of the equation is satisfied provided
m* + Km + n 2 = o.
This is evident if we substitute = m 2 Ae mt , = mAe mt .
„ — K ± VK 2 — An*
Hence m = 3— = m x or w 2 , say.
Thus the complete solution is : = Ae m i* + ~Be m 2*.
A and B are the two constants necessary in the complete solution.
If we substitute the values of m x and m 2 we find :
e = e iAe 4 +Be * 1 (12)
K 2
When — > n* the indices of the exponentials in the bracket
are real and is not periodic. merely changes in value with t
according to the exponential law. Of course K and n depend
on the particular problem considered. K measures the friction
and n depends on the nature of the restoring force. If friction
were not present the body would vibrate with period — »so
n
that this may be said to be the natural period of vibration if
there is no friction.
The case with which we are concerned is when the motion is
oscillatory, and this requires that the indices of the exponentials
within the bracket should be imaginary,
i.e. £1 <n i m
4
Write * = V — 1, we then have for the terms in the bracket *
DIFFERENTIAL AND INTEGRAL CALCULUS 29
By altering the constants we can put this in the i orm :
C sin
or more simply :
>p
K 2
.*+ Dcos
yjn^-
5!
4
t,
Esin(^J« a - — -* + <x)
■(13).
Thus the solution is now to be written :
6 = Ee ^sin [yjn* - — • t+ a)
where E and a are the constants. \
The student may, if he wishes, take 6 as given by this value,
substitute in the equation and verify that this satisfies it.
This form of the solution shows the similarity with the last
case in which we had
== A sin {pt -f a).
_Kt_
The amplitude is now Ee a instead of A. It thus varies
— K£
with the time, and since K is positive, the value of e a is less
than unity, and the amplitude diminishes with the time. The
motion is said to be damped, and the damping depends upon K.
does not maintain the same amplitude during the motion,
as in the undamped case. The curve illustrating the motion is
drawn in fig. 13.
B = Ee a where t has
t —
Fig. 13
When t = o, O = E sin a,
and when t = - -r- \\ n 2 — -— - — a,
2 \ 4
this particular value.
Other pairs of values (0, t) may be obtained in the same way,
and the curve drawn as above.
It will be noted that the curve crosses the 2-axis at the
points AxA 2 . . . and that these points are equidistant.
The interval k x k z is called the period of the motion.
30 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
It will be noticed that 6 is zero when
sin Ujn »- — * + a)
vanishes.
For simplicity write q =» J n * _ J£ a
The n sin (qt+ a) =o if'
j* + a = o, n, 2tt, 3tt, etc.
that is when t = - - n ~ a . 2 * ~ a » 3* - « ^ .
?' <1 ' q q ' *
On the curve these times correspond to the points A, A v
a 2 , A3, etc.
In the case of point A x the oscillator is at the zero position,
but is moving in a direction opposite to that at A or A 2 .
The difference between the above values of t at A 2 and A 3 is
3 TC ~ « _ " — a _ 2n
? q q'
Thus the period is L« _ j^2 . . . , / x >)
Had there been no damping, the period would have been —.
We shall call these T x and T respectively.
At such points as B,, B 2 , B 8 , etc., the value of sin (qt + a )
is numerically equal to unity, and the amplitude is then measured
by Ee 2 with the appropriate value of t.
At B 2 the value of t is^- a and at B a half a period later the
value is — - a + -T, = ^- - a.
2? ^2 x 2.q
The corresponding values of sin (qt + a) are +1 and -1 • this
means of course that the displacement is on the opposite side
of the mean position in the second case.
Thus SA =tf +-^
X>2 < -'2
In the same way
B 2 C 2 JSIi B3C.
B£~r e * = B^ = etc (15)
Thus the ratios of successive maximum displacements are
constant.
KT
The value-— 1 is denoted by x which is called the logarithmic
decrement.
CHAPTER I
MEASUREMENT OF LENGTH, AREA, VOLUME,
AND MASS.
The measurement of the length of a body may be made by
using one of the usual vernier or micrometer devices, such as the
vernier calliper, the micrometer screw gauge, or the spherometer.
The measurement of small objects may also be carried out by
use of the travelling microscope or the micrometer microscope.
In using either of these instruments care must be taken, when
viewing the image of the ends of the object, that this image is
in the same plane as the cross-hair or the small scale in the
eyepiece, otherwise parallax errors may be introduced. The
image seen should not show any relative movement with the
cross-hairs when the eye is moved across the field of view.
When focussing the cross-hair, as a preliminary adjustment,
the eye should be unstrained. The microscope is turned to a
bright distant object and the adjustment of the eyepiece should
be made so that the distant object viewed by the one eye is in
focus when the cross-hair as viewed by the other eye is also
clearly focussed.
The Comparator
When a length exceeds a few centimetres, the travelling
microscope or micrometer microscope are not used individually
as measurers, but are replaced by an arrangement of two such
instruments arranged at a variable distance apart on a fixed
graduated bed. Each microscope may be moved in the usual
manner in a direction parallel to the length of the bed. Each
microscope is provided with a scale or fine cross-hair in
the focal plane of the eyepiece. Each eyepiece is adjusted
so that the scale or cross-hair is in sharp focus for normal
vision, and is replaced in the carriage.
An object to be measured is fastened rigidly along the bed,
in the groove provided for it. The image of one end of the object
as seen by the first microscope is brought into coincidence with
the image of the intersection of the cross-hairs and the second
microscope is moved until the image of the other end on the
object coincides with cross-hair intersection in that microscope.
Thus the images of the two ends of the object as seen by the
31
32 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
two microscopes normal to its length are formed at the inter-
section of the two pairs of cross-hairs.
The object is removed and a standard rule substituted. The
readings of the standard scale seen opposite the two cross-hairs
obviously enables the length of the object to be ascertained by
subtraction.
As an example of this Substitution method we will consider the
experimental details of the following experiment.
The Comparison of the Yard and the Metre
Standard rules, engraved as finely as possible with inch and
centimetre graduations, and of lengths one yard and one metre,
are employed in this experiment.
Place the yard rule on the bed of the comparator and focus
the one microscope on a scale division near one end of the yard,
moving the latter and the microscope until this is accomplished.
Then move the second microscope until a scale division near
the other end of the rod is sharply focussed : the number of
inches included between the two being noted — 36 if the scale is
sufficiently well graduated to allow of this. Then, taking care
not to upset the arrangement of the microscopes in any way,
remove the yard scale and substitute the metre, so that the
graduations are in good focus. Move the scale so that the
image of a division near one end is in coincidence with the cross-
hair intersection in the one microscope. Under these circum-
stances the second microscope will not be opposite a division.
Coincidence of the cross-hair and the image of a scale division
is brought about by a movement of the microscope, parallel to
the length of the bed, which is measured on the vernier scale
attached to it.
The size of the gap between the two cross-hairs ± the move-
ment of the microscope is then read off in cms.
Care is taken in noting the movement of the microscope to
see exactly the unit used in these graduations. In this way we
obtain two measurements, one in each system, for the same
distance and may calculate the number of inches to the metre,
or cms. to the yard.
The vernier scale movement on the microscope is often replaced
by a micrometer screw capable of a much shorter range of move-
ment. With such a screw traverse, the movement of the micro-
scope may be readily measured to «ooi of a cm. Using sucli a
comparator the values given below were obtained :
(1) z yard = 91-5 cms. + -008 inch.
/. 1 metre s= 39/331 inches.
MEASUREMENT OF LENGTH 33
(2) 50 cms. = 19*6895 inches,
i.e. 1 metre = 39*379 inches.
(3) 1 metre = 39*375 inches + -002 inch
= 39'377 inches
mean value
1 metre = 39*362 inches.
!
The Planimeter
The estimation of the area of a plane figure may be carried out
by one of the many geometrical methods or by the use of a
planimeter, an instrument designed to measure such areas
directly. /
Of this class of instrument the Amsler planimeter is generally
used.
Fig. 14
It consists, essentially, of two arms AC and EB, hinged at A,
fig. 14; AC is of fixed length and is provided with a needle
point loaded above by a small weight as shown in the diagram.
The second arm may be varied in length by sliding that portion
of it which carries the tracer, B, into the slot provided in the
other half, EA. By means of fine adjustment, S, the length BE
may be set accurately at any division along the graduated face
of BA.
In addition to the tracer, B, this arm is provided with a small
wheel to which is attached a graduated drum which moves past
a fixed vernier scale, V. By means of the graduated drum and
vernier, the rotation of the wheel may be measured to Tff ta of
a complete turn. The axle of the wheel and drum is arranged
parallel, to the length of EB and is provided with a worm gear,
which moves a horizontal indicator, D, one division per revolution.
When placed on a plane the instrument is supported at three
points : the needle point C, the point of the tracer B, and the
point of contact of the wheel with the plane.
34 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
To measure an area the needle C is placed at a point outside
it, such that B may be moved round the boundary of the area.
For the measurement of large areas this will be impossible,
but the details for such a case will be seen later. Starting at
any point on the boundary of the area, the tracer is moved
carefully over its contour until it is finally in the starting position.
During this operation the wheel will have rotated in general a
definite number of revolutions plus a measurable fraction of a
revolution. From this observation, the area of the figure may
be calculated.
The method of calculating the area will be best understood
by first considering the theory of the instrument.
— X
Fig. 15
When the needle C is fixed in the plane of the area to be
measured, any movement of B along the boundary of the figure
will result in a movement of A along the arc of a circle with C as
centre and radius CA = a cms. (see fig. 15). Further, the
wheel, W, will roll a distance equal to the total displacement
when the movement of EB is at right angles to its length :
movement parallel to the length causes no rotation, the forces
acting on the wheel due to contact with the plane, under these
latter circumstances only produces a couple tending to move
the axle parallel to its length about the pivots.
So for any intermediate form of displacement, the rotation
produced in the wheel will correspond to the component of the
displacement at right angles to the length EB, and if n revolutions
occur, 2ttwR will measure this normal displacement, R being the
radius of the wheel.
Now it will be shown below that the area £0 be measured is
equal to 2imR x AB : since the distance of the wheel from A
MEASUREMENT OF AREA
35
does not occur in this expression we must first show that this
distance has no effect on the number of revolutions the wheel
makes.
In fig. 15, let B and B 1 be two positions of the tracer a very
small distance apart on the boundary of the area to be measured,
B B 1 K ; A and A 1 being the corresponding positions of the hinge.
Let AB = b, and suppose that the centre of the wheel is at P
(fig. 16). Draw AN 1 and PN normal to AfB' 1 from A and P and
let AN 1 = ds : PN = ds' : AP = c.
AL being parallel to A X B*, let the angle LAP = dtp.
Fig. 16
Since BB 1 is a small distance, ds, ds', and d<p are also small.
Now PN = NL + LP
or _ ds' = ds -f c- dq>. '
The distance moved by P as B traces the boundary of the
area BB X K is S ds'
or S ds' = S ds + S cd<p.
B finally returns to the starting point, and therefore 2 dtp = o
i.e. S ds' = Sis.
Thus a wheel placed at A would indicate the same movement
as the one at P, or at any other point along EB as could be shown
by the same process as above. The position oj the wheel on the
arm AB does not, therefore, affect the reading of the instrument.
To show that the area of a plane figure which does not include
the needle point C is equal to (2imR)b let us refer the figures
to rectangular axes with C as origin, as in fig. 15 •
Let CAB be one position of the planimeter and CA X B X a second
position such that BB 1 is a small displacement of the tracer.
Let the area to be measured be BB 1 K.
Referred to these axes let x, y be the co-ordinates of B.
36 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The area BB*K may be conveniently referred to polar co-
ordinate r and 0, i.e. let CB be r, /_ BCX = then if /_ B X CB
= dd, the area B*CB = \r • rdd = \rHB, i.e.
S \rW = area of BB^ . (i)
For as the radius vector moves round the figure on the boundary
remote from O, the small area contains, in turn, each element
of the area to be determined, + the external triangle from the
boundary of the figure on the side near to O ; this latter area
is deducted from the sum as the radius vector travels along this
near boundary, for here 66 is negative.
Now x = r cos y = r sin
dx = — r sin • dd -f cos dr, dy = r cos • do + sin • dr
whence xdy — ydx = r* dd.
Or, the area of the figure, £ S rHB from (i)
= \ S (xdy — ydx) (2)
Let /__ ACX = a ; A*CA = dx ; AN X A J = 90 ; CA = CA 1
= a ; AB = A^ 1 = b.
The co-ordinates x and y of the point B may be expressed as
under :
x = a cos a -f- b cos q>, y = a sin a + b sin <p.
.*. dx = — a sin a dx — 6 sin 9? dg? d[y = a cos a da + & cos yip.
.*. x dy — j/d# = (a cos a + 6 cos 93) (a cos a-da + 6 cos tp-dcp)
-f (a sin a +b sin 9?) (« sin a da -f& sin 9>-ig>
= a 2 da 4- & 2 dg? + ab cos (a — <p) d (a + 9) (3)
But (a + <p) = 2 a — (a — 9?) or d (a + <p) — 2 dec — d (v. — <p).
,\ a& cos (a — ?>)*i (a + 9?) = a& cos (a — 9?) {2 da — d (a — ?>)}
— 2 ab cos (a — • 9>) da — ab cos
(a — 9?)-d (a — <p)
and
AN 1 = ds = AA 1 cos A X AN == a • d acos (a — 9?).
so that
aft cos (a — <p)-d (a + <p) = 2b' ds — ab cos (a — q>) • d (a — 9?),
and equation (3) becomes
* • dy — y • d# = a 2 dat.+b 2 d<p + 2b'ds — ab cos (a — ?>)• d (a — ?>).
Now, when the trace moves round the curve, Sdy = o, Sda = o
for the planimeter returns to the exact position of starting :
also we have then
2 cos (a — 9) • d (a — 95) =0, for S^cos (a — <p) • d (a — 9?)
« £sin ( a - g>) J
MEASUREMENT OF AREA 37
But limits (i) and (2) are identical so that this = o (see page 22),
i.e. S (x-dy — ydx) = E zbds = ibYds.
Now Sis = 27twR, as already shown, and we saw in equation
(2) above
Z(xdy — ydx) = 2 x area enclosed in BB 1 K,
i.e. 2 area of figure = 2& 2tcwR.
/. Area of figure BB X K = b • 27mR.
Thus to measure any area sufficiently small, C is fixed in the
paper at a point outside the area and B is taken round the
boundary. The rotation of the wheel is measured, as is the
distance from the hinge to B, and the area is thus equal to the
product of BA and 27mR. »
To carry out the calculation, R and AB must be measured.
The distance from the tracer point to the hinge may be estimated
by holding the instrument parallel to a scale or squared paper,
and estimating as nearly as possible the position of the axis of
the hinge on the scale. The difficulty is of course in estimating
the true position of this axis.
The value of R may be obtained by measurement with a screw
gauge, which must be used with great care, as the edge of the
wheel is easily damaged by screwing up the gauge unduly.
Another and safer way of finding R is to note drum-reading
on the wheel, and move the wheel along a straight line ruled on
paper, until the wheel has made several complete revolutions.
The distance moved and the number of revolutions enable the
value of 2wR to be measured directly.
The two measurements described above cannot be made with
very great precision, but are performed much more accurately
in the construction of the instrument. The graduations on the
arm BA, ' 100 d cm,' etc., are made in the construction, and
signify that when this graduation is adjusted to the fixed mark,
one revolution of the wheel corresponds to an area of 100 sq. cms.
Thus, if the graduations are to be trusted, i.e. unless the instru-
ment has been subjected to rough handling, the area of the
figure is equal to«x (the number on the graduated arm opposite
the fixed mark.
The Case when the Needle is Inside the Figure.
Now let us consider the case when large areas are to be
measured, e.g. BEjE^ . . . En, fig. 17. The needle support
is fixed at a central point C, and the tracer may then be made to
trace the boundary line of the area.
But in this case it will be seen that 2nriR • b does not give the
true value of the area of the figure ; for in this case ~Ldq> = 2w
and is not zero.
38 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Take a point B on the boundary, such that BA is at right
angles to CW, the line joining C to the centre of the wheel, W ;
and draw a circle with C as centre and CB as radius, shown in
the broken line in fig. 17. Then if B is moved round this circle
the wheel W will move round a second circle of radius CW ; the
relative positions of the two arms remaining constant so that
the axle of wheel is always at right angle to the radius CW. Thus,
while B is moved round the broken-line circle, W is moving
always parallel to its axis around the second circle. So that
during the whole revolution the wheel will not rotate. That is
to say, the circle BFjF^ etc., so drawn, is such that when traced by
/
E,
=7
1
/^v
*<>
B 9
Fig. 17
the tracer, the wheel indicates zero movement. The area of the
circle is u(CB) 2 and is quite definite in size, depending on the
setting of the graduated arm. This is sometimes called the zero
circle, or datum circle.
If now we consider our area EjE^a . . . E 1]L and commence with
the pointer at B : passing from B to E 2 via E x , B is moved
outside the circle. W will move towards C, and a definite
rotation in one direction is made by the wheel. We could
bring the tracer back to B along the path E^B without altering
the reading of W. This reading corresponds, in the way previously
considered, to the area EaFBE!.
However, having reached E 2 , continue along the boundary
MEASUREMENT OF AREA 39
E2E3E4. To do this the tracer moves inside the zero circle and
W will therefore move outside its circle, i.e. in the opposite
direction to the previous movement.
Similar movements occur round the figure. The planimeter
therefore adds algebraically the area of the curve outside the
zero circle. Having carefully noted the direction of rotation
of W when B traverses such a part as E 2 , we can tell from the
final reading of the dial D whether the figure is of less or greater
area of the zero circle. Suppose n revolutions of the wheel are
indicated and the arm is set at/the ' 100 □ cm. ' mark. If the
indication of the n revolutions is in the same direction as the
indication of the wheel when moving outwards, e.g. along BEiE 2 ,
the area^of the figure is
100 • n + nCB 2 sq. cms. if CB is measured in cms.
W
Fig. 18
The same difficulty as before is met with in finding CB. The
instrument may be set along two lines drawn at right angles so
that the point of contact of the wheel is at the intersection of
the lines, and B and C are each on one line, fig. 18 ; CB is measured
directly or WB and WC are measured and CB calculated. How-
ever, the value of this zero circle is inscribed on a second face of
the arm BA. Adjacent to the '100 □ cm.' mark is a number
which gives a value of the area of the zero circle, not usually in
sq. cms., but in revolutions.
Thus, if in the case taken above, there are n revolutions in-
dicated and the second scale gives m, as the equivalent area of
the zero circle, the area of the figure is
100 (n + m) sq. cms.
To become acquainted with the instrument and familiar with
the method of using, draw several small regular figures, calculate
the areas, and then find them, using the planimeter at different
graduated scale-settings.
Draw a circle of known radius, calculate the area. Move the
tracer round the circumference when the movable arm is set
at various graduations. Note the number of revolutions in each
case. From the calculated area and the number of revolutions
40 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
observed find the area corresponding to one revolution, and
so check the graduations. Repeat this process with several
measured areas, and obtain a calibration of the instrument.
Measure b and R and again calculate a value for the area
corresponding to one revolution. This is not so accurate as
the above method, but shows the value of the construction as
described.
Draw irregular figures on squared paper and find the areas
by adding the squares. Compare these results with those
obtained by the planimeter as calibrated.
Measure the radius for the zero circle at each setting, and
check the value of the graduated scale, by calculating the area
and dividing by 2nRb. The quotient should agree with the
graduations on the scale.
The instrument may subsequently be used for area measure-
ments when required, making use of the calibration if errors
are found by these experiments.
Note.
When the boundary of the figure does not cut the zero circle
the process is identical ; for, suppose ABCD be such a figure,
and the zero circle is completely inside the figure as shown.
The process is as before. Start at any point A, and, with the
fixed point inside the figure, trace round the boundary in one
direction, say along the path ADC. Join AC. Then, if the
tracer be taken along CA the value of the area ADC may be
calculated on the foregoing theory. If now the tracer be brought
back along AC and thence via B to A, the wheel will indicate
precisely the same as if the path ADCBA had been taken, for on
reversing along AC the record on the wheel for the path CA will
be neutralized. The foregoing theory shows that the sum of
MEASUREMENT OF AREA 41
the readings for the two paths record the number of revolutions
corresponding to the two areas, and is equal to that for the
boundary of the figure. Hence, starting at any point and
tracing the complete boundary gives a record of the number of
revolutions n, which when added to the zero circle number
enables the area to be evaluated.
The Graduation and Calibration of a Tube
Graduation. The method of graduating a glass capillary tube
described below is one which could be employed generally in
the etching of scales on glass.
A length of glass capulary'tube is coated with a thin layer of
paraffin wax, by warming the glass and applying a small block
of wax to the heated surface. The waxed tube is again gently
heated, in the Bunsen flame, and rotated so that when cooled it
is evenly coated with a thin wax layer.
The tube is then clamped to a board at such a height that the
upper surface is approximately on the same plane as a metal
scale which is fixed in line with the tube, and at the other end of
the board.
A beam compass is arranged so that the two needle points
are from 50 to 100 cms. apart, depending on the length of the glass
tube. One needle point is placed in a cm. graduation of the
scale and the other needle point is drawn across the wax coating
of the tube, removing a straight line of wax.
The beam compass is moved 1 millimetre, and a short scratch
again made on the wax coating. This process is repeated until
the required length is marked in this way, the whole cm. marks
and 5 mm. marks being made larger than the rest.
By means of a steel point the cm. divisions, o, 1, 2, etc., are
scratched on the wax coating.
To etch this scale on the glass, a swab of cotton wool, fastened
at the end of a stick, is dipped into hydrofluoric acid, and then
applied to the wax-coated tube. Where the scratches of the
graduations and numbers have removed the wax, the glass is
etched by the acid.
In this process care is taken, of course, to avoid any of the
acid touching the skin or clothing. ,
One end of the tube is also given several scratches, and is
covered with acid at the same time as the scale. After ten
minutes or so examine one of these scratches near the end of the
tube, by scraping away the wax : if the glass is sufficiently
etched, the whole process is stopped ; if not, leave for a few
minutes and then examine a second test scratch near the end,
and so on until the etching is complete. Wash off the acid with
42 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
tap water and remove the wax, the last traces may be removed
with turpentine or xylol.
A little rouge, or lamp-black and shellac, rubbed over the
etched scale brings out the markings.
Calibration of the Tube.
The inside of the tube is cleaned either by the usual method,
using caustic soda, alcohol, ether and tap water, or by immersing
it for about twelve hours in a solution of potassium bichromate
and strong sulphuric acid (equal parts). The tube is finally
washed well with tap water and dried.
A thread of mercury, about one-third the length of the tube,
is drawn into the bore, e.g. ip cms. for a tube 30 cms. long. By
means of a small length of rubber tubing attached to one end of
the graduated tube, the position of the thread may be varied at
will by altering the pressure within the tube.
The mercury is first adjusted so that one end comes near the
zero graduation. By eye estimation the position of the other
end may be determined to £ of a subdivision ; hence the length
of the thread when in this position is obtained. The mercury is
then moved, by gently blowing through the rubber tube, until it
occupies the central third of the tube, and its length is again
estimated in scale divisions. The length taken by the mercury
is finally measured in a similar manner in the remaining third
of the tube. The volume of mercury is constant, so therefore
the length indicated will depend on the average cross-section
of that part of the tube filled with mercury. Thus this pre-
liminary test will show the general form of the bore.
A short thread of mercury, about 1 cm. long, is next introduced
into the tube, replacing the 10 cm. length. This is moved to
occupy approximately the space between the o and 1 cm. gradua-
tion. The tube is arranged horizontally on a sheet of mirror
glass on the platform of a travelling microscope. The readings
on the scale engraven on the glass tube, corresponding to the
two ends of the thread, are taken by means of this microscope.
For this, the cross-hairs in the eyepiece of the microscope are
turned so that one is parallel to, and the other at right angles to,
the length of the tube. The intersection of the cross-hairs is
brought into coincidence with the meniscus, so that one cross-
hair appears tangential to it. The vernier reading of the micro-
scope is noted. It is then moved towards the middle of the
mercury thread until a graduation on the tube is seen. The
difference between the vernier reading under these circumstances
and the last readings gives the distance between the end of the
thread and the glass tube scale reading. The distance between
adjacent scale readings on the tube is measured in like manner.
MEASUREMENT OF VOLUME 43
From these readings the length of the mercury projecting beyond
an engraved division on the tube may be calculated in terms of
the graduation of the tube, and the length of the thread of mercury
may be measured in these units.
The mercury is moved to occupy the space between the i and
2 cm. graduations, and again measured. This is repeated along
the length of the tube, and the result entered as in column 2 of
the table. The mean value of this length is obtained, and in
the third column the difference between the observed and the
mean value is tabulated for each part of the scale (p. 44).
From these observations we may calculate the correction to
be applied at each part of the scale to convert the scale readings
to the corresponding volume readings. Thus, if a tube is divided
into 20 cms. the mean value of the thread length as calculated
from column 2 gives the reading for all parts in a uniform tube
which is 20 cms. long, i.e. the difference between the length of
the thread between o and I as observed, and the mean value
gives the correction to be applied to the scale reading to correct
it to the true volume of thread equal to ^ of the total volume of
the bore.
For the bore between 1 and 2 cm. graduations, column 3 again
gives the correction to be applied to this small part of the bore.
To correct the total length from o to 2, to give -£$ of the volume,
the sum of the first two terms in column 3 must be added alge-
braically. So, to find the correction for any scale reading, the
sum of the third column must be taken up to and including the
difference term for that reading.
This is done with the figures for a tube tested in this manner
and the correction entered in the last column.
With the corrections so obtained a correction curve should
be drawn, plotting the correction as ordinate and the scale-
reading as abscissae.
This kind of calibration is often required in thermometry,
where equal volume expansion of mercury is employed. In
such a case the scale-readings will not be uniform. In calibrating
the thermometer the instrument maker fixes a few points along
the stem by comparison with a standard thermometer. This
method automatically makes some allowance for the change in
the cross-section of the tube.
In the example taken the stem was divided into 1 cm. lengths
for calibration : it is doubtful whether much is gained by such a
small subdivision, but is used in the example above to bring out
the principle involved. In practice, the use to which the tube
is to be placed determines the number of points tested along the
tube. With a thermometer 10 points along the length will be
ample for most purposes, i.e. with a o° to ioo° thermometer,
44 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the thread detached should be the length of io°C, or about
3 cms. This is described in greater detail later ("Thermometry ").
POSITION OF
THE THREAD
IN THE TUBE
LENGTH
OF
THREAD
DIFFERENCE
FROM MEAN
CORRECTION
— I
I*040
—0*044
- -044
I — 2
I*O00
— *004
- -048
2— 3
I '000
— -004
- -052
3— 4
0-975
+ *02I
- -031
4— 5
0-950
+ -046
+ -015
5-6
1-000
— -004
+ -on
6-7
0-980
+ -016
+ -027
7-8
I -000
— -004
+ -023
8-9
I -000
— -004
+ -019
9 — io
1-000
— -004
-f -016
10 — II
1*000
— -004
+ -on
II — 12
1-020
— -024
- -013
12—13
I-OOO
— -004
— -017
13—14
0-980
+ -016
— *00I
14—15
I-OIO
— -014
- -015
15—16
0-975
+ -021
+ -006
16 — 17
I-OOO
— -004
+ -002
17—18
1-000
— -004
— -002
18—19
1-000
— -004
— *oo6
19 — 20
0-990
+ -006
o-ooo
mean
0-996.
The Balance
The balance, as seen in fig. 20, consists of two pans
suspended by knife-edge supports, KK, from the ends of equal
arms of the beam ST, which is pivoted on a central pair of knife-
edges.
The central knife-edge is made of agate and rests on small
plates of the same material, and KK support such plates.
The free and sensitive movement of the beam depends upon the
sharpness of the knife-edges. It is therefore important that
they should support weight only when in use. To release the
knife-edges the central pillar Q supports an ' arrestment ' A.
The fixed arm A carries at each end two points which fit into
a pair of cups on the upper agate plates at K. When the beam
is lowered the weight is taken from the knife-edges by this means.
MEASUREMENT OF MASS
45
When releasing the beam, the latter should be in the horizontal
position, i.e. the beam is only arrested when two outer knife-
edges are opposite the supporting point of the arrestment. In
such circumstances all three knife-edges are released with an
absence of jolting.
H.L.CRimTIU.
Fig. 20
In the ordinary way the centre of gravity of the beam, etc.,
is just under the line of support. The lower the centre of gravity
the less sensitive the instrument. The position of the centre
of gravity may be varied by an adjustment of the 'gravity
bob ' B.
46 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The two masses W may be adjusted to change the rest positions
of the pointer which moves over the small scale at the base of
the pillar Q. >
A fuller account of the balance will be found in " The Theory
of the Physical Balance," by J. Walker.
In using the balance, several simple things should be
remembered, viz., before using, dust the pans with a camel-
hair or similar soft brush ; arrest the beam before changing
masses in the pans ; use the rider to start oscillation. Never
touch the pans or ' weights ' with the fingers, or place chemicals
or wet vessels on the pans ! Final observations should be
performed with the case closed.
METHODS OF WEIGHING
The Oscillation Method of finding the Rest Position of the Pointer
The arrangement of knife-edge supports ensures that the
friction is reduced to a minimum. Consequently, when the beam
is oscillating it will have a long period of swing which is very
slightly damped, i.e. the pointer moving just clear of the scale
will take a long time in which to come to its final position of rest.
This rest position may be estimated by the method of oscilla-
tion as follows : Suppose the scale be graduated from left to
right into, say, 20 graduations. The end of the pointer, being
arranged as near to the scale as possible, is viewed by the eye or
by the aid of a lens, taking care to avoid any slight parallax
error. The position of the turning point should be estimated
to at least J of a small division — if possible to T V of a division.
Seven such turning points should be obtained — three on one side
and four on the other side of the zero position. The mean value
of the three on the one side and also the mean of the other four
should be obtained. Then the mean value of the two means
gives the position of rest. (See example in the table below.)
For such damped oscillation the mean of an even number of
observations on each side of the zero would give a value biased
towards the side of the zero of the first observed turning point ;
so an odd number is taken as stated. For if we assume that
the damping is small and that the first swing through the rest
position is to the left, we have, from a consideration of such a
lightly damped oscillation, the angular deflection, 0£, the first
T
swmg to the left is after a time - and is (see page 28),
4
8l = e- A *
where A is a constant, O the undamped oscillation.
MEASUREMENT OF MASS
This is, under the conditions of very small damping,
01 = O (i - A ~) to the left.
The deflections are therefore :
47
TO THE LEFT OF ZERO
TO THE RIGHT OF ZERO
, (x_aT)
».(x-A3J)
eo (x-A ? )
4"^
mean O (i - A^) O (* - A^
i.e. the mean is o since the mean position left and right is an equal
distance to each side of the zero.
Zero Position of the Unloaded Balance
As an example of the method of oscillation, the following
determination of the zero position for the unloaded balance is
taken.
The beam is released from the arrestment and given a slight
oscillation over, say, about 5 divisions of the scale on each side
of the zero. After a few swings the oscillation should be steady,
and seven readings, are taken, as under :
READING TO THE LEFT
READING TO THE RIGHT
I. 8-8
3. 9-2
5- 9'5
7. 9-9
2. 12-0
4. II'l6
6. n-2
Mean of left hand reading = 9-35
Mean of right hand reading = n-6
~ . . t , Q'^5 + H'6
Zero positions unloaded = ^-^ = 10-475
A mean value of three such determinations gives the rest point.
In general, call this rest position for no load X.
48 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Sensitivity of the Balance
The sensitivity of the balance is denned as " the angle through
which the beam will turn for one milligramme difference in load in
the two pans."
It is usual in practice to measure the sensitivity in terms of
the movement of the potater over the scale.
This should be measured for no load, by adding one milligramme,
by means of the rider, to one pan and noting the position of rest
by the oscillation method as before.
Load each pan with increasing equal masses up to the limits
specified for the balance, and for each mass find the sensitivity
i.e. change of rest position for one milligramme on one pan.
If the beam were rigid and the knife-edges truly in the same
line the sensitivity-load curve would be a straight line parallel
to the load axis.
For increasing loads there may be, however, a slight depression
of the knife-edges at the pan supports, and a change in sensitivity
as a result.
Find the sensitivity-load curve for the balance and note the
region, if any, of maximum sensitivity. The value of the load
at which there is maximum sensitivity depends upon the use for
which the balance is designed.
Method of Gauss or Double Weighing
As an example of this method we will consider a determination
of the number of grammes equivalent to one ounce troy ; the pro-
cess can naturally be repeated with any other unknown mass.
Place the ounce troy in the left-hand pan and add ' weights '
(grammes) to the right-hand pan until the pointer remains on the
scale when the beam is released ; the rest position of the oscillat-
ing pointer is estimated in the manner previously used. Let
this be y on the scale. y
Now find the sensitivity of the balance at this load, by adding
one milligramme and proceeding as before. Suppose the sensibility
for this load is s.
The mass of the ounce troy in grammes is the mass (to the nearest
centigramme) in the scale pan + ( ^7*) m^S 1 "* 11111165 = M i»
say, when x is the zero reading for no load.
The ounce troy is then transferred to the right-hand pan and
the process repeated by adding ' weights ' to the left until, to
the nearest milligramme, M 2 , the mass as so compared is obtained.
Suppose that the two arms of the balance are of slightly different
length, the left-hand arm being a cms. and the right b.
MEASUREMENT OF MASS 49
Then W being the true mass of the ounce troy, neglecting
buoyancy, >
aW = M x 6
«M 2 = Wft
Hence W = VMA
The difference between M x M 2 and W will be very small, so that
we may take as an approximation
W = i(M, + M,).
Borda's Method, or the Substitution Method
A second method, quite as accurate as the double weighing,
is a simple method of substitution. It eliminates equally well
the errors due to unequal length of the arms, etc.
The ounce troy (the unknown mass) is placed on a scale pan
and lead shot is used to counterpoise it. The position of rest
when the counterpoise is complete is noted by the method of
oscillation.
The ounce troy is now removed and replaced by standard
masses until balance is again obtained. From the sensitivity
of the balance for this load we may estimate very readily to a
milligramme the mass which has exactly substituted the ounce
troy. Thus the mass is obtained, avoiding errors due to faulty
construction of the balance, etc.
Buoyancy Correction.
When discussing methods of weighing'no account was taken of
the buoyancy of the air on the ' weights ' and the mass to be
compared with them.
Suppose, as before, we find that W is the mass of the body, by
one of the methods above ; the true mass allowing for buoyancy
we will denote by M.
Suppose that in the determination of W we used copies of
standard masses made of a substance of density D.
Let p be the density of the ' unknown ' mass and o the density
of the air.
We have really compared (1) (the true mass of the body M —
the buoyancy on the mass) with (2) (the mass W of the ' weight ' —
the buoyancy on the ' weights '). These two quantities are equal,
(m-;..)-w-;W..
i.e. M =w(i -~\
a
I
-M' + -;-»--)
50 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
neglecting -= cf . i.
= W + W ( I p-D>
The observed value of W has therefore to be corrected by the
factor W g -i)..
This correction depends on the density of the ' weights ' and
the substance and the density of the air.
For most purposes the density of the air may be taken as •0012
grm./c.cm. and for general use a table may be calculated giving
the value of the correcting factor I =- J o for the two common
materials used in the manufacture of ' weights,' brass and
aluminium, taking the density of brass = 8-4 grms./c.cm.
Aluminium = 2-65 grms./c.cm. Thus :
DENSITY OF
SUBSTANCE
WEIGHED (p)
CORRECTION FOR BUOYANCY
BRASS ' WEIGHTS 'D =8-4
ALUMINIUM
' WEIGHTS ' D =2-65
V p 2-65/
•5
•55
•60
etc.
•00226
etc.
etc.
•00195
etc.
etc.
CHAPTER II
MOMENTS O^ INERTIA AND DETERMINATION OF ' g '
Kinetie Energy of a Body Rotating about an Axis
Let ABC (fig. 21) be a section of a body by a plane at right angles
to the axis about which it is rotating, O being the point of inter-
section of this plane and the axis.
,FlG. 21
If we imagine the body to be subdivided into a large number
of very small particles of mass m x , m 2 , m 3 , etc., distant r v r 2 , r 3
cms. from the axis, it is evident that when the body rotates
each particle will move with a velocity which depends on the
distance r from the axis.
Consider one such particle at P, of mass m and distant r cms.
from O. If the body rotate with a uniform angular velocity w,
in the direction of the arrow, and if v is the velocity of P in the
path, we have
v
- = w.
r
The kinetic energy of this particle is %mv z = \mw % r i . For
all such particles the total kinetic 'energy of the body is therefore
^MjW^r 2 + ^m z w 2 r 2 z + J^gze'Va 2 + . . . .
i.e. Kinetic Energy = H\w 2 mr*
= \w* Emr*
The sum of such quantities as mr 2 , taking every particle
throughout the body, is denned as the moment of inertia of the
Si
52 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
body about the axis through O. If we denote this by I , then the
kinetic energy of the body is
ll wK.... (i)
thus I replaces the mass, and tfre angular velocity replaces
linear velocity in the corresponding case for linear motion
where K.E. = %mv 2 .
In the same way if the axis passed through the centre of
gravity ; I being the moment of inertia about an axis through
the centre of gravity, K.E. = $Iw 2 , 1 being of a different magni-
tude from I .
To express the moment of inertia of a body about an axis in
terms of the moment of inertia about a parallel axis passing through
tfee centre of gravity, we proceed in the following manner :
In fig. 22 let ABC be a section of the body at right angles to
either axis. Let G be a section of the axis passing through the
centre of gravity and O be corresponding point of intersection
for a parallel axis, the distance OG being fixed and equal to
a cms.
Fig. 22
Consider at any point P a small particle of mass m gm., OP
being r cms.
The contribution of this particle to I is mr 2 . Now produce
OG to D, and from P drop a line perpendicular to OD meeting
it at D.
f 2 = OP* = OD 2 -f- PD 2 = PD 2 + DG 2 + GO 2 + 2OG • GD
= PG 2 + a 2 + 2a • GD
Thus I = Em (PG 2 + a 2 + 2a • GD)
= SwPG 2 + Swa 2 + X2ma • GD.
Now SwPG 2 = I, the moment of inertia about a parallel axis
through the centre of gravity.
Sma 2 = a 2 Zm = a 2 M,
MOMENTS OF INERTIA AND DETERMINATION OF 'g ' 53
where M is the total mass of the body. Further, the expression
2aSGD«w = o, for by the definition of the centre of gravity the
sum of the moments {GD-m) throughout the body, about an
axis through G, is zero.
Thus , . I = I+^ 2 m (2)
Radius of Gyration.
We have defined the moment of inertia of a body about an
axis as 2#w 2 . Now if the whjole of the mass of the body were
concentrated at one point distant K from the axis, we should have
I - K 2 M.
If the distance K were s o chosen that
K 2 M = Emr 2 = 1 v -
it is called the ' Radius of Gyration ' of the body about the
axis of rotation taken.
As with the moment of inertia K has different values depending
upon the axis chosen.
Moment of External Forces.
Considering a rotating body as before, we may readily deduce
an expression for the moment of the external forces applied to
the body to impart a definite angular acceleration.
Imagine a force applied to such a body as shown in fig. 21
Suppose the body to be subdivided into small particles as before,
of which one at P has mass m and is r cms. from O. Then
v, the velocity of the particles, is given by
dd .
v = r • -r. or rO
at *
the acceleration of the particle in its path is
dv d*B ■■
dt= r aW ==rd
this is occasioned by a force mr§, whose moment about O is
mr 2 0. For the whole body to rotate with this angular acceleration
the total external couple applied is thus
Sw 2 = QXmr 2 = 10 (3)
Thus the moment of external forces applied to the body is I'd.
Calculation of the Moment of Inertia for a Solid about any Axis.
The numerical value of the moment of inertia of a solid about
any axis may be readily obtained by integration. Having
calculated this value for an axis passing through the centre
of gravity, the corresponding value for the case of the body
suspended through a parallel axis may be obtained by adding
the term Ma 2 as shown on page 52.
54 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We will consider an example of such calculations which is
often employed, especially in magnetism, and which illustrates
the points already considered.
Calculation of the Moment of Inertia of a Rectangular Rod about
an Axis at Right Angles to its Length and passing through the Centre
of Gravity.
Let ABCD be the rectangular bar, with centre of gravity at G,
and supported by an axis KK 1 passing through G, normally
to the face AD, fig. 23. Let
M be the mass of bar (assumed to be uniform)
p the density of the material of the bar
ol the length of the bar
2& the breadth of the bar
2d the depth of the bar.
-y b
Fig. 23
Let G be the origin of a system of co-ordinate, the z axis coincid-
ing with the axis of rotation : the x and y axes being at right
angles to this, the x axis being parallel to the length, the y
parallel to the breadth.
The most convenient method of finding I the moment of
inertia about KK 1 is to consider firstly a very thin section of
the bar cut at right angles to the x axis, and of thickness 3x.
Through the centre of mass G 1 of this section, imagine an axis,
LL 1 , parallel to KK 1 . TJie moment of inertia of this section
about KK 1 is equal to the moment of inertia about LL 1 plus the
product of the mass of the section and x 2 , where x is the
distance between G and G 1 . Imagine a very small rectangular
portion PQRS of EFHI, y cms. from the axis and of width dy
(fig. 24).
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 55
The mass of this parallelepiped is [dx dy 2cC\ p. The moment of
inertia of the section EFHI about LL 1 is therefore
fzddxdy gy* = 4^- p • dx.
J -h 3
About KK 1 the moment of inertia is therefore
***'"** + 2d2b 9 xHx.
3
Hence the total moment of inertia of the whole body about the
axis KK 1 is *
3
8bdlp
pdx + 4 bd x
(6 2 + I 2 )
'■pdx J
M
- j(b 2 + 1%
EXPERIMENTAL MEASUREMENT OF MOMENT OF INERTIA
When the moment of inertia of a body cannot be conveniently
calculated it may be found experimentally by imparting to it
a known amount of energy and observing the resulting rotation,
or, if the body is small, the method of the moment of inertia
tableonay be employed, whereby the change in moment of inertia
in a given system, due to the body, may be directly calculated.
The following typical" experiments will make these methods
evident.
Moment of Inertia of a Fly-wheel
To find experimentally the moment of inertia of a fly-wheel
about the fixed axis of rotation a mass is attached to the axle
56 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
of the fly-wheel by a cord which is wrapped several times round
the axle. When the mass descends, it causes a rotation of the
wheel. The mass in its descent loses a definite amount of
potential energy. Neglecting friction for the moment, this loss
is equated to the gain of kinetic energy of the mass and the fly-
wheel, and an equation results in which all the terms are known
or measurable except I, the moment of inertia of the wheel and
axle about the fixed axis.
Fig. 25
The wheel might be supported on a horizontal or vertical
axis. The two usual types met with are seen in fig. 25. The
process to find I is the same, so we will consider one of them —
the vertical axis type.
The mass m is attached to the axle at a point where there is
either a hole or a pin. If there is a small hole in the axle as
at P, then to the end of the cord a small ' pin ' is attached. This
can be made from a short length of suitable-sized brass wire
to fit easily in the hole, or if the axle has a pin projecting, a loop
is made at one end of the string. The length of the string is so
adjusted that when m is on the floor, or whatever solid object
is to arrest it in its descent, the other end of the string may be
just attached to the axle. So that when the mass descends,
the moment it is arrested, the string leaves the axle.
If w be the angular velocity imparted to the wheel, and r
be the radius of the axle, the velocity of the mass m just
before striking the floor is rw. So that, neglecting friction, we
have mgh = fynrV -f £Ize> 2 (4)
where h is the distance through which m has fallen.
To measure the angular velocity, a chalk-mark is made on the
circumference of the wheel, in a position which can be seen
the moment the mass touches the floor. The number of revolu-
tions, n, made by the wheel after the mass becomes detached
is counted by observing the chalk-mark. The time taken for
the wheel to come to rest whilst completing the n revolutions is
also observed (t sees.).
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 57
The wheel is finally brought to rest by the frictional forces
acting against it. If this frictional force is constant, the wheel
is uniformly retarded. It commences with a definite angular
velocity and finishes with zero angular velocity, so that the initial
velocity is double the average velocity.
Now the average angular velocity = —7- radians per second,
Ann
i.e. w —- L -r*
The linear velocity of the mass m is - —
Another method, though inferior to the above, is to observe v
directly by timing the descent of the mass. If the mass descends
the distance h in t 1 sees, the average velocity is — ' and the final
2h
velocity is -pr-
As we have noticed above, the frictional forces are not always
negligible, so that, for a more accurate determination of I,
allowance must be made for the energy lost in overcoming friction.
Let there be n x revolutions of the wheel during the descent of
the mass, and let / ergs be the energy per revolution used in
overcoming the frictional forces, then the total energy expended
in this way is n-J.
mgh = fwu 2 + \lw 2 + nj.
Now we already know that the energy possessed by the rotating
wheel, \Iw z , is used up in overcdming friction in n revolutions,
i.e. fn = %lw 2
1
f = ^Iw\
n
i.e. mgh = \mrhiQ*+ |I*Wi +— M (5)
Experimental Details.
Arrange the cord round the axle so that throughout the whole
of the unwinding the cord from the axle to the pulley, T, is
practically horizontal, or at right angles to the axle.
Bring the mass m so that the bottom of it is level with a fixed
point, and the string of such a length that it fulfils the conditions
already stated.
The distance h from the fixed point to the floor is directly
* measured. The number of revolutions the wheel makes whilst
the mass is descending may be determined by making a chalk-
58 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
mark on the axle and allowing the mass to descend slowly,
counting the number of revolutions (%) during the descent.
The mass is once more wound up and allowed to fall freely.
When it is heard to strike the floor a stop-clock is started and
the number of revolutions of the wheel before being brought
finally to rest is counted, i.e. n and t are observed : m the
mass is known and w = ^-7—-, hence the value of I is calculated
by the aid of equation (5).
The experiment is repeated two or three times with the same
mass and the mean value of — taken.
I is further checked by repeating with two other masses m 1
and m 2 .
The cord used should be of small diameter compared with the
diameter of the axle, otherwise the value of r in equation (5) is
the sum of the radii of the axle and cord.
Rolling Bodies
The two following experimental methods of finding the moment
of inertia of a body about a given axis depend upon observations
of rolling bodies.
The energy of a rolling body may be very simply obtained.
Consider, for example, a cylinder rolling with a uniform linear
velocity v cms. per second (fig. 26).
AB, the line of contact of the cylinder and the plane on which
it rolls may be regarded as the momentary axis of rotation of
the cylinder.
The kinetic energy of the body is therefore given by
£ze> 2 (moment of inertia of the cylinder about the axis AB)
where w is the angular velocity of rotation.
Now, if a is the radius of the cylinder, m its mass, and I the
moment of inertia about a parallel axis through the centre of
gravity; the kinetic energy is :
\ w 2 (I + ma % )
= |lrc; 2 + \mv* (6)
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 59
That is, the kinetic, energy is equal to the sum of the kinetic
energy of rotation and translation.
Wheel and Axle on an Inclined Plane
The moment of inertia of a wheel and axle about an axis
passing through the centre of gravity and parallel to the axle
may be obtained by observing its descent down an inclined plane,
and applying equation (6).
W
7&
n>£acm3.
/// / /
$
m
Fig. 27
The method of rotation will be apparent from a consideration
of the section, fig. 27. R and R^are rails supported on a hollowed
inclined plane. The axle of the* wheel rests upon the rails. The
whole plane may be inclined at any angle to the horizontal.
For each inclination the wheel and axle is allowed to roll down a
measured length, I cms., of the plane, in a time which is measured
by means of a stop-clock (t sees.).
terns'
Fig. 28
If the vertical distance between the starting position and the
finishing position be h cms., fig. 28, and v be the final velocity
acquired in the descent, we have, equating potential energy lost
to kinetic energy gained,
mgh = %mv 2 + £1 M (7)
where m is the mass of the wheel and axle, and a the radius of
the axle.
The body starts from rest and moves with a constant accelera-
tion ; the final velocity is therefore twice the average velocity.
This latter is equal to -, i.e. v = —•
The plane is adjusted by suitable means to one fixqd inclination.
The wheel and axle is placed at a convenient marked starting
60 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
point on the rails. The position of the centre of the axle is noted
by means of a vertical-reading simple cathetometer. The position
of the centre of the axle is also noted when the wheel is against
the stop at the other end of the plane. The length I along the
plane between these two points is measured directly. The value
of the mean time of descent for three experiments is obtained.
The mass m is also obtained by means of a spring balance or an
ordinary balance which is capable of weighing such a mass.
a is obtained in the usual way by means of vernier callipers.
2.1
Hence, from (7), substituting the value - for v,
'ghP
I^ff-x)
The experiment is repeated for several values of h and the
mean I is obtained.
A
D
B
1 w
H-H*
,
Fig. 29
Moment of Inertia 0! a Disc Supported on Strings
A disc, usually made of wood, is suspended by means of a metal
axle on two strings, as shown in fig. 29. The string is wound
evenly on the axle AB on both sides until as much string as
possible is wound up. If now the axle and disc, of mass m
grammes,' is released, it will descend until the whole of the cord
is unwound ; it will then rise again due to the string being wound
on the axle in the other direction.
Suppose that from the starting point to the lowest point
reached the distance the centre of the axle moves is h cms.,
and that the linear velocity at the moment when all the string
is just unwound is v cms. per sec, then, if r is the radius of the
axle, we have as the energy equation (equating potential energy
lost to kinetic energy gained) :
mgh = %Iw* + \mv*,
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 6l
I being the moment of inertia about an axis passing through the
centre of gravity and parallel to the axle.
v
w the angular velocity at the lowest point ^s = -,
i.e. mgh —-S-j- + fmv 2 .
I = (mgh - frnv 2 )^-
If the time of descent of the disc is t seconds, the average
velocity is - cms. per sec.
2.h
v, the final velocity is therefore — .
i — '*(£•- *)
Experimental Details.
Weigh the disc, then measure the distance between the position
of the centre of the axle in the starting position and the final
lower position.
The value of r is equal to the sum of the radii of the axle and
the cord which supports it, unless the cord has small radius
compared with the radius of the axle. These radii are measured
by means of a micrometer screw.
The cord is wound evenly on the axle until the disc is at the
starting point. Care is taken to ensure that the axle is horizontal,
otherwise the disc fouls the cords in descent.
The time of descent is measured several times by means of a
stop-clock, and the mean value taken.
The Bifilar Suspension
In order to determine the moment of inertia of a body about
an axis passing through its centre of gravity, we may make use
of a bifilar suspension of known or measurable dimensions.
The body is suspended with the axis of rotation vertical, and the
time of vibration of the system, T, obtained by observing the
time of 40 or 50 complete swings.
If, for example, we wish to determine the moment of inertia
of a cylinder about an axis through the centre of gravity, the
cylinder is supported in two wire stirrups CE and DF (fig. 30),
which hang at the ends of two very thin wires AC and BD, which
are fixed at A and B, AB being 2d cms. The distance between
C and D remains fixed and equal to 2d 1 cms.
When the body is displaced slightly in the horizontal plane
62 ADVAN£ED PRACTICAL PHYSICS FOR STUDENTS
it will perform oscillations whose periodic time T may be ascer-
tained as already shown.
If m is the mass of the cylinder and / is the length of the wire
AC or BD, we may readily see that
t = 2x /te:
^ mgdd 1
For, let the tension in the string be t dynes and O be the mid-
point between A and B.
j — i
3
Fig. 30
When viewed from above, fig. 31 (a) represents the relative
positions of the four ends of the wires. When displaced, the state
of affairs is seen in fig. 31 (b). Where A 1 and B 1 are projections
of A,B, fig. 30, on the horizontal plane through CD, and C 1 , D 1
are the displaced positions of C, D.
Consider the forces at D. Due to the tension on the string
there is a force t, which has a horizontal value t cos a, where a is
the angle between the string BD 1 and the horizontal. Now
B^ 1
cos a = — - — » for, in reality, the point of suspension, B, is above
B 1 , BB^ 1 being a right-angled triangle.
So that along D^ 1 in the horizontal plane there is a force
/-«^ (8)
Of this force the component at right angles to the displaced
body, i.e. C^ 1 , is effective in restoring the body to its original
position, the component along the direction OC having no
turning moment.
From B 1 draw B*E at right angles to C 1 D 1 . The component
MOMENTS OF INERTIA AND DETERMINATION OF *g ' 63
EB 1
normal to OD 1 is / sin EDW =%i^r = / 1 ' sa y»
from (8) f 1 = EB 1 '- (9)
A similar force acts at C 1 , constituting a restoring couple of
moment/ 1 - OD 1 .
The restoring couple is, substituting value of f 1 from (9),
EB 1 *
OD 1 .
A O.
2d-
1T
D (a)
(b)
Fig. 31
If the small angle of displacement, D 1 OB 1 = 0.
EB 1 = OB sin = d sin 0.
OD 1 = 2d 1 .
2dd x
/.Moment of the couple = —j- t sin 0.
is usually made very small, so we have for the value of the
restoring couple:
2dd 1 mg ^
I
= **%%. e (10)
V
wis,
for when d is not very different from d 1 1 =£= — .
We have already seen (p. 53) that the moment of external
d 2 6
forces acting on a suspended system is I -^ ; this quantity is
equal and opposite to the restoring couple,
i.e.
T d*d _ mgdd 1
dt* I
This will be recognized as the equation of simple harmonic
motion (p. 25), as we have the angular acceleration = It— X
the angular displacement, ° . being a constant.
64 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The time of vibration is therefore
27C / II f TJ \
It is interesting to note that for bodies of the same dimensions
and of uniform density the value of T is the same. For,
let p be the uniform density, then I = Ewr 2 . Consider the
body divided into small volumes v, : Smr a = Xvpr 2 = pE vr % .
Also m = Spv = pSv
—tt£ ♦ » i.e. independent of the density.
The above experiment may be carried out using a metal and
a wooden cylinder as the supported body.
If the dimensions of the wood and metal cylinder are practically
identical, the time T will be found to be the same within small
limits. Suspend each in turn and find T by timing 50 swings,
or by the method of page 118, and, from the formula above,
calculate I.
For such a regular solid an independent calculation gives a
second value of I, which should agree very nearly with that
already obtained.
/ja *2\
For a cylinder, about the axis taken, I = M( 1 — J where
2L is the length of the cylinder, r the radius.
The formula may be further tested by varying d, d 1 and I. It
will be found that T 2 is proportional to t-^.
Moment of Inertia Table
Fig. 32 shows the essential features of the moment of inertia
table. AB is the table suspended by a fairly stout wire, K, from
the overhead frame. The circular table supports three or more
masses which just fit into a groove, concentric with the circum-
ference. When the wire support is vertically above the centre
of gravity the masses, W, may be moved round the groove into
any position without altering the moment of inertia of the whole.
The masses are arranged so that the table lies horizontally. In
that case the axis is through the centre of gravity and is normal
to the surface of the table. If now the table is given a slight
twist, a restoring couple is called into play in the wire, equal
to, say, t per unit angular displacement, and the result is
oscillations about the axis of support, of periodic time
T = 2 ,yx
^ T
where I is the moment of inertia of the system about the axis
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 65
of rotation ; for) suppose the table be twisted through an angle 0,
the restoring couple due to torsion is t0. We have already seen,
page 53> that the moment of external forces on such a rotating
body is 10" where 9 is— 57a • This couple is opposed by t0 : t0 and
10 are equal and opposite ; i.e.
I 9 = - T0
T0.
T
fQ
or
=
£
K
This was shown in the introductory chapter (p. 25) to repre-
sent simple harmonic motion whose periodic time, T is given by
T= /-=V-
yi
If now a regular shaped body is placed symmetrically on the
table at C, so that its centre of gravity is vertically above the
centre of gravity of the table and therefore in the previous axis
of rotation, the time of oscillation for the loaded table is
\= 2rt y
(i+k)
where k is the moment of inertia of the regular body about the
axis of oscillation : k may be calculated directly from the mass
and dimensions of the body.
5
66 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We have, therefore, from the two equations above :
8 _ 4" 2 (I + k) . Ts _ 4 * 2 (I)
„ T, a I + k . k
Hence -^ = —~- = i + j>
or I = k T 2 _ Ta (12)
If, then, a bcdy of unknown moment of inertia about a given
axis is placed centrally on the table with the centre of gravity
in the axis of oscillation, we may find the value of I lt its moment
of inertia abbut this vertical axis passing through the centre of
gravity, by timing the oscillations of the table when loaded by
the body. If T 2 is the time of complete swing,
Ts _ „ yr±n,
and we have
t =2«y-»
i.e. from these two equations :
T a T2
i. = il -r^' 1 (I3)
Substituting, from equation (12) above,
T 2 T* T2 T 2 T2
x i Y 2 x a — T 2 ~ T 2 — T 2 ^™'
Experimental Details.
The time T for a complete swing of the table is obtained by
timing as many swings as possible, or by the method of p. 118.
To find the value of I, the moment of inertia of the unloaded
table about the wire as axis, a regular solid, such as a plain
cylindrical ' weight ' is employed. The mass should be fairly
heavy so as to cause as big an alteration in T as possible. A
two-kilogramme ' weight ' is of the order to employ with the
apparatus described.
To ensure that this standardizing mass is arranged with its
centre of gravity over that of the table, the lead weights W should
be adjusted so that the table swings horizontally. Any alteration
in the position of the masses, W, will not alter I, so long as the
table is horizontal and the axis of oscillation is vertically through
the centre of gravity.
Having obtained T and T v using a cylindrical regular ' weight/
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 67
the value of k should be calculated. For such a flat cylindrical
object, k = — - where a is the radius and M the mass of the
' weight.'
By equation (12) I is calculated and the table standardized.
Other regular solids may now be used and I, obtained by
equations (13) or (14), and then by calculation a check is obtained
on I x as previously obtained.
The following results were obtained in the above manner:
Table unloaded Table loaded with Table loaded with
T 2000 grammes unknown body
Time 9-375 sees. 97 sees. i5'5 secs -
radius of 2000 gm. ' weight ' = 675 cms.
k _ 2000 x 675' =45 . 5e2gm , cm *,
2
Ii = 45*502 X
i5*5 2 - 9'375 s
97 2 _ 9 . 375 2
Ij = i'ii x io 6 gm. cm 2 .
I, by approximate calculation, assuming a regular shape to
the body = 9-86 x 10 5
The Compound Pendulum.
Let ABC, fig. 33, be a section of a body, passing through the
centre of gravity, and at right angles to an axis about which
it may turn, the point O being the intersection of the axis with
this plane section.
The body is at rest when the centre of gravity, G, is vertically
under O.
68 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If the body is given a small displacement so that GO makes a
small angle with the vertical, then, m being the mass of the
solid, the restoring force has a moment mg OG sin = mm sin 6
(putting OG = a.)
We saw (p. 53) that in such a case the moment of the forces
is equal to Io^.
*2 2
i.e. I -=- = — mga sin = — mgad for small angular
displacements.
This represents a simple harmonic motion (p. 25), whose
periodic time, T = 2w \J — — •
I = I + ma 2 , where I is the moment of inertia about a
parallel axis through centre of gravity and is equal to mk 2 where
k is the radius of gyration about this axis.
Thus T = 2* Jft> + a2fH = 2nJ« a (l 5 )
\ mag > g v 3/
This result is similar to that obtained for a simple pendulum :
k 2
in fact, a simple pendulum of length I = — + a would have
the same periodic time, T. Such a simple pendulum is called the
' Equivalent Simple Pendulum.'
In the case taken, if all the mass of the body were concentrated
k 2
at a point, P, along OG produced such that OP = \- a, we
should have a simple pendulum with the same periodic time.
The point, P, is called the ' Centre of Oscillation,' O being
called the ' Centre of Suspension.'
Now since
I = \- a
a
or a 2 — al + k 2 = o,
the length a is not the only value for OG, which has / as
the equivalent simple pendulum, for the above equation has two
roots, a x and cc 2 , such that,
<«1 + «2 = l \ (l6)
ajOCa = k 2 )
Since a is one value, a x say,
k 2
we have a + oc a = I or oc 2 = I — a = — •
Thus if the body were supported on a parallel axis through the
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 69
former centre of oscillation; P, it would oscillate with the same
time T as when supported at O.
From what has been seen above it is evident that there are an
k 2
infinite number of points distant, a and — from G, for any
k 2
point on a circle drawn from G as centre and radius a or —
will satisfy the condition given ; so that any axis parallel to the
normal at G on the curved surface of two cylinders, of which
the dotted circles are sections, will be axes of suspension which
give the same time.
If the body were supported by an axis through G, the time of
oscillation would be infinite. From any other axis in the body
the time is
+&
g
a 2 _1_ £2 .
This has a minimum value when is minimum.
a
a 2 +k 2 a* — 2ak + k 2 + 2ak {a - k) 2 + 2ak
Now = ■=
a a a
This is a minimum when a = k.
The corresponding minimum Tj is T t = 2*:^-— (17)
o
and will occur for a series of axes parallel to that through G,
and on the surface of a cylinder whose axis is the axis through
the centre of gravity and radius, k.
An experiment which brings out these facts may be performed
by using as the body a rectangular rod of brass about 1 metre
long. This may be suspended on a knife-edge at various points
along its length. To facilitate such suspension it is convenient
to have a series of holes drilled along the bar at about 2 cms.
intervals (fig. 34),
Level the knife-edge, and suspend the bat at, say, every other
hole in turn, and time 50 swings at each hole, which is a measured
distance from the centre of gravity of the bar (which may be
obtained by simple balancing). Or the holes may be measured
from one end of the bar.
Having obtained a set of values for T, and the distance from
the centre of gravity, plot a curve with the periodic times as
ordinates and the distances as abscissae. A curve such as shown
in fig. 35 will be obtained.
The values of T near the minimum points, MM 1 , should be
further investigated by taking the time for vibrations in every
•See also page 118 for a method of timing.
70 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
hole, three each side of the approximate position, and the graph
completed.
Let the line CG, fig. 35, be drawn from C, which represents
the centre of gravity oX the bar.
Draw any line EABFD parallel to the axis. This cuts the
curve in four points, which have the same periodic time, T = BC.
It will be found that the lengths, FB, BA, and BE, BD, are equal ;
i.e. FB and BA correspond to radii GQ, GP, and BE and BD
to radii GO and GR, in fig. 33.
Take either set in pairs, say BA, BD, these are corresponding
lengths to a x and <x 2 in equation (16), i.e. AB + BD = /, the length
= 2 *Vi
Fig. 34
of the equivalent simple pendulum. Its periodic time T is
numerically equal to BC.
Hence from equation, T
all factors except g are known, whence g may be calculated.
If now a tangent is drawn to the curve, such as line LMM^L 1 ,
HM = HM 1 = radius of gyration about an axis through the
centre of gravity : this may be measured directly.
Further, by equation (16), k = V^ol^ = VAB • BD, so
a second value of k may be found. The corresponding periodic
time is, numerically, the length of HC = T lf say.
Hence, once more, by equation (1 7),
V g
g may be evaluated or, as the direct exact measurement of
k (MH or M^H) is difficult, as there is some doubt in the general
case as to the exact location of M and M 1 , this formula be used
to calculate on third value of k,
Ti 2
8^
ft — 75 — 2 ■
MOMENTS OFlNERTIA AND DETERMINATION OF V 71
The mean value of k may be taken and the moment of inertia
about a parallel axis through the centre of gravity calculated, for
I = km,
where M is obtained by direct weighing.
L- -
- C
Ois^ortce. fromC.G D'wsfaoce Tfom OG
Fig. 35
Hater's Pendulum
From the preceding experiment it is obvious that if it were
possible to obtain, for a rigid body; two parallel axes of suspension,
along any line through the centre of gravity and on opposite
sides of it, which have exactly the same time of swing,
then the value of g could be very well determined by
measuring the distance between such axes. This distance
would be equal to the length of the equivalent simple pendulum,
/. If the equal periodic time about these axes were T,
Then
= *■<$•
I and T being measured directly.
The Kater pendulum is one by means of which this may be
realized in practice to a very close approximation.
It consists of a long rod which is provided with two fixed
knife-edge supports, K and K, and terminates at each end in
a ' bob,' B and B. Usually, the one bob, B, is made of brass
and the other of wood.
M and m are two adjustable masses which may be fixed in any
72 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
position between the knife-edges. Their adjustment serves to
move the centre of gravity to such a position that the time of
swing is approximately the same from either knife-edge.
The pendulum is supported on knife-edge, K and K, in turn,
and the approximate periodic time, T and T lt is obtained by
counting swings, timed by means of a stop-clock.
The large mass M is moved until these times are approxi-
mately the same. The small mass m serves as a fine adjustment
to this purpose.
Having adjusted the masses so that the time for a complete
vibration is very nearly the same from both knife-edges, it will
be realized that to obtain exact agreement for T and T x would
be a most tedious experiment.
Fig. 36
However, we can see in the following way that such exact
agreement is not essential.
Let a and a x be the distance from the centre of gravity of the
pendulum to K and K.
> ag y a
+ k 2
ag > a x g
T*ag == 4*2 (a* + k*) T x 2 a lg = 4* 2 (a^ + k*)
Subtracting
(T z a-T 1 *a 1 )g=4n*(a*-a 1 *)
£! = a T2 ~ <*i T i a _ I/H±Il* j. HjzIA 1 a\
g a*- ai * ~2\a + a 1 + a - a x ) '-"P*)
a and a x may be made to differ by a fairly large amount by
suitable adjustment of the masses, M and w. With a little care
*p2 np 2
T and T x may be very nearly equated, and so the term —
a ■— a x
becomes small. T and T x may be measured very accurately by
the method of coincidences, (a -f a-^ may be measured directly
as the distance between the knife-edges, by a comparison with a
metal metre scale by means of a comparator (p. 32). The
important first term in equation (18), (R.H.S.) is thus carefully
evaluated.
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 73
The second term is small and no serious error is involved if
a and a x are measured from the knife-edges to a point at which
the pendulum may be balanced horizontally on a knife-edge.
By this method a very reliable value of g may be obtained.
Method of Coincidences.
This method of timing a pendulum consists in hanging it
by a knife-edge from a rigid support, in front of, say, a seconds
pendulum of a standard clock, the height of the support so
arranged that the tails of both pendulums are on the same level.
At rest, viewed by a telescope from in front, the Kater coincides
with the seconds pendulum.
If both pendulums are of the same period and start oscillating
together, when viewed through the telescope, they appear to
move as one. If the periods are not the same, they will be
seen to get ' out of step,' and at one point both will pass a fixed
reference point together and going in the same direction. This
will not again occur until one pendulum has gained or lost a
whole swing.
Suppose the seconds pendulum makes n complete vibrations,
each of period T (2 sees.), and the experimental pendulum makes
(n + 1) complete swings, of period T lt
Then Tn=T 1 (n + i) (19)
Tl _ n 1 _ 1 _i_ , v
T — n + 1 ~" i — n n 2 * '
+ n
Suppose n = 500.
T *-i-^ +
T 500 250000
further terms are negligible.
Hence T j is obtained in terms of T ; which in the case taken
is 2 sees.
Similarly the time about the other axis may be checked.
In the coincidence method it may be observed that one is
never quite sure within a few (say m) passages of the pendulum
which is the correct coincidence. We can easily see that the
error introduced by this cause is not appreciable when n is fairly
large. Thus we know that in equation (19) instead of n we may
put (n ± w),
i.e. T (» ± m) = T t {n ± m + 1).
T*_ 1
T X+ 1
n ± m
74 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
— : i _ _ _ approx.
i i
= i — -
"(*±2)
n n''
In practice the coincidence may usually be limited to one of
about 6, i.e. m = 3.
A small value for n = 500.
• ii-.i_.i-i 3 .
T 500 ^ 5oo 2>
i.e. an error of about 1 in 100000 is introduced, due to the
uncertainty.
In practice a cross-hair in the focal plane of eyepiece of the
telescope is a useful reference point against which to estimate
coincidence.
Sphere on a Coneave Mirror. An Approximate Method of Deter-
mining *g', the Acceleration Due to Gravity
A concave mirror is arranged horizontally, facing upwards,
so that a small steel ball may be allowed to perform oscillations
on its surface, in a line through the lowest point.
The time of oscillation of the steel ball is obtained by timing
as many oscillations as possible on the surface. The observation
is repeated, and from these results a mean value of the periodic
time T is calculated.
Then if
R is the radius of curvature of the upper face of the concave
mirror, as measured by a spherometer,
m the mass of the sphere,
r its radius,
g the acceleration due to gravity,
it will be shown that
T= 2, j5^- r ' (21)
whence g = _? . !£. (R _ r ) (22)
Consider the sphere in its position of equilibrium to be with
its centre at B (fig. 37), and when displaced to the extreme
position, with the centre at C.
MOMENTS OF INERTIA AND DETERMINATION OF «g ' 75
We will consider the case of a mirror of large radius of curvature
and the displacement BC to be small compared with R.
The potential energy of the sphere at C is mg • AB. Now
AB = OB — OA = (R — f) (i - cos 6), i.e. the potential energy
a
is (R — r) 2 sin 2 - • mg, or when is small as specified :
P.E. = 2(R - r) (~fmg = \ (R - r ) 02 m S-
The centre of gravity describes a circular path, BC in the
BC
diagram, so that ~ • = 0-
R -r
Hence the potential energy is
i /T> . BC 2 i
- (R - r) 75 i^wg = -
mg
(R-r) 2
2R — r
-•BC 2 .
At this point there is no kinetic energy.
At B the whole of the energy is kinetic, and equal to
^mv m * + ^Iw m *
where v m is the maximum linear velocity of the centre of gravity
and w m the maximum angular velocity of rotation, and I is the
moment of inertia of the ball about an axis through the centre of
gravity, at right angles to the plane of the paper, i.e.
K.E. at B is - mv m 2 + - 1 ?*£•
2 2 r 2
At any intermediate point P distant x cms. from B along the
76 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
arc, the total energy is equal to either of these quantities and is
therefore a constant,
1 1 1 i x z
i.e. - mx z H -x z + - mg-j=- . = constant,
2 2 r 2 2 & (R — r)
x being the velocity at that P along the path.
Differentiating we have :
mxx + - X 'X + (R _) ) * , * = °-
Dividing by x and rearranging
mg
* = f- =-. x,
(R-,)(m+l)
i.e., the acceleration is a constant times the displacement ;
this was shown (p. 25) to be S.H.M., whose periodic time T is
given as under :
|(R _,)(„+!_)
T = 2TC = 27^
I mg y mg
!(R - r) (m + I)
Now, I, the moment of inertia about the axis described, is
2
equal to — mr*.
(R -r).Z
Hence T = 2n W ^ (21)
o
T is obtained by observations as indicated above ; r is measured
by means of a screw gauge, R, by means of a spherometer, not
by an optical method, unless the front surface is silvered. Hence
all the terms in (21) are known except g, which may be
calculated. The above method does not yield accurate values forg.
Atwood's Machine
A modern form of Atwood's machine is illustrated in fig, 38.
The two masses, M x and M 2 , are equal, and are connected over the
pulley, N, by a strip of white paper in the form of tape, while
an equally long strip, M^Ma, connects the other ends of the
weights, so that for all positions of M x and M a there is an equal
mass of paper at each side of the pulley.
The vibrator, V, carries an inked brush, B, and as^ the paper
passes below it a trace, somewhat in the form of a sine curve,
is drawn on the tape. The complete period of the vibrator
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 77
is usually about one-fifth of a second ; but this should be carefully
tested by means of a good stop-watch before beginning the
experiment.
The curve then provides a time record. The apparatus is
used to provide an exercise in the determination of the accelera-
tion due to gravity ; but even in its best form the experiment
has no claim to great accuracy ; it does, however, provide an
instructive exercise in Mechanics.
Mj stands on a platform, as shown in the diagram, and the
mechanism of the apparatus provides for the release of the
Fig. 38
vibrator and of M t simultaneously. Small weights are provided,
which rest on the top of M x , and on release of the platform cause
an acceleration of the masses. The rider can be removed by a
second platform, P, after a velocity has been acquired.
The trace on the tape records the acceleration and velocity
beyond P, and from the former of these the value of g
may be determined.
There is always a frictional resistance to be accounted for,
although this is reduced by making the pulley light and mounting
it on ball-bearings.
It is best to get rid of the retardation due to friction by making
78 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
loops of wire, which may be placed on the top of Mi and remain
when M 2 passes through P.
The necessary addition can be judged approximately by
observing the fall of Mj after it has been given a small velocity.
If Mj moves down uniformly without appreciable loss of speed,
the frictional error is nearly corrected. A finer observation may
then be made by allowing the brush to make a trace. If the
line consists of uniformly spaced waves the velocity is uniform.
When this has been adjusted the wire loops are left in position
and are not taken into account in the calculations.
Suppose a rider of mass, m, lies on M x (in addition to the wire
loops).
When Mj is allowed to fall, suppose it does so with an accelera-
tion, /.
Let Tj denote, the tension in the paper above M x . The lower
paper strip is loose and is not supposed to exert any force on the
two masses.
Let I denote the moment of inertia of the pulley, and let T 2
denote the tension in the paper on the left of the pulley, i.e. the
tension acting on M 2 . Denote the radius of the pulley by a
dw
and its angular velocity at any instant by w. Then -j~ is the
angular acceleration, and we have :
dw .
From the forces on (M x + m) we have :
(m + MJf = (m + UJg - T x
and from considering M 2
Ma/ = T 2 — Mag,
while the motion of the pulley is expressed by ;
I^=(T 1 -T ! )a,
from which we have :
I/=(T 1 -T 2 )^.
We may therefore eliminate T x and T 2 and find :
j = g™
m + M x + M a + \
The quantity —is called the ' equivalent mass ' of the pulley,
and its magnitude in grammes is engraved on the pulley.
We may therefore find the value of g from this equation from
observations which give the value of/.
MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 79
This may be determined from the trace. By
removing P, the trace may be made long, M t
being allowed an extended fall.
The line drawn by the brush will consist of /^*B
waves which open out uniformly. Mark these off in
groups of five, as at A, B, C, etc., beginning at a
point A, where the trace is opened out sufficiently
to be distinct. Measure carefully the distances AC,
BD, CE, etc., and divide by the time interval
which elapses between these points. This will give
the average velocity over the strips measured,
and this velocity is the velocity at the points
B, C, D, etc. The differences between these
velocities should $all be the same, or very
approximately so. Take the average of all the
determinations and so obtain the average increase
in velocity during five periods of the vibrator. Hence
deduce the acceleration byj dividing by the time of
five vibrations.
This is the value of /.
Repeat the experiment with the various riders ^q j
provided.
Fig. 39
CHAPTER III
ELASTICITY
All bodies, when acted upon by forces, are deformed a certain
amount. The magnitude of the deformation produced by a
definite applied force enables a value of the elastic constant of
the material used to be calculated.
We may, in a general manner, call the forces applied ' stresses,'
and the deformations produced 'strains.' However, these two
terms have, more often, a more precise meaning, depending on
the mode of application of the forces. We shall recognize three
ways of producing a deformation : (i) by uniform compression
or extension, (2) by applying equal and opposite forces in one
direction, i.e. stretching, (3) uniform shear. Deformation may
be produced in any of these ways or by a combination of them.
(1) Uniform Compression or Extension.
If a body of volume V be subjected to a uniform pressure of
p dynes per sq. cm., a contraction will ensue. This corresponds
to a change in volume of <5V, say. The fractional increase in
, . <5V
volume is ^ •
In this case the stress applied is p dynes per sq. cm!, and the
strain is -= numerically.
(2) Stretching.
The most direct example of this type of deformation is seen
in the case of a wire fixed at one end and supporting masses at
the other end. In this case the force acting on the wire is the
weight of the suspended masses and the reaction at the point
of support. These are equal and opposite, acting in a direction
which coincides with the length of the wire. Due to their
action the wire will increase in length and at the same time will
be reduced a very small amount in cross-section. The reduction
in cross-section for a wire will not be of a sufficiently large amount
to be readily measured directly.
If 81 is the increase in length of a specimen whose original
8l
length is /, the fractional increase in length, the strain, is -=■•
I
80
ELASTICITY
81
The stress producing this strain is denned as the force acting
on each unit of area normal to it, i.e. if a is the area of cross-
section of the wire and the total mass applied is m grammes,
the stress is — - dynes per sq. cm.
(3) Shear.
Consider a cube of material ABCDEFGH (fig. 40) fixed
at the lower face and acted upon by a tangential force F at the
upper face. As a result of this force the cube will take up a
position shown in an exaggerated manner by the broken lines
in the figure, the vertical sides being sheared through an angle
of <p radians from AH and BG to AH 1 and BG 1 .
For such a shear — — „„ „ is the stress, i.e. the tangential
force per unit area.
The strain is measured by <p, i.e. the ratio
EE 1
AE
if EE 1 is small
compared with AE 1 .
In all the above cases the stress is measured as a force per
unit area ; in the c.g.s. system in dynes per square cm. The
strain in each case is a ratio of like quantities, and has therefore
no dimensions.
Hooke's Law.
If the stresses are below a certain limiting value which depends
on the material of the body to which they are applied, the strain
disappears when the stresses are removed. If the limiting value
is exceeded, the material is strained beyond the elastic limit,
82 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
and such strain is permanent ; as the stresses are still further
increased the result is a fracture of the material.
For stresses below the elastic limit, it was established by
Hooke that the strain produced is proportional to the stress
applied, i.e. under such conditions we have
stress
-7 — — = constant.
strain
The constant has a definite value which depends on the
material, and which, in the three cases taken, is called, (i) the
' bulk modulus of elasticity,' (2) ' Young's modulus,' and (3) the
' modulus of rigidity.'
Young's modulus is the most readily obtained directly by
experiment.
The following notation will be used throughout in dealing with
these elastic constants.
(1) Bulk modulus = K = £
oV
A
(2) Young's modulus = Y = — =
61
T
F
A
(3) modulus of Rigidity, n = —
In addition to the above three elastic constants, we may add
a fourth, which is concerned with stretching. We noticed that
during stretching there is a lateral contraction of the specimen.
The fractional lateral contraction produced is proportional to
the longitudinal stress applied and the ratio of
Fractional lateral contraction
Fractional longitudinal extension
is called ' Poisson's Ratio ' fp) . Thus, if the specimen is a cylinder
of radius r and length I, and the changes produced in these
dimensions are 6r and 61, we have
r
""IT
/
The following relations between the elastic constants may be
ELASTICITY
83
readily deduced (see for example, Poynting and Thomson's
" Properties of Matter ").
* -2( 3 K-f*) * (2)
Thus, if any two of the constants are found experimentally, the
remaining two may be calculated from the above equations.
Determination of Young's Modulus for the Material of a Wire
A direct method of finding Y is to support, vertically, a long
length of the wire, load it with definite masses, and observe the
extension produced.
Fig. 41
It will usually be most convenient to obtain such a length
that, when supported at the ceiling, the wire extends almost to
the floor. A second wire, C, is supported in like manner from
the same support, and carries a fixed load of sufficient magnitude
to keep the wire taut. The wire D carries a platform P. At V
a vernier scale is attached to the wires, one half fixed to one
wire, the other half to the second wire.
The wire D, whose Young's modulus is to be determined,
should be free from ' kinks ' and should carry sufficient load to
make it taut, so that any further load added merely causes a
stretching of the wire, and does not simply straighten out bends
and kinks. If a heavy platform is employed at P, this weight
84 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
may be sufficient. Read the vernier ; place one kilogramme on
the platform and notice the extension. If this is due solely to
the stretching of the taut wire, the vernier reading, on removing
the kilogramme load, should be once more the same as at commence-
ment ; if this is not so increase the load until on adding a further
kilogramme the readings on the vernier scale have a definite value,
which is reduced to another definite value on removing the load.
Having obtained satisfactory repeats for this adjustment, the
load on the scale pan P should be increased by equal increments
and the vernier reading for each load noted. Having arrived
at a safe maximum load, the latter is reduced by the same
increments and the vernier readings again noted.
The values obtained may be tabulated as under.
LOAD
VERNIER READINGS.
EXTENSION
LOAD INCREASING
LOAD DECREASING
MEAN
FOR 6 KILOS
2
4
6
8
10
12
mean extension
for 6 kilos =
The mean vernier reading for each load being taken and
tabulated as shown, we may obtain several values of the extension
of the wire for a definite load.
Thus, in the case taken, the loads were o, 2, 4, 6, 8, 10, 12 kilos.
The difference between the vernier readings for and 6 kilo load
gives the extension for 6 kilos. In the same way the difference
between the readings for 2 and 8 kilo load, 4 and 10, 6 and 12,
also gives the extension for 6 kilogramme increase in load. This is
entered in the last column, the mean value, I say, of which is
used in the calculation of Y.
The radius of the wire is measured in at least six places, using
a micrometer screw, and the mean value taken, r cms. say. The
original length of the wire, about 7 metres, is measured directly
(L cms.).
6000 x 981
Hence
TVT'
I
in the case taken.
ELASTICITY
85
An alternative way (due to G. F. C. Searle) in which to measure
the extension of the wire is illustrated in fig. 42.
The standard wire terminates in a frame A which supports a
mass M, sufficiently large to maintain the wire in a stretched
state. The wire to be investigated is also fastened to a similar
framework B. The two are fastened by cross-pieces C and D,
which prevent relative rotation of the frames, but allow the
frame B to be depressed relative to A, when masses are added
to the scale pan S.
Fig. 42
A spirit-level L is supported at one end on a rigid cross-bar
of the frame A, and at the other on the point of a micrometer
screw V, which moves vertically through a rigid cross-bar. The
micrometer screw has the usual circular division, which enables
the movement of the head to be estimated to ts or rita of
a complete turn, enabling the movement of the point of the
screw (and hence the end of the spirit-level) to be measured to
shs or ths of a millimetre.
The level is first adjusted, when the wire is suitably stretched
free from ' kinks/ so that it is truly horizontal. The load of,
86 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
say, one kilogramme is added to the scale pan S. The micrometer
screw is moved a suitable distance over to scale G, so that the
spirit-level is once more horizontal.
The amount of movement required to bring this about is
obviously equal to the elongation of the wire by the load added.
The results may be tabulated as in the former method and
the value of Y calculated from the mean of a set of observations.
Bending of Beams
The value of Young's modulus may be found by less direct
measurements for substances not in the form of a wire.
Consider a rod of any uniform cross-section, say rectangular,
Fig. 43
bent into the form of a circular arc of fairly large radius. Take
a section of the rod by a plane passing through the long axis of
symmetry and parallel to it, and passing through the centre of
curvature (i.e. the plane of bending). The layers of the material
of the bar in the lower half will be compressed and the upper
half extended. There will be one plane, therefore, at right angles
to the plane of bending, which will remain of the same length as
before the bending took place. This plane is called the neutral
surface, and it will be shown to pass through the centre of gravity
of the bar. It is represented in fig. 43 by NS.
If we imagine the bar to be made up of a number of filaments
along the length, then such filaments, as stated above, will be
extended or compressed according to their position above or
below the neutral surface. One such as shown at EF, fig. 43,
or at P in the section diagram (fig. 44) a distance y above the
neutral surface is extended.
ELASTICITY
87
The strain in such a filament depends upon y, for, if R is the
radius of curvature of the neutral surface, and the angle sub-
tended at the centre O by NS, the unstretched length of the
filament EF is the same as the present length of NS =R0.
Also EF = (R + y)Q.
Hence the elongation is yd and the strain is therefore ~L.
Now if / is the force acting on the filament of cross-section a,
we have from the definition of Young's modulus
/
or
N
Y
a
. /
R
3 R
-y»
i.e. fccy
•O « SO Cf
s
3
______
"-t=
Fig- 44
Thus, the arrows in the lower part of fig. 44 show the type of
forces acting on all such filaments into which we have subdivided
the bar.
This system of forces on the bar must have an algebraic sum
of zero.
Y
le.
S/ = oor-^- Say = o.
Since ^ is not zero,
R
Say = o.
Thus, the neutral surface passes through the centre of gravity
of the bar.
The forces have a definite moment about the neutral surface.
Y
The moment for the smgle force is fy = = .- ay 2 .
For equilibrium the sum of such moments is equal and opposite
to the external couple which set up the internal forces. If C is
the external couple, we have
Y „ Y .
s r «r
R*'
88 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
where * = Say 2 . From the similarity between this case and
the corresponding sum in considering masses in connexion with
moment of inertia, i is sometimes called ' the moment of inertia of
cross-section.' It may be calculated in the same way as moment
of inertia if area replaces mass.
We have therefore for such a bent beam
*Y = CR -....(3)
Cantilever
Consider a light beam fixed horizontally at one end and loaded
with a mass m at the other. If the mass of the beam is small
compared with the load m, the whole depression may be taken
as due to the load.
In fig. 45 let AC 1 be the unloaded position for the neutral
surface, and AC the position taken when the load is applied at C.
To obtain an expression for the depression of the end in terms
of the dimensions of the bar, m and Y, it is convenient to refer
to a system of axes with the end A as origin ; AC 1 being the x
axis, and a line at right angles to AC 1 from A in the plane of
the paper the y axis.
We will assume that the curvature of the loaded rod is small,
i.e. the total depression at C is small.
Consider a section at B, fig. 45, x cms. from A. As already
seen, across the face of such a section a system of forces exist.
These forces on the segment BC are extensions above the neutral
surface, and compressions below, constituting a counter-clockwise
t"Y
couple C = -p on BC.
At the same time the force mg at C has a clockwise moment
equal to mg(l — x) on BC.
For equuibrium these two opposite couples are equal in
magnitude,
iY
i.e: -^ = mg{l -x) (4)
ELASTICITY 89
where R is the radius of curvature at B.
{■+■©•}'
Now R = jw-
d*y
dx*
But in this case ^ is small — for the total depression is
dx
assumed to be small — so (~\ is negligible compared with
unity, and therefore we have for such small curvature
R=JL,
d*y
dx*
d 2 v W / x 1
i.e. ^ = -W? (I — x) from equation (4) above.
dx % *Y x
The value of the total depression at the encf of the bar may
be obtained by integration, and is the value of y when x = /,
after such a process.
Integrating we have
BO -£( fc -3 (5)
dy
the constant of integration being zero, for when #=0,^=0.
A second integration between the limits o and I gives
L^Jo *Y L2 6 Jo
where y is the end depression,
i - e - y° iY 3
For a bar of rectangular cross-section, i about the neutral axis
NS (fig. 44) is — , where b is the breadth and d the depth of the
bar.
12W I*
Hence ^» = 6^Y*3
v _4¥ 3 (6a)
If the mass of the beam is not negligible, see treatment on
page 94.
90 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Beam Supported at Two Knife-edges and Loaded in the Middle
If the beam is now supported at the two ends and loaded
with a mass m at the mid-point we have a reaction ^ at each
2
knife-edge. If / is the total length of the bar, the depression
produced at the centre will be exactly the same as the depression
at the end of a similar bar of length - and loaded at the end with
m
rrii ..
a mass — , i.e. if we imagine the bar clamped at the mid-point
and a force ^ applied at one end. The movement of the end
is precisely the same as the depression in the middle in the
actual case taken.
Ar
Fig. 46
I
Such a depression may be obtained by substituting the length
- and force -£ in equation (6a), giving for the depression
y ° ~ *Y ■' 48
or Y==-^!_ ." >>(7)
4bd 3 y
The value of Young's modulus for the material of a beam may
be obtained in this manner.
The beam, say of iron, and of about 1 cm. square cross-section,
and about 1 metre long is supported on two knife-edges near its
ends, and a load is applied at the mid-point by placing masses
on a pan which is suspended from a knife-edge which rests on
the bar at this place. The depression is measured on a vernier
scale, one scale of which is fixed, the other moves with the beam.
The load is increased, and the vernier reading (y ) for each
load is tabulated as under. The value for — is obtained in each
case, the results being tabulated as under.
ELASTICITY
91
load m
VERNIER READING
DEPRESSION (y ).
m
2000 gms.
3000 „
4000 „
5000 „
mean
m
Jo~
m
A mean value of — from the series of observations is taken.
y Q
The distance I between the knife-edges is measured directly,
being of the order of 1 metre ; this can be done with a good
degree of precision.
Now d occurs in the third power, and is only a small quantity,
therefore many observations must be taken and the mean value
used. For d approx. 1 cm., an error of *i mm. means 1 per cent
error, and this is magnified in d 3 to 3 per cent.
Substitute the values found, in equation (7)
^/m\ gl*
W4W»
Koenig's Method
, Fig. 47
Another method, due to Koenig, of determining the value of Y
for the material of a beam, is by means of the type of apparatus
shown in fig. 47.
92 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The bar carries a knife-edge which supports the load on a
pan P.
At the ends of the bar are two mirrors, M x and M 2 , almost
normal to the bar, but slightly displaced to enable a scale S to
be seen in the telescope T, the light from S having suffered two
reflections.
The telescope carries a cross-hair in the eyepiece, and the
apparatus is arranged so that a scale division, as seen in the
telescope, coincides with the cross-hair.
If now the bar is loaded with, say, i or 2 kilogrammes, the
mirrors will be turned towards each other as a result of the
depression produced, and the scale division viewed in the telescope
will be altered* This difference is noted = x divisions, say.
Then we will show that
Y ~ WdH ••••• (8)
where W = Mg, M being the load in grms.,
I = distance between knife-edges,
D r= distance between scale and the more remote
mirror, M 8 ,
m = the distance between the mirrors,
b and d having the same values as before, b
the breadth, d the depth of the bar.
For a bent cantilever we saw, equation (5)
or between limits o and I,
b
Now -2- is tangent of the angle through v/hich the beam
has been bent. Let 9 be this angle.
ti, W l %
Then tan <p — -=- T
»Y 2
Now for in the present case, I being the whole length of the
bar supported by two knife-edges, we obtain the angle through
/ W
which each end is turned by substituting V for I and — for W in
the last equation,
i.e. in the present case
WZ*
tan * = i67Y
dxj i Y 2
ELASTICITY
93
For rectangular bar i =
bd*
12
tan 9 —
3WZ
4bd*Y
For small depressions the angle <p is very small and so
tan <p = q>.
<p =
3W
4bd*Y
®
Now a value of <p may be obtained from a consideration of
the movement of the mirrors.
Let m x and m t be the original positions of the mirror (fig. 48).
In the first case the image of the division at D is in coincidence
with the cross-hair ; when the mirrors move, each through an
angle <p, F is then seen in coincidence with the cross-hair.
Fig. 48
Let us imagine the rays of light reversed, ABCD being the
original path : when m x moves through q> and takes position
m\, BC is moved through 2q> striking w 2 at E.
Obviously m being the distance between the mirrors CE =
m • 2q> very nearly.
EG is a line drawn parallel to CD. The ray EF is swung round
through an angle 49, for in addition to BE having moved through
2<p, m 2 itself has rotated through <p.
i.e. since GEF = 49 FG = 49* D, very nearly.
But DG = CE = 2mp.
;. x = DF = 2<p{m + 2D)
<p =
2(m + 2D)
.(10)
94 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
from (9) and (10)
x 3W/ 2
2(w -f 2D) 4&i 3 Y
2&i 3 #
y== 3 W/ a (m + 2D) ^
T P W
In performing this experiment a mean value of — is obtained
as in the last experiment, from observation of x corresponding
to several loads and the other terms measured as before. Hence
by substituting in (8) Y is obtained.
Determination of Young's Modulus of the Material of a Bar by the
Vibration Method
We have already seen that the depression due to a load W
at one end of a beam which is rigidly fixed at the other end is
given by equation (6), page 89.
This strain sets up an equilibrating internal stress equal to
3«Y
* L fs—y which acts as a restoring force.
If the beam be allowed to oscillate in a vertical plane the
restoring force when depressed y cms. is therefore
3*Y
Expressing the load as W dynes (m grms.) we may write, if y
W
is the acceleration of the mass m = — at the end of the bar,
g
W .. 3*Y
3»Yg
ovy = -wi* > y
This is a periodic motion, whose period T is
-*■>£
w
T -">fe£- <">
'3*Yg
or, as the mass at the end of the rod is m grammes,
\ nd r
'3*'Y
A more complete treatment of this case may be seen under.
In fig. 49 let O be the point of support, OA a vertical section of
the unloaded beam, and OB the section of the beam when
depressed by a load m at the end, so that the end depression
AB = z , and the depression of the centre of gravity shown
ELASTICITY 95
is u , whereas any point P on the bar is depressed a distance # ,
P being a distance s from O ', the total length of the bar is
I cms. ,
Now let us imagine the bar to be homogeneous, and of mass
6 per unit length ; so that its total mass is M = 13.
We have, if i is the 'moment of inertia of cross-section,'
considering the length (I — s), following the treatment of page 88,
tY = R[W(Z - $) + (I - s)<5-g^^]|,
taking into account the weight of the section of the bar concerned,
Fig. 49
which acts through the C.G. of the section. Since — = —
if the beam is not greatly bent,
,-Y g- = W(Z - s) +~(l - s)» 6-g,
whence integrating
for when s = o, x = o, i.e. the constant of integration is zero.
Integrating between the limits s = o, s = /, x = o, to x = z
=,(1^ ............. ...« I2 )
since M = d'l.
This deals with a case of a steady depression. When the bar
vibrates it is depressed below the position OB to some extreme
position OC. To arrive at a value of the forces acting in the
material of the bar in such a case let us assume that Wis increased
to W\ causing the bar to be depressed into the position OC,
where the extra depression of the end is z, of the centre of gravity
of the projecting bar u, and of the point P is x.
96 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
XT -V 73 W . M ^ &
Now tYz =l 3 \- o~
3 8
and iY^ + ^/'W^M^
3 o
i.e. »Y* = — (W 1 - W)
*=5^(W»-W). (13)
In this case W — W is just balanced by the internal stresses
in the beam which are thus equal to
(W*-W)==-^s (14)
This gives the value of the restoring force, hence the equation
W
of motion of the mass — 'neglecting the mass of the beam, is
W dH__ 3*Y
g dt* I 3 '*
or * = -&*S.z
or Z*W
X - e - Ni**" ^5)
This is identical with the result expressed in equation (n),
which was deduced from very simple considerations. It does
not, however, take into account the mass of the beam itself.
This is most conveniently done by considering the energy of the
system.
As the beam oscillates about OB, the steady deflected position
of equilibrium, we will express the potential energy of the parts,
using OB as the reference position of zero potential energy.
Thus, in the position OC the Potential Energy of the system is
— VJz — Mgu + (Energy stored as strain in the beam).
The last term is obviously equal to the work done in straining
the beam, i.e. is
/ (straining force) dz
(W 1 — W)dz = I ^tj — dz by equation (14) above
3*X* 8
* 2l*
ELASTICITY 97
.-. P.E. is
3*Y a_ W 2_ M gw _ ^6)
2 I 3
If instead of obtaining equation (12) by integrating the previous
equation between the limits o and I we had integrated between
the limits o and - . we should have obtained the depression u
2
of the centre of gravity as under.
*° *Y 48* +384 *Y
. u may be obtained in a similar way to z (p. 96) to be
u =
W '- W ._L_Z3
and since by (13)
»Y 48
16
So that in similar terms (16) becomes
1 75-** - Ws - f 6 Mgz (17)
Now the kinetic energy of the system is the sum of the K.E.
of the mass-— I = - — £*) and the total K.E. of the vibrating
g V 2 g /
beam.
To obtain the K.E. of the beam consider an element of length
ds at P. The K.E. of this element is
£ d.ds(x) a
x may be obtained by precisely the same method as z and u.
* = ,4(t-iD(")
so that x = -\---} z
The total K.E. of the bar is
2 v '
I.
98 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
2 / 6 L V20 36 ^ 252/J
280 280
Thus the tote/ Kinetic Energy in the system is
iT 1 '-'-^ 11 ** (l8)
The sum of the Potential and Kinetic Energy is constant.
Thus, adding (17) and (18) we obtain
I/W + 33_ M Ua + 3*Y 2 _ m _ 4 M^ = constant.
2\g I40 / 2l 3 16 &
Differentiating
I /W + _33 M \ 2i g + 3 *Y 2 ^ _ Wi _ 5 M q
2\g 140 / 2 I s 16
Dividing by £ we have
(7 + ^ M )'" + 3 , J«-(w + 4lfe)-o....(i 9 )
This represents a harmonic medium whose period T is un-
affected by the constant term W + -4 Mg as seen below.* The
period is the same as that of
\g 14° / '
or T
-«W V « 3( y° ; •; (20)
* We may write equation (19) :
put * = *-^( w + i! M s)
/. 'p = Z, "P=" z
= 2*^1
W + -33_m > )/»
3^
ELASTICITY 99
Now, i, the ' moment of inertia of cross-section,' depends, of
bd 3
course, on the form of the rod. For a rectangular rod, i — —
(about the neutral surface), where b is the breadth and d the
depth of the cross-section in cms.
The Experiment may very well be carried out using an ordinary
boxwood metre rule.
A definite length, /, of the rod is projected from the top of a
table, to which it is rigidly clamped, as seen in fig. 50.
f /////////y///////y/yyy^/y/^/^y^
Fig. 50
w
A mass — is rigidly attached to the end of the metre scale,
so that it has no 'play/ i.e. there is only the one vibration,
W
namely, that of the scale itself. The value of — should be such
o
as to cause but small depression. The rod is made to vibrate,
and the time T is obtained by timing 50 vibrations in the usual
manner. The length of the vibrating rod is altered, and the
periodic time again determined. This is carried out for several
W
lengths of rod, and also for several masses — at the end of the
o
scale.
It must be remembered that M in the expression (20) for T is the
mass of the vibrating part of the scale, and must be obtained for
each length / employed.
Of course, if m' is the total mass of the scale and L the total
vn'l
length, then M = -y- for the uniform rod.
The results of the various experiments may be tabulated as
under.
joo ADVANCED PRACTICAL PHYSICS FOR STUDENTS
/
/».
w
1 '
M
W + ,33 M
8 140
T
T*
Mean
11 (YL
T 2 U
33M\
140/
_ i6it a (/ 8 /W
+ 33M\
g 140/
Note.
W.
g
is the mass attached in grammes.
The method works equally well for such substances as wood,
where the correction for the mass of the beam is small, and for
brass, etc., where— M is comparable with the values of W.
RIGIDITY
The modulus of rigidity is determined by observation of the
twist produced in a wire by a definite couple, either statically,
or less directly by torsional oscillation.
ELASTICITY xot
Measurement of the Rigidity of a Wire by the Static Method
Let us consider a long wire fixed rigidly at the upper end and
subjected to a couple, C, turning it in the direction shown in
fig- 5i. .
We may consider a section of the wire (fig. 52), and of this
section an annular ring as seen. Consider a small rectangular
segment EFGHIJKL in the undisturbed wire. When the couple
acts, this rectangular parallelepiped will shear as we have pre-
viously seen, the faces making an angle <p with the original
vertical direction. This happens throughout the ring, and
adjacent sections will behave in the same way throughout the
length of the wire.
Fig. 52
A vertical line, MN, along the wire (fig. 51) is moved by each
ring through an angle <p, and so finally takes up a position MP
when MP is inclined at an angle g> to MN.
If I is the length of wire from the fixed end, and R is the radius
of the wire, then since N is moved to P when the couple acts,
the radius ON moving to OP, where the angle NOP =6 radians,
we have
arc NP = h = R0 • • --(21)
Considering again the annular ring of fig. 52, of radius r and
width &r, we have the shear produced by the couple C.
Let /be the value of the shearing force over the area, 2-nr-Sr,
of its upper and lower surfaces. From the definition of rigidity n
f
2-nr ' dr
n == «^—
<p
or f = 2isnqrrdr.
Since this force has a moment fr about the axis of the wire,
we have, replacing <p by the more easily measured term 0, from (21),
' 2w»0
ft — T
102 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The total couple throughout the solid section is therefore :
Jo Jo I 2l
a couple which is equal to the applied couple C for equilibrium,
hence
C-=£-' (»)
This may be re-written :
C . ttR*
-'I — • n.
c
-, the couple required to produce unit angular twist, is called the
' coefficient of torsion ' = t, say.
We thus have :
*R*
2
or
«R*
2
tZ = «-"-^- .(23)
xl = i • n.
=i the moment of inertia of cross-section for the circular wire,
and in equation (22) is measured in radians.
Fig. 53
Experimental Details.
The method of finding the coefficients is to fix a wire specimen
rigidly at one end, apply a measured couple at the other, and
measure the twist produced at a given distance I from the fixed
end.
The two types of apparatus usually employed are shown in
figs. 53 and 54.
ELASTICITY
103
Using the vertical wire type, pointers are fixed to the wires
at different distances from the fixed end, and a couple is applied
to the free end by adding weights to the scale pans S, S 1 . The
amounts of twist, V a > 3 , at the distances, l lf Z a , Z 3 , from the
clamped ends are observed on the circular scales shown. It will
f) ft f)
be found that -~ = ~ = -~ as we may expect from equation (22).
L x / 2 / 3
Now consider a fixed length of wire I cms. from the support.
If M is the sum of the masses applied in S and S', and the diameter
D
of the wheel at which the couple is applied is D, then Mg— = C
in equation (22).
3
A series of values for 0, corresponding to various masses on
the pans, is obtained. These results may be tabulated as under.
MASS ON PANS
ANGLE OF TWIST
M
M
6°
6
_
100
15
6-6
200
30
6-6
300
46
6-52
400
61
6-55
500
77
6-50
104 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The last column of the table shows the ratio — to be a con-
stant — or nearly so. For such a variation as the one shown in
the table of results above, the mean value of — is taken and
substituted in equation (24). Re-writing (22) we have
„ D TmR* B . „.
M-g • — = — j — 6 e in radians,
M ,gV>l
or n = —• • -s— 6 in radians.
tcR 4
The length of the wire is measured directly. The arm of the
couple, D, is measured by means of callipers.
As R, the radius of the wire, occurs in the fourth power, especial
care is taken to obtain its true value. Determinations of R
are made with a micrometer screw gauge at several points along
the length of the wire, and a mean value taken.
^ M
Substituting the mean value of — from the table of results,
all the unknowns of the equation are ascertained. However,
as is usually measured on the circular scale in degrees, the
M
value of the ratio — must first be converted to radians.
Assuming therefore now that is measured in degrees, the
end result which converts to radians, etc., is
M I g(i8o)D , x
The same type of observations are necessary with the horizontal
apparatus, which makes use of a shorter length of wire specimen.
Maxwell's Needle
If a bar, AB, is suspended horizontally by means of a wire
whose modulus of rigidity n is to be determined, the value of n
may be obtained in terms of I, the moment of inertia of the
bar about the axis of suspension, and T, the time taken for the
bar to make a complete horizontal oscillation.
Consider a small displacement from the position of rest of
the bar. We have already seen (p. 102), that in such a case the
couple called into play in the wire to equilibrate the displacing
couple is t = -j- per unit angular displacement, where
tcR 4
i = , so that the couple exerted by the wire for an angular
displacement 6 is t0 for that particular wire.
ELASTICITY
105
It has been shown (p. 53) that the moment of such external
forces on the oscillating bar is
T dH
Hence
I '-!£- - rS
dt*
i.e. these couples are equal and opposite at any point.
^ = "I
This will cause vibrations whose periodic time, T, is given
by (p. 25)
Hence, knowing T by timing a number of swings, if I can be ,
ascertained t may be evaluated and hence n determined since
*R 4 n
Fig. 55
/ In Maxwell's needle (fig. 55) the bar is replaced by a hollow
tube, of length D cms. Fitting in the tube are four equal-sized
cylinders, each of length/ — j, two of solid brass, and two
hollow.
Let I„ be the moment of inertia of the hollow cylinder of
length D, about the wire as axis,
I ! be the moment of inertia of the solid brass cylinder
about a parallel axis through, its centre of gravity,
1 2 be the similar moment of inertia for the hollow brass
cylinder,
tn x the mass of each solid brass cylinder,
m 2 the mass of each short hollow brass cylinder,
I the length of the wire in cms.
In the first case place the cylinders in the order shown in
106 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
fi g- # 55 (a) and find the periodic time T x . Then arrange the
cylinders as in (b), once more obtaining the periodic time
by timing, say, 50 complete swings. Let this time be T 2 .
Suppose that the moment of inertia of the complex bar in
the first case is I', and I" in the second.
We have T t = 2* JX
T 2 = 2tc <J —
Ti* - t,« =^-Vr -r)
Now we have
for the moment of inertia of each of the four units is given by
the ' law of parallel axes,'
I' = I + 2l 2 + 2 m 2 (^ + ali + zm t (^) 2
whence
n D 2 D 2 / \ D 2
= 2w x -g - 2w a -g- = f m x — m 2 J —
By such an arrangement I' — Y may be evaluated and hence
t may be calculated.
To enable an accurate measurement of T in both cases, the
hollow frame carries a small mirror. A beam of light from a
lamp is focussed on the mirror, and the reflected beam is directed
on a scale. As the reflected beam, shown as a spot of light on
the scale, passes a certain mark in one direction, a stop-clock is
started, and, starting counting, 1, 2, 3, etc., as the spot again
passes the scale reading in the same direction, the time for fifty
complete oscillations is obtained, when the four short cylinders
are arranged as in fig. 55 (a) and (b), whence T t and T 2 are
obtained. The length I of the wire from the rigid support,
and the length D are measured directly : m x and m 2 are obtained
by weighing, and therefore all the factors for the determination
of n are measured, for,
2 I' — F . .m, - m, D 2
and n
4w 'TV=Tr» = 4w TV
_ 2tZ
~^R*
ELASTICITY 107
Determination of the Modulus of Rigidity of the Material of a Flat
Spiral Spring
In this experiment the value of n for the material of the wire
of a flat spiral spring is deduced from a knowledge of the periodic
time of vertical oscillations of the spring when loaded with a
known mass, and the dimensions of the spring.
Let us consider a flat spiral spring, whose turns are closely
wound with a wire whose radius r is small compared with the
radius of the spring itself ; such a spring may be made by winding
the wire on a wooden cylinder of suitable diameter.
Let the ends of the spring be bent twice at right angles, the
free ends, such as BC (fig. 56), being therefore along the axis of
the cylinder.
If the spring is clamped vertically at one end, and loaded with a
mass, M, at the other end, as shown in the figure, the force, Mg,
along the axis exerts a couple tending to twist the wire in the
direction of the arrow.
It was shown* on p. 102 that, under such circumstances, if I is
the length of the wire from the fixed end, and R the radius of
the wire,
C = f^ 6 (25)
2 I
In the case of the spring we may apply this formula when /
represents the total length of wire, and is equal to 2ttRN, where
N is the number of turns.
If r = radius of the wire,
R = radius of the cylinder on which the spring is
wound -f «r,
I = 2«RN,
M = the load in grammes,
to c -'r-iaar (25a)
108 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Take a section of the spring at A, shown in fig. 57 enlarged.
When the couple is applied the arm AB is twisted through the
angle 0, given by equation (25a) above, and takes up the position
AB 1 .
Fig. 57
The depression x, of the end B, is obviously equal to R0,
approximately, when R is large,
i.e.. * = R.***!C = 4NR!C
If / be the restoring force on M due to the wire, the
couple C =/R.
ie - *5RF=' "• (26)
The equation of motion of the moving mass is therefore
M dt* ~ 4NR 8 ' *"
This was shown (p. 25) to represent S.H.M., whose period is
given under.
T = ->/^ r w
From observation of the periodic time of the spring oscillating
in a vertical manner, n, the rigidity coefficient of the material
of the spring can therefore be obtained.
However, in the foregoing we have neglected the mass of the
spring itself, and have also assumed that the total depression
produced by M was due to twist alone.
We must now consider these factors.
Shear in the Spring
We can very easily see that the shear effect in the type of spring
taken above is of a negligible order when the radius of the
wire is small compared with the radius of the spring. For this
purpose again neglect the mass of the spring.
If / is the force acting along the axis, at any point A in
the spring there is an equal and opposite force,/, for equuibrium.
Thus, for the small segment AD, shown enlarged in fig. 58,
at A there is a downward force /, and at D an upward force /,
constituting a shearing couple which would cause a depression
ELASTICITY
109
of A to some lower position A 1 , through an angle 9', say, where
from the definition of rigidity
v An
The area A is the area over which the forces / act, which, in
this case, is the area of cross-section of the wire, nr 2 ,
„> - f
i.e.
icrht
Fig. 58
The total depression due to this shear for a length / of the wire
(28)
Now, using the expression for the depression due to twist,
given in equation (26), and substituting I = 2«RN in the ex-
pression (28) above, we have
/2tcRN
Depression due to shear _ nrhi r^
Depression due to twist /4NR 8 ~~ 2R a
r*n
Take an average case, r = -05 cm., R = -8 cm.
r* = 1
2R* 512
i.e. depression due to shear is about -2 per cent of the depression
due to twist.
no ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Thus, when r is small compared with R we may neglect the
shearing effect.
The Mass of the Spring
The expression for the time of vibration is modified if account
be taken of the mass of the spring itself.
The total mass moving is greater than M. We may regard
the problem as being similar to the case of an ideal massless
spring, loaded with (M + m'), where m' is an additional load
which just has the same effect as the mass of the actual spring
used ; m x may be called the ' equivalent mass of the spring ' and
T = 2tt l (M + m') 4 NR 3 ~
\ r*n
Thus, using this formula, we may find n and m' by obtaining
T for two loads, and solving for m' and n.
However, we may treat the problem in somewhat more detail,
and deduce an expression for the equivalent mass m', in terms
of the actual mass m of the spring.
It has been shown that the twist of the wire of the spring at
the end is = f— , since C, the couple, is equal to MgR.
The depression of the end due to this twist is R0.
If this depression is x,
* = 2 ™^! (29)
In the vertical oscillations, the mass M at the end of the
spring is depressed further than this. To arrive at a value of
the forces called into play due to such further straining of the
spring, suppose a bigger mass, M 1 , be applied, causing an increased
depression z, i.e. total x -{• z.
Then . x + z = ~^-
* = ^-(M*--M)*» (30)
To allow for m, we may consider the energy of the system
with regard to the position of rest of the loaded spring.
•The internal forces equilibrating the force
(M 1 — M)g, axe therefore = —r^i-
Therefore the equation of motion of the mass M is
Mz= —
= 2 *Y
2M/R»
■K**n
as previously shown, equation (27), neglecting the mass m of the spring itself.
ELASTICITY in
When displaced a distance z from the position of rest (which
is x cms. below the unloaded end)
Potential Energy is
—Mgz—mg (displacement of centre of gravity) + (energy stored
as strains in the spring).
The centre of gravity is lowered a distance which is equal to
z
half the depression of the end= - .
The energy of the spring is equal to the work done in straining
the spring, i.e. since the internal forces are equal to (M 1 — M)g
= — rp-'^i from (30), the work done in depressing the end a
distance dz is l—p~\zdz,
i.e. energy in the spring is
•ttrhi , _ nr*n z 2
2mF' z ' dz " iJR 2 ' 2
Total P.E. is therefore
7tr% 1
___*2_Mg Z - -mgz.
The Kinetic Energy is
\ M(i) 2 + K.E. of the moving spring.
If 6 is the mass of unit length of the spring wire, the second
term above is
/ — (mass of a small element) x (velocity of the element) 2 .
At a point P s cms. from the point of support measured along
the length of the spring, consider a small element ds. The
velocity of that element is j • z, its mass is dds,
i.e. K.E. of moving spring is
. 2J
dds • j 2
= i*Ly-=I(™W since m-M.
2 l z 3 2 \ 3/
The total energy of the system in the position considered,
and which is constant by principle of conservation of energy, is
therefore
^ -(^ + ^, + 1(11+5),.- constant.
H2 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Differentiating and dividing by i we have
5£.-(* + «.) + („ + !).-.;
as shown on footnote, p. 98, such a system is a periodic motion
whose periodic time T is the same as for
or
nr*n
2.1 t
M + ™W
i-e. T = 27r-JJi f (31)
The effect of the mass of the spring is therefore the same as
though M were at the end of a massless spring together with a
load equal to | the mass of the spring.
Experimental Details.
A flat spiral spring is chosen, having a radius R which is
fairly large compared with the radius of the wire. One end is
clamped firmly in a heavy retort stand ; a mass is attached to
the lower end of the spring, and the time of vibration of the
vertical oscillations is obtained by timing 50 vibrations with a
stop-watch. This is repeated for various loads.
/, the length of the wire in the spring, may be obtained from
a knowledge of N, the total number of turns in the spiral. If
there are exactly N turns, the length is 2tcNR where R is the
average value of several observations of the mean radius of
the spiral, i.e. the value of the outside radius of the wire spring,
minus the radius of the wire of which it is made.
r occurs in the fourth power, so should be measured with
extreme care. A number of values are obtained with a screw
gauge at points along the length of the spring, and the mean
value taken.
/ being equal to 2tcRN, we have from equation (31)
i6k 2 R 8 N + 3 ..
n = — jT-'—fr* - (32 )
The results of the several experiments may be conveniently
tabulated as over.
ELASTICITY
"3
M.
(M + tn/3.)
T.
T*.
M + w/3.
"£2
mean value of
M H —
3
The mean value of
— ^ for the series of loads (M) taken
is obtained and substituted in equation (32), thus giving the
mean value of n for the complete set of observations.
Determination of Young's Modulus of the Material of a Spring
The value of Young's modulus for the material of the wire
of which a spring is made may be obtained by supporting the
spring vertically and allowing a bar, which is firmly fastened
to the lower end, to perform horizontal swings as in the case of
Maxwell's needle (p. 102).
It is shown below that in such a case the time of a complete
horizontal oscillation is
8RNI
where
Y
R
r
I
is Young's modulus for the wire,
is the radius of, the spiral, measured from the centre
of the wire,
the radius of the wire,
the moment of inertia of the bar about the axis of
suspension,
N the total number of turns in the spring.
The above assumes, as in the last experiment, that the spring
is a flat spiral ; the layers are very close together, and each may
therefore be regarded as horizontal.
Let fig. 59 represent a horizontal section of a small length s
of the spring, with NS the section of the neutral surface by the
plane of the diagram. This neutral surface will be normal to
the diagram. ABCD is the section of the element of the spring
when the bar is in an undisturbed position.
8
H4 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
When the bar is turned in a horizontal plane through any
angle y>, the section taken, in common with every other element
of the spring, will become more curved, as shown in the broken
lines.
Consider a single ' filament ' of the material, as was done in
the case of the bending of beams (p. 86). If a is the area of
cross-section of the filament, and / is the force required to bend
up the filament through a small angle
Y =
Strain of the filament
Now, if R is the normal radius of curvature of the neutral
surface, and PQ is a distance x cms. beyond NS, the original
Fig. 59
length of PQ = (R + x)6 , where O is the angle between the
radii from the original centre -of curvature O to the ends AC
and BD of the element considered.
Now, due to stresses similar to /, the element has a greater
curvature, and consequently the neutral surface has a smaller
radius of curvature R. If O 1 is the new centre of curvature,
the new length of PQ is (R -f x) 0.
However, the neutral surface is Of the same length as before,
so that R O O = R0.
The strain is therefore (R + x)S — (R + x)d -f- original length
== — — f — — approximately.
S = Rn0fl = R0-
ELASTICITY H5
Strain is therefore
* \R0 "" *W = * \S ~ Ro/
/
4Z
• Y =
The moment of this force on the filament about the neutral
surface is fx. The total moment of the couple acting on the
element considered is therefore
**»-/*«■(£-£)
If the angle subtended at the centre by i cm. of the wire
be q> and <p Q , respectively, the above couple is
"C = Yi{v - <Po)-
If I is the length of the wire in the spring, the total change
of angle, i.e. the angle through which the inertia bar moves is
y, = l(<p — <p Q )
i.e. C-Yt*-
This is the value of the- couple due to the internal stresses,
called into play by the strains in the wire,
i.e. Iv = — j- v
The periodic time T is given by
i = ^1, 1 = 2*RN
4
or
I8INR
T = aTc^J^Y -
32k5NR
1 ~" r*T*
n6 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Experimental Details.
The spring is set up as in the last experiment, and a rod, say,
of rectangular cross-section is clamped to the end of it, so that
there is no free play between the end of the spring and the rod,
and the centre of gravity of the rod is under the suspension.
The ' inertia rod ' is then given a displacement in the horizontal
plane, and the subsequent horizontal oscillations are timed. T is
obtained by timing 50 complete swings in the usual way.
r and R are carefully measured as previously described, and
the total number of turns of wire in the spring, N, is counted.
The value of I may be calculated from a knowledge of the
dimensions and the mass of the bar (see p. 54).
Determination of the Modulus of Rigidity and Young's Modulus for
the Material of a Wire by Searle's Method
To find Y, Young's modulus, or n, the coefficient of rigidity,
of the material of a wire specimen by this method, the wire is
Fig. 60
fastened to two identical rods, &J& lt A 2 B 2 , at their mid-points,
as shown in the diagram at C^ C 2 . These rods are usually
square or circular in cross-section, and are supported by threads
from points T 1 and T 2 , such that the axis of suspension and the
axis of the wire intersect at the centre of gravity, and the sus-
pended rods are a distance apart, which allows the wire to be
stretched in a straight line, the whole assuming an H formation.
If now the ends, B x and B 2 , of the rods are drawn together
and fastened by a loop of thin cotton, the wire will be bent into
the arc of a circle, subtending an angle 2<p at the centre ; each
rod will make an angle q> with its original direction.
The suspension of the two rods being such that the torsion
is negligible, the only couple acting on the bars is that due to
ELASTICITY 117
the bending of the specimen, CjCj. With this method of suspension
we have a couple produced equal to ^ in the wire (p. 88),
where * is the ' moment of inertia of cross section/ and R is the
radius of curvature of the arc C^.
Fig. 61
Now, if I is the length of the wire between the rods,
R~i.
2q>
If r is the radius of the wire (assumed circular),
• = *^
4 '
whence the couple acting is
4 * I ~~ 2/
Let I x be the moment of inertia of the rod, kj& x , about the
axis of suspension, -57J the angular acceleration. The moment
of the external force is, by the theorem given on p. 53, 1-j^-
This is equal and opposite to the restoring couple exerted by
the bent wire. The equation of motion for the rod is therefore
This is simple harmonic motion (p. 25), and
T =2K J5T
\*Yr«
ii8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If the rod AjBj has a length 2L and width 2a and mass
M grammes,
M
L 2 + a*
whence
Y =
8tt/ L 2 + fl2
M.
y 4 T 2 3
To evaluate Y for a specimen of wire, the arrangement
described is fitted up. The thin cotton loop holding the rods
together at B X B 2 is burnt, and the resulting oscillations are
timed with a stop-watch, T being obtained by timing 50 complete
oscillations. An alternative method of finding T, and which
may be used in many experiments where T is required, is seen
under.
A reference point having been chosen — say, a chalk -mark
under the oscillating bar — the time is noted at which the counting
of the swing is commenced. After 5 swings the time is again
noted, and the times at the end of every 5 is recorded in a table
as seen below.
(I)
NO. OF OSCIL-
LATIONS
(2)
TIME
(3)
NO. OF OSCIL-
LATIONS
(4)
TIME
(5)
TIME FOR 30
OSCILLATIONS
U
30
a
a — u
5
V
35
b
b -V
10
W
40
c
c — w
15
X
45
d
d — x
20 sm
25
y
z
50
55
e
f
e -y
mean time for 30 oscillations. =
Tabulating in the second column the time for the starting point,
5, 10, 15, 20, 25 oscillations, and in the fourth column the time
for the 30, 35, 40, 45, 50, 55 oscillations, the difference in any
line between the fourth and second column value will give the
time for 30 complete oscillations. The mean value of the last
column being taken as the time for 30, T may be evaluated.
The radius of the wire occurs in the fourth power. The value
of the diameter is therefore measured in several places, say, six,
and the mean radius, r, calculated. The other measurement for
the evaluation of Y are straightforward.
It will be noticed that in this method the value of Y is obtained
by timing, and not by observing deflections as in some of the
ELASTICITY 119
other methods. Another advantage of the method is that it
requires only a small specimen of the material.
To find the modulus of rigidity of the specimen, the rod A 2 B 2
is clamped rigidly in a horizontal plane and the wire acts as a
support for the rod AjBj, which may therefore be made to
perform horizontal torsional oscillations. The rod is displaced
slightly from its position of rest causing a restoring couple to
act on it, due to the twist of the wire.
For a displacement 0, the restoring couple is
as established on p. 102.
We therefore have as the equation of motion of the rod
T d 2 d _ _ nnr 4 *
dt* ~~ 2.1
This is once more a case of simple harmonic motion, whose
periodic time, Tj, is given by
\ = ^
2/1
If the rod is of square or circular cross-section, the moment
of inertia about the axis taken in this experiment is the same
as the last case. If of rectangular section, the moment of inertia
will be obtained by calculation as before.
Tj may be found as described for the first experiment; the
length I and the radius r are already known.
Hence n may be found by substitution in the formula below :
_ 8tc/I
Determination of the Bulk Modulus for Glass
The bulk modulus, as shown on page 82, may be expressed as
k--L
V
where p is an increase in pressure causing the small volume
change 6V.
In general, it will not be convenient to apply a uniform pressure
to a body, but it is a simple matter in most cases to apply an
extending force per unit area, or a pressure, in one direction
only. If, as a result of such a pressure, the change in volume is
<$V l , then 6V 1 — — • 6V where <5V is the change in volume when
the same pressure is applied uniformly over the body.
If a cylinder be supported at one end, and an extending force
120 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
be applied to the other, in a direction coincident with the axis
of the cylinder, we have the extending force at one end and the
equal reaction at the other. The change in volume in this case
is of the type 6V 1 above. So that if the change in volume of a
body be observed in such a case, the value of k may be calculated.
The apparatus used to determine the bulk modulus of, say,
glass is seen in fig. 62. A glass tube g is cemented to two caps
of brass, A and B. The upper brass cap A is provided with
two pegs, which act as a support for the whole apparatus. The
lower cap B carries a hook to which a pan S is attached.
I?
a
B
S
Fig. 62
Fitting in the upper end of the tube is a rubber cork, carrying
a capillary tube, CD, which is graduated and calibrated in
the manner described on page 41.
The tube g is filled with water. Care is taken to avoid
trapping air, and the cork, etc., is placed in position, resulting
in a little water rising in the capillary tube, i.e. the whole of the
glass tube and part of the capillary are filled with water to a
definite level. If now the pan S is loaded with, say, 5 kilogrammes
the volume of the glass will increase by a small amount. The
difference in the levels on the calibrated tube enables the value
of the volume change to be determined if the temperature
remains constant throughout the observation.
However, the apparatus is, by the construction, very much
affected by small temperature changes ; it is a water thermometer
with a very large bulb. To obtain a good approximation to the
value of 5V 1 , the volume change, the reading on the tube is
noted when S has no load ; 5 kilos are applied and the reading
ELASTICITY
121
noted, the load is removed, and once more the scale reading is
taken and the process repeated with 5 *& os ^ no load for
10 observations, each one being taken at a regular time interval
as under.
TIME FROM
TIME FROM
COMMENCE-
LOAD
READING
COMMENCE-
LOAD
READING
MENT
MENT
o min.
o kilos.
5 min.
o kilos.
*
5
5*
10
I
6
I*
5
6*
IO
2
7 •
a*
5
7*
10
3
8
3*
5
8*
10
4
9
4*
5
9*
etc.
10
A similar process is repeated with a load of io, 15, and 20 kilos.
The results are plotted as a graph, using a fairly large scale
for the ' scale readings ' as ordinates.
O ^ Time in minute© .
Fig. 63
The no-load curve will be a continuous line, and the short
curves for 5, 10, etc., load will be at a vertical distance above
it, depending on the load employed.
The form of curve obtained in a particular experiment, using
one type of glass, is seen in fig. 63.
AA X , BB 1 , CC 1 are then drawn parallel to the axis of scale
readings, and their values measured : the mean value of this
122 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
length gives the scale reading increase in volume of the glass
tube g. The actual volume change is then obtained from the
calibration curve of the capillary tube.
This process is repeated for the curves giving the results for
each load, the mean of the three ordinates being taken in
each case.
From each value of <5V the increase in volume SV per kilo
load may be calculated. A mean value for all loads therefore
gives the mean for all the observations taken, for the value
of SV per kilo.
The original volume of the glass tube may then be found by
obtaining the product of the length between the brass caps and
the area of cross-section.
To obtain the cross-section, a little water is poured into the
empty tube so that the surf ace becomes visible above the lower
cap B. A measured volume of water is then run into the tube,
so that the level is just below the upper cap. From a measure-
ment of the increase in the level and the volume introduced
from a graduated flask or measuring jar, the average cross-section
may be calculated. Hence V is obtained.
To find the stress applied to the tube we must find the internal
and external radii of the tube. The load applied at the end
causes a stress, a force per unit area, equal to /T -. m ^ — r-, where
7U(R 2 — f 2 )
m is the mass, R the external and r the internal radius of the tube.
Now, Ttr 2 was determined above, when obtaining V. To find
R, the external radius is measured with callipers at about 6 or 8
places, and the mean taken.
Thus, if 6V is the mean value of the volume change per kilo-
gramme load, i.e. for a force of 981 x 10 3 dynes in one direction,
the change for a uniform stretching force would be $dv,
981 x io 3
and K = -r— — dynes/cm a .
Poisson's Ratio for India-rubber
k
A length of solid india-rubber is supported at one end and
loaded at the other. The rubber, of. circular cross-section, is
marked at about six places along its length, and the diameter
at these selected places is measured with a micrometer screw
for each load placed in the scale pan at the free end of the india-
rubber.
ELASTICITY 123
The length of the specimen is also measured for each load.
Then, since Poisson's Ratio has been denned as
Lateral contraction /
_ Original diameter
^ ~ Longitudinal extension
,. Original length
(i may be calculated for each load, the mean value from the six
measurements in the lateral direction being used.
For a rubber cord of about f-inch diameter, suitable loads
would be 500, 1000, and 2000 grammes.
It will be found that, to some extent, the values obtained
depend on the history of the specimen. The results when the
load is increasing will be somewhat different from the values for
the same load when decreasing.
In all observations the readings should not be taken until about
ten minutes after the adjustment of the load.
CHAPTER IV
SURFACE TENSION
The surface of a liquid acts in many respects in a manner
analogous to a stretched membrane. The well-known example
of mercury resting on a clean wooden surface shows the effect
to a marked degree. The mercury takes the form of a globule,
as if it were surrounded by a membrane supporting it in this
form.
The examination of a water drop slowly formed at the end of
a glass tube or tap from which it emerges provides another
example of this phenomenon. The water in this case accumulates,
as though it were collected in an invisible membrane, until of
a definite size, when it is detached as a spherical drop.
D i B
A.
tff
Fig. 64
B
These effects are due to forces existing in the surface of separa-
tion of the liquid from the air and the other media in contact
with it. The effect is generally known as surface tension. This
term may be defined in two ways, depending on the point of
view taken.
If we imagine the surface of the liquid to be cut by a plane,
there is a definite force per cm. acting on the line of intersection,
at right angles to its length and parallel to the surface. This
force, expressed as dynes per cm., is defined as the surface
tension of the liquid. The value of this force depends on the
liquid and the surrounding medium, but, unless otherwise stated,
will be taken throughout this book to represent the force per
cm. when air is the medium.
For example, consider a frame ADEB, fig. 64, of width I cms.,
across which a film of liquid, whose surface tension is T dynes
per cm., is stretched.
If the film terminates at the lower end on a light rod AB,
124
SURFACE TENSION 125
since the liquid film has two surfaces, it will exert an upward
force of 2T/ dynes on the rod. If the rod has a mass of m grammes
it will be in equilibrium when
217 = mg.
Now, suppose the rod be displaced a small distance 8x, to A*B 4 ,
against the surface tension forces, the work done is 2TI • dx,
and the resulting increase in area of the surface is 2I • Sx.
Thus, the work done per sq. cm. of surface is
2Tl6x „
^^= Teigspersq.cn,.,
which may be taken as defining surface tension, i.e. the surface
tension is the work done in enlarging the surface by one cm?"
When a liquid is placed on a horizontal plane surface, the
form it takes depends, for a given liquid, on the material of which
the plane surface is made. Thus, if water is placed on a clean
glass surface it spreads over it, whereas if the glass is greasy
the water takes the form of globules.
Bl
u v-— <
1 •
A
Fig. 65
The angle contained between the plane surface and the liquid
surface is different in each case. If we measure this angle in
the liquid we have a measure of the angle of contact. Thus, in
fig. 75, showing a section of a mercury drop on a glass surface,
is the angle of contact, whereas for a liquid like water which
' wets ' the glass the angle of contact is zero — the water spreads
over the surface.
METHODS OF MEASURING THE SURFACE TENSION OF A
LIQUID
(1) Wilhelmy's Method
To determine the approximate value of the surface tension
of such a liquid as water, paraffin oil, or turpentine, the following
method may be employed.
A clean wire, preferably platinum, is bent into the form shown
in fig. 65, making three sides of a rectangle of breadth / cms.
126 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
It is then suspended from a beam of a balance by means of
a thin wire, and counterpoised when the upper horizontal arm, B,
of the frame is almost immersed in the liquid.
When a balance is obtained the frame is dipped under the
surface of the liquid by lowering the beam of the balance, and
then withdrawn again. A film of the liquid will be formed in
the frame as seen in the shaded part of fig. 65.
Due to the downward surface tension pull on each side of the
film of length I there will be an apparent increase in weight = 2IT.
This can be found experimentally by adding ' weights ' to the
other scale pan> until on raising the beam it remains horizontal.
Fig. 66
To perform this approximate experiment clean the frame by
holding it in a Bunsen flame until red hot.
Repeat the experiment as described above several times, and
take the mean value, whence if m is the mean value of the added
mass
2T/ = mg,
T = — j- dynes per cm»
Take care that the frame is the same height above the water
surface when the balance is made, before and after immersing
in water, to eliminate buoyancy errors. The surface tension
effect on the two vertical limbs is eliminated, but the experi-
ment may be regarded as one which gives an approximate value
of T as described above.
(2) By Weighing Drops
The liquid whose surface tension is to be measured is allowed
to form drops at the end of a narrow tube, C, fig. 67. If m is the
mass of the drop, we have
mg = KT,
where K is a constant.
SURFACE TENSION
127
From a simple approximate investigation of the case we see
that mg = 7tfT as under.
When the drop is about to break away from the tube we will
assume it has the cylindrical form shown in the firm lines, D.
The broken line indicates one of the subsequent forms.
If r is the radius of the orifice, we have, at this level inside
T
the drop, an excess of pressure equal to — due to the cylindrical
curvature of the liquid surface. This is equivalent to a downward
T
force — • T?r 2 =Tnr.
J y
The weight of the drop being mg, the total downward force is
T7rr + mg.
This is equal to the upward surface tension force over the
circle of contact, i.e. T • 2nr,
or T2rcr = jzTr + mg,
or Tnr = mg.
Fig. 67
This is deduced, assuming static conditions to hold in the
actual case. To a closer approximation Lord Rayleigh showed
that
mg = 3- 8/T
gives an expression of the relation between T and m.
In practice the uncertainty may be avoided by taking the
expression T = \f- and finding the value of K for the particular
tube used.
A tube is drawn out to a fine capillary of about *5 mm.
at the end. It is connected to a burette by means of a
short length of rubber tubing. The burette and tube are of
course first made thoroughly clean, as in all the surface tension
determinations, by the method given on page 42. A liquid of
known surface tension, say water, is placed in the burette and
128 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the tap is opened to an extent which allows the drops to form
at the rate of about one every second. A large number, say
ioo or more, are collected in a weighed vessel and m per drop
is obtained, hence, knowing T, K may be calculated. The same
tube is now used for the liquid of unknown surface tension, say
paraffin oil. Some of the liquid is run through the apparatus to
remove all traces of water, and the burette is then filled with an
uncontaminated sample *of oil. The collecting and weighing is
repeated, and knowing K, T is calculated.
Find by the above means the surface tension of alcohol,
paraffin oil, benzine, etc.
Fig. 68
(3) Determination of the Value of the Surface Tension of a Solution
in the Form of a Film
The value of the surface tension can be obtained for, say, a
soap solution film in the following way.
Two pieces of copper wire, ABC and DEF, are bent as seen
in figs. 68 and 69, so that AC = DF = about 4 or 5 cms.
At the points A, C, D, and F a length of cotton thread is fastened,
so that when the whole arrangement is suspended at B the
thread takes the form of a rectangle about 5 cms. by 10 cms.
(fig. 68).
If now a film of soap solution be stretched across the thread,
the latter will be pulled into the form shown in fig. 69, AD and
CF taking the form of arcs of circles.
Let AC = DF = 20 cms., GH = 26 cms., AD = CF = 2A
cms., and the mass of the lower copper wire, DEF = m grammes.
Let T be the value of the surface tension in dynes/cm. When
stretched in this manner, there will be a tension in the thread
equal to, say, / dynes.
Suppose a is the angle which is included between the horizontal
and the threads as shown in fig. 69.
The film being two-sided, the vertical force upwards due to
surface tension on the length 2a at the bottom of the film = 4«T.
SURFACE TENSION
129
The resolved part of the tension / at D and F is / sin a upwards,
i.e. total upward force of 2/ sin a.
The downward forces are mg, the weight of the lower copper wire.
We thus have for equilibrium :
mg = 2/ sin a + 4«T (1)
Considering the equilibrium of half of the film as obtained by
a vertical dividing line through the mid-point of the film, we
have the following equal and opposite forces acting :
Fig. 70
(a) A force '/ cos a ' at *^ e to P an ^ an e(ma * one at * ne bottom
= 2/ cos a.
(b) due to the two-sided film a surface tension effect
= 4™,
i.e. 2/ cos a = 4TA (2)
From (1) and (2) we have tan a =
_ mg — 4aT
4Th
(3)
130 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
From the geometry of the case (see fig. 70), if O is the centre
of the circle of which CHF is arc, a = z. OCIC
So that tan a = ^§»
and if HK — d = a — b,
r -d
tan a =
h
Eliminating r since h 2 = (2r — d)d,
h 2 + d 2
r =
2d
h 2 -d 2
We have tan a = ,,
2dh
Equating this to the value given in (3) we get
mg — 4aT ^ h 2 - d 2
4T& ~~ 2dh
2 d(mg) = 4T(A a - d 2 ) + 4*T • 2d,
whence T = /T . ,f -,
2(h 2 — d 2 + 2ad)
Putting d — (a — b)
Using the very simple apparatus described, a fair value of T
maybe obtained simply from a knowledge of a, b, h, and m.
A film is stretched across the string very easily by placing
the frame horizontally in a flat dish containing the soap solution.
Care must be taken to avoid excess of soap solution spreading
over the part of the apparatus other than the thread. None
must be allowed to remain on the lower copper wire, for obvious
reasons.
The measurement of the dimensions of the film can be made
by means of ordinary dividers. The method does not justify
the use of a travelling microscope. The measurements should
be taken fairly rapidly so that they are all obtained before the
condition of the film changes appreciably.
In an example 2a = 3-5 cms., 26 = 1-2 cms., 2h = 11-7 cms.,
m = 1-51 grms., which gives T = 23 dynes/cm.
(4) By Measurement of the Rise 0! a Liquid in a Capillary Tube
If a clean, fine-bored capillary tube is depressed into a liquid
Which ' wets ' it, and is then clamped vertically, the lower end
SURFACE TENSION 131
of the tube being just below the surface of the liquid, it will be
found that a column of the liquid remains in the tube, so that
the surface in the latter is a height h cms. above the free surface
of the liquid in the vessel which contains it.
Suppose r is the radius of the tube and p the density of the
liquid.
The forces acting on the liquid in the tu£>e are :
(1) The weight of the liquid. This is equal to (the volume of
the liquid) X gp.
Now the volume of the liquid is equal to V = nr 2 h + (volume
of the meniscus) for a uniform tube. If r is small the meniscus
is practically hemispherical, hence
i.e. the downward force is
V = *r*h-+ \{nr 2 ) r --t^ 3 '
3 j
7tr
'(* + 0*p (5)
(2) The Upward Surface Tension Force
The line of contact is the intersection of the glass wall
and the liquid surface, i.e. a circle of radius r. If a is the angle
of contact the upward force, from the definition of surface
tension, is equal to
2-nr • T cos a (6)
For equilibrium, the forces (5) and (6) are equal and opposite ,
2tztT cos a = itr 2 f h + - Jgp,
i.e.
or
T = gpr
=H>
2 cos
when the liquid wets the glass as in the case of water and clean
glass a = o,
^ T -££(»+£) (7)
To find T, using a capillary tube, a glass tube is cleaned
thoroughly. This may be done by using nitric acid (in which
the tube is boiled) and caustic soda ; the tube is washed in
tap water and dried in alcohol and ether, or the glass tube is
allowed to stand for several hours, overnight if possible, in a
concentrated solution of sulphuric acid (one part) and potassium
bichromate (one part). It is then washed in tap water and
dried. It 4s not advisable to use distilled water.
132 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The tube is then heated and drawn out to a capillary. A
length of uniform bore is chosen and clamped vertically in a
vessel which is brimful of the liquid whose surface is just above
the top of the containing vessel, as shown in fig. 71. Care is
taken to avoid touching the tube or the liquid in this adjustment,
for even small traces of grease cause a large variation in the
value of the surface tension.
The tube is viewed by means of a travelling microscope,
provided with a vertical traverse, the lower end of the meniscus
is focussed and the vernier reading of the microscope noted. The
free surface of the liquid in the containing vessel is next focussed.
If the liquid surface is just above the top of the vessel, this
level may be viewed very readily, and from the vernier reading
on the microscope scale in this position the value of h may
be obtained.
1
1
—4
Fig. 71
To find r, the tube is broken at the point at which the meniscus
rested, and viewed horizontally by the microscope.
By arranging the cross-hairs »in the eyepiece to be tangential
in turn to the two ends of a diameter, the internal radius may
be measured on the vernier attached to the traverse.
Alternatively, a weighed amount of mercury (Density, D =13-6)
may be introduced into the tube and its length observed by
means of the microscope, when the tube is horizontally on the
bed of the microscope.
Hence r, the mean radius of the tube, may be found if I is
the length occupied by the mercury of mass m, for
{nr 2 )lD = m.
The experiment is repeated, using tubes of various diameter,
and a mean value of T obtained.
An alternative method of measuring h, which also overcomes
the difficulty of viewing the liquid through the glass beaker,
is one which makes use of a pin bent twice at right angles so
SURFACE TENSION
133
that the point is displaced about one centimetre from its original
position. The pin is attached to the capillary tube by means
of a rubber band (see fig. 72). The point is in this way well
removed from the curved surface of the liquid round the tube
itself. It is adjusted to coincide with the free surface of the
liquid. , . . .
The image of the upper end of the pin is brought mto coinci-
dence with the cross-hairs of a microscope, and the vernier reading
on the vertical traverse is noted. The meniscus is next viewed,
and the distance between it and the pinhead determined by the
subtraction of the vernier readings. A subsequent measurement,
by means of the microscope, of the vertical distance between
the pin point and head enables the value of h to be obtained.
Fig. 72
(5) Surface Tension Determination from Measurements of Bubbles
This method is suitable for measurements of surface tension
of soap solution.
Inside any curved film in equilibrium there is an excess of
pressure over the outside, by an amount which depends on T,
the surface tension, and R, the radius of curvature.*
Consider a bubble with excess pressure p inside : take a
section through the centre ; then there is an equivalent force
over this section = p X (area) = ^>tcR 2 .
The opposite surface tension forces, since there are two surfaces
to the bubble, is 2T x (circumference of section)
= 2T-2tuR.
* It can be readily shown that inside a cylindrical./*/** (which has two surfaces)
2T
this pressure excess == -j^-
If now the film has an equal curvature in the other direction at right angles—
2T
making spherical bubbles, there is a further pressure excess of ^, making a
total ±£ =
p the excess of pressure.
134 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
For equilibrium these are equal and opposite,
i.e. 4tcTR = ttR 2 £
P =
4T
R
A suitable form of apparatus with which to obtain a measure-
ment of p and R is seen in fig. 73.
This consists of a fairly wide U-tube, containing water, sealed
to a T-piece, T. One arm of the T-piece is bent at right angles
and terminates at B, where AB is parallel to the U-tube limbs.
To the other arm at C is attached a piece of rubber tube, R,
having a glass rod, P, which just fits it, and which can act as
a piston.
P c^*
/ /^ T " A
V->
D
F
Fig. 73
B
-D-
The piston P is withdrawn and the end B immersed in the
soap solution ; then P is advanced slightly, causing a bubble
to form at B. The excess of soap solution is drained off this
bubble by touching any excess with a pencil, or the side of the^
dish which contains the solution. The rod is then further
advanced, so that a bubble of, say, 1 to- 2 cms. diameter is
blown. It will be noticed that the water in the U-tube takes
up a position as in the diagram indicating a pressure inside the
bubble in excess of the atmospheric pressure.
This difference in level may be measured, using a microscope
with horizontal and vertical traverse.
Having read this, the microscope is moved until the bubble
is focussed, first the image of one side, and then the other being
brought in coincidence with the cross-hair. The difference in
reading of the vernier of the horizontal traverse giving the dia-
meter D, of the bubble,
p = hg dynes per sq. cm., for a water manometer.
R-5.
Hence T =
teD
SURFACE TENSION
135
(6) Quincke's Method
This method is most readily followed for the determination
of the surface tension of mercury. A large flat drop is formed
on a horizontal platform, and from its dimension, as measured
by means of a travelling microscope, the value of T, the surface
tension, and 0, the angle of contact of mercury on the platform
may be calculated.
£
$
--B--AB
Fig. 74
We will assume that the radius of the flat drop, R, fig. 74, is
large, so that the drop is truly flat, i.e. the pressure at a point
just above and just below the upper surface is the same.
Consider a section as in fig. 75, obtained by cutting it by a
vertical plane passing through the centre, LMRN, and two
parallel vertical planes, AINL and ESRM normal to LMRN.
Let ACDE be the horizontal plane of maximum area in the
drop, so that a tangent plane at AE is vertical.
M
T
;«
^*^s
L
sfff///
/Ve
/^T "--
C
VV 'ff " "
I N
Fig. 75
By considering the forces on the upper portion, AECDLM,
which is in equilibrium, a value of T may be obtained in terms
of the dimensions.
The horizontal forces acting on this portion of the drop due
to the remainder of the drop are
(a) T-LM, left to right,
(p) The hydrostatic pressure over the surface LMDC, from
right to left varies from o at the upper surface to gph 1
at CD, and is of a mean value " — , i.e. total force is ^—
2 2
X area LMDC, where p is the density of mercury,
i.e. equating (a) and (p)
T-LM = £- p - l • JMh\
2
T== ^i 2
2
or
.(8)
136 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Considering now the equilibrium of the whole slab, we have
in addition t© the corresponding terms, T • LM and ^--, where
2
A is now the total height (fig. 74), a term T- IS • cos a= -T« LM cos
(fig. 75), from left to right.
or T-LM(i -cos«) =££^LM
2
or, substituting the value of T from (8),
. 6 h
sin — = — -==— - (q)
The experimental details necessary to carry out such a deter-
mination are as follows : A clean glass slab, provided with three
levelling screws, is first of all arranged horizontally, by means
of a spirit-level. Mercury is placed on its surface in the form
of a circular drop (fig. 76).
-*- ^
Fig. 76
N.B. — The mercury should be as free as possible from impuri-
ties. To ensure this it should, if possible, be submitted to one
of the methods of purification described on page 532.
To obtain the values of h 1 and h, use is made of a travelling
microscope, which is provided with a cross-hair. The cross-hairs
are arranged so that one is vertical in the eyepiece ; this can be
done by viewing a thin wire plumb line. The edge of the drop is
then focussed. When the vertical cross-wire is tangential to
the image of the side of the drop and the intersection of the
cross-hairs is at the point of contact, such as A, figs. 74 and 76,
the position on the vernier of the microscope is noted. Then
by making use of the vertical movement the upper and lower
surfaces of the drop can be focussed and the corresponding
vernier reading will enable the values of h x and h to be ascertained.
Note
For liquids which wet the surface of the glass a simple modifi-
cation enables the method to be utilised.
SURFACE TENSION 137
A concave lens of over one metre radius of curvature is
supported concave side downwards, on three legs inside a glass
box, having one plane glass side. The box is filled with the
liquid, and an air bubble is blown under the concave surface,
using apparatus similar to that shown on page 141. A narrow
bent tube is connected to R in place of the U-tube, etc. shown
in fig. 79. The slight concave surface enables this air bubble
to be blown without much trouble. The section of such a
submerged bubble would be similar to that shown in fig. 74,
when inverted. Then, if h 1 is the distance between the lower
surface of the large air bubble and the plane AB, and if h is
the total thickness of the bubble as measured from the lower
surface to the plane of contact with the lens, all measured through
the plane glass window by means of a microscope, equations
(8) and (9) will give T and for the liquid in the box.
(7) Rayleigh's Method
This method depends on a measurement of the wave length
of ripples formed on the surface of the liquid whose surface
tension is to be determined.
The velocity v of a harmonic disturbance on the surface of any
liquid is given by*
„= !A g+ ^I, (10)
N2« 5 ^ Xp
when x is the length of the wave, T the surface tension, p the
density of the liquid, that is
» = ^+£- T ) (I0 «>
The surface tension is therefore seen to increase the effective
4.7C 2
value of g. For large waves the term ^--T can be neglected,
x p
and the velocity of propagation is <J-s.
For waves of less than 1*5 cm. wave length, the value of the
term involving T becomes more important, and when X is suffi-
ciently small the velocity becomes more nearly equal to
-^
Xp
i.e. the first term in equation (10) becomes negligible cf. the
second when X becomes very small.
Equation (10) shows that when x = o, v = 00 , and when
* See Poynting & Thomson's " Properties of Matter."
138 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
X = oo , v = oo , between these values of x there is a minimum
value of v, corresponding to a value of x, which is obtained when
_X _ 2ttT
2n S ~~ Xp '
(T~
X = 2n-J— ; (n)
this mmimum velocity is therefore
'-GJ
V2
For water x OT = 17 cms. from equation (11), for T =75, p=^i,
g = 981, and hence
v m = 23 cms. per sec.
Waves having a smaller wave length than this critical value
which corresponds to the minimum velocity are called ripples,
and the more important term is the one involving T.
Such ripples cannot be viewed and X measured directly, but
the following method, first used by Lord Rayleigh, may be
employed with success.
The liquid is placed in a large flat dish, say, a large porcelain
developing dish. As in all surface tension experiments, the dish
is cleaned thoroughly before introducing the liquid.
The ripples are made by having a tuning fork electrically
maintained, arranged at one end of the dish as in fig. 77. Attached
to one prong of the fork and dipping in the liquid is a thin
sheet of aluminium foil, D. If the frequency of the fork is
n, and X the wave length of the ripples we have
v = n\.
But the ripples cannot be directly observed and measured, as the
phase change is rapid, and a general illumination results.
If these ripples are viewed by intermittent light, the frequency
of the flashes of illumination being the same as the source of
the ripples, then they appear stationary and may be measured.
(Cf. stroboscope, p. 404.) Alternatively, if the observer has
intermittent views of the surface of the liquid, such views being
of same number per second as the vibration number of the
tuning fork which causes the ripples, between each view the
ripples will have moved forward a distance equal to a wave-
length, and if illuminated by a constant source of light they
will appear stationary.
To obtain such intermittent glimpses of the surface, the latter
is observed through the prongs of a second tuning fork of the same
frequency, and maintained in vibration by the same circuit.
Two thin pieces of aluminium foil, A and B, are attached to the
SURFACE TENSION
139
prongs of the second fork, so that direct vision is impossible when
the fork is at rest, but is obtained when the prongs of the fork
are at the position of extreme separation. Thus, for each
complete vibration of the fork a view is obtained of the surface,
and an apparently stationary train of waves is seen when viewed
in this way. The wave length can be obtained by direct measure-
ment of the longest possible number, m say, by dividers
adjusted over the surface.
When performing this experiment with water, the measurement
will be found to be somewhat complicated due to shadows cast
by the water, and the exact setting of the dividers over the
surface for the stretch of m waves will not be as easy as in the
case of a more opaque liquid.
Fig. 77
A method which has been found to give satisfactory results
is to suspend an incandescent bulb about two metres above the
surface of the water. This casts a series of shadows on the
bottom of the white porcelain dish. The dividers are arranged
also to cast a shadow and adjusted so that the two ends of the
dividers' shadow coincides with the corresponding parts of the
first and last of the m waves. In such a case the magnification
of the shadow of the waves due to the obliquities of the incident
beam is compensated by a corresponding distortion of the shadow
of the ripples, and the kind of thing shown in fig. 78 results :
the length of, say, ten or more ripples may be obtained, whence x
may be calculated.
Now, if n is the frequency of the fork,
or
T=^(2M a X*-£).
140 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
A suitable experiment by such a method is to find the variation
of surface tension of a salt solution with concentration.
Use first of all pure water. Make sure that the aluminium
plate is clean, and that no soft wax, by which it is fastened to
the tuning fork, adheres to it, and so contaminates the water.
Measure the value of X and calculate T, n being known.
A suitable frequency for the two tuning forks is about 60.
Then repeat the experiment with a sodium chloride solution
having \, 1, ij, to 5 grm. molecules per litre, and plot a graph
showing the increase in the value of T with concentration of
the solution.
Fig. 78
(8) Jaeger's Method
In this method air bubbles are formed in the liquid under
investigation. The bubbles in this case have one surface only,
and therefore the excess of the internal pressure over the external
assuming a hemispherical form is
* 2T
The value of R is fixed, and equal to the radius of the orifice
from which the bubbles emerge (approximately).
The following apparatus, as seen in fig. 79, is used.
A thin glass tube, C, drawn out to a fine capillary of less than
*5 mm. ('2 to *5 mm. is satisfactory), and is fixed vertically in
the liquid. It is connected either rigidly, as in the diagram,
or by means of a piece of india-rubber tubing, to glass tube, AB,
which terminates in another rubber tube, R, which is clamped
in a screw pinch cock, T.
Sealed into AB is a glass T-joint which leads to a manometer U.
Air is sent in at D at a steady rate, and consequently, bubbles
are formed in the liquid. To ensure that they are always formed
at the same depth, h cms., below the surface of the liquid, a
scratch is made on C, and this is adjusted to coincidence with
the liquid surface.
When the bubble is of the same diametei as the orifice at G,
SURFACE TENSION
141
it becomes unstable and breaks away. During the formation
of the bubble the manometer rises and the maximum difference
in heights between the two columns, H cms. , is observed for many
bubbles, the pressure of the incoming air being constant ■ *
When a good agreement for H for a series of bubbles is obtained,
the tube C is taken out and the value of R determined by
viewing the end with a microscope. If the capillary is very
small, a microscope carrying a $mall scale in the eyepiece is
used, and the value of the scale divisions in the eyepiece scale
determined by comparing with the image of a small mm. scale
viewed by the microscope.
B
R
Dtf
/*%&
d^
T
CL-jl
U
>
\
B i
Fig. 79
The tube is rotated and other values of the ' diameter *
taken. These measurements should be taken before commenc-
ing the experiment to ensure a regular tube.
A good method of producing the stream of air is to allow
water to enter, one drop at several seconds interval, into a
Winchester, closed at the top by a cork, which is waxed into
the neck of the bottle and provided with an entrance and exit
tube. The types shown in figs. 79 and 80 serve very well : the
air is trapped inside the bottle by water in the bottom of the
bottle in the form shown in fig. 79, and in the U-tube entrance
in the type illustrated in fig. 80.
In this way the pressure gradually rises until of sufficient
magnitude to form a bubble of maximum diameter. This
detaches and an interval then occurs, during which the pressure
again rises, as is seen in the manometer U, and the whole process
is repeated.
By slowly increasing the pressure, its maximum value corres-
ponding to H cms. difference in the manometer may be easily
*It -might be. found advisable to draw out the open end of the manometer U,
to a capillary tube, to damp the oscillation of the liquid.
142 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
measured either directly on a scale or by means of a travelling
microscope.
If only one large bottle is available, air is compressed into it
and the outflow tube of rubber connected to B. By varying
the compression on this tube by means of a screw clip the condi-
tions described above may be obtained.
Let H be the maximum pressure difference established in the
manometer, and h the depth at which the orifice is placed below the
free surface of the liquid, then the pressure of the air inside the
^bubble, when the latter is about to leave the orifice, is n + gaH,
where n is the atmospheric pressure and a the density of the
liquid in the manometer.
"3
\
/
k
VW/
Fig. 80
The pressure outside the surface of the bubble is n+ gph,
where p is the density of the liquid under investigation. Then,
P being the excess of pressure inside the bubble over that on
the outside
2T
p = (n -f-goH) - (n + g?h) = -^ ,
or* T = ?£(aH-p/*) (12)
The values of R, H, and h, are obtained as already described,
and T is calculated.
The method is a good one for obtaining relative values of
surface tension. As an example, find the variation of surface
tension with concentration for a salt solution containing from
o to 5 gramme molecules per litre.
* The result expressed in equation (12) above is developed assuming the bubble
formed is hemispherical and of the radius of the tube.
Ferguson ("Phil. Mag.," No. 28, 1914, page 128 et seq.) has deduced an
expression for T without making this assumption and arrives at the result :
where
I »Vb/
SURFACE TENSION
143
By altering the temperature of, say, water in the beaker, the
change of surface tension with temperature could be obtained.
(9) Capillary Tube Method (Sentis).
A capillary tube is drawn out to about *5 mm. bore as in
Jaeger's method. It is immersed in the liquid under investiga-
tion, and then withdrawn and clamped vertically. Some of the
liquid will emerge at the lower end and form a drop as shown
in fig. 81 (1), so that the distance from A, the meniscus in the
tube, to B, the lowest point of the drop, is h x cms., and MN is zr.
If now the lower end of the tube is surrounded by a vessel, C,
containing the liquid, the column will fall in general, but the
meniscus may be brought to the original level by raising C until
the free surface of the liquid in the beaker is h 2 cms. below the
meniscus, fig. 81 (2).
Fig. 81
From a knowledge of h x , h 2 , r, and p, the density of the liquid,
T, may be Calculated from the formula
T=f{r(/> 1 -^)-^..
To establish this formula we may assume that the portion
of the drop, shown in section as MONB, is hemispherical. This
approximation is a safe one when the radius of the capillary tube
is small, and r small compared with h v
Consider the forces acting below the horizontal plane of
maximum area shown in section as MN.
The length OB = r, and hence the column from MN to the
meniscus is (h t — r) ; of this a length, h 2 , is supported by the
upper surface tension forces as shown in fig. 81 (2), and hence
the pressure at MN due to the liquid column is g?{h 1 — h 2 — r),
which contributes a downward force g9{h 1 — h 2 — r)nr 2 . The
2
weight of the hemisphere - T:r 3 gp also acts downwards.
144 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The surface tension acts vertically in the circle of section of
the drop and plane MN, and has a value T • 2izr ; hence
T27tr = gpfa — h t — r)*r* + -*r 3 gp,
To make a determination of T for a liquid, the cleaned capillary
about 25 cms. long is drawn out to a radius of about «5 mm.
at the one end, and is almost entirely submerged in the liquid,
so that the latter fills the greater part of the bore. It is slowly
withdrawn and is clamped vertically. By means of a travelling
microscope, the length MN = 2t is measured and the microscope
is then focussed on the meniscus. A small glass beaker is
raised under the tube, until the liquid in the beaker just touches
the drop. The reading of the micrometer screw, which raises
the platform carrying the beaker, is noted. The beaker is then
further raised until the meniscus, as viewed by the fixed micro-
scope, again acquires its original level. The micrometer screw
reading is again noted. The difference between these two
readings is equal to (h x — h t ).
P is determined in the usual manner, whence T may be calcu-
lated from the formula deduced above.
If the form of adjustable table with micrometer or vernier
attachment is not available, some simple convenient method
may be devised for the measurement of (h x — h t ). For example,
two microscopes may be used. With the first the value of MN
is observed, and then the point, B, is viewed and its image
brought into coincidence with the cross-hairs; the second
microscope is adjusted until, viewing the tube conveniently at
right angles to the first, the image of the meniscus is in coincidence
with the cross-hairs in the eyepiece.
The beaker, C, is then introduced and adjusted until the
meniscus is again as before, producing an image in coincidence
with the cross-hairs of the second stationary microscope. A pin
is adjusted to coincidence with the free surface of the liquid in
the beaker which is then removed. The first microscope is then
moved a distance which is measured on the vernier scale attach-
ment, until an image of the point of the pin is in coincidence
with the cross-hairs, the vertical distance moved by the microscope
is {h x — h 2 ).
Care is, of course, taken that the capillary tube does not move
during the experiment.
SURFACE TENSION
*45
(10) Anderson and Bowen's Method i
A method by which to determine the value of the surface
tension of a liquid, and the angle of Contact with glass was
described in the " Philosophical Magazine," April, 1916. Another
method not so readily adaptable to general laboratory use is
seen in the same magazine, February, 1916.
A small rectangular sheet of thin cover glass is cleaned (by
standing it in concentrated sulphuric acid and potassium bichro-
mate, etc. as described previously) and dipped into the liquid
whose surface tension is to be measured. It is withdrawn and
clamped vertically. The liquid takes up the form shown in the
diagram, fig. 82.*
Ar
N
O
i
O l
h
Fig. 82
The drop has two curvatures, making the equivalent of a
cylindrical lens, concave at the upper half and convex below,
the centres being at O and O 1 .
The upper limit of the drop may be at O or any point, N,
above. The drop is tangential to the glass plate at O.
If A is the focal point of the concave lens, OA the axis, f x the
focal length, B the focal point of the convex lens, and OB the
axis of the lens of focal length, / 2 ,
OO 1 =
r, =
P
Pi
P*
n
T
surfaces
h cms.
the radius of curvature of the concave
(assumed symmetrical and equal),
— the radius of curvature of the convex surfaces,
= the refractive index of the liquid,
= the density of the liquid,
= the pressure in the liquid at O,
= the pressure in the liquid at O 1 ,
= atmospheric pressure,
= surface tension of the liquid.
* It was established by the original experiment that for water, glycerine,
olive oil, and turpentine, the angle of contact is zero, and hence the form of
fig. 82 represents the section of the drop.
146 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We have, as the lens is a thin one, using the lens formula
/■>(!*- *) (7 ~ 7). (page 278.)
7i = { *- x) k (I3)
rr { *- J) k --m
Since the pressure inside a cylindrical surface is greater than the
pressure outside by p = T Q-), (see Poynting and Thomson's
" Properties of Matter"), and here we have a cylinder of radius r,
i.e. R x = r
^ = U -J- 1 (i5)
^ = n+l .. (l6 )
Now p2-pi= gph,
Pt ~ Pi = T ( - 4- - J by (15) and (16) above.
Hence t(J- + ±-J = g9 h.
But by (13) and (14) above
1 _ 1 1 _ 1
*1 2/lOx-l) ' y2~2/ 2 ((X -I) 1
also
T
i.e.
2(|*
^)(7x + 7 2 ) = ^
(17)
or T - 2gp ^^ ~ I ^ 1 ^ 2
(/i+/ 2 )
If one side only of the glass sheet is wet, using the same notation
we have
T _ gP%-i)A/ 2
(/!+/■) (l8)
The apparatus used to obtain T, is a collimator illuminated
by a sodium flame, and adjusted to give a parallel beam. The
light passes through the cover glass and the liquid lens, and is
viewed by a low-power microscope provided with a vertical
traverse, and a traverse parallel to the axis of the microscope.
The usual type of travelling microsope will need a littie
modification to make this latter condition possible.
If a parallel beam of light be sent from the collimator from
SURFACE TENSION 147
left to right, and normal to the plate, a virtual image of the
horizontal slit will be formed at A by the upper half of the
liquid lens. The distance OA may be measured by using the
low-power microscope, arranged with its axis parallel to the
direction of the incident beam. The microscope is first focussed
on the image at A, and then moved backwards a measured
distance until the glass plate is in focus. The distance moved
being = OA = f v In the same way OB and/ 2 may be measured
by noting the difference in reading of the microscope when the
glass sheet is focussed and then when the image of the slit is
coincident with the cross-hairs.
It will be found most satisfactory to use one side of the cover
glass only, i.e., dry the other side before making the observation,
and allow the incident beam to fall on the dry side.
As the incident beam is parallel, it will be found, of course,
that the observing microscope must be moved in a vertical x
direction to enable a focus of first A and then B to be made.
This distance, h, is noted.
The refractive index and the density may be obtained from
tables, or by one of the many methods available.
Thus, having measured / x , / 2 , and h, knowing jx and p, T may
be calculated for the liquid used.
Measure in this manner the surface tension of water and
turpentine.
The Variation of the Surface Tension of a Liquid with Temperature
The variation of surface tension with temperature may be
obtained by Jaeger's method, which enables a good comparison
of the relative values of the surface tension at different tempera-
tures to be made.
The details of the experiment are as described on page 140.
The bubbles in this case are formed in the liquid at different
steady temperatures, and H is determined for each.
A large beaker is rilled with the liquid, say water, and heated
to about 90°C, and then allowed to cool. The value of T being
obtained every io°C. The liquid is well stirred before each
observation, and if a large volume is taken will remain sensibly
at the same temperature throughout the observation.
A curve is plotted, showing the decrease of T with temperature.
Capillary Tube Method
The decrease of the value of the surface tension may also be
investigated by a capillary tube method. Either form of appara-
tus shown in fig. 83 or fig. 84 may be employed. One
form of apparatus is filled with the liquid and immersed in a
water bath whose temperature can be regulated either by a
148 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
thermostat or by manipulation of the Bunsen burner and a
stirrer. The bath is raised to boiling point and allowed to cool,
so that at about 90-95 °C. the whole of the apparatus and contents
are at the temperature of the bath. The difference in level
TJ
Fig. 83
B
between the two surfaces, A and B, is measured as quickly as
possible with a travelling microscope. This process is repeated
at different temperatures, say, every io°C, and from a knowledge
of this difference in level, h and the radius of the tube as measured
B
Fig. 84
by the method given on page 132, the value of T at each tempera-
ture may be calculated and a graph representing this relation
plotted.
The value of the density (p) is obtained, for each temperature,
experimentally, or from tables, and is used in the evaluation
of T by the above methods.
The essential to success in this and every other surface tension
experiment is that the glass, etc., is clean.
CHAPTER V
VISCOSITY
WpEN adjacent layers of a fluid move with a relative velocity,
forces, known as viscous forces, are brought into play tending
to reduce this relative movement.
If we consider a fluid whose upper layer is moving with a
velocity v in a fixed direction, the state of affairs shown in fig. 85
will be reached, where intermediate layers, between the upper
layer AB, which has a velocity v, and the lower layer CD, which
is at rest, have a velocity shown by the arrows.
L*ve» .in N/v©-no«N»
Fig. 85
The force, F, acting on any area in a plane at right angles
to the diagram, and parallel to EF, is proportional to the area A ;
v
and to the velocity gradient, in the case taken -y 1 i.e. at constant
temperature.
F oc A x (vel. gradient).
Taking the normal to EF, in the plane of the diagram, as a y
axis we have
F-'.A*. W
where t\ is a constant for the liquid and is called the coefficient
of viscosity.
In the case of a liquid flowing down a tube, the axial stream
is moving with a definite velocity and the layers in contact with
the walls of the tube are at rest and, provided that the pressure
difference which is causing the flow is not too great, the result is
the regular type of motion already considered.
If the pressure exceeds a certain limit, the liquid no longer
proceeds in this regular manner, i.e. no definite stream-line flow
takes place. The result in this case is called turbulent motion.
149
150 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We will assume, in the experiments that follow, that the
pressure applied is below this critical pressure, and that the
motion is therefore regular.
The Determination of the Coefficient of Viseosity for a Liquid by Obser-
vation of the Flow of the Liquid through a Tube
The value of the coefficient of viscosity of a liquid, such as
water, may be obtained by measuring the quantity passing per
second x through a tube of uniform radius, when a definite
pressure difference exists between the ends of the tube.
Consider, as in fig. 86, a section of such a tube, whose radius
is R cms., and imagine a thin tube of liquid in it of radius, r,
and thickness dr ; the area of cross-section of such a tube is 2nr -dr.
Fig. 86
If P be the pressure difference between the ends, then the
force actmg on the tube ABCD, due to this pressure, is P • 2nr • dr.
Over the curved area of such a hollow cylinder there are
viscous forces which are of a magnitude dependent on the value
of the distance r from the axis of the tube.
Fig. 87
Fig. 87 shows the form of distribution of velocity of the layers
for various values of r ; as shown later, equation (3), the curve
of fig. 87 is a parabola, and hence j- is proportional to r, and
therefore the viscous force, F, is also proportional to r by (1).
dr ex P ress the rate of cn ange of this force with r, then
VISCOSITY 151
the value of the difference between the magnitudes at the two
dF
surfaces of the thin tube ABCD, since the thickness is dr, is j- • dr.
In steady flow we have, therefore, the equal and opposite
forces, which may be expressed as :
-($->
2izrP - dr,
-^=27^ (2)
dr
Now equation (1) gives
The area A is the area over which F acts, i.e. is 2itr • I, I being
the length of the tube. Equation (2) becomes
r [ -117 2tzH J = 27crP.
it V dr J
dr
Integrating
,dv r % „ , n
— 2l\Ttrl -j = 27C — • P + C,
dr 2
C being the constant of integration.
But when r = o, -j — .*. C = o,
i.e. 7 dv _,
— 2vjZ 3- = rP.
dr
Integrating once more
— j 2t\1dv = j r¥dr,
— 2lf\V = — -f B,
PR 2
When r = R, v = o /. B =
Hence » = -— (R 2 - r 2 ) (3)
In one second, a column of liquid, v cms. long, and 2-nr • 87 cross-
section flows, down such a hoUow tube, i.e. the volume passing
per second = j-y- (R 2 — r 2 ) I 2itr • dr.
Thus, for the whole tube of radius R, we have the sum of
such expressions.
152 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If Q is the total amount of liquid passing per second
/.r p
27c — (RV - r*) 8r
o
4h
TQ
p
I
|~RV
4 J<
W 8 I r>
(4)
A suitable form of apparatus to use in an experimental deter-
mination of T) by this method is seen in fig. 88. The liquid, say
water, is contained in a large bottle, B, standing a suitable
distance above the level of the table. The water flows from
Fig. 88
this reservoir to the union, X, thence through a capillary tube
of known length to the union, Y, and so on, via a length of
india-rubber tubing to a graduated jar, J, where the thermometer,
C, measures the temperature of the emerging water.
From the unions, X and Y, two lengths of india-rubber tube
make connection to the manometer M. The difference in the levels,
E and F gives in cms. of water the value of the pressure difference
between the ends of the experimental tube, K.
A pinch-cock L enables the flow of the liquid to be regulated.
In order to maintain a constant difference of pressure between
the two ends, X and Y, whilst the water is flowing, the bottle B
is closed by means of a tight-fitting india-rubber cork through
which a glass tube passes, to a point well below the surface of
the water. 'This end being open to the atmosphere allows the
entrance of air bubbles as water flows through the tube. The
lower end of the tube remains at atmospheric pressure, and so,
VISCOSITY | 153
until the whole of the water above this point has passed through
the tube, the manometer will remain with a uniform average
difference of level.
The flow should be so arranged that the emergent water
issues as a slow trickle or succession of drops, avoiding a rapid
stream of water, which might cause the flow in the tube to become
turbulent, in which case the formula which is developed, assum-
ing a regular flow, will break down.
In adjusting the apparatus, the water passing is not collected.
When everything is steady the tube is inserted into J, as seen
in diagram, and the time is taken in seconds for a definite amount
of water, say 500 c. cms., to pass. From this, Q is obtained.
If h is the difference in level in the water manometer, P = g?h,
approximately, or as shown later, equation (n),
more exactly.
Before determining the remaining unknowns, I and R, several
values of Q, corresponding to different values of P, should first
be obtained, for we have
_kR^ P
^"8 l"Q'
p
By varying P and Q, the mean value of -~- can be obtained,
and this, not one value of P and corresponding Q, used for the
computation of 73, provided that the temperature of the room remains
constant during the experiment.
The value of / may be obtained by direct measurement of K.
Now R, wjiich occurs in the formula in the fourth power,
must be obtained as accurately as possible. A suitable method
is the measurement of the length and mass of a column of mercury
in the dried tube as described on page 132.
Unless the liquid is passing through the tube with a very
small velocity, equation (4) must be modified to allow for the
kinetic energy imparted to the liquid. This reduces the effective
pressure P.
To arrive at such a correction, consider the case of a liquid
flowing through an irregular tube whose two ends are at different
levels.
Take two points, B and C, in the tube such that the cross-
sections are A t and A 2 ; the velocity of the centre of the liquid,
v x and v 9 ; the pressure in the liquid, P x and P 2 ; and the dis-
tances of the centre at these places are L x and L 2 cms. below a
fixed horizontal plane.
154 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If the liquid moves a distance dx x at B, it will advance a dis-
tance dx 2 at C, such that
A 1 dx 1 = A 2 dx 2 (5)
The work done on the liquid between B and C is therefore
PiAji*! — "P 2 A 2 dx 2 (6)
This work must be the equivalent of the energy gained in
the liquid.
Gain of Kinetic Energy
The mass (A^xjp of liquid enters at B (p being the density
of the liquid, assumed constant). This has a K.E.,
I A x dx x ^v x 2 .
Similarly for the mass of liquid leaving, and since condition (5)
holds, the g ain of kinetic energy is
Aiteiftpw," - IpvS) (7)
Fig. 89
Potential Energy
The P.E. of the mass of liquid at B is gL 1 (A 1 ^ 1 )p, and at
C is gLgAjjtaaP.
For a gain of P.E., C must be higher than B, L 2 <Lj,i.e.
the gain of P.E. is
A 1 dx 1 (pgL 1 - P gL 2 ) (8)
Equating (6) to the sum of (7) and (8) we have, using (5)
A x dx 1 (P 1 - P 2 ) = A l dx 1 {$ 9 v 2 * - i pVl *) + A 1 ^( P ^L 1 -p^L 2 ),
re-arranging terms,
Pi + *P» x 2 ~ gtU = P 2 + |pV - £pL 2 ,
i-e. P + f p*> 2 — g P L = constant (9)
Applying to the case shown in fig. 90, where the liquid is
contained in a wide tank, BC, and flows .through a uniform
tube, CD, with a constant maximum (central) velocity, v, we
may find the effective pressure difference between the two ends
of the tube using equation (9) above.
VISCOSITY
155
If II is the atmospheric pressure we have, applying (9) to the
point B and C, where C is just inside the tube and is at pressure p',
H + ipv v i-gpL B = p' + l ? v* - £ P L C
For such a wide cross-section as the tank we may assume the
surface to be fixed. This may be further brought about by the
device shown in fig. 88.
i.e. v B = o.
We will further use this surface as reference plane for measure-
ments of L, i.e. L B = 0, L c = L cms.
We therefore have
II =P' +$ p v*-g P L,
or p' =11 +g p L-|p*;2,
the pressure at the open end, D, being n, we have the pressure
difference between the ends of the tube, CD, p, = p* ~n
i- e - P =£pL-| P t/ 2 ( I0 )
Thus, the effective ' head ' is reduced due to the gain of kinetic
energy by the liquid.
TT
C D
Fig. 90
It was seen (p. 151) that the velocity of the stream at a distance
r from the axis of the tube, was
2ril\ 2 J'
i.e. when r = o, we obtain the value of v used in equation (10)
above, i.e. the maximum velocity,
PR 2
v
Now we saw (p. 152),
473/
P 7C R*
Hence
_2Q
rR2 '
and the effective pressure, p, is given below,
*=* p ( L -i^r.) («)
156 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
An alternative apparatus for the determination of tj is seen
in fig. 91. The horizontal capillary, DE, is fixed in a cork
which closes the lower end of the tube, CF, which has con-
strictions at A and B, on which are scratches on the glass. The
whole vessel is filled with the liquid and the ground-glass stopper,
S, placed in the neck above A. The volume between the scratches
is calibrated, so that its capacity, V, is known. The stopper, S,
is removed, and the time, t cms., is taken for this volume to flow
Fig. 91
through the tube, i.e. when the level reaches A a stop-clock is
started, and when the level reaches B, the clock is stopped.
p, the pressure, is taken as due to the average height, h, of
the liquid above the capillary tube level, i.e. p = gph, or as
shown on more precisely in equation /n),
The other terms are measured as for the first form of apparatus
described.
Determination of the Viscosity of a Liquid by the Coaxial Cylinder
Method
The value of the coefficient of viscosity for a liquid such as
glycerine may be obtained, using the apparatus shown diagram-
matically in fig. 92. The liquid, say glycerine, is placed in the
cylinder, AB, which may be rotated by hand or by a small motor.
A belt driven by either means passes over the pulley, P, and the
VISCOSITY
W
rotation is imparted to the cylinder by the crown bevel wheels,
W. The revolutions may be counted by the revolution counter, R.
Hence, if a number of revolutions, n, be timed, the angular
velocity, w, may be calculated.
Immersed in the glycerine is a second solid cylinder, CD,
which is suspended on a phosphor-bronze suspension, which
carries a mirror, M, and which is maintained central by the pivot S.
Due to the viscous forces in the liquid, the inner cylinder will
experience a couple, C, which turns it through an angle, 0, such
that the torsional restoring couple just balances the turning
moment due to the liquid.
If t is the restoring couple per unit angular twist in the sup-
porting wire, and 6 be the constant deflection in radians, we
have t0 = the restoring couple due to the torsion of the wire.
To obtain a value for the couple due to the viscous forces, we
will first consider the co-axial cylindrical surfaces.
Let fig. 93 represent a section normal to the axis of these
cylinders, of radius R 2 and R, as shown.
Take any thin cylindrical ring in the liquid having points,
A and B, on the same radius. If the liquid rotated as a whole,
there would be no velocity gradient — no relative motion. Thus,
for no relative motion, when A moves to A 1 , B moves to C. If
w is the constant angular velocity, AO = r, and BO = r + Br ;,
AA 1 = rw ; BC = (r + 8r)w. Now, in the actual case,
the outer cylinder moves with an angular velocity, Cl say„
and the layer in contact with the inner cylinder is at
158 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
rest, i.e. the liquid does not have this constant angular velocity.
Actually, the particle at B moves with some larger angular
velocity {w + dw), and B moves to B 1 where BB 1 = (w + dw)
(r + dr), thus having an excess over the no-relative motion
velocity equal to (w + dw) (r + dr) — w(r + dr), i.e. = 6w(r+dr)
i.e. the velocity gradient is — ^~ — '-, or since dr is negligible
cf . r, the velocity gradient ~ = r • — in the limit.
Thus, considering the forces on the cylindrical shell, if / is the
length of the inner cylinder from the surface of the liquid to
Fig. 93
the lower extremity, the area of the cylindrical shell is (2^)/,
and therefore from (1),
F = yi(2*rl)£>
where F is force due to the viscosity over the curved surface
of such an imaginary shell. The moment of this about the common
axis is
C = Fr — 2itrHt\r ^—»
dr
substituting the value for -j-.
Integrating we have
J2nlridw =J~-JT>
i.e. 2ir/i)tt> =* - — % + B,
where B is a constant.
Now, when r = R lt w = o,
B
r = R,, w = H, 2R X *
VISCOSITY 159
i.e. 2 ^n=§[^-^ 5 ],
Also we have
L== R^-R** (I2)
C = T0,
Wh6nCe " - ^OR.'R. 1 ' " - (I3)
In the actual case taken (12) must be modified to include the
couple due to the viscous effect on the under surface of the
inner cylinder. If this effect were truly due to the parallel
circular plates, we might readily deduce an expression for it.
But the stream-lines in the liquid will not be composed of two
such regular patterns. The effect at the end of the cylinder,
CD, will be a gradual transition from the one to the other.
Some end correction is therefore needed. Let us assume that
the couple exerted on the cylinder, CD, due to the under surf ace
and the lower portion of the curved surface, and also the "effect
due to the support, S, which rests on AB, for an angular velocity,
CI, is K CI, then the total couple will be
_ WRi 2 R a 2 Ka
C ""(R,»-R 1 *)- Q + K0 '
where V is the length of the curved surface of the inner cylinder,
measured from the liquid surface to the lower limit of the regular
stream-line between the cylinders ; the second term deals with
the rest of the curved surface and the lower surface, etc., as
stated.
If now we have the space between the Cylinders filled to a
level, such as L 1} we have, if l x ' is the corresponding length and
B x the deflection
^=°mw+«i- <->
For a second case, where the length of the cylinder immersed
is Z 2 ', corresponding to a level, L 2 , in the figure, if 2 is the
deflection
*- pg£g! + K] ( I5 )
K may therefore be eliminated from (14) and (15) and, if the
angular velocity, Cl, is the same for both determinations, we
have by subtraction
T
(0
i- e *>- fi R 2 2 -Rx 2 L i 2 J ( }
160 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The length to be measured is therefore the difference between
the two levels, Lj and L 2 .
Thus, by maintaining the speed of rotation constant, and
observing the steady deflection corresponding to two levels of
the liquid surfaces, knowing L^ = // - /,', and the dimensions
of the cylinders, tj may be calculated in terms of t, which itself
may be evaluated by observation of the twist of the wire, when
the liquid is not in the cylinders, and loads are applied at G, as
indicated in the diagram.
For liquids whose viscosity is fairly large, the torsion control
will not be sufficient. In this case the restoring couple is in-
creased by adding masses, m, m, thereby increasing the restoring
couple by m D, where D is the diameter of the wheel G.
In many cases it is advantageous to eliminate the torsional
restoring couple entirely. This may obviously be done by
adding masses, m, m, of such magnitude that the inner cylinder
is brought to its rest position, as observed by the reflected beam
from the mirror, M, on the scale, which was previously used to
measure B x and 2 .
In that case, if masses of total value, M x , are supported on
the wheel, G, C t = Mj • — , and for the second case with the liquid
at level, L 2 , C 2 = M 2 -.
Then equation (16) simplifies to
(M, - M a )| - "gp%^ (// - V).
This latter method of working will be found most satisfactory
for glycerine and similar liquids, whilst the former method, using
the torsion of the fibre, will be satisfactory in the case of less
viscous substances, such as water.
Viscosity of a Liquid (Oscillating Disc Method)
The viscosity of a liquid may be determined by timing the
period of oscillation of a flat circular disc in air, and finding
the logarithmic decrement in the liquid and in air.
O. E. Meyer has shown that with such a disc, the coefficient
of viscosity for a liquid, tq, is
7u P T(y 4 + 2r 3 ^) 2 \\ 7c °j + V 7c °j j " * ' * (I7)
where I is the moment of inertia of the disc, and attachments
about the axis of suspension,
p the density of the liquid,
T the time of a complete swing in air,
■n =
VISCOSITY x6i
r the radius of the disc,
d the thickness of the disc,
x*the logarithmic decrement in the liquid,
x the logarithmic decrement in air.
The development of the above formula is beyond the scope
of this book, and may be found in " Poggendorf Annalen,"
No. 113, page 55.
Without any discussion of the development of the result, we
will use it as an empirical formula which agrees with determina-
tions by other methods. It is an excellent method whereby to
study the determination of the logarithmic decrement of an
oscillating system.
Logarithmic Decrement
Consider a suspended body to oscillate about the suspension
as axis in a simple harmonic manner. If I is the moment of
inertia of the body about "the axis of suspension, F the restoring
force per unit angular displacement, we have the equation of
motion,
IB -f F0 = o.
If now a frictional resistance acts on the body so that K is
the resulting opposing couple per unit angular velocity, the
above equation is modified into the form
I'd + Kb + F0 = o.
By the method given, p. 28, trying as solution = djt,
we arrive at the result,
-J£« / /F K 2 \
e=6 .e *i cos^yjy-— t + aj,
when O and a are arbitrary constants, being the angular
displacement at any time t.
The period time, T, of such a motion is
T 2"
I F K*
\ I 4I 2
K
The amplitude is o e al * , and decreases exponentially with
the time. O is the amplitude when the friction is eliminated,
i.e. when K = o.
In the case where the friction resistance, K, is very small,
K 2
the value of K 2 is correspondingly much smaller and -p- becomes
F
negligible compared with j .
162 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The periodic time becomes
The oscillating body therefore performs vibrations in equal
times, but the amplitude gradually dies out (see p. 29).
Starting from the undisturbed position of the body, let a x
be the angular displacement measured to the first turning point
(measured to the right, say).
Let a 2 be the angular displacement following, on the other
side of the zero (to the left, say), also measured from the zero
position.
a 3 the next swing in the original direction measured from
the zero.
• a/ of+fit* +ir«/i *"* _ •
The first deflection 0^ is after a time t = — ; <x 2 after time -^- ;
4 A
5T
0.3,-^— ' and so on,
4
Thus
K T
To the right : a x = d e " 2l ' 4
_K 3T
To the left : a 2 = e »*■ ' 4
_K 5T
To the right : a 3 = d e »i ' 4
K 7T
To the left : a 4 = 6 e ~»i ' 4
a 2 a 3 a 4 a 5 a n+i
So that
Log^=log^...=log^-=g.± = x,say, (18)
K T
x = -y , — is called the logarithmic decrement.
Now we have
logfe • «t . ?• . . . X -*^) = log (^) n
\ a 2 a 3 a 4 a »»+iy V a 2y
Since — 2 = - 3 . . .-^- =-°^
a s a 4 a n+I a.
*. n • log ^i = log ( -^— )
Again, since
VISCOSITY 163
OCi Oa . a l _ a l + « 2
a 2 a 3 ' a 2 a 2 + a 3
2L* _ <^3_ <*2 _ «2 + a 3
a 3 a 4 a 3 a 3 + a 4
Hence
. a x 1 , (x.i + a 2
log — = r lo g * « „
«2 + a 3 x a » + a .n+l_
n+2
w a 2 + a 3 a 3 + a 4 ' ' a n+1 + a
x== Li og *i + " 2 (20)
n a n+1 + a B+2
(i.) The value of x, the logarithmic decrement, may be obtained,
as in (18), by observing the value of the first swing, say to the
right, and the successive swing to the left, whence
x = log, -i,
oc 2
or if logarithms to base 10 are used
x = 2-303 log 10 j 1,
a 2
X here depends on the observation of swings.
It will be seen that a x and a 2 cannot be very accurately observed,
hence the method of equation (19) may be used.
(ii.) By observing the 1st and the (n + i)th J swings, the error
is reduced to - of that in the first method corresponding to
n
equation (19).
(iii.) The third method of finding x, set out in equation (20),
depends on the observation of swing from left to right (|
periods). Here no knowledge of the zero reading is required.
This last method is obviously the one to be recommended, for
most purposes, of the three discussed above.
It will be seen that, if we replace <x. 1 + <x 2 by p 1? <x 2 + a 3 = p 8 . . .
a n + a n+1 = p„ that equation (20) becomes
>=>£; (2I)
Since T, the time for a complete swing, is constant when the
damping is small, we may find X from equation (21) by observing
Pi directly, and then allowing the system to oscillate a measured
time until the amplitude is appreciably reduced, say, from half
to quarter of the original, then measure p B+1 . The number of
swings between the observations may be found from the time,
for the number of complete swings may be obtained by dividing
the time by T; hence (n + 1) is known, p refers to half* a
164 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
complete swing, so that twice the above quotient will give the
number of swings between observations.
An application of the above may be seen in the following
method of finding the logarithmic decrement, which is especially
advantageous if there is uncertainty in measuring the ends of
swings, and hence p v
Suppose we read an even number of successive left and right
deflections, cc v <x 2 , a 3 , a 4 . . . a m .
Then Pi = «i + a 2 >
P3 = «3 + a 4 ,
Ps = <*s + a 6 , etc.,
omitting p 2 , p 4 , etc., for convenience in tabulating.
After a timed interval, when the deflection is reduced to about
half the original deflection, obtain the same number of left
and right readings. From a knowledge of T, suppose the first
of the second set of readings is a n , and a«, a B+1 , a B+2 . . .
are observed, we may obtain
Pn = *»+*n+l»
Pn+2 = a n+2 ~t~ a n+3> e ^C.
We have since h = li = hzj = ^
P 2 P 3 P»
Pi _ Pi P
Pn \U
and in same way for the other pairs giving
Pi _ P 3 P 5
Pn P«+2 Pn
etc.
+4
01 h = Pi + V* + Pi • ■ • = ,*(«-!)
P« P»+ P« +2 . . . Pn+4
Hence (»_i)x = log, Pi + P 3 + P 5 • • •
2-3026 x Pi + P 3 + P 5 ■ „_. , }
(n-i) gl ° P„ + P B+2 +p a+4 . . . ~l 29)
The practical details of this are given below.
Experimental Arrangements
A suitable form of apparatus with which to make a deter-
mination of t) for a liquid is seen in fig. 94
The flat disc is suspended horizontally by a phosphor-bronze
suspension which is attached to a rod rigidly fastened to the
centre of the disc. This rod carries a cross-bar whose ends
VISCOSITY
165
have a serew thread, along which two masses may be screwed
to balance the disc horizontally.
A small concave mirror is fastened to the rigid support.
The time of oscillation of this system is first obtained in air.
This observation is carried out by aid of the usual lamp and
scale arrangement. A beam of light from a lamp is directed
on to the mirror, and is brought to a focus by the latter on a
scale about a metre away. As the spot of light passes its rest
position on the scale, during its oscillation, a stop-watch is
started and stopped after 50 complete swings, hence T, the time
for one swing (i.e. the interval between the spot of light passing
the zero at consecutive times in the same direction).
Determination of I. The value of I may be obtained from
the time of swing. If t is the restoring couple per unit angular
u
■jm
Fig. 94
displacement due to the suspension, we have, in air, where the
logarithmic decrement is small
2n
£
(23)
The value of T has been obtained as above.
A small ring of thick copper wire is now placed symmetrically
on the disc and T 1 , the time of a complete swing, is obtained
from a determination of the time of 50 swings, for the loaded disc.
We have
T'
2tc
P
+ r
(24)
where V is the moment of inertia of the copper wire ring about
the axis of suspension.
If a is the radius of the ring of wire, and if the centre were
in the axis of suspension, I x f = ma 2 where m is the mass of the
wire ring.
166 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Squaring (24) and (23) and dividing, we have
-Z 2 = t + v = ,r_ , ^i 2
Hence I = T2 ' ma \
TV2 <-p2
r, d, and p may be readily found by the usual methods.
The logarithmic decrement is now found for the disc oscillating
in air. Use is made of the method embodied in equation (22)
above.
The disc is shielded from draught in an empty glass dish, as
in the figure. A straight vertical wire is placed in front of the
lamp and by the mirror, and a lens an image of the wire is
focussed on the scale.
The disc is given a displacement, and the oscillations are
observed by means of the light spot. As the spot passes the
position of rest a stop-clock is started. The first turning point
of the spot of light is read off, on the right, say, then one turning
point on the left, and so on, tabulating as under 8 or 10 readings
on each side. The disc is then allowed to perform the oscillation
until the amplitude has decreased to half or quarter (depending
on the time) of the original amplitude. At this stage the reading
at the turning of the light spot at the right is observed, and the
stop-clock stopped. The next reading to the left is taken, and
so on for 8 or 10 on each side.
The time being t seconds between the starting of the clock
and the first observation of the second set of swings, T being
the periodic time of the system in air, there have been —
it
complete swings or ^ half periods, hence n as shown under. The
above process is repeated in the liquid whose coefficient of
viscosity is to be determined, say paraffin oil.
Thus, in the table opposite, which gives the observations for
the disc in air and paraffin oil, we have, if T = 3*07 sec, n, the
number of the £ swing at the commencement of the second
set of observations for air = 2 ( — — j = 400,
\ 3'07/ *
i.e. for air
_ 2-3026 An-A
x °- ( 4 oo-i) logio Wiy
399
ss '0027525.
VISCOSITY
i67
AIR
PARAFFIN OIL
SCALE READING
LEFT
RIGHT
2-4
45
2-5
45
2.8
44.8
3-o
44-5
3*i
44'3
3*3
44-2
3-3
44-o
3'6
43-8
37
437
4-o
43-6
AMPLITUDE
42-6
42-5
42.0
4i-5
41-2
40-9
407
40-2
40*0
39-6
Total 411-2
Time Interval = 615 sees.
16.5
30-4
16-5
3o-4
16-5
30*4
16.5
30-4
16-55
30-3
16-6
30-2
167
30-2
167
30-2
16-6
30-2
16-6
30-2
13-9
13-9
13-9
13-9
137
13-6
13-5
13-5
13-6
13-6
Total 137-1
SCALE READING
LEFT
5'i
6-o
6-5
7-2
7-8
8-5
9-1
97
10-2
II-O
RIGHT
47-2
46-6
46-0
45-2
44-5
43*8
43-2
42-6
42-0
4i-5
AMPLITUDE
P
42-1
40-6
39*5
38-0
367
35-3
34*i
32-9
3i-8
30-5
Total 361-5
Time Interval = 123 sees.
20-7
20-9
21-0
21-2
21-4
21-5
21-6
21-8
22-0
22-5
31-3
31-2
31-0
30-9
307
30-6
30-5
30-2
30-1
30-0
io-6
10-3
io-o
97
9*3
9-1
8-9
8-4
8-i
7*5
Total 91-9
In the same way the value of the logarithmic decrement in
paraffin may be calculated,
2 - 3026 log 10 3 6l -5
X =
(n -1)
91-9
n being obtained as above from the time interval measured in
that case from the commencement of the first set of observations
to the commencement of the second set.
168 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Thus, by the methods given, we have a value for I, p, T, r, d, X
and x ; hence substituting in equation (17) tj may be calculated.
Repeat the experiment, using water and paraffin oil, and
calculate the value of y\ for each.
FLOW THROUGH A VERTICAL TUBE
The variations of the coefficient of viscosity of water with
temperature may be investigated by means of the apparatus
shown in fig. 95.
A
T
■
i
,
1
1
— -
■ C
B
iH
i____lL____i
Fig. 95
AB is a copper cylindrical vessel into which the capillary tube,
CD, is fitted by means of a rubber cork.
Water is heated in a boiler to a temperature 5 or 6 degrees
above the one at which a determination is to be made, and is
transferred to AB, the end, D, being closed by the rubber stopper
shown. The water is stirred and the temperature noted by the
thermometer, T. It is not very difficult to arrange that the
water cools slowly and regularly about the desired temperature.
The water is then allowed to flow into a graduated vessel and
the time taken for a measured amount to pass. This process
may be repeated, using water at various temperatures.
Care must be taken that the temperature, throughout the
flow, is practically constant, otherwise unreliable results will be
obtained.
The radius and the length of the tube may be measured as
before.
VISCOSITY 169
Then, as shown under, we may obtain the value of tj for each
temperature from the following formula : —
using the same notation as before.
In the case of a vertical tube, such as the one described,
account must be taken of the gravity attraction on the .liquid
in the tube ; the direct application of the results of equation (4)
dealing with a horizontal tube being thus inadmissible.
Consider a vertical tube of length, / cms., and in the liquid
imagine a thin cylindrical tube of the liquid of radius r, and
thickness dr.
Let p be the pressure difference between the two ends of the
tube, then we have in the steady state of flow pzitr • 8r-\- 2-kt • drlpg
the downward forces, opposed to -=— • 8r due to viscosity
where F = yjA • -j-,
dr
i.e. 2-*r(p + gpQdr = — rf( tjA^V
or integrating r(p + gpV)dr = — 2-qldv,
since A = 2twI.
Hence, once more integrating,
r 2
R 2
where C, the constant of integration = — r(lgp + p), for when
r = R, v — o,
i.e- V = (R'-r*)^±^ (25)
Through such a tube the quantity of liquid passing per second
s= v (2izr • dr) ,
i.e. the total quantity per second, Q, is
27W • v dr.
Substituting from (25) above and integrating,
7U R* p+gtl
In the case taken we may apply the result of equation (9),
page 154, p + %pv 2 — gph — constant.
If AB is the upper surface of the cylinder which contains
the liquid (fig. 95), the cross-section being large, the velocity of
«/
170 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
AB will be small compared with that at C. Talcing AB as the
reference plane for measurement of L, h being the distance between
C and AB, we have,
i.e. the pressure, p' , at C is
P' = n +g?h -|pt> 2 ,
where v is the velocity at the centre of C.
The pressure at D is n, therefore p = p 1 — n ;
hence p = gph ~ \pv 2 ,
and thus we have the value below
O - - JL 4 Z ph -jpfl a +g?l
* ~ S' 7] I
Q.-^fl^leL- (26)
The correction %pv 2 may be obtained from (25) above, i.e.
To a first approximation, putting p = gph,
R 2 — r 2
or at the centred where r — o,
R 2 gpH
41?
To the same approximation
n _ « R* g P H
2Q
Hence we amend equation (26)
,R* ^ 2Q 2 \
This equation is used to calculate -t\ in the experiment already
described.
Viscosity of Air
Consider two parallel circular plates, one suspended by a fibre,
and the other rotating at a constant speed. If the space between
the plates be filled with any gas a velocity gradient will be set
up in the layers of the gas parallel to the rotating plate. The
layer in contact with the rotating disc will move with the latter.
VISCOSITY 171
Due to the viscosity of the gas, the adjacent layer will also
acquire a velocity comparable with the former. Thus throughout
the space, the air strata will acquire a motion, just as in the
case of a liquid flowing through a tube. The layer of the gas in
contact with the suspended plate will therefore experience a
force tending to rotate it in the same direction as the constantly
rotating parallel plate. Due to the force, a couple will act on
the plate, which will therefore turn through a definite angle of
such a magnitude that the restoring couple due to the torsion
in the suspension just balances the displacing couple of the
viscous drag.
A & o! Qd b
R B
* 1
Fig. 96
It can be shown that each stratum of air moves as though it
were a solid, i.e. it moves as a whole.
To obtain an expression for the deflection of the suspended
plate in terms of ij, the coefficient of viscosity of the gas, etc., let
us assume that the edge effect is negligible, and the gas between
the plates behaves as though the plates were of infinite dimensions,
an assumption which is justified by using a guard ring round
the suspended plate.
Let d be the distance between the discs,
w the angular velocity of rotation of the moving plate,
R the radius of the suspended plate.
Consider a stratum shown in fig. 96 by the horizontal broken
line, between EF, rotating, and CD, which is suspended. (The
parts, AB in the diagram represent a guard ring to eliminate
the end effect, CD being of radius R.)
In the stratum considered take a point, P, on a circle of radius
r cms. about the axis of rotation.
Points Q and R are the projections of P on CD and EF. The
velocity of R is r - w ; the velocity of Q is zero. The velocity
slope is therefore -3-. If uniform, the value of the velocity of
a
P is ~ • wr, y being the distance of the stratum considered, below
the plate CD.
We have seen that by the definition of viscosity
F ~ At) Tr ~ " T A '
172 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In the stratum considered, let a second circle of radius, r+dr,
be drawn ; between the two circles there is an annular ring of
width, dr. The area of the ring is 2nr • dr. The viscous force
acting on such a narrow ring is, from the equation above, equal
to F, where
f = 2^.<*.^t.
a
The turning moment about the axis is
a
Such a moment acts on the suspended disc on the projection of
this area.
Fig. 97
The total couple is the sum of such couples taken over the
entire area. Let this couple be C, then
C = A*r» &dr*= ^ *-* = ^R^.
Jo & » 4 2d
Let the suspended plate be turned through an angle, 0, due
to this couple. The equilibrating couple due to the torsion of
the suspension is t0, where t is the restoring couple per unit
angular displacement. This gives
7crjR%;
t9 = ^3- < 2 7)
A suitable form of apparatus is essentially of the form shown
diagrammatically in fig. 96. A brass plate is rigidly connected
to a shaft which may be rotated by means of a belt drive on a
pulley, which is on the shaft. A small counting indicator
serves to record the revolutions of the disc, which is steadily
turned by hand or by a small motor.
At a distance, which may be adjusted, above the plate is a
mica disc, suspended by a phosphor-bronze suspension, and
arranged inside a guard ring of brass. The suspension carries
a small mirror which serves to measure the deflection by the
VISCOSITY 173
usual method of lamp and scale ; the whole is enclosed in a
brass case which serves as a shield. A preliminary experiment
will give an indication of the most suitable speed with which
to rotate the plate, for any given distance, d, between the plates.
The zero of the spot of light having been determined, the
lower plate is rotated until a full scale deflection is obtained.
The speed of rotation is maintained constant. When this is
steady, the counting gear, having been read, is thrown into
action, and a stop-watch started. Maintaining the spot constant
by steady rotation for as long as possible (at least several minutes)
the value of the deflection is noted. The counter is then thrown
out of action and the stop-watch stopped. In that way the
number of revolutions n, in a known time t, is obtained, hence
As R occurs in the fourth power, several values of the diameter
are obtained in all directions, as accurately as possible, and the
mean value calculated.
d is measured by means of a cathetometer. From the value
of the steady deflection and a knowledge of the distance between
the mirror and the scale, may be calculated in radians. is
half the value of the angle subtended at the mirror by the length
of scale, moved over by the spot of light, i.e. tan (20) may be
obtained, knowing the linear deflection and the distance between
the mirror and the scale.
Thus, from this experiment all the terms in equation (27) are
known except t and 13.
To obtain t the suspended plate is twisted from its equilibrium
position, and the simple harmonic oscillations set up are timed.
If T is the mean value of the periodic time for, say, 50 complete
vibrations, we have (p. 105)
= 27t \7
(28)
where I is the moment of inertia of the mica plate about the
axis of suspension.
If now the mica plate is loaded by placing on it, symmetrically
with regard to the suspension, a circle of brass wire of radius
a cms. and mass m grammes, the moment of inertia has
been increased to T, where I' = I + ma 2 .
The time of oscillation, T', of the loaded plate is next deter-
mined by timing 50 oscillations, then
- 2 *VT
■<«9)
174 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Squaring (28) and (29) and subtracting, we have
T' 2 - T* = 4 TC2 ( r - 1 ) ,
Anhna 2
T = — .
^'2 ^2
Thus, by equation (27)
2d An 2 ma z t
i.e.
7rR 4 (T /2 — T 2 )
2nn
_ qmaH d
73 -~" -R4(T' 2 - T 2 ) n '
Having obtained the deflection in degrees, <p° say, we have
= i8o"'
so that
tc ma 2 t<p d
•n
45R 4 (T' -T>
The Determination of the Viscosity of a Gas (by the Flow though a
Capillary Tube)
A simple method for determining the viscosity of a gas which
may be described as ' The Constant Volume Method,' has been
described by Prof. A. Anderson (" Phil. Mag." Dec, 1921,
pp. 1022—3). The apparatus is illustrated in fig. 98. It consists
of a bulb, V, from which a tube, DO, projects downwards and
is connected by rubber tubing to a glass tube, AB. These tubes
contain mercury up to certain levels marked at A and O.
Just below the bulb a capillary tube leads from the tube, DO,
as shown at D. This capillary tube is provided with a piece
of rubber tubing which may be closed by the pinch-cock, P.
The whole apparatus is mounted on a stand and is of about
the same size as the constant volume air thermometer.
The arm, AB, is mounted on a carriage which is readily adjusted
by means of a rack and pinion which is regulated by the turn-
screw, C.
The volume of the gas within V, down to some mark, O,
and extending to the end of the capillary tube is determined.
The pinch-cock is then opened to the air and the mercury brought
well beldw O by properly adjusting AB. The pinch-cock is
then closed, and the air compressed by raising AB until the
mercury stands at O.
A few minutes' interval is allowed to elapse so that the tem-
perature of the gas, which may have been disturbed in this
VISCOSITY
175
compression, once more attains the temperature of the room.
The difference in level of the mercury in the two tubes is observed.
The pinch-cock is opened for a definite accurately measured
time and then closed. During this interval the level of AB is
continually adjusted, so that the mercury always stands at O,
and the gas consequently maintains a constant volume. At
the end of "the time the difference in level of the mercury is
again noted. From the observations made, viz, from a knowledge
of the volume, V, of the gas, the two pressure differences, p x
and p 2 respectively, the time interval, t seconds, and the baro-
metric pressure, P, the viscosity is determined from the formula :
71 =Vt
ttR 4
-7- !Og (
(*» + P) ifil ~ P)
8/V ^ ~ 5e (p t + P) {P2 - P) '
where R = radius of capillary tube and I = its length.
r
KJ
Fig. 98
The above formula may be obtained from an extension of
the result obtained for the flow of a liquid through a capillary
tube.
Suppose, in the first case, that the gas enters the capillary
at a fixed pressure P x , and leaves at a pressure P 2 . Let V x and
V 2 be the volume entering and leaving per second; then
p x V x = P 2 V 2 . The volume of gas passing any point in the
tube per second depends on the pressure at the point. The
velocity is consequently variable along the axis of the tube,
and therefore the method of the liquid flow cannot be applied
to the full length of the tube.
Consider a small element, dx of the tube. Let p be the mean
*-"rf*-8
176 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
pressure in the element and dp the difference in pressure at
the ends.
Equation (4), page 152, for the liquid flow, is
yi ~ l 8 „'
If q x is the volume of gas passing through the element, the
above formula may be applied (dx is very small), and since in
this case — = — ^- (since the pressure decreases with increase
in x),
dp it R*
Now, PjVj = P 2 V 2 = qjp, since the mass of gas passing any
point is constant, i.e. q x = * * ; therefore
PiVi^ _dp nJR*
p ~~ dx'S 7) '
F 1 V 1 dx = -~^ pdp.
Integrating over the limit of the tube
P 1 V 1 /'W= -Z R * T'p-dp,
Jo ° 1 J Pi
i.e. W = ¥ -^r^-T6-T (30)
If V is the volume available to enter the tube at a pressure P x ,
we may write
dV
We may apply Boyle's Law to the instantaneous values of
P and V at entry,
i.e. PV = constant,
„dV , „dV . ^dV „dP
*di +Y dt =°> he - v Tt=~ Y W
whence from (30)
(P x » - P 2 2 ) * R* _ _ dP
/ 16 71 ~ dt ^
dP
where -=- is the rate of change of pressure at the end of entry,
dt
and V the fixed volume. s
P a in the experiment is the constant atmospheric pressure
VISCOSITY 177
at the end of the capillary, and Pi has values, say, p x and p 2
at the beginning and ending of the period of observation of
t seconds duration. Re-writing (31), we have
ifiWVo* " + 2P J >x V^+P, p,-p J '
* R4 * - * rion Pl + p - 2 T a
i6^V 2P a L g Pi - P 2_L
~ 2P 2 t 10g £ 2 -P 2 10 \^-Pj»
which becomes, on writing P 2 = P = atmospheric pressure,
_/tcR*\ PJ
71 " \«V J. (p* + P ),( fr - P)
lg, "¥rrp)(^ 2 -p)
The term ^%. may be calculated once for all for the apparatus
and is the ' constant of the apparatus.'
The Constant Pressure Method
The determination of yj may be made with constant pressure
difference between the ends by maintaining the levels of the
mercury columns in the two tubes at a fixed difference. The
point, O, would be chosen at the commencement near the bottom
of the tube, and the time taken for a volume corresponding to
a length of tube between the original and final positions of O
measured. If p is the total constant pressure inside V, we have
v » = tf l6 V (30) '
in which all terms except tj are known.
The difficulty in this modification lies in keeping the mercury
levels a fixed distance apart as the gas is driven out.
CHAPTER VI
THERMOMETRY AND THERMAL EXPANSION
The Comparison of a Thermometer by means of a Standard
It is recommended that each student before beginning his
experiments on heat should choose a thermometer, test its
accuracy, and use it when required throughout his experiments.
The method of comparison is to immerse it together with a
standard thermometer in a bath and observe temperatures over
a suitable range simultaneously by both instruments.
It is convenient for the following experiments to have two
thermometers, one reading from o° C. to 35 ° C. and the other
from o° C. to ioo° C. Both should be calibrated in this way.
A large water or oil bath should be carefully heated over a
Bunsen flame and constantly stirred. The thermometers should
be placed in the liquid so that the mercury thread shows just
above its surface, and with their bulbs close together.
A record of temperatures at intervals of 5 should be taken
over the range, and a curve drawn with temperature corrections
as ordinates and with readings from the thermometer to be
calibrated as abscissae.
By means of this curve the readings of the thermometer in
later experiments can be reduced to that of the standard.
A better method of heating the liquid is to place it in a bath
standing in a box lined with cotton wool and to supply the heat
by passing an electric current thrbugh a resistance coil immersed
in the liquid. If the current is drawn from storage cells, and a
variable resistance included in the circuit, it is possible to adjust
the current so that the bath is maintained for a long time at a
constant temperature. The most convenient form of stirrer is
a small propeller driven by a small electric motor.
The Calibration of a Mercury Thermometer
It is impossible in practice to obtain a perfectly uniform bore
in the stem of a mercury thermometer, so that it is not sufficient
in an accurate instrument to divide the interval between the
fixed points into a number of parts of equal lengths. The
makers of thermometers usually attempt to make some
correction for this lack of uniformity by adjusting the distance
178
THERMOMETRY AND THERMAL EXPANSION 179
between consecutive divisions to suit the bore of the stem at
the various points. But, in spite of this, unless the thermometer
is exceptionally carefully constructed, errors remain and a cali-
bration has to be made if accurate observations are required.
In a given thermometer, as a rule, the divisions will be unequally
spaced at different parts of the tube, and the bore will vary from
point to point.
The first step is to divide the tube into segments consisting
of five or ten degrees each, and over each to find the average
distance between each division. We assume for the sake of
definiteness that we are considering intervals of ten divisions.
Measure each of these, beginning at o° by means of a micrometer
microscope, and deduce the average length per degree for each
of the ten intervals up to ioo°. We are assuming that the
thermometer is divided into degrees Centigrade from o° to ioo°.
When this has been done a thread of mercury is broken off
from the main column of length equal to that of about io° on
the thermometer scale.
This thread may be obtained by connecting a small jet, made
by drawing a glass tube to a narrow bore, to a gas-pipe, and
lighting the gas at the narrow end, adjusting the supply to
produce a flame about half a centimetre long. If this flame be
applied cautiously at the point of the thread where it is desired
to sever it, the thread will divide. During the application of
heat the thermometer must be rotated to avoid fracture due to
unequal heating.
The thread is moved by gently jerking the thermometer until
one end is at o° and the other near io°, and its length measured.
The thread is then moved so that one end lies at io°, and the
other near 20 , and so on up to ioo°.
If it is difficult to get the detached thread down to zero,
owing to the projection of mercury from the bulb past the zero
mark, the bulb should be cooled by wrapping it in wool and
moistening with ether. This will clear the tube and allow the
thread to be moved down without its re-joining the main mass
of mercury.
Denote the ten lengths of the thread by l lt Z 2 , l z , etc.
Let these be reduced to their equivalents in degrees. This
is readily done since the average width of one degree is known
in the different parts of the scale. Denote these equivalent
lengths by t x , t 2 , t z , etc., and the mean of these by t.
If the tube had a uniform bore of the same length as that of
the actual instrument between o° and ioo°, the readings would
be t, 2t, 3*, etc., instead of t lt t x + 1 2 , t x + 1 2 + t z , etc.
Let the values added respectively to t x , t 2 , t 3 , etc., to make
them equal to / be d v 6 S , <S 3 , etc.
i8o ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Then
t = t x + 8 X
t = t z + <5 a , etc.
The corrections to be applied near io°, 20 , 30 , etc., are therefore :
or
t-t x ,
2t — t x — t 2 , 3* — h — h — *a> etc.,
<5i + d 2>
^1+^2+ 5 3-
The correct temperature corresponding to t x is (t x + d x ), and
corresponding to the point [t x + 1 2 ) it is {t x + t 2 + 8 X + <5 2 ),
and so on.
We have assumed up to the present that the fixed points
at o° and ioo° are correct.
A table is drawn up as shown below. The correction to be
applied near io 9 is — -023, for the thermometer column is too
long at this point, near 20° the correction to be applied is
(_ .023 + -032) or + '009. The correction would have been
+ -032 had the point near io° been correct, but since that was
not the case both errors come in. We correct similarly for other
points by adding errors algebraically, and recording in the last
column the amount to be added to the thermometer reading to
obtain the correct temperature.
Note that in the example given the thread was not very near
to the mean length of io°. It is convenient to arrange this as
closely as possible to io°. Strictly, the error —023 ought to
be applied to the recorded temperature of 10-231, but the error
will probably not vary very rapidly in the neighbourhood of any
given point. Hence we take the corrections in the last column
as applied at io°, 20 , etc.
Mean length of thread, as deduced from column 4, 10-208.
REGION
OF TUBE
. MEAN
LENGTH
PER SCALE
DIVISION
LENGTH OF
MERCURY
THREAD
EQUIV-
ALENT OF
THREAD IN
DEGREES
DIFFERENCE
FROM
MEAN
CORREC-
TION TO
APPLY TO
UPPER
READING
O - IO°
IO°- 20°
20°- 30
3O - 40
4O - 50°
50 - 6o°
6o°- 70
70 - 8o°
8o°- 90
90°-ioo Q
•2385
•2389
•2391
•2389
•2398
•2350
•2355
•2353
•2347
•2367
2-442
2-431
2-449
2-435
2-438
2-426
2-404
2-393
2-392
2-408
IO-23I
10-176
IO-243
10-193
10-167
10-323
10-208
10-170
10-192
10-173
8 X — 023
<5 a +'°32
<5 3 —035
<5 4 +-oi5
<5 6 +-041
6* —"5
0- -ooo
<5 8 +-038
6 9 +-016
<5io+'°35
023
+ .009
— •026
— •Oil
+ •030
085
085
—047
03I
+ •004
THERMOMETRY AND THERMAL EXPANSION 181
Draw a curve with thermometer readings as abscissae and the
corrections to be applied to obtain the corrected readings as
ordinates. The curve should pass through the #-axis at o° and
ioo°, since these points have been assumed to be correct.
If, however, the fixed points are incorrectly placed, the errors
must be found in the usual manner with ice and steam.
Suppose that the zero correction is d , while that at ioo° is
^100-
The upper fixed point must be corrected for pressure, latitude,
and height above sea-level.
The barometer must also be corrected, owing to the fact that
it is probably not read at the temperature at which the instrument
was standardized.
This last correction may be made by the following formula :
If h denotes the height at o°, and h t that read at t°,
h = h t {i — -0001622).
For latitude x and at a height d feet above sea-level, the
length of the column which produces the standard pressure at o°
and at sea-level in the standard latitude of 45 ° is :
L = (760 + 1-94562 cos 2X -f- -000045466^) mm.
Under this pressure, L, the boiling point is ioo° at the height
and latitude of the place of observation.
h h
Thus, h is equivalent to -j- atmospheres or ^- x 760 mm.
under normal conditions. From this and the following table
the correction to the boiling point may be made.
CORRESPONDING
MAXIMUM VAPOUR PRESSURE
TEMPERATURES
720 mm.
98-493° c.
725
98-686
730
98-877
735
99*067
740
99255
745
99443
750
99.630
755 ,
99815
760 1
ioo-ooo
765
100-184
770
100-366
775
100-548
780
100-728
The zero on the scale actually records the reading
the ioo° records (100 — 5 100 ).
Thus a correction is required for this.
d n , and
182 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Plot on the curve the two points (0,0), (100, <5 100 — <5 ),
and join them by a straight line. The ordinate of this line at
x
x° is (3 100 — <5 ) • — , since each degree, even if correct as
regards bore, would register only = — 10 ° ° , we must add
J 100
to it an amount, — — , on account of the errors at the
100
fixed points
To x° we must add
100
Thus, if we draw on the same graph on which the first set of
results was plotted this second curve the difference between the
two ordinates taken algebraically will give the true reading
corrected for errors due to bore and fixed points.
Re-draw a new curve, showing as abscissae temperatures as
recorded by the thermometer and as ordinates the difference
of the orcjinates of the two curves, and the resulting curve will
give the amount to be added to any recorded temperature to
give the true temperature.
Throughout the taking of measurements the temperature of
the detached thread should remain constant ; and in order to be
sure that this condition holds, place another thermometer close
by and observe whether it varies or not. Do not handle the
thermometer under examination more than is necessary, and
only do so by holding it at the tip away from the bulb.
Newton's Law of Cooling
The object of this experiment is to verify Newton's Law of
Cooling, which states that the rate at which a body cools is
proportional to the difference of temperature between itself and
the enclosure in which it is placed. The constant of proportion-
ality depends on the surface exposed and the thermal capacity
of the exposed body, and the law is true for small differences of
temperature only.
The apparatus required is a small metal thimble, into which
water at about 8o° C. can be placed, provided with a cork through
which a thermometer passes for noting the temperature of the
liquid.
The enclosure consists of two calorimeters, one inside the
other, and containing water in the space between, to provide
an enclosure at nearly constant temperature. A thermometer
THERMOMETRY AND THERMAL EXPANSION 183
placed in this water, which should be stirred occasionally, gives
the temperature during the experiment.
Observe the temperature recorded by T x and T a (fig. 99), at
Fig. 99
intervals of half a minute during the initial stage of the fall,
and, as the rate decreases, the interval between observations
may be increased. The record of T 2 should not vary very much.
Tabulate the results thus :
TIME (MINS.)
RECORD OF T ,
RECORD OF T 2
72-5° C
12-2° C.
•5
7i-3° C
12-2° C.
1
70-1° C.
12-2° C.
i-5
69-0° C.
12-2° C.
2
67-8° C.
12-2° C.
2'5
667 C.
12-2° C.
3
65-6° C.
12-2° C.
3'5
6 4 -5° c.
I2'2° C.
4
63-5° c.
12-3° C.
4-5
62-5° C
12-3° C.
—
—
—
Draw a curve showing the relation between the temperature
Tj and the corresponding time.
Make the temperatures the ordinates and times the abscissae.
The rate of fall of temperature may be obtained from this
curve by measuring the tangent of the angle of inclination of
the tangent to the curve to the axis of t. As explained in the
introductory chapter this measures the value of -^ at the
various points of the curve*
it
184 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
According to Newton's law these values should be proportional
to the differences between T x and T 2 . If T 2 does not vary very
much its mean value may be regarded as the mean temperature
of the enclosure. x
If T 2 varies too much to permit this approximation, draw a
curve showing the relation between T 2 and the time on the
diagram which shows the relation between T x and the time.
Then at each time we can determine the value of (T t — T^
and the corresponding value of -— from the same graph.
Make another table containing two columns, one for values
of (T x — T 2 ) and the other for the corresponding values of — j^-
Draw a curve with the values of (T 2 — T 2 ) as ordinates, and
those of -~ as abscissae, when, if these two quantities are
proportional, the result should be a straight line.
The Use of the Weight Thermometer
The weight thermometer consists of a glass bulb, B, drawn
out at the upper end into a capillary stem, A.
/
1
\
B
KJ
Fig. ioo
A convenient size is obtained by making B about 6ne and a
half to two inches long, and between a quarter and half an inch
wide. It forms a good exercise to make the apparatus from a
piece of glass tubing. If this is done, care must be taken to
get rid of any blob of glass likely to accumulate at the bottom
of B, otherwise in the subsequent heating B will be very likely
to crack.
The apparatus measures the expansion of any liquid placed
in B relative to glass. In order to deduce the real expansion it
is necessary to know that of B.
The coefficient of expansion of mercury has been found with
great accuracy, and may be taken as -0001818.
THERMOMETRY AND THERMAL EXPANSION 185
We may therefore use mercury to find the expansion of B,
and then use B to determine the expansion of other liquids.
We shall consider the determination of the expansion of water.
(1) Determination of the Expansion of the Weight Thermometer
Carefully weigh the apparatus.
Surround B by wire gauze and warm carefully in order to
drive out air. Place the end A under clean mercury contained
in a dish and allow B to cool so that a little mercury enters.
Then, when sufficient has entered, boil the mercury so that the
space above it may become full of mercury vapour. Once more
place A below the mercury, when B will fill as the vapour
condenses on cooling.
It is best to carry out the process gradually, heating and
cooling B several times, and allowing a little mercury to enter
at a time.
Warm up the mercury first to prevent the bulb from cracking
when it enters.
Allow B to cool gradually, and finally surround it with ice,
keeping A immersed all the time.
Bring up a small weighed, empty dish, and remove the mercury
at A, replacing it by the empty dish. Remove the ice and allow
B to acquire the temperature of the room. Mercury will, of
course, flow over into the dish. As soon as the flow ceases,
weigh the bulb, B, and the dish. Let the total weight of mercury
within B at o° be denoted by W .
Immerse B and as much as possible of the stem in boiling
water, and catch the mercury that flows out in the little dish.
After the bulb has remained in the water for a quarter of an
hour, to allow it to assume the temperature of the boiling water,
remove it, dry and carefully weigh it. Suppose that w is the
amount that has flowed out. As a check re-weigh the dish and
again determine w.
Let p denote the density of mercury at o° C. and p t that at
temperature T.
If V denote the volume of the apparatus at o° and V x that
at T, while p denotes the coefficient of expansion of glass,
£ = * + P T.
Wl
But V =-^,
Po
- 1 + Pi - Wfl pi W() l^^'
186 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
• s =, W ° ~ w * +ocT x
" W ' T T
W — ze> te>
— a —
W * W T
« =H7^ P + *
i.e.
W — w W Q —w T
W W I
p ■
W -a>; p T W -ze> r
The ratio ^ _ ^ is small, and since p is also small we may
often neglect the first term on the right and use simply
a - p = w ,1 .
P W -ze> T
(2) The Coefficient of Expansion of Water
The value of this coefficient varies considerably throughout
the range o° to ioo°. In this experiment the average value
between two temperatures, say, 20 C. and 6o° C. will be deter-
mined. If both temperatures are above that of the room the
experimental difficulties are not so great. We assume this to
be the case though the method is quite general, and exactly the
same precautions have to be taken as in (1) if water flows out
on moving B from the lower temperature enclosure to the balance.
Carefully fill the weight thermometer with water that has been
recently boiled to get rid of contained air, and place it completely
immersed in water at temperature t x . Remove it, dry, and weigh.
Place the thermometer in water at t 2 and repeat.
Use the formula given above to deduce a for water, making
use of the value of p previously determined.
Determination of the Density of Water at Various Temperatures
by means of a Glass Sinker
In this experiment a solid is weighed while totally immersed
in water at different temperatures, so that by the principle of
Archimedes the weights of the fluid displaced by the solid
corresponding to the various temperatures are known.
Let V denote the volume of the solid at o° C. and a the co-
efficient of expansion of the solid, so that at temperature t°
the volume is V (i + at). If Pt is the density of the water at
temperature, t, the loss of weight due to immersion is
V (i + a*) p t . This value is observed by the balance ; let it be
W«. Then p ,=- — ^i
THERMOMETRY AND THERMAL EXPANSION 187
In order to carry out the experiment, a long Wire is attached
to the scale-pan of a balance and passed through a hole in the
base of the balance to support the solid, which hangs in the
water. Since during some part of the experiment the water
will be at a temperature considerably above that of the immediate
surroundings of the balance, it is necessary to use a wire about
40 cms. long, so that convection currents may not disturb the
equilibrium of the balance.
The wire should have a diameter not greater than T V mm., so
that surface tension may not cause any appreciable effect where
it enters the water. In practice thin copper wire is often employed,
though it is preferable to use a short length of platinum wire,
specially treated to diminish surface tension effect for immersion
in the water. ' #
A stirrer is necessary to keep the temperature of the liquid
uniform, and as soon as the whirls, due to stirring, have died
away, the balance is made and the temperature taken by a
thermometer immersed in the water with its bulb as near as
possible to the solid.
It is preferable to heat up the water to the highest desirable
temperature and allow it to cool down. In this case the weights
in the scale-pan will require to be continually diminished. The
weights should be adjusted before an observation so that the
solid appears a little too heavy. After a short interval the scale
pointer will cross the zero position, and at this instant the tem-
perature of the water should be observed.
In this way a series can be obtained very conveniently for
temperatures above atmospheric.
The solid is first weighed in air so that the values, W*, corres-
ponding to different temperatures may be obtained by
subtraction.
As a sinker it is usual to employ a glass bulb containing lead
shot, and a may then be taken as -000025.
When it is not possible to reduce the temperature right down
to zero it may be taken down to some convenient low temperature,
t . By finding the weight, W tQ , of the submerged vessel at this
temperature, the volume, V tQy may be deduced by the help of
the density table appearing on p. 188, taken from Kohlrausch's
" Physical Measurements,"
v - W ">.
since V '»~7r
The densities at the other temperatures may then be deduced
by the formula
W«
p < V t0 {i + «(*-g}'
188 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
DENSITY OF WATER BETWEEN o° AND 20 C. PER CC.
TEMP.
DENSITY
TEMP.
DENSITY
0°
0-99988
11°
0-99965
1°
0-99993
12°
0-99955
2°
0-99997
13°
0-99943
3°
0-99999
14°
0-99930
4°
1-00000
15°
0-99915
5°
0-99999
16 6
0-99900
6°
0-99997
17°
0-99884
7°
0-99993
18
0-99866
8°
0-99988
19°
0-99847
9°
0-99982
20°
0-99827
10°
0-99974
In the laboratory it is convenient to begin at a temperature
of about 8o°, and make observations about every io° C.
A curve should be drawn exhibiting the relation between
temperature and density.
Alternatively the experiment may be made in order to deter-
mine the coefficient of expansion of water for varying intervals
of temperature from the formula : p = p*(i + p*).
The Constant Pressure Air Thermometer
In this form of thermometer temperature is denned by means
of the equation
V< = V (i + at) ;
t denotes the temperature, V t and V the volumes occupied of
a certain mass of gas at two temperatures, the former at t°, and
the latter at a convenient fixed point : the zero of the scale.
By choosing the melting point of ice as t = o°, and the boiling
point of water as t = 100 °, under standard conditions we can
find the value of a.
We may therefore say that the equation is assumed, and that
t is defined by it,
or
-w-y
It is assumed that the pressure remains constant throughout.
The diagram shows a simple form of constant pressure thermo-
meter.
THERMOMETRY AND THERMAL EXPANSION 189
The mercury reservoir is adjusted so that the mercury stands
at the same level in the two tubes, CD and EF. When this
is the case the pressure is equal to that of the atmosphere in
both tubes.
Suppose it is desired to determine a certain temperature with
this instrument, say the melting point of wax. First surround
the bulb completely with powdered ice and turn the three-way
tap, T, so that B and CD are open to the room, and adjust G
so that the mercury stands at the zero division on the graduated
scale of CD.
Fig. ioi
Wait for about ten minutes to allow the bulb to cool exactly
to o° C, and turn the tap so that B and CD are connected to
each other but cut off from the atmosphere. In this way the
zero on the scale is made to correspond to o° C.
Now immerse the bulb in boiling water, and lower G until
the mercury stands at the same level in the two tubes, and
observe the scale reading.
Read the barometric height, and deduce the boiling point of
the water.
Surround the bulb with warm water and adjust its temperature
to the melting point of the wax. To do this, put a small piece
of the wax in a small test tube and immerse it in the water.
Heat the water until the wax melts, and then let it cool a few
degrees, and then warm up very slowly, keeping the water
igo ADVANCED PRACTICAL PHYSICS FOR STUDENTS
stirred until the wax begins to melt again and then take the
reading of the thermometer. Of course G must be adjusted so
that the level of the mercury is the same in both tubes.
From the observations made we can deduce the melting point
of the wax. Note the temperature also by means of an ordinary
mercurial thermometer.
Theoretical Considerations
The fundamental equation of gas thermometry, whatever may
be the form of thermometer, is simply :
Total mass of gas in the instrument = constant.
In practice it is not possible to maintain all the gas at the
same temperature ; some of it is necessarily remote from the
point of application of the body examined. These remote
regions are described by the term ' dead space.'
In our apparatus the dead space extends from fl to the level
of the mercury in C.
We shall suppose that the scale readings are in ccs., beginning
at the zero and extending downwards.
Let the reading corresponding to the case when boiling water
surrounds B be denoted by x b , and let x w be the reading when
the wax is melting.
Let the volume from the top of the bulb at the point where
it is immersed to the zero of the scale be denoted by v, with a
suffix to indicate the temperature at which it is measured.
The temperature of the dead space, which has a total volume
(v -f x), will vary from one end to the other ; but we shall make
our calculations by assuming that this temperature is uniform
throughout and equal to that measured by placing a thermometer
in a position approximately midway between the two ends of®
this space.
We shall denote this by the letter t, and when the bulb
temperature is t we shall write t t for the corresponding tempera-
ture of the dead space.
Let the volume of the bulb together with that part of the
tube which is immersed be V at the temperature zero, let p
denote the coefficient of cubical expansion for glass, and p the
density of air at zero.
The mass of gas in terms of the quantities measured when the
bulb is at o° C. is :
V Po + «V Pr •
The temperature of the dead space is, of course, not necessarily
at o°, it has some value t .
Or we may write for this mass :
THERMOMETRY AND THERMAL EXPANSION 191
••(
T -*SHH3) '■'
When the bulb is at a temperature, t, let x t denote the reading
on the scale. This denotes the volume between the zero of the
scale and the mark x t at the temperature at which the apparatus
was graduated. This is often done at about 15 C, and the
laboratory temperature is usually in this neighbourhood. We
shall not introduce any great error into our calculations if we
regard this as measuring the true volume of this part of the
apparatus under the conditions Of the experiment.
The temperature of the dead space is now t t , and the total
volume is therefore :
v (i + p«r t ) + *t
The mass of gas in the dead space is :
v (z + PT f )p x tPo
+
I + OCTf 1+ OCT
Hence the total mass is measured by :
V (i + Pflpp , v (i + Pt,) Po x tPo
i+att "*" 1 + <xt, "t" 1 + <xt, {2)
If the expressions (1) and (2) be equated, since they denote
the same quantity, it will be found that :
*. = v o(I + .Tof^i* + f«{i±^ -f±^n . . o)
l_I+otf V (i+aT 1 + aTjJ w;
The second term in the square bracket may be written :
?o(« - P) ( T f - T o)
V. (i + oct ) (1 + ax t )'
where we have neglected the product a p in the numerator.
In practice the difference between t t and t is not very large,
and the construction of Jhe apparatus provides that ^is small.
We may thus neglect this term in comparison with the first
term in the bracket without introducing any great error. The
student is recommended to find an approximate value of the
two terms, so that he may better appreciate the effect of this
neglect of the second term.
We then have :
- *• = v «<« - «« • TT#- (4)
If we apply this to the case when the bulb is surrounded by
boiling water, of which the temperature is b, corrected of course
192 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
for any variation of the barometric height from normal, we
have from (4)
*» - V (a - p)6 Z + aT&
Hence from (4) and (5)
1 + ub
I + a6 £
*6
(5)
.(6)
I + a^o I + orf
-» + T - lr + » ,
a a
The value of — may be taken as 273*1.
j t and t h may be observed on a mercury thermometer, although
this is introducing into the experiment the mercury scale.
Equation (6) is a linear equation in t, which is thus determined
from the reading on the scale of the air thermometer.
This enables us to deduce the temperature of the melting wax.
Fig. 102
The Constant Volume Air Thermometer
The diagram illustrates a common form of the apparatus.
The bulb, B, is connected by a capillary tube to rubber tubing,
DE, and to the glass tube, EF.
THERMOMETRY AND THERMAL EXPANSION 193
EF slides against a scale, SS, by means of which the height
between the mercury levels in the tubes, CD and EF, can be read.
EF can be clamped in any desired position, so that the level
of the mercury in the tube CD stands always at a definite mark, C.
In this case the temperature is denned by assuming the relation :
p t =p (i+<rt), or t=l.t±Zl±*.
p t denotes the pressure within B at a temperature t, and po
that at a standard zero position — the temperature of melting
ice, while the volume of gas remains constant.
The pressures are measured by adding the atmospheric pressure
to that due to the mercury column of length, h.
In practice the part of the apparatus containing the air expands
with rise of temperature, and there is the dead space, HC, to be
alloweoVfor as in the last experiment.
Let V denote the volume of the bulb and immersed portion
of the apparatus, v that of the dead space at o°.
Suppose that the dead space remains at temperature t during
the experiment when B is surrounded with ice, and t& when it
is surrounded by steam.
We shall show how a may be determined experimentally.
If p denote the density of air under normal conditions, and
p that at pressure, p, and temperature, t,
_ Po P_
9 - I + at ' 76
The mass of gas contained in the thermometer is :
-^-•V + 76 ^o (l+aTo) (7)
expressed in terms of the conditions prevailing when B is at
the temperature of melting ice.
Similarly the mass of gas expressed in terms of the conditions
prevailing when B is at temperature, t, is :
Vq(i + PQ-/>iPo ■ Vo(t + t*t)P#o ,™
76(l+0C/) ^ 76(1+ OCT,) W
Hence on equating (7) and (8) :
Again on account of the smallness of the ratio ~ we have as
*
an approximation :
*-*'T+* (I °)
J 3
194 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
When the bulb is at the temperature, 6 i of boiling water,
i + $b
Po =Pb
I + <x&
(II)
We may assume the value -0000232 for p and thus calculate a
from (11) by observing the pressures when the bulb is surrounded
by melting ice and by steam respectively.
Equation (10) then enables us to deduce the temperature
corresponding to any pressure, fi t .
Take a mercury thermometer and immerse it close to the
bulb, observing its readings and the corresponding pressures.
From the latter deduce the temperatures from (10) and draw
up a table recording these in one column opposite to the records
of the mercury thermometer in a second column.
Draw a graph with air temperatures as ordinates and mercury
temperatures as abscissae, exhibiting the deviations between the
two temperature scales.
In order to calculate the value of —^ if this is necessary, first
adjust the mercury to the mark C, immersing B in watei that
has come to the room temperature.
Fig. 103.
Read off the pressure, P, to which the air is now subjected.
Carefully raise EF so that the mercury approaches the bend
at H, and so fills nearly all the dead space. Let the pressure
within the bulb be now P.. Then, since the conditions are
THERMOMETRY AND THERMAL EXPANSION 195
isothermal and the whole volume, v , has been very nearly filled
with mercury in the second case,
P 2 V = PxCVo + v ),
*V P x
We may read the temperatures t and t t by means of a mercury
thermometer placed close to the dead space and obtain a closer
approximation to the value of, a.
In the second form of apparatus (fig. 103), into which the first
may be readily converted, the volume of the dead space is made
negligible. _
In order to measure the difference of level between C and F,
two tubes, WW, connected by a rubber tube containing water
are adjusted so that the level on the left is the same as that
at C, and consequently this is the same on the right at C 1 .
The distance, C*F is readily observed on SS.
CHAPTER VII
CALORIMETRY
The Specific Heat of a Solid by the Method of Mixture
The student will be familiar with the principle of the method
of mixture. The main object in this description is to give an
account of the method of making a correction for the error
arising from radiation.
If W is the water equivalent of the calorimeter and contents,
m the mass of the solid, and s its specific heat, and if t x is the
initial temperature of the calorimeter, t % the final temperature
and T that of the solid initially, then if there has been no loss
of heat we have :
ms (T - jQ = W(* 2 - t x ).
In practice there is a loss or gain of heat from or to the calo-
rimeter, which should be added on the right-hand side of this
equation, since all the heat from the solid has not been retained
in the calorimeter.
We may make the correction by the consideration that the
final temperature, t 2 , would have been t 2 + At, where At is an
interval of temperature which must be small if the experiment
is to be successful.
Hence the corrected equation is :
ms{T - t 2 ) = W(t 2 + At - t x ).
At will be small if during half the experiment the calorimeter
gains heat, and in the other half loses heat. This can be arranged
by adjusting the initial temperature so that the room temperature
is approximately a mean between it and the final temperature.
A preliminary experiment is made to find out roughly the
temperatures that will be attained during the experiment.
The same amounts of the materials are used in a second case,
but the calorimeter is cooled down by adding small pieces of
ice or warmed up, as may be necessary, so that t x and t 2 may
lie at nearly equal temperature intervals below and above the
temperature of the room.
The temperature of the calorimeter is noted immediately before
immersing the hot body, and then at quarter or half minute
intervals until the maximum temperature is attained, the
196
CALORIMETRY
197
observations being continued beyond this point at definite
intervals.
These results should be plotted on a graph (fig. 104).
The curve obtained will be similar to ABC. Had there been
no losses or gains on account of radiation, the curve would have
been similar to ADE, the final -temperature remaining constant
at the level DE.
Ms
Fig. 104
We can, by applying Newton's Law of Cooling, derive the
curve ADE from ABC.
Suppose the axis of t to be divided into small intervals, OM x ,
MjMji, M 2 M 3 , etc., of magnitudes 8t x , 6t z , dt z , etc., and let the
average temperature during these be : X , 2 , 3 , etc.
Then the loss by radiation in 8t x is kd x 8t x , where k is a constant.
.-. p^i = QxMi + ke x dt x .
Similarly the loss during MiM 2 is kd 2 8t 2 , and consequently
P 2 M 2 = Q 2 M 2 + kd x dt x + &0 2 # 2 .
This process may be continued to any extent, and if we make
the intervals sufficiently shorty M x , Q 2 M 2 , etc., are not sensibly
different from 6 X , 2 , etc.
Thus, denoting by © the ordinate of the upper curve at a
time, t, and by that of the lower at the same time :
©! = ©! + ke x 8t x ,
© 2 = 2 + ke x 8t x + kd 2 8t 2 ;
and generally :
© rt = 0„ + k{6 x dt x + 2 # 2 + . . . + B n 8t n )
= n + &(area of lower curve from OA to ordinate B ),
i.e.
© = + k I ddt
198 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We can thus make the correction by carefully drawing the
lower curve, calculating the area up to the point 0, which it is
desired to correct, and add its product by k to 0.
We require in this experiment the ordinate corresponding to
any point P 1 along DE, beyond the point B.
We therefore determine graphically OAQ x M and apply the
correction to Q X M, thus obtaining P*M.
In order to calculate k, find th6 rate of cooling along BC,
corresponding to a mean temperature 0.
k = — • j- (Newton's Law.)
It will usually be convenient to note the fall for four or five
minutes, and make the deduction from it.
Reynault's method of making the correction is actually to
make the correction of the ordinates and draw the curve ADE.
For this purpose we require to know the rate of cooling at any
particular temperature.
Find the rate at one period, as above, and plot jr against
on a curve. In doing this it is assumed that the relation is
linear in accordance with Newton's Law.
At the temperature of the room ~ vanishes ; for if the calo-
at
rimeter and surroundings were at the same temperature then
there would be no loss due to radiation.
Fig. 105
We therefore have two points on the graph, and by joining
dd
these by a straight line we can determine -j- at any value of 6
at
(fig- 105.)
Divide ABC into sections very nearly straight, as AR, RQ, etc.
dd
Note the mean temperature over AR and from the graph for-r
dd
note the rate -5- for this temj^erature. Multiply this by the
CALORIMETRY
199
time, ON, and add the result to the ordinate, NR, thus obtaining
NR 1 . Let this correction be <50 v
In the same way find the amount <50 2 , lost during the interval,
NM. Add the sum (<50 x -f <50 2 ) to MQ and so obtain MQ 1 .
Continue this process until the maximum ordinates along DE
are attained.
The curve obtained in the experiment should attain this
horizontal branch very nearly, and the ordinate is the quantity
Speciflc Heat of a Liquid by the Method of Cooling
The rate of loss of heat of a body depends only on the tempera-
ture of the body and that of its surroundings, on the area, and
on the nature of the surface exposed.
If the difference of temperature between the body and its
surroundings is not large, the rate of emission of heat is propor-
tional to the temperature difference. This is Newton's Law of
Cooling.
Suppose a mass of liquid, M lf is enclosed within a calorimeter
of mass m x , and let S x and s x denote the specific heats respectively.
The thermal capacity of the system is (M^ + m^^. If the
temperature fall from t t to t 2 in n x seconds, the average rate of
loss of heat is (MjSj + m x s x ) . — -
In the case of the second liquid under the same conditions,
let n 2 denote the number of seconds required for a fall of tem-
perature from t x to t 2 , and the loss of heat per second is
200 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We have by Newton's Law
(M X S 1} + m x s x )
h-U
= (M 2 S 2 + m x s x )
h-h
:. s x =
n x (M 2 S 2 + m x s x )
(i)
_ m x s x
n 2 M x "M7
The apparatus for carrying out the determination consists of
a small calorimeter fitted with a rubber stopper through which
a thermometer may pass (fig. 99.) A calorimeter of aluminium
of about an inch diameter and three inches high serves the purpose
very well. The calorimeter should be supported by threads, or
should stand on a non-conductor within a double-walled enclosure,
the thermometer passing through a cork in the lid of the enclosure.
In order to secure a uniform temperature, the space between
the walls of the enclosure may be partly filled with water. In
this case care must be taken that the inner box does not float,
or it may happen that it will touch the calorimeter and there
will be loss of heat by conduction.
The enclosure may consist of two calorimeters — an outer large
one fitted with a lid, and an inner smaller one standing on blocks.
First fill the aluminium calorimeter about two-thirds full of
water, and warm it to a temperature about 70 C. by immersing
it in hot water. Place the apparatus in the position shown in
the diagram, and take readings of the thermometer at intervals
of half or whole minutes down to a temperature below 30 C.
Note from time to time the temperature of the enclosure,
which should hardly vary during the experiment.
Fig. 107
Some time must of necessity elapse between the observations
on one liquid and those, on another, and although it is not difficult
to maintain a constant enclosure temperature throughout each
set of observations, it often happens that the mean temperatures
recorded by the thermometer, T 2 , are appreciably different in
the two cases.
CALORIMETRY
201
This difficulty may be avoided by using a second aluminium
container similar to G, and suspending it by the side of G inside
C. The records of the temperatures of the two liquids are then
made almost together, and the enclosure temperature is the
same for each.
Make up a table containing the liquid temperatures opposite
the times of observation, and in a third column record the
enclosure temperatures.
Draw on the same graph as illustrated in fig. 108 the curves,
one for each liquid, with the differences of temperature between
liquid and enclosure as ordinates, and with the times for abscissae.
Let AB denote the curve for paraffin (say) and A B that
for water.
Draw the horizontal lines, T X AA and T a BB to cut the curves
at A, A and B, B as shown.
Let t x denote the temperature of the liquids above that of
the enclosure in the first case, and t 2 the corresponding
temperature in the second.
Then in the time that elapses between the instant measured
by TjA until that measured by T 2 B the paraffin cools down the
interval (f r — t 2 ), and the water cools down the same amount
during the interval between T^o and T 2 B . Denote the two
periods of cooling by n x and n z . In this case the value of S a is
unity, and for aluminium the value of s x is -219.
By weighing the liquids and calorimeters we obtain sufficient
data to give the value of s 2 for paraffin by means of the formula (1).
Determination of the Specific Heat of a Solid by means ol Joly's
Steam Calorimeter
A metal jacket, J, enclosed in a casing of felt surrounds a
platinum pan, P, suspended by means of a fine wire attached
to one arm of a balance, whose base is shown at BB,
202 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The upper end of J is closed by a light metal disc, D, through
a hole in which the wire passes. This disc is free to move,
and when oscillations occur in P it finally settles down so that
the wire passes through the hole without contact with the disc.
Just above D, a small coil of wire carrying a heating current
round the suspension prevents condensation on it and also on D.
Fig. 109
In the first place, J is allowed to attain the temperature of
the room, and the inlet and outlet pipes are then closed.
The pan is balanced in the usual way, and in the meantime
water is boiled in a container ready to supply J with steam by
means of I.
When a good supply is obtained O is opened and steam passed
through I. When the steady state is attained it will be found
that additional weights are required to counterpoise P on account
of the condensation of steam on it. Suppose w grammes are
condensed and let the initial temperature of P, which has been
observed by a thermometer placed in T, be t lf and that of steam t z .
Once more allow the apparatus to dry and weigh the solid S.
When steam is again passed into J with S in the pan, a greater
amount of steam will be condensed on account of S. Let this
now be W and suppose the initial temperature now is T v
Previously the scale-pan condensed w grammes of steam and
rose in temperature through the interval (t 2 — *,). In the
second case it rises from Tj to t%. The amount of condensation
w '
per degree rise of temperature is -r- — r~, hence the weight of
(t 2 — *i)
w
steam condensed in the second case is : -j- r-r x {t 2 — Tj).
CALORIMETRY
203
Denote the mass of the solid by m and the specific heat by s.
The mass of steam condensed by the solid is :
w(t t - TJ
W-
('• ~ h)
Hence we have the equation :
ms(t t - TO =fw - H' !
Ti)
]
L,
(*, - tj
by means of which the value of s may be calculated from a
knowledge of the value of L or, conversely, L may be determined
if the specific heat of the solid is known.
L denotes, as usual, the latent heat of steam.
A correction to account for the differences in apparent weights
of the solid in air and steam has been neglected. The temperature
of the solid is supposed equal to that of the apparatus and
surrounding air. An interval of certainly not less than twenty
minutes is required to allow the solid to acquire this temperature.
Bunsen's lee Calorimeter
Description and Preparation for Use
A diagram of the apparatus is shown in fig. no. The calori-
meter is represented by ABCFED. It consists of a test tube, B,
Fig. 1 10
fitted into the glass jacket, A, which is drawn out at its base
into the tube, CFE. This tube ends in a cup, E, closed by a
cork, through which passes the narrow glass tube, D.
A is filled partly with clean mercury and partly with distilled
water containing no air.
204 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The surrounding jacket, J, is a calorimeter closed with the
cork or wooden stopper, S, which supports the apparatus.
In order to keep J and its contents at the freezing point,
it is placed in a larger vessel, standing on non-conducting blocks
and packed round with a mixture of ice and snow or with flaked
ice.
By cooling the inner surface of B sufficiently a layer of ice
may be formed round the outside, as indicated at I.
On melting one gramme of ice the volume diminishes by
•0907 c.c, so that if heat be added at B the amount may be
determined by noting the change of volume as a result of the
partial melting of I. The change of volume is observed by noting
the movement of the end of the mercury column at D along
the capillary tube. If this has been previously calibrated the
change can be observed directly.
In order to fill the apparatus, remove the capillary tube, D,
and the stopper in E, and introduce into A sufficient distilled
water to fill it to about half. Invert the apparatus with the
open end of the test tube downwards and carefully boil the water,
continuing until A is about one-third full.
While this is proceeding, boil some distilled water in a large
beaker, and towards the end of the evaporation of the water in
A, place the end, E, well under the surface in the beaker.
Cease boiling the water in the calorimeter, and allow more to
flow over from the beaker. In this way the inside of the calo-
rimeter and the tube, CFE, become filled.
Clean mercury must now be passed in to lie below the water
in A.
Introduce it gradually from a pipette held under the surface of
the water in E, allowing displaced water to overflow. Take care
that no air bubbles are introduced with the mercury, particularly
when it becomes necessary to tilt the apparatus to allow water
to pass over the mercury in A towards the tube. Fill up with
mercury to E, place the stopper in position, and by carefully
adjusting it make the end of the thread coincide with any
desired position along D.
The apparatus should then be placed in a calorimeter containing
water and ice to reduce the temperature as nearly as possible
to zero.
This will probably take an hbur at least, and the progress
of the fall may be tested by placing a thermometer in B.
When the temperature is about 2° C. introduce cooled ether
into B. The ether may be cooled by placing it in a cooled test
tube, and standing it in the calorimeter with the apparatus.
Draw air through the ether and cause it to evaporate, con-
tinuing until a cap of ice surrounds B.
CALORIMETRY
205
The solidification will cause D to move farther along the
capillary, and enough ice should be formed to cause more
expansion than is likely to be required in succeeding experiments
with the apparatus.
The evaporation of the ether may be brought about by some
such device as that indicated in fig. 111.
T
n
J=&
Fig. 111
When sufficient ice has been formed the remaining ether is
evaporated, and a current of air further drawn through to
remove all traces of it.
Now place the apparatus within the jacket, J, and stand it
in the vessel, K, packing it round as described above with ice
and snow.
Leave this standing for an hour or two until a steady state
has been reached, and the movement in the capillary tube is
only slight and steady.
It is not possible to maintain the end of the mercury thread
quite steady with this arrangement, so that the slight motion
must be accounted for in determining results from observations.
The capillary tube may be calibrated by the method described
on p. 41.
Calibration of the Apparatus
By placing warm water within B of known mass and tempera-
ture, we may note the movement of D at various parts of the
capillary tube for a known absorption of heat.
In performing the experiment it will be sufficient to calibrate
the tube for one particular region, and in using the instrument
again a slight pressure on the stopper or a slight easing of it will
drive the end of the thread into the calibrated strip.
Heat up pure water to about 25 C, and transfer carefully
to B. Allow the apparatus sufficient time to become steady,
and note the displacement of the end of the mercury thread.
Let this be /, and let the time be noted between the insertion of
the water and the return to steady conditions. This will be
denoted by t.
206 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In order to correct for the small creep of the thread, observe
the rate of motion just before adding the warm water, and also
just after the absorption of heat.
All these measurements are to be made with a travelling
microscope mounted conveniently opposite the capillary tube.
If the rates of creep are respectively p and p 1 , then the average
rate may be taken as \{f> + p 1 ) during the time t.
Thus the displacement of the thread due to absorption of heat
from the water is / — %(p + p x )t. We shall call this quantity L.
If m is the mass of water added and its initial temperature,
then a motion of L units corresponds to the absorption of md
calories.
Determination of the Specific Heat of a Substance
Let M grammes of the substance be heated to a temperature 1 ,
and let s denote the specific heat.
The substance when placed in B will transfer to the ice M0 x s
calories.
If the mercury thread moves a distance, L 1 , in the calibrated
region, the heat absorption is -j— L 1 calories.
Tu ,,., m0L l
Thus M0 x s = — j — ,
md L 1
S= Wi'L-
The correction for Creep must again be applied by observing
the motion just before and just after the insertion of the mass.
L 1 is the corrected length.
When a solid is put into B, a pad of cotton wool should be
placed at the bottom of the tube to prevent breakage when it
falls. In order to facilitate removal it is a good plan to tie a
light thread round it. This will introduce only a slight error.
During the absorption of heat, and generally while the apparatus
is in use, the end of the tube, B, should be stopped with a plug
of cotton wool.
In the case of determining the specific heat of a liquid the
experiment is almost exactly a repetition of the calibration.
In order to dry the tube after liquid has been put in, a roll
of clean blotting paper may be used.
The Determination of the Density of Ice
In this experiment we require to know the volume per unit
length of the capillary tube. We have assumed that this has
been previously calibrated. It thus remains to determine the
CALORIMETRY 207
shrinkage due to absorption of a definite quantity of heat.
The experiment may be performed in conjunction with the
calibration just described.
Let L denote the latent heat of fusion of ice, and suppose
warm water added to B imparts k calories.
k
The amount of ice melted is -=- grammes.
Let dv denote the shrinkage as measured by the movement
of the mercury thread.
Then -=- grammes of ice have become ■=- grammes of water
L L,
at o° C.
Let d denote the density of water at this temperature.
k
The volume of water is j-^ c.c, and the volume of the ice
is thus :
Gi + »)«*
Hence the density of ice is :
A
L
. grammes per c.c.
&+*
The Latent Heat of Fusion of Ice
It is assumed that the student is familiar with the principles
of the determination of the latent heat of fusion and has carried
out the experiment without making corrections for radiation.
We are concerned in this description chiefly with an account
of how this correction may be made. Care is taken, as in the
determination of specific heat, to adjust the initial and final
temperatures so that the room temperature is the mean of the
two. *In addition, care must be taken that the calorimeter is
not cooled down so low that the dew point is reached, otherwise
there will be a deposit of dew on the apparatus, and a liberation
of latent heat in consequence.
We may make the correction for radiation as in the experiment
on specific heat, but an alternative method will be described.
Note the temperature when the ice is placed in the calorimeter,
and at intervals of half a minute until it is melted, and finally
at intervals of one or two minutes during which the calorimeter
is absorbing heat by radiation.
208 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Let the temperatures observed in this second period be
1 1> 1 2> I 3 • • • ■*• n+V
Then the change of temperature due to radiation during these
intervals will be :
»e t = AftCTi + T 2 ) - t ]
<50 2 = AftCT, + T 3 ) - t ]
<50 o = A[£(T n +T B+1 ) -*„],
by Newton's Law of Cooling, where A is a constant depending
on the calorimeter but not on its temperature, and t is the
temperature of the surroundings.
Thus the total change of temperature A0 is given by :
A0
= A{ Tl + 2 Tw+1 +(T 2 +T 3 + ..+T n )-^ J
But A0 = T B+1 - T 1( so that A can be calculated.
In the first part of the experiment during the melting of the
ice, let the observed temperatures be : t x , t 2 , . . . t n+1 .
Then we have :
At
AJ * 1 + J n+1 + (*, + t 3 + . . . + t n+1 )-nt Q }.
Thus, since the observed minimum temperature is t n+1 , the
corrected minimum is (t n+1 -j- At).
In the above the <50's and dt's are to be treated algebraically,
for some will be negative and some positive if the initial tempera-
ture is adjusted in the manner described.
Thus, if W is the water equivalent of the calorimeter and
contents, and m the mass of ice melted, we have :
wL + mt n+1 = W{*i - {t n+1 + At)\.
The Latent Heat of Vaporization. (Berthelot's Apparatus)
One form of this apparatus is depicted in fig. 112. Its essential
feature is the condenser spiral, C, with the receptacle below.
This is immersed in water in a calorimeter and the calorimeter
shielded by packing it round with a non-conductor and placing
it in a convenient vessel, or better still by standing it on non-
conducting blocks in an empty larger calorimeter and enclosirig
within the larger vessel (see fig. 112).
The condenser is dried and weighed. It is placed in the
water and allowed to stand until the temperature becomes steady
as recorded by the thermometer, the uniformity of temperature
CALORIMETRY
209
throughout the calorimeter being procured by means of a
stirrer.
A quantity of vapour is introduced into the condenser, and
the spiral provides a large area of contact with a cold surface,
so that liquefaction takes place and the liquid collects in the
receptacle. The water is constantly stirred, and its temperature
observed by means of the thermometer.
Fig. 112
When a suitable rise is obtained the supply of vapour is cut
off and the thermometer watched until the maximum temperature
is reached. In this interval, immediately after cutting off the
supply, the end of the condenser outlet tube must be closed by
a cork to avoid convection effects.
The condenser is removed, dried, and weighed, and the mass
condensed thus determined. Denote this by m, so that if L
denotes the latent heat the supply of heat on condensation is
mL calories.
Let T 2 denote the temperature of vaporization, and T 2 the
final temperature of the calorimeter and contents, while T
denotes the initial temperature of the calorimeter.
Let s denote the specific heat of the liquid, and W the water
equivalent of the calorimeter, condenser, and remaining calori-
meter contents. Then : W(T 2 — T ) =wL + m(Tj — T^s ; so
that L may be determined.
One of the weak points of this form of apparatus is the mode
of introduction of the vapour.
210 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
As the diagram shows, the liquid is vaporized over a small
gas ring in the reservoir. It thus easily happens that the
vapour gets superheated, and, in addition, it is difficult to
shield the calorimeter effectively from the heat of the flame.
The screen, S, is introduced to reduce this effect.
It is preferable to use an electrical method of heating, and this
is done in the more recent forms of the apparatus.
Kahlenberg's heater is illustrated in fig. 113. The liquid is
contained in the vacuum flask, H, and is heated by passing
an electric current through the platinum wire, PP.
For ordinary laboratory practice a test tube may take the
place of H without introducing much difficulty in shielding the
calorimeter.
Fig. 113
Correction for loss due to radiation in the calorimeter, C, must
be made by one of the usual methods (p. 197).
The Heat of Solution of a Salt
When a salt is dissolved by a liquid the solution is accompanied
by an absorption or liberation of heat. The amount varies
with the proportion of the salt to the solvent, i.e. with the
resulting concentration, and with the amount of salt dissolved.
The number of calories absorbed or liberated when one gramme
of a salt is dissolved in a certain amount of solvent is said to be
the heat of solution for the particular concentration.
The experiment described below is designed to measure this
quantity.
CALORIMETRY
211
The apparatus necessary is illustrated in fig. 114. It consists
of an outer protecting calorimeter of metal, C, which carries a
cork through which the inner vessel, A, is supported.
This vessel also carries a cork through which pass a thermo-
meter, stirrer, and thin test tube, B, into which the salt may
be placed.
Fig. 114
The vessel, A, and stirrer, S, are usually of glass since many
solutions attack copper. There is uncertainty about the specific
heat of the glass, so that the water equivalent of the vessel, A,
and its contents should be determined in a separate experiment.
For many purposes, however, we may assume the specific heat
of the glass to be «i6. The pure solvent is placed inside A, and
it should surround the lower part of the tube, B, which contains
the salt.
The apparatus is allowed to stand so that the salt may acquire
the temperature of the solvent. Half an hour should be allowed
for this, and during this interval the salt may be occasionally
stirred by a clean dry glass rod.
The weight of solvent is obtained in the usual way before
the insertion of B. Let this be denoted by W.
Suppose the water equivalent of A and its contents is w, and
the specific heat of the solution s. This quantity must be
determined later by the method of cooling (p. 199), or by any
other convenient experiment.
It will be necessary, also, to make a correction for radiation
by one of the methods previously described (p. 197).
Let the initial temperature of the salt and solvent be denoted
by t and the final temperature, corrected for radiation, by t v
212 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
and suppose that the weight of the salt dissolved is q. The
heat of solution being Q, we have :
qQ= {(W+q)s+w}(t-t ),
for (W -f q) is the weight of the solution.
The value of Q will be positive for the liberation of heat, and
negative for absorption.
In order to mix the salt and solvent, the glass rod used above
for stirring should remain in the salt until just before mixing.
At this instant observe the temperature, t , recorded on the
thermometei^and push the rod through the bottom of the
thin tub<T?T
This tube must, of course, be clean and dry, and the salt will
then fall into the solvent and be dissolved. The rod should
then be removed, and care taken not to carry with it any of the
solvent. The solution is assisted by the stirrer, S, and observa-
tions of T taken every quarter or half minute in the initial
stages, and later at longer intervals.
A graph is drawn showing the variation of temperature with
time from the instant of mixing, and this curve corrected for
radiation as explained above, or we may use the non-graphical
method.
To find the Ratio of Specific Heats at Constant Pressure and Constant
Volume for Air. (Clement and Desorme's Experiment)
Apparatus and Experimental Details
A glass reservoir, provided with a tap, T, giving a wide opening
to the air is connected to an oil manometer, G, and to a pump;
Fig. 115
an ordinary bicycle pump is convenient. A small excess pressure
is applied to A, the difference between it and the atmospheric
pressure being measured by the manometer (fig. 115).
CALORIMETRY 213
In the first stage of the experiment the temperature under
these conditions is allowed to become steady.
In the second the tap, T, is opened and closed suddenly by
giving one half-turn.
By this means the pressure falls to that of the atmosphere
in so short an interval that we may suppose there is no passage
of heat to A during this expansion.
The condition of the expansion is therefore adiabatic.
Finally the temperature is allowed to return to that at the
beginning of the experiment, during which process the pressure
in A increases, though it does not recover its original value.
Theory.
Suppose that gas occupying the volume below the dotted line
remains in the flask all the time.
Denote its volume by V while that of the flask is.V.
Let the initial pressure be P , and let that immediately after
the adiabatic expansion be B.
Then BV' = P V ',
where r — ratio of specific heats, the value of which we require.
The flask is open to the air so that B is the atmospheric pressure.
It is in order that we may ensure a fall of pressure from P
to B during a short interval that the tap is wide.
In the final stage let the pressure become P when the tempera-
ture has attained a steady value.
We have now passed from volume, V , and pressure, P , by
an isothermal process to volume, V, and pressure, P.
/. PV = P V .
/. P'V «= P 'V '.
Hence by dividing this by BV f and P V f , we have :
B
or
(0-
p r-x
B .
p *
log
r —
log
P_
Po
Let the difference in heights of the manometer be h initially,
and h finally.
Then if pressures be measured in terms of heights of the liquid
columns :
p = B + K, P = B + h,
214 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
r =
j—X (approx.),
Po
by expansion in logarithmic series and neglecting higher powers of
~ and -4g — than the first.
This is permissible since the values of h used are only a few
centimetres. 5 cms. is a convenient value for h .
Hence r — -, — 2-r-
«o — A
The result may also be obtained by another method.
In fig. 116 let AC denote any curve relating the pressure and
volume of a gas. The elasticity is denned to be the ratio :
stress
strain
The stress will be measured by a slight change in pressure,
and the strain by the corresponding slight change in volume
per unit volume. Let us consider the volume, v, represented
by BC. Let a change of pressure, dp, denoted by FA, bring
about the change in volume denoted by CF. We shall record
this by &o, but 6v = — CF on account of the diminution of
volume on the addition of pressure.
>/OU.C»KA«:
Fig. 116
CF CF
Thus, the elasticity, E, is measured by AF -7- — » for —
J v v
denotes the change of volume per unit volume.
.-. E-— flt
dV
When we proceed to the limit and make the changes very
small we have :
E = — v-r--
dv
CALORIMETRY 215
For a gas the value of E depends on how the change is made,
and we shall consider two cases, first the case of an isothermal,
and then that of an adiabatic change. Let AC denote the
adiabatic curve and AB the isothermal ; the former is steeper
than the latter. Let E* denote the adiabatic and E«, the
isothermal elasticity.
From the formula for E we have :
E = — v x slope of curve.
Thus E^ == — v x slope of adiabatic,
and E fl = — v x slope of isothermal.
If we start at A with a particular volume, v, measured for each
curve by P A, we have :
E$ slope of adiabatic
E^ — slope of isothermal
The changes of pressure and volume in the experiment are
small, so that the curves, AB and AC, are approximately straight,
and the slopes can be measured by :
p£ and gg respectively.
E*__AF
•* E„~~AE
Now, AF is the change in pressure during the adiabatic part
of the expansion, viz., h , and AE is the change during the isother-
mal part. In our case the atmospheric pressure is that at the
end of the adiabatic expansion, i.e. that at C, and since the
point, B, on the curve represents the final state, CB denotes
the pressure, h.
Thus AE = h Q - h.
~,, Ea h n
i.e.
xxiua
E, ~
' h — h
For the adiabatic
Differentiating we
expansion
log^> +
have :
idp
pdv
we have :
pv r — constant,
r log v = 0.
+■^-0.
V
dp
dv ~
P
- — r.
V
In our case p and v denote the values of the co-ordinates at A,
id -£- i:
dv
axis of x.
dj>
and j- is the tangent of inclination of the curve AC to the
216 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
. Similarly, for the isothermal case we have :
pv = constant,
and £»_£
dv v
p and v are the co-ordinates of A, but the slope is now for AB.
Thus the ratio of the slopes is r.
■ • - -* = r = h °
E«j h — h
In carrying out the experiment make about six independent
determinations, and increase the pressure cautiously so as not
to expel oil from the manometer.
CHAPTER VIII
VAPOUR DENSITY AND THERMAL CONDUCTIVITY
Vapour Density. (Victor Meyer's Method)
The vapour density of a substance can be found by measuring
the volume of the vapour produced from a small quantity of the
solid or liquid whose weight is known. In Victor Meyer's method
this volume is found from the volume of air displaced by the
vapour.
The apparatus (fig. 117) consists of a vertical glass tube pro-
vided with a bulb at the lower end, A, and a side tube, ST. The
side tube dips under water in a beaker and a rubber cork closes
the upper end of the tube.
Fig. 117
As it is necessary, on introducing the vapour, to allow a small
bottle, D, to fall the length of the tube, it is advisable to place a
little asbestos at the bottom of A to prevent breakage.
The tube, A, is surrounded by a larger tube containing a liquid
which boils at a higher temperature than the substance to be
experimented on.
217
218 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In the case of the determination of the vapour density of
ether, water may be used in the bath.
The various parts of the inner tube should be kept at a constant
temperature during the experiment, and in order to maintain
this condition the outer tube is screened from draughts by
surrounding it with a cylinder of asbestos or cardboard which
fits it above the bulb.
Before beginning the experiment the inner tube must be quite
dry. If necessary it should be warmed over a Bunsen flame
while a current of air is blown through it.
The apparatus is set up as shown in the diagram, and heating
is kept up until no more bubbles come out from the side tube.
The substance is weighed and enclosed in the small bottle, D,
and suspended close to the upper end of the inner tube. When
everything is quite steady the bottle is allowed to fall ; vaporiza-
tion takes place, and the cork of the bottle is blown out. Air
T
B
Fig. 118
passes over into the side tube and may be collected in the burette,
B. It is better to collect it by the method illustrated in fig. 118,
in which case the gas collected can be brought to atmospheric
pressure by raising or lowering the burette. In the other case
it is necessary to correct for the height of the water column, L.
In order to cause no disturbance on introducing the substance
to be vaporized a piece of thread or thin wire should be passed
through the cork and be held by a stop-cock, E, which pinches a
piece of tubing, F. The bottle is allowed to fall by opening E,
and then closing it immediately.
A better method is to use the apparatus shown at G. By
turning the wire, H, through 180 the bottle will be caused to
slip off the hook.
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 219
The air collected at B is over water at a temperature, T, and
will be saturated with water vapour at this temperature. Let B
be the saturation pressure at this temperature. If v is the
measured volume of air, and H the total pressure, then v the
volume under normal conditions is given by
H - B 273
v = v x — 7 — x L ^f
76 273 + T
If w — weight of substance enclosed in D, the density of the
vapour is :
w __ w 76 273 4- T
v ~ ~v H -B 273
1 c.c. of hydrogen at o° and 76 cm. pressure, weighs -0000900
grammes, and its molecular weight is 2.
Hence the molecular weight of the substance examined is :
2 w 76 ^ 273 + T
•00009 v H — B 273
At the conclusion of the experiment remove the stopper from
the end of the inner tube to prevent any sucking back of water
from B into the bulb, A, as the apparatus cools. It is important
to cause the bulb, A, to be heated by the steam from the bath,
and it should be adjusted to prevent actual contact with the
water in the bath, and to be out of reach of splashes when boiling
takes place.
Vapour Density. (Dumas's Method)
A large flask is cleaned and dried, and fitted with a cork provided
with a bent piece of glass tubing drawn to a fine point so that
it can be easily sealed by the application of a Bunsen flame
(fig. 119).
The flask is suspended from the arm of a balance and weighed.
Let the observed weight be W lt and let w x denote the weight of
air displaced by the wall of the flask, while w 2 denotes the weight
of air displaced by the closed flask, so that w 2 — w x denotes the
weight of air within it. This will be denoted by w a . Hence
if W denote the real weight of the flask :
W 1 = W-ze' 1 (i)
Now introduce a small quantity (5-10 c.c.) of the liquid to
be vaporized into the flask. The actual quantity will depend
on the size of the flask and will be discovered by trial. The
amount given is of the right order when a flask of capacity about
a half litre is employed to find the vapour density of chloroform.
This liquid is very suitable in the case of a laboratory exercise,
since it has a high vapour density. The flask is placed within
220 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
an enclosure, J, over a sand bath, with the tube, T, projecting
through the lid.
Heating is continued until the liquid is vaporized and no more
issues from T. This may be tested by placing a polished surface
near the end of T. It will become dimmed if vapour is still
coming out.
When the steady state is reached, T is sealed off.
Fig. 119
Let this happen at a temperature, t°, measured by means of a
thermometer hanging close to A, within the enclosure, J, and let
the volume of the vapour be V t , and density ? t . Allow the flask
to cool, and weigh. Break off the end of T under water. The
flask will fill with water, the space occupied by the condensed
vapour becoming negligible under the new conditions.
Preserve the broken pieces from T, and after drying the outside
of the tube re-weigh the flask and water, noting the temperature
of the water, t °.
Let W 2 denote the weights in the scale-pan when the flask
and vapour are weighed, and let w v denote the weight of the
vapour.
Then W 2 = W + w v - w 2 ;
.-. W 8 — Wj = w v — {w t — Wj) by equation (1)
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 221
Let V t0 denote the volume of the flask obtained from the
weight of water it contains at t °. Then if p denote the coefficient
of cubical expansion of the glass, which may be taken as '0000232,
V t = V i0 {1 + ?(<-«„)},
If the barometric pressure be P, we can reduce p t to normal
conditions by the formula :
(273 + *) , 76
273 P
(273+J) . 76 w %
H =~m T' Pt
2 73 P V #0 {1 + fi(t-t Q )}'
w a denotes the weight of air filling the flask at the temperature
of the air within the balance. Let this be t 1Q C.
Then if d a denote the density of air under these circumstances,
w. = d.V* = da V f() {1 + p (V- - t Q )\
-^•^.^f^r-v^i + p^-^)},
d — -001293 gm. / c.c.
From these two formulae, since w v = w a + W 2 — W 1( we may
now calculate p .
One of the difficulties of the experiment is to drive out all the
air from the flask and replace it by vapour. It frequently
happens that on attempting to fill the flask with water some air
is left behind. More of the liquid is required for driving off the
air in this case.
If the volume of air left over is small, we may apply a correc-
tion. At the temperature t , of the water, let the volume of air
be v. This may be determined by filling up the flask with water
from a measuring flask.
The total pressure to which the mixture of vapour and air is
subjected is the sum of the partial pressures, P 1 and p\ of the
vapour and air respectively. The weight of the vapour is now
w v \ obtained by subtracting the weight of air of volume, v, from
w v , determined as above :
wj> = p%
V <0 denotes as before the total volume of the flask, i.e. the
volume of the water after the bubble of air has been replaced.
We can find P 1 by remembering that the air of volume, v, and
at atmospheric pressure occupied a volume, V t , under the partial
pressure, p 1 , the temperatures being respectively t ° and t° C.
222 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Thus
aP = V t p l
273 +* 273 +r
p 1 = P -P 1 ;
.-. P x V t = PV t - 273 1 1 • Pv,
* * 273 + 1
273 + * 76 I 273 + 1 J
or p =
w J
273. P
L 273 + 1 273 + *oJ
76 L 273 + 1
The term, — » will be small if the experiment is to be
273 + 1 y
successful at all, so that it will only be necessary to change the
value, w v , to the actual weight of vapour, to, 1 .
It should also be remarked that the volume, v, is a mixture of
air and water vapour, whereas it has been assumed to consist of
air only. We shall not, however, further consider this correction,
which is small and affects a term which must be already small.
The discussion shows how the error affects our formula, and it
would be sufficient to measure the volume of the bubble, multiply
by the density, -001293 gm./c.c, and subtract from w v .
After a few attempts the bubble will usually be sufficiently
small to be neglected altogether.
Conductivity of a Copper Bar
The apparatus consists of a bar of copper, CC (fig. 120), with
two holes bored well into it to carry thermometers, T x and T 2 ,
mercury being placed in the holes to ensure good thermal con-
tact. At the ends of the bar are two metal boxes, through one
of which, A, is passed steam, and through the other, B, a steady
stream of water from a constant pressure head.
The shelves, LL, within B, serve to prevent any flow of water
straight from inlet to outlet. The temperature of the water is
taken just before it enters and leaves B by the thermometers,
T 8 and T 4 . The apparatus is allowed to attain a steady state,
when all the thermometers will record steady temperatures.
It is usual to pack loosely round CC and the boxes some
cotton wool, the whole being enclosed in a felt-lined wooden box,
through which T t and T a project. T 3 and T 4 are kept as close
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 223
as possible to the box, and it is a good plan to wrap the T-pieces
loosely with wool.
In this way the heat from A is transmitted by conduction
to B, and the heat passing across in any time, t, is noted by
collecting water as it leaves B and weighing it. If the mass
collected is m the rate of transmission of heat is
*» (T 4 - T 3 )
t '•
T. T, T,
f 1
j-L— 3
B Tc
-L— }
C A
Fig. 120
T 3 and T 4 denote the initial and final temperatures of the
water. Since this quantity of heat is transmitted from T x to T a
—a distance, d, say, the amount is k • (T 1 — T a ) • j, where A is
the area of section of the bar, and k the conductivity
A is measured by finding the diameter of the bar if it is cylindrical,
or by measuring its breadth and height if rectangular.
We have, therefore :
k (Tj — T a ) • j = m
(T 4 - T 8 )
In another form of apparatus the cold water is passed through
a metal spiral wound round the end instead of through the metal
box. Good thermal contact is made between the spiral and bar,^
and the temperature of the water measured at entrance and exit
as before.
Thermal Conductivity of Rubber Tubing
The apparatus required consists of a length of rubber tubing, B,
a copper heater, A, for producing steam, a calorimeter, C,
thermometer, T, and a measuring glass, as illustrated in fig 121.
The method of procedure is as follows :
A quantity of water is introduced into the calorimeter, C
and weighed.
224 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Steam is passed through B until a rise of temperature of the
calorimeter and water of from io° to 20 ° C. has occurred. The
initial and final temperatures are noted and also the time of
passage of the steam. The tube is removed and the length that
has been immersed is noted — I (say). Two pieces of cotton
should be tied round the rubber at the points where it enters
and leaves the water.
Fig. 121
Let the initial temperature be T and the final T 1 . If there
had been no loss of heat by radiation the final temperature would
have been some other value, T 1 + A T, and it is necessary to make
the correction AT.
In order to do this, observe temperatures, at intervals of half a
minute or some convenient period, beginning at T and ending
at T 1 , during the passage of the steam.
These will be denoted by t v t a , ... t n . x .
If the temperature of the room be t we have for the change
of temperature due to heat radiated or absorbed, as the case may
be, according to Newton's Law of Cooling :
T +t x
^c^-4
dd 2 = C J — t [,
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 225
dd n = C
. . \ ... c
*oj,
where 3d denotes the change in any interval, the suffix denoting
which interval.
Thus
AT = S.50 = C | T ^ T1 - w* + (*i + *i + • • • + *«-i)|.
C is a constant depending on the calorimeter and contents,
and must be determined if M is to be calculated.
When the final temperature, T 1 , has been attained, cut off the
supply of steam and allow the calorimeter to cool, observing
temperatures, T^ T 2 , . . . T B , at equal intervals.
Then as before if 8t denotes the loss of temperature in each
interval we have :
S# = C |3i±I-« - (»-i) t + (T a +T 3 + . . . + T.-0
But S# = T x — T n , so that C can be calculated and may be
employed in the above case.
The ranges of temperature in the two cases should be as
nearly as possible the same.
If M denote the water equivalent of the calorimeter and con-
tents the heat transmitted through the tubing is
MfP+AT-T).
We can connect this with the conductivity, k, in the following
way :
Let the outer and inner radii of the tubing be r x and r 2 , and con-
sider a portion of unit length of the tube between radii, r and
r + 8r, at which the temperature is t.
The rate of change of temperature at the distance, r, is — -j-.
ar
The negative sign expresses the fact that the temperature
diminishes as r increases.
Thus if Q denote the quantity of heat transmitted per sec. per
unit length, i.e. across an area, 2nr,
r» t, dt
J 2-ak J r
is
226 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
where the integration is to be taken between the limits, r x
and r 2 for r, and between the inner and outer temperatures of
the tubing for t.
Let the steam temperature be t, the outer temperature is
taken as the mean of T and T 1 ,
/. ,- i<T+T') = 2 -filog/J.
But Q = y (T 1 + AT - T),
. . M T> + AT - T , r,
■■ * - 55 x 2 ' 3 ° 3 x utt ' lo e» f,
2
(Change being made to logarithms to base io.)
The value of r x may be determined by means of a screw gauge,
and in order to find r 2 , place a length of the tube of 5 to 10 cms.
in water in a measuring glass and note the volume, v, displaced.
Then if L denote the length of tubing :
v = tcL (rS - r 2 *).
All the quantities except r 2 are known, so that this value can
be determined.
Another way of determining the radii is to cut the tube clean,
normal to its length and use it as a rubber stamp, pressing it
lightly on a clean sheet of paper.
The impress of the outer and inner circumferences will be
distinct and the diameters may be measured by means of a
travelling microscope.
The Conductivity of Glass
The conductivity of glass in the form of a tube may be found
by the method described in the last experiment. A different
arrangement of apparatus is required, but the theory is identical
in both cases.
Steam is passed through a jacket, J, round the tube, B. Within
B a stream of water is caused to flow from a supply which provides
a constant head of pressure.
Within B is a spira> made of cord or rubber so that as it pro-
gresses up the tube the water is caused to traverse it spirally.
This is important as the temperature at any cross-section of B
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 227
must be the same throughout the section. The rate of flow is
adjusted to cause a difference of temperature of about 20 be-
tween T and T 1 . The thermometers are enclosed in T-pieces
as near as possible to the ends of B, the T-pieces being covered
with felt or cotton wool to prevent loss of heat by radiation
before the temperature is taken.
Fig. 122
In order to measure Q, water is collected on exit for a measured
time.
Conductivity ol Cardboard by the Method of Lees and Chorlton
The apparatus consists of a retort stand provided with a clamp
(fig. 123) and metal ring, AB, from which hangs a cylindrical
slab of copper or brass, DE, of diameter about 12-5 cms. On
this rests a hollow cylinder, C, of the same diameter, provided
with inlet and outlet tubes, G and H, through which steam may
be Dassed.
Towards the base of C and into DE holes are bored so that the
thermometers, T x and T t , may be inserted and the temperatures
read
The two cylinders are nickel-plated in order to produce a
surface of uniform emissive power.
Suppose a thin slab of material of the same diameter as the
cylinders is placed between them, and let the loss of heat that
is radiated from the edge of the slab be small enough to be
neglected in the calculation. Then all the heat transmitted
across the slab is radiated from DE during the steady state.
Let A denote the area of cross-section of the slab and d its
thickness. Let the thermal capacity of DE be denoted by W,
and let the thermometers, T x and T 8 , record temperatures, T x
228 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
and T 2 in the steady state, while the temperature of the surround-
ing air is T . The heat transmitted through the slab per sec. is :
k.
Tx
d
•A,
where k is its thermal conductivity.
A B
\
H
G
E
'T 2
F
12 I
Fig. 123
The heat radiated per second from DE is
C • (T 2 - T ),
where C is a constant.
Thus
.T,
A = C (T 2 - T ).
We may readily use the apparatus to give comparative results.
If two sheets be cut of the same diameter as DE, one of glass
and the other of cardboard, about 1 mm. thick, and if they be
inserted between the cylinders we may observe two sets of
temperatures, one for each. Let the letters without dashes be
used to describe the experiment with glass while those with
dashes correspond to cardboard.
In the latter case we have :
h'~
T/ - T 5
d'
•A = C(T 2 '-T / ).
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 229
On dividing we have :
k' _ T 2 ' - T ' d'
k T 2 — T n d
T t -T s
T 2 - T„ d T/ - T 2 ' '
d and d 1 are measured by means of a screw gauge.
In order to find T place the thermometer below a sheet of
cardboard, FF, to protect it from direct radiation from DE, but
it should be placed directly below DE, so as to give the tempera-
ture of the air which rises upwards to DE.
An absolute value may be found for k by determining the rate
of fall of temperature of DE at the temperature, T 2 .
This may be found by removing C and allowing a Bunsen
flame to play on DE until T 2 registers a temperature about io°
above that recorded during the steady state.
Observe the temperature recorded by T 2 as the slab cools to
about io° below that recorded during the steady state. The
observations should be made every half-minute, or more fre-
quently if the change is rapid, and a graph drawn relating the
time and temperature.
From the graph determine the value of the rate of change at
various temperatures.
Plot a second graph with the rates, -^-as ordinates and the
corresponding temperatures, T, as abscissae (fig. 124). From
this determine the particular rate for the temperature, T a .
Denote this
"&\
Weigh the slab and determine its water equivalent, using for
the specific heat of copper the value -094, or of brass the value -09.
Denote this by W.
Hence the loss of radiation is also expressed by W( — J
and we have the equation :
by means of which k is determined.
/
230 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Make a table of the observations during cooling as follows :
TIME (MIN.)
TEMP.
(T.)
AVERAGE TEMP.
RATE— .
dt
*
I
I*
2
2*
3.
Forbes's Method of Determining the Conductivity of a Metal Bar
This experiment consists of two parts, in the first of which a
metal bar is heated at one end until the steady state is reached.
In the second part a bar of the same material and cross-section,
but shorter, is allowed to cool under similar external conditions.
In order to maintain constant external conditions the experi-
ment should be performed in a part of the laboratory sheltered
from draughts.
3
Fig.
125
The bar (fig. 125) usually has one end curved and dipping into
a convenient molten metal contained in a vessel on the other
side of a screen, which protects the bar from direct radiation
from the source of heat.
The metal may conveniently be molten lead or solder. The
bar is provided with a series of holes which lie regularly along
its length into which thermometers fit.
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 231
If these are small and contain mercury or the molten
liquid, it is found that the process of conduction is not
disturbed by their presence. The holes near the hot end should
contain the molten liquid and the remainder mercury.
By observation of the temperatures at various distances along
the bar measured from the scifeen, the temperature slope may
be found. This is best done by plotting a curve (fig. 126) and
calculating the slope, -=-, from the inclination of the tangents
at various points.
e
3£
Fig. 126
A thermo-junction may be used alternatively to find the
temperatures by dipping one junction successively into the holes.
For the calibration of the junction the reader is referred to p. 545.
The importance of maintaining steady external conditions
will be appreciated in this part of the experiment, and great
care will be required to maintain the whole length of the bar
simultaneously steady.
The bar is assumed to be sufficiently long that its end is at the
external temperature, O . A convenient length is one metre and
its section may be about 2 cms. square.
In the steady state all the heat passing a section, B, escapes
from the surface between B and the end.
The rate of flow at B is :
dx>
where k is the conductivity and A the area of section.
This rate of flow is calculated from the second part of the
experiment by determining the rate of cooling of the portion of
the bar beyond B. The equation thus obtained serves to find k.
The second bar may be conveniently 10 cms. long. It is heated
to the temperature of the molten metal, but so that its surface is
not damaged and remains similar to that of the first bar. To
do this and at the same time to prevent sudden cooling of the
molten mass, the rod is wrapped in several layers of paper and
completely immersed.
232 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
It is provided also with a hole to carry a thermometer and is
suspended under the same conditions as prevailed round AB,
and its temperature observed at successive times so as to include
in the range those values which prevailed along the bar.
A curve is drawn showing the relation between the temperature,
6, and the time, t, for the short bar.
This bar is similar to the long bar and cools under the same
conditions, so that the rate of cooling for both is the same.
d6
From the curve we can deduce the rate of cooling, — , in the usual
manner.
We require for the purpose of the calculation the rate of cooling
for points along the bar in the first part of the experiment. We
dd
must correlate the values of x and —=- , x denoting distances
measured from the screen.
This may be done from the curves.
Take a series of values of x from the curve illustrated in
fig. 126 and observe the corresponding value of 0. From the
curve relating and t take the values -=- for these particular
values of 0.
dd
We then have the corresponding values x and -=-
at
Plot these on a curve as illustrated in fig. 127.
Fig. 127
de
This curve will cut the axis at a point, C, where -^ vanishes,
at
or where the temperature of the bar is equal to that of the sur-
rounding air.
If 5 is the specific heat and p the density, the rate of loss of heat
between two points separated by distance dx is
AdXpS -57-
dt •
The total loss per second between B and the end of the bar
is thus :
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 233
i
the
Aps 5— ax
b «
* Ap Vb -s-
= Aps x shaded area of fig. 127,
graph measures the distance from the hot
dx
where OP on
end up to B.
This area may be found by the planimeter or calculated if
carefully drawn on squared paper.
•■• * A GD B =Apsxs -
S = shaded area and ( •-=- ) denotes the temperature slope at
the point B.
■••*•(&- *
We can thus determine k, the values of p and s being given as
constants of the apparatus or determined in the usual way.
These values may be taken to be 8-93 gm. per c.c. and -094
respectively when the bar is of copper.
The Determination of the Conductivity of a Bar of Metal by Angstrom's
Method
In this method heat is supplied to a long bar by alternately
heating and cooling it at one region in regular periods.
In this way the temperatures at points along the bar fluctuate
periodically, and on account of surface radiation the temperature
Fig. 128
amplitudes diminish as the distance from the region of supply
increases, while the maximum and minimum values occur at
later timefe with the increasing distance.
The bar must be chosen so that it is sufficiently long to allow
234 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
us to neglect the effect of the terminal faces. The fluctuations
should die away at a short distance from the cooler end, which
thus has the same temperature as the air surrounding it.
When the heating is continued long enough the periods develop
themselves completely, in which case the mean temperature at
any point of the bar preserves some constant value.
We first consider the theory of the experiment.
Let us consider a bar, AB (fig. 128), with one end, A, exposed
for a definite time, T x , to a current of steam and for a succeeding
time, T 2 , exposed to a current of cold water. Suppose that
this process is continued regularly until the steady fluctuations
throughout the bar are developed. These fluctuations will have
a complete period of (T x + T 2 ) which we may denote by T, and
for the sake of convenience we shall write
Let the conductivity of the bar be, k, p its density, s its specific
heat, A its cross-section, P its perimeter, and its temperature
at a point, P, distant x from a convenient origin. We shall
choose this to be at A.
Consider a point, P 1 , distant dx from P, as in the diagram, and
let its temperature be (0 + 66). The heat flowing into the
element, PP 1 , of the bar across a plane at P drawn normally to
its length is equal to :
, P — P i
dx A P ersec ->
where the subscripts denote the points at which is measured,
i.e. the heat flow amounts to :
- k '~Sx~' A
-- k 'Tx' A = F ^ s ^'
in the limit when dx is made infinitesimal.
Thus this expression denotes the flow from left to right at the
point P, of the bar. Since depends on x, in the general case
this flow will also depend on x, i.e. F depends on x.
Again consider the element PP 1 . We have calculated the
flow, F, into it at P, and the flow out at P 1 will be
F + — • dx.
dx
Hence the total flow of heat into PP 1
^ _d¥
dx
dx
J, d * 6 A *
= ft ' -j— z • A • 6x per sec.
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 235
This heat is used up, partly in warming up the part of the bar
concerned, and partly in radiation from the surface.
If the temperature is changing at the rate, -j-, t denoting the
time, the first part amounts to :
and the second to :
h • P • 6x (0 - O ).
h denotes a constant for the surface and O is the external
temperature.
We shall, however, suppose that is measured in degrees
above the surrounding temperature, so that we may write the
last expression simply :
h-V'Sx-B.
We, therefore, arrive at the equation :
kAdx ^ = h • P -dx • 4- A -gsdx -=?
ax* at
or
2 -*&-=•■ <*>
where K =~ , H = -^- (3)
ps Aps
Any function which satisfies certain conditions can be expanded
as a Fourier Series.
Students of Physics should make themselves acquainted with
the Fourier Analysis and they may be referred to Carslaw's work
on this subject. For the present purpose we need nothing more
than the statement that the expansion is possible.
Fourier's Series is the following :
f(t) = A + A x cos pt + A 2 cos 2pt + A 3 cos 3pt + . . . ) , *
+ B x sin pt + B 2 sin 2pt + B 3 sin ipt + . . . J w
Here f(t) is any function existing over some interval from
t = a to t = b and satisfying certain conditions with regard to
continuity, etc., which we need not enter into here. We merely
mention that the series is applicable to the function we shall use
in our experiment.
With regard to the values of the A's and B's the rule is
that:
236 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
A B =^ / /(«) cos -7 — — u du;
b — aj a b — a
B„ = T -/ t(u) sin T w. du;
b — a J t JK ' b — a
n has the integral values, i, 2, 3, etc.
2tc
The value of p in the above series is in this general case r
When f(t) is periodic and has the complete period, T, we take the
limits, a and b, at the ends of this period and write a = o, b = T,
so that the series represents f{t) from t = o to t = T, and on
account of its periodic character represents it also from T to 2T,
etc. In this case :
. 2tc
P = x '
I I /* T
Aj L-J f{u)du;
2 /* T
A B = =r / /(w). cos npu du ;
2 f T
B„ = — / /(w) sin w^)w • ^w.
Ic/
The periodic function with which we are concerned is that
which expresses the fact that from t = o to t = Tj the temperature
has some value, $ lt and from t = Tj to t = T 2 it has the value 2 .
Such a function can be represented by a Fourier Series and we
shall assume that the particular expression is (4), with coefficients
calculated according to the rule. These, however, are not
required for the experiment.
It is shorter and convenient for our purpose to write (4) in the
form :
a Q + a x sin {pt + r x) + « 2 sin (2pt + r 2 ) + (5)
And let us suppose that this is the expansion which expresses
the temperature at A.
Now the temperature, B v at the point whose distance is x t
from A is a fluctuating function and so is that for any other
point, %.
We may therefore write for 6 lt some such value as (5), viz.
B x = C ' + C/ sin (pt + a/) + C 2 ' sin { 2 pt + <V) + • • • • (6)
and for the point, x %
6 2 = C ' + C ± ' sin {j>t + d{) + C/ sin ( 2 pt + «■*) + . . (7)
The quantities C ' C/ . . . , <V 6 2 ' . . . , will differ from
C " Cx" . . . , 5/ 5/ . . . , since they correspond to different
values of x.
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 237
■■**'.
We have to solve the equation (2), and we have a clue from the
experimental observations of diminishing amplitudes and lagging
maxima and minima. Moreover, the value of is known at the
point A where x is zero. This value is given by (5).
The solution is in fact :
6 = a,?-*** + a 1 e- a ^ sin (pt + p,* + r x ) +
a#-** x sin {2pt + M + r 2 ) + etc (8)
Where : a » = ? K (a n * - p n *) = H, \ {g)
2Ka n p tt = -np J
When x = o this expression reduces to that of equation (5),
so that the solution satisfies the end condition.
The reader maj' verify that the solution satisfies equation (2)
by substitution. He will observe that the coefficients of the
separate sine and cosine terms and the terms independent of
trigonometrical functions all vanish if (9) holds.
Now if we consider the two points, x x and x 2 , at a distance, I,
apart, by equations (6), (7) and (9) we have :
C/ = a x e~ a **\ C x " = a x e-"***, d x " - <V = p^.
For S x = $ x x x -f r x and d x = $ x x 2 + r x
Hence «. x l = log ~i and a^ = ' ^ x log ^V
But by the last of the conditions (9)
TW-Olog^
K = ■?— = TT-s «. (10)
and in the same way :
K = pr-, (11)
T(d n ' -d n ")\og^ fl
In this formula we note that the constant of radiation, h, does
not appear and we are not troubled with the difficulties always
associated with it. The conductivity, k, is given by Kps so that
if our experiment is performed carefully and as a consequence
the quantities, C n and 6 n , accurately known, we have to rely on
the accuracy of the knowledge of the density and specific heat.
Both these are accurately known. This is the reason of the
importance of Angstrom's Method. The student is recommended
to refer to a translation of the original paper in the " Philosophical
Magazine," series 4, vol. 25, p. 130, 1863.
From (11), we observe that each coefficient, C n , gives rise to a
value of K. As the carrying out of the experiment will show, the
coefficient, C x ; gives the most reliable result but the value corre-
238 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
sponding to C 2 should also be worked out. It remains to describe
the experiment and to show how to find the C's and <5's.
The bar, which may be of copper, iron or brass, has one end, A,
inserted in a chamber so arranged that steam and cold water
may be passed in alternately as described above. The total
period is to be measured and care exercised to reproduce the
conditions exactly in each successive period.
Steam from a conveniently large tin flask and water from the
tap will set up the right conditions.
T\me or the f>o»r>r« =c , 8- oc^
CX.\
&, So 3G
Fig. 129,
The steady temperature fluctuations will be the more easily
attained if the bar is sheltered to avoid draughts in its neighbour-
hood and consequent troublesome convection effects. The
temperature at points on the bar should be observed by means
of a thermo-junction placed in a small cavity in the bar. Two
such cavities will be required and should contain mercury, or
it will be sufficient to hold the junction in contact with the bar
at the points concerned. The thermo-junction must be calibrated
in the usual way (see p. 545).
When this is done read the temperature at convenient intervals
by means of the galvanometer in the circuit and obtain as many
readings as possible throughout two complete periods, or more.
Plot a curve for 6 against the time very carefully on squared
paper as illustrated in fig. 129. Suppose that this refers to the
point, x v Repeat the process for x v
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 239
We shall determine the necessary quantities from the graphs.
In order to see how to use the graphs multiply the equation
(6) by sin pt and integrate both sides over a complete period
from any arbitrary time, t=t x to t=t x +T t where, of course T= —
P
We thus have :
0! sin pt dt = Co' / sin pt dt + C/f sin (pt + <5/) sin pt dt.
+ C 2 ' f sin (2pt + <V) sin pt dt + etc. (12)
In doing this it is well to remark that we are integrating the
series on the right-hand side term by term, adding all the in-
tegrated terms and equating to the integrated function on the
left. This process is not always legitimate, but on account of
the properties of the particular series we may apply it in the
present case.
On performing the integration the only term on the right-hand
side which does not vanish is the second, and this has the value :
^TC/cos <V-
Denote the integral on the left by S,.
Then S, = £ TC/ cos d x '.
In the same way if S« denote the value of
^,+ T
B x cos pt dt,
J t x
we find
S^iTC/sin V-
We can determine C x and 6 X from these two equations, for
tan V=|^ .(13)
and C t '» = -^-(S,* + V) (14)
In the same way if we multiply the series by cos 2pt and
sin 2,pt and integrate we find :
X cos 2pt dt = £ TC a ' sin <5 2 ',
and as before we can express C a ' and «5 2 ' in terms of the quantities,
S ac and Sg,, the meaning of the latter being of course
J]
0j sin 2pt dt sm $ TC a ' cos <5 3 '.
Jto. j* .
240 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
This suggests a graphical method of determining the necessary
quantities.
Measure as many ordinates of the curve, fig. 129, as is con-
veniently possible and multiply each by sin pt or sin -=r .
T is the complete period of the periodic heating and t is the
value of the time appropriate to each ordinate.
Plot a new curve with these new quantities as ordinates and
times as abscissae and thus obtain fig. 130,
Cutue sWxjoinQ \V»e uatia\tor> of 0sinSTTP ,jj\tVt
\ke rime -fa* the \x>\r\X X.| T
Fig. 130
Draw any ordinate A X B X for some arbitrary time, t v and
construct C X D X the ordinate at one complete period later.
Measure the area between these ordinates, the curve and time
axis carefully with a planimeter.
This gives S,.
VAPOUR DENSITY AND THERMAL CONDUCTIVITY 241
Go through the similar process to find S c .
Multiply Bx by sin 2pt and cos 2pt to find S 8 , and S 2C . All
this has to be repeated for the second point, x 2 , and we then have
all the data necessary to deduce k from formulae (10) and (11).
C?otoe sWou/mq tW uwriatlons of ■& coe -=^r UJ»m
the Hme at Fhe ^pomf 0C t
Fig. 131
Another way of making the calculation is to make two ex-
periments with different periods, T' and T". If <x n ' and <x„" are
the values of a n in these cases respectively we find from (9) the
result :
K
|t/2 „ /i
'\ „ /2
V 2 «» 2 .
2a « a n * a n " — a,
a n ' and a n " are derived as before, and
16
242 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
It is to be noted that we do not in this case require the values d.
The density and specific heat may be taken from tables or be
measured by any of the usual methods.
In drawing the curves it is a good plan to have two complete
periods shown to give a means of testing the accuracy of the
areas, S, and S c .
Always test the end of the bar remote from the heat supply
to determine if it remains at the air temperature. It will suffice
to place a thermometer close to this end. A longer bar must be
used if the fluctuations continue right to the end.
Angstrom used a square bar of side 2-375 cms., and the length,
I, between the two points, x x and x 2 , was 5 cms.
It has been found, however, that a cylindrical bar of diameter
from 1 to 2 cms., and of length about 60 cms., will suffice,
with such a range of temperature as described above. In the
original experiment Angstrom heated his bar at the central
region and, of course, had similar conditions on both sides
CHAPTER IX
MISCELLANEOUS EXPERIMENTS IN HEAT
Determination of the Radiation Constant. ("Phil. Hag.'*, ser. 6,
1905, p. 270)
By Stefan's Law the total radiation from a black body is propor-
tional to the fourth power of the absolute temperature, or
R = oT*.
It is the object of this experiment to determine a.
Apparatus
The diagram (fig. 132) illustrates the arrangement of apparatus
which consists of a blackened hollow metal hemisphere, B, about
ten inches in diameter, fitted into a wooden box, W, lined with tin.
Fig. 132
This fits on to a table, of which the top, DE, is shown, containing
a small hole at S, which lies at the centre of the hemisphere.
B is heated to a uniform temperature measured by the thermo-
meters, T x and T t , by passing steam through the box above
the hemisphere. The black surface of B is the radiator, and
the heat is received by a small disc of silver placed at S, and
blacked on the upper surface to prevent reflection. It is better
to fit the disc in a vulcanite frame rather than to allow it to
touch the table directly.
From the disc are lead away two wires, one of constantan,
and the other of silver, to a galvanometer and second junction.
343
244 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
S is thus one of the junctions of a thermo-electric couple,
the other is placed in a tube containing oil standing in a calori-
meter, C, containing water or ice.
The junctions to the galvanometer are kept in a canister, A,
packed with cotton wool to prevent any electrical effect due to
difference of temperature at these junctions.
It may be further necessary to include a resistance in the
circuit to keep the deflection on the scale if the galvanometer
is too sensitive.
A rise of temperature due to absorption by the disc is thus
recorded on the galvanometer.
Theory
Let Rj = radiation absorbed by the silver disc per unit area
per second, and R that emitted. Let the temperature of the
radiator be T 1} and of the disc T.
If the whole enclosure, the disc included, had temperature T lf
there would be equilibrium, and the disc would both emit and
absorb R x in unit time. The energy absorbed would arise, of
course, from B. This same energy falls on the disc when at the
lower temperature and is absorbed, but the energy emitted is
now R. Thus the gain of energy per sec.
= (R t - R)A,
where A denotes the area of the disc.
/TV
Let m denote the mass of the disc, s its specific heat, and -57
at
its rate of change of temperature.
Then we have :
ws ¥ = ^f-- A = X (Tl4 " T4) '
or Jms _ dT
AfV-T*) dt*
where J = Joule's equivalent (4*2 X io 7 ergs per calorie).
All the quantities on the right are measured, and hence a is
calculated.
Experimental Details
It is first necessary to ascertain the" relation between the
readings of the galvanometer scale and the difference in tempera-
ture between the two junctions.
In order to make this comparison the disc is surrounded with
cotton wool, and the cold radiator placed above it.
The calorimeter, C, is then heated and the difference in
temperature between it and the disc recorded on a graph against
the readings of the deflection.
MISCELLANEOUS EXPERIMENTS IN HEAT 245
We thus obtain : difference in temperature per scale division =
AB
BC' J
By measuring the temperature of C, we can then deduce from
the galvanometer deflection the temperature of S from this graph.
V
f
u Oivmie*-*? on Scale
Fig. 133
Secondly, we require the rate of rise of the temperature of the
disc when the radiator is put on.
As soon as the temperature in the enclosure, B, has become
steady, the box is placed over S, keeping the latter at the centre,
and the galvanometer is read at equal intervals of time ; these
may conveniently be every 5 or 10 sees.
The table shows a record of observations made every 10 sees.
TIME
SCALE DIV.
TIME
SCALE DIV.
O
227
40
190
10
218
50
l8l
20
208
60
173
30
199
70
165
These results are plotted on a graph.
Scale Divisions
Fig. 134
dT
Draw a tangent to the curve and measure the value of -=- as
at
close to A as possible, since errors soon arise by conduction from
the silver disc.
246 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
m and s are found in the usual way. It will be convenient
to regard these as constants of the apparatus, and to record
them on the occasion of making the apparatus. It is inconvenient
to make determinations at each experiment.
Or s may be measured by Regnault's Method from a piece
of silver identical, with that in the apparatus.
The Rise of Boiling Point of Solutions
The object of this experiment is to determine how the tem-
perature at which a solution boils depends on the concentration,
to determine the rise of the boiling point above that of the
solvent, and to compare experimental results with those given
by the thermo-dynamic formula.
If P denote the osmotic pressure of a solution, i.e. the pressure
due to dissolving one gramme molecule of the substance in one
litre of the solvent, S, the molecular weight of a salt, and s, the
number of grammes in ice. of the liquid, we have :
P _ 2R0
s S
where R is the value of the gas constant per gramme molecule,
and is the temperature of the solution, i.e. for any particular
salt and solvent and for a temperature 0,
P ocs.
All this is true for dilute solutions only.
A calculation based on the second law of thermo-dynamics
shows that the rise of temperature of the boiling point, T, is
given by the formula :
T- £?'
where P is the osmotic pressure, L = latent heat of vaporization
of the solvent, p = its density, and = temperature of boiling of
the solvent.
For purposes of the calculation it may be assumed for the
case of water, p = — at boiling point.
P = 22-3 x 10 6 dynes per gramme molecule per litre at o°C.
If = B.P., the approximate value for P at temperature is :
— - x 22*3 xio 1 dynes.
273 ° y
MISCELLANEOUS EXPERIMENTS IN HEAT 247
e is in absolute units, and the value for any particular
concentration may be deduced by simple proportion.
L, for water at ordinary pressures, may be taken as 537 X 4*2
X 10 7 ergs, in which units it must be expressed for the equation.
Suppose w grammes of solute are dissolved in W grammes of
solvent.
Fig. 135
. W W
The volume of the solvent is — c.c. or
P 1000 p
litres.
If M denote the molecular weight of the solute, we have
... . 1000 pw .
litres, i.e. w — gramme mole-
cules per litre.
w , , • W
— gramme molecules m
M 6 1000 p
248 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Thus the depression according to the theory should be :
1900 pw JP0 ioooze>P0
MW ' L P ~ MWL '
Verify this expression by taking several values of w.
A modern form of apparatus for carrying out this experiment
will be described. It is represented diagrammatically in the
figure (135). A is a glass tube which holds the solvent from
which proceed two tubes, B and C, provided with corks.
The thermometer passes down B, and the bulb is immersed
m the solution or pure solvent in A.
Weighed quantities of solute may be added through C.
Through the cork, H, passes the metal tube of the condenser, F,
which is cooled by passing in water by one of the tubes, T, and
allowing it to flow out by the other.
The small tube, G, carries garnets, which are placed at the
bottom of A, and which together with a short wire through the
base of A, assist the commencement of boiling and tend to prevent
superheating. A stands on a cylinder of asbestos, D, which
rests on a plate, E, of pipeclay, and the liquid is heated by two
burners placed below E.
The thermometer is set and its reading taken when the
pure solvent is boiling, and the difference noted when
the solute is added and the solution boils.
i We have first to weigh the vessel, A, when detached
from F, and not carrying the thermometer. Add a
weighed amount of solute, w, to a convenient quantity
of solvent. After solution, and after taking the boil-
ing point of the solution, F is again removed and
also the thermometer, and A is again weighed.
Since w is known, we can obtain W, the weight
of solvent. Care must be taken not to remove any of
the liquid with removal of the thermometer. The
thermometer bulb must be placed in the solution ; if
it is above it attains the temperature of the steam
coming off from the solution, and this is not at the
temperature of the solution, but at the temperature
of steam appropriate to the prevailing pressure.
The Beckmann Thermometer #
The magnitude of the change of temperature is not
very great, and great care is necessary to measure
B it accurately. The most convenient form of thermo-
meter for this purpose is one of the Beckmann
Fig, 136 type.
MISCELLANEOUS EXPERIMENTS IN HEAT 249
A common form of this thermometer is made of Jena glass,
having an apparent coefficient of expansion for mercury of
magnitude—^—. It is illustrated in fig. 136.
A special feature is the large bulb, B, containing a compara-
tively large quantity of mercury from which a fine capillary
tube extends, bent as shown at A, where the capillary is widened.
The mercury in the bulb being of considerable volume gives
a large additional volume on expansion, and this causes a large
movement in the fine capillary tube when the change of
temperature round the bulb is only slight.
If, however, the quantity of mercury in the bulb were fixed
the range of temperature for which the instrument could be
used would be very small unless the capillary were inconveniently
long.
The thermometer is used to measure a smalt change in
temperature above or below some particular temperature.
Suppose that we require to measure a small change of temperature
above T .
It would be best to arrange so that the mercury extended
from the bulb up to the zero of the scale when at the temperature,
T . A slight increase in temperature would then drive the
mercury along the capillary, which is marked in hundredths of
degrees at intervals of sufficient width to enable an experimenter
to estimate to thousandths.
Suppose now that it is required to read temperatures slightly
above another temperature, T lt which is greater than T . If
we could withdraw sufficient mercury from the bulb it would
be possible to arrange that the mercury extended up to the zero
mark at the temperature, T v and slight increases would drive
the thread along the fine tube as before.
Of course, the expansion per degree rise of temperature in the
second case is not strictly the same as in the first, since the
initial volumes are not equal in the two cases.
In order to see how this effects the observations, let us suppose
that the graduations on the stem are correct at t ° C, i.e. with
mercury filling the bulb and extending to the zero mark, when
the surrounding temperature is t °, a rise of temperature of i° C.
would cause an expansion to the first degree mark above the zero.
Let the volume of mercury at t ° under these conditions be
denoted by V , and let the coefficient of apparent expansion of
the mercury in glass be a.
Then the volume of the capillary per i° is V oc.
Now suppose that mercury is drawn off until at temperature,
/, the mercury fills the bulb and extends to zero. The volume
drawn off is V a(rf — t ).
250 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The remaining mercury had a volume at t ° of
V {i-a(*-* )f,
and on further heating of one degree would expand by an amount
V o^i -ce(* -t )}.
Since the volume of the capillary per i° is V a this expansion
will be registered as {i — <x.{t — t )\ of a degree.
The degrees on the seale are too large for the temperature t.
A correction could be made by dividing the scale readings by
{i — cc(t — t )\, and it would Abe sufficient to add to the
observations an amount <x{t — 1 ) per degree registered on the
scale to obtain the corrected rise.
As has been stated a has the value - — , so that if we suppose
the thermometer correct at 90 C. and we record a rise just
above ioo° C. we should add to each degree an amount
? = -0016 .
6370
This is an appreciable amount since we can record up to
thousandths of a degree.
The correction can be neglected when we are recording to
hundredths of a degree.
The widening of the capillary at A enables the change in the
quantity of mercury in the bulb to be made.
Suppose it is necessary to read a temperature between ioo°
and 101 C. The thermometer is placed into a bath at a tempera-
ture of about 102 C. to 103 C, and so that the mercury extends
from the bulb up to the widened part when at this temperature.
If a quick jerk is made the thread will break at the point,
where the capillary enters the wider part.
When cooled to about 100 C, i.e. when held in steam, the
temperature of which may be determined by observing the
atmospheric pressure, and referring to a book of tables, the top
of the thread should sink to the lower end of the capillary. It
will not lie exactly at zero, and the position occupied by the end
at this known temperature is recorded. It may be necessary to
vary the upper temperature when the thread is continuous before
the end will sink conveniently towards the zero at the temperature
of steam.
When this condition has been attained the small rise of
temperature is readily recorded.
It is not necessary to bring the mercury thread exactly to zero
before reading the small temperature change, and if the correction
is to be applied it may be made by giving the value to t which
corresponds to the temperature for which the apparatus is set,
just as if it had been set at the zero.
MISCELLANEOUS EXPERIMENTS IN HEAT 251
If it is more convenient to observe a diminution of temperature
below a certain point, it is only necessary to bring the top of
the mercury thread at this temperature to a point high up on
the scale. A slight depression will cause the top of the column
to sink, and the interval may be measured as before.
In such accurate temperature measurements it is necessary to
make a correction for the emergent column. This is done by
reference to Grutzmacher's tables taken from the " Zeitschrift
fur Instrumentenkunde " (1896), p. 220.
A reference to this table will show how important such a
correction is.
GRUTZMACHER'S TABLE
VALUE OF ONE
SCALE DIVISION
WITH THE THREAD
CORRESPONDING
TEMPERATURE
IN DEGREES
ALL OUTSIDE AND
VALUE IN DEG.
INTERVAL.
CENT. WITH
AT MEAN TEM-
ENT. PER SCALE
THREAD TOTALLY
PERATURES BELOW
DIVISION.
IMMERSED.
°c.
°c.
°c.
°c.
—35 to —30
•982
•977
to 5
•997
15
•995
45 to 50
I-OII
26
1-015
95 to IOO
I-02I
32
1-032
145 to 150
I'027
138
1-045
195 to 200
1*028
44
1-053
245 to 250
1-024
50
1-055
The second column gives the correction due to the variation
of mercury within the bulb, and it shows that the thermometer
reads correctly at a temperature between 5 and 45 , for in this
region lies the point where the value of one scale division is i° C.
The fourth column gives, in addition, the correction for the
fact that part of the mercury is at a different temperature from
that in the bulb. v
In order to make a correction, choose the part of the table
which describes most nearly the condition of the experiment,
e.g., at a temperature in the neighbourhood of ioo° with a
mean thread temperature about 30 , one degree recorded by
the apparatus is approximately 1-032°.
Two thermometers will be required, one adapted for use in
the neighbourhood of o°C. and the other adapted for use near
ioo° C.
Molecular Weight by Depression of Freezing Point
The thermo-dynamic formula for the depression of freezing
point is
T-f
252 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
where L is the latent heat of fusion of the solvent, p its density,
and is the temperature of fusion.
P is the osmotic pressure.
Let W denote the weight of solvent used, and let w grammes
of solute be added of molecular weight, M.
Let P denote the pressure for one gramme molecule per litre,
i.e. 22-3 xio 6 dynes/sq. cm. By using this value and p = i,
L = 80 x 4*2 X 10' ergs, 6 = 273 Absolute, calculate the
corresponding depression T .
W
In the present case the volume of solvent is litres, and the
r 1000 p
w
number of gramme molecules dissolved is^rv m
The corresponding depression is T 1 , where
looop w Pe __w_ . I000 .Pf .
1 " W M L P ~ WM lou L
It will be the object of the experiment to calculate M.
The vessel, D, with the stirrer, S, is weighed and a quantity
of the solvent introduced, and the vessel again weighed.
This gives W.
The thermometer is placed with its bulb in melting ice, and
the top of the mercury thread adjusted so that it stands near
the upper end of the scale. This temperature is o° C. and small
differences from this point may be read from the thermometer
graduations. These may be corrected by Griitzmacher's Table.
A weighed quantity of the solute is introduced into the solvent
through the side tube, C, and solution brought about by help
of the stirrer.
The vessel is placed in an enclosure, E, and surrounded by a
freezing mixture, F.
The thermometer, B, must be kept in ice until D has cooled
down to o°, and then the transference must be quickly made
to D. Otherwise the thread will rise beyond the scale, and
mercury will flow into A.
During solidification the mercury will stand at a definite mark
if the solvent is pure, and the difference, T 1 , can be measured.
In order to set the thermometer ready for use in the experiment
it must be placed in water cooled to within a degree or two of
the freezing point. The thread of mercury which must be
continuous from B to A is broken by a quick jerk at the top of
A, when the temperature of the water is attained.
In this way the thread will extend nearly to the top of the
scale, when the bulb is at the temperature of melting ice.
MISCELLANEOUS EXPERIMENTS IN HEAT 253
For the temperature at which the solution freezes the fall
below the freezing point of water is obtained by subtracting the
second reading with the bulb in the solution from that with the
bulb in ice. The degrees thus read will require correction on
account of the emergent column.
/
Fig. 137
In order to make the correction, use Grutzmacher's Table,
e.g., if a temperature difference recorded is 1-5° C. with an
average room temperature 15 C, the degree as read on the scale
is really only -995°. Thus the difference is :
i-5 x -995 = 1-492°.
I ° T °
The thermometer reads to , so that we can estimate— —
100 1000
and it is therefore necessary to take account of such an error
as that arising above, viz. of -008° C.
Determination ot the Mechanical Equivalent of Heat by Friction
Cones
The apparatus consists of two vessels of gun-metal of the
shape of truncated cones, one fitting within another. The
inner cone contains water, and is helcj in position by a measured
254 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
couple applied by means of a weight hanging over a pulley and
connected to a disc attached to the cone.
The outer cone is rotated by means of a cord to a large wheel
rotated by hand.
Fig. 138
It is possible by steady turning of the handle to cause the
weight to hang almost steady, so that a steady couple is applied
to the inner cone by friction, and its amount is measured by
multiplying the force and the diameter of the wheel to which
the cord is attached.
A counting device is fixed to the outer cone so that the number
of rotations- is recorded.
If n is the number of rotations, w the weight in dynes applied
to the disc, and a its arm, the work done is
2tfnwa ergs.
The rise in temperature of the water is measured by means
of a thermometer, so that if M represents the total water equiva-
lent of the cone and contained water, and if the rise in temperature
is T, the heat developed is MT calories.
If no heat is lost by radiation and conduction, the mechanical
equivalent, J, or the number of ergs necessary to produce one
calorie is given by :
2ntW>a
J =
MT
A common form of apparatus in use in laboratories is shown
in the figure, it was designed by Dr. G. F. C. Searle.
An improved design is described in the " Phil. Mag.," Sept.
1920, by H. P. Waran.
In order to prevent radiation losses it is a good plan to cause
a rise in temperature of not more than io° C, and to cool the
water about 5 below the room temperature initially. The two
MISCELLANEOUS EXPERIMENTS IN HEAT 255
cones are mounted in a metal case lined with cork to diminish
conduction losses.
The heat is conducted through the walls of the inner cone to,
the water, and it is necessary to stir the water during the experi-
ment to assist the flow of heat throughout its extent. This
may be done by means of the thermometer.
One of the features of Waran's improvement is the automatic
stirring of the liquid which consists of an oil with good
conductivity and known specific heat.
The Determination of Joule's Equivalent of Heat by Callendar and
Barnes's Electrical Method
The principle of the experiment is to supply electrical energy
to a wire surrounded by water, and to measure the heat developed
in the water by noting the rise of temperature.
Fig. 139
If we express the electrical energy in ergs or joules We can
thus deduce that required to generate one calorie.
Inside the glass tube, H (fig. 139), is fixed a helix of manganin
wire of resistance about 9 ohms. The ends of the wire are
joined to the terminals, C and C 1 .
Water from the tank, B, enters H by a side inlet tube at
one end and after flowing round the wire comes out from a
similar outlet tube at the other. Thermometers, which enter
the ends of H as shown, penetrate the inflowing and outflowing
water, and serve to record the temperatures of the water before
and after it has received heat from the wire.
The vertical tubes, AA, allow air bubbles which may flow in
with the water, to escape. The level of B is variable for the
tank is movable up and down its stand.
The makers of the apparatus recommend that the rate of flow
256 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
should lie between 55 c.c. and 65 c.c. per min., or about 1 c.c.
per sec. B should be adjusted to produce this rate.
The tank consists of an inner and outer chamber.
Through the tube, E, water enters the outer chamber from the
supply, and comes out through M to enter H.
The water flows over from the outer chamber to the inner
and escapes to the drain by means of F. Thus water is supplied
at a constant pressure to H, and the flow is made steady. The
outflowing water may be collected in a measuring glass,
or, of course, it may be collected in a weighed beaker and the
mass per second flowing out deduced.
The apparatus gives best results for a current of approximately
2 amperes, so that a voltage of from 20 to 25 should be applied.
To obtain a steady source ten or twelve accumulators should be
used and connected through an adjustable resistance to C andC 1 .
In this case, with the rate of flow indicated, a difference of
temperature of the ingoing and outgoing water of about 8° C.
is maintained.
If possible the mean temperature of the water should be that
of the room, to avoid errors due to radiation.
When it is not possible to arrange this exactly, it is necessary
to apply a correction. Let the temperature of the room round
the apparatus be t , and the mean temperature of the water at
entrance and exit be t.
Then the number of calories lost per second in radiation is
m{t — Q x -05,
where m is the outflow per second. Or we may make the
correction by adding -05(t — t ) to the difference of temperature
recorded by T and T 1 . This correction has been obtained
experimentally by the designer.
Measure the resistance of the spiral by means of a Post Office
Box.
Connect the resistance, R (fig. 140), in series with the supply
of current and the ammeter, A, which has a range up to 3 amperes.
If a voltmeter is available there is no need to measure the
resistance of the helix, for the difference of potential between
C and C 1 may be measured by connecting it through the key, K,
in parallel as in the diagram.
Adjust the current to about 2 amperes, and if necessary make
slight variations from this to adjust the temperatures so that
radiation losses are made as small as possible.
Let m denote the rate of outflow per second, C the current in
amperes, E the voltage supplied between C and C 1 , and T the
difference in temperature of the two thermometers, corrected if
necessary according to the rule given above.
MISCELLANEOUS EXPERIMENTS IN HEAT 257
The supply of electrical energy is EC joules per second, and the
heat developed is ml calories per second.
Thus, J, the heat equivalent, is given by:
JwT = EC,
EC
or J =^ joules"
EC
-f_fl.g-ftfla.il ».tuuuuuuub>-
++■-+- GLh
Fig. 140
If the resistance of the wire is measured, instead of EC we
must write C 2 R. The accuracy to be expected is between
one-half and one per cent.
The following is an example of an experiment carried out in
the laboratory, and indicates the order of the quantities :
Temperature of room, 17-3° C.
Temperature of water at inlet, i6*8° C.
Temperature of water at exit, 25-35° C.
Mean temperature of water, 2i'07° C.
Temperature difference of water hi the two cases, 8-55° C.
Radiation correction = -05 x (21*07 ~~ I 7*3) = ' x 9° C.
Corrected temperature difference, 8-74° C.
Volume of water flowing out in 2 mins., 115 c.c.
m = — - gms. per sec.
120 ° r
Current, 2 amperes. Resistance of wire, 8-92 ohms.
,\ J = -J 2— = 4-260 joules.
— x 874
120 '^
*7
CHAPTER X
REFLECTION
The Sextant
The instrument consists of a graduated arc, SS (fig. 141), with
two radial arms, A and C.
A third arm, B, moves about an axis through one of its ends
at right angles to the plane of SS. It is fitted with a clamp
and tangent screw, so that it can be accurately adjusted, and
carries a vernier at its end which moves over the scale of SS.
Fig. 141
A plane mirror, M lf is attached to B, lying with its surface
in the direction of B, and in a plane normal to that of the scale.
The axis about which N B turns lies in the surface of this mirror,
which is called the index glass.
The second mirror, M 2 , fixed to the arm, C, is called the horizon
glass, and its plane must also be perpendicular to the scale.
It consists of a plate fif glass only one-half of which is silvered.
At T, on the arm, A, is fixed a telescope with its axis parallel
to the plane of SS and passing through the centre of M 2 .
Suppose the movable arm is turned so that the mirrors are
parallel, and T directed towards a distant object, on which it
is focussed.
258
REFLECTION 259
*
Only one image will be seen of the object, for the light from
M x and M 2 is brought into the telescope in the same direction,
M 2 T. The rays by the two reflections may not he along the
same line as those seen directly through M 2 , but since they are
parallel there is only one image formed by the telescope (see
fig. 142).
^vfi
Fig. 142
If the index glass be rotated through an angle, A, by turning
the arm, B, then the rays reflected by it into the telescope no
longer come from the* same object as that which supplies the
ray, QM 2 . Two superposed images are seen in the telescope,
and the angle between PM^ and QM 2 is 2A, since on turning a
reflector through any angle a beam of light is rotated through
double this angle. Thus every degree on the scale, SS, corres-
ponds to a difference of direction of two degrees. The scale is
marked to give directly the angle between the rays, PMj and
QM t , i.e. to measure the angle subtended at the instrument by
two distant objects.
In order to measure this angle one of the objects is observed
directly through M 2 , while the other is made to produce a super-
posed image on that of the first by rotating M x into the proper
position. The angle through which the index glass is turned
from the parallel position is then one-half of the angle subtended.
Before making any measurement the instrument must be tested
to see that it satisfies the following conditions :
(1) The plane of the index glass must be normal to the plane
of the scale,
(2) The axis of the telescope must be parallel to the plane of
the scale, and
(3) The index and horizon glasses should be parallel, and at
the same time the vernier should read zero.
It will not be necessary, as a rule, to adjust for the first of
these, but in order to see that the instrument is satisfactory in
this respect look at the image of the scale in M^ Since M x
passes through the centre of this scale the latter and its image
will appear to intersect at the edge of the mirror and, if the-
adjustment is satisfactory, to lie in the same plane.
Both M x and M a are attached to frames which can be turned
26o ADVANCED PRACTICAL PHYSICS FOR STUDENTS
through small angles by means of screws. If necessary Mj may
be adjusted until the test is satisfied.
The second condition is tested by observing two objects and
causing them to coincide at the centre of the field of view. The
axis of the telescope is a line joining the centre of the object
glass to the centre of the eyepiece, or to the centre of the field
of view. Perpendicular to this axis lies one of the cross-wires.
Tilt the instrument until the images lie near the edge of the
field, and note if they still coincide. Then tilt it so that they
lie near the opposite edge. If coincidence persists the axis is
correctly adjusted. If this is not the case the telescope can be
adjusted by means of the screws.
Observe an object through the telescope and make its image
appear in the field of view by reflection in M x . If it is possible
to cause them to coincide, the two mirrors are parallel, and on
account of the first adjustment this will mean that the third
condition is partly satisfied. By means of the screw attached
to M 2 the two images can be made to coincide if they do not do
so at first.
When these conditions are satisfactorily arranged it will
probably happen that the pointer does not read zero when a
distant object is viewed. To correct for this it is only necessary
to note the zero error and apply it in all the observations.
Coloured glasses are provided for diminishing the brightness of
any object such as the sun. These can be made to intercept
the light immediately before falling on the mirrors.
Experiment i
Place two candles at as great a distance as is convenient, and
measure the angle they subtend at the instrument.
Also find the angle by measuring the distance to each candle
and their distance apart.
Let the candles be C a and C 2 , and let S denote the sextant.
The distances to be measured are a, b, and c, and if
s = \{a + b + c)
REFLECTION 261
tan J-J(* -»>('-*)■
2 M S ( S _ a )
Check the values of obtained by the two means.
Experiment 2 \
Let a trough of mercury or a carefully levelled mirror be
placed so that the image of a lamp can be seen directly, and
also by reflection, and measure the angle subtended at the
sextant by the object and its image by holding the plane of the
instrument vertically, at as large a distance from the lamp as
possible.
The elevation of the lamp is half this angle.
Li„-""
Fig. 144
The diagram (fig. 144), illustrates that the angle, CBD, is
measured since the instrument is of necessity above the surface,
AE.
Actually, we require the angle, LAL 1 , but since we use a
distant lamp, the two angles do not appreciably differ.
Thus, the elevation may be measured by half the angle CBD.
Measure the horizontal distance between L and A, and deduce
the height of L above the floor.
Check the result by actual measurement.
Measurement of the Angles of Crystals by Wollaston's Goniometer
The goniometer is a convenient instrument for measuring
accurately the angles between the faces of small crystals which
are too small to be examined by means of a spectrometer.
It consists of a circular circle, S, which may be rotated by
the large milled head, B, and its position read off by means
of a fixed vernier, V.
The crystal is fixed by soft wax to a plate, P, carried by an
adjustable support, D, which may be rotated by the smaller
head, A.
262 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The edge of the crystal formed by the two faces between which
it is desired to measure the angle is adjusted so that it lies parallel
to the axis of the circle.
This adjustment is first made approximately by eye. In
order to make the adjustment accurately, view the upper corner
of a distant window in both faces. On rotation the images will
move in a vertical plane if the edge is parallel to the axis. This
may be tested by noting if each image moves in a direction
parallel to the edge of the window as seen directly by the eye.
Place the instrument so that the axis is parallel to a tall,
distant window, and turn the screw-head, B, until the graduated
circle comes against the stop.
Fig. 145
The eye is placed close to the crystal so that an image can be
seen by reflection in one of the two faces.
The axis is then rotated by the smaller screw-head, A, until
the top of the window, as seen by reflection, appears to coincide
with the bottom, as seen directly.
When the adjustment has been made the angular position of
the circle is noted.
By means of B, the circle is now turned until the top of the
window, as seen by reflection in the other face, coincides with
the bottom, as seen directly. The second face now occupies
a position parallel to the first, and if is the angle between
them, the circle has been rotated through its supplement. This
angle (180 — 0) is read off from the circle and deduced.
It should be noted that the crystal must lie as close to the
axis of the goniometer as possible, for the motion of the crystal
from the first position with reflection in one face to the second
position, with reflection in the adjacent face, consists both of
translation and rotation, unless the crystal is on the axis. The
amount of translation may be sufficient to cause an error in the
angular measurement.
If the window is a long way away the error is only small.
REFLECTION 263
The Determination of the Radii of Curvature of Spherical Mirrors
(A) Concave Mirrors
The most convenient method of determining the radius of
curvature of a concave mirror is to place a pin point in front of
it and to locate the position in which the image of the pin appears
to coincide with the pin itself. The method of parallax is
employed to ascertain when coincidence is attained.
" The rays from the point of the pin falling on the mirror are
reflected back from the surface along their original paths and
must therefore strike it normally ; consequently, the pin point
lies at the centre of curvature of the surface.
Another method consists in locating a series of pairs of con-
jugate points for the surface and using the formula :
z -+i = * (I)
v u r v '
A pin is set up as object and another pin adjusted until the
image of the first coincides with it. We can then measure a
pair of values, u and v.
Several pairs of values are obtained, and the above formula
then gives r. Take the average of four or six observations.
(B) Convex Mirrors
Method 1
In the case of a convex mirror the image is virtual, and it is
not convenient to locate it by a pin placed in a particular position
since the image lies behind the mirror.
Q
C
Fig 146
A pin, P, is set up in front of the mirror, CC, and in between
them is placed a plane mirror, M, so that the image of P in both
can be observed. The mirror is adjusted until the two images
coincide (fig. 146.)
By the simple law of reflection in the plane mirror, M, we
know that the image of P in M, say, Q, lies at the same distance
from M as P does, but on the other side of it.
We can thus calculate the distance, AQ, for
AQ = MQ - MA = MP - MA.
264 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Then v = — AQ, adopting the usual sign convention, viz.
directions measured from A towards the object are positive,
and in the opposite direction they are negative.
Thus, using the formula (i) we can again deduce r by measuring
AP and AM.
Several pairs of values should be obtained, and they should
give the same value of r.
Method 2
Another method is to set up a pin and form a real image of
it by a convex lens. The image is located by placing a second
pin, Q, so that there is no parallax between it and the image.
Then place the convex surface between the lens and second pin,
and move the surface until an image is formed coincident with
the first pin, P. The rays after passing through the lens are
directed to the point, Q, but strike the surface normally, and are
therefore reflected back along their path. The radius of curvature
of the convex surface is MQ (see fig. 147).
Fig 147
Method 3. By means of a telescope, metre rule, and small
millimetre scale
The diagrams (figs. 148, 149, 150) show the arrangement of
apparatus. S is a small scale placed horizontally in contact with
the surface of a convex mirror, along a line dividing it into two
equal parts.
E
e
Jhr-
3-
Fig. 148
The scale, RR 1 , is mounted at some convenient distance away,
usually about 60 or 70 cms., and below its middle point is fixed a
telescope, T, focussed on the image of the scale, RR 1 , in the mirror.
The apparent length of the image is read off by means of SS 1 ,
which will be sufficiently well focussed to make this possible.
The distance from the centre of the mirror, P, to the middle
REFLECTION
265
of RR 1 , is measured, say, d, and from these two measurements,
together with the length of RR 1 , 2I, it is possible to calculate
the radius of curvature of the surface of the mirror.
Let rays from R and R 1 strike the mirror at L and L 1 , and be
s
± T
J
Fig. 149
reflected down the telescope at O. Then LL 1 will denote the
extent of the image, and the point, B, at which these two lines
meet OP will be conjugate to O for reflection in the mirror.
We may say that a point source at O will have a point image
at B, so that if PB = x, we have :
— J -uJ — _ 2
% a r
r denoting the numerical value of the radius of the mirror.
R Fig,. 150
But if we take LL 1 as approximately straight, since the image
is of small dimensions, and denoting LL 1 by 2c
I _ d + x
C X
Hence
so that
x d\c J'
or
2 _
r
r =
1
I — 2C
The result may be verified by means of a spherometer.
266 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The Focal Lines formed by a Coneave Mirror
When light diverges from a point and falls on a mirror, it is
supposed in the elementary theory that after reflection the rays
pass through a single point or appear to proceed from a point.
This is approximately true if the dimensions of the mirror are
small compared with the distance from the source. A closer
approximation to the truth is that the rays after reflection pass
through two lines or appear to come from two lines, situated
in parallel planes and perpendicular to one another.
These are the focal lines and it will be shown how to calculate
their positions theoretically, while it will be the object of experi-
ment to verify the result obtained.
We shall take the case of light falling obliquely on a concave
mirror.
If the point source lies on the axis of the mirror we have
symmetry about this axis, and the two lines degenerate into a
point or circle through which the rays pass.
Fig. 151
In the diagram, MM 1 denotes the concave mirror, and C its
centre of curvature. The complete circle of which the section,
MM 1 , forms a part is drawn for convenience.
P denotes the position of the point source of light, and the
diameter is drawn through P.
The extreme rays, PM and PM 1 , are drawn and the reflected
rays, MB, M X A, are drawn intersecting at F 1? and cutting off
from the diameter the strip, AB.
REFLECTION 267
The mirror is a part of a sphere, so that rays falling on the
mirror from P, whether in the plane of the figure or not will
pass through AB. AB is thus a focal line; and is denoted by F 2 .
If we imagine the figure to be rotated about the diameter, the
rays, M X A and MB, will still intersect at a point but now out
of the plane of the diagram. For a small rotation the point of
intersection would be on a line through F x normal to the figure.
Thus, a second focal line is through F x perpendicular to the
plane of the figure.
Let the angle of incidence at the point M 1 of the mirror be i,
and let PD = p.
Denote the distances of the focal lines from D by p ± and p 2 ,
i.e. DF X = Pl , and DF 2 = p 2 .
Let the radius of curvature, CD, be R.
Let the mirror subtend an angle, r, at C, a at P, and p at F v
We shall regard its dimensions as small in comparison with R,
P and pi, so that these three angles will be small and may be
measured by drawing perpendiculars from M on to the corres-
ponding lines, and dividing this perpendicular by the distance
from the point concerned,
MM 1 cos i
e.g. a = ,
p
since the normal from M, on the line, PM 1 , makes an angle
very nearly equal to i with MM 1 , and MM 1 is small and is
regarded as straight.
In the same way
_ MM 1 cos i
P ~~ "*
Pi
(for cos i = cos (i + di) when 6i is small),
,., MM 1
while r = — — •
R
From the triangles OMFi and POM 1
^OPM 1 + ^OM x P = ziOFiM + z.OMF x
.'. a + 2* = 2(» + di) + p,
or. 2<5*' = a — 0;
and in the same way from triangles, CQM and PQM 1 ,
M ' = a — r,
;. 2f = a + 0,
2MM 1
or — — - = cos 1. MM
ix.
f+i),
\P Pi/
(- 1 + ~)
\P Pi/
i.e. I - -\ 1 cos * = ^
268 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
This is the formula concerning the position of the first focal line.
In order to determine p 2 , we note that :
A PM X A = A PM J C + A CM^A,
i.e. | • pp 2 sin 2,i = \ pR sin i + | Rp 2 sin *',
which may be rewritten as :
2 cos i i i ^
R p p 2
So long as i remains constant, p x and p 2 vary with p just as
u and w vary together in the formula :
- + -=-•
v u f
When * is made zero we obtain the usual formula,
- +— = * ,
p Pi r'
for both Pi and p 2 , so that the two lines coincide.
To find p x and p 2 experimentally, use a small hole in a screen,
with a lamp behind as a source of light.
Find R first by adjusting the mirror so that an image is formed
on the screen at the side of the hole as in the experiment on
concave mirrors, page 263. The distance from screen to mirror
will then be R.
Allow the light to fall on the mirror at angles of about 20 ,
30 , and 40 , and find the positions of the focal lines by means
of a sheet or card or white paper held in a clamp.
Measure the distances from the mirror to these lines, thus
obtaining p x and p 2 .
In order to find i, it is convenient to mount M on a stand
carrying a pointer moving over a scale of degrees.
Read off the mark against the pointer when the image of the
hole is thrown back near the object, and turn from this position
to any required incidence.
If no scale of this kind is provided, measure the distances,
PF X and PF 2 , say, a and b.
We then know three sides of each of the triangles, PMFi and
PMB, and can determine i from the usual trigonometrical formula
for the tangent of half the angle of a triangle.
Compare the values calculated in this way with those deduced
from the formula derived from the above theoretical considera-
tions.
Searle's Methods of Determining Optical Constants
Accurate methods for the measurement of radii of curvature
of polished surfaces, of focal lengths, and for the localization of
the cardinal points, have been described by Dr. G. F. C. Searle.
REFLECTION
269
For the original accounts the reader is referred to the " Philoso-
phical Magazine," Feb., 1911, pp. 218-224, or to the "Proceedings
of the Optical Convention, 1912," pp. 161-172.
The Determination of the Curvature of Spherical Surfaces
For this purpose a table is mounted on a tripod stand (fig. 152),
two of the feet of which carry screws for levelling. The table is
horizontal, and can rotate truly about a vertical axis. It carries
a millimetre scale on the top which can be clamped in any position,
and a carriage bearing the surface slides along it. The figure
shows a lens system in the place where the carriage slides.
Fig. 152
It is essential that, as the carriage slides along the scale, the
centre of curvature of the surface should move along a line which
intersects the axis of rotation of the table.
In the figure the scale is shown, and also the wooden slider
which acts as the carriage. On the slider is a metal mount
which may be screwed to the carriage. This mount carries a
horizontal spindle, each end of which is turned to a conical point.
One of these ends is provided with a screw thread, so that it
will fit a brass plate carrying three screws.
The screws fit into a second brass plate to which the lens or
mirror to be examined is fixed by a little wax. The arrangement
thus provides a convenient means of adjusting the surface.
The edge of the carrier is smooth and straight, so that it can
slide along the scale, and an index mark serves to record the
position.
In preparing the apparatus for use the spindle is first set
accurately parallel to the edge of the carrier.
The tip of a pin is held just in contact with one of the conical
270 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
points of the spindle, from which the brass plate is removed.
The carrier is taken from the table top and replaced so that the
other conical point lies near to the pin.
If it is possible to bring this point and the pin into contact,
the spindle is parallel to the edge of the board and scale.
The spindle is rotated until this adjustment is possible, and
the mount then firmly clamped to the carrier by means of the
screw. It is also necessary that the axis of the spindle should
intersect the axis about which the table turns.
The scale is adjusted and the spindle moved along it until one
point lies as nearly on the axis as can be judged by eye.
A microscope is then brought up and focussed on the point,
with the scale lying normally to the axis of the microscope,
The table top is turned through 180 , so that if the axis of the
table passes through A (fig. 153), and P denotes the first position
of the spindle point, the second position will lie at P', where
PM = M'P'. The slider has to be moved through the distance,
P'P," in order to bring the point once more on to the cross-wires.
If this distance is recorded and the slides moved back half this
amount, the point will be at M', the foot of the perpendicular
from the axis to the line P'P".
X-
M 1
— 1 —
X
p
u
iM
.M u
Fig. 153
The microscope is then traversed so that it is focussed on the
point. Its axis is thus directed along MA. Turn the table
through 90 , so that M' moves to M" ; and move the scale along
a direction perpendicular to itself until the point once more
comes on the microscope cross-wire. The point then lies on the
axis of rotation of the table, and the axis of the spindle will
intersect that of the table as it slides along the scale.
Mount the surface to be examined on the brass plate, and
attach it to the spindle. If the surface is one of the faces of a
lens, the side not under examination should be blacked by
covering it with vaseline and lamp black to absorb rays striking it.
Set up an object in some convenient position and view its
image in the surface. If on rotating the spindle about its hori-
zontal axis the image does not move, the centre of curvature
lies on this axis, and the preliminary adjustments are complete.
Since the image remains stationary the rotation of the surface
serves to bring up fresh parts of the sphere exactly into the
place occupied by the part moved away, i.e. the centre lies on
the axis of rotation.
REFLECTION 271
The object may conveniently be a fine brightly illuminated
line drawn on a piece of ground glass.
Set up this object again so that rays fall nearly normally on
the surface, and examine the image formed by reflection in it
by means of a microscope. By moving the carriage a position
can be found in which a rotation of the table to and fro produces
no displacement of the image. This occurs when the centre of
curvature of the surface lies on the axis of revolution of the
table, for rotation then has the effect of replacing One part of
the surface by another, and reflection occurs as before.
The position of the index mark is noted.
If the lens is moved until the vertical axis of the table is a
tangent to the surface, a rotation to and fro will not displace
the centre of the lens. Place in the centre a few grains of lyco-
podium powder, and focus the microscope on one of these.
The lens is moved and the focussing repeated until the movement
ceases. The index reading is again noted. The difference of
the two gives the radius of curvature.
Preliminary observations may be made by eye until the motion
appears to cease in the two cases, and only the final exact adjust-
ments need be made by the aid of the microscope.
CHAPTER XI
REFRACTION
Introductory Remarks
Rays of light in passing from one medium to another usually
undergo a deviation from their course in the first medium.
On the wave theory this is accounted for by the fact that the
light travels with different velocities in the two media. The
refractive index with respect to two media is defined to be the
velocity of light in the first divided by the velocity in the second,
and we denote this by x yL 2 .
When light travels from air to another medium, say glass or
water, we shall write a \L g or n«, unless there is no doubt that we
are dealing with air and some other stated medium when we may
write simply \x.
We have by definition,
I
and i^s = ~'
3^1
In particular
■"" = ^T
a result which will shortly be found useful.
In many experiments in this and following chapters it will be
necessary to furnish a bright source of monochromatic light.
The most convenient way to provide such a source is to use a
Mecca burner, which consists of a Bunsen burner rather larger
than the ordinary type of burner provided with a wide end over
which is stretched a gauze with a wide mesh.
If a small bead of soda glass is placed on this gauze and the
Bunsen made to roar as much as possible a quite satisfactory
yellow flame will be produced.
It is a great advantage that there is no crackling in the flame
as in the case of the use of common salt, when small pieces of hot
salt are thrown about falling on the bench and on the slits of
spectrometers. In the case of the latter serious damage to their
shape may result.
The slit may be illuminated directly, or better still, an image
of the brightest part of the flame may be thrown on to it by a
short focus convex lens.
272
REFRACTION
273
The student will do well to pay attention to the small point of
illumination of the slit. It is important always, but assumes
greater importance in the case of experiments on interference
and diffraction which will be described in the next chapter.
The difficulty of discovering Newton's rings, interference fringes,
and diffraction bands is almost always due to a lack of care to
obtain the best possible illumination of the slit or whatever may
be used as a source of light.
Determination of the Refractive Index of a Plate by its Apparent
Thickness
The apparatus necessary is a good travelling microscope and a
plane cover glass.
Set the microscope with its axis vertical and focus it on the
metal platform.
Note the reading on the vertical scale.
Insert the cover glass over the point on which the instrument
was focussed, again focus on the metal and observe the scale
reading.
Usually the metal surface, though dark, is easy to observe, but
if desired a thin sheet of white paper may be placed over it and
the surface of the paper used instead of the metal surface.
Raise the microscope until the upper surface of the cover glass
is sharply focussed. There will usually be specks of dust on the
surface to assist this setting of the microscope, but if any difficulty
arises place a small drop of ink on the surface and focus the
extreme edge of the drop. The drop need be no larger than that
made by a sharply pointed pen.
Note the scale reading when this third adjustment is made.
If we describe the scale readings by (1), (2), and (3) respectively ;
the difference between (1) and (3) gives the actual thickness of
the glass and that between (2) and (3) the apparent tfrickness.
Consider a point, P (fig. 154), situated at the lower surface of
*8
274 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the glass and from which rays of light originate. Only those
making small angles with the normal to the surface will enter
an eye of microscope placed above P.
If one of these rays, PO^ 1 , makes an angle, *, with the normal
in the air and an angle, r, in the glass,
sin*" OO 1 0*P O x P
H =
so that \l = 7^7 = , ., . f
sm r O 1 ? 1 OO 1 ~~ Q 1 ? 1 '
For small values of * and r we may write :
O 1 ? 1 = OP 1 , O x P = OP,
OP _ thickness of glass
OP 1 — apparent thickness'
for all rays inclined at such small angles appear to come from P 1 ,
so that OP 1 is the apparent thickness.
Determination of the Refractive Index of Liquids by Total Reflection
When a ray of light passes from a medium of refractive index,
Pi, to another of refractive index, \t 2 , with an angle of incidence, *',
and of refraction, r, we have :
\x 2 sin *
ll * 2 — nj ~~ sin r
We may not always find a corresponding value of r for a given i
unless — is less than unity.
In the case when — > i, the value — sin % must not exceed
{*2 t*2
unity. In the limiting case when
sm * = — ,
the corresponding value of r is 90 , and i then measures the
critical angle. For values of i greater than this critical value
the surface acts as a perfect reflector.
If i is slowly increased a value is finally attained when the
refracted ray suddenly disappears.
In this case if the second medium is air so that \l 2 = 1 we have :
1
— = sm *.
Hi
This formula may be used to determine y. f by mounting a small
rectangular trough, CD (fig. 155), with sides of plane glass on
the table of a spectrometer, so that a parallel beam of light may
be passed through it from a collimator, GH, and received in the
REFRACTION
275
telescope, AB. The light is suddenly cut off when the air cell,
EF, consisting of two plates of glass mounted parallel to one
another and cemented together with a thin air space between,
is turned so that the light falls on the air at the critical angle.
8
*
It will be noted that the critical angle is that for air and glass,
but the apparatus is used to determine the refractive index of
the liquid in the trough.
Let the ray, ABCDE, be incident from the water on the glass
and be totally reflected at the glass-air surface at the critical
angle. Then if FBG and FTOG 1 be the normals to the glass
at B and D, we have :
— = sin z BCH,
where CH is the normal at C and aV . g denotes the refractive
index from air to glass.
Also
V)V-g
^aV-a _ sin ABF sin ABF
av. w ~ sin CBG ~~ sin BCH'
*'• olXw ~"sinABF '"( J )
We have thus to measure the angle ABF, and from it we deduce
the value a [i Wf the refractive index from air to water.
As a source of light use a Bunsen flame containing sodium*
and illuminate the slit of the collimator, which must be adjusted
for parallel light (p. 279). Focus the telescope on the slit and
turn EF (fig. 155), until the light just appears. EF is attached
276 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
to a pointer which moves over a scale of degrees. Note the
position of the pointer. Turn EF from this position into another
where the image again disappears.
Let AA 1 denote the axis of the telescope and EF the first
position. The second will be E X F X if L AOE = L AOE 1 . The
Fig. 157
second position is reached either by turning through the angle
EOE* or EOF*, and ^EOF* = * - iLEOE*.
When the apparatus is used there is no uncertainty concerning
which angle is measured.
The angle we require is that between the ray and normal to
EF or ET 1 , i.e. the angle between OA and the normal to EF or
E*F*. This angle is half the angle between the two normals, and
this is the same as half the angle E x OF.
Thus we have to note the angle through which EF turns and
take one half of it to find the angle ABF of formula (1).
To Find the Refractive Index of a Liquid, using a Lens and a Plane
Mirror
In the experiment a plano-concave lens is formed of the liquid
under examination (fig. 158), and its focal length found experi-
mentally. The refractive index and radii of curvature of the
two surfaces enter into the formula for the focal length, so that
it is possible to deduce the index, ja, from a determination of the
focal length, /, and the radius of the curved surface of the lens.
The liquid lens is made by placing a drop of the liquid on a plane
mirror and laying a convex lens of from 10 to 20 cms. focal
length on the drop. The liquid is squeezed into the space between
the mirror and lens and we have a combination of two lenses
one of glass and the other of liquid, giving a combined focal
length of, say, F.
REFRACTION 277
If the convex lens has a focal length, /', then :
1=1+1:.
We may therefore deduce / from a knowledge of F and/'.
To determine /', place the lens horizontally on the mirror and
adjust a pin, held in a stand above the lens, until the inverted
1
Fig. 158
image and object appear together and there is no parallax
between them. The distance from lens to object gives /' (com-
pare p. 263). Now place the liquid and lens on the mirror and
again find the position of coincidence of object and image.
This gives F, so that we now have /.
For the purpose of substituting in the above formula, F and/'
must be given their appropriate negative signs, and / will turn
out to be positive.
If r is the radius of curvative of the curved liquid surface,
and n the refractive index of the medium,
~f-~T~ < a >
since the second surface has zero curvature.
r can be found by measuring the radius of the surface of the
lens in contact with the liquid by means of a spherometer, or
by any of the methods described in the last chapter, so that n
can be calculated from the equation (2).
If it is preferred, the determination of r may be avoided.
If water be used and its index regarded as known and having
the value, 1-33, we may make a water lens as above, and calculate
its focal length, /*.
But 7=T->
or ix = 1 4-
•33/"
/ '
where p is the refractive index of a liquid other than water.
278 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
A convenient liquid to use for the experiment is aniline.
Care must be taken to prevent it from getting at the back of the
mirror since it dissolves the varnish protecting the silvering.
Determination of the Refractive Index of a Lens by Boys' Method
This method of finding the refractive index of a lens consists
in measuring its focal length and the radii of curvature of both
surfaces. The value of {* may then be determined from the
equation :
T- <*-<}-!>
where / denotes the focal length of the lens and r and s the radii
of curvature of its surfaces, with the usual convention regarding
the signs of the quantities in the formula. In a convex lens let
v x and r 2 denote the radii of curvature numerically and F the
numerical value of the focal length. Then :
**-o-*(Z + k)
Determine F by any of the methods described below, p. 293.
In order to find r lt set up a pin with its point on a level with the
centre of the lens. Two images will be seen by reflection in the
faces of the lens, one erect from the front surface which acts as a
convex mirror and one inverted by the concave back surface.
It is the latter which is required for the experiment. Move the
pin until its inverted image is coincident with it, as judged by
the method of parallax. When this is the case the rays must
strike the back surface normally and be returned along their
incident course.
If an eye be placed on the side of the lens remote from P
(fig. 159), it will receive the transmitted part of the ray, PB, and
will see the image of P in the direction, CBQ.
C __
A
Fig. 159
Q will thus be the image of P in the lens, and OP and OQ
are conjugate distances. Thus, writing u = OP and v = OQ, we
have :
1 i_ _ 1 , .
OQ OP F * [3)
REFRACTION 279
This enables us to determine OQ from OP and F.
Q is the centre of curvature of the left-hand surface of the lens
since QAB is normal to this surface. Thus OQ gives the value
of the radius, say, r x .
Turn the lens round and repeat the process to obtain r 2 , the
radius of curvature of the other face of the lens.
Let OP = d x in the first case, and let d 2 denote the corresponding
distance in the second. Then from (3) :
I
*x
1
~1 x
1
~~F
1
1
d 2
1
~F
I
F
= (*-
-1)
U7 +
1
d z
■1)
Hence
so that we may calculate ^ from the experimental determination
of F, d v and d z .
It is sometimes difficult to see the image by reflection at the
back surface of the lens, but by holding it over the surface of
mercury or floating it in the mercury, the image may be made
to stand out and be easily located.
The Spectrometer
The spectrometer consists essentially of a telescope and colli-
mator. The latter is a system of lenses mounted in a telescopic
tube with an adjustable slit at one end, and it serves the purpose
of rendering rays from the illuminated slit parallel on emergence.
Both are mounted on a rigid stand, the collimator being fixed,
and the telescope rigidly attached to an arm which rotates about
the centre of the stand. Both are mounted horizontally with
their axes in the same plane.
P
I
. I _
a .' t
A 1 B
P
Fig. 160
The centre of the instrument is occupied by a table provided
with screws for levelling. Underneath the table is a metal scale
of degrees on which can be read off the positions of the telescope
and of the table. In the diagram, T denotes the telescope, AB
the table, and C is the collimator. PP denotes the axis of the
instrument and the table and telescope rotate round it. The
telescope is fitted with a Ramsden eyepiece carrying cross-wires.
28o ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Before using the spectrometer for any experiment certain
adjustments have to be made in addition to those made in the
construction of the apparatus by the maker.
In the first place, the eyepiece of the telescope is adjusted so
that the cross-wires are distinctly focussed. The telescope
should then be taken to an open window and focussed on a distant
object such as a distant telegraph post, care being taken that
there is no' parallax between the image and the cross-wires.
When this is done the telescope must not be readjusted again
during an experiment or it will be necessary to repeat this process.
It may happen that a second observer whose sight differs from
that of the first is unable to focus the cross-wires easily. He may
readjust the eyepiece provided that he does not alter any other
part of the telescope, for then he is not causing it to be out of
focus for parallel rays. He merely gives himself convenience
in focussing easily and leaves the cross-wires and image without
parallax.
The telescope is now turned towards the collimator, the slit
being made vertical and illuminated with monochromatic light,
and the collimator is adjusted until a distinct image of the slit
falls on the cross-wires.
The apparatus is now adjusted so that parallel rays pass from
collimator to telescope.
2*^
6
"ill -
F
\
" -T-
1 i -*—^
Fig. i6i
When a prism is used in the spectrometer it is necessary to
adjust it so that its refracting edge is vertical.
The screws on the table enable this to be done. They are shown
at D, E, and F, placed at the corners of an equilateral triangle,
and the table is usually ruled with lines as shown (fig. 161), to
assist in setting the prism with one face normal to one of the
sides of the triangle, for example EF.
The table should be made as nearly horizontal as possible by
the use of a spirit-level, and the prism then placed with one face
perpendicular to EF.'
REFRACTION 281
Now suppose light from the collimator falls on this face and is
reflected into the telescope. If this face is vertical the slit will
now appear to lie in the same part of the field of view of the
telescope as when it is seen directly.
The three screws should be adjusted to restore the image to
its direct position if necessary.
Fig. 162
* Let the light be reflected into the telescope by the other face
bounding the refracting edge. If further adjustment is necessary
it must be done by the screw, D, for this will not disturb the
previous adjustment, since it does not turn the face perpendicular
to EF out of its vertical plane.
The two faces are now vertical and the instrument is adjusted.
It is sometimes necessary to arrange one face of a prism so
that it lies normal to the collimator or telescope.
This may be done by turning the telescope from the position
in which the slit is seen directly without the prism, through a
right angle, so that the axes of the telescope and collimator are
perpendicular to each other.
The prism is now placed on the table of the instrument and the
table rotated until, by reflection in the face concerned, an image
of the slit is thrown on the cross-wires. The face now lies at 45
to the axes of the collimator and telescope and a further rotation
of 45° will bring it either perpendicular to the collimator or
telescope.
In making measurements with the spectrometer the slit should,
as a rule, be narrow, and the cross-wire should lie accurately
down the centre of the image of the slit before the position of
the telescope or table is noted on the metal scale.
282 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Tangent screws help in the accurate final setting of the telescope
and table, and verniers on this scale serve to measure the angles
to an accuracy of one minute of arc.
A modern form of the apparatus is illustrated in fig. 162.
The Refractive Index of a Prism by the Method of Minimum Deviation
When a prism with a refracting angle, A, causes a minimum
deviation, D, in light passing through it the refractive index is
measured by the formula :
sin £ (A + D)
sin I A
Thus the determination of jx consists in measuring A and D.
There are two methods in common use for measuring an angle
of a prism.
One of these consists in allowing the light from a narrow slit to
fall partly on one of the faces of the prism bounding the angle and
partly on the other. An image of the slit can be seen in the
telescope when the latter lies on either side of the prism.
Thus in the case illustrated (fig. 163), the telescope will receive
rays from the direction AO on one side and from AK on the other.
The telescope has to be turned through the angle KAO in turning
from one direction to the other.
F D L
Fig. 163
Since DA and AO make equal angles with AC, we have :
^- EAC = z. CAO,
and similarly, /_ EAB = ^ BAK.
Thus £_ OAK = 2 ^ BAC.
To measure A we need only focus the image of the slit on the
cross-wire in the two positions and halve the angle through which
it is rotated.
REFRACTION 283
Suppose AB is the face which reflects the light into the telescope
along AK. If the prism is rotated until the face, AC, now lies
parallel to AB the rays will once more be reflected in the direction,
AK, but now by AC. Some of these will enter the telescope if
there is a sufficiently broad pencil of them. But the prism has
been rotated so that AC moves round through the angle CAB 1 .
Thus we measure, by means of the table, the angle (180 - A),
and A can be deduced.
Measure the angle of the prism by both methods.
It now remains to find D.
Set the prism so that with A as refracting angle, light is refracted
through it and received in T.
It will be found that as the prism is rotated the telescope has
to be rotated to keep the image of the slit in the field of view.
Rotate the prism until T is as close to the position directly
opposite the collimator as possible, with the slit in the field of
view. When this is the case the angle between the telescope and
direct position is as small as possible.
Note the position of T and then remove the prism and observe
the slit directly. Again note T's position, and hence find D.
It is, of course, necessary to adjust the spectrometer and edge
of prism m the way described in the section on the adjustments
of the spectrometer.
The final movements of the telescope or table must be made
carefully with the tangent screws.
The Dispersive Power of a Prism
The dispersive power of a medium is measured by :
t*B — v.*
w =-
v- — 1
t* B and n R are the refractive indices for blue and red rays and u
has the value, £ (^ + ^). J *
The refractive indices may be found by the method of minimum
deviation.
As a source of blue and red rays a discharge tube containing
hydrogen may be used.
The^ tube should be held vertically, and the slit illuminated
by it directly, or an image of a bright part of the tube thrown on
to it by means of a short focus lens.
284 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Three well-marked lines can readily be seen, one in the red,
a second of blue-green colour, and a third in the violet. Use the
first and third of these — they are known as the C and H r lines
respectively, while the second is the F line.
The Refractive Index of a Liquid by Total Internal Reflection within
a Glass Prism
A glass prism with one face unpolished is mounted on the table
of a spectrometer. The table is levelled and the edge between
polished surfaces set normal to its plane (see p. 281). The
angle between these faces is measured in the usual way by allow-
ing light from a wide collimator slit to fall on both faces and by
measuring the angle between the two reflected beams.
Light from a sodium flame is allowed to fall on the unpolished
surface, or an image of the flame is thrown on it by means of a
lens to cover the whole matt surface.
It is first necessary to find the refractive index of the glass
prism and then to coat one of the bright faces with a thin layer
of the liquid. Glycerine is a very convenient substance with
which to carry out the experiment.
The theory of both parts of the experiment is the same.
Theory
ABC denotes the prism, of which AB is the unpolished side.
This side acts as a collection of point sources of light of which S
denotes one. Rays from it strike AC and are reflected and re-
fracted there. Those like SD falling at an angle of incidence
less than the critical angle are partially reflected and partially
refracted so that the ray, EF, issuing from BC is less intense than
the incident ray, SD.
A
Fig. 164
Rays falling at angles greater than the critical angle suffer no
refraction at AC, so that the emergent ray from BC is scarcely
less bright than the incident ray. In any case there will be a
REFRACTION 2 85
marked difference between the former and latter group. The
ray, SG, is drawn for critical incidence so that the direction of
the emergent ray, HK, stands between those for the faint and
bright rays. *
We get a similar state of things for any other point in AB, and
for each point the rays striking AC at critical incidence give rise
to rays parallel to HK on emergence.
Such a group of rays is brought to a focus in the focal plane of
a telescope placed to receive them.
Similarly, a group of rays, parallel to EF, will correspond
to all rays falling on AC parallel to SD, and these will fall on the
telescope in a direction different from that of HK, and will form
a line in the focal plane not coincident with the former. This is
true for all the directions of rays from BC. We shall thus have a
multitude of parallel lines in the field of view divided into two
groups of different intensity by the critical direction, HK. The
effect will produce a field sharply divided into bright and dark
halves by the direction, HK, making a with the normal to BC.
If the telescope is turned to face AC the field will be similarly
illuminated on account of the reflections that have taken place
on BC. The issuing critical rays will make an angle, a, with the
normal to AC.
If the cross-wires of the telescope are set on the dividing line
of the field when it is directed towards BC, and again when
towards CA, we can deduce the value of a by observing the angle
through which the telescope has been turned, since the telescope
is rotated through
180 — C + 2a = (say).
Thus 6 + C ~ l8o °
a =
2
where it is assumed that the telescope is moved from side, BC,
towards CA in a counter-clockwise direction as seen in the
diagram.
From C and a we can calculate \l from the formula given below.
Suppose that AC is coated with a substance of refractive index,
Pi, and that the index of the prism is p.
We have on referring to the diagram :
v-i = v- sin c,
p + c = C, where = z. MHG
since the points C, G, M, H, are concyclic,
sin a = {x sin 0,
.*. l*i = V- sin (C — p) = {i sin C (i — sin 2 p)* — cos C sin a
= sin C (n 2 — sin 2 a)* — cos C sin a.
286 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
When (*! = i, i.e. when there is no layer on face, AC, we have :
„ /r + sin a cos C V ■ • 2
^2 = 1 -T- ) + sm 2 a>
V sinC J
This is the formula from which n may be calculated.
Fig. 165 shows the state of affairs when the critical angle is
large. This will be the case when the media on either side of
AC are nearly of the same optical density.
A
Fig. 165
The formula above will still apply if the value of a has its sign
reversed.
The above method of measuring a can be used in finding {*,
but when one side is coated it would be necessary to clean that
side and coat the other when the telescope is turned. This would
make it difficult to keep the prism fixed and would be
inconvenient.
It is therefore best to use an eyepiece, such as the Gauss eye-
piece with cross-wires that can be illuminated.
Fig. 166
The light from these may then be reflected in the face opposite
the telescope and the image made to coincide with the object.
When this is the case the telescope stands perpendicularly to the
face and the angle between this direction and that in which the
division of the field is viewed measures a.
REFRACTION 287
After finding jx for the glass, place a few drops of the liquid on
one of the polished faces and press over it a thin plate. This
ensures that the face is covered with liquid,
It is better for the sake of definition of the two halves of the
field to allow light to fall at grazing incidence on the prism
surface, say, AC. Then the rays entering the prism make angles
less than c with the normal so that the field is now only half
illuminated and the edge corresponds to the direction, HK.
AB should be kept dark by covering with a sheet of dark paper.
If the light is incident externally on the liquid film, it must enter
by the edge, AD (fig. 166), any ray such as P would not reach AC
at grazing incidence.
Calibration of a Spectroscope
The spectroscope already described had only one prism, but
it is an advantage sometimes to use two mounted together on
the spectrometer table. The experiment may be carried out
with one only, but a wider spectrum is obtained with two.
S .L 0. i %\ ^rj>k *
Fig. 167
If the prisms, P 2 and P 2 , are used to produce a series of spectral
lines due to some source, S lt the telescope, T lf will have a definite
position on its scale corresponding to each particular wave length.
To calibrate a telescope is to find the values of the wave lengths
corresponding to the different parts of the scale. If a curve is
drawn whose co-ordinates are the wave lengths and corresponding
scale readings, such a curve is a 'calibration curve,' and may be
used to determine any wave length from the division of the
telescope scale at which it is seen.
288 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
First make the usual adjustments for parallel light. Place
one prism, P x , in the position corresponding to minimum deviation
and then put in P 2 also in the position of minimum deviation.
Examine the spectrum of a Bunsen flame containing sodium
light. Fix the telescope so that the sodium line lies on the cross-
wire and note the position on the metal scale of the spectrometer.
A graduated scale is provided consisting of close rulings cut on
an opaque screen so that the lines are transparent.
It is fitted at the end of a second collimator, C 2 , which is
adjusted to direct rays down the telescope after reflection on
one face of P 2 .
The telescope is already set for parallel rays so that by
adjusting C 2 so that an image of the scale, ss, appears at the
cross-wires, it is ensured that parallel rays emerge from C 2 .
The position of the sodium line is taken as a point of reference
and scale divisions are read so many to the right or left. If the
scale, ss, is moved accidentally no error is then caused and the
calibration curve will still be of use.
In a good spectroscope the sodium lines will be separated and
will appear as two very close together. Use one of these as the
reference line.
Measure the scale positions of a series of lines of known wave
length extending over the visible spectrum.
Use only sharply denned lines.
The table gives the wave lengths of lines which may be pro-
duced conveniently in the laboratory.
In order to produce the lines from the metals use a spark
between poles made of these metals connecting them to opposite
terminals of a Leyden jar charged by an induction coil.
The salts are volatilized in a Bunsen flame as with sodium.
Examine the inner cone of a Bunsen flame ; it is due to carbon
monoxide, and contains several series of lines. The spectra of
light from discharge tubes containing various gases should be
examined as well.
Lines from a Neon lamp should be examined also. The
field will be observed to contain many lines, the yellow line of
wave length 5853 units is a bright line and it is a good exercise
to determine from the curve the wave lengths of the other lines,
afterwards comparing with standard tables.
Plot the curve on a large scale on squared paper.
The Auto-Collimating Spectrometer
In this instrument the telescope acts also as the collimator.
The apparatus is almost identical with the ordinary spectrometer,
REFRACTION
TABLE OF WAVE LENGTHS
289
SALT OR METAL
Lithium Chloride
or any Salt of
Lithium
Any Salt of
Sodium
Salt of Potas-
sium
Strontium
Chloride
DESCRIPTION OF LINE
Thallium
Chloride
Helium
Hydrogen
Red
Double Yellow
Red
Extreme Violet
Blue
(Not to be confused with
the bands)
Green
WAVE LENGTH
TENTH METRES)
Red (C)
Blue-Green (F)
Violet (H r )
6708
/5890
I5896
7668
4044
4047
4607
5351
6678
5876
4471
3889
6533
4861
4340
except that there is no separate collimator and the telescope is
modified.
On looking into the eyepiece of the instrument, the field is
seen to be divided into two parts, the lower half is dark and the
upper bright, but crossed by a pin which extends from the upper
edge downwards across about half of the bright part of the field.
Near the eyepiece end the tube of the telescope is provided
with an opening by means of which a slit just within may be
illuminated.
Just below the dividing line of the two halves of the field of
view a small right-angled prism is placed which deviates the light
from the slit down the centre of the telescope tube. Its position
is denoted by the dotted rectangle, P, in fig. 168,
19
290 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If a polished surface such as the face of a prism is placed in
front of the object glass and set at right angles to the axis of the
telescope we may, by focussing, make the rays leaving the object
glass parallel, so that they will be returned by reflection at the
face of the prism and come to a focus in the focal plane of the
telescope.
*S
ET?
Focus the eyepiece carefully or^ the pin. Illuminate the slit
by means of a small electric lamp — a four-volt lamp, supplied
by two accumulators, is convenient for the purpose. The slit
will be observed to be of the shape of an inverted T. The image
must be brought so that the horizontal bar lies along the edge
bounding the two halves of the field, when the end of the vertical
bar will coincide with the point of the pin. In order to obtain
this result the face of the exterior prism must be accurately
normal to the emergent rays. The table is provided with three
screws in order to level to the prism, and the process described
on p. 280 must be followed.
The positions of the slit and pin are such that when the image
of the slit lies at the same distance from the object glass as the
pin, the latter is at the principal focus.
The prism is slightly tilted to throw the image of the slit on
to the upper half of the field of view.
To Find the Refractive Index of a Prism by means of the Auto-colli-
mating Spectrometer
Place the prism on the table and adjust the faces bounding the
refracting angle as described above so that the light reflected
normally by both throws an image of the slit into the field of
view just below the pin.
When this is so the faces are vertical and consequently so also
is the edge of the prism.
REFRACTION 291
Use monochromatic light by placing a sheet of yellow glass
between the bulb of the electric lamp and the slit.
Observe the position of the table when the face, AB, reflects
the light normally. Rotate the table until the light falling on
AB and refracted there strikes AC normally and is returned along
Fig. 169
its path. The table has been turned so that the face, AB, has
turned from a direction normal to the rays from the telescope
into the position at which refraction takes place, i.e. it has turned
through the angle of incidence, i. The angle of refraction in this
case is the same as the angle, A, and is so marked in the diagram.
This angle may be measured by setting the face, AB, normal
to the rays and reading off the position of the table. Then by
rotating the table until the rays strike AC normally we turn the
table through an angle (180 — A) .
We may therefore calculate ja, since we know the angle of
incidence and refraction in a particular case and
_ sin i _ sin i
sin r sin A"
The Pulfrich Refractometer
This apparatus is shown in figure 170. It is designed to
measure refractive indices of solids and liquids to an accuracy
of about r~ per cent.
The principal part of the apparatus consists of a prism having
two plane polished faces at right angles to each other. One of
these is horizontal and the other vertical. On the horizontal
face is placed the substance whose refractive index is required.
If this substance is a solid it must have two faces cut perpendicu-
larly to one another, both of which are cut accurately plane, so
that one may rest on the horizontal surface of the prism and the
other stand vertically. Optical contact is brought about by
placing a few drops of a liquid on the horizontal surface of the
2Q2 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
prism, which has a refractiye index higher than that of the solid
to be experimented upon, and standing the solid on it. The
makers recommend monobromonaphthalene for this purpose.
In the case of a liquid, it is contained in a glass cell cemented to
the prism.
Fig. 170
Light is directed into the liquid in a direction almost parallel
to the horizontal surface so that light entering the prism makes the
critical angle, c, with the normal.
Let it emerge from the prism at an angle, i. (Cf. fig. 171.)
Suppose that the refractive index of the substance to be examined
is n and of the material of the prism, ii -
Then
sin c = — »
sin *
sin (| - c)
REFRACTION
293
i.e.
cose =
sin*
sin 8 c -f cos 2 c = 1 = ^- +
sin 2 i
(V_JV
t* — V(Jt 2 — sin 2 *'
The substance must have a refractive index less than that of
the material of the prism if it is to be examined by this method.
The apparatus measures the angle, *. For this purpose a
telescope is attached to a circular scale and the rays are received
by it. Since no rays within the prism make an angle greater
Fig. 171
than c with' the normal on entrance to the prism, the angle,
\i ~~ c ) ' measures the minimum angle at which rays strike the
vertical face. Corresponding to this, i measures the minimum
inclination of the emergent rays to the normal. Thus in the
telescope the rays emerging in this direction bound the field of view.
The apparatus is arranged so that the rays are deflected down
the telescope, and when its cross-wires lie on the dark edge of the
field the scale reads off the angle, *', to an accuracy of one minute of
arc. no, of course, is an instrument constant and has the value 174.
In addition a microscope screw is divided so that the value of
the refractive index may be measured for different wave-lengths
to a still higher degree of accuracy.
The prism and specimen are surrounded by a metal water
jacket and thermometers are provided for reading the temperature.
A table is supplied with the instrument giving the value of t*
corresponding to different values of *.
Determination of the Focal Lengths of Thin Lenses by means of Pins
Convex Lens. Method, 1
Support the lens vertically and place a vertical pin behind it
so that its point lies on a level with the centre of the lens and on
the axis of the lens.
294 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The adjustments are rendered easier by placing a sheet of
white paper behind the pin.
Set up another pin so that it coincides with the image. When
this is the case the image and second pin will not appear to move
relatively to each other when the eye is moved horizontally from
side to side.
Arrange the lens and first pin so that the image is not at the
same distance from the lens as the object. Then without moving
either pin displace the lens until a second position is found at
which the pins occupy conjugate points.
Fig. 172
Let the lens have been moved a distance, d cms., while the
distance between the pins is D cms.
In the figure we have evidently :
F^i = u = — — ,
and C^Pa = v = — ^— (numerically).
Thus from the equation :
1 _ 1 1
v u ~~ f
we have : / — — — ^n — '
since in the formula we have to write :
D +d
v = — — .
2
Determine several different values of D and d, and calculate the
value of /from each ; the result should give a constant.
Method 2
Put up a plane mirror immediately behind the lens and parallel
to it. Then place a pin in front of the lens as in the first case
and adjust it to make it coincide with the image formed by
refraction in the lens and reflection in the mirror. The distance
from the pin to the lens is equal to the focal length of the lens,
REFRACTION
295
for the rays are reflected back along their path and must therefore
strike the mirror normally. They leave the lens las parallel rays,
and must therefore originate from the principal focus.
Fig. 173
Method 3
Set up a plane mirror, P^ 1 , at some distance from the lens,
and a pin on the other side of the lens. The pin should be
mounted so that its centre is on a level with the centre of the lens
and should be adjusted until the position of the image of the j>in
is made to coincide with the pin. The way this is brought
about may be seen from the diagram. The image formed by the
lens evidently lies on the surface of the mirror.
B
Fig. 174
If the distances be measured we have the positions of a pair of
conjugate points.
Make several observations for different distances and calculate/.
In making experiments with a convex lens it is useful to know
the focal length approximately before making an accurate
determination of it. Sometimes time is wasted in trying to locate
a real image when the object is so placed that a virtual one is
formed. It should be noted that if the object is at a distance
from a lens, which is less than the focal length, the image is virtual.
Focus a distant object — a lamp or window, if not too close,
is suitable — and measure the distance from the lens to a well-
defined image thrown on to a sheet of paper. Unless the lens
has a very long focal length this distance will be approximately/.
Concave Lens
Method 1
Put up the lens and place a convex lens of known focal length
in contact with it so that their axes coincide.
296 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Note if the combination will form a real image of a distant
object, and measure approximately the distance from the lens to
image so formed. This will indicate whether the combination
acts as a convex lens. If it does not the convex lens is too weak,
and a stronger must be chosen.
Determine the focal length, F, of the combination by any of
the above methods, and let / denote that of the convex lens, f 1
that of the concave.
Calculate f 1 from the formula :
Fig. 175
Method 2
Set up a concave mirror behind the lens and a pin in front.
An inverted image of the pin will be seen on looking through the
lens, and it may be made to coincide with the pin by suitably
adjusting the mirror.
The diagram illustrates that this is brought about by the normal
incidence of the rays emerging from the lens on the mirror.
The virtual image of P is thus at P 1 , the centre of curvature
of the mirror.
Remove the lens after noting its position and the distance of P
from it, and again adjust the pin until it coincides with its image
formed by the mirror. We thus locate P 1 and can measure v,
the distance from the position of the lens.
On substitution in the formula :
1 __ 1 _ 1
v u ~J
we calculate /.
Repeat this for several different cases.
The Focal Length of a Lens Combination
In the case of a thin lens, in which no account of the separation
of the two surfaces is taken, there are two points on either side
of it called principal focal points at equal distances from the
lens if it is situated in a medium which is uniform all round it.
REFRACTION 297
These two points are distinguished by calling bne the first and
the other the second focal points. The former is the position
of an object corresponding to an image at infinity while the
second is the position of an image corresponding to an object
at infinity.
A plane perpendicular to the axis at the position of the lens is
called the principal plane and if / denote the focal distance, we
have the relation :
v u f*
connecting the distances of the conjugate points from the principal
plane.
If the principal plane and focal points are known it is a matter
of simple geometrical construction to determine the image of
any object.
When the lens is not thin, or when it is necessary to deal with
a system of lenses, Gauss has shown that the formula is the same
but that in this case there are two principal planes, the first and
second, separated by a finite distance.
The distance of the first principal plane to the first principal
focus is called the first focal length, and the distance to the second
principal focus from the second principal plane is the second
focal length.
If the lens system is situated in a medium the same on both
sides the two focal lengths are the same and we shall denote
either by /.
When the object is at a distance,- u, from the first principal
plane and the image is at a distance, v, from the second, the formula
is still :
111
v u ~~~ f*
A very important property of the principal planes is that if
a ray is incident on the system towards a point on the first
principal plane, the emergent ray is directed from a point in the
second principal plane which lies on the same side of the axis
and at the same distance from it as the first.
It is now possible by a geometrical construction to determine
the position of an image corresponding to any object. This is
illustrated in the diagram (fig. 176).
The ray, BI^, parallel to the axis must pass on emergence
through the second focus, F a , and it appears to emerge from L 2 ,
where PiL x = P 2 L 2 .
Pj and P 2 denote the principal points and PiL^ and P 2 L 2 are
the principal planes.
2 g8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Again the ray through F 2 emerges parallel to the axis from M 2 >
where P 1 M 1 = P 2 M 2 .
We thus find B 1 and can draw in the image, ATO 1 .
If the media on either side are different, F 1 F 1 is not equal
to P 2 F 2 .
Two other important points for the lens system are the nodal
points. They are defined to be points on the axis such that a
ray directed to the first nodal point emerges as a parallel ray
directed from the second.
]
B
h
^
V **
\F<, A 1
A
F^
Mi
^X!
^
B 1
Fig. 176
When the medium is the same on both sides of the system the
first and second nodal points coincide with the first and second
principal points.
We have, therefore, the line, BP^ parallel to PaB 1 , and the
magnification, m, is
A^ 1 = A X P 2 = v
AP X
AB
u
just as in the case of a thin lens.
A direct determination of / requires the location of a pair of
conjugate points, but we do not know the position of the principal
points so that it is not possible to find v and u for substitution
in the formula :
v u f
We may, however, determine / by measuring the magnification
of an object by the lens for two different positions.
From this formula we have :
— = 1
Thus if u x and m x denote the distance of an object from the
first principal plane and the corresponding magnification, we have :
m x f
REFRACTION
299
and for a second position :
1
Wo
/. / =
u.
I I
m x m z
Thus the actual lengths, u, are not required for the calculation,
but only the distance between the two positions of the object,
which is given by the difference between u x and u 2 .
If several values of — are observed and positions corresponding
to u noted on the optical bench, a graph may be plotted of —
against u.
Fig. 177
The relation is linear, and if two points, P t and P 2 , be chosen
on the line as far apart as is convenient :
P 2 N 2 - P X N X
NoN, <
It is convenient to use a transparent glass scale as object and
to illuminate it by monochromatic light. A similar glass scale
is used as a screen so that the apparent length of a certain number
of divisions of the first scale seen in the image can be read off at
once on the second scale.
The screen is adjusted as usual by noting when there is no
parallax between the image and scale.
For accurate work it is necessary to use a low-power microscope
to measure the image.
Another method which is instructive is to adjust the positions
to obtain a magnification of magnitudes 1 and 2.
300 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
When the object and image are equal in size their distances
from the corresponding nodal points are both 2/.
This may be verified from a diagram or from the formula.
Now keep the screen fixed and move the object and lens until
the magnification is 2.
It may be verified that in this case the object is — / from the
first nodal point, and the image is 3/ from the second. The lens
system has thus been moved through a distance, /, and this may
be measured directly.
Suppose the diagram (fig. 176), denotes the case for a magnifi-
cation unity.
In this case AP X and A X P 2 are both equal to 2/. We know the
value of /, so that since A is a definite known point we can locate
Pj and similarly P 2 is known from the point, A 1 .
The lens system is sometimes encased in a tube, and in that
case the distances from P x and P 2 to the ends of the tube should
be recorded.
If the lens is merely a thick lens of glass the distances of the
principal points from the faces of the lens should be recorded.
Determination of the Principal Planes of a Thick Lens
Let DAEB denote a thick lens and let C and C 1 be the centres
of curvature of the faces, DBE and DAE respectively.
There exists a point, O, such that all the rays which pass
through it emerge from the lens parallel to their original direction.
This point lies on the axis, CC 1 , and is called the centre of the
lens.
The rays passing through O after their first refraction, were
directed originally towards a point, P. Thus P and O are
conjugate points with respect to the first surface.
REFRACTION 301
The rays after the second refraction appear to come from P 1 ,
so that P 1 and O are conjugate points for the second surface.
These points, O, P and P 1 , are fixed points for the lens and P and
P 1 are the principal points.
If the distance of an object is measured from P and denoted
by u, while the image is measured from P 1 and denoted by v,
we have the relation :
1 _£ _ 1
v u ~ f'
as in the last experiment.
The quantities, v and u, are to be given positive signs if measure-
ment is made towards the source of light, and negative if it is
made in the opposite direction.
Parallel light incident on the lens on the right is brought to a
focus at F 1 where PT 1 =/.
Light originating at F is parallel on emergence where PF = — /.
Let a source of light be located at a point, S, where FS = x,
x is to be regarded as a number of cms., and is not given a sign.
The focal distance is likewise measured by the number F. The
lens, when convex, has a negative focal length, i.e. in the formula
/ = - F.
The image will be formed at I, say at distance, y, from F 1 .
Then u = (F + x), and v = - (F + y).
Thus
^ »
F +y F + x F
or X y = F 2 .
The disposition of the conjugate points shown in the diagram
is such as occurs in the case of a convex lens. The object and
image both lie within the focal distance or without it. In this
case x is taken positive when measured from F towards the right,
and y is positive when measured from F 1 towards the left. This
relation enables us to find F, for although it is not easy to locate
P and P 1 , which are required if the first formula is used, it is easy
to locate F and F 1 , and hence to measure the distance of object
and image from them.
To locate F, set up a plane mirror on the left of the lens and
put up a pin in such a position that its image coincides with it.
The rays emerging from the lens must in this case be parallel
and strike the mirror normally so that object and image lie at F.
Repeat this for the point, F 1 , leaving the former pin in position.
Leave pins at F and F 1 and do not move the lens. Place a
third pin at a point such as S, farther from the lens than F, and
locate its image, I.
302 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The distances, x and y, are now determined and we can calcu-
late F.
Hence since P and P 1 lie at distance, F, from F and F 1 respec-
tively, the principal points are located.
Repeat this for several values of x and y.
"Piano-Convex Lens
In this case any ray, R, striking the lens at P in the figure,
where P is the pohit of intersection of the axis and curved surface,
is not deviated but gives rise to the parallel emergent ray, R 1 ,
for it is just as if refraction took place in a slab of glass bounded
by the face, A, and the tangent at P. "R 1 appears to Come from
P l , so that P and P 1 are the principal points. These may be
located as above.
Fig. 179
We may, also, calculate the distance AP 1 .
For AP 1 is the apparent thickness of the widest portion of the
lens,
AP _ t
•'• '""AP 1 d
This gives d if \l and t are known.
It forms a useful exercise to determine P and P 1 by the method
described above, and then to deduce y. from the observations.
If PP 1 = 5,
t
* = *— *•
Determination of the Principal Points by Rotation of the Lens
If the lens is mounted on a stand which can be rotated about
a vertical axis we may use the property of the nodal points to
locate P and P'.
It is convenient to place the lens in a holder which can slide
along a scale fixed to the rotating stand. The arrangement of
the apparatus is illustrated in fig. 180.
A mirror is fixed on one side of the lens normally to the axis
and does not rotate with the lens.
"t
REFRACTION 303
On the other side a pin is set up and moved until its image,
by two refractions through the lens and a normal reflection at
the mirror, coincides with the object.
The pin then lies at a principal focus.
The lens is moved along the scale and the pin adjusted so that
image and object coincide until a position is found, when slight
rotations of the stand fail to cause displacement between the
pin and its image. When this is the case the rotation takes place
about a vertical axis through the nodal point nearer the pin.
The figure explains this, for if P' denote the other nodal point
in the symmetrical position and P" the position of this nodal
point in a slightly displaced position a ray from F falling on P,
first passes along PP'M and is reflected back along its path after
incidence on the mirror at M.
Fig. 180
When the lens is rotated into the position indicated by the
dotted lines the ray, FP, is refracted and emerges from the lens
parallel to its original direction, and directed from P', i.e. it takes
the course, P'M', and is reflected back to P', emerging once more
along PF. The image of the point therefore keeps the same
position.
The position of the axis thus fixes the nodal point, and the
axis is usually clearly indicated on the stand. This finds the
point, P, and, by turning the lens, P' may be found in the same way.
The Measurement of the Focal Length of an Optical System by means
of a Goniometer (Searle's Method). (Proc. Optical " Con-
vention," 1912, p. 165.)
A simple form of goniometer devised by Dr. G. F. C. Searle,
in conjunction with Messrs. W. G. Pye and Co., provides a*very
instructive method of determining the focal length of an optical
system by means of the properties of the nodal points.
The goniometer is illustrated in fig. 181, and consists of a
wooden base provided at one end, with a spherical pivot, con-
sisting of a ball of phosphor-bronze, and carrying at the other a
scale marked in millimetres.
A movable arm rotating about the pivot carries an achro-
matic lens of focal length about 35 cms., a vertical adjustable
frame, across which a vertical wire is tightly stretched, and a fine
wire passing across an opening which serves as a scale index.
304 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The ball is adjusted at a distance of 40 cms. from the scale,
which can be read to an accuracy of one-tenth of a millimetre,
so that a rotation of the arm through one-seventieth of a degree
is able to be measured. This is about —5- radian.
4000
Fig. 181
In fig. 182, let HiB and H^B 1 denote the principal planes of
an optical system, the points, H^ and H 2 , denoting the first and
second principal points.
In practice it usually happens that the medium on each side
of the system is the same, viz. air. As the general case presents
no additional difficulties and can be examined also by this method,
we shall write jx x for the refractive index of the medium on the
right of the system and i* 3 that of the medium on the left. Then,
if U is measured from H t and V from H 2 we have the relation,
V U F
between the object distance, U, and the image distance, V.
F is a constant for the system, and if the value of U is F x for
emergent parallel light, i.e. Fj denotes the first focal distance,
Fi = - i*iF,
and similarly the second focal distance, F 2 , is given by :
F 2 = [x 3 F.
The usual sign convention is adopted in applying these formulae,
directions from the lens towards the source are positive and those
in the opposite direction are negative. In fig. 182, F x and F 2
denote the first and second focal points respectively.
Nj and N 2 are the nodal points, and with the same medium
on either side of the system they coincide respectively with H^
and H 2 .
In the general case the first nodal point lies at the distance
(F x + F 2 ) from K v and N 2 is at the same distance from H 2 .
REFRACTION
305
Thus
and similarly :
NJF, = H.Fj - H 1 N 1
= F 1 -(F 1 + F 2 ) = -F 2 ,
N 2 F 2 = H 2 F 2 + H 2 N 2 = F v
As drawn in the diagram the value, F t = F^, is positive,
since it is measured from H lf while F 2 is negative.
Any ray incident on the system in a direction passing through
N x gives rise to an emergent ray in a parallel direction passing
through N 2 .
Fig. 1 8a
Thus the ray, ANj, gives rise to CD, and CD if produced would
pass through N 2 .
If the point, A, lies in the focal plane a divergent pencil from
it gives rise to a parallel beam.
Two rays, AB and ANj, are drawn and the corresponding ravs
are BT. and CI).
Since CD is parallel to AN X the angle between the axis and the
emergent beam is equal to angle, ANjF v
Thus if we measure the distance, AF t , denoting it by d, we can
find the length, N^, provided that we measure the wangle
between the axis and the parallel beam.
If this is denoted by a we have :
NjFi = dcot a;
or when a is small, N^ = - = F 2 (numerically).
The goniometer is a very convenient apparatus for carrying
out this measurement.
It is first adjusted so that the vertical wire lies in the focal
plane of the lens. This may be done by adjusting the wire so
that the image of a distant object seen through the lens falls on
the wire; or a plane mirror may be placed near the lens on the
side opposite to that on which the wire lies and the latter adjusted
3<>6 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
until there is no parallax between the wire itself and its iihage.
The frame is then clamped.
The optical system is now brought up so that its axis lies
collinear with that of the goniometer lens when this lens lies so
that the horizontal wire index is central.
A glass scale is placed in the focal plane of the optical system
and may be represented by AF r A denotes one of the marks
on the scale. The scale may be adjusted in the focal plane by
looking through the goniometer lens and the system and by
placing it so that its image is seen coincident with the vertical
goniometer wire. The method of parallax is used to obtain an
accurate adjustment. When this is made as closely as possible
by means of the unaided eye a lens should be used as a magnifying
glass to make it still more exact.
Note the reading on the glass scale when the apparatus is in
the symmetrical position, so that the point corresponding to Fj
is observed. Note also the indication of the goniometer scale.
Turn the goniometer through some convenient small angle
and note the mark of the scale seen coincident with the vertical
wire. Again note the goniometer reading. We thus have the
values of AF X and of the angle, a, and can consequently deduce
F 2 . By turning the system end for end, so that H 2 lies to the
right of Hj, we find F x .
We can thus locate the positions of N x and N 2 with respect
to the outer surfaces of the system. These should be recorded,
and since HjNj = F! + F 2 = H 2 N 2 we may also locate the
principal points. Record these also.
In order to carry out this experiment it is convenient to employ
two thin lenses situated at a known distance apart. These form
an optical system of the type described, and having the same
medium on each side.
Hence N x and N 2 are both nodal and principal points, and F!
and F 2 are numerically equal but of opposite sign.
Find the focal length and the position of the principal points
for this case.
It is shown in treatises on Optics (see, for example, Houstoun's
"Treatise on Light," p. 45), that for two lenses separated by a
distance, d, the position of the first nodal point is at a distance
from the lens on which light is incident and whose focal length
is/i, given by
-fid
h+h + d>
while the second nodal point is at a distance :
__M_
fl+ft+d
REFRACTION 307
from the second lens, f a , denoting its focal length, while the focal
length of the system is :
F = A/e .
fx+f % + *
Measure f 1 ,f 2 , and d, and verify these results,
/i aim / 2 may be readily measured by the goniometer by re-
placing the optical system by the lenses in turn.
Another property of this system, which is of importance in the
theory of Optics, concerns the lens equivalent to this system.
This lens is such that it produces an image of the same size, but
not in the same position, as that produced by the system, when
situated at a point at a distance from the first lens equal to — -?-,
/a
i.e. at the first nodal point.
Fig. 182, shows the relative position of the nodal points in the
case of two convex lenses.
TJie Focal Length of a Microscope Objective
The goniometer provides a useful means of measuring a short
focal length, such as that of a microscope objective*.
The screen, AF 1} is replaced by a micrometer slide provided
with a magnifying reading glass. The slide is provided with a
small engraved scale with divisions at each tenth of a millimetre.
The objective is placed to receive parallel rays emerging from the
goniometer lens and focusses them on the scale. Thus an image
of the goniometer vertical wire is received and is viewed by the
magnifying glass.
The position is read for the symmetrical position and again
after a slight displacement of the goniometer arm. The image
is now focussed at a new point corresponding to A (fig. 182).
The distance between the two images is read off, and from a know-
ledge of the angle of rotation of the arm, the focal length is
calculated as before.
The Determination of the Foeal Length and Principal Points of an
Optical System by means of the Revolving Table
The principle of the method is the same as that described on
page 302, for it depends on the same property of the nodal points.
The revolving table affords, however, a much more convenient
and accurate means of carrying out the experiment. (See figs.
152 and 183.)
Each nodal point is determined by the principle that when a
small rotation is made about a vertical axis through the second
nodal point, the image produced of a distant object is not dis-
placed. Thus the position of the axis of rotation of the table
locates the second nodal point.
308 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The lens system is mounted on a slider with straight edges,
and is placed on the table so that the edge slides along the scale,
An index mark on the slider serves to locate positions with respect
to the scale.
The design secures that there is no side shaking.
It is essential in this experiment as in the former (p. 268), to
make the axis of the system intersect the axis of revolution.
Fig. 183
In order to arrange this the image of a distant object is made to
coincide with a pin held in a clamp and the system is moved on
the table until, with this focussing exactly made, a slight rotation
of the table causes no displacement between the image and pin.
Suppose that the axis of the system is represented by N X N 2
(fig. 184), and that S denotes the distant source, while I is the
image, S is so far distant that it may be supposed to lie on either
of the lines, IN lf N^L, I 1 N V
I Nt_N, s
I ! L ZZT. *
A n! l
l'
n, n z
Fig. 184
Let A denote the point of intersection of the axis of revolution
and the horizontal plane.
If AN 2 is normal to the direction in which the light is travelling
a slight rotation does not displace N a at right angles to the axis.
Thus the light incident at the first nodal point still emerges along
N a I, and I is not displaced.
REFRACTION 3<>9
Nj 1 denotes the displaced position of N x and the dotted line
LNj 1 denotes an incident ray from S.
Thus with this setting the nodal point, N a , lies in a plane through
the axis of revolution and normal to the direction of the rays.
The table top is now turned through two right angles, and the
lens system moved along the scale until the distant object is
focussed on a second pin as before.
In general this pin will not coincide with the first, but will be
displaced in a direction normal to the direction of the light.
This is illustrated in the figure. Nj in the second case become
the second nodal point.
If I 1 denotes the second pin it is clear that the axis lies on a line
drawn parallel to the light and passing midway between the pins.
The scale on the table top is undamped and moved through
half the distance between I and I*, so as to carry the axis of the
system into the correct position.
The apparatus is now once more carefully adjusted so that
the image does not move for slight rotations.
The axis then passes through the nodal point. The system is
then turned end for end and the second nodal point found.
In order to find the focal length readily, a clamp is fixed to the
table which carries a scale (fig. 183).
The pin is left in position and the table turned until the scale
just touches it and the scale reading is noted. The table is turned
so that the other end of the scale just touches the pin and the
scale reading is noted for this case also. The distance between
the two marks on the scale is determined by subtraction ; one-
half of this gives the focal length.
A diagram is then made showing to a convenient scale the
positions of the cardinal points with respect to the first and final
surfaces of the optical system.
The apparatus may be employed to examine such a system as
that described in the previous experiment with the goniometer.
In this case the apparatus is not very convenient for the measure-
ment of short focal lengths. It is better to use the goniometer
for this purpose.
When a distant object is not available in the above experiment
it is only necessary to place a plane mirror on the side of the lens
system away from the pin and adjust the pin until image and
object coincide.
The focal length of a thick lens, the radii of the surfaces being
r and r 1 , and the thickness t, is given by :
F== ¥*
>-!).{ nfr 1 - r) - (n - 1) t\ '
Here r and r 1 must be given their proper signs.
310 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
It is instructive to determine the value of the refractive index
of the glass forming a thick convex lens.
Determine r and r 1 by the revolving table method (p. 269),
and F by the method just described, while t may be measured
by means of callipers.
Solve the resulting equation for i*.
The experiment is intended to provide practice in working with
a thick lens and becoming familiar with the formula for F.
The distance between the principal points is given by
t Qi - 1) (r 1 - r - *) ,
(x (r 1 - r) - ((i - 1) t
This should be verified by the results of the experiment which
locates the position of the nodal points, which are the same as
the principal points in this case.
The student should verify the two expressions given above.
Spherical Aberration with a Thick Lens
In the simple theory of lenses it is assumed that all the rays
starting out from a point on the axis are brought to a focus after
refraction through the lens to another point on the axis.
A,
Fig. 185
This is only approximately true for rays lying very close to
the axis. A ray, PA X (fig. 185), will be made to cross the axis
at F, and if PA X makes a small angle with the axis, F will be the
focus for all such rays. But a ray, PA 2 , will cut the axis at C 2 ,
and PA 3 at C 3 , after refraction, these points lying closer to the lens
as the incident ray is more inclined to the axis.
The distances, FC 2 and FC 3 , are called the longitudinal aberra-
tions of the corresponding rays.
The distance, FC 2 , depends on the inclination of the emergent
rays to the axis. If this distance is x and the tangent of the
acute angle at C 2 is m, we may say that x is some function of m.
We do not know the way in which x depends on m, but whatever
be the form of this dependence we may write x in powers of m, or
x = a + bm + cm 2 + dm 9 + . . . ,
where a, b, c, d, etc., are constants.
REFRACTION 311
Since the lens is symmetrical about the axis, FC 2 is the same
for positive and negative values of m.
Thus if we write — m instead of m in the above equation we
have the same value of x, so that x depends only on even powers
of m.
.'. x = a + cm 2 + em 4 -f . . .
When the emergent ray has a very small value of m, the' point,
C 2 , is at F, or x vanishes. Thus a vanishes and we have :
x = cm 2 -f em 4 -f . . .
If m is not very big, m 4 and higher powers of m, will be very
small, so that we may write :
x = cm 2 .
Let F be taken as origin and FP as axis of x.
Let v denote distances measured at right angles to FP.
The line, B 4 C a , has an inclination to the x axis whose tangent
is m, and it passes through the point, C 2 , with co-ordinates (cm 2 , o).
So that its equation is :
y = m (x — cm 2 ).
The emergent rays cross one another at points B 2 , B 3 , B 4 , etc.,
and the points at which consecutive rays cross lie on a curve
known as the caustic of the lens.
Suppose that to a ray consecutive to that through C 2
the corresponding tangent is m 1 .
Its equation will be :
y = m 1 (x — cm 12 )-
If we solve these two equations we can find the co-ordinates
(x, y) of the point common to both lines. We do not actually
require these but only the relation between these two co-ordinates,
which will give a locus of the intersections of consecutive rays
or the equation to the caustic.
Subtracting one equation from the other :
x (m 1 —tn)=c (m l3 — m 3 ),
x = c (m 12 + mm 1 -f m 2 )
= 3cm 2 ,
since m and m^are so nearly equal ;
m
'3c)» C ' \3C J ~ 3 ' (3cY
(3c)* \3cJ 3 ' (3c)* *
or 2j,cy 2 = 4# 3 .
312 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Thus for values of m, such that m* is negligible the form of
1 the caustic is a semi-cubical parabola.
In the figure, DCE 1 and D*CE are the extreme rays from the
lens and DEF, D X E X F the two branches of the caustic curve.
Fig. i 88
All rays inside the extreme rays touch this curve at points
which are nearer to F, the smaller the inclinations of the rays to
the axis. The rays are consequently crowded together just
inside the caustic curve, and a screen placed between C and DD 1
will show a circular patch of light with a brighter illumination
round the circumference. At points between C and EE 1 , a
brighter patch appears in the centre which increases to EE 1 where
the illuminated area is a minimum. EE 1 is called the circle of
least aberration.
Beyond EE 1 the first circle diminishes until at F it becomes a
point.
Set up a large lens on the optical bench with a small hole
illuminated by a Bunsen flame containing common salt as a source
of light. The light may be concentrated on the hole by means
of a small condensing lens.
It is best to have the distance from the hole to the large lens
about twice the focal length of the lens.
Examine the image by means of a micrometer eyepiece, when
the appearances described will be seen if the source lies on the
axis of the lens.
Find the position F, and read off the position of the eyepiece
on the bench scale. Move in towards the lens and measure the
diameters of the circular rings at a series of distances from F.
Plot the results on squared paper and verify the equation of the
caustic curve.
The radius of the circle of least aberration is r = J cm 3 ,
where c is constant in the equation and m Q is the value of m for
the extreme rays.
REFRACTION 313
tn is nearly proportional to the aperture of the lens. Vary
the aperture by putting diaphragms over the lens and measure
the corresponding diameters of the least circles. Verify that the
diameter is proportional to the cube of the aperture.
The above value of r is readily calculated by remembering that
it is the value of the ordinate of the caustic at the point where
it is cut by the extreme ray.
It is therefore necessary to solve the equations :
v = m (x — cm 2 ) {4)
2ycy 2 = 4a; 3 .
The solutions are
x = ^cm^, y = ± I cm *.
In order to find the value of FL, we note that by equation (4),
which is the equation of the line, CL, the value of v corres-
ponding to x = o is — cm 3 .
This negative sign occurs because we have measured the angle
of which m is the tangent from the x axis in a clockwise direction.
In the case of the line, CL, the value of m is negative since the
angle is obtuse.
The lateral aberrations of the extreme ray, i.e. the length of
FL, is thus — cm *.
But the radius of the circle of least aberration is £cw 3 (nu-
merically).
FL
Thus the ratio 7^=; = 4.
VjrJi
Verify this result.
CHAPTER XII
INTERFERENCE, DIFFRACTION AND POLARIZATION
Introduction
The theory that light consists of waves in the ether leads us to
expect, by the principles of superposition, that class of phenomena
described by the term 'interference.' Particles of a medium,
when simultaneously displaced by the arrival of several disturb-
ances, have a resultant displacement obtained by adding together
the vectors representing individual displacements. In particular, if
a particle is subject to displacements in directly opposite directions,
it will be displaced a smaller amount than if it were subject to
either separately, and if both the displacements are equal, but
oppositely directed, there will be no displacement of the particle.
On the other hand, displacements arriving at a particle in the
same direction will cause it to be moved a distance equal to the
sum of the separate displacements.
The arrival of oscillatory disturbances at any point may,
therefore, cause larger or smaller displacements of an ether
particle than would occur as a result of each separately.
The intensity of illumination at any point of a medium is
proportional to the square of the amplitude of the vibrations
executed by the ether particle at that point.
It can thus be seen how it is possible to produce in the ether
places of large or small intensity as a result of the arrival of
two trains of waves. It may even be possible that there will
Fig. 187
be darkness at certain points since it is possible that as a conse-
quence of adding vectorially the displacements we get no net effect.
This point will be considered in detail.
Let Sj and S 2 denote two sources of monochromatic light.
From each is thus emitted a train of waves of a particular wave
314
INTERFERENCE, DIFFRACTION, POLARIZATION 315
length, the same for each. In addition, suppose that the distur-
bance starting from S x is in the same phase as that starting
from S 2 . By this we mean that the ether particles at S x and S a
are in exactly the same state of vibration, they are moving in
the same direction, and are displaced the same amount from
their central positions. These disturbances are propagated with
a particular velocity in the ether, and at points equidistant from
S t and S 2 the displacements will be in the same phase when they
arrive there. In other cases where points are at different dis-
tances from S x and S 2 , it may happen that the displacements
are in different directions on reaching the point. Draw AB at
right angles to the middle point of S^g and let it cut a screen,
MjMa, normally at B.
We shall inquire as to the illumination at different points on
the screen. The disturbances will reach B in the same phase,
the displacements will be in the same direction, and will thus
unite in increasing the illumination at B, which will always be
bright.
Of course, we think of the waves as transverse, i.e. the motion
of the particles is normal to the direction of propagation.
Take any other point, P, distant x n from B.
The light from S 2 travels over the path, S 2 P, and that from
S x over SjP. Thus those disturbances, which reach P simul-
taneously, started at different times from their sources.
The vibrations will thus be in different phases on reaching P.
If, however, the difference between S 2 P and S X P is a whole
number of wave lengths, the disturbance from S a left that point
a complete number of periods before the disturbance from S x
set out towards P. The displacements are thus in the same
direction, and will unite at P to give brightness, just as they do
at B.
If the difference of path is an odd number of half wave-lengths,
the displacements on reaching P will be directed oppositely, and
will tend to destroy the disturbance at P. If the amplitudes
are equal there will be no displacement, and consequently no
illumination at P. In any case there will be a marked falling
off in brightness. We ought, therefore, to be able to distinguish
on either side of B as we proceed outwards alternate bright and
dark places.
Suppose the distance AS! = AS 2 = d, while AB = D.
Then PMj = PB -MjB
= x n — d,
and similarly PM 2 = x n -f- d.
(Mi and M 2 are the feet of the perpendiculars from S x and S 2 on to
the screen.)
316 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
.♦.S 2 P={D* + (*„+<*)*}* =D|i+^^j (i)
x -\-d
in which we neglect higher powers of -—■ — than the square.
This means that we suppose that x n and d are of small magni-
tude relatively to D. This will evidently be the case if we
confine our attention to a few only of the alternations in intensity
about B, for the distances apart of adjacent bright points on
the screen are of a magnitude comparable with the wave lengths
of the light, and this is very small.
In the same way
S t P = D{l+i2iL^)!j; ( 2 )
Thus for a bright point :
2d
and for darkness
D x n = «x, (3)
2d 2W + I ^ , ,
D*« = ^-- X; * (4)
or writing 8 = 2d, i.e. SjSg = <5,
we have x n • yz= «xor(w + |)x, (5)
according as P is bright or dark.
In the experiments to be described (5) is of fundamental
importance.
The first three experiments are three examples of obtaining
sources S 2 and S 2 of the kind described.
It would be useless to set up two slits and illuminate them by
a sodium flame, for we have in such a flame a multitude of
sources in different phases.
The method adopted is to cause light from a single source to
travel to the points of which P is typical by two different routes,
and to unite on arrival. We thus have the equivalent of two
sources emitting vibrations of the same phase.
In making experiments on interference and diffraction accurate
measurements have to be made ; for this reason the optical
bench is used. It consists of a strong, rigid metal frame, provided
with levelling screws on which it stands. The frame consists
of two metal rails, one of which is graduated accurately in
millimetres. Along the rails slide metal uprights, each of which
is attached to a vernier at its lower end so that its position on
the bench may be accurately read off.
-INTERFERENCE/DIFFRACTION, POLARIZATION 317
The uprights serve to carry a slit, micrometer, microscope,
lens, or whatever piece of apparatus is necessary. If it is
necessary to move any piece of apparatus transversely across
the bench, it is placed in an upright fitted in a support provided
with a transverse micrometer screw.
Fig. 188
Although it is possible to read off positions of the uprights on
the bed of the frame, since the slit or cross-wire may not be
exactly above the indicator mark we cannot read off directly
the distance from slit to cross-wire. A correction must always
be applied. In order to find this correction a stand, carrying
a carefully measured rod is placed on the rails, one end of the
rod is placed in contact with the slit while the other end is viewed
in the micrometer. Let the length of the rod be I, and suppose
the distance between the slit and micrometer uprights is I 1 , as
observed on the scale. Then to convert the readings as obtained
from the upright to the distances required we must add to the
observed readings the quantity (I — I 1 ).
Determination of the Wave Length of Sodium Light by means of
Lloyd's Single Mirror
The simplest way of obtaining interference bands is by means
of a mirror silvered on the front surface or blackened at the
back in order to avoid multiple reflections.
The diagram illustrates how the interference is brought about.
Light from a slit, S lt travels directly to a screen, PB, and
also by the alternative path after reflection at the mirror, MM.
For example, a ray may reach P by the direct path, SiP, or by
the path, SjCP. The latter ray produces the same effect at P
318 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
as would arise from a ray, S 2 P, S 2 denoting the position of the
image of S x in the mirror. We thus have the equivalent of two
sources, S x and S a , emitting vibrations in the same phase, except
for any change of phase that may occur on reflection at MM.
MM is mounted vertically on an upright of an optical bench,
and the slit, S r , is carried vertically on another upright.
1 P
- B
S» Fig. 189
MM is mounted as accurately parallel to S x as possible, and the
fringes are looked for by the micrometer microscope. They will
usually come into view, if not already there, after slightly rotating
the mirror, and are rendered distinct by adjusting the width
of the slit.
The mirror should be parallel to the length of the bench, and
we shall see how to make this adjustment accurately shortly.
When the fringes appear as definite and bright as possible,
it is necessary before proceeding further to make sure that the
distance, SjSg, can be found.
The method adopted is to use a lens and obtain the magnified
and diminished images of S^Sg, but we require a lens whose focal
length is less than a quarter of the distance between S X S 2 and PB.
In Chapter II, p. 293, we described a method of finding the
focal length of a lens in which the object and screen are kept
fixed while the lens is moved. In the diagram in connexion
with this experiment, suppose that Pj denotes the position of
the object, S X S 2 , and lt one of the positions of the lens, while
P 2 denotes the position of the image of S a S 2 .
By moving the lens to 2 , the image is formed at the same
place, but it now differs from the former in magnitude.
Let d lt d 2 denote the respective distances between the lines
S x and S 2 in the image, while 8 is the actual distance, S X S 2 , between
the slits.
Then li_L2l ^! _££_*.
lhen 8 ~ P^' 8 - OtPj
But P 1 1 = P 2 2 and P^ = PaO^
.-. 8* = d x d z (6)
Thus, by measuring d x and d z we may deduce <5.
INTERFERENCE, DIFFRACTION, POLARIZATION 319
If MM is too close to PB it may be impossible to obtain both
images with the lens between the mirror and screen. The mirror
must be placed so that this is possible. This is the reason for
performing this part of the experiment first since it is useless
to measure the distance between consecutive fringes unless we
can find the distance, S^. The distance apart of the fringes
does not depend on the position of the mirror, but if the mirror
has to be replaced in another position it is easy to throw out its
adjustment. Both images of Sj and S 2 will be in focus in the
plane of the cross-wires only provided that MM is normal to
this plane, i.e. to PB.
It is important that MM should be normal to PB. We have,
therefore, to rotate MM slightly until the two are accurately in
focus, the lens being adjusted so that its centre lies on a level
with the centre of Sj and opposite the edge of the mirror.
On removing the lens, MM should be rotated slightly about a
horizontal axis to get the position where the fringes are brightest,
when S x and MM are parallel.
Set the cross-wire accurately down the Centre of the first bright
fringe, and move it always m one direction by means of the
screws giving the transverse motion, stopping at every three or
four fringes to note the position. From the observations deduce
the distance between consecutive fringes.
Finally measure the distance from S^ to PB.
From (5) we have :
N x n = n\ -,
and proceeding to the next bright band :
D
*«+i = (« + 1) x- ;
.*. % n +l x n = ~7~ (o)
Let s denote the distance between consecutive fringes, and
we then have :
x =
p y/d x d z .
As a rule it will be best to make use of the full length of the
bench, so that D is large, also <5 should be as small as convenient.
At the same time d x or d 2 must not be too small or it will be
difficult to make the determination of 6 accurately.
In recording the results of this and the two following experi-
ments it is a good plan to begin, as mentioned, at one edge of
the field, and make out a table as shown below.
320 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We can then readily obtain a series of independent readings
for the determination of the separation of the bands.
In this experiment the first fringe does not lie at the point
corresponding to B, fig. 187. This is because the reflected ray
undergoes an abrupt change of phase on reflection at the mirror.
This has the effect of displacing all the fringes towards B by a
certain distance leaving the separation between successive
fringes unaltered.
NO. OF
1 BAND
MICROMETER
READING
(a)
NO. OF
BAND
MICROMETER
READING
(&)
SEPARATION FOR 15 BANDS
(b) - (a)
I
4
7
10
13
16
19
22
25
28
lyiean Separation for 15
Bands
Interference by means of FresnePs Double Mirrors
Interference fringes may also be formed by the use of two
mirrors silvered on the front surface or blackened at the back,
very slightly inclined to one another.
The diagram illustrates the arrangement. 0M X and 0M t
denote the two mirrors which may be mounted on the optical
bench and placed accurately vertical.
S denotes the source of illumination and may most conveniently
be a Bunsen flame coloured by a sodium salt.
The fringes are again observed by means of a micrometer,
and AB denotes the position of its cross-wires.
The screen, CD, protects AB from direct light from S.
The source is placed on one side of the bench as the diagram
indicates, and may be the slit of the optical bench supported
in a clamp. Sj and S 2 are the two virtual images in the mirrors.
SxSg and S lie on a circle with centre at O.
INTERFERENCE, DIFFRACTION, POLARIZATION 321
The calculation, mode of measurement, and adjustment, are
identical with those of the previous experiment.
The shaded area of fig. 190 shows where rays from both mirrors
interfere, and the section of this region with the plane of the cross-
wires of the microscope is the position within which the fringes lie.
M;
n/ ^ ^~rr rTT7Trr7r777ti
B
Fig. 190
Fresnel's Biprism
The biprism is a prism with one of its angles only a little less
than two right angles, and with two equal small base angles.
The figure illustrates its action. The biprism is represented by
CDEF, and it acts like two prisms placed base to base.
Rays from a slit, S, are deviated in each part of the prism and
unite on the screen, as in the case of the rays, SGP and SHP.
We have virtually two sources, S x and S 2 .
Fig. 191
The details of the experiment are very similar to those described
in the experiment with Lloyd's mirror.
The biprism is held in a stand which can rotate about a hori-
zontal axis parallel to the central line of the bench. By this
means the edge, D, of the prism can be rotated to bring it parallel
with the slit.
The adjustments are first made roughly by the eye, and usually
the fringes will be observed even with the rough adjustment.
Slight rotation of the biprism will, as a rule, improve the appear-
322 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
ance of the bands, and still better definition will be obtained by
narrowing the slit.
The distance, S^, is measured with the aid of a lens as before.
In adjusting D parallel to S the following device is of great
assistance. If the eye is placed on the side of the prism away
from the slit, and moved across the bench, the slit will appear
to cross from one side of the prism to the other. As it crosses
D, unless the slit is parallel to the edge of the prism, the top or
bottom will cross the edge first, while if parallel it will appear
to make the transition suddenly. The prism must be rotated
until this sudden jump occurs.
Be careful after arranging the apparatus to give good fringes,
to make the determination of the length, S l S 2 > before measuring
the separation of the fringes.
The Determination of the Radius of Curvature of the Face of a
Convex Lens by means of Newton's Rings
Newton's Rings are formed as a result of interference between
the incident and reflected rays from a source of monochromatic
light on the air film between a plane glass plate and a convex
lens in contact with it.
G
A 1 A
The diagram shows the lens, L, and the plate, P, in contact
at the point, Q.
Two incident rays are drawn, AB and A 1 B 1 . The first is
partially reflected and refracted at the points, B, C, D, E, and F.
INTERFERENCE, DIFFRACTION, POLARIZATION 323
A ray is drawn which is represented as suffering refraction,
except at D, where it is reflected. The neighbouring ray, A 1 B 1 ,
which follows the course, A 1 B 1 EFG is shown, and these two rays
being brought together along EFG will interfere and may produce
greater or less illumination than each separately, according to
their phase difference. In calculating this it is to be remembered
that on account of the reflection at D, at a medium optically
denser than that in which the ray travels before reflection, a
phase change of half a wave length is imparted to it.
The total phase difference between the two united rays is thus :
(MD + DE + Jx) as reckoned in path length ; for had the rays
both left the lens, the wave front would have been EM, so that
the phases are the same at E and M.
Each ray, such as AB, has a corresponding ray, such as AjB^
arising from the same point in the source of light with which it
can interfere, so that an extended source may be used and a
Bunsen flame containing sodium acts very well ; in fact, an
extended source is necessary in order to obtain a large area
containing rings.
The rays entering the eye or an optical instrument will be
contained in a small cone, and for the rays in this pencil the
change in path difference is very small, so that all the rays
from the small region of the lens will be naturally reinforced or
caused to interfere.
There will be brightness or darkness according as :
MD -f DE + |x = »x or (n + £)x,
where n is a whole number, i.e. according as MD -f- DE = an
odd or even number of half -waves.
Let CD fall on the plate at incidence 6.
Let t be the thickness of the film at this point.
The diagram is drawn with t large, or D a long way from Q,
for convenience, but in the formation of the rings the film is
very thin, and the part of it with which we are concerned is
very close to Q.
Just near D we may regard the film as an element with parallel
faces separated a distance, t, as in the enlarged element.
MD + DE = 2CD - CM
2 CN ~, T . . ~ XT cos 2
= — — - — 2 CN sin = 2 CN . ■ = it cos 0.
sin sin
Thus the condition for brightness is :
2t cos 6 = (2» + 1) -,
324 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The angle, 0, may be measured by the angle between the
normal to the plate and the incident rays before striking the
lens, since near Q the lens acts as a parallel plate of glass very
approximately.
It is usual to cause the rays to fall normally on the lens, when
we have :
2t = (n + £)x.
If we consider a plane normal to that of the paper through
Q and D, we can see that the circumference of a circle with
centre, Q, and radius, QD, will pass through points at which the
air film has a uniform thickness, and consequently all round
this circle rays incident vertically will undergo the same phase
change, and thus alternate bright and dark rings are formed
about Q.
Let the radius of the ring be r n = QD, and let the ring be the
(» + i)th from the centre.
K
The centre itself will be black, for the air film is infinitely
thin at this point. If this is not the case at first it is because
some dust particles lie between the surfaces, and these should be
removed.
DQ 2 = DN • DK = t(2R - t), (fig. 193)
R == radius of lower surface of the lens,
/. DQ 2 == y„ 2 = 2M,
approximately, since t 2 may be neglected.
Thus for brightness :
r w 2 = R(n + £)x.
The first ring corresponds to n = ; the second for n = 1,
and so on.
Thus the radii are proportional to Vi, V3, V57 etc.
For the purpose of the experiment a convex spectacle lens of
about 100 cms. radius of curvature is suitable, and the light
INTERFERENCE, DIFFRACTION, POLARIZATION 325
from a sodium flame is reflected down on to it by means of a
sheet of plane glass held at 45 ° to the vertical (see fig. 194).
The rings are viewed by means of a travelling microscope.
Fig. 194
In order to focus quickly on the rings, remove the lens and
focus on the top of the glass plate. On replacing the lens, and
adjusting the microscope over the point of contact the rings
which lie in the air film should be distinct. A good bright
sodium flame is necessary, and often the difficulties disappear
if this point is attended to.
RING
MICROMETER
READING (l)
MICROMETER
READING (R)
DIAM.
(diam.)*
20
19
18
5
4
3
2
1
■ 1
(l) denotes the reading on the left of the centre, (r), that on
the right.
/
326 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Move out the micrometer to about the twentieth ring from the
centre, and then, moving back again, turning the screw always
one way to avoid any errors due to backlash, set the cross-wire
carefully down the centre of each bright ring and observe the
micrometer reading. In this way, by observing the rings on
both sides of the central dark patch, the various diameters are
determined.
Make a table as shown above.
Draw a curve with the square of the diameter as ordinate*,
and the number of the ring as abscissa. The graph should be
a straight line.
P 2 N 2 = (D„ 2 )*,
P X N X = (D n] )*,
D tt 2 - 4R*(«+i),
P 2 N 2 - P t N x _ (Pg* - (D Wl )«
Then if
Since
N X N S
n,
4RX.
If sodium light is used, x may be taken as 5890 x io~ 8 cms.
In this description we have not taken account of the fact
that the rings are seen, not directly, but after refraction through
tlae lens. They are formed in the air film in the space between
the lens and plate of glass. This difficulty is avoided by placing
the plate above the lens and in contact with it, for in that case
we view the rings through a plane sheet of glass.
Fig. 195
It is, of course, more convenient to place the glass below the
lens, for reasons of ease in keeping them steady. The error in
this case is not great if a thin lens be used, for then the object
— the rings — is at the surface of the lens and consequently at
its principal plane. The image is in the second principal plane,
and of the same size as the object. For a thin lens these planes
and lens surface are nearly coincident. In practice we have an
INTERFERENCE, DIFFRACTION, POLARIZATION 327
image of magnification slightly differing from unity, or the
diameters in the formula above are to be multiplied by such a
fraction. For the purpose of the experiment this factor is
omitted.
Jamin's Interferometer
The apparatus consists essentially of two glass plates, AB and
A 1 !? 1 , of the same dimensions and optical character. The plates
are very carefully worked and are of the best optical glass.
They are mounted parallel to one another, standing on tables
on an optical bench at a distance apart of about one metre.
The first, AB, is set at 45 to the bench with its surfaces vertical,
and it is illuminated by rays from a sodium flame.
B
Crv
ft
E
flD S l «
V
Fig. 196
In the diagram a ray, RS, is shown. It is partly reflected
and refracted at S, and the refracted beam again partly reflected
at T. The divided ray takes the paths, RSS^HPR 1 and
RSTUU 1 R 1 , so that it is reunited by the second glass plate.
This plate is mounted parallel to the first, and set vertically.
It rests on a table and may be slightly rotated about a vertical
axis. If both plates are exactly similar and are parallel, the
lengths of the two paths will be the same for all rays, but by
slightly rotating one of the plates a difference in path may be
introduced, differing for different directions so that alternate
bright and dark bands will be obtained.
The tube, CD, is then placed in the path of one of the divided
rays between the plates, the other being allowed to pass clear
of CD.
CD is a hollow glass tube with ends of plane, optical glass.
It is fitted with a tap, E, by means of which it may be exhausted
or filled with gas.
328 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The tube is filled with air at the atmospheric pressure, and
the fringes found by eye. A telescope provided with cross-wires
is then focussed on the fringes.
In order to ensure that the fringes arise from interference
between rays that have passed, one through CD and the other
outside it, we may cover up the end of CD, and note if they
disappear as they should if due to this cause.
In order to be sure that the rays producing interference do
not both pass down CD, it is necessary to intercept the light at
the sides of the tube. If they are still present they will arise
from rays passing down CD.
When the correct fringes are obtained the cross-wire of the
telescope is focussed as accurately as possible down the centre
of one bright band, after exhausting the tube as much as possible
with an air-pump. The pressure should be brought down to i cm.
of mercury at least, and a manometer connected up to read
the pressures. Air is now allowed to pass very slowly into the
tube by the tap, and the fringes watched. They will appear to
move across the field of view, and the number passing the cross-
wire must be counted. Allow about 5 or 6 to pass, and then
stop the inflow and read the manometer. Repeat this, step by
step, until the tube is filled with air at atmospheric pressure.
Draw a graph showing the relation between the pressure within
CD, and the number of fringes that have passed from the initial
stage. By an exterpolation deduce the number that pass
between the limits of complete exhaustion and the attaining of
atmospheric pressure.
Let this number be «.
This means that the difference of optical path in the tube,
when completely exhausted and when filled with air at
atmospheric pressure, is wx.
If the tube is of length, /, and the refractive indices are y. and
Ho, when the pressure is atmospheric and when the tube is
exhausted, respectively :
l(v- — y-o) = ^x.
But Ho = *»
wx
so that i* = 1 -\--t-*
A convenient length for the tube is 30 to 40 cms., when for
the exhaustion produced by a good air-pump the value of n is
of the order 200.
The compensator described in connexion with the experiment
with Rayleigh's refractometer may also be used with Jamin's
interferometer, and the calculation of the refractive index at
normal temperature and pressure may be made with the help
of Jamin's interferometer also.
INTERFERENCE, DIFFRACTION, POLARIZATION 329
A cylindrical lens with the cylindrical axis vertical is often
placed just in front of the point S. This has the effect of widen-
ing the source light in a horizontal direction, leaving it
unchanged vertically. The source in this case is a slit.
The Refractive Index of Air by means of the Rayleigh Refraetometer
A diagrammatic representation of the apparatus is given in
fig. 197.
* 0*1 ■ F JEE 3 B aTc~»
SB p CG
Fig. 1^7
ABCD is an airtight metal box, divided into two separate
chambers, each of which may be connected to a manometer.
The pressures in the chambers are varied and measured by means
of the manometers. The chambers are closed at the ends by
means of parallel plates of good glass.
The collimator is provided with a slit at K, and provides a
parallel beam of light which falls on a screen, carrying two fine
slits, LL, placed in front of the air chambers, so that one slit
lies adjacent to the end of one, and the other slit adjacent to
the end of the second chamber.
These slits are prolonged so as to extend higher than the top
of the box, ABCD. Thus, light from the slits passes over the
box as well as through it, and finally enters the telescope, T.
These two fine slits produce interference fringes in the focal
plane of the telescope, and if the pressures in the chambers are
equal, the set of fringes in the lower half of the field appear to
be continuations of the fringes in the upper half which arise from
rays that have passed over the top of the box.
In order to deviate the upper set of fringes down, to make
comparison with the lower set easy, a prism, P, is provided which
intercepts the upper set of rays, and deviates them downwards.
When white light is used, two sets of coloured fringes are
obtained with a white central fringe.
If there is a difference of pressure between the two chambers,
there is a displacement of the lower set of fringes owing to the
resulting difference of optical path.
G consists of two plates of glass, inclined at a small angle,
placed to intercept the lower set of rays. When it lies symmetri-
cally with respect to the rays striking it, it introduces no
additional path difference, but on rotating it the rays through
one plate traverse a longer path within the glass than do those
in the other.
330 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In this way the lower central band can be moved about, and
we may, by a rotation of G, bring back the fringe to its central
position after it has been displaced on account of the difference
in air density in the two chambers. G carries a pointer moving
over a scale, and we may calibrate the scale so that the difference
of path introduced by setting G is known. Thus, by altering
the pressure within the chamber and then moving G to counteract
the displacement of the central fringe, we can read directly from
the calibrated scale to how many wave lengths the path difference
in the two chambers amounts, and a graph is plotted with the
pressure differences as ordinates and the number of wave lengths
as abscissae.
The calibration is performed by making the pressure in both
parts of the chamber equal and illuminating K with mono-
chromatic light, e.g. by a sodium flame.
The scale is set so that the bright bands of the upper set of
fringes lie over those of the lower. The pointer is then moved
over the scale until a certain number of bands pass a fixed point
in the upper series. The following table will illustrate how this
result should be recorded :
CALIBRATION
OF SCALE IN WAVE
LENGTHS
WAVE
SCALE
WAVE
SCALE
DIFFERENCE FOR 40
LENGTHS
READING
LENGTHS
READING
WAVE LENGTHS
1-48
40
4*50
3-02
5
1-85
45
4'8 9
3-04
10
2-24
50
5*24
3*oo
15
2-62
55
5'6i
2-99
20
3-oi
60
5*99
2-98
25
3-38
65
6'37
2-99
30
374
70
6-75
3-oi
35
4-12
75
7-12
Mean for 40
3-00
24-03
wave lengths
Mean value of
3-004
1 wave length
in scale divi-
sions
•0752
After the calibration of the scale the collimator slit is again
illuminated by white light, and the central fringes arranged one
above the other.
334 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The apparatus is shorter since the length of liquid traversed
need not be so great as that required for a gas.
If a liquid is placed in one of the vessels, and the same liquid
containing a solvent in the other, we may determine the effect
of the solution on y., or inversely, we may estimate the amount
of solvent contained from its refractive index.
Fig. 202
In this instrument the movable plate is in the upper beam
•instead of the lower, but the method of use is otherwise similar
to that of the previous experiment.
Two thermometers project into the liquid for recording their
temperatures.
Examine the changes produced by adding small quantities of
a salt to pure water, and placing in one cell the solution so
obtained and water in the other.
We have, if y. and tx 1 denote the refractive indices of water
and of the solution respectively, d the thickness traversed by
the light, and x the wave length :
where n is the number of displacements of the central fringe.
The experiment is performed with white light so as to have a
definite central fringe, but x is the wave length of the light used
to calibrate the scale. By moving the pointer over the scale
until the two central fringes are coincident, the scale reading
INTERFERENCE, DIFFRACTION, POLARIZATION 333
the tubes containing the gas, which would normally separate
the lower interference bands from the comparison upper bands,
produce no image in the field of view on account of the refraction
in H. The two sets of fringes stand one immediately above the
other.
They are focussed by the achromatic lens, Q, and examined
by the cylindrical lens, R. This provides a large horizontal
magnification of the fringes, which lie close together because
Fig. 201
the slits are of necessity rather wide apart (cf. formula 6 of
this chapter, d is large in the present case). The lens, however,
does not give magnification in a vertical direction, so that the
shadow cast by the obstacle is not broadened vertically, and
there appears a sharp dividing line between the two sets of bands.
The Rayleigh Refractometer for Liquids
Fig. 202 shows a Hilger apparatus based on the foregoing
principles for comparing the refractive indices of liquids.
332 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If T is the absolute temperature, we have also :
—, = constant.
pT
Thus — - — . T is constant.
P
If jx is the refractive index of air at N.T.P.
{* — i . T _ t*o_— i
x 273.
P 76
We shall see that y. can be determined by observations in this
experiment.
Suppose the length of the tubes containing the air to be I,
and the refractive indices, jx x and [t it corresponding to pressures.
p x and p 2 . Let the wave lengths of monochromatic light be X,
and X, respectively.
The excess of waves in one tube over those in the other is :
' (t t) ". G£ ~ ■£) = i ( "' - " i)
This number is deduced from the scale over which the pointer
moves. Suppose it is m.
Then :
T ■ 76Tx
273l(p2 - Pi)
The pressures are measured in centimetres of mercury, and the
m
ratio, -r — , is deduced from the graph as described, while the
length of the tubes, which is usually about 25 cms., may be
measured by means of a metre rule
B C
A
D * E
Fig. 199
In figs. 199 and 200 a modern form of apparatus is shown
diagrammatically. The plates, L,K, correspond to G, and the
prism, P, is here denoted by H, and its action shown in the
fig. 200. It can be seen how such obstacles as the upper edge of
INTERFERENCE, DIFFRACTION, POLARIZATION 331
The pressures in the chambers are varied and measured by
the manometers.
The central fringes are kept one above the other by means
of G, and the pressure differences recorded along with the position
of the pointer on the scale.
Draw a graph with scale readings as ordinates and pressure
differences as abscissae (fig. 198).
O 5 lO t5 ZO
Pressure Difference.
Fig. 198
White light is used in order to provide a definite central fringe.
The position of this fringe is independent of colour while all
other fringes have positions which depend on the wave length.
If monochromatic light is used, all the fringes are alike, and the
central fringe is indistinguishable from the others.
Since the screw attached to the compensating glasses has been
calibrated in wave lengths of sodium light, the observations give
the path differences in terms of so many wave lengths of yellow
light, and the refractive index deduced will be that for this
particular wave length.
Fig. 201 gives a general view of the apparatus.
From the slope of this graph may be deduced the difference
in path in wave-lengths for a difference of pressure of 1 cm. of
mercury.
Observe the temperature of the air in the tubes by placing a
thermometer close to them and noting its indication throughout
the course of the experiment. When the pressure is varied by
means of the manometers, the changes should take place slowly,
and time should be allowed for the air to take up atmospheric
temperature.
Theory of the Experiment
For a gas the relation,
u — 1
= a constant,
is very approximately true. We assume it in this experiment,
(i is the refractive index, and p its density.
INTERFERENCE, DIFFRACTION, POLARIZATION 335
will give the number of wave lengths which one ray has fallen
behind the other in traversing a path of different nature from
that of the other.
The calibration is carried out in a preliminary experiment
with both liquid cells containing water and with a sodium flame
illuminations, as in the last experiment.
Michelson's Interferometer
The apparatus is illustrated diagrammaticaUy in fig. 203.
It consists of two plane mirrors, M x and M 2 , silvered on their
front surfaces and mounted vertically on a heavy, firm, rigid
stand. The stand consists of a metal bed provided with a large
micrometer screw of very fine pitch. Rotation of the screws
causes M t to slide along the bed and its position may be read
off at the screw-head. A general view of the apparatus is given
in fig. 204.
M,
W
The second mirror, M 2 , is fixed at the end of a metal arm
mounted at the end of the bed, and at right angles to it. This
arm also carries the two sheets of plane optical glass, P a and P 2 ,
which are equally thick, and are mounted at an angle of 45 to
the arm.
336 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In order to produce interference fringes, a source of light, e.g.
a sodium flame, is placed at the focus of a lens, L, and a parallel
beam of light thrown on to the glass, P x , the direction of the
beam being along the arm carrying M 2 . One of the rays of such
a beam is shown in the figure as SN. On striking P x it undergoes
partial reflection and refraction at N, and the refracted part is
again divided at K, aDd later further division takes place at P 2 .
Fig. 204
We consider one of the ways in which division can take place,
the recombination of two parts, which have started from one
ray and have different optical paths, producing interference.
A marked difference in this case from that in some of the
foregoing is the production of interference between rays differing
in phase by very many wave lengths. In the case of Newton's
rings or Fresnel's biprism, the path difference amounts to a few
waves only.
The ray, SN, is refracted in P x and reaches K, where it is partly
reflected so that it gives rise to KQT and partly refracted to
give KM 2 . These rays are both returned along their paths and
reunite to produce KW. In the course of their journey each
passes through the glass a distance equal to 3NK, and change
in phase is brought about by the difference in the path in air.
This may be varied by altering the position of M x .
The ray, KW, may be observed by eye, and usually curved
INTERFERENCE, DIFFRACTION, POLARIZATION 337
>
interference bands similar to those observed in the experiment
on Newton's rings will appear. The curves will not be seen
closed, only parts of the closed curves can be seen.
We use here, as in the experiment on Newton's rings, an
extended source of light, for the rays from each point of the
source are divided and re-combination takes place between rays
which originally belonged to the same ray. Rays falling on P x
in the same direction all suffer the same phase change and emerge
parallel to KW, so that an optical instrument will focus them
together in its focal plane.
Other rays in a direction slightly inclined to SN, emerge as
a set of parallel rays, slightly inclined to KW. These undergo
a different phase change, and also lie in the focal plane displaced
from the image due to rays parallel to KW.
It is necessary that the mirrors, M x and M 2 , should both be
vertical, and in order to allow this adjustment to be made, at
the back of M 2 are three screws pressing against springs that
cause it to rotate.
If a sheet of tin carrying a fine hole is placed in the path of
the incident light, four images are seen as a rule, when the eye
looks in the direction, WK. The reason for this is best seen by
reference to a diagram (fig. 205).
The emergent rays are marked p lt p it p s and p lt and it will
be noted that p 2 passes through the glass plates three times;
so also does p x .
Thus if the rays are made to coincide they will be in a condition
to annul or reinforce each other according as the path difference
is an even or odd number of half wave-lengths.
The rays, p a and p x , traverse the plates five times and once
respectively.
338 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If a card is placed in front of M x , the rays p x and p s still
appear ; and p x will be the brighter since it traverses the plates
only three times, so that it can be distinguished.
By placing a card in front of M 2 , the rays are cut out except
p a and pi, so that by this means the images can be distinguished,
and those due to p x and p 2 caused to overlap by adjusting the
screws behind M. x and M 2 .
When both mirrors are vertical the images coincide in pairs.
One constituent of each double image lies behind the other.
Those which produce interference lie along the direction, WK.
-P
Fig. 206
Fig. 206 illustrates the mode of production of the fringes from
another point of view. The interferometer acts as if we had a
source, S, from which rays could be reflected by two parallel
mirrors, M x and M 2 , in which they would produce images, I x
and I 2 . The distance, MjM 2 , is equal to the difference of the
distances, NMj and NM 2 , of fig. 203.
At any point, P, rays, SAP and SBP, would unite, and if their
paths differed in length by wx, a bright point would arise.
This is equivalent to stating that in this case :
PI 2 - PIj = »x.
Thus we obtain the particular interference band at all points,
P, for which this relation holds, i.e. P lies on a hyperboloid of
revolution with SM X M 2 or with WK, of fig. 203 as axis. In a
plane perpendicular to that of the figure we have a circular
section of the hyperboloid which explains why the fringes are
circular as seen on looking along WK.
The Determination of the Frequency of Light from a Sodium
Flame or any Monochromatic Source
A striking feature pi the Michelson Interferometer is the screw
which displaces M v In the apparatus illustrated in fig. 204 the
screw has a length of 200 mm. and a pitch of 1 mm. The head
of the screw is furniihed with a scale divided into one hundred
parts, each thus corresponding to one hundredth of a millimetre.
The screw may be rotated by the handle seen in front of the
apparatus. The sm^ll lever on the right of the front of the
apparatus puts into action the slow motion screw, one turn of
INTERFERENCE, DIFFRACTION, POLARIZATION 339
which corresponds to one division on the head of the main screw.
As the head of the slow motion screw is also divided into one
hundred parts, it is possible to record a motion of the mirror of
one ten-thousandth of a millimetre.
In order to obtain the fringes, set up the sodium flame at S,
and place immediately in front of it a sheet of tin with a small
hole in it just opposite the bright part of the flame, and adjust
the flame and hole to lie on the level of the centre of the mirrors
and plates.
Mount the lens, L, also so that its centre is at the same height
as the hole, and place a piece of plane mirror between L and P a
and move L until an image of the hole is thrown back on to the
tin close to the hole. The light is then parallel as it leaves L.
Remove the plane mirror and look in the direction, KW, when
usually four images of the hole appear. Adjust the mirrors so
that these images coincide, two by two, as indicated above.
Then remove the tin sheet ; allow the light from the flame
to fall through L on to the apparatus.
As a rule very slight movements of the mirrors will bring the
fringes into view if they are not already to be seen. Sometimes
a slight motion of M x along the bed of the apparatus helps to
discover the fringes.
Set up in front of P 2 on the side towards W a sharp pin point
to mark the position of the centre of one fringe. Rotate the
screw slowly by the slow motion screw, and watch the movement
of the fringes across the field of view, counting the number which
seem to pass the point, and observing from the scales how far
M x has moved.
When Mj moves back a distance £x the path difference between
the two rays which unite to interfere has been increased by x,
so that where a particular fringe originally appeared the neigh-
bouring fringe now apparently lies.
Thus if Mj moves a distance, I, the number of fringes which
2I
appear to move past the point is correspondingly — . If these
are couhted, since / can be measured, we can find x.
Do this for such monochromatic extended sources as are
available.
Note also that the intensity of the fringes appears to alternate
as the distance of M x varies. We shall make use of this fact in
the next experiment.
The Determination of the Difference of Wave Length for the
Sodium D Lines.
Adjust the mirrors Mj and M 2 so that their distances from N
are equal, as nearly as can be judged by eye.
340 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
This adjustment may be brought about accurately by observing
the images of the holes formed as above in the two mirrors, and
adjusting M x until there is no parallax between them. In this
case they are equally distant from the observer, and the two
mirrors are consequently equidistant from N.
The light from a sodium flame, though for many purposes
considered monochromatic, contains two fairly intense waves
whose frequencies are close together.
In the present case both sets of fringes which arise from the
two waves overlap, but if M x is slowly moved away there is a
gradual separation of the two sets, and finally the bright band
of the one lies over the dark band of the other. This happens
when the distance that M x has moved from the first position
contains one more quarter of a wave of the one than of the
other, for then a difference of phase, corresponding to one-half
wave length has been added to one more than to the other.
Or, to put it otherwise, let I denote the distance moved by M lf
the additional air path added to each wave incident on M x is
thus 2.1.
Suppose 2.1 contains n x waves of length x x , and n 2 of lengths x a .
Then the difference between n x and n t is one-half.
Suppose for the sake of definiteness that n x > n t and conse-
quently x x < x t .
Then, since n x = —and », = — ,
Xj x 8
we have : 2l( J = -,
\X X Xj/ 2'
i i_ i .
Xj x a 4'
If the two waves had equal intensity the field would become
uniformly illuminated, and the fringes would disappear.
In this case, since one of the lines is more intense than the
other, we get an alternation in distinctness, the brighter fringes
still stand out in contrast with the adjacent less bright ones.
Note the positions of Mj at the beginning and successively at
positions where the fringes become least distinct and again
distinct as M x goes further away and the path difference contains
one complete wave more of one colour than of the other. Do
this for as many cases as possible, and if d denote the distance
between the positions of M x in which two successive distinct
sets of fringes occur, we have
j_ i_ __ i
Xj X a 2d'
Assume the shorter light in the sodium light to have a length
INTERFERENCE, DIFFRACTION, POLARIZATION 341
5890 x 10- 8 cm., and deduce the difference between x x and x 8
in this case.
The Production of Coloured Fringes
These can be obtained when the mirrors are set for equal
paths. Do this as in the last experiment with sodium light and
then replace it by a white source. The fringes due to the different
colours overlap in this position.
With sodium light, as M x is moved, it will be noticed that the
circles change the direction of their curvature ; the position
required is just when the transition takes place. It is difficult
to decide just when this occurs, and it is a good plan to note
the two positions of M t in which the fringes are definitely curved
in one way and then in the other. Put in white light when M x
occupies the position corresponding to one of these directions
of curvature, and then slowly move it back to the other position.
The coloured fringes will appear during this movement.
M, must be moved slowly or the fringes change their places
too rapidly to be noted.
Fig. 207
The Hilger Wave Length Spectrometer
This apparatus, illustrated in fig. 207, consists of a cast-iron
stand with two arms at right angles on which are held rigidly
the telescope and collimator.
342 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The telescope may be fitted with a high-power eyepiece with
adjustable cross-webs or with a shutter eyepiece which can be
adjusted to cut out any part of the field except that under
particular examination with an adjustable metal pointer. The
pointer has a brightly polished fine point and is illuminated by
reflecting light from outside by means of the mirror shown in
the figure in position above the eyepiece. Thus any point in the
field may be taken as a reference point by setting the pointer
to it (fig. 208).
Fig. 208
In addition the shutter eyepiece may be employed with light
filters which impart any desired colour to the bright point.
This adds to comfort in reading and consequently to accuracy.
A
Fig. 209
The vertical collimator slit may also be reduced in length by
means of a cross horizontal slit, so that a small rectangular
source is obtained.
INTERFERENCE, DIFFRACTION, POLARIZATION 343
The principle of the apparatus is based on the constant devia-
tion prism which is illustrated in fig. 209. The faces particularly
concerned in the deviation of the ray are inclined at the angles
marked in the figure, and total reflection occurs at the face AC.
The prism is mounted on a turntable in a position marked
for it. The mean deviation of the rays is a right angle, and in
order to pass through the spectrum the table is turned by means
of a screw to which a drum is attached provided with a milled
head (fig. 210). On the drum is a scale so that the wave length
of any line under observation and appearing in the field of view
of the telescope may be read off directly.
Fig. 210
Before taking any observations of wave length it is necessary
to adjust the prism accurately so that the correct wavelength
is indicated when the corresponding line appears at the eyepiece
indicator.
To make this adjustment, illuminate the slit with light of a
standard wave length, set the drum so that the appropriate
wave length is indicated at the index of the drum and adjust the
prism so that the line appears under the eyepiece index.
Clamp the prism in position with the screw provided.
Other wave lengths of light illuminating the slit may then be
determined by rotating the prism by means of the drum until
the line appears at the eyepiece index and reading off the number
against the drum index.
Fig. 209 indicates the course of a ray in the prism.
In the apparatus designed for use with certain accessories —
the Lummer-Gehrcke Parallel Plate, the Fabry-Perot Etalon or
the Michelson Echelon Grating, the collimator arm is of greater
length than that illustrated in the figure to permit of interposing
the accessory between the prism and collimator.
344 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The method of fixing the collimator and telescope and of
obtaining different parts of the spectrum by rotating the prism
is very convenient and accurate ; moreover, the drum can be
rotated while looking through the eyepiece, and one is saved the
inconvenience of moving round with a rotating telescope.
A suitable standardizing wave length is the red line of the
helium spectrum, which has a wavelength of frequency 6678-1
Angstrom units (1 Ang. unit = io -10 metre). It is the shorter
of the two red helium waves. Or the sodium lines may be used.
These are separated by the prism and have wave lengths 5890*2
and 5896*2 Angstrom units respectively. It is a good plan to set
the instrument on one of these lines and check the setting by
turning to the other and noting if the reading gives the correct
wave length.
The slit may be illuminated by throwing an image of the
source on it. In the case of the sodium lines the source is
obtained in the us,ual way, and for a helium line throw the image
of the bright part of a helium discharge tube on the slit.
A protective metal cover for the prism table is provided.
When a photograph of the spectra is required the eyepiece is
removed and replaced by a camera with a suitable focus lens.
This is shown in fig. 211. It is capable of adjustment by tilting
so that the whole spectrum can be photographed and a vertical
Fig. 211
displacement enables the same plate to be used giving photo-
graphs one above the other. There is, of course, also a shutter
for exposure.
In addition to the usual vertical slit there is a horizontal
adjustable slit attached to a hinge so that it can be swung out
of the way or in a position covering the vertical one.
INTERFERENCE, DIFFRACTION, POLARIZATION 345
Thus we have crossed slits, and a small rectangular source can
be obtained by closing down both the slits as much as is required.
The Lummer-Gehrcke Parallel Plate
This piece of apparatus 1 s described in the " Annalen der
Physik," vol. 10, 1903, p. 457, and a very complete account of
its use and theory given.
The reader is recommended to refer to the original paper,
but we shall give below as much as is necessary for our purpose.
The plate can be used in connexion with the Hilger Constant
Deviation spectrometer, and is of quartz of refractive index 1*544.
It has the following dimensions :
Length 130 mm. ,
Width 15 mm.,
Thickness \\ mm. (approximately).
Like the echelon grating it produces spectra of a high order,
and has consequently a high resolving power. The approximate
resolving power in the present case is about 200000.
It is therefore well adapted for the study of complex lines
in the spectrum which in lower orders appear as single lines.
It may be used also to measure such small displacements as
occur under the influence of magnetic fields, and in the section
immediately following the present description we describe how
it may be used to determine the value, — , from observations on
J m
the Zeeman effect.
N
F3
Q
O
Fig. 212
The plate itself is shown in fig. 212 and mounted on its stand
in position on the spectrometer in fig. 213. In this figure the
screws permitting adjustment in the various directions are shown.
The action of the plate is illustrated in fig. 214. Here LM X
denotes an incident ray of monochromatic light making an angle,
i, with the normal to the plate. After refraction the angle is r.
The figure shows the production of two beams emerging from
the plate on opposite sides. The rays, M x i, M 2 2, M33, etc., and
Nil 1 , N 2 2 x , Njtf 1 , etc., on account of their different courses due
to successive reflections and refractions, are in different phases
when they reach the position denoted by 123 and zH 1 ^ 1 respec-
tively. These two traces mark out wave fronts and the emergent
beams produce interference bands. The upper beam consists of
a system of bands with alternating intensities, the maxima
346 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
having an intensity which may be measured by 2 J, while the
minima have intensity, J. Thus the effect is the same as is
obtained by imparting to the whole field an intensity, J, and
drawing bands across it of double the intensity. The transmitted
system, however, consists of a set of maxima of intensity, J,
with alternating minima of nearly zero intensity. For con-
venience in observation the second system is to be preferred,
and, by using light at grazing incidence so that i is a right angle
and r the critical angle for quartz the sharpness of the lines is
increased.
Fig. 214
It is not possible to give here the reasoning which leads to
these statements ; the discussion is given very clearly in the
original paper.
These points find application in the Hilger pattern.
A slot will be noticed on the right of the carrier of the plate,
fig. 213. Just opposite this is a prism of such dimensions that
the light along the directions MiNj, M a N 2 , etc., is in the critical
direction.
The prism lies on the under surface of the plate, so that the
beam emerging from above is the transmitted beam.
The other beam is absorbed by the black lining of the stand
on which the lower surface of the plate rests.
In fig. 212 the prism is represented at OQ.
The Theory of the Plate
Corresponding to every direction, *", of incidence there will
be in the wave front, i^^ 1 . . . , a particular variation of
phase from point to point on account of the different courses
I'UV :,.:i.
INTERFERENCE, DIFFRACTION, POLARIZATION 347
taken by the rays. These rays are received by some optical
instrument, for example, the telescope of the spectrometer, and
focussed in the focal plane. Corresponding to the waves drawn,
we shall have a point image in the focal plane. These rays,
however, are those lying in the plane represented by the paper.
Above and below them lie rays coming from rays above and
below LMp and parallel to LM X , whose courses are exactly
similar, so that in the focal plane above and below this point
image lies a series of points forming a line. We shall work out
the intensity corresponding to this line. If we consider a slightly
different direction of incidence, i + 8i, the emergent beam is
also slightly different in direction, and in its course through the
plate. The phase differences in the corresponding wave front
will thus be different, and the corresponding line in the focal
plane will have a different intensity.
A wave travelling along a direction denoted by r will at a time
t, be represented by :
a sm 2tc
(t -{)
Here T is the period of vibration in the wave, x the wave-length,
and a the amplitude, t is the time measured from some con-
venient instant, and r the distance from a convenient origin.
In our case the origin will be conveniently the point, N x ,
and we shall consider the wave motion corresponding to the
position, i^^ 1 , at which the time is measured by t.
On refraction into the plate there is a diminution of intensity,
and since the intensity is proportional to the square of the
amplitude of the wave motion we can express this by regarding
the refraction as causing a diminution of amplitude, so that
amplitude a in the air becomes sa in the quartz, where s is a
fraction. On passing out at Nj, the amplitude is again reduced
by a fraction, s 1 . The amplitude for the ray Nji 1 is thus
We may not suppose that s and s 1 are the same, since in one
case the refraction is from air to quartz, and in the other from
quartz to air.
There is also a change of intensity on reflection at the points
M 2 , M 3 , M 4 , etc., and N x , N 2 , N 3 , etc. We shall suppose this to
diminish the amplitude each time by a fraction, a.
Thus the ray, N^ 1 , has undergone refraction at M x and N 2 ,
and reflection at Nj and M 2 . The incident amplitude, a, is thus
reduced to oHsfy ; and similarly the amplitude of N33 1 is
The disturbance which gives rise to the ray, N 1 i 1 , starts out
348 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
from N x with an amplitude, ss^, and at i 1 , since it has travelled
a distance, r lt we may represent the displacement by :
ss^ sin 2tz
«-?>
Denote the foot of the perpendicular from N 2 on N^i 1 by A,
and from N 3 on N 2 2 x by B, and denote the equal distances,
N 2 A, N 2 B, by e. Let the distance, MjNj be denoted by <5,
and the refractive index of the quartz by [*. Then the distance
traversed in quartz is equivalent to the distance, [l6 in air.
Thus the point 2 1 is at an equivalent distance [r 2 + 2jx<5)
from N lf and since r t = r 1 — e we may denote the displacement
at 2 1 by :
„ , . ft r x — c + 2fi<5\
aHsH sm 27tf ^ * — J ;
and similarly the displacement at 3 1 is :
. . ft Y x — 2e + 4tx<5\
o^s 1 ^ sin 2rc( — 5 x )
There is, of course, an indefinitely large number of such terms
as the three given above, and since all these rays are focussed
at a point in the focal plane of the telescope, the total effect at
this point is obtained by adding together all the terms.
In practice we shall have a small parallel bundle of rays
falling at M lf giving rise to small parallel bundles at N x , N 2 , N 8 ,
etc. Throughout the bundles there is, however, the same phase,
as may be seen by considering a ray parallel to LM X , falling at
any angle, i, to the normal. Thus the total effect is merely
multiplied by some constant factor on account of the incidence
of more light than that we have supposed is represented by the
ray LM X .
Thus we have a total displacement, Y\ where :
Y = SS %sin 2,(| -£)■+ «Wsin a, (I- r - ~ ;+ 2 »> )
, . . ft r x — 2e + 4tid\
-|- ss x a<s* sin 2rc(=r - 1 + ...
A typical term may be written :
»>..» sin a, j (i - &) + p ■ i^| ,
where p has the values, o to infinity. *
If we write a = 2ir (f ~ ^) ' P = T ^ ~~ ^'
we have :
Y = Xss l a{o& sin (a - p$)} (7)
* In practice the upper limit to the value of P is about 15, the dimensions of
the plate permitting about 15 reflections. The terms beyond the fifteenth are
email and contribute but little to the value of Y.
INTERFERENCE, DIFFRACTION, POLARIZATION 349
The sum of this series is finite since a is a fraction, and has
the value
sina — o 2 sin (a + ft)
1 — 2a 2 cos ft + o 4
For the proof of the summation the reader may be referred to
Hobson's " Plane Trigonometry," §76, p. 91.
Thus the resultant displacement is :
_ r , sin a — a 2 sin (a + ft)
1 — 2a 2 cos ft + o 4
It will be noted that for any particular direction through the
plate the quantities, e and 6, are constant, so that ft is a constant.
s, s 1 and a are also constant terms so that it is only a that
contains the variable time, t.
sin a — a* sin (a + ft) = sin a (1 — o 2 cos ft) — cos a • a 2 sin ft.
We may write this
= A sin (a. — <f>)
■u a. c 2 sin ft
where tan <t> = _— ,
1 — o 2 cos ft^
and A — Vi — 2a 2 cos ft 4- 0*/
===== ' sin (a -
2a 2 COS ft + o 4
Thus Y = . • sin (a — tf>)
Vi —2a 2 COS ft + o 4
sra lT-* — r)
Vi — 2a 2 COS ft + o 4
This denotes a simple harmonic vibration of amplitude,
55 x a
Vi — 2<J 2 cos ft + a 4 '
and it measures the amplitude on the line in the field of view
of the telescope due to the reception of the emergent beam
along the direction which makes an angle, i, with the normal
to the plate. The intensity is proportional to the square of this
amplitude, and is therefore a maximum when (1 — 20 2 cos ft+o 4 )
is a minimum, and a minimum when this expression is a maximum.
But 1 — 2« 2 cos ft + c 4 = (1 — a 2 ) 2 4- 4a 2 sin 2 Jft.
This has a minimum value when sin \ ft has the value zero,
and a maximum when sin \ ft has the value unity.
Thus lines of maximum intensity correspond to the value :
ft = o, 2k, 47c, etc.,
and lines of minimum intensity correspond to :
ft = *> 3*> 5"» etc.
But ft = (2(i<J — e).
350 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Thus for a maximum,
2[x<5 — e = n\,
and for a minimum,
2y.8 — s = (n + £)x,
and n may have any integral value.
If d denote the thickness of the plate :
8 — d sec r,
and c = MiMa sin *
= 2d tan r sin i = 2^ tan r • {/. sin r.
Thus 2{A<5 — e = 2yd cos r,
and the bright bands lie in directions given by :
2yd cos r = n\.
By proceeding further with this discussion, Lummer and
Gehrcke have drawn the conclusions, to which we have referred
above, concerning the intensity of the bands.
Students who prefer it may substitute the following proof
which enables the same formula to be derived shortly, but in a
way that does not lead to any expression for the intensity of the
lines.
The path difference between any two of the rays, say Nil 1
and N 8 2 x is equal to :
2(X(5 — e = 2yd cos r.
We may take all the rays emerging from the plate in pairs
which are separated by. the same distance, equal to N X N 2 .
These therefore interfere and will produce darkness or bright-
ness, according as this phase difference is equal to wx or (w+|)x,
where n is a whole number as before.
Thus for bright lines :
2yd cos r = n\.
If we refer to fig. 213 and note the position of the plate it is
clear that the different orders come out from the plate above
one another. As the path difference increases the angle of
emergence decreases, so that the higher orders will lie higher in
the field of view than the lower in the case of a telescope with
an erecting eyepiece.
In our case the mean angle, r, is the critical angle and there is
grazing incidence. The different orders correspond to angles
very slightly differing from grazing incidence.
By substituting the value, y. = 1*544, and the value for the
critical angle, r = 40°22 ', together with the value of d given above,
we find that for X = 5890 tenth metres (10- 10 metre), the order
is approximately 18000.
In order to set the instrument in position, the spectrometer
is first set up in the manner previously described, and the slit
illuminated by means of some convenient monochromatic light.
Paga 351
INTERFERENCE, DIFFRACTION, POLARIZATION 351
The plate is placed in position and adjusted by the screws until
the brightest image is obtained in the eyepiece. Since the
orders lie one above another, a vertical slit cannot be used, for
the different orders will appear overlapping the image of the
slit. Thus the crossed slits must be used with a small rectangular
source. The length of the slit need not be very small, for on
account of the dimensions of NO, fig. 212, only a fraction of the slit
is effective in producing bands. The slit should be small enough
to avoid overlapping, but wide enough to produce intense bands.
The images may then be viewed or photographed as desired.
We have throughout disregarded the possibility of a change of
phase on reflection at the two surfaces.
The Fabry-Perot Etalon. (" Annates de Chimie et de Physique"
1897, Ser. 7, p. 459.)
This apparatus is another means of obtaining a high resolving
power, and is in many respects similar to the Lummer-Gehrcke
Plate. Fig. 215 shows the apparatus in position on the arm of
a constant deviation spectrometer.
Fig. 216 shows the two plates, ABCD and FGHE, of which
it is composed. These are placed accurately parallel, and are
separated by a distance piece consisting of a hollow cylinder of
fused silica. This substance has an extremely small coefficient
of expansion, so that the distance between the plates may be
regarded as independent of the temperature.
A B E r
D C H G
Fig. 216
The faces, BC and EH, are silvered by cathodic deposition
in order to increase their reflecting power but so as to leave
them partly transparent.
Light entering the plates therefore undergoes multiple reflec-
tions between the silvered faces and produces also partial
transmission through the opposite face.
The faces, AD and FG, are inclined to CB and EH in order
to avoid interference effects that would occur through multiple
reflections and refractions if all were parallel, but AD and FG
•
352 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
are parallel, so that light incident on one side leaves the other
undeviated.
Fig. 217 illustrates the action of the plate. The air space
between the plates is shown, and the course of a ray as it is
multiply reflected between the plates. The refracted portion
leaving EH is shown, but the refraction at the other face is
omitted for convenience. We consider an incident ray, LM X ,
from the instant of its arrival at a point, M v within the first
wedge and immediately before refraction into the air space.
L
^ Y
Fig. 217
The wave emerging within the second plate is shown, and
XY denotes a wave front.
The wave is refracted out at the surface, FG, into the air,
but as no further change of phase occurs after the light leaves
the air film, we do not need to consider the wave beyond XYZ.
Let the wave on starting out from the point, M x , have, in the
air space, an amplitude, a, and let its period and wave length
be T and x, respectively. Whenever the wave passes from the
air space into the second plate, let the amplitude be reduced in
the ratio, 0, and on reflection at the partially silvered faces let the
amplitude be reduced in the ratio, /. Both these quantities are
positive proper fractions.
Thus the wave proceeding along M X N X may be denoted by :
y = a sin 2n ( - — -J,
where r measures the distance from the point, M x , at which the
displacement, v, is considered.
Let the equivalent path difference between M x and the points,
X, Y, and Z, differ successively by an amount, 6.
INTERFERENCE,, DIFFRACTION, POLARIZATION 353
Then proceeding as in the theory of Lummer-Gehrcke Plate,
we have :
8 = 2e cos i,
where e is the distance between the plates, and * the angle
of incidence at the surface, EH.
Thus the displacement at X may be written :
da sin 2n
\T x>
where r x denotes the equivalent air path between M x and X.
The displacement at Y, which after two reflections has been
reduced by the ratio, / 2 , and for which the equivalent path is
('1 + <5), is given by :
and similarly at Z we have :
tf% ana. (i - i±»?).
Of course there are many more points such as X, Y, Z ; and
if we continue the series of terms until the contributions become
negligible, and if we view the rays in a telescope, we have, for
the total displacement in the field of view of the telescope, a
quantity :
Y = da |sin27c (y~^) +/ 2 sin27r (i-^ 1 - a)
+/ 4 sin 2k (| — £ - 2AJ + . . .J, (8)
where A is written for - •■
x
Strictly, A ought to include a quantity due to change of phase
on reflection at the air-glass surfaces ; if this is w then A — - + w.
In order to compare this with equation (7) above, we may
note that the quantity, a, is the same for each, that 6a takes
the place of ss^, and that / takes the place of o, while 2* A
corresponds to p.
If, therefore, we write p = 2tcA, equation (8) may be written :
Y = S Oaf* sin (a - p$) (9)
where p has the values, o to infinity. *
Thus as before, the value of Y is :
, sin j 2ir ( — — -^ ] — </> I (10)
Vi -2/ 2 cos p+/* ( \T x/ / K '
2 3 *Strictly p has an upper limit between 10 and 20. See p. 348
354 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
where :
. . /a sin , x
tanft = J ~ a .. . (ii)
i — / 2 cos p x '
Thus the intensity of the line appearing in the focal plane of
the telescope, corresponding to this particular direction, is propor-
tional to the square of the amplitude of this expression, or we
may measure the energy by the expression :
<W
(i -2/ 2 cosp+/*)'
This may be written :
2 fl a
(i -f 2 )''
1 + ( T^p sin2TCA
Thus the intensity fluctuates between a maximum and mini-
mum value, the former of magnitude :
fl 2 a 2
(i -/ 2 ) 2 "
This may be large provided that the value of / is not far from
unity.
Denote this by I .
Then we have for the intensity :
I„
I =
I + (Wr sin27cA
The minimum of I corresponds to the value for A which makes
sin 2 reA = i,
and the magnitude is consequently :
(h^ < a >
Now with / not very different from unity, this may be a very
small fraction of I , so that there is a great contrast between
the maxima and the minima.
Another point may most conveniently be brought out by a
numerical example.
Suppose / = '87 = ( y^ ) — this value has been chosen
for convenience.
Then
1 = h
1 -f 48 sin 2 7cA
INTERFERENCE, DIFFRACTION, POLARIZATION 355
A maximum occurs when A = m, and the next when
A = (m + 1), tn denoting an integer.
Consider the case when A = m -f ^,
i.e. when we have gone over ■- of the interval between the two
values of A.
In that case it follows that :
I =
Io
1+48 sin 2 —
^10
= i I , approximately.
This means that the intensity falls off quickly as the maximum
position is left, so that we have bright lines in the field of view
separated by a comparatively long dark interval.
Thus, if there is a second ray in the field of slightly ^different
wave-length, its lines will not overlap those of the other ray
unless there is very little difference indeed between the two
wave-lengths.
From the above theory we see the influence of the partial
silvering in producing sharp bright lines on a background that
is almost black. If a layer of air is used between two plates
without any silvering, the reflecting power is small, and we have
fringes produced and superposed on a field of uniform illumination
with no comparative broad spaces between the fringes.
We can explain these points quantitatively by the aid of
some numerical examples given in the original paper.
It is usual to speak of the reflecting power of a surface and
not of the quantity we have denoted by/. But since the intensity
of a ray is proportional to the square of the amplitude, and since
the amplitude on reflection is reduced by /, we have, if the
reflecting power is denoted by R,
R=/»,
so that the expression denoted by (A) above gives for the ratio
of the minimum to the maximum intensity :
p
If R = -042
P = -84,
so that for a small reflecting power the maxima and minima
have nearly equal intensity.
If, however, R = -74,
P = -02,
and the minima are very feeble.
356 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The way the intensity falls off rapidly as the maximum is
left has already been shown. The value for / taken above,
corresponds to a value of R = 75, approximately.
Fig. 218 shows graphically the difference in the two intensity
curves corresponding to displacement in the field of view. The
continuous line is the curve for the Fabry-Perot silvered plate,
while the dotted curve is for plates of low reflecting power.
Displacemcnr in Field of VieuJ
Fig. 218
Note in the one case the broad intervals of practically no
intensity and contrast with the other in which the intensity falls
off slowly, leaving comparatively small dark intervals.
Lines due to a second ray of slightly different wave length
from that for which the second curve is drawn would produce
crests over the dark intervals and leave the field nearly uniformly
bright, and it would not be possible to distinguish the separate
fringes.
The determination of the Ratio, — , for an Electron by means of the
m
Zeeman Effect
When a monochromatic source of light is placed in a magnetic
field, and rays are received in a direction parallel to the direction
of the field, it is found that the normal frequency is changed,
and two or more lines appear symmetrically displaced from the
usual position in each direction.
The light in each case is circularly polarized ; the component
of higher frequency is polarized in the opposite direction of
rotation to that for the lower.
Reference for the theory of this phenomenon should be made
to " The Electron Theory of Matter" (O. W. Richardson).
If H denotes the intensity of the magnetic field, and its direction
is parallel to that of the light, the increase and decrease in
1 eH.
frequency of the two components is of magnitude — • — >
and H is measured in ' gausses,' the name given to the electro-
magnetic unit of magnetic field.
In the experiment we measure the difference in wave length
INTERFERENCE, DIFFRACTION, POLARIZATION 357
of the two components, i.e. we measure <$x, corresponding to a
change of frequency,
*» = — --H.
2-nm
Now &v = -g <5x, (numerically),
since c = v\
1 X 2 e
so that we measure <5x = — — H . —
2.tz c m
tt e c 5\ , '
Hence: -.^.g.- (l2)
is measured in electromagnetic units.
An electromagnet with adjustable pole pieces and taking a
current of three amperes gives good results with an ordinary
vacuum tube.
A hole is drilled in the pole pieces so that the source may
be observed along the direction of the magnetic field.
The field is measured by means of a fluxmeter, by which the
total flux through an exploring coil placed between the pole
pieces is measured. This will be read directly on the fluxmeter.
Suppose it is B, Maxwell's. If A is the effective area of the
coil and H the field, on placing the coil into the position, where
•p
H is required, the flux is HA, so that H = -^, and H is in gausses
(see experiment on Grassot Fluxmeter, p. 482.)
We have now to determine <5X by means of the Lummer plate.
First align the apparatus and drilled hole as nearly as possible.
Place a sodium flame on the far side of the hole, and obtain the
shorter sodium lines as sharply as possible with the drum set at
5890.
Place a helium tube or other convenient source in position
between the pole pieces, and obtain the yellow line. The yellow
line .was used in an experiment and satisfactory photographs
were obtained, but it would be preferable to use the blue line
so that the exposure is not long, and ordinary photographic
plates (special rapid) may be used. We shall, in what follows,
consider the yellow line to have been employed, and the wave
length is then 5875-6 A.
Adjustments will be carried out as described above, and the
best possible definition and illumination obtained.
On applying the magnetic field it will be observed that the
lines broaden, and in some cases actual separation will occur,
as in fig. 219.
358 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The state of polarization may be tested by intercepting the
light just before it enters the object glass of the telescope by a
quarter- wave plate.
The plate must, of course, be suitable for the particular wave-
length concerned — the usual plate met with in laboratories will
be suitable to examine the yellow light. This part of the experi-
ment is introduced so that the student may take the opportunity
of verifying the character of the vibrations in the lines.
Fig. 219
The quarter-wave plate reduces the circular vibrations to two
linear vibrations at right angles to one another, and on examining
these with a Nicol it will be found that when one line is cut
out the other is present in the field.
After carefully adjusting the apparatus so that the effect is
seen by eye, attach the camera and photograph the spectrum
without the magnetic field. By means of the rack and pinion
raise the plate, put on the field and take a second photograph
below the first.
The plates when developed will give lines as in fig. 219.
Separation of the lines is seen in the higher orders, e.g. at A
and B. Let a denote the line when no field is applied to the
source, and A the corresponding line with the field applied
Measure the mean displacement of the components of A by
means of a microscope, and measure the separation between the
two successive orders a and b.
Denote these distances by I and L respectively.
We proceed to work out a formula, showing how <5x may be
. /
derived from the ratio j-
By the equation for the Lummer-Gehrcke plate :
2,\sd cos r = n\, (13)
we see that in proceeding to a neighbouring order (n -f- 1), the
angles of emergence and refraction are (*' -j- 51) and (r + <5R),
INTERFERENCE, DIFFRACTION, POLARIZATION 359
where: 2d\i cos (y -f <5R) = {n + 1) x (14)
From (13) and (14) we have by subtraction :
— 2d tan r cos * <5I = x •••(15)
In this step we use the relation : sin (*' + 61) = y. sin (r + <5R),
and consequently cos idl = y. cos y<5R (16)
There is no variation in n since the wave length remains
constant ; we pass merely to a new angle of emergence.
<5I represents the angle between the rays which produce two
consecutive lines as a and b on the photographic plate (fig. 219).
Now consider the difference in direction, 5i, for two lines in
order, n, of wavelengths, x and x + <5X. ,
It is an angle, 8i, which corresponds to the displacement
between the components of A.
Since the plate is fixed in the sjfectrometer and the angles are
small :
8i I . .
sT=L <M
Referring once more to equation (13) we have in the order, n,
a wave length, X, corresponding to a direction measured by *,
and a wave length, x + 8\, corresponding to * + Si.
By differentiating (13) we find :
— 2d[i sin rdr = w<5x — 2d cos r -=- <5x (18)
and from the equation,
sin * = [t sin r,
. . . du.
fji cos rdr = cos 1 St — sm r j~ <5X (19)
In this case it is necessary to take account of the variation
of (a since the wave length changes.
By eliminating 8r from (18) and (19) it is found that :
— 2d tan r cos i 8i = (n 4- J <5x ; (20)
\ cosy d\J v '
and on substituting for n from (13) :
— 2d tan r cos i 8i — 2 ( - cos r — ^ \ 6\ (21)
\X cosy d\J v '
Thus from (15), (17), and (21) :
I 6i 2d/y. I d\x \ . , .
=-= 7r= — ( —cos r • — - ) <5x (22)
L 81 x\x cosy d\J K '
The value of y is the critical value for quartz, and it remains
to determine j- for the particular wave length used in the
360 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
experiment. The table below showing values of n corresponding
to different values of x should be used to draw a graph from which
~ may be obtained by measuring the slope at any point where
it is required.
From the equations (12) and (22) we find :
2.TZC I
W *L
e
m
2,&
(>
cosy
COS Y
di)
(23)
and the units are electromagnetic
If possible take the ratio-y- for more than one order — several
orders will usually show sufficient separation for this purpose.
REFRACTIVE INDEX OF QUARTZ IN THE VISIBLE
SPECTRUM AT io°
WAVE LENGTHS (IN (Jt[x)
REFRACTIVE INDEX.
396
I-558I
410
I-5565
434
1-5540
486
1-5497
508
1-5482
533
1-5468
589
1-5442
643
1-5423
686
1 -5410
760
1-5392
768
1-5390
Measurement of Wave Length by Diffraction at a Straight Edge
The theory of this experiment will be found in Schuster's
" Theory of Optics " in Chapter V.
In the first few sections of the chapter it is shown that on
observing a straight edge illuminated by monochromatic light
from a narrow slit a series of alternate bright and dark bands
will be seen.
If S denotes the slit (fig. 220), EF the straight edge fixed
parallel to it, and PQ a plane perpendicular to SE, we
shall have the series of bands along PQ. Let Q be the nth
bright band from P, where P is the point of intersection of SE
and the plane.
INTERFERENCE, DIFFRACTION, POLARIZATION 361
Denote PQ by x n , then it can be shown that :
(24)
Xn=s ^PH4n~i)(p-hq)
In this formula p is the distance, EP, and q is equal to SE,
the distance between source and slit.
F c
Q
Fig. 220
The bands are not regularly spaced as are interference fringes.
To carry out the measurement set up in one of the stands of
the optical bench a straight edge parallel to the slit which is
illuminated by sodium light. First adjust the parallelism by eye,
and finally make a slight rotation by the screw of the holder until
the bands are most distinct ; then make the slit as narrow as is
compatible with sufficient illumination. Observe the fringes
with the travelling microscope, and measure the distance between
the first or second bright band and one of the most distant
that can be seen plainly.
If this is the nth and the first observed is the mth we have :
x n -x m = ^(£+g) |v^=l - V^t^i] . . . .(25)
The student should take the opportunity of observing the
diffraction fringes arising from the light passing a narrow wire,
needle point and narrow slit
In the case of the narrow wire diffraction bands unequally
spaced will be seen outside the geometrical shadow. Within it
a series of equally spaced bands will be observed. These may
be described as interference fringes due to the two parts of the
wave, one on* either side of the wire. The effects of these are
equal to those of the two half period zones which lie at the
edges of the wire so that they act like two sources at a small
distance apart.
362 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Determination of the Wave Length of Light by means of a Plane
Diffraction Grating
A diffraction grating is made by ruling a large number of
equidistant parallel straight lines on glass. The lines are ruled
by a diamond point moved by an automatic dividing engine
containing a very fine micrometer screw which moves sideways
between each stroke. A photographic replica of a plate made
in this way is often used in its place.
The number of gratings to the inch is marked on the glass.
In handling the grating do not touch the faces of the glass,
hold it between thumb and finger by the edges.
Adjust the collimator and telescope for parallel rays in the
usual way and observe the direct image of the slit noting how
it lies in the field of view.
Set up the grating with its face normal to one side, EF, of the
triangle formed by the levelling screws (cf. fig. 161). Throw an
image of the slit into the telescope by reflection from one face
of the grating and adjust the screws to bring it into the same
part of the field as that occupied by the direct image. This
makes the faces vertical.
In order to set the grating at right angles to the rays adjust
the collimator and telescope at right angles to each other, and
turn the table until the slit is reflected on the cross-wires of the
telescope. Then turn the table a further 45 .
It now remains to tilt the grating so that the lines are parallel
to the slit.
View the first diffracted image, making the slit as narrow as
is convenient, and adjust the screw D until the best image is
obtained. The lines and slit are then parallel.
Find the diffracted images on both sides of the line of direct
vision. It will be easy to observe two orders, and if a bright
light is used and an image of the brightest part of the flame
thrown on the slit by a short focus lens the third may be seen also.
If (a + b) is the width of the grating element the formula is :
(a + b) sin = wX,
for normal incidence, being the angle of diffraction, X the
wave length of the incident light, and n the order of the spectrum.
Obtain by taking half the angular distance between the
corresponding images on each side.
(a + b) is deduced from the number of rulings on the grating.
Use any source of monochromatic light or light giving well-
marked lines as that from a discharge tube containing hydrogen
which gives three well-marked lines, red, green, and violet, known
as C, F and Hy.
INTERFERENCE, DIFFRACTION, POLARIZATION 363
The above formula is obtained by considering rays in pairs
passing through adjacent clear spaces of the grating.
For example, consider the two spaces AB and CD.
We may consider the rays passing through them in pairs,
taking together those rays which are symmetrically situated, as
for example, QLQ 1 and TMT 1 .
These reach the grating in the same phase, and the line ABCDE
represents a section in the plane of the figure of the wave front
incident on the grating.
Fig. 221
Consider a wave front, WW, at a later interval, and suppose
that it makes an angle, 0, with the plane of the grating.
Draw the rays AP 1 , BR 1 , etc., perpendicular to this wave front
and making the angle, 0, with the normal, AO, to the grating.
These rays are all received by a telescope and united in the
focal plane. Their paths from the grating to the instrument
differ, so that they reach the focal plane with different phases.
This phase difference is due to the difference of path traversed
after leaving the grating.
Take the case of the two rays, AP 1 and CS 1 . Their path
difference is AN, where N is the foot of the perpendicular from
C on AP 1 . Any other pair of rays, e.g. LQ 1 and MT 1 , situated
symmetrically in AB and CD have the same path difference.
This holds for all pairs of rays symmetrically situated. Thus
the phase difference on arrival in the apparatus that receives
them is the same for each pair.
We shall thus have reinforcement if AN is equal to a whole
number of wave lengths.
Now AN = AC sin 0.
364 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If a is the width of a space and b that of a line,
AC = a + b,
(a + b) is called a grating element. Thus a bright line occurs
in a direction 0, provided that
(a + b) sin = n\.
The Plane Reflection Grating
This reflection grating consists of a plain polished sheet of
metal across which parallel lines are drawn very closely together
as in the transmission grating.
It is necessary to mount the polished face vertically and this
is done as in the previous case.
The light does not fall from the collimator normally on the
grating, but at a measured angle, *.
In order to measure i, the grating is first set with the polished
face normal to the rays from the collimator which, together with
the telescope, has been adjusted for parallel rays.
The table carrying the grating is turned through a definite
angle, say io°, and is then fixed.
Suppose rays from any direction make with the normal to
the grating and are received in the telescope.
Consider any three rays, MA, NB, and OC (fig. 222), incident
on the grating element ABC. AB is polished while BC is the
position of the line where the polish is scratched and where the
incident rays are absorbed.
A B C D
Fig. 222
The path difference between the extreme rays MAQ and OCS is
AF—CE, where the dotted lines, AE and CF, denote incident
and reflected wave fronts respectively. If AB = a, BC = b, this
difference is equal to
(a + b) (sin 0~sin *).
It is possible to divide up the bundle of rays falling on AB
and CD into pairs, one from each bundle, having this same
INTERFERENCE, DIFFRACTION, POLARIZATION 365
difference of path. In order to do this it is only necessary to
choose rays occupying the same relative positions in the two
bundles.
If this path difference is a whole numbsr of wave lengths,
and the emergent parallel rays AQ, BR, CS, etc., are brought to
a focus by a telescope, we shall get a bright image due to reinforce-
ment of the rays. We thus have :
(a + b) (sin 0*~sin i) = n\,
where n may have the values 1, 2, 3, etc. The corresponding
values of 0, say, B lt 2 , 3 , etc., give the directions in which
the different orders of diffracted images are seen.
If the observation is made from a direction on the same side
of the normal as i, the formula is :
(a + b) (sin i -f sin 0) = nx.
Obtain first the reading of the scale marking the position of
the telescope when it is directed towards the slit with no grating
intervening. As above, set the normal to the grating at some
definite angle, *, to the incident rays. Move the telescope round
to receive light in a direction corresponding to = i on the
opposite side of the normal to *. This corresponds to order zero
— it is the position where the ordinary reflected beam is received.
Now move round to the next position where a distinct image
can be found. This will correspond to n = 1. Carry out this
process as long as it is possible to observe images at all. The
higher orders get fainter but resolve the lines more than the
lower.
The number of lines to the inch is given for the particular
grating, so that it is possible to deduce (a + 6), which must be
expressed in centimetres. From the values of *, and n, it is
then possible to evaluate X. It should be possible to obtain
separation of the sodium lines in the second order, and the
wave length for each constituent should be calculated.
Use three or four different values of i and compare the value
of x obtained, finally take the mean of the series.
Resolving Power of a Telescope
Theory
Let a parallel beam of monochromatic light fall on a slit, AB,
and let us examine the intensity of the light along a direction
CE, inclined at an angle, 0, to the normal to the slit. Let the
light in this direction be brought to a focu$» at E. From A draw
ADF perpendicular to the beam so that z. BAF is 0. The rays
issuing from the slit on arrival at E will have different phases
on account of the varying length of their paths from AB to E.
We may measure these differences by such lengths as PM, varying
in amount from zero at A to a maximum of BF for the extreme
ray from B.
366 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Divide the slit, AB, into a very large number of equal parts,
such as PQ. If we make each of these small enough, all the rays
in the element of which PQ is a section, may be supposed to be
in the same phase on arrival at E. Suppose that the rays from
each small element differ in phase by an amount, 3, from the rays
of the adjacent element. Each element will be supposed to
contribute an amplitude, A, to the whole beam so that if there
were n slits, and all had the same phase on reaching E, the
amplitude would be«A.
Let the disturbance at A be denoted by ' A sin pt.
This is an expression which represents the displacement in
any simple harmonic vibration, and rays of light afford an example
Fig. 223
of this type of vibration. The rays from the next element,
differing in phase by 6 must be represented by A sin (pt + <5),
from the next by A sin (pt + 28), and so on for all the elements
of number n.
Thus the total effect at E is obtained by adding up these
separate elements.
Let T measure this effect so that :
T = A {sinpt + sin (pt + d) + ... + sin (pt + n -1 <5)}.
BF
The maximum phase difference is measured by 2n -— radians.
INTERFERENCE, DIFFRACTION, POLARIZATION 367
Let <p denote this phase difference ; it is also measured by n — 1 8.
The above series for T can be expressed by the more convenient
formula :
T =
A sin (pt + - — — 3 J sin - <5
. 6
sm-
2
as is shown in textbooks on Trigonometry.
When 6 is very small, sin \d = £<5 ;
n A sm —
= «Asta l£sin( ^ + W)
in which |<5 is neglected because of its smallness.
When 9 is zero there is no difference of phase, and the direction
CE is normal to AB. In that case the amplitude at E is nk,
and we denote this by B. Thus the amplitude in any direction,
EF, is , *? , the factor sin {pt + \<p) denoting the oscillating
character of the disturbance.
The intensity is proportional to the square of the amplitude,
or the intensity at E is measured by : 4_ — ' sm t L.
rrw - • 1- sin 2 £? .
This is a maximum when — ^~ is a maximum.
9 2
If this function be examined for its maxima and minima by
means of the differential coefficient, it will be found that the
maxima occur at points where the values of - are :
o, i-437r, 2-4675, 3-4775, etc.,
and the minima at
rc> 27r, 375, 47c, etc.
At the latter values the intensity is zero.
The graph drawn in fig. 224 shows the relation between intensity
and values of \<p.
When the light is examined at H the intensity corresponds to
the ordinate at I, and on moving round we come to a place, E,
say,where there is darkness corresponding to the zero ordinate at 75.
In this case :
1 1 BF
i.e. BF^= X.
368 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Thus a minimum occurs in the direction in which BF == X.
Now the angular distance between CE and CH is the same
as angle BAF, and since this is small it is measured by
or x
(breadth of incident beam).
1
AB
^ M
bAstt
Fig. 224
If another beam were incident on AB in the direction along
CE, there would be a maximum for this beam at E, and it would
overlap the minimum of the former.
The intensity curves might then be represented on the same
diagram as in fig. 225. The curve on the left represents the
intensity curve for the second direction while the upper dotted
curve shows the two compounded.
Fig. 225
The resultant has two pronounced maxima with an appreciable
dip between. It will thus be possible by the aid of this falling
off in intensity between two bright regions to distinguish the
two beams or they will be 'resolved.' Moreover, the angle
between the two directions is ECH or X ~ (width of beam). This
is taken as a limiting case, and the resolving power is measured
by this ratio. If the angle between the incident beams is less
than this the two maxima cease to be distinguishable as two
and blend into one.
In order to verify this theory a telescope is fitted with an
adjustable slit which is placed as close as possible to the object
INTERFERENCE, DIFFRACTION, POLARIZATION 369
glass. The width of the slit is carefully measured by means of
a micrometer microscope.
A suitable object for this experiment may be made by coating
a sheet of plane glass with tin-foil and cutting two fine parallel
lines in the foil with a razor blade at a distance of two or three
millimetres apart. When these are illuminated with a sodium
flame they provide two bright slit-sources. This object should
be placed at different distances from the object glass of the
telescope, the aperture of which may be varied by means of an
adjustable slit placed immediately in front.
A certain minimum width of this slit will be found for which
the two lines appear as separate lines. This width varies with
the distance from the object glass to the two slits, and for smaller
widths the lines appear as one.
A table is made of the minimum widths of the slit and the
corresponding distances.
The angle subtended by the two fine lines at the object glass
is measured by== > where d is the width of the slit and D the distance.
The theory above described shows that the value of this angle
is-> where x is the wave length of the light, and a is the width of
the aperture.
The object of the experiment is to compare the theoretical
and practical resolving powers, the former determined by - and
the latter by ~*
An examination of the practical resolving powers measured in
the Wheatstone Laboratory of King's College, London, during
the past session has shown that the values obtained were about
20 per cent, greater than the theoretical value.
Repeat the experiment with two point sources and circular
apertures of different diameters.
In this case the field of view contains a bright small central
circle with concentric alternate bright and dark rings. If two
sources close together are such that the bright centre due to
one falls on the first dark ring of the other, the two sources just
cease to be distinguishable as separate.
Sir G. Airy has shown that if d is the diameter of the aperture,
the angle subtended when separation ceases,
e = i-22-j.
d
The point sources should be two fine holes in a sheet of tin.
They should be illuminated by monochromatic light and placed
at such a distance from the object glass that they just cease
to be seen as two holes.
24
370 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Place the telescope at different distances from the holes and
focus it on them. Adjust the aperture diameter until the holes
just cease to be distinguishable as two separate sources of light.
Measure the distance between the centres of the holes by means
of a micrometer microscope, and the distance between the plane
of the holes and the aperture by a metre rule. From these
measurements deduce 6 and compare it with the theoretical value,
x • v
i'22-j, in each case.
a
Polarization by Reflection. Verification of Brewster's Law
When light is reflected from surfaces the reflected beam is
partially polarized, that is to say, that the transverse vibrations
constituting the light have, on the whole, a greater component
along a particular direction than in any other.
Ordinary light is supposed to consist of a transverse vibration
which changes its direction in space, though of course always
in the wave front, so rapidly that on the average in any appreci-
able interval of time the component in one direction is the same
as that in any other.
The reflected light has lost this property and is polarized so
that it has a greater component in one direction. This direction
is normal to the plane of incidence.
The transmitted light has a greater component in the plane
of incidence.
8
S
M
Fig. 226
In the diagram (fig. 226), a ray, SA, is represented as being
reflected at a glass sheet, MM, so that AB is polarized. In
order to test the polarization the ray is received in an analyser,
which in some instruments consists of a Nicol prism. This- is
a prism of Iceland spar which is cut into two along a diagonal
plane and cemented together with Canada balsam. Iceland
spar has the property of dividing a ray of light into two rays
refracted in different directions and one polarized perpendicularly
to the other. The layer of Canada balsam serves to reflect one
INTERFERENCE, DIFFRACTION, POLARIZATION 371
of these rays to the side of the prism where it is absorbed by.
the blackened walls of a case, and the other ray is transmitted.
In this way the emergent beam is made to consist of light
vibrations all in one direction. Incident light with its vibrations
in this direction passes through the Nicol, while if the vibrations
are perpendicular to this direction the light is unable to get
through. Light with vibrations in any intermediate direction
has only the components parallel to the direction of trans-
mission passed on.
The vibrations transmitted are parallel to the shorter diagonal
of the end of the prism.
On examining AB with this prism it will be found that as
the prism is rotated there is a change of intensity in the trans-
mitted light. This means that AB consists of vibrations with
the components in one direction greater than in another or it is
partially polarized.
On altering the inclination of MM the alternations can be
varied in extent, and in one position the change between
brightness and darkness is a maximum. The angle of incidence
when this occurs should be noted by the help of the scale of
angles attached to MM. Theoretically the light should be
completely polarized in one position, so that for one particular
S
Fig. 227
setting of the Nicol darkness should be complete. In practice
it will be found that the light is not quite all cut out, though
with care a position will be found when, this is very nearly true.
The apparatus is represented diagrammatically in fig. 227.
Brewster's law is that for complete polarization :
tan t = {*,
372 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
where * is the angle of incidence and ^ is the refractive index
of the reflecting material.
This law should be verified. It will be found best to allow
light from a window on the opposite side of the laboratory to
fall on MM, and to adjust for the maximum effect, and afterwards
to set up a sodium flame as at S, and make the exact adjustment.
In another form of apparatus the Nicol is replaced by a second
glass plate which can be rotated about the direction AB as well
as inclined at different angles to the horizontal.
When the mirrors are parallel the polarized light in AB is
readily reflected by the second mirror, while on rotating it through
a right angle from this position about AB this polarized light
will not be reflected.
We shall thus obtain alternations in intensity on rotating the
mirror about the vertical axis, and there will be a maximum
effect for a particular inclination of MM, in which case tan i. =\j.
Rotation of the Plane of Polarization. Laurent's Saccharimeter
The essential parts of the saccharimeter are two Nicol prisms,
N x and N 2 , illustrated in fig. 228, one of which serves to polarize
a beam of light passing through it while the other analyses the
transmitted beam and detects its plane of polarization. These
Nicols are spoken of, respectively, as the polarizer and analyser.
6
» t^y o
L N, H
Fig. 228
r\7
N 2
When N x has reduced the light vibrations to a particular
direction, viz. parallel to the short diagonal at the end of the
prism, all the light transmitted by N 4 can pass through N 2 if
N 2 is oriented exactly in the same way as Nj, i.e. if its shorter
diagonal lies parallel to that of N lf and its length lies parallel
to that of N ; . We are here neglecting the diminution of intensity
due to absorption, which always goes on, since actual bodies are
not perfectly transparent. We mean that no light is cut out
in this case on account of polarizing effects of N 2 .
In this position the Nicols are said to be parallel. If, however,
N t is turned from this position through a right angle no light
from N! can get through N 2 , since N 2 is now so oriented that
the light vibrations falling on it are in a direction perpendicular
to its short diagonal, and such vibrations are not transmitted.
In this position the Nicols are said to be crossed.
Certain substances like quartz, and solutions like that of sugar,
INTERFERENCE, DIFFRACTION, POLARIZATION 373
possess tiie property of rotating the plane of vibration of light
as it passes through them, so that if N x and N 2 are crossed when
the active substance is not placed between them (in which case
no light will get through N 8 ),on inserting the active material,
on account of the change in direction of the vibration, some
light will pass through N 2 .
It is found that a rotation of N 2 in one direction or the other
will bring it into a position when the light is once more stopped.
This shows that the light is still polarized, but its vibrations
have changed direction in traversing the medium. We ought,
therefore, to be able to measure the amount of this rotation by
measuring the angle through which N 2 is turned ; but, unfor-
tunately, N 2 can be turned through an appreciable angle when
the light is cut out without any apparent return of the light.
This lack of sensitivity is overcome in the saccharimeter by a
special device.
Just in front of the polarizing Nicol on the side towards the
analyser is placed a semicircular sheet of quartz cut parallel
to the optic axis. The complete circle is made up by a semi-
circle of glass of such thickness that it absorbs the same amount
of light as the quartz. The position of this circle is at H, and
it covers the open end of N x completely.
F A
O ~ Q
Fig. 229
When the light falls on the quartz it is separated into two
components, polarized normally to one another, which travel
through the quartz with different velocities. Let one component
be represented by OI and the other by OQ just as the light reaches
the quartz plate, these are the components of a vibration along
OR ; It is here supposed that the ether particle is at O but
just moving in the direction OR, so that its component directions
are OQ and OI. As the disturbance passes through the plate
374 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
there will be a gradual change of phase between these components
on account of the differing velocities of transmission. After a
time the disturbance will reach a point in the plate where one
component displacement is along 01 while the other component
is along OQ 1 . These combine to give a displacement OR 1 .
The quartz plate is cut so that, as the disturbance just leaves
the plate on the other side, this difference of phase exists between
the components. The difference is one-half of a period, and the
plate is called a half-wave plate or half shade.
Of course the light which traverses the glass side proceeds
undisturbed, and its oscillations are still along the direction, ED,
shown in the upper part of the diagram, parallel to OR.
If N 2 receives this light when its short diagonal is at right
angles to OR 1 this component is not transmitted, and no light
from the quartz side gets into the eye, while part of the com-
ponent OR gets through and the glass side appears illuminated.
Generally, light passes in from both sides of the plate, but both
sides are not equally illuminated. When both sides present the
same illumination the principal plane of the Nicol is either along
AB or normal to it, for it is clear that in either of these positions
the components transmitted are the same for both sides.
It happens that the eye can readily detect a change from
the equality of illumination in both halves of the field, parti-
cularly if both halves are equally dark, i.e. when the Nicol is so
placed that the smaller components are transmitted.
If the Nicol is set for equal iUumination on both sides, and
an active substance is interposed, it will be necessary to rotate
the Nicol to find once more the position of equal intensities.
The amount of rotation measures the angle of rotation of the
plane of polarization.
S
Fig. 230
Another common method of bringing about an increase in
sensitivity is to use the biquartz. This consists of two semi-
circular discs of quartz fitted together to form a complete circle.
One of these rotates the plane of polarization of the incident light
in a clockwise, and the other in a counter-clockwise direction.
The amount of rotation per unit thickness varies with the colour
INTERFERENCE, DIFFRACTION, POLARIZATION 375
of the light. For a particular thickness the rays from a sodium
flame will be turned in opposite directions through a right angle,
so that if the short diagonal of the Nicol lies parallel to this
direction the yellow rays get through, and if the diagonal is
perpendicular to this direction these rays are cut out.
When white light is used it is robbed of the yellow constituent
when the Nicol lies in this latter position, and the colour observed
is greyish and is called the tint of passage.
It is easy to detect a slight change from this uniform colour,
for an appreciable change takes place to a partly blue and partly
red field, one colour belonging to each side.
Let LMNS denote one end of the analysing Nicol, and let UV
denote the direction of vibration of the light.
This light will be cut out if MS— the shorter diagonal— lies
normally to UV. A small rotation of MS counter-clockwise will
bring it into a position to cut out the light and so will a larger
rotation in the opposite direction, the sum of these rotations
being 180 .
It is thus not easy to decide which way the plane has been
turned. But if two lengths of the rotating substance be used,
one slightly longer than the other, the rotation for the longer
must be greater than for the shorter.
The direction of rotation of MS which shows a larger angle
in the case of the longer is the direction in which the rotation
has taken place.
Tubes of glass with carefully worked end-pieces are used to
carry the solution to be examined. The ends are held in position
by metal caps screwed against them. It is necessary to have
rubber washers between the glass ends and the tube to avoid
strain when screwing up ; for a strained end will produce rotation.
Find the amount of rotation for a solution of sugar in water
and deduce its specific rotation. This quantity is denned to be
the amount of rotation produced by one decimetre of solution
divided by the weight of dissolved substance In unit volume.
Let w grammes be dissolved in ioo ex. and suppose a length,
/ cms., produces rotation, 0. The specific rotation is :
6 w e
iot -. = iooo T'
I ioo lie
Repeat for various strengths of solution, and for different
lengths of tubes.
When several tubes are obtainable it is interesting to observe
the effect of causing transmission through different lengths of
a solution of a particular strength.
By this means it may be verified that the amount of rotation
376 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
is proportional to the distance traversed by the light in the
solution.
Another instructive experiment is to make solutions of different
known concentrations, which may be measured by the number
of grammes of substance dissolved in ioo c. c. of solvent, and to
measure the amount of rotation in traversing a particular distance
through the solution. A curve showing the relation between
the rotation and concentration should be plotted.
The Lippich Polarizing System
In modern polarimeters the half shade is replaced by a more
convenient method of dividing the field. In the most recent
polarimeter the field is divided into three parts (fig. 231), the
two outer similarly illuminated for all positions of the analysing
Nicol and the central portion which may be differently illumin-
ated from the neighbouring regions and which has to be matched
with the outer parts of the field.
Fig. 231
l n l
///
M
1 j
Fig. 232
The mode of producing the divided field is illustrated in
fig. 232. N is the polarizing Nicol and LL are two small Nicols
fixed in position and mounted in a brass cylinder in front of N.
The directions of vibration of the light emerging from the
cylinder and falling on the optically active substance are repre-
sented by the arrows, I, n, I. The central portion passes through
the Nicol, N, only, while the outer parts pass through N and L.
When the analyser, M, is turned so that it transmits vibrations
along a direction bisecting an angle between n and I the whole
field is uniformly illuminated. For all other positions of M the
field is not uniform.
Thus, to measure the amount of rotation of any substance
placed at A, the analyser is first put into the position correspond-
ing to uniformity of field. The substance to be examined is
then put into position and M again rotated until the field is
once more uniform. The angle of rotation measures the rotation
due to the active substance.
INTERFERENCE, DIFFRACTION, POLARIZATION 377
The optical system by which the field is examined is not shown
in the diagram. It lies to the right of M and is focussed on the
plane through the right-hand ends of the Nicols, LL. This
system has the advantage that it is suitable for the examination
of all wave lengths, whereas the half shade has to be constructed
for one wave length only. Fig. 231 shows at OCO how the field
is divided into three parts.
The Half Shadow Angle
The various devices employed to enable accurate observations
to be made in polarimetry, which have been described, produce
two beams of polarized light with vibrations in directions inclined
to one another.
In the Lippich system we have denoted the two directions
by I and n.
In fig. 233 these directions are denoted by B/ and Bw respec-
tively, and the angle between them is 20. This angle is called
the ' half shadow angle,' and the magnitude of this angle has
an important bearing on the question of sensitivity.
E
Fig. 233
Suppose that BC bisects the half shadow angle, and that DBE
denotes the direction of vibration of the light which traverses
the analyser.
When this lies at right angles to BC the intensities of the two
beams are equal as seen through the analyser, for the components
of the displacements transmitted, viz. BF and BG, are equal,
and the intensities are in the ratio :
BF 2 : BG 2 .
Suppose the analyser is turned through an angle, a.
Then the transmitted components are :
BZ cos ZBE 1 and Bw cos nED 1 ,
or BZ sin (0 — a) and Bn sin (0 + a).
Thus the ratio of the intensities is :
sin 2 (0 — a) : sin 2 (0 + a).
In photometric work it is assumed that the eye can detect a
difference of intensity of one per cent.
Thus, if we regard as a given angle, we may say that the
378 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
change in setting of the analyser of the amount a will just be
detected when
sin 2 (0 — a) = «99 sin 2 (0 + a).
From this equation it follows that when a is a small angle,
so that we may write :
sin a = a, and cos a = I,
a = -0025 tan 0.
If a has a small value the apparatus is sensitive, and it would
appear that the sensitivity is improved by making as small
as possible.
But as gets small difficulties arise on account of the fact that
the light is never plane polarized, it is always elliptically polarized
in practice.
may not be indefinitely diminished.
Polarimeters are usually fitted with a small movable arm
projecting from the tube which carries the polarizer. This arm
carries an index mark which moves over a scale. By means
of it the polarizing Nicol can be rotated so that the half shadow
angle can be adjusted within limits. A modern form of the
apparatus is illustrated in fig. 234.
The sensitivity is thus to some extent under the control of
the observer, who will discover as he becomes familiar with his
instrument the best adjustment for sensitivity which suits him.
The reader may be referred for a more detailed and complete
account of all these questions to the article on " Polarimetry "
in the " Dictionary of Applied Physics."
Soleil's Compensator
Sometimes the saccharimeter is fitted with a piece of apparatus
consisting of two quartz wedges. This is known as Soleil's
Compensator. Fig. 235 illustrates the apparatus. The wedges
are ABC and DEF, and these are mounted in metal holders
which can be moved by means of a rack and pinion, so that the
wedges are translated in either direction parallel to AC or DE.
Thus if a ray of light is passed perpendicularly to AC it is
possible to place varying thicknesses of quartz in its path. The
quartz wedges are cut so that the optic axis lies perpendicular
to AC and DE, and polarized light passing through them suffers
rotation of its plane of polarization.
The amount of rotation can be varied by moving the wedges
by means of the rack and pinion.
The quartz wedge is placed just in front of the analyser.
An index mark moves along a scale as the wedges are displaced,
so that a record can be made corresponding to each thickness
of quartz interposed.
i'aae 378
INTERFERENCE, DIFFRACTION, POLARIZATION 379
The analyser is first rotated until with the quartz wedges
occupying a convenient zero position the field is equally dark
on both sides, supposing that the polarimeter is fitted with a half
shade. The quartz wedges are then moved a small amount by
the rack and pinion, and the amount of rotation of N 2 necessary
to restore the uniformly dark field of view is recorded.
Fig. 235
By making a number of observations a curve can be plotted
which shows the rotation corresponding to the various dispositions
of the wedges.
When light has undergone a rotation before passing through
the wedges, this rotation may be counteracted by interposing
the correct thickness of quartz.
Thus suppose the Nicols and quartz occupy the zero position
described above, and that an optically active material is inter-
posed, the rotation resulting may be counteracted by displacing
the quartz wedges either so as to increase the thickness traversed
or to diminish it. When the appearance in the analyser is the
same as that of the zero position, we know that the wedges have
caused a rotation equal in magnitude, but opposite in direction
to that of the active substance.
Thus by observing the record opposite the index mark we can
deduce the amount of rotation due to the substance.
In plotting the graph, rotations in one direction will lie on
one side of the origin up to 180 °, while those in the other direction
will lie on the other side up to 180 .
CHAPTER XIII
PHOTOMETRY
Introduction
The light emitted from a small source is absorbed very little by
the air through which it passes, so that we may say that any
surface, surrounding the source completely, will receive the same
total amount of light.
Let this total amount be denoted by M.
= luii
Fig. 236
Imagine a cone with its apex at a small source of light and let
its solid angle be w. All surfaces receive the same amount
of light on the parts lying within this cone.
If w is small, say 6w, the cone may be regarded as denning a
particular direction and the intensity will be regarded as the same
for all rays within this cone. If L denote the total amount of
light emitted within this cone per second, we write : L = Kdw.
This equation defines K, which is sometimes called the
'candle power ' of the source.
In general, K is dependent upon the direction, but when K is
the same for all directions
M = 4ttK,
where M is the total amount of light emitted per second by the
source.
Let the small cone cut a surface in the element, <5S, and
let the mean direction of this cone make an angle, 0, with the
normal to &S.
380
Then aS =
PHOTOMETRY 381
r 2 8w
cos0
, T K cos ^
and L = — -—- — <5S.
r 2
The intensity of illumination is defined to be the amount of
light falling on unit area per second,
i e I — — — ^ cos ®
~ ss 7*
The intensity, I, varies inversely as r* and directly as cos 0.
When I is the same for two similar surfaces they appear to the
eye to be equally bright, and on this principle the use of photo-
meters depends.
The efficiency of a source of light may be denned as the ratio
of K to the material or energy consumed per second.
The average candle power, in the case when K is not constant,
divided by the amount consumed per unit time is called the mean
spherical efficiency. This term is employed because the average
M
value of K is denned as — , and this denotes the average amount
of light falling on 1 sq. cm. of a unit sphere placed with its centre
at the source.
The efficiency of a candle is :
K -^- weight of wax consumed per second,
of a gas flame :
K -r- cubic feet of gas consumed per minute,
and of an electric lamp :
K 4- watts supplied to it.
The watts are measured by the product, volts x amperes, one
watt denoting the rate of working when a current of one ampere
falls through a potential difference of one volt.
The unit in which K is measured is the candle power of a
standard candle in a horizontal direction when the flame is
50 mms. high.
If another source is used and compared with the standard the
two are adjusted to the same height and placed at such distances,
r 2 , and r it from a conveniently placed screen that each makes it
appear equally bright. In this case by what has been said above
concerning intensity we have :
t _ Kjcos T K 2 cos
Ii - -JT- - I. - -jt—
382 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
since the angles are the same,
power its value is unity, and,
If K, is the standard candle
It is assumed that the student is familiar with the simple forms
of photometer such as Rumford's and Bunsen's. It is difficult
to make accurate comparisons of the illuminating powers of
sources with these instruments.
We shall be concerned chiefly with the more accurate types of
instruments in this chapter. Even with these care and practice
are required, but a skilled observer can obtain accurate results.
The Efficiency of Sources of Light
Take a gas-burner provided with an indicator registering the
quantity of gas supplied, and adjust the flame until it stands at
the same level as a candle flame and the grease spot of a Bunsen
photometer.
The candle may be taken as the standard with the value of K
unity, and it must be shielded from draughts and must burn
steadily.
Determine K for the flame corresponding to different rates of
supply of gas, by adjusting the distances between the sources,
until the photometer screen appears alike on both sides.
Plot a curve, showing the relation between the efficiency and
the supply per hour.
It will be found that the efficiency increases with the supply
up to a maximum and then diminishes.
It is most economical to adjust the supply to the value
appropriate to the maximum.
•
r v "
•
A
R
1"
■
1
Fig. 237
We may similarly measure the efficiency of an electric lamp.
Arrange the lamp, L, on the same level as the grease spot and
candle flame, as before, and measure its candle power when the
current is supplied at different voltages.
Fig. 237 illustrates the arrangement of apparatus.
V is a voltmeter joined to the terminals of L, and A measures
the current in amperes. The resistance, R, is adjustable and is
PHOTOMETRY
383
used to vary the current supplied to the lamp.. The power is
obtained by connexion to the main through a plug. A con-
venient voltage is 100 volts.
Care must be taken not to short-circuit the mains.
Plot a curve, showing the variation of candle power, with
rate of supply of energy as measured in watts.
The Flicker Photometer
One form of this instrument is illustrated in figs. 238 and 239.
The essential part consists of a white wheel, W, of which the edge
is about 1 cm. wide, and is cut to the shape of a ridge running
spirally. This is shown at RR, in fig. 239. The wheel is mounted
in a box, black on the inside, provided with a rod, so that it may
be supported in the carrier of an optical bench.
J
Si
s,
Fig. 238
The wheel is provided with a central axis and is rotated by a
spring within the box, which is wound up by the key, K.
The two sources to be compared are placed one on each side
of the apparatus, as at S! and S 2 . These are carried in stands
on the bench, so that the distances from the wheel are measured
accurately and their heights are adjusted with the aid of two
lenses fixed on the box.
The small doors EF and GH, are closed, and the lenses used to
throw an image of the sources on marked points, P 1 and P a , on
the doors. When the images fall on these points the heights of
the sources are correctly adjusted.
384 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The edge of the wheel is observed by means of an eyepiece, J,
and usually unequal parts of the edge are seen on the two sides.
When the wheel rotates the widths of these vary, and unless the
illumination is equal on both sides a flickering effect is observed.
If the illumination is the same on each side the ridge character
disappears.
S x and S 2 are moved to or from the apparatus until the flicker
effect ceases, when the illuminations are equal, and if I x and I 2
represent the illuminating powers of the sources, and r x and r 2>
their distances from the wheel, we have :
\*
The distances should be measured from the middle of the wheel.
It is not easy to decide when the "flicker ceases, and the degree of
accuracy obtainable is not very high.
Compare in this way a standard candle and a lamp.
The Lummer-Brodhun Photometer
This photometer is one of the most accurate. It is illustrated
in fig. 240, and consists of a box, LMNO, containing the prisms,
P lf P„A, and B, by means of which light is reflected and trans-
mitted into the telescope, T, placed at 45 to the sides of the box,
with its object glass in one corner.
The two sources to be compared are at S x and S 2 , from which
light falls on the slab, DD, which consists of magnesium car-
bonate. From the diffuse sides of the slab rays are scattered and
absorbed by the sides of the box, except those that cut the
sides of V x and P 2 normally. These are reflected by the hypot-
enuse faces into the two right-angled prisms, A and B.
These prisms are the principal part of the apparatus.
The hypotenuse of A is rounded off, except for a circular
central portion which is placed in optical contact with the
PHOTOMETRY
385
iarger face of B. The reason for this is, that rays of light may
pass from A to B at this junction just as if the prisms formed one
solid medium. Rays falling on other parts of the hypotenuse
faces are totally reflected.
In this way rays from P x pass on through B, forming the central
bundle of rays in the beam emerging from the right and entering
T. The rays outside this circle are reflected and are absorbed by
D
Fig. 240
the sides of the box. In the same way, rays from P 2 are
reflected outside the circle, while those falling on the circle are
transmitted. Thus the field of view of the telescope is illuminated
by a central circle of rays, originating at S 2 , while the outer rays
come from S x . Generally these two parts will be of different
brightness, and on moving S x or S 2 , the two parts may be made
equally bright. The eye can judge this easily and readily
appreciate a slight deviation from equality. It is on this fact
that the sensitiveness depends.
From DD both sets of rays follow similar paths and light is
absorbed equally. When the field of T is uniform, the slab is
25
386 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
illuminated equally on both sides. If Lj and L 2 denote the
illuminating powers respectively, and d x and d 2 denote the
respective distances from DD, we have :
L x L a
df d
2
Thus any two sources may be compared with accuracy.
Compare two sources as in the last experiment.
In order to eliminate errors arising from inequality of the
reflecting power of the two surfaces of the slab and a possible
difference in optical paths, DP X A and DP 2 B, it is usual to mount
the box, LNOM, on a central horizontal axis lying perpendicu-
larly to SiSa. Readings are first taken with the telescope as
in the figure, and then with the box turned through two right
angles so that the telescope lies on the left.
The Nutting Photometer and its use for the Determination of the
Absorption of a Solution
This form of photometer is the most accurate instrument of
its kind. It is based upon principles described by P. G. Nutting,
but has been modified by Messrs. Hilger and Co., whose apparatus
is here described.
It is used in combination with the Constant Deviation
Spectrometer (pp. 341 to 345), and the apparatus is illustrated
in fig. 241, in the position ready for use.
The essential features of the photometer are illustrated ki
fig. 242.
The box, A, is of aluminium, blackened inside, provided with
two small windows, Q and Q 1 , by means of which light is admitted
along the two directions, QR and Q X S.
Light from a suitable source is deviated by two prisms, P and
P 1 , carried in the plate, C. On the plate a number is inscribed,
indicating the distance that the source must be placed from it,
in order that the necessary deviation may be produced. This
distance is usually 19 cms.
Behind the window, Q 1 , lies a Nicol prism, Nj, which polarizes
the light entering at Q 1 ; but behind Q in the form of apparatus
shown no Nicol prism is placed.
In another form a Nicol lies behind Q also, and its purpose is
to counteract as much as possible the elliptic polarization that
occurs on reflection at the surfaces marked R and S. It is
found that this is reduced to a minimum by a particular orienta-
tion of the Nicol.
The prism, RS, is composed of three slabs, the two outer are
alike and are cut at the ends, R and S, at an inclination of 45 ° to
the length and to the incident light.
Pm& 3^5
PHOTOMETRY
387
Thus the beam, QRST, is totally reflected at R and S, by these
two end faces. Between these slabs lies a central one of the
same composition and thickness cut at R at 45 like the others,
but cut square at the other end, and projecting as the diagram
Fig. 242
indicates. Thus the central portion of the beam, QR, is totally
reflected at R, and transmitted out at the other end, and is ab-
sorbed by the blackened walls of the box. The beam, Q X S, is
totally reflected at the end, S, by.the upper and lower slabs,
and the light absorbed by the walls of the box, but the central
portion is transmitted in the direction, ST, by the central slab.
Thus a tripartite field is produced and may be observed from
the end of the tube, F ; the outer portions are illuminated by
light which has entered at Q, and the central portion by light
from Q 1 .
The instrument is constructed and the prism and Nicol, N 1 ,
chosen to cause the light within the box to suffer approximately
the same absorption along the two paths, QRS and Q X S.
The tube, F, carries a second Nicol which is not shown, which
acts as an analyser and which can be rotated by means of the
divided metal circle, G. This circle carries two scales, one
marked in degrees and the other giving ' densities ,' a term which
will be explained below.
When the instrument has no zero error the degree scale reads
zero when both Nicols are parallel. In this case the light already
polarized by N x is transmitted by N 2 , while the unpolarized
outer portions of the field are polarized by N 2 in the same way
as the central portion by N r
Thus when there is no absorbing medium between the source
and one of the windows, the field appears under these circum-
stances of the same intensity in the outer and inner portions.
F also carries a condensing lens which may be adjusted by the
rod, R, projecting downwards from the tube.
The purpose of this lens is to converge the light so that on exit
all the light may enter the pupil of a normal eye. This is an
important condition with which accurate photometric apparatus
must comply in order that intensity comparisons may be of any use.
388 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The rod carries a scale past an index mark in the slot, B, on
which are engraved two sets of numbers which have to be as-
sociated in pairs. The upper scale records the distance of the
source from the front of A in cms., and the number below this
record on the lower scale denotes the maximum breadth of the
source which is permissible if the photometric condition is to be
satisfied.
Beyond the circle, G, the tube carries a lens system which
focusses the light on to the slit of the spectrometer, and if it is
desired to examine the field of view directly by eye an additional
eyepiece is fitted into the tube.
The apparatus requires careful adjustment which may be
carried out as follows :
Remove the prism from the spectrometer, illuminate the colli-
mator slit and place on the prism table a piece of plane mirror or
a right-angled totally reflecting prism, and adjust it until an image
of the slit lies on the cross-wire of the telescope. Now place the
source to be used with the photometer at the distance from the
collimator slit at which it is to be situated during the experiment,
having regard to the photometric condition which limits the
distance to some extent on account of the size of the source.
Adjust the source until an image of it lies in the centre of the
field of view of the telescope, the collimator slit being now wide
open and the eyepiece removed from the telescope, so that the
source appears to lie at the centre of the object glass. Slide
the photometer into position with the end of the tube, F, as
nearly as possible 1*4 cms. from the slit, and with the window,
Q 1 , directly between the collimator slit and source. Cover up
the window, Q.
Adjust the photometer by means of the three screws on which
it stands, and by rotation about a vertical axis until the image
of the source lies once more in the centre of the objective of the
telescope.
Place the plate, C, into position at a distance of 19 cms. from
the source. The distance between P and P 1 is 3-8 cms., so that
by moving the source a distance 1*9 cms., it can be brought to
lie opposite the middle point of PP 1 and it should then be in the
correct position. -
This may be judged first by observing if bright circular patches
of light lie symmetrically round the windows, Q and Q 1 .
Make sure that, with the rod, R, adjusted so that the index
lies opposite the mark denoting the distance between the front
of A and the source, the width denoted by the lower reading is
greater than that of the source. If this is not the case the source
has either to be displaced* farther from A or diminished in size.
Place the constant deviation prism and eyepiece of the telescope
PHOTOMETRY
389
in position and adjust the spectrometer correctly for sodium
light as indicated on p. 343. In doing this a strongly coloured
Bunsen flame may be placed just closer to C than the light source.
Make the line as sharp as possible by rotating slightly the
milled head at the end of F.
Now remove the Bunsen flame and open the slit to let in a
convenient quantity of light. It will be probable that the
Fig. 243
central part of the field of view is slightly displaced with respect
to the outer parts. This is exaggerated in the upper part of
fig. 243. This may be corrected by a further small rotation of
the photometer about a vertical axis. The three parts of the
field should be separated by fine dark lines. If these are too
wide they may be made narrow by adjusting the base screws of
the apparatus.
Correction of Zero Error
In practice it is usually found that when the apparatus is set
at zero the three parts of the field are not uniformly bright, and
that the error is not the same for different wave lengths. In
order to correct for this the readings of the apparatus are recorded
when the field is uniform for different wave lengths. A shutter
eyepiece is fixed to the telescope to cut down the light except
over a narrow central strip, the wave length for which is recorded
on the drum. When the central portion is the brighter, rotation
of the analyser cuts it down, and the readings on both sides of
the zero are observed and the mean taken for a series of different
wave lengths and a curve plotted, showing the relation between
error and wave length.
390 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
These readings are observed on the degree scale or density
scale as may be required.
If the central part of the field is the darker it is not possible
by merely turning the analyser to bring about uniformity. In
this case a weak absorber, e.g. a plate of glass, is put into the path
of the beam just before it enters Q. The thickness is chosen
so that in the zero position the central field is just stronger, when
the correction may be made as before. This should not often
occur because the instrument is made so that the central portion
is brighter than the outer, so that as the apparatus deteriorates
with time the central portion may still be the brighter and the
interposition of the weak absorber may be avoided.
The substance of which the absorption is required is placed
in the path of the light before it enters Q.
In the case of a liquid it is necessary first to measure the
absorption of the vessel containing it.
The observations consist of noting the readings on the scale, G,
when the field is uniformly bright for both directions of rotation
of G.
Let I denote the intensity of the light entering at Q 1 , and
let I denote the intensity of that which enters at Q. The density
scale records the values log ~, i.e. the logarithm of the ratio of
the intensity of the light entering the medium to that transmitted
by it.
If this number be divided by the thickness of the material
traversed a quantity known as the ' extinction coefficient ' is
obtained.
Let a denote the amplitude of the polarized light transmitted
by Ni and suppose that the analyser is turned through an angle, B,
from the position in which it is parallel to N r The amplitude
of the vibrations transmitted by the analyser is therefore a cos 0.
When the field is uniformly matched the intensity is I, and
hence
Io_ a 2 .
I a 2 cos 2 '
.-. log ^-= log sec 2 6.
This shows the relation between the density scale and scale
of degrees.
A solution which gives a characteristic absorption curve is
one of eosin in alcohol. Eosin may be obtained from a bacterio-
logical laboratory, and the solution must be very dilute, or so
much absorption takes place that the transmitted light is very
feeble.
PHOTOMETRY 391
Carry out the determination described above and plot on the
same curve the values of the ' density,' log —, and the cor-
responding wave lengths, for the zero error, for the vessel alone
and for the vessel and solution.
From these it is possible to obtain the value, log^ for the
solution alone.
Fig. 244 shows the result of an experiment with eosin. Some-
times the variations occur very rapidly and a curve is obtained
like that drawn in diagram 243. Unless frequent observations
are made in the neighbourhood of AB the variations at C and D
may be missed. Whenever the curve shows any sign of change
the neighbourhood where this occurs must be examined with
care.
AlosoY^Hon Curue for a u>cak
Solution of Eostn >n AlcoKo\.
the. o|p|per cwtye. \s ■&><•
tElo&n -v- Glass etc.
"The \oujer curve. i» •¥<*
/toro covrecC\or\ <wA cjVass
cxbsor^Hon .
■4400 *=\BOO 5EOO B600 6000
WaveWr^Yhe x \o cms
Fig. 244
It may not be necessary to determine exactly the value of
log y for the solution, and in this case it is sufficient to make
the zero correction and that for the containing vessel together,
the whole appearing as a combined correction curve.
CHAPTER XIV
SOUND
To Find the Frequency of a Note by means of the Siren
In this instrument a musical note is produced by puffs of air
following one another in rapid and regular succession. The
series of puffs is produced by blowing air through a number of
holes in a rapidly rotating plate.
The diagram (fig. 245) illustrates the instrument. It consists
of a cylindrical metal chamber provided with a tap through
which air can be blown. In the upper end is cut a series of holes
lying regularly spaced on a circle with its centre on the axis of the
Fig. 245
cylinder. Above this lies a circular metal disc provided with a
similar series of holes which fit above the former. The disc is
mounted so that it can rotate about the axis of the cylinder
and so alternately cover and expose the lower series of holes.
392
SOUND 393
The two series of holes slant in opposite directions as the figure
shows, and when a current of air is blown into the chamber and
the disc given a slight rotation, the puffs of air on escaping produce
a pressure which drives the disc.
By adjusting the influx of air the regularity and speed can be
controlled so that notes of varying frequencies can be produced.
If N is the number of revolutions made by the disc per second,
and the number of holes is n, the frequency is N». In order to
measure the number of revolutions the disc is provided with a
metal bar, provided with a screw at one end, which works two
dials, one registering units and tens, and the other hundreds of
revolutions.
This form of apparatus is due to Cagniard de la Tour ; but it
has the disadvantage that the speed can only be increased with
greater air pressure and a consequently louder note. It is also
difficult to keep the speed uniform.
It is preferable to drive the disc with an electric motor, of
which the speed may be regulated by including a resistance in
the supply circuit, and the holes should be cut normally to the
disc in order to avoid air pressure in the direction of the rotation.
The siren gives a large number of harmonics, and it is necessary
carefully to single out the fundamental note.
Let it be required to find the pitch of a given note, as, for
example, that produced by an open organ pipe.
Carefully adjust the speed of the siren until beats are heard
between the note it gives and that of the organ pipe by blowing
at a particular pressure from a bellows connected to the chamber.
The blower should then endeavour by a slight change of pressure
to produce from the siren a note giving no beats. To some
extent the frequency may be controlled by the tap, but it is
important to keep a steady pressure on the bellows.
At the same time a second observer should measure the speed
of revolution by observing the number of revolutions recorded
on the dials in a definite time (30 or 60 seconds).
Verify the result of the determination of frequency by measur-
ing the length of the pipe and its diameter. For an open organ
pipe emitting the fundamental the wave length is approximately
twice the length.
The correction necessary to obtain a more accurate result
is to add «6 radius for the open end and 2*8 radius for the flute
mouthpiece. If the length of the pipe be I, and the radius of
the pipe, r, the half wave length is given by :
- = I + 3'4^
or X = 2(1 +■$•#).
394 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The velocity of sound in air for ordinary temperature may be
taken as V = 33,300 cms. per second, or make the accurate
correction for temperature by formula (A), p. 416.
V
Thus the frequency is -••
Determine the frequencies of several organ pipes theoretically
and experimentally, and draw up a table recording the speed of
rotation of the siren disc, and the observed and calculated
frequencies in each case.
The Tonometer
In Scheibler's tonometer a number of tuning forks are arranged
in ascending order of frequency, each of which gives the same
number of beats with its neighbour. The forks thus form a
series in which the frequency increases by equal steps, and they
are arranged so that the highest frequency is twice that of the
lowest.
In Appunn's tonometer the forks are replaced by reeds set in
vibration by a blast of air from bellows of large capacity, and the
apparatus has the appearance of a small harmonium provided
with a series of stops by means of which any note may be
sustained.
This form of apparatus is not so accurate as the original one,
for Lord Rayleigh has shown that the frequency of a vibrating
reed is to some extent affected by the vibrations of its neighbours.
As it is necessary to vibrate two successive notes in the experiment
we have no longer a constant register of frequency as in Scheibler's
instrument.
It is first necessary to find the absolute frequency for each note
on the instrument. Suppose there are (k + 1) notes, and con-
sequently k intervals between them, and that the frequencies
are N x , N 2 , . . , N* +1 ,
beginning from the lowest.
If the number of beats be observed between all the successive
notes, and be denoted by
n x , w 2 , . . . n k ,
respectively, we have the following relations :
N a -N 1 =» 1 ,
N 3 — N 2 = n 2 ,
N* +1 — N* = n k .
Then adding both sides :
N* +x -N 1 = # 1 + » l + ...+» 4 .
SOUND 395
But N* +1 = 2N X ;
/. Nj = n x + n 2 +. . . + n k .
We thus find N x by counting the successive numbers of beats,
and we can then deduce from the equation the frequencies of
all the other notes of the series. In practice, of course, the
numbers of the beats will vary between the successive notes in
different parts of the scale to a slight extent.
In order to determine the frequency of the lowest note, count
the number of beats at five or six different parts of the range and
deduce the average difference of frequency.
Let this be denoted by n. Then the frequency of the lowest
note is k x n.
After this determination has been made the frequency of any
given note coming within the range may be determined. For
example, suppose the frequency of a fork is required. Find by
trial the note nearest to it in pitch and count the number of
beats when the notes are sounded together.
If this number be denoted by x, and N/ is the frequency of the
note nearest the unknown, the frequency of the latter is N, ± x.
The frequency of the note next above N, is N, + n, and on
sounding this and the unknown note together the frequency of
the beats will be less than n if its pitch is higher than that of N/,
and greater than n if its pitch is lower. This enables a distinction
to be made between ± x.
In making this last part of the experiment it is advisable not to
rely on the accuracy of the average number of beats for each
interval, but to measure separately the frequency of the beats 6i
the notes immediately above and below N/.
It will then be easy to decide on the exact position of the
unknown pitch above or below N/.
The Determination of the Frequency of a Tuning Fork by the
Method of the Falling Plate
A smoked glass plate, P, is suspended vertically by a piece of
thin string or thread over two nails, QQ, the thread being attached
to the upper edge of the plate by means of sealing wax or by any
other convenient method.
The fork is held in a clamp, H, and carries a light style of
bristle or thin aluminium wire attached to one prong by as little
wax as possible. The style is just in contact with the plate so
that when the plate falls it removes some of the soot and leaves
a trace.
In order to prevent breakage a padded wooden stand, AB, is
placed just below the plate.
The fork is stroked gently by means of a violin bow and the
/*'
396 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
plate is allowed to fall straight down by burning the thread
between the nails, QQ.
A wavy line is traced by the style similar to that shown in
fig. 247, but with more waves, and usually of small amplitude.
A point, O, is chosen just clear of the indistinct portion drawn
when the plate was moving down in the first stage of the motion,
and consequently before its velocity had sufficiently increased
Fig. 247
to open out the waves, and from it is counted a number, n, of
complete waves to the point S. Again, n waves are counted
to the point S r Let the spaces, OS and SS X be of lengths, s and s lt l
respectively. Then the time taken to fall over these two lengths
is the same, let us say t.
If N denotes the frequency of the vibrations :
n = N*.
SOUND 397
Let u denote the velocity of the plate at the instant correspond-
ing to the mark, O.
Then by the equation for space described under acceleration, g,
s = ut + \gP,
and s -f- s x = zut + 2gt 2 ,
for the time of description of s + s t is 2t.
Hence Sl — s = gt 2 ,
or
-4
N = n • J- g
vs x — s
"1 "
The distances, s and s v are measured carefully by means of a
travelling microscope, and the value of N obtained is that for the
fork vibrating with the load consisting of the style and wax and
affected by the friction of the style against the plate.
This should be allowed for by taking a second fork of nearly
the same frequency as that under examination, before attaching
the style, but of slightly higher pitch. Carefully load this fork
by adding wax until no beats are heard when both sound together.
Then when the first fork is loaded and has the style touching
the plate as in the experiment, again sound the two together
and count the beats. The number of these per second gives the
number of periods lost per second on account of the loading and
friction.
This number added to the value of N, determined in the
experiment, gives the corrected frequency.
In marking off the points O, S, and S v be careful to choose
them at corresponding points of the waves.
If O is at the summit of a crest, S and S x must lie in a similar
position n and 2w waves later, respectively.
The plate may be conveniently blacked by holding it just over
a turpentine flame which gives a good deposit of soot ; the
flame of a paraffin lamp gives also a satisfactory deposit.
Chronographie Methods of Determining the Frequency of a Fork
In both the methods to be described under this title a fork is
set in vibration and, electrically maintained, with a style attached
to one of the prongs lightly touching the blackened surface of a
cylinder which can be rotated about its axis.
On rotating the cylinder the track made by the style appears
as a wavy line which can be opened out so that each vibration is
distinctly separated from its neighbours by adjusting the speed
of rotation.
398 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The drum may be rotated by the handle, H, or by a falling
weight not shown in the diagram, which rotates the drum at a
convenient speed.
In this case the rotation is regulated by a governor, and by
releasing a catch, the handle, H, can be employed for rapid
movement of the drum. This also is not shown in the figure.
The drum moves along an axis cut with a screw thread, so that
the track drawn by the style forms a wavy helix round it.
Fig. 248
It is then necessary to have some record of time with which to
compare the vibrations. The two methods differ in the way
the time is obtained.
In the first a stand is mounted conveniently near to the fork,
as illustrated in fig. 248, which carries a pointer actuated by an
electromagnet, M. The pointer, P, is carried on a lever one end
of which consists of a strip of iron or steel, which is attracted by
the electromagnet core when a current flows through the exciting
coils. When the current is off the lever is held in a position with
the strip a short distance from the magnet by means of a spring.
Thus, as the cylinder rotates, a line is drawn round the surface.
On passing the current in the coils the pointer moves and makes
a kink in the line. If the current is put on at regular and known
intervals, a time record along the side of the waves is produced
on the drum. Thus by counting the number of waves between
consecutive kinks the frequency of the fork may be deduced.
The regular intervals are obtained by connecting the wires,
AA, to a battery, and completing the circuit by means of the
mercury cup, M, and the pendulum, P, as shown in fig. 249.
Mercury is poured into a hole cut in wax, W, so that it stands
just above the wax surface, and the wax so placed that a strip
of wire hanging down from the pendulum just touches the surface
as it passes its lowest position.
The circuit is completed twice in each complete period, and at
each instant the pointer makes a record on the drum.
It is improbable that the interval between successive records
will be one half a period, since this would require exact coincidence
of the point of contact with the mercury and the lowest point of
SOUND
399
the swing. The time between alternate records will, however,
measure the time of a complete period.
Thus, in counting up the vibrations, find the mean number
between alternate records.
Obtain a long helix, begin at the first stroke of the pointer,
and count the number of vibrations up to some later odd numbered
stroke. Find the mean number per complete period. Repeat
this, beginning with the second stroke and ending at some later
even numbered stroke, and find the mean number again. The
two values should agree, but if there is a slight variation take
the average value of the two results. .
Fig. 249
To obtain a blackened surface, take *a sheet of smooth white
paper, and wrap it round the drum, one layer thick, holding it
in position by means of gummed paper. To blacken it, rotate
it over a turpentine flame, or coat it with camphor smoke.
The coating of soot should not be very thick, the style will
then remove the soot and leave a white wavy trace.
The paper may be smoked again when once used and the track
covered up.
The time of a complete period is measured in the usual way,
by timing the pendulum.
The frequency obtained is, of course, subject to a correction
similar to that of the last experiment, and this should be
determined and applied in the manner described.
Fig. 250 represents a convenient method of carrying out the
determination in an alternative way.
The cylinder is held vertically, and is rotated by a handle or
string round a drum as shown.
The fork is maintained electrically as before, but the axle of
the drum is connected to one secondary terminal of a small
400 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
induction coil, while the second terminal is connected to the fork.
The primary circuit is completed through a pendulum and
mercury cup as before. Each time the primary circuit is com-
pleted the induction coil is excited and a spark passes to the drum
from the style, knocking off a little soot and leaving a white dot
to record the instant of closing of the primary circuit. The
same procedure and precautions are adopted as before. A thin,
flexible copper wire will be found suitable for suspension of the
pendulum, and with a small induction coil there is no incon-
venience on account of shocks obtained when the apparatus is
touched.
Fig. 250
This method was employed by A. M. Meyer. It should be
noted that uniformity of motion of the drum is not necessary.
The methods are both inferior to those in which an optical method
is employed for determining frequency as in Rayleigh's method.
The Frequency of a Tuning Fork by the Stroboscopie Method
The fork is fitted with two very light plates fastened at the
extremity of the prongs, one to each, and so that one may vibrate
freely past the other. These plates may be of thin cardboard or
aluminium, so that the loading affects the fork to the smallest
possible extent, and they are stuck on to the fork by means of a
little wax. A disc is taken provided with a number of dots
placed at equal intervals round circles concentric with the disc.
Each circle has its own interval length.
SOUND 401
The fork is maintained electrically (p. 138), and the disc
placed behind it with its dotted surface brightly illuminated.
Each plate attached to the fork is provided with a slot, and
when the fork is at rest the slots lie directly behind one another.
Thus the disc can be seen through them.
The slots and one of the circles of dots are so placed that the
dots can be seen by looking through the slot, as the disc slowly
rotates. If the fork is vibrating it is possible to see through the
slots twice in each complete period, and thus 2,n times per second,
where n denotes the frequency.
The disc is caused to rotate uniformly by means of an electric
motor provided with a resistance in circuit to vary the speed.
The speed is gradually increased by adjusting the resistance
until when the fork is vibrating the dots in one of the circles
appear to be at rest.
In this case a dot moves up as the disc rotates, so that each
time the slots are in line a dot is just in line with them. The
eye sees apparently one stationary dot, and the effect of rotation
is lost by looking through the slot.
In order to count the number of rotations per second made by
the disc, a counting arrangement is attached and the time taken
over a definite number of rotations, when the dots remain
apparently steady, by means of a stop-watch.
Let there be N rotations per second in the case when the dots
belong to a circle containing p of them.
In this case the time taken by a dot to take the place of the
one preceding it is : ~-r second. This is equal to one half the
period of the fork.
Hence — = — —
N^> 2 n '
or n = - Np.
Now further increase the speed of rotation until once more the
dots appear steady. In this case the speed of the disc is such
as to cause a dot to take the place of the dot two intervals in
front. The time taken to do this is ^n, where N 1 is the
N 1 ^
number of rotations per second
From this we may calculate the value of n once more.
On further increasing the speed until a dot takes the place of
another three intervals in front we can obtain a third calculation.
Repeat this for several series of dots.
With the fork loaded it is necessary always to correct for the
loss of pitch due to loading as before.
26
402 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We may, however, avoid this by making one prong of the fork
bright over a small area, and by rotating a disc provided with
several series of concentric holes in front of it. If the fork is
well illuminated and the disc carefully mounted, it will happen
that for some particular speeds the fork appears stationary when
seen through the holes.
The experimental details and the mode of deduction of the fre-
quency of the fork from the observations are similar to the former.
The observations in stroboscopic experiments can nowadays
be made much more conveniently by the use of a neon lamp than
by the method described above.
This lamp has the property that it lights up immediately a
voltage is supplied to it without any appreciable lag, and is
extinguished immediately when the voltage is taken away.
The lamp consists of a flat piece of aluminium with a rod of
the same metal lying a short distance from it and parallel to it
in a bulb containing neon at low pressure.
When included in the secondary circuit of an induction coil,
it lights up each time the current in the primary is broken. There
is no effect at ' make ,' because in the construction of the usual
type of induction coil the effect at make is suppressed and that
at break of the primary circuit intensified.
If the primary coil is put into the circuit which actuates the
fork the primary current is made and broken once per vibration
of the fork and the lamp flashes out once per complete vibration.
Of course, the make and break attached to the primary must
not be allowed to work, the fork takes its place. All that is
necessary is to make the connexions in the usual way and to screw
the platinum-iridium point close up to the clapper to prevent
separation and to place some object, e.g. a small block of wood,
between the soft iron on the clapper and the end of the armature
to prevent oscillation as the current fluctuates.
The lamp is used to illuminate the rotating disc provided with
a series of dots on a white background and the speed adjusted
until one row of dots appears stationary.
When this is the case one dot just moves up to take the place
of a dot somewhere in front of it during the interval of darkness
between the flashes of the lamp, i.e. during the period of vibration
of the fork.
The calculation of frequency is made as before.
It is a great advantage to be able to avoid loading or marking
the fork and to have the disc in any convenient position where it
may easily be observed.
Instead of a series of concentric circles with regularly spaced
dots, it is more convenient to draw on a circular disc a series of
concentric regular polygons. A small triangle is drawn just
SOUND 403
about the centre of the disc, about this a square, then a pentagon,
and so on. c The triangle may be coloured white, the space
between it and the square blackened, the space between the
square and pentagon coloured white, and so on alternately.
When the/speed is adjusted exactly one of these figures appears
stationary and the frequency is easily calculated. By varying
the speed the figures may be made to appear stationary in turn
and several determinations of frequency made.
The Composition of Two Simple Harmonic Vibrations in the Same
Direction (Beats)
An apparatus which will combine graphically two simple
harmonic vibrations in the same direction has been invented by
Koenig. It consists of a large fork mounted on a stand and
provided with a clamp by means of which a strip of glass can be
held horizontally and fastened to one of the prongs (fig. 251).
B
Fig. 251
AB denotes the strip and C a weight, attached to the other
prong for the purpose of balancing. The glass is coated with a
thin layer of lamp-black, by means of a smoky flame.
A second fork is mounted above the first and carries a light
style adjusted so that it just touches the glass plate.
This fork is fixed to a sliding base by means of which the style
can be drawn along the smoked plate.
If both style and plate are vibrating a curve can thus be
traced which represents the motion of the upper fork relative to
the lower.
Thus if at any instant the lower fork is displaced a distance, y,
from the standard position, and the upper is displaced a distance,
y\ the displacement of the style over the plate is {y 1 — y).
The forks are made to vibrate with nearly equal amplitudes.
This may be done by bowing or by using a strip of metal which
is wide enough to open out the forks to a convenient extent ;
the metal is then quickly removed. The slider is drawn along,
not too quickly, and the trace obtained examined.
> When the forks have nearly the same frequency this will con-
sist of a wavy line with waves of varying amplitude. In this
case, with the amplitudes of the forks nearly equal, the smallest
waves will have almost zero amplitude, and the fluctuation
expresses graphically what the ear recognizes as beats.
4 o 4 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The time between consecutive minima measures the interval
between the beats.
If the glass plate is long enough three or four intervals extending
between consecutive minima will be obtained. The speed of
motion of the slider must be adjusted so that the individual
vibrations are drawn out to an extent which enables them to be
counted easily.
It is best to count the number of intervals on the plate and the
total number of vibrations between the first and last minimum
points. This enables the mean number of vibrations between
consecutive beats to be obtained.
Let this number be x.
We shall suppose that the two vibrations have frequencies n
and (n + m) per second, so that they may be represented by :
y 1 = a sin 2% (n + m) t
and y = asm. (2nnt + a),
where a is included to take account of any difference in phase
that may exist when the vibrations begin.
The resultant displacement recorded on the plate is :
Y = (yi — y) = a {sin 2-n (n + m) t — sin (2nnt + a) }
= 2a cos \2rtln -\ — \t +- isin (271— t — -V
In the cases when beats are heard, m is much smaller than
n or in + -\ so that the simplest way of regarding this expression
is to consider it as a S.H.M. of amplitude :
2a sin (2tc • \mt — |a) = A (say),
and then Y = A cosj 2* \n + ~J t + £<x •
represents a S.H.M. of amplitude, A, and frequency, \n + ~J.
The amplitude, A, attains a maximum value, 2a, and sinks
to zero alternately. It is zero for the values of t given by :
imt — |a = o, re, 2tc, etc.,
i.e. for values of t = ^ — ^ + -. ^ + - , etc.,
i.e. at intervals of— sees., or m times per second.
m
Thus the number of beats is m per second, the same number as
the difference between the frequencies of the two notes. The
frequency of the note heard is In + — V
SOUND 405
(tn\
n + — J vibra-
tions per second and m beats. The number of vibrations between
two beats is :
H» + -)•
m\ 2 /
But this is counted on the plate and found to be x.
Thus — In -\ — ) = x,
tn\ 2/
or n — m (x — \).
The beats can be timed by means of a stop-watch.
When the notes are sounding, count them for as long as possible,
and take the time of the interval. If the number of beats counted
is N and the stop-watch is started on the first and stopped at
the Nth, the interval between the beats is :
T
N^T
where T is the time interval recorded on the stop-watch.
Thus -is known, and the above equation gives n. The
M
frequencies of the notes are thus, n and (n + m).
The method must be regarded as an illustration of the
phenomenon of beats — it is not an accurate method for the
determination of frequency.
B
rf
.P
Fig. 252
The Composition ol Two Simple Harmonic Vibrations Perpen-
dicular to one another. (Lissajou's Figures)
Let the co-ordinates of a point, P (fig. 252), be (x,y) and let P
move so that
x = a sin pt
y = b sin (pH + a).
The motion of P then consists of two simple harmonic motions
along two perpendicular directions.
406 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
When t is zero, x is also zero ; but y has the value b sin a.
Thus the two S.H.M.'s are in different phases.
From the equations it follows that :
.1 . x
t = — arc sin-
p a
and t = — arc sin \ — — •
p 1 o p 1
Hence between x and y there exists for all values of t the
relation :
i . x I . y a
- arc sin - = -— arc sin \ — -:•
/> a p 1 b p 1
This represents the curve on which P lies.
When the relation between p and p 1 is simple, as for example :
p =p\ p = 2p\ 2p = 3P 1 , etc.,
the point, P, describes the curve completely in a short time, and
afterwards retraces it.
The case, pj= p\ is very simple, for we then have :
x = a sin pt,
y — b sin (pt + a).
In the general case this represents an ellipse with its centre
at the origin.
If a = o, then :
x y
a = b'
and P describes a straight line.
1Z
If a = —, P describes the curve :
2
a 2 ^ 6»
This is an ellipse whose principal axes lie along OX and OY.
In the case when the amplitudes of vibration are equal along
both directions, a = b and the locus of P is the circle :
x* + y 2 = a*.
Thus when the periodic times are equal, in general P describes
an ellipse ; special cases of this general case are the straight line
and circle. All these curves are therefore appropriate to the
case of equal frequencies.
If OB is equal to b sin a, B represents the position of P at the
time t = o, or at instants later by a complete period than this
initial time.
SOUND
407
If the frequencies are not quite equal, let us say that the
frequency along OY is a little the greater, then when another
period is complete the displacement along OY will be a little
different from that in the first case, i.e. P will lie at some point, B 1 .
If OB 1 = b sin a 1 ,
a 1 is slightly different from a,
and there has been a slight change in phase on the part of the OY
motion. Thus a will continually vary, and will pass through all
the values from o to 2n. This will cause a continual change in
the shape of the curve described by P. It will sometimes be a
straight line and sometimes an ellipse. If it happens that the
two amplitudes are equal, we shall have a circle sometimes.
The closer p 1 is to p, the slower will this change take place,
so that by watching the movement of P we can test the closeness
of the two frequencies. If the figure is maintained steady,
without change, the frequencies are equal.
Tfre same argument can be applied to the cases where one of
the other simple relations exists between p and p 1 .
When the shape of the curves corresponding to these relations
is known the approximate ratio of the frequencies can be recog-
nized and the exactness of the ratio tested by observing the
rate of change of shape throughout the series.
We can deduce the ratio of the frequencies by examining one
of the curves, e.g. fig. 253. It cuts the Y axis in four points and
the X axis in three, so that the frequencies are in the ratio, 4 : 3,
for the vibrating point makes four vibrations parallel to OX in
the same time that it makes three parallel to OY.
Fig. 253
This principle may be employed to investigate the way the
period of vibration of a rod, fixed at one end, varies with the length
of the rod. The apparatus consists of a vertical flat rod or spring
with a lens fixed at its upper end and a horizontal spring carrying
a screen with a small aperture (fig. 254).
408 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Light from a source, S, is focUssed by a lens, L 1? on to the
aperture, A, while the lens, L 2 , carried by the vertical spring, Vr
focusses an image of the aperture on the screen so that we have
a bright point, P.
Motion of the horizontal spring alone, causes P to trace a
vertical line, and represents the S.H.M. of A.
Motion of the vertical spring alone, produces a horizontal
S.H.M. on the screen. When both springs vibrate together, the
path of P represents the combination of the twb motions.
The vertical spring is of fixed length, but the horizontal spring
can be clamped at various points, so that the vibrating length
can be adjusted.
Make the first adjustment so that P describes an ellipse, straight
line, or circle without change of form.
In this case the periods are the same.
Now change the length until another steady curve is obtained
without change of form. Draw it carefully and deduce the
ratio of the periods of vibration.
Make several determinations of the ratio of frequencies and
corresponding lengths, and draw a curve showing the relation
between the length and frequency, taking the vertical spring as
a standard.
Fig. 255, shows the curves described by P for a few frequency
ratios which will serve for reference.
The Vibration Microscope
The essential features of this apparatus are the same as those
described in the last experiment. The two vibrating springs are
replaced, one by a fork and the other by a vibrator, the frequency
oi which is to be compared with that of the fork.
The second vibrator is sometimes another fork or violin string.
SOUND
409
A bright source of light such as a speck of chalk is attached to
the second vibrator, while the fork carries, attached to one prong,
a lens which forms the object glass of a small microscope.
If the lens alone vibrates, on looking through the eyepiece the
motion of the chalk is simple harmonic, on account of the vibra-
'/.
2
*>
%.
Fig. 255
tions of the lens. If the other vibrator is in motion and the lens
is at rest, the motion observed is, of course, that of the vibrator
alone. These two motions are arranged to take place in two
perpendicular directions so that a figure of the type described in
the last experiment is observed. It is steady if the frequencies
are exactly adjusted, but goes through the appropriate series if
the frequencies are not identical. The rate of progress through
the series may be observed and determined by means of a stop-
watch. After one completion of the cycle there has been a gain
of a whole vibration by one vibrator over the other. If the
frequencies be N and N 1 , and the time for completion of the
series is t seconds, then
N~ N l =
410 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
for N2 and N 1 ^ are the numbers of vibrations made respectively,
and these differ by one.
If the fork be slightly loaded we can find which is the greater
of the frequencies by again observing the rate of progress through
the cycle. If this time is shorter than before, the time of gaining
a period is less than before and the frequency of the fork is the
smaller.
We shall describe how the apparatus may be used to find out
the character of the vibrations performed by a stretched string
in the manner in which Helmholtz used the apparatus to examine
the vibrations of a violin string.
P L P
T
Fig. 256
8
The string, SS, is stretched below the prongs of the fork, PP,
to one of which the lens, L, is attached. The part of the string
to be examined is slightly blacked by ink, rubbed with wax, when
dry, and powdered with starch or chalk. A few white particles
will remain sticking to the string, and one of them is illuminated
by a lamp and focussed by the microscope and its movements
observed. The tension of the string is adjusted until the figure
apparently described by the chalk, as seen in the microscope,
remains steady. The vibrations of the fork are electrically
maintained while the string may be bowed or plucked.
The frequency of the fork is known, so that, as its motion is
simple harmonic, we can find the displacement due to it at any
of the instants during the vibration. The string vibrates at
right angles to the direction of vibration of the fork, so that
Fig. 257
from the curve obtained we can subtract the vibrations of the
lens and draw a curve showing the displacement of the string
at different times.
The white speck is first obtained in the centre of the field, and
its mean position represented by the origin, O (fig. 257).
SOUND
411
The curve is drawn to a convenient scale accurately from
measurements observed by the scale in the eyepiece. If the
microscope is not furnished with a scale, throw an image of an
illuminated scale to lie coincident with the string and view the
white speck and image together.
In the figure, OA denotes the amplitude of vibration of the
fork, its motion being assumed to take place along AOA 1 .
If its frequency is n, the motion is given by :
x — OA sin 2nnt.
Thus for any time, t, we can find the displacement, ON, along
the x-aods. At this instant the displacement along the other
direction, i.e. due to the motion of the string, is NL. Thus by
observing several values of the ordinates and the times corres-
ponding, we can plot a second curve showing the time-displace-
ment for the string. Its shape will indicate the character or
quality of the note emitted.
Fig. 258 shows the observed curve and the time-displacement
diagram for a stretched string when bowed. The first curve was
observed while the string was being bowed. The bow is drawn
slowly and regularly across the string. Slight fluctuations are
liable to occur during this process, but they appear as slight
variations of a figure remaining on the whole permanent. The
dimensions of this figure were obtained.
Fig. 258
The deduction of the diagram for the string is made in the
following way :
Draw on squared paper the figure observed (fig. 258), and
draw the extreme vertical tangents, AP and A 1 ? 1 . We are
assuming that the vibrations of the fork are executed along AA 1 .
With the middle point of AA 1 as origin, describe a circle on
AA 1 as diameter, and divide the circumference into a convenient
number of equal parts. In the diagram the number is twelve.
If a perpendicular be drawn from any point on the circum-
ference of this circle on to AA 1 , the displacement of the foot of
this perpendicular from O will represent the displacement of the
prong of the fork from its mean position.
Along the line, A*A, produced, beginning at o, twelve equal
412 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
intervals are marked off, as shown, to represent intervals of time
corresponding to the points on the circle. Beginning at A, draw
ordinates to the curve passing through the points marked on the
circle. For convenience in drawing only one of these, that
through the point, 8, is drawn. Let this cut the curve at the
point, Q, and the line, AA 1 , in N. Then NQ denotes the dis-
placement of the string from its central position in magnitude
and direction.
This displacement is plotted at the point, 8, on AA 1 produced,
and we thus obtain a point, q, on the displacement diagram for
the string. This process is carried out for all the points on the
circle, and the diagram plotted.
It will be noted that the ordinate, NQ, cuts the curve in a
second point, B. There is no doubt as to which point is to be
taken when actually drawing the curve, for we begin at A and
pass along one branch of the curve and back along the other.
In this discussion we have associated the upper part of the
curve from P to B and then to P 1 , with the times corresponding
to the points from o to 6.
Transverse Vibrations of Strings. (Melde's Experiment)
The object of the experiment is to verify the laws of vibration
of a string under tension. In such a case a disturbance travels
along the string with a velocity, v, given by :
\ m
T denotes the tension expressed in absolute units, i.e. poundals
or dynes, and m is the mass per unit length.
When the string is fixed at both ends, there is a node at each
end in its fundamental mode of vibration, with a loop or antinode
in the middle. The corresponding wave length is twice that of
the string. If this wave length is denoted by \ Xt we have :
ni== t \S'
«! denoting the frequency of this note.
This mode is illustrated in fig. 259. The string may also vibrate
to produce the overtones or harmonics as illustrated in figs. 260
and 261.
In these cases, if the frequencies are n 2 and n z , and the cor-
responding wave lengths x 2 and x 3 , we have :
SOUND
413
n.
x 2 \w'
and Wo =
and if / denote the length of the string :
Thus
n x : n 2 : n s :
Fig. 259
In the experiment the string is set in resonant vibration by
impulses having the same frequency as one of its modes of
vibration. For this purpose ordinary string is unsuitable. It
is not uniform and does not divide into equal segments; but it
will be found that a length of fishing line is satisfactory, as a
rule it is sufficiently Uniform.
One end of the cord is attached to the prong of a tuning fork,
by tying it to a small wire hook soldered on to the prong, or to a
small screw which is held in a hole bored in the prong.
The other end passes over a small pulley and carries a weight
which produces and measures the tension.
The vibrations of the fork are electrically maintained (see p. 138)
and by properly adjusting the length and tension the string can
be made to break up into stationary undulation with well-defined
nodes.
The fork may be placed so that the motion of the prongs is in
the direction perpendicular to the string (fig. 262), or along it
(fig. 263).
414 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In the former case the frequency is the same as that of the
fork, in the latter it is half as great.
For when in the second case the prong is in the extreme position
on the left the string is slack in the first vibration, and when in
Fig. 263
the extreme position on the right it is horizontal and tight. The
inertia of the string carries it onward so that when the prong
returns to the extreme left position, and thus completes one
vibration, the string completes a half vibration.
The student should examine the interesting effects produced
when the prong moves in a direction between these two and
thus produces in the string a combination of the two modes of
vibration.
We shall consider the former case only in this description of
the experiment.
The frequency of the fork is denoted by N, and when the string
is in resonant vibration, this is also the frequency of the mode of
vibration of the string, corresponding, let us say, to a wave
length, x.
Then : N = £ J.2,
x \ m
£ =_!__ constant.
Thus by varying the tension and consequently the wave
X 2
length, we should find ~ constant.
To determine X, measure the distance between the first well-
defined node on the right of the string and the last on the left.
Let this be d and suppose there are k loops between, then,
_ 2 d
X_ k '
Note that the ends of the string at the fork and the pulley
should not be taken. There is a certain amount of movement
at these two points.
x"
Draw up a table showing values of X, T, and „•
SOUND 415
KUNDrS TUBE
(a) The Determination of the Velocity of Longitudinal Waves along
Rods
Kundt's apparatus consists of a glass tube about a metre long,
and of diameter about 3 cms., provided with an adjustable
piston near one end. The tube is supported horizontally on a
table by resting it on two wooden V-shaped stands.
Near the other end of the tube is a second piston, Q, attached
to the end of a metal rod, DQ (fig. 264). This rod is clamped at
its middle point, C.
A _B c D
— * 3TFT I"1 r~7 IS~P z
Fig. 264
For the purpose of the experiment the tube must be quite dry,
and a light powder, such as lycopodium powder, is placed in a
line at the bottom of the tuj>e extending along its length between
the two pistons. A convenient way of inserting the powder, is
to spread it along a metre rule, place the rule in the tube and turn
it upside down.
If the tube is not dry it must be warmed above a Bunsen flame,
and a current of air blown through it.
The metal bar, which may be of brass with a diameter of about
•5 cm., can be set into longitudinal vibration by stroking it along
CD with a piece of wash-leather and powdered resin.
In the fundamental mode of vibration the ends, D and Q, are
antinodes, and the fixed point, C, is a node. The wave length
is twice the length of the rod.
x = 2l.
If the frequency of the note is n, and the velocity of the waves, v,
n\ = v.
If the distance between the pistons is L, the air between them
will have a fundamental wave length, 2L, and overtones with
corresponding wave lengths:
L, f L, I L, etc
The corresponding frequencies are :
v y v_
2L' L' |L ' etC "
where V denotes the velocity of sound in air.
If one of these frequencies is the same as that of the rod, the
air will be set into strong resonant vibration, and will move the
light powder. This will settle down at and near the nodes where
416 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the air is least in motion. As many as possible of these should
be used to find the average distance between two nodes. Choose
as carefully as possible the position of a node at one end of the
tube and locate the node nearest the other end. Measure the
length between these two points, and divide by the number of
spaces, such as NM (fig. 265).
-ijiSis-.iiijjii-.ijjjjii.
N M
Fig. 265
The^pattern will be somewhat similar to that in this figure,
and the longest line of each set marks the nodal position.
It will probably happen that there is not strong vibration of
the air at first, but by slowly moving the piston, P, forward or
backward, the length between the pistons may be adjusted so
that resonance occurs.
Twice the distance between the nodes is the wave length of
the sound in air. The velocity of sound at o° in air is 33,060 cms.
per second, and at a temperature, t :
v, = 33060(1 +^y (^
Thus the frequency, n x — — ', where x 1 is the wave length in
A
air. Since resonance occurs,
n = n 1 ;
1 _ Y«
**• 2/ ~~ x 1 '
or v =^V t -
In carrying out the experiment it is a good plan for one
observer to continue stroking the rod, while the other carefully
adjusts the piston until the powder moves violently and settles
down into the pattern of fig. 265.
(b) The Veloeity of Torsional Vibrations in a Rod
If instead of stroking the rod longitudinally with the resined
cloth, it is held near the end, D, and the cloth turned so that it
slips over the surface in a direction that would cause the rod to
rotate round its axis, a note is emitted of different frequency
from that given when the rod is in longitudinal vibration. This
note corresponds to torsional vibrations and will set the air in
resonant motion as before.
SOUND 417
Find in this way the velocity of these waves.
It is not easy to obtain a loud note by this method — the force
applied should not be great, but, with a little practice, it should
be possible to produce the note.
(c) The Determination of Young's Modulus and the Modulus of Rigidity
These constants may be determined from a knowledge of the
velocity of longitudinal and torsional waves in the bar.
The formula for the former is :
and for the latter :
«-£.
*->£■
where E is Young's Modulus, n the modulus of rigidity, and p
the density of the material of which the bar is made. Its value
may be taken from a table of physical constants. When ex-
pressed in C.G.S. units, E and n are expressed in dynes per sq. cm.
Find the values of E and n from the determination of the
velocities in the previous experiments.
{£) The Velocity of Sound in Carbonie Acid Gas
Kundt's tube may also be used to determine the velocity of
sound in gases. Suppose the tube to be filled with a gas in
which the velocity is V and the wave length, \.
Then the frequency is
Vj V.
\ X 1 '
• Yi _ h. — distance between nodes in the gas
V ~~ X 1 distance between nodes in air.
The procedure is, therefore, first to obtain resonance between
the rod and air column, and to find the mean distance between
the nodes, then to drive out the air and fill the tube with the gas,
and again obtain resonance, and measure the distance between
the nodes in this case.
\ a is then found from the last equation.
The gas must be quite dry or the powder will stick to the glass
and fail to respond to the motion of the gas when resounding.
It may be necessary to pass it through drying tubes before filling
the tube.
A slight modification of the apparatus is necessary for this
purpose (fig. 266).
27
4x8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In the former case the adjustable piston may fit loosely, but
for the present purpose it must be both adjustable and gas-tight.
It may consist of a cork round the outside of which a rubber
band or piece of cloth is stretched.
f*T*
Q
Fig. 266
The cork carries a tube to admit the gas and the piston may be
adjusted by means of this tube.
The metal rod passes through a tightly fitting cork, also pro-
vided with a tube which can be opened and closed by means of a
stop-cock.
We shall suppose that the velocity of sound in carbon dioxide
is to be measured.
Connect the source of gas to the inlet tube, I, and open the exit
tube, E, to the air. Allow the gas to flow in steadily so that it
will fall to the bottom of the tube and the air will flow over at E,
also, so as not to disturb too much the powder which we assume
has been spread out along the bottom of the tube.
The experiment should be performed close to an open window
or in a draught cupboard to prevent escape of the gas into the
room.
Continue the passage of gas long enough to ensure that the air
is driven out and then close the inlet and outlet tubes by means
of stop-cocks, and proceed as in the last experiment.
Care must be taken that the temperature of the gas is the
same as that of the air, unless the temperature of the gas is
measured in some way. If the gas is delivered from a cylinder
it will be colder than the air and it must be allowed to acquire
the air temperature before closing the stop-cocks. Otherwise
we shall be measuring the velocity in the air at one temperature,
and that in the gas at another. Reduce the velocity to that
at zero.
To do this, note that : ^ = ^ = ratio of nodal distances,
where the affixes t and o denote temperatures.
In order to determine when the air is all driven out from the
tube, collect a little of the gas issuing from E, in an inverted glass
cylinder over mercury, and introduce on to top of the mercury
column a little of a solution of caustic soda or potash. By
noting how much of the gas is absorbed it can be seen if any air
is left. The C0 8 is all absorbed by the solution and no gas should
be left.
SOUND
419
(e) To Calculate the Ratio of the Specific Heats of a Gas
From the result of the last experiment we may determine the
constant, r, for carbon dioxide, i.e. the ratio of the specific
heat at constant pressure to that at constant volume.
For the velocity of sound in a gas at o°, is given by the formula :
Po
where p is the pressure and p the density of the gas at o° C.
The value of p may be determined by the barometer since the
tube has been filled at atmospheric pressure.
p must be expressed in dynes per sq. cm. To make the cal-
culation, take the density of mercury as 13*60 gm. per c.c, and
P = '001974 gm. per c.c.
The measurement of the distances between nodes may be
performed simply by the ordinary use of a metre scale ; but a
slight addition to the apparatus will add to the accuracy.
A metre rule is fixed just below the tube and parallel to it,
and sliding over the rule or along one of its edges is a wooden
base, B, carrying a metal disc, D, with a hole, H, at the centre,
and a frame, F, with cross-wires, C (fig. 267).
Fig. 267
J
H serves as an eyepiece and HC is aligned on the nodes marked
out by the powder.
An index on the base indicates the position of the stand.
If the apparatus is aligned consecutively on the nodes, and the
420 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
positions of the index recorded, a table may be made out as
follows :
NO. OF
NODE
READING
AT INDEX
NO. OF
NODE
READING
AT INDEX
LENGTH OF
5 HALF WAVES
I
2
3
4
5
6
7
8
9
10
Average Length of 5 Half Waves . .
Mean Half Wave Length ...
Mean Wave Length
Chladni's Figures
The nodes of a vibrating stretched string are points of its
length where, theoretically, there is no motion. On either side
of the node the string moves simultaneously in opposite directions.
In the case of a vibrating plate there exist nodal lines, i.e. lines
in the plate where there is no motion. They divide the plate
into segments so that the parts on either side of the nodal line
at any instant are moving in opposite directions.
The point of support of a plate is necessarily on a nodal line.
Round and square plates are usually supported centrally in the
experiment, but interesting results arise when they are supported
elsewhere.
By touching any point on the edge or surface of the plate
with the finger nail, the plate is prevented from vibrating at the
point, and if it lies on a nodal line the corresponding mode of
vibration may be excited.
In the case of a circular plate clamped at the centre the nodal
lines are radial, and the fundamental vibration gives two per-
pendicular diameters. These may be obtained by sprinlding
white sand over the surface of the plate, touching two points on
SOUND
421
the edge separated by one quarter of the circumference, and
bowing vertically across the edge at a point one-eighth of the
circumference from one of the fingers. A square plate clamped at
the centre gives a large variety of figures. Some of these are
shown in the figure in order to assist in producing them.
111
J
V
^
>
r
13)
.
<l3l
Fig. 268
( 14)
Some of the nodal points on the edge and surface of the plate
should be touched, and the plate bowed along the edge at one of
the points midway between the nodes. The fundamental vibra-
tion gives two perpendicular lines through the centre parallel to
the edges of the plate.
The simpler figures are easy to obtain, but skilful manipulation
of the bow is required to produce the more complicated ones.
422 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The student should obtain as many as possible, and clamp the
plate at other points than the centre to investigate other modes
of vibration.
The Relation between Pitch and Volume In the ease of a Narrow-
necked Resonator
Consider the case of a bottle containing air closed by a piston,
P, without friction in the neck of the bottle.
Jil
B
Fig. 269
Let v denote the volume of the bottle below the piston, m the
mass of the piston, and A its area of cross-section. Let the piston
be originally in the position of equilibrium and let the pressure
outside be p 9 , while that inside is p. Then we have :
pA = ptA + mg.
If now the piston be displaced downwards a distance, x, so
quickly that the change may be regarded as adiabatic, a new
pressure, p 1 , will be generated, such that :
p 1 (v — A*)* = pv\
We shall suppose that x is small, so that we may write :
The total force downward is now :
mg + p<A - p x A == (p - p x ) A
v
Thus the equation of motion is :
d % x pyA''
x.
m W =
x.
SOUND
4*3
This is a simple harmonic motion and the time of vibration
about the position of equilibrium is :
2w
J
mv
(see p. 26)
In the calculation we have assumed that the pressure is the
same throughout the gas during the oscillations.
This is not true since time is required for the transmission of
the pressure. The other assumption concerning the adiabatic
character of the compression is very approximately true, es-
pecially as the neck is narrow and heat will not easily escape
from the bottle or be transmitted to it. The piston will thus
behave only roughly according to the formula, and will have an
approximate period of oscillation given by the above value,
i.e. it will have a frequency :
T 27T \
£yA s
mv
Thus we have approximately :
rih) = constant.
It is the object of this experiment to verify to what extent the
gas behaves according to this formula.
The piston, in practice, is the layer of air in the neck of the
bottle and by pouring water into it, resonance is produced be-
tween it and a series of forks of known frequency. A medicine
bottle will be found convenient and must first be calibrated by
pouring in water from a measuring flask to various depths and
measuring the height of the water surface above the bottom of
the bottle. This should be done for a series of intervals up to
the base of the neck.
Make a table thus :
HEIGHT OF
VOLUME
VOL. OF AIR ABOVE
WATER
POURED IN
HEIGHT IN COL. I.
*1
Vl
v n -v x
K
v*
f„-t> 2
K
*>s
v n -v 3
K
t>n
424 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
v n denotes the capacity of the bottle.
Draw a curve showing the relation between the heights of the
water and the volumes of air above it, i.e. draw a curve with the
values h as abscissae and the volumes in column 3 as ordinates.
By means of this curve we can deduce the volume of air in the
bottle corresponding to various heights of water.
Forks of pitch varying from 256 to 512 should be used and a
curve plotted with volumes as ordinates and values of — as
abscissae. The resulting curve should be a straight line through
the origin if the theory is correct, but in practice it will be found
not to pass through the origin although the relation is very
nearly linear. The curve will obey closely the law :
n % (v -f- c) = constant.
The value of c is a correction to be applied to v.
This may be regarded as a neck correction, and the ratio of t
to the volume of the neck should be recorded.
The agreement of pitch between the fork and bottle should be
tested by blowing across the neck and noting beats between the
note obtained and that of the fork.
The Interference of Sound Waves. Determination of a highly
pitched Note by means of a Sensitive Flame
When a continuous note is produced in front of a smooth wall
the reflected and incident trains of waves produce stationary
undulation in front of the wall, and consequently there exist
nodes and antinodes in the air. The experiment consists in
locating the positions of consecutive nodes from the wall out-
wards. Thus the wave length is determined, for the distance
between two consecutive nodes is half a wave length. By
noting the air temperature and deducing the velocity of sound
in air appropriate to it (eq. (A), §10 of this chapter), we may
thus deduce the frequency of the note by the relation :
V =wx.
In order to locate the nodes a sensitive flame is used. This
flame is produced by supplying the gas under pressure to a pin-
hole burner. The requisite pressure can be obtained by leading
the gas from the main into a large gas bag and placing a board
on the top of the inflated bag to carry a weight.
From the bag a pipe leads the compressed gas to the burner.
When the bag is full, turn off the main tap and put weights on
the board so that a tall flame from one to two feet high is pro-
duced and there is no flaring. It will be found that in the
sensitive state on making slight noises as, for example, by jingling
SOUND 425
keys in the neighbourhood of the jet, the character of the flame
changes. It flares and shortens, recovering its former state
when the sound ceases. This is caused by the motion of the air
as the sound wave passes.
Start with the flame close to the wall and sound a note from a
highly pitched whistle, or other suitable source, and move the
flame slowly outward from the wall.
It will be found that at certain points the flaring ceases and
the flame increases in length. At these points the air is still,
and the points are at nodes of the stationary wave motion.
If a smooth wall is not conveniently situated, set up a large
sheet of glass or smooth board at the end of a table, and move the
flame outward from the surface along the table. The flame will
probably need adjusting before it will respond readily. To do
this use the tap leading from the bag to the burner, and also
vary the weights producing the pressure. It appears to be
necessary to use a rather long gas tube to convey the gas from
the bag to the jet.
In obtaining the most sensitive flame it is to be noted that the
orientation of the flame is important and the burner should be
turned about a vertical axis, so that different sides of the flame
are presented towards the direction of the sound. The flame
appears to have different degrees of sensitiveness on different
sides.
CHAPTER XV
MISCELLANEOUS MAGNETIC EXPERIMENTS
Measurement of the Pole Strength of a Bar Magnet, using a Grassot
Fluxmeter
The search coil of the fluxmeter (see p. 482) is placed on the
bar magnet as shown in the figure so that it encircles the mid-
point of the magnet. At this stage the reading of the fluxmeter
is noted. Then if the bar magnet is uniformly magnetized,
the coil, when withdrawn, cuts all the lines due to the pole
past which it moves.
The fluxmeter indicates the flux change in units which are
specified ; in the case of the instrument described on p. 482 each
division corresponds to a change of flux equal to 10000 maxwells.
If there are x divisions change during the withdrawal of the
search coil, and there are n turns of wire in the search coil, then
since from a pole of strength, m, there are 471m lines, we have :
iooooa;
ioooo*
or m = «
The experiment should be repeated, using search coils having
different values for n.
This gives quite constant values for m as seen in the following
experimental results.
Coil A : n = 100.
Initial reading of Fluxmeter 3
Final reading of Fluxmeter 46
Deflection, first experiment 49
Deflection, second experiment 50
Mean deflection 49-5
40-5 x 10000
m = -^-^ = 394-1.
47c x 100 J ^
Coil B : n = 8.
Deflection, 4 divisions.
10000 x 4
m = -^8 " 3 ^4
mean value of pole strength, 394*25.
426
MISCELLANEOUS MAGNETIC EXPERIMENTS 427
Magnet was 1*58 cms. by -75 cm., i.e. 1*23 sq. cms. in cross-
section, i.e. intensity of magnetization, assumed uniform, is :
394^5 _
1-23
320-5.
Distribution of Magnetism along a Bar Magnet
This may be determined by using the fluxmeter in a manner
very similar to that of the last experiment. The magnet is
marked off in centimetres along its length, and the search coil
is placed at the mid-point, around the magnet. The coil is then
advanced in centimetre steps and the deflection of the instrument
noted, i.e. for o to 1 cm., 1 to 2 cms., 2 to 3 cms., etc. The
deflection in each case is proportional to the magnetization in
the space moved over by the coil.
The variation of magnetization along the length is seen by
plotting deflection against distance from the centre.
Gauss's Proof of the Law of Force
The most satisfactory proof that the force between two magnetic
poles varies inversely as the square of the distance between them
was first given by Gauss.
The method consists of a comparison of the magnetic force
at a point on the axis of a magnet with the force at a point on
a line drawn at right angles to it, at its mid-point.
Let us first calculate the value of the force at two such points,
assuming that the force between poles varies inversely as the
l\
S,
m£
Xk
■*
~JP
Fig. 270
«th power, so that the force between two poles of strengths,
m and m', r cms. apart is :
mm'
F =
r"
' End on' position ('A ' position of Gauss)
Let NS be the magnet (fig. 270) and P a point along the axis
produced, such that the distance from P to O, the centre of the
428 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
magnet, is r cms. If the length of the magnet is 2.1 cms., we have
the magnetic force at P, which is the force on unit pole placed
at P, is :
m m t,
.= r A .
(r - l) n (r + iy
This is the net repulsion due to N and S on the unit pole at P,
and is equal to :
F A =*m
F =-^ ^ ~ \ ~r)
If we use a bar magnet such that / is small compared with r,
we have :
expanding the two expressions.
Neglecting the powers of - higher than the first, this expression
becomes :
_ 2mnl _ Mw .
** A r n+l Jn+i' W
where M is the magnetic moment of the magnet.
' Broadside on ' Position (' B ' Position of Gauss)
Again in fig. 270 let Q be a distance r cms. from O, measured
along OQ, which is drawn at right angles to the magnet at the
mid-point.
Due to the pole at N, there is a repulsion at Q along NQ equal to
m ,
{\/(y a + / 2 )}'
and a similar attraction due to S, along QS.
The resultant of these forces may be obtained by considering
MISCELLANEOUS MAGNETIC EXPERIMENTS 429
the isosceles triangle, QNS, as triangle of forces. The sides
the resultant
QS and QN each representing a force ^
2ml
+ IH*'
NSis
i.e.
F R =
M
{r>
M
f a + / 2
n+>
yn+lK-j-
0'
rf
.(2)
or when I is small compared with r.
F --*
Comparing equations, (1) and (2) we see that for such a short
magnet the magnetic force at the two points, P and Q, are as n : 1.
The numerical value oin is obtained by comparing the magnetic
force at two such positions.
A magnetometer is employed. This consists of a small magnet
fastened in a light frame which carries a small mirror, the whole
being suspended by a thin silk fibre in a cylindrical brass case
as shown in fig. 271. Fig. 272 shows the usual form of small
suspended magnet.
Fig. 271 Fig. 272
The magnetometer magnet is allowed to come to rest, care
being taken that all the torsion is removed from the silk suspension.
Two boxwood scales, C and D, are arranged as shown in
fig- 273, one in the magnetic meridian, and one at right angles
to it.
A lamp, L, and scale, S, are arranged in the usual way, as
shown, so that an image of the lamp is reflected and focussed
on the scale S. * The distance of S from the magnetometer mirror
should be about one metre.
When the magnetometer needle is arranged to swing freely in the
centre of the case by means of the levelling screws shown (fig. 271),
the position of the needle with respect to the ends of the boxwood
scales can be very readily obtained, if the radius of the cylindrical
case is measured.
430 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If now a very small bar magnet is placed at any point, such
as B, fig. 273, in an E. and W. position, a magnetic field will be
set up at M, corresponding to the ' broadside on ' position. The
distance, r, between the magnet centres being obtainable by
§
1 1 1 1 1 1 1 1 1 1 u 1 1 ifcgfr
A
direct measurement on the boxwood scale. The deflection of
the magnetometer is found by observation of the movement of
the reflected beam of light.
The magnet is then placed, again E. and W., at an equal
distance from M, on the scale D (A, fig. 273), and the deflection
again obtained.
It is desirable to eliminate the zero reading of the magneto-
meter ; the double deflection, left and right, may easily be
obtained by reversing the magnet at the two positions A and B.
The experiment is repeated for various values of r. For
example, with a magnet 4 cms. long the values of the deflection
at 50, 60, 70, 80 cms. may be obtained and tabulated as under.
DISTANCE
OF MAG-
' END ON ' (A) POSITION
' BROADSIDE ON ' (B)
POSITION
NET FROM
CENTRE
MAGNET-
OMETER
IN CMS.
VALUE
A
- = n
B
SCALE READING
MAG.
DOUBLE
DEFLEC-
TION A
SCALE READING
MAGNET
DOUBLE
READING
B
DIRECT
REVERS-
ED
DIRECT
REVERS-
ED
50 cms.
I7'6
— l6'7
34-3
8.2 •
-9.O
17.2
2.0
60
10-3
— IO'I
20-4
4-6
-5*1
97
2.1
70
6-5
- 6-2
127
2-8
-3'3
6.1
2-08
80
4.4
- 4-i
8-2
i-8
-2-3
4-i
2-07
90
3'3
- 2-9
6-2
i"°
—2*2
3-2
i-94
mean
2-04
MISCELLANEOUS MAGNETIC EXPERIMENTS 431
The deflected position of the magnetometer needle depends
on the relative strengths of the horizontal component of the
earth's field, H, and the field due to the deflecting magnet.
If 6 is the angular deflection produced, we have,
H sin = F cos 0,
or F = H tan 0.
So that for the same value for r
F\ _ n __ tan t
F B ~ 1 "" tan 0, '
where X is the deflection when the magnet is at A, and 2 when
the magnet is at B, in the ' broadside on ' position.
If the distance from the scale S is fixed and the corresponding
deflections are d 1 and d 2 ,
tan t _ d x
tan 0, ~ d 2 '
d x
i.e. n = -—-
This ratio is tabulated in the last column of the table, and a
mean value obtained.
The Variation of Residual Magnetism with Temperature
The object of this experiment is to investigate the behaviour
of a magnetized carbon steel rod when subjected to temperature
changes.
Fig. 274
A rod of carbon steel of about 8 cms. in length and -5 cms.
in diameter is magnetized between the poles of an electromagnet.
The magnet so formed is set up inside a copper or brass tube,
which is of slightly larger diameter, as seen in the upper portion
of fig. 274.
432 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The magnet is drilled with a sma^lhole which receives a thermo-
junction, as shown. This thermo-junction is calibrated as
described on page 545 and serves to register the temperature
of the magnet. Some simple heating device is arranged to vary
the temperature of the magnet. In the diagram a second brass
tube provided on the upper surface with a series of holes is
shown, fitted on the end of a brass Bunsen burner. If a brass
Bunsen burner is not available a simple heater may be made
by taking such a second tube as the one shown, perforated with
a series of holes, having one end closed and at the other a device
for admitting an air-gas mixture similar to that of the common
form of ' gas ring.'
The magnet is set up at right angles to the meridian, and at
a distance d cms. from a mirror magnetometer, of the form shown
in fig. 271 ; the distance, d, should be such that at room tempera-
tures the magnet produces a full scale deflection. The magnitude
of the deflection is noted, and the temperature is obtained from
the thermo-junction balance point on the potentiometer.
The temperature of the rod is gradually raised by means of
the heater, and the magnetometer deflection and the thermo-
junction balance point are noted for every 20 C. rise, until the
temperature is about i8o°C. At this point the observations
are made much more frequently as the critical part of the varia-
tion is being observed. At about 250 C. the readings may be
again taken at about every io° C. or 20 C. intervals, and observa-
tions continued until the temperature is at the maximum
obtainable value for the heater employed.
The results are tabulated and the value of the temperature
obtained for each point ; the tangents of the corresponding
angular deflections of the magnetometer are also tabulated.
The magnet is then allowed to cool slowly, and the observations
are continued until finally the specimen is once more at room
temperature.
The magnetometer is at a constant distance from the magnet,
and in a fixed control field, H, hence the moment of the magnet
is proportional to the tangent of the angle of deflection.
This quantity is plotted against temperature, and a curve
somewhat similar to that of fig. 275 is obtained. The firm line
shows the relation between tan a and t° C. as the temperature
rises, and the broken line the relationship as the temperature
decreases.
For carbon steel in general, it will be found that at about
200 C. the magnetic moment is reduced to zero. An increase in
temperature causes a reversal of the polarity as shown in the curve.
This negative moment is a maximum at about 210 C. and then
decreases to zero at about 8oo°C. In general such a high
MISCELLANEOUS MAGNETIC EXPERIMENTS 433
temperature will not be available with the heater described, but
the most interesting part of the curve may be obtained. On
cooling, the negative moment is a maximum at D, at a temperature
about io° C. or 20 C. above the original position, and on regaining
room temperature will have a small positive moment, corre-
sponding to E in fig. 275.
ioo C
Fig. 275
The above experiment which demonstrates an interesting
feature of the magnetization of such an annealed carbon steel
rod is due to Prof. S. W. J. Smith and was described by him in
the " Proceedings of the Physical Society of London," No. xxiv,
15 Aug., 1 91 2. Reference should be made to that paper, which
gives an explanation of the observed results in terms of the
magnetization of the iron and iron carbide molecules which
compose the bar.
Briefly, the iron carbide molecules are set in line on magnetiza-
tion and exert a demagnetizing effect on the iron molecules
which are reversed in the internal field due to the carbide. At
about 210 the carbide is totally ineffective, but the reversed
iron molecules being more retentive have a maximum external
effect. As the temperature rises, the iron molecules approach
their neutral temperature, and finally at about 800 entirely lose
their magnetic effect. (See the original account in the place
stated above.)
The Increase in Length 0! a Bar on Magnetization
To investigate any change in length produced in a rod of iron
when magnetized, the following apparatus is set up. Fig. 276
shows a section of a brass case wrapped with two solenoids and
provided with a water jacket, WW, through which water at a
steady temperature may be circulated via tubes, T : the two
coils, C, are wound together so that their magnetic effects
28
434 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
are identical at the axis of the cylinder. The free ends of each
coil are connected to separate terminals, P, Q, R and S. The number
of turns and mean diameter of the two coils are as nearly as
possible the same, so that if a current were sent through from
T
w
IZL
W
zdT
PQRS
Fig. 276
P to Q, the magnetic effect is the same as when the same
current is sent from R to S through the second coil, or if the two
coils are connected in opposition there would be no magnetic
field along the axis.
The method of winding and the resulting magnetic field
should be tested at the outset by sending a current through the
coils, and noting the magnetic effect on a compass needle held
near the end of the solenoid.
At the top of the brass case is a circular brass end-plate to
which are soldered rigidly three pins, S, which terminate in
points on which a sheet of plane glass, G, may rest.
The iron specimen to be investigated is first thoroughly de-
magnetized by heating, etc., or by reversals of a diminishing
current through a solenoid surrounding it (as described on pages
MISCELLANEOUS MAGNETIC EXPERIMENTS 435
449-451) . The specimen should be a cylindrical rod with plane ends
made to fit the cavity provided along the axis of the solenoids.
The end effects of magnetizing solenoid are overcome by first
placing inside the apparatus a cylindrical length of brass, B,
with plane ends, the lower end resting firmly on the thick brass
base of the instrument. Above the iron rod is placed a second
brass cylinder, A, also with plane ends ; the upper end projects
slightly above the level of the upper surface of the end plate.
A long focus lens is fastened firmly to this brass cylinder by
means of soft wax or plasticine. The length of the supports, S,
is such that a small gap is left between the upper face of the
lens and the lower surface of the glass sheet, G.
Above the parallel walled sheet of glass, G, is placed a second
sheet, inclined at 45 to the normal. This reflects light from a
sodium flame towards the lens and plane surface.
The reflected beams are viewed by a microscope M, and the
Newton rings formed are observed in this way. The microscope
contains a cross-hair which is used as a point of reference.
When a current is sent through the solenoids arranged in
series, and to add their magnetic fields, the rod may be gradually
magnetized by increasing the current strength. If any change
in length occurs, there will be a movement of the Newton ring
system, which may be observed in the microscope.
It will be seen (by reference to page 323), that if the rod
changes in length by - where X is the wave length of sodium
4
light, that one particular part of the field which was formerly
dark will now be bright or vice versa. The direction of move-
ment of the fringes will show whether the rod increases or decreases
in length.
Now, due to the current circulating in the coils, a certain
amount of heat will be developed. Thus for a rod of 50 cms.
length a rise in temperature of about -03 will cause an increase
in length more than sufficient to move fringes corresponding to a
change in size of the gap equal to -. It therefore becomes
necessary to ensure that the rod is maintained constant in
temperature throughout. Water at constant temperature is
circulated through the jacket, WW, for some time before com-
mencing observations and continuously during the experiment.
Under these circumstances tap water from a supply removed
from any hot water pipes will be admissible.
As a further check on the observations, to ensure that the
results are not spurious due to the heating, each observation is
preceded by one in which the current to be used is sent through
436 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the twp coils arranged in opposition, i.e. the heating which may
affect the dimensions of the bar will be present, and any movement
of the fringes is due to that cause, since there is no net magnetic
effect.
The coils are rapidly changed over, and the same current
causes a magnetization of the bar. The true effect on the
length of the bar due to magnetization may therefore be obtained
To facilitate the rapid change in the direction of the current,
a Pohl commutator may be employed : the ends of one coil,
say, R and S, are connected to one pair of the terminals of the
commutator ; the other coil is connected in series with a battery
and variable resistance to the middle terminals. Thus by throwing
over the movable arm, the coils may be changed from a condition
of producing fields in opposition to one in which their magnetic
fields are in the same direction.
Carry out observations using currents up to a maximum value
which is determined by the current-carrying capacity of the
coils, CC.
• Plot the results showing change in length in terms of the
wave length of sodium light, against the field strength measured
in Gauss.
CHAPTER XVI
TERRESTRIAL MAGNETISM
Determination o! the Horizontal Components of the Earth's Magnetic
Field (H)
The method due to Gauss, described below, is usually employed
to determine the horizontal component of the earth's magnetic
field ; it may also be used to measure any magnetic field which
is uniform over a sufficiently large volume.
The method involves two experiments. In the first a magnet
of known moment of inertia is suspended freely in the earth's
field, at the place where H is to be found, and from observation
of the time of swing, the product MH where M is the moment
of the magnet, may be calculated. In the second experiment
the field due to the same magnet is compared with the earth's
field by means of a magnetometer.
Determination of MH
Let the magnet be suspended in a light stirrup and perform
oscillations whose periodic time is T seconds, it being supposed
that the suspension has no initial twist. Then if I is the moment
of inertia of the magnet about the axis of suspension, and i the
moment of inertia of the suspending frame about the same axis,
and if t be the restoring couple per unit angular displacement
due to torsion of the suspending fibre, we have :
= 2*J
'(MH + t)' {1)
for (I + i) is the moment of inertia of the system, and (MH + t)
is the total restoring couple per unit angular displacement, when
small displacements are considered, whence :
MH = A, (say) (ia)
M
To find jr
A magnetometer is set up in the place at which MH was
found, and the magnet placed with its centre d cms. from the
centre of the magnetometer needle, east or west of it and lying
east and west. If be the deflection of the magnetometer,
and F the value of the field due to the magnet, we have :
437
438 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
F = H tan 0,
and F = ^ 2 _j 2)2 >
where il is the distance between the poles of the magnet.
Alternatively, if the magnet were placed so as to be in the
€ broadside on ' position, i.e. E. and W. with its centre in the
meridian through the centre of the magnetometer needle, we have
F lf the field produced at the magnetometer, in this case is
M
~ (d t * + /*)*'
where d t is the distance between the centre of the magnet and
the magnetometer in this case, and I is the distance between the
poles of the magnet.
Whence for the ' end on ' position
Htan6 = (i2 _ /2 p (2)
or for the ' broadside on ' position
M
Htan^ = p^47iji (3)
M
in either case the values of g may be found.
Let this value be B.
Whence from the two experiments :
H
-£
In performing these experiments special forms of magnetometer,
etc., are advisable. All iron is removed from the neighbourhood
at the outset, and care taken to maintain the magnetic conditions
the same throughout the two experiments.
The magnet used may very well be a true cylindrical bar
magnet of not more than a few centimetres in length.
Let the length as determined to & of a mm. be 2l x cms., and
the radius, measured with a micrometer screw, be r cms, the
mass of the magnet, weighed to at least one in a thousand, being
m grammes. I, its moment of inertia about an axis through
the centre of gravity and normal to its length, is
—ftr+i)
To find T, use is made of a light stirrup suspended by a fibre
from a torsion head. A brass cylinder of the same mass as the
magnet is first placed in the stirrup, and all the twist is then
removed from the fibre. The torsion head is then turned so
TERRESTRIAL MAGNETISM
439
that the brass rod is in the magnetic meridian. It is then
replaced by the magnet.
The frame is provided with a small mirror, otherwise one end
of the magnet is polished to act as a mirror, and a lamp and
T
Fig. 277
scale is arranged so that a beam of light reflected from the
mirror on to the scale enables the movement of the suspended
system to be observed.
Set the magnet swinging through a very small angle, and
obtain the time taken for 100 complete swings. Repeat this
and take a mean value, from which T may be obtained.*
To find i in equation (1) weigh the stirrup and estimate the
moment of inertia from the dimensions.
The restoring couple due to the torsion, t, may be obtained
by rotating the torsion head through, say, a complete turn.
Note the resulting angular deflection of the magnet, y> radians,
whence (2* — y>) t = MHy,
MH V
or T = 7 r-
(27C — y>)
It will be found that, if the fibre is thin and the stirrup a very
light one, t and i are negligible compared with MH and I.
Including these corrections we have :
T = 2*
I+*
^H(x +_£_)'
* Alternatively, the method of timing set out on page 118 may be used.
440 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
whence
MH
4* 2 (I + i)
T 2 ^r~
27t — \p
The magnetometer experiment is performed, using a special
form of magnetometer (Kew type magnetometer). This consists
oi several small magnets rigidly fastened in a metal frame,
which also carnes a small mirror. The whole is encased in a
brass case, provided with suitably arranged glass windows so
that a beam of light may be reflected from the mirror on to a
scale S cms. away from it.
The magnetometer frame carries a long bar which has four
pegs, Pj, P 2 , P 3 , p 4) so arranged that when a small carriage is
placed on any of them, the cylindrical magnet, which the carriage
supports, lies with its axis in the same plane as the magneto-
meter needle. Figs. 278 and 279 show diagrammatically the
arrangement of the magnetometer.
Dl
^a.
-1
J
S
"i 1 1 1 1 ri 1 1 1 1
Fig. 278
M
?4.
_J
Fig. 279
The magnetometer and bar are first levelled (using a spirit-level)
by means of the levelling screws on the base of the instrument.
Pj and P 4 , P 2 and P 3 are fixed, in the construction of the
instrument, equidistant from the needle when the system is
level. Measure P X P 4 = D x cms., say ; measure P 2 P 3 = D 2 cms.
The scale is then adjusted so that the reflected spot of light
is at its centre when the magnetometer is under the influence
of the earth's field alone. The arm which carries the pegs is
set first of all in the direction of the magnetic meridian, and the
small carriage in which the cylindrical magnet is carried is
placed on V x and turned round so that the magnet is at right
angles to the meridian. The true position for the magnet at
right angles to the meridian is obtained by noting the deflection
of the magnetometer. This will be a maximum when the normal
position is acquired.
TERRESTRIAL MAGNETISM 441
In this manner the deflection in cms. on the scale may be
measured. The magnet is then rotated through 180 until a
maximum deflection in the opposite direction is obtained in the
magnetometer. Let these deflections be <5 X and <V cms., and the
corresponding deflections when the magnet is supported on the
other pegs be 6 2 , 6 2 ', etc.
Obtain the mean value of 6 t d t ', <5 4 6 J — 6 say, that is the
mean deflection for distance £D X — d lt ; also take the mean of
#2 <V> ^3 <V — sl > sa Y> corresponding to a distance £D 2 = d 2 .
If q> and <p' are the corresponding angular deflections of the
magnetometer, we have :
St St
tan 29 = ^- ; tan 29' = ~-»
Hence we may calculate q> and q>' from the observed deflections.
The cylindrical magnet was placed as described above so that
the field at the magnetometer was that due to the ' broadside on '
position ; thus from equation (3) above
H tan * = w+w
M
i.e. g = (df + l*f • tan g> = (d 2 * + If tan 9'.
H
Now
(i» + jy = rf»(i+!j!-. . .),
neglecting the fourth and higher powers of -5.
i.e. H = d, 3
I+ !(j,)^ tan "'
M ■•'^.fCr i )fw.
M
H
H
Hence / may be eliminated, giving
( * - * A =d 2_ d ,
V^tany d 2 tan <p'J x 2 '
M
Hence =5. is determined, and the value of H may be calculated
using this and the 1 previously obtained value of MH.
The bar carrying the four pegs may be alternatively arranged
at right angles to the meridian, and the carriage containing the
magnet again placed on the pegs with the magnet at right angles
to the meridian, producing a field of force at the magnetometer
corresponding to the ' end on ' position. The process described
above is repeated, and the average deflections obtained enable
442 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the calculation of the mean deflections, x and 2 , for the distances,
d x and d v
Hence again :
TTi „ 2M I
H tan X = ^r-T r x
D x 3
2£ 2 )
d^ ' ' ']
TT A „ 2M
Htan 2 = ==r-. X
D 2 S ( T _ i 2 ) '
to the same order as before.
Once more eliminating I, half the distance between the poles
of the magnet, we have
2 .M(_I j * ^ =^-^,
H V*! tan ©j d % tan 2 /
M
whence ^ may be again calculated. This second method gives
larger deflections and is therefore preferable to the first.
Fig. 280
The Dip Circle— Measurement of the Angle of Dip
The dip circle consists of a long magnet supported on a hori-
zontal axis which passes (approximately) through its centre of
gravity and is at right angles to its length. The magnet is sup-
TERRESTRIAL MAGNETISM 443
ported in wheel bearings or on agate edges, so that it may turn
freely and with a minimum of friction. On the same base as
the support for the knife-edges is a vertical graduated circle
which enables the position of the ends of the needle to be
obtained. The whole structure is supported on a circular table,
which is capable of rotation around a vertical axis. This rotation
may be measured on a horizontal circular scale about which the
table may rotate about its axis, and, by means of a suitable
vernier, may be measured to at least one minute (see fig. 280).
The position of the ends of the magnet may be read on the
vertical scale by means of microscopes carrying verniers, which
move round the scale. If the plane of the vertical scale be turned
in the direction of the meridian, so that the horizontal axis of sup-
port of the magnet is at right angles to it, the magnet sets in the
direction of the earth's lines of magnetic force, and the angle
included between the horizontal and the position of the needle
is the angle of dip. '
However, there are many sources of error . The axis of rotation
/ . of the magnet may not quite coincide with the centre of the
scale. One end, therefore, of the magnet would read too small,
and the other too large a deflection.
2 The axis of rotation may not be truly through the centre of
" gravity, in which case a couple is exerted, tending to turn the
magnet so that the centre of gravity comes vertically under the
axis of support.
^ The axis of the magnet does not usually coincide with the
geometric axis of the needle, so that the centres of the ends of
the magnet do not give a correct reading for the magnet when
set along the earth's line. The possible errors are fully set out
in any textbook of magnetism.
A consideration of the observations taken and described below
will show that these errors may be eliminated.
To determine the angle of dip, the dip circle is set up and
by means of the levelling screws on the base of the instrument,
using a spirit-level, the 'horizontal ' scale is made truly horizontal.
The plane of the meridian must then be found, so that the
magnet may set freely. To do this, the instrument is first
turned so that the upper, S, pole is at the 90 mark, and the
reading of the vernier on the horizontal scale is noted.
• It will in general be found that the lower, north, pole of the
magnet is not quite at the lower 90 scale reading. A slight
turn of the screw which rotates the instrument about the vertical
axis is then made until the north pole is at the 90 reading, and
the horizontal vernier scale is once more read. The circle is
next turned through 180 , and the two more observations are
made. The needle is now reversed in its bearings and the four
444 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
observations repeated. The mean value of the horizontal scale
readings for these eight positions is taken. This is the mean
position for the plane of rotation at right angles to the meridian.
The case is then moved through an angle of 90 from this
mean position so that the plane of the needle now coincides with
the magnetic meridian.
The following 16 readings are then obtained :
(1) Read the position of each end of the magnet.
(2) Rotate the instrument through 180 , and once more read
both ends of the magnet.
(3) Reverse the needle in its bearings and read both ends on
the graduated circle.
(4) Turn the instrument through 180 and again read both ends.
(5) Remagnetize the needle and repeat the above process.
The needle is remagnetized so that its poles are intercharged.
The north pole for the observations, 1 to 4, above now becomes
a south pole. Under these new conditions eight more values
are obtained. The mean value of the 16 readings gives the
true value of the inclination. The observations may be con-
veniently tabulated as under.
MARKED
SIDE
OF
MAGNET
FACING
MARKED POLE DIPPING
(NEEDLE READINGS)
magnet remagnetized
(needle readings)
UPPER
POLE
LOWER
POLE
MEAN
UPPER
POLE
LOWER
POLE
MEAN
East
West
a
P
a'
P'
Needle reversed in bearings.
East
West
Y
<5
Y'
d'
mean
q+ P + r + 8
4
== A
Angle of Dip =
mean
A +B
a' + P' + Y' + <r
= B
Note. — It may be necessary to obtain corresponding 16 values for the
determination of the zero instead of the 8 described on pages 443-4'
CHAPTER XVII
PERMEABILITY OF IRON AND STEEL
Measurement o! Permeability by the Magnetometer Method
Consider a long thin specimen of iron wire placed in a magnetic
fiield of uniform strength H. Due to the fields the soft iron
specimen becomes magnetized, having poles of strength, ± m.
Let ,
2/ be the distance between the magnetic poles,
s be the cross-section of the wire.
The number of lines coming out from the north pole end is
47cm, and an equal number enter the south. Due to the poles
developed, there are a number of lines of magnetic force passing
from the south to the north in the specimen, equal to ^nm.
This is equivalent to - — per sq. cm. of cross-section of the
iron, assuming that the lines enter and leave at the ends only.
In addition to the above there are H lines per sq. cm., due to
the field, making the total number per sq. cm. of H + - — •
This is the induction, B, in the specimen. Thus N
The magnetic moment of the wire is 2,1m, its volume is s • 2/.
Hence the intensity of magnetization, which is the magnetic
moment per unit volume, is
2.1m _ m _ T
2.1s ~ s ~
Hence B = H + 4* 1 ( x )
The permeability (v) for any value^of H is defined by the ratio — •
We therefore have :
[L = 1 + 4« jj
The ratio -=y is usually defined as the susceptibility or coefficient
XX
of induced magnetization, so we have, putting k for this quantity,
jx = 1 -f- 4nk.
445
446 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We have imagined the length of the magnet to be large. Other-
wise there is an effect due to the poles at the end of the specimen
to be considered, in expressing the value of H. For, due to the
north and south poles developed in the specimen, there will be
a magnetic field in the opposite direction to H, reducing the
value to H 1 , say. This demagnetizing field can be neglected,
however, if 2.1 is made large compared with s. A suitable length
is obtained when 2X is greater than 200 times the diameter of the
wire.
If the specimen is arranged horizontally or vertically near a
small magnet, suspended by an unspun silk fibre, the magnet
will be deflected by the induced magnetism in the specimen.
From a knowledge of the deflection and of H, the value of the
constants, B, \k, I, k, may be calculated.
To find the above constants for a given wire specimen, the
type of reflecting magnetometer described on page 429 is used.
The inducing magnetic field is obtained from a solenoid,
through which a current of known magnitude may be passed,
and the deflection of the magnetometer may be observed.
Iron Wire Specimen arranged Vertically
The arrangement of the apparatus is seen in fig. 281. L is
a lamp and scale, M the magnetometer which is clamped to the
wooden base shown, at a convenient distance from the specimen
which is placed in a vertical solenoid, S. The wooden base is
arranged in an east and west direction so that the magnetic
effect of the specimen may be measured.
Fig. 281
Before introducing the specimen, the solenoid is connected
in series with a small coil, C, and several accumulators, a variable
resistance, an ammeter reading to 3 amperes, and a reversing
switch.
The coil, C, is so arranged that the magnetic effect produced
by it on the magnetometer is the opposite sign to that produced
by the solenoid, S.
PERMEABILITY OF IRON AND STEEL 447
The position of C is adjusted so that whatever is the value
of the current in the circuit there is no movement of the needle.
In that way the whole movement of the magnetometer will be due
to the magnetism induced in the iron wire.
The current is then switched off and, having noted the position
of the reflected spot of light on the scale, the unmagnetized
specimen is introduced. If there is any movement of the spot
of light it may mean that there is a certain amount of magnetiza-
tion in the specimen. It should be demagnetized by first of all
heating to red heat and then hammering vigorously. Having
once more replaced it in position in the solenoid, the current
should be adjusted to about one ampere. The direction of the
current should then be constantly reversed, whilst at the same
time its magnitude should be reduced to zero. In this way
any small residual magnetization may be neutralized.
A preliminary experiment on a second sample of the same iron
wire enables a value of the maximum current to be determined.
The distance MS is adjusted so that a full scale deflection is
obtained when the sample is saturated. The original specimen
is then introduced so that its lower end is level with the axis
of the magnetometer magnet.
Starting with the soft iron unmagnetized specimen, the zero
position of the magnetometer on the scale is observed. The
value of the current is then raised to, say, '2 ampere and the
deflection noted. Then without breaking the circuit the current
strength is increased by increments of m 2 to a final value of about
3 amperes, a limit which depends upon the thickness of the wire
in the solenoid winding, and also upon the result of the preliminary
experiment, whichever limit is the smaller fixes the maximum
current.
Having increased the current to a suitable maximum value,
it is then decreased by similar steps, again taking care that the
circuit is not broken in the process, until zero current circulates.
The current is then reversed and increased to the same maximum
in the other direction. The current is again brought back in steps
to zero, and having once more reversed the current it is raised
to the maximum in the original direction.
Tabulating these results a connexion is found between the
value of the current in amperes and the magnitude of the deflec-
tions or the value of the scale reading, which, when plotted, will
show the general form of the curve below.
The results are then converted to the corresponding C.G.S.
values for B and H.
In fig. 283 let n s represent the magnetized specimen which
will produce at P the position of the magnetometer, a field
448 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
strength, F, at right angles to the Earth's horizontal component,
and which causes the deflection, 0, say.
We have F=H'tan0,
where H y is the magnitude of the Earth's horizontal component.
Fig. 282
Let m be the pole strength induced in the specimen,
I be the length in cms.,
r the distance from the lower end of specimen to the
magnetometer in cms.
I
I
\
I
I
1,
\
\
\
u» r -
Fig. 283
\
Then
F = — — ^r-. cos SPn
r* SP 2
m
m
r 2 r 2 + /* (r 2 +l 2 )*
/i r \
V 2 (r 2 +/ 2 )v
PERMEABILITY OF IRON AND STEEL 449
Whence :
H'tane
m =
/L I \ '
\y2 ( y 2 + l^ij
and I = -, B = H + 47rl.
Further, using a solenoid of n turns per cm. having a
current of c amperes, the uniform field H = - — is effective on
10
the specimen.
Hence, for each reading in the table of results, I, B and H,
also (x and k, may be calculated. If the curve is drawn for I and
H, the area enclosed is equal to the work done on the specimen
in the cycle (i.e., the heat developed).
The values of H and B are in C.G.S. units, i.e. Gausses (see
table on page 633).
(2) The B-H Curve for a Sample of Iron (using a Ballistic Galvanometer
This method is specially applicable to the determination of
the B-H curve for a specimen in the form of an anchor ring, or
a very short hollow cylinder. For such a specimen, magnetized
by a magnetic force of, say, H gausses, no free poles will be
developed, and therefore no demagnetizing field will be set up.
The value of the induction, B, will therefore correspond to H,
and not some smaller field, H', as in the previous case.
The experimental arrangements of this method, as shown in
fig. 284, are such that a variable field, H, may be set up, by
passing a current through a primary winding, P, which is closely
wound on the anchor ring, and some means of measuring B in
the specimen.
The method employed to measure B is to wrap a few turns
of wire, S, round the anchor ring and primary winding and
measure the quantity of electricity which passes through a
ballistic galvanometer, BG, due to the change in the induction
in the specimen for a known change in H. Since the ballistic
galvanometer must be standardized, the secondary circuit is
completed through a second small coil of a mutual inductance,
M, used in the standardizing experiment. Thus the ballistic
galvanometer is in a fixed resistance circuit in all measurements.
The current from an accumulator may be regulated by resist-
ances R x and R 2 and measured by an ammeter, A. By means
of the Pohl commutator the current may be sent directly or
reversed through the primary windings, P, when K 2 is to the
left, or through the standardizing mutual inductance, M, when
the key K 2 is closed on the right.
29
450 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
By closing the key K lt the resistance R 2 may be cut out.
The value of H may be calculated, as explained later, from
the value of the current strength as obtained from the ammeter ;
and B may be calculated from the observed throw of the ballistic
E -p;
Fig. 284
galvanometer, BG. Since the galvanometer circuit is composed
of a comparatively low resistance, a moving coil instrument may
be too heavily damped to be serviceable (see page 481). A
moving needle instrument is used if this is the case.
As a preliminary experiment, the key K 2 is closed to the left,
and R x and R 2 decreased, until, on closing the Pohl commutator,
the galvanometer gives a full scale deflection from the zero,
which should be the central graduation of the scale. The
current required to do this is noted and is used as a maximum
value in the main experiment. Usually from 2 to 3 amperes
will suffice, the sensitivity of the galvanometer being adjusted
to measure the throw produced.
In general, the past history of the iron anchor ring will be
such that residual magnetism in the specimen is almost certain.
This must now be reduced to zero.
To do this the galvanometer circuit is broken and the resistances
R x and R 2 reduced to a minimum. The current passing in P
is then reversed many times and R x and R 2 gradually increased
until finally the current which is reversed is very small. It
should be noted that R x and R 2 are such that changes in resistance
may be brought about without breaking the circuit.
PERMEABILITY OF IRON AND STEEL
451
Another method of demagnetizing the specimen is to pass an
alternating current through P and a liquid resistance in series.
The alternating current is gradually reduced to zero by with-
drawing the electrodes of the resistance.
The galvanometer is once more put in circuit with S, etc.,
by closing the key in the circuit (not shown in the diagram) :
K x is closed, and R x given the value corresponding to the maxi-
mum current as determined by the preliminary experiment.
The commutator is then closed to the right, and the throw of the
ballistic galvanometer, <5, noted. The reading of the ammeter
is also noted. The throw will correspond to an induction, Bj,
and the ammeter reading to a magnetizing force, H x , represented
by some point such as S in fig. 285. We now use this as a point
of reference. The galvanometer circuit is again broken, and the
commutator is reversed rapidly some 20 to 25 times, and finally
left on the right, i.e. the iron is taken several times round the
cycle represented in fig. 285 and is said to be in the 'cyclic state.'
BG is again put in circuit. When all is steady, R 2 is given a
small value and K x is opened. The magnetizing field is thereby
decreased and the throw of the galvanometer, d 2 , noted. This
throw corresponds to a decrease in induction, Bj — B 2 . The
value of the current corresponding to H a is noted on the ammeter.
Fig. 285
Kj is now closed, the galvanometer key opened, the Pohl
commutator reversed 20 to 25 times, and finally left to the right.
The galvanometer is put in circuit ; R 2 is given a larger value ;
K x is then opened and the throw due to the change in the induc-
tion, say, Bi — B 3 , is noted ; the ammeter is again read.
This process is repeated with the commutator to the right,
until R 2 is infinite and consequently the current and H zero,
i.e. the relation of B to H represented by the path, SA, of fig. 285,
is investigated.
After each measurement, the iron, by the reversal of the
maximum current, is returned to the state represented by S,
which therefore becomes the reference point.
452 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The key, K lf being closed, etc., the commutator is reversed
some twenty times once more and finally left to the right ; R 2 is
given a large value and the galvanometer is again put in circuit.
The commutator is then thrown over to the left, and at the
same time K x is opened, i.e. the current is reversed and at the
same time made of small value. This gives a point on the part
AB of the curve. The same starting point is taken (S), the
drop in induction measured, and the change in H from a maximum
position to a small negative value obtained.
This process is repeated in many steps until finally R 2 = o,
i.e., until the change in field, H, and induction, B, corresponds
to a reversal of the maximum positive to maximum negative
values.
At this stage we may assume that the curve is symmetrical,
and obtain the full hysteresis loop having obtained SABS 1
experimentally, or we may proceed to find SfA^S by experi-
ment, using S 1 now as our reference point. The process is as
before except that the commutator is now left on the left-hand
side. K x is opened in turn to R 2 having values from o to infinity.
CURRENT C
IN E.M. UNITS
FIELD H
IN
E.M. UNITS
THROW OF GALVO. (<5)
CHANGE OF
INDUCTION
FROM S OR S 1
INDUCTION
(B)
ASSUMING
SYMMETRY
(I)
(2)
(3)
(4)
(5)
FOR
CURRENT
CHANGE
Drop
from S
•24
to -24 = 2-o cms.
10000
•22
•24 to -22 = 1-5 cms.
500
9500
•20
•24 to «20 = i*8 cms.
700
9300
•18
•24 to- 18 = 2-4 cms.
1000
9000
. . .
etc.
+•02
•0
— 02
— 20
— 22
—24
•24 to — 24 =
20000, say
— 10000
PERMEABILITY OF IRON AND STEEL 453
At this stage (point A 1 ) R 2 is given values from infinity to o.
and Kj opened as the commutator is thrown over to the right-
hand side.
It should be found that the value of the throw for the reversal,
when the full current is passing through the primary is identical
for the change from S to S 1 and S 1 to S, and that either value
is approximately twice the first recorded throw, <5 X .
The results may be tabulated as shown. The original direction
of the current in the coil, P, is taken as positive and the current
value may be recorded in column (1) and the corresponding throw
in column (3).
To Calculate H
Imagine a unit magnetic pole to be taken round the axial
circle of the solenoid. If H is the value of the magnetic field
strength due to a current, C, in the solenoid, the work done on the
pole for such a complete circular path is 2i*r • H, where r is the
mean radius of the anchor ring.
If there are N turns in the winding, the unit pole is linked
with each winding in the complete path, therefore doing work
47rC for each or a total of 4nNC,
e.g. 471NC = 2TtrK
H = ?^,* <D
r
where C is the current strength in E.M. units (not amperes).
The value of the current in E.M. units (not amperes) shown in
column (1) may be converted to gausses (lines per sq. cm.) by
multiplying by the factor — as seen in equation (1).
To calculate the induction, B, corresponding to the observed
value of 0, an auxiliary experiment is necessary. The two-way
switch, K 2 , is closed on the right, so that a current maybe sent
through the long straight solenoid, M (about 40 cms. long and
4 cms. in diameter), which has m 1 turns per cm. Inside M is
the second small solenoid which has been in series with S and
the galvanometer throughout the preceding experiment.
For a current of C E.M. units flowing through M, since the
solenoid is long, the field strength at the centre is ^imfi gausses.
*The result of (i) may be obtained from the general formula giving H inside
a solenoid of n x turns per unit length, i.e.
H = 4irtt]C.
N
In our case n x — -—-,>
H r
3NC
or
454 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Let the radius of the inner coil be r 2 cms., and N a the total
number of turns in this coil, then the total flux in the inner
coil is :
4izm 1 C(izr 2 e N 2 ) maxwells.
If now the current be reversed in M the change in induction
in the central coil is :
8n*m 1 Cr t , N 2 lines.
Let R be the total resistance in the galvanometer circuit.
dN
Since the electromotive force is numerically ~rr#
at
r> <*N f ,, fdN
Q = **n£*S mC (2)
For an instrument of the moving-needle type :
e = *G sm 2 ~( I+ S>
Q = K 1 sini( I+ ^).
T H
where K t = — . ^-and is constant.
7C (jr
Now for small deflections, if d is the scale deflection in cms.
a
sin oc d oc sin - and x is constant in the present case, even using
a moving coil instrument, for the circuit is constant,
i.e. Q = K 2 i;
where K 2 is a constant (i.e. Qccd for either type of galvanometer)
substituting in (2) above
^ = iL^a,
or Kd = Stt^jN^C,
where K = K 2 R == a constant.
The last equation, which connects the deflection, d, with the
total change in flux, enables the calculation of the value of B
corresponding to each field H.
The change in the number of lines, threading the entire circuit,
corresponding to 1 scale division deflection on the galvanometer
scale is :
K ^ d
PERMEABILITY OF IRON AND STEEL 455
The deflections 8 V <5 2 , etc., in column (3) therefore correspond,
to a change in flux equal to Kd lt K<5 2 , etc. This is a change in
the total flux in the space of the secondary winding S. If r x is
the radius of cross-section of the iron anchor ring, approxi-
mately that of the secondary winding, S, and m is the number of
turns in the secondary, and (Bj — B 2 ) in the change in induction
to be entered in column (4), we have, since B is the number of lines
per square cm. of cross-section :
(B x - B 2 )m • Ttf-j 2 = K<5,
Bl-B « = ^'*
( a m r x 2 )
The term in the brackets is evaluated and used as a reduction
factor, converting 6 to induction for column (4).
To obtain column (5), assume a symmetrical form for the
curve, i.e. assume that the induction for the maximum positive
and negative current values are equally placed from the zero
induction line, i.e. equal to half the induction for a full current
reversal, which is the mean of the extreme values in (4). The
method of calculation may be understood from the numbers shown
in the suggested table.
Note
The B-H curve could also be obtained for an anchor ring as
above, using a fluxmeter in place of the ballistic galvanometer.
The calculation of H is as before. When the induction is
large the number of turns in the secondary coil can be diminished,
so that the fluxmeter reading is not excessive. The calculation
of B from the observed deflection, 0, for any current change in
the primary, is more direct than for the ballistic galvanometer.
Unit scale deflection on the fluxmeter corresponds to a flux of
10000 maxwells. So that if the secondary coil used is of m
turns, and the radius of each turn of the secondary is r lt the
induction change (Bj — B 2 ) corresponding to the deflection 0,
is given by :
WX(Bi — B 2 ) Xtt/! 2 = 100000,
„ -. 100000
i>i — Jt>2 = r«
The experiment is carried out in a manner similar to that
described above for the galvanometer as flux measurer.
CHAPTER XVIII
AMMETERS, VOLTMETERS AND GALVANOMETERS
The most convenient form of current measurer is the ampere
meter or ammeter, which indicates directly, on a graduated scale,
the strength of the current passing through the circuit in which
it is placed.
As a result of the method of construction the range of useful-
ness of the instrument is limited, in the usual form, to measure-
ment of current not less than a milliampere. Recently, however,
by improving the suspension, etc., the two-pivot instrument
has been manufactured capable of measuring one microampere,
for example, a Weston ammeter is made with an open scale
division equal to one microampere.
Similar conditions hold for the voltmeter.
Fig. 286
The Voltmeter
This instrument consists essentially of a coil of thin copper
wire, C, which is supported on an axis pivoted on jewelled pivots,
P, P 1 , and is free to move in the cylindrical gap between the
soft iron pole piece, DD\ of an aged steel permanent magnet, NS.
456
AMMETERS, VOLTMETERS AND GALVANOMETERS 457
The field is strengthened and made approximately radial by
the insertion of a soft iron cylinder; I, in the space inside the coil.
This cylinder, as Seen in the lower fig. 286, is fixed to a brass
bar, B, which forms a bridge between the pole pieces. The
air gap is thus reduced and the coil moves freely in this
space.
To establish a restoring couple, the two hair-springs, H, H 1 ,
are attached to the axis, and to points on the framework which
are insulated from each other. These springs also serve as
leads for the current, to and from the coil.
Attached near the upper spring is a light counterpoised pointer
which moves over a scale G. The centre of gravity of the whole
system is arranged to coincide with the axis of suspension.
When in use the instrument is placed in parallel with the
points whose potential difference is to be measured. The
internal resistance of the instrument must, therefore, be large
in order to avoid any appreciable rearrangement of current and
potential drop in the circuit. The current passing through the
voltmeter is therefore very small for such a high internal
resistance, and therefore the heating of the coil is not very great.
The internal resistance is not made up entirely of that of the
copper coil. In most forms the greater part of the internal
resistance consists of a resistance in series with it (fig. 287). The
chief reason for this is to avoid any error due to heating in the
moving coil.
Such heating may be due to (a) atmospheric rise in temperature,
( p) the Joule (C 2 R) effect. An increase in resistance of the moving
part would occur in either case unless the temperature coefficient
of the wire were small. Manganin has a small temperature
coefficient, but a higher specific resistance, i.e. for the same coil
resistance a less radiating surface is available for manganin
than copper.
The effect of (a) in raising the temperature of the coil may
be best eliminated by making the moving coil resistance fairly
low, and hence the percentage change of the whole is
reduced.
Hence the usual compromise is a copper moving coil of com-
paratively small resistance and a series manganin coil of high
resistance. This series resistance is constructed of thicker wire
than would be possible for the moving coil.
Having in this manner secured the best approximation to
constancy for the internal resistance, it will be seen that the
small current, C, passing through the instrument is proportional
to the potential difference between the terminals. Now the
deflection produced is proportional to the current for a large
range of deflection, when the field is radial, and hence the deflec-
458 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
tion produced is proportional to the potential difference between
the terminals.
The instrument is made ' dead beat ' by winding the moving
coil on a copper frame.
Seia Rfesisra^ce
To >->pPer >b \ouo«r v
P
?AVWVVWWVWWWV-1
B
-3 Volts
Fig. 287
H
-»50 Volte
The same instrument may be used to measure different ranges
of potential. This will be seen from fig. 287. Thus if a potential
of 3 volts, when applied to PL, produces a full scale deflection
in the instrument, whose internal resistance (AB + coil) is
345*5 ohms, it will be seen that the current in the coil is -00865
ampere.
Now, if a higher potential, say, 150 volts, is to be the new
value corresponding to a full scale deflection, this may be con-
nected to H and P so that a bigger series resistance, BC, is
included, such that the total internal resistance, R, is given by
•00865 = ^,
or R is 17270 ohms.
The same current will flow through the coil and hence the
deflection will be again a full scale deflection.
The graduations on the scale will therefore subdivide the
o to 150 into equal increments, and each division corresponds to
fifty times the value which corresponds to the lower voltage
applied to PL.
For the lower range voltmeter, measuring potential of the
order of a millivolt, the value of the internal resistance is
smaller, for the deflection is proportional to BC, where B is the
magnetic flux in the gap and C the current. B is constant and
so C to produce the deflection, when a low potential is applied,
is obtained by decreasing R.
AMMETERS, VOLTMETERS AND GALVANOMETERS 459
In such a moving-coil instrument the direction of deflection
depends upon the directions of the current. The higher potential
should be connected to P which is marked +. An accidental
reversal of this order is apt to strain the needle.
Ammeters
To measure the current in a circuit, the measuring instrument
used should be of low resistance, unless some account be taken
(as in galvanometers) of the resistance introduced in this way.
For example, if a small resistance of known magnitude, r, be
included in the circuit, the current strength, c, may be found
if the value of the potential drop, V, along r, be determined by
V
means of a milli voltmeter, for c = — . This arrangement of a
T
voltmeter shunted with a low resistance is utilized in the am-
meter. The fixed-range ammeter usually contains the shunt
inside the case. The value of the shunt resistance is small,
and therefore the resistance of the whole instrument is of the
same order. The moving coil of the ammeter is often, also,
provided with a series resistance as in the voltmeter to minimize
temperature variations as described above.
Many of the better forms of ammeter are not provided with
fixed shunts but require an external one. The value of the
shunt resistance determines the range of the instrument.
For bigger ranges the smaller is the resistance of the shunt.
Suppose, as before, the maximum scale reading is obtained for
a current, c, through the coil ; this is proportional to the potential
difference between the ends of the shunt. It will be obvious,
that if the external current, C, be doubled the drop of
potential along the shunt will be doubled, so that if a shunt of
half the original resistance replace the first, the potential drop
will be equal to that which is required to send a current, c, through
the coil and series resistance of the instrument, and so produce
a full scale deflection.
The shunts are made of manganin, which has a low temperature
coefficient. The dimensions required in using a definite manganin
strip may be calculated. If it is found that to produce no
appreciable heating the shunt width has to be excessive, it is
usual to construct the shunt of several strips in parallel.
The instrument should never be used without the appropriate
shunt for the current to be measured.
The ammeters and voltmeters described above have the
advantage of being direct reading on a calibrated scale ; they
are robust and do not require any adjustment. But as indicated
at the outset, the general type of instrument is not sufficiently
sensitive to measure currents of less than a milliampere or
460 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
potential less than one millivolt ; but by more delicate con-
struction and general improvement they can be made to measure
to one micro-ampere and micro-volt. In such a case the extra
sensitivity entails^ very precise work, and makes the cost of the
instrument somewhat higher than for the ordinary range (i.e.
one millivolt or ampere).
Some forms of instrument are available which combine the
voltmeter and ammeter. The necessary shunts and series
resistance are contained inside the case, and by connecting to the
proper terminals, the instrument may be used either as an
ammeter, or as a voltmeter of several ranges (see, for example,
fig. 288).
Unipivot Instruments
To increase the sensitivity of the above types of instruments,
a modification of the support of the moving part was introduced
by R. W. Paul. The pivot friction was reduced very considerably
by the use of the one-pivot method of suspension, and at the
same time all the advantages of the form of double-pivot sus-
pension were retained, so that a sensitivity corresponding to
one subdivision per micro-ampere is obtainable for the uni-pivot
instrument.
The construction is shown in figs. 289 and 290. A circular
coil is suspended about a spherical core of soft iron between
the poles of a permanent magnet.
Fig. 290
Fig. 290 shows the detail of the coil support. A vertical
spindle carries a light counterpoised pointer, and rests on a
polished jewel at the bottom of a cylindrical hole drilled in the
soft iron sphere.
1-'JG. 288
Fag* 4(10
AMMETERS, VOLTMETERS AND GALVANOMETERS 461
The cylindrical spring at the upper end has a very, slight
lifting effect on the coil, and produces a restoring couple when
the coil is deflected ; it also serves as a lead for the current to
the coil.. The current leaves the coil by the flexible wire shown
at the lower extremity.
The centre of gravity of the moving part is at the point of
support.
A simple device is included to raise the point off the jewel
when the instrument is not in use. This is shown under the
coil in fig. 289, which gives the general appearance of the instru-
ment when one pole piece and the magnet are removed.
For many purposes it is necessary to be sure that the ammeter
or voltmeter used is reliable to a fair degree of accuracy. The
instrument may be calibrated in the laboratory (e.g. by potentio-
meter and standard cell), but it is here suggested that each
laboratory be provided with one form of ammeter or voltmeter
(or both) which has been tested at the National Physical
Laboratory, and is provided with a correction certificate.
Such instruments should be retained at laboratory standard,
and the working instruments checked against such standard
instruments.
Galvanometers
When smaller currents are to be measured, use must be made
of some form of galvanometer, an instrument which is not as
robust as the above, it must be levelled before use, and further
the current must be calculated from the observed deflection
produced by it.
Thus it is not as convenient and simple to use as the ammeter,
but has a sensitivity which is impossible to attain in the latter.
The increase in sensitivity is largely produced by a more
sensitive method of suspension. The friction of the pivot is
entirely removed by the use of a fine suspension of silk or
phosphor-bronze. The suspension carries a small concave mirror
by means of which small movements of the moving part may be
magnified. The two common methods of producing such
magnification are by use of
(1) a lamp and scale,
(2) scale and telescope.
(1) Lamp and Scale Method
In this method a beam of light from an incandescent lamp or
Nerst filament is directed by means of a lens on to the concave
mirror, which reflects it to a scale some distance away.* The
greater the distance, D, between mirror and scale, the greater
*The scale should be placed a distance away from the mirror equal to its
radius of curvature. The condensing lens, over which is stretched a vertical
wire, acts as an illuminated object whose image, a circular patch of light with
vertical black line, is used to measure deflection.
i _,. ^
462 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the magnification produced. When the mirror rotates through
an angle, 0, the reflected beam moves through twice that angle,
causing a movement of the spot of light, say, d cms. on the scale.
Hence tan 20 =■=>
To measure such deflection it is essential that the mirror
should produce a clear image. For this reason, with the size
and character of the mirror available on such a suspension, the
usual maximum value of D is one metre.
(2) The Telescope and Scale Method
A scale is set up horizontally at about one metre from the
galvanometer mirror, and a telescope, usually under the central
graduation of the scale, is turned towards the mirror. When the
mirror, which should be a plane one for this method, is parallel to
the scale, the latter may be seen in the telescope. The reading
of the scale in coincidence with the cross-hair in the eyepiece of the
telescope is noted. When the mirror moves a second scale reading
will coincide with the cross-hairs. The difference for an angular
rotation, 0, of the moving system may be accurately measured.
If this is equal to d cms. then, once more, tan 2 = ^-
It is advisable, for simplicity of reading, to use either a telescope
with an erecting lens or prism or to use a scale provided with
' inverted ' graduations, so that the scale appears the right way up.
Sensitivity
For many reasons it will be apparent that the instrument
cannot be calibrated once for all. The scale distance is variable ,
the suspension may of necessity be replaced, and so on ; there
is, therefore, no permanent direct reading scale as in case of
ammeters and voltmeters.
The usual method of converting scale readings to the corres-
ponding currents is to obtain the ' sensitivity ' of the instrument.
The sensitivity of a galvanometer may be defined in many
ways. In addition to being a ' reduction factor,' converting
scale divisions to current, as indicated above, the sensitivity also
gives an indication of the possibilities of the instrument. How-
ever for a comparison of two instruments another factor will be
discussed later.
Current sensitivity may be defined in either of the two following
ways :
(a) The number of mms. deflection produced on a scale one
metre away by one micro-ampere, i.e. by io- e ampere, or
( p) the number of micro-amperes required to produce one mm.
deflection on a scale one metre from the galvanometer mirror.
The second definition is perhaps from some points of view
AMMETERS, VOLTMETERS AND GALVANOMETERS 463
the better, but (a) gives a value which varies directly with the
property measured. It must be understood that a sensitivity
of 1000 mms. per micro-ampere means only that a very small
current produces a deflection which would correspond to 1000 mms.
deflection when a micro-ampere passes, assuming the deflection
to be still proportional to the angular movement of the coil.
It is purely a theoretical mode of expression. The deflections
measured never actually exceed from 5° to 8°.
Volt sensitivity may be similarly defined, substituting micro-
volt for micro-ampere in the above.
It will be seen that for a fixed current strength the current
sensitivity is greater for a bigger coil resistance, i.e. a bigger
number of terms.
Volt sensitivity, which deals with a fixed potential applied to
the galvanometer, is the greater the smaller the value of the
resistance of the coils.
Determination of Current Sensitivity
The galvanometer whose resistance, G, has been determined,
is connected in series with a megohm (10 6 ohms) and a cell of
known electromotive force, E. The deflection on the scale one
metre away is noted = d x mms., say.
E x 10 *
Now the current passing is-; — micro-amperes; from
which the number of mms. deflection produced by one micro-
ampere may be calculated.
An alternative method is shown in fig. 291.
Fig. 291
A steady accumulator whose electromotive force, E volts,
is known accurately — either by a potentiometer comparison with
a standard cell or by measurement with a calibrated voltmeter —
is connected in series with a high resistance, R ohms, through a
commutator, C, to a low resistance, S, which is in parallel with
the galvanometer of resistance, G.
464 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The mean deflection d, say for both positions of the com-
mutator is obtained.
The current causing the deflection is readily calculated, for
SG
the effective resistance of the shunt and galvanometer is u . - ;
the main circuit current is therefore equal to*
^Wo W
where B is the battery resistance which, if an accumulator is
used, is negligible compared with R. Hence neglecting B in (i),
the current through the galvanometer is
E S
C« = cT — x /o _i_ r\ X io 6 micro-amperes
(S+G)
E'Sxio 6
micro-amperes (2)
R(S + G) + SG
. . . . d
Hence the sensitivity is ^- mms. per micro-ampere.
Thus the sensitivity having been obtained the deflection
produced in any case may be converted to the corresponding
current values so long as the suspension remains the same and
the scale is at the same distance, one metre, from the galvano-
meter mirror.
STEADY CURRENT MEASUREMENT
Galvanometers for the measurement of steady direct current
may be subdivided into two general classes according to the
nature of the moving part: {a) moving needle, (b) moving coil.
(a) Moving-needle Galvanometers
The student will be familiar with the simple tangent galvano-
meter. In this form a small magnet is suspended at the centre
of a coil of wire of n turns. When a current of C, expressed in
electromagnetic units passes through the coil, placed in the
magnetic meridian, the magnetic field set up causes a deflection, 0,
such that the restoring couple due to the horizontal component
of the earth field, H, balances the couple due to the magnetic
field of the current, and we have :
Hr
C = — tone, (3)
where r is the mean radius of the coil.
In general the assumptions upon which the formula is developed
■of*
* N.B. — If S is very small, c.f. R, the factor „ , ^ may be omitted as this
S + G ES 106
is a little less than S. If further S is small, c.f. G, (2) becomes j^ s ^tg)
AMMETERS, VOLTMETERS AND GALVANOMETERS 465
are not justified, and corrections should be applied to allow for
the width of the coil, etc. ; such corrections are but seldom used,
and the galvanometer is not used as a small current measurer.
A modification, known as the Helmholtz galvanometer, consists
of two similar coils placed on a common axis at a distance apart
equal to the radius of either.
A short magnet is supported at the mid-point between the
centres of the coils. As with the tangent galvanometer, the
plane of the coils is placed in the magnetic meridian, then using
the same notation as before,
C^-^tanS; (4)
327m
n is tne number of turns in each coil.
These two forms are such that the absolute value of the current
may be simply calculated, but they have no claim to great
sensitivity.
Sensitive Current Detectors
The result expressed in (3) for a simple tangent galvanometer
shows that, for this form of instrument, the deflection for a given
current may be increased by
(a) decreasing H, the control field,
(p) decreasing r,
(y) increasing n.
There is a limit to methods ((J) and (y) which is prescribed by
the practical problem presented.
Further, as r is decreased and n increased, the conditions
upon which (3) was developed no longer hold.
(a) may be best carried out by the use of an external control
magnet which neutralizes the value of the horizontal component
of the Earth's field. Further, by using an astatic pair of magnets
or groups of magnets the control effect may be reduced without
interfering with the magnitude of the deflecting couple.
The above conditions are embodied in the Thompson (or
Kelvin) and Broca galvanometers.
The Thompson Galvanometer
The Thompson galvanometer is illustrated in fig. 292. The
control magnets, EE, enable the galvanometer to be used in
any position independently of the Earth's field, the suspended
astatic system is shown at the right of the diagram. Each set
of magnets is placed at the centre of a pair of coils, FF. The
magnets are mounted on a light rod, the whole being supported
from a torsion head by an unspun silk or a quartz fibre.
39
466 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The magnitude and direction of the control field may be varied
by alteration of the positions of EE along the vertical rod shown.
Sets of coils of different resistances are usually supplied.
Using high-resistance coils a sensitivity of about 600 mms. per
micro-ampere may be attained, i.e. the instrument will detect
currents of the order of io~ 9 ampere.
The movement of the magnet is detected, using the small
mirror, B.
Fig. 292
The Broea Galvanometer
This instrument is shown in fig. 293. The moving ' astatic
pair ' is made up of two vertical magnets having consequent
poles as shown. The control field is due to the magnet, B, which
moves in a ball socket to any desired position.
The coils, EE, may be of any suitable resistance, say, 10, 100
or 1000 ohms as required for sensitivity in the circuit. They
are connected in series, one at each side of the spaCe occupied
by the centre magnet, and arranged to produce a field in the
same direction.
As in the Thompson galvanometer, an aluminium vane, G,
moves between two parallel plates which may be adjusted by
the rods terminating in the metal knobs, CC. The damping
of the system may be altered by an adjustment of the distance
between these plates. Currents of the order of io -10 ampere
may be detected,
AMMETERS, VOLTMETERS AND GALVANOMETERS 467
The Thompson and Broca galvanometers are most advan-
tageously employed as very sensitive current detectors whose
sensitivity may be rapidly adjusted over a wide range by the
control magnets.
Both forms require levelling before using.
Fig. 293
(b) Moving-coil Galvanometer
The moving-coil galvanometer is constructed in a manner very
similar to the ammeter already described. A phosphor-bronze
strip, F, fig. 294, acts as suspension to the coil, C, which is free
to move in the gap between the pole pieces of a permanent
magnet, one of which is not shown in fig. 294, and a soft iron
cylinder, I, which is screwed to a brass plate as seen in the
lower figure.
In this case the magnetic field in the air gap is approxi-
mately radial, as shown by the broken lines in the lower figure
(fig. 294).
The current enters the coil via the phosphor-bronze strip, F,
and leaves at the under end by means of a helix of phosphor-
bronze, S. The control is mainly due to the twist of F in such
a case.
If B is the magnetic flux in the air gap, A the area of
one turn, n the number of turns, and t the restoring couple per
unit angular twist of the suspension, F, we have, for the case of
468 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
a radial field, when a current, c, passes, causing a deflection, 0,
BnAc = t6, (5)
Gc = t6
G = BnA and is the couple on the
coil for unit current and is called
the galvanometer constant,
or c = kQ,
when k is a constant and equal to =^-i .
BwA
era
Fig. 394
For small deflections, 6 = tan 0, i e. © is proportional to d, the
deflection on the scale.
From (5) above it will be seen that to increase the sensitivity,
t must be made as small as possible, and B, n and A as large
as possible.
To decrease t we may decrease the cross-section of the strip
or increase its length.
For a circular wire it was shown on page 102
R*
Ttt 7'
The most profitable method of decreasing t is therefore to
reduce the cross-section, i.e. use a fine suspension. This is
limited by the fact that very fine suspensions are also very
fragile and somewhat difficult to use. From this point of view
Fig. 295
Pagt 469
AMMETERS, VOLTMETERS AND GALVANOMETERS 469
phosphor-bronze is the most satisfactory material and is almost
universally employed . The usual type of moving coil is supported
by a fibre of the greatest convenient length.
The increase of B, n, and A must be considered together as the
terms are interdependent.
Assuming the magnet to be fully saturated, the value of B
in the air gap depends on the size of the gap. The smaller the
air space, the larger is B. The effect of increasing nK is to
increase the size of the coil, and consequently the air gap, and
hence decrease B.
In practice it is usual to use a standard size of air gap which
allows a frame of such size to be wound with wire, and move
freely in the available space, that the best compromise between
these two effects is obtained for a maximum value to the product
B-ff'A.
An example of galvanometers of this type is shown in fig.
295-
Such instruments may have a sensitivity as great as 1500 mms.
per micro-ampere, i.e. the instrument will measure currents of
the order io -10 ampere.
Adjustment
The moving-coil instrument must be adjusted before use.
The coil is first released, then the instrument is levelled by means
of the levelling screws, so that the coil does not touch either the
pole pieces or the iron core, and is thus able to swing freely.
The instrument may be used in any position, and the coil
is turned by means of a torsion head, which carries the suspension,
until the plane of the coil is approximately parallel to the sides
of the magnet. It is inadvisable to make this adjustment, unless
the reflected beam does not fall on the scale, for there is a danger
of breaking the suspension.
Onwood Moving-coil Instruments
The Onwood galvanometer differs somewhat from the above
general types in that it requires no levelling, is not so fragile,
and is more convenient to move.
The difference is in the method of suspension as seen in fig. 296.
NS is the permanent magnet with a soft iron core, d, which is
drilled to the centre. From this point, h, is a short suspension
c, which is of sufficient length to support the coil, a, from the
point, g, clear of the fixed parts. A small mass, e, at the end
of the rod, /, keeps the system vertical.
470 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The current is led through the frame to the iron core, and
thence through the suspension, and the tube, b, to the coil
which it leaves by means of a flexible ligament at the base of
the coil, but not shown in the diagram.
Fig. 296
Since the coil is supported at the centre it will be equidistant
from pole pieces and iron core alike for all positions of the
instrument not too far removed from the horizontal, i.e. the
instrument does not require levelling.
A small spring, not shown, at the upper end of the suspension
protects the latter against sudden shocks.
The galvanometer is a very small one and the sensitivity
claimed ranges from 20 to 500 trims, per micro-ampere according
to construction details.
THE CHOICE OF A GALVANOMETER FOR THE MEASUREMENT
OF A STEADY DIRECT CURRENT
To select the galvanometer most suitable for a particular
experiment many factors must be considered. To decide firstly
between the two general types described above, their relative
advantages and disadvantages are discussed below.
Moving-needle Type
(1) The value of the control field is affected by the proximity
of external magnetic fields. The instrument may register a
deflection when no current circulates through the coils when
AMMETERS, VOLTMETERS AND GALVANOMETERS 471
magnets are moved in the laboratory, or dynamos set in motion,
or even by a passing electric tramway car. Thus there is
some uncertainty unless such stray fields are eliminated.
(2) The moving magnet may have its moment altered by the
field set up when a current passes through the coil.
(3) Unless the instrument is made ' dead beat ' the needle
takes a long time to come to rest and is therefore somewhat
troublesome to use. This may be overcome to some extent by
using an external damping circuit, consisting of a solenoid
through which a current may pass when the circuit is closed.
This solenoid is placed near the case of the instrument and the
circuit closed momentarily as the needle swings, in such a way
as to produce a field which opposes its motion.
(4) The damping of the instrument is independent of the
resistance of the circuit in which it is placed.
(5) The needle may be supported by means of a quartz sus-
pension which has the property of returning after deflection to
its original position. However, the slight torsion in the fibre
is usually neglected, and the expression for the current is therefore
not strictly accurate.
Moving-coil Type
(1) The moving-coil galvanometer has a large permanent
magnet, and is therefore practically unaffected by stray external
fields.
(2) No demagnetizing effect is possible on the moving part.
(3) The movement of the coil may be very rapidly arrested,
even when the instrument is not ' dead beat,' by connecting the
ends of the coil together through a very low resistance as the
coil passes its rest position.
(4) The electromagnetic damping varies with the value of
the resistance of the external circuit. If used in series with a
low resistance this damping is very high, and the coil may take
several minutes to attain the full deflection, a fact which is very
often overlooked when the instrument is used.
(5) The zero-keeping quality of the suspension depends upon
the degree of sensitivity attained. The torsion at the fibre is
taken into account in the expressions for the current.
Having decided, from the conditions of the experiment, the
type of instrument most suitable, the next problem is, What is
the most satisfactory galvanometer resistance ?
The Order of Resistance of a Moving-coil Galvanometer which is
most sensitive in a Given Circuit
In general terms we may state that if the current is to be of
a fixed value, independently of the galvanometer, considering all
472 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
factors, the sensitivity will be proportional, approximately, to
VG, i.e. will increase with increasing resistance.
If a fixed potential difference is to be measured the sensitivity
will obviously be greater the smaller the resistance ; approxi-
mately, the sensitivity is proportional to —j^, i.e. high resistance
i VG
for detecting current, and low for detecting small potential
differences.
The problem usually presented is, given an external resistance,
R ohms, what is the best value for G, the galvanometer resistance
for measuring the current due to a fixed electromotive force.
We saw (pages 467-8, equation (5)) that the couple due to the
current c is nABc, and if E is the electromotive force in the circuit
-ETG (6)
Further, it was shown on that page that in a galvanometer
there is but a limited space between the pole pieces to obtain a
maximum sensitivity. Fig. 297 shows a cross-section of the
frame which will just move freely in this space. Let the cross-
section of the whole of the windings be a sq. cms. and assume
that the windings entirely fill the space with copper. If p be
the mean perimeter of the coil windings, we have length of wire
used = np, cross-section of the wire = —
r n
Therefore G = — *- = ■ * •
a a
n
where o is the specific resistance of the wire, say, copper,
i.e. ft = VT (7)
ypa
Hence the couple due to the current
= BwAc
T> A 155 _E
from (6) and (7).
AMMETERS, VOLTMETERS AND GALVANOMETERS 473
The condition for the couple to be a maximum and therefore
produce a maximum effect is that
BAEVoT. VG
Vp^ R + G
be a maximum.
We have seen that a and p are constant, so that the condition is
p , „ is to be a maximum,
K -f- (j
•p
or that —7= -f- y^Q be a minimum,
i.e. R = G.
That is, under the circumstances stated, the maximum sensitivity
is obtained when the galvanometer resistance is equal to the total
eocternal resistance.
It should be noted that the galvanometer resistance in this
discussion refers to the copper coil resistance only. The resistance
of the suspension should be included in the value R. This being
so, it is apparent that there is a lower limit beyond which the
resistance of a coil may not be reduced with any advantage.
If the galvanometer is to be chosen as a detector or measurer
of small differences of potential the most suitable instrument
will be one of low resistance (e.g. for thermo-couple work).
Other qualities of galvanometers to be considered when
making a selection of galvanometers are : —
(1) Damping
The damping of the moving part in a galvanometer apart
from external artificial agency may be considered due to two
separate causes.
(a) The damping due to the viscosity of the air. This is present
in moving coil and needle alike, and is approximately proportional
to the angular velocity of the system. It is always present, but
is usually small.
(b) Electromagnetic damping. In the case of a moving magnet
the amount of damping due to this cause is very slight when the
magnet is in a non-metallic case, e.g. when the coils are wound
on wood or ebonite.
This is the reason for the long and troublesome wait which
occurs before the needle returns to its rest position. This may
be reduced as described under. However, in either case the
amount of damping is obviously independent of the external
circuit.
In the case of the moving-coil instrument, the suspended coil,
when closed by an external circuit, is moving in a strong magnetic
474 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
field. Under such circumstances the electromotive force
induced in the circuit sets up a reverse current in the closed
circuit, which is therefore brought to rest.
The value of the damping current depends on the magnitude
of the external resistance, and may become very great for a low
series resistance.
For many purposes it is necessary or convenient to have a
galvanometer such that the moving part very rapidly returns to
the zero position after being deflected. A galvanometer having
this property is said to be ' dead beat.'
In both types of instrument this may be brought about by
increasing the electromagnetic damping. In the moving-needle
type this is accomplished by encasing the needle in a copper, or
similar metal, case. The movement of the needle sets up eddy
currents in the copper and the magnet is rapidly brought to rest.
The moving coil may be rendered dead beat by winding it on
a metal ' former ' or frame. This constitutes a closed metallic
circuit, and the desired result is obtained. Alternatively, if the
galvanometer is wound, for ballistic purposes, on a non-metallic
frame, e.g. bamboo, the same result is obtained by facing the
coil with a thin sheet of copper foil (cut into a ' picture frame '
which is the same size as the edge of the coil).
(2) Period and Constancy of the Zero
A galvanometer which is very sensitive has a long time of
swing ; and also, due to a very fine suspension, some trouble may
arise due to the ' creep ' of the zero. However, the type of
instrument used in experiments in this book will not be of the
extremely sensitive order at which this trouble arises. The main
cure for the trouble lies in the selection of suspension, and that
really involves the selection of an instrument maker who will
take the trouble to minimize this fault. Beyond this the correc-
tion of residual effect must be solved by the ingenuity of the
experimenter as applied to the particular experiment involved.
Quartz for moving magnets and phosphor-bronze for moving
coils cause least trouble in this respect.
For average work 5 to 10 seconds per complete swing will be
best value for direct steady current measurement or for use
in ' null ' methods.
MEASUREMENT OF QUANTITY OF ELECTRICITY
The Ballistic Galvanometer
A galvanometer suitable for measuring a quantity of electricity
is called a ballistic galvanometer, and has the following essential
features :
AMMETERS, VOLTMETERS AND GALVANOMETERS 475
(1) The periodic time of swing, T, of the moving part is fairly
large.
(2) Damping of the moving part is very small.
The first condition is fulfilled by making the moment of
inertia of the needle or coil which forms the moving part as large
as practicable and by reducing the controlling forces, for
— £
where I is the moment of inertia of moving part, and t is the
restoring couple per unit angular displacement.
Thus by increasing I and decreasing the restoring forces, T,
the time for one complete swing, is increased.
The second factor, damping, is reduced in a way which depends
on the instrument (needle or coil).
As seen when considering damping (page 473) the electro-
magnetic damping only may be reduced. The air damping is
usually small.
(3) A third condition is, that when used to measure a quantity
of electricity, the whole of the transient current shall pass before
the needle or coil moves from the zero position. Should there
arise a case in which the quantity of electricity to be measured
takes longer time to traverse the instrument, due, for example,
to inductance in the circuit, the time of swing of the needle must
be increased, by loading it, so that this third condition is fulfilled.
As indicated above, the galvanometer may be of the moving-
needle or moving-coil type. We shall develop a relation between
the throw or angular deflection in either type, and the quantity
of electricity which passes.
Moving-needle Type
This type of ballistic galvanometer consists of a needle sus-
pended by a fine quartz or unspun silk fibre, at the centre of
two coils, through which the quantity of electricity, Q, passes.
The control in this case is either the Earth's field or a control
magnet. The needle in its zero position is arranged at right
angles to the axis of the coils, so that when a current passes a
field is set up at right angles to the control field.
Let G be the galvanometer constant, i.e. the field due to the
coils for unit current circulating through them,
H the value of the control field strength,
I the moment of inertia of the magnet about the axis
of suspension,
M the magnetic moment of the magnet.
476 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Suppose a current of strength, c, to pass through the coils
for a very small interval of time. Since the third condition above
holds, the needle will be at right angles to a field of strength Gc,
and will experience a turning moment, GcM.
This couple will produce an angular acceleration in the
needle. Hence (page 53)
IB .= GcM,
16 = GM fcdt = GMQ (8)
At the end of the swing, the needle having turned through an
angle O , the kinetic energy of the moving needle, \1Q % , has
been reduced to zero in doing work against the magnetic force,
Hra (m being the pole strength of the needle) at each pole.
The work done is 2Hm f cos O J
= Kml (1 — cos O )
= 2MHsin 2 -^>
2
where I is the distance between the poles.
Thus 2MHsin 2 -^ = -I0 2 ,
2 2
l0 2 = 4MHsin 2 ^ (9)
Hence squaring (8) and dividing by (9)
T Q 2 M 2 G 2 , x
4HM sin 2 ^
We have also for the period, T, of the suspended needle in
the control field, H,
t T'MH , ^
I= -^- («)
Equating (10) and (11) :
Q 2 MG 2 = T 2 MH
„ . ' O "" 4T5 2
4H sin 2 -^ ^
^ 2
Hence :
^ T H • e o
Q = «'G' sm i •••• < I2 >
AMMETERS, VOLTMETERS AND GALVANOMETERS 477
In developing this result we have assumed that the original
kinetic energy of the needle is wholly used in moving the
magnet against the field, H, through an angle, O . If, however,
there are any frictional forces, i.e. the needle is slightly damped,
some energy will be required to overcome this force, and the
result will be that the observed angle of swing, say, a, is not
truly of the magnitude given in the undamped case. The true
value for Q is therefore dependent not on the observed value, a,
of the angle, but on some slightly bigger angle O .
It will be shown that O = a(i +-)> where xis the logarithmic
decrement of the suspended system,
o-"-B(-+i)] ™
Moving-coil Ballistic Galvanometer
There are several disadvantages in the moving-needle instru-
ment as in the case of measurement of steady direct current,
e.g. (1) and (2), pages 470-1* These defects are overcome in the
moving-coil type of ballistic galvanometer.
The electromagnetic damping is reduced by using a coil
wound on a bamboo frame ; T is increased by using a fine phos-
phor-bronze strip suspension.
To obtain the connexion between Q, a quantity of electricity
discharged through the galvanometer, and O , the first throw,
we may first of all assume that there is no damping.
Lei: G be the galvanometer constant, i.e. the couple acting on
the coil when unit current passes through it.
t the restoring couple due to the suspension, for unit
angular displacement.
I the moment of inertia of the suspended system about
the axis of suspension.
As for the moving magnet galvanometer (p. 475) let c be the
value of the current at a time t, then
Gc = I0 (13)
Integrating
Gfc . it = GQ = 10 (14)
Considering the kinetic energy of the coil, we have, if is the
original angular velocity given to the coil by the impulse due to
the discharge of a quantity of electricity Q,
K.E = £I0 2
478 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
This energy is used in twisting the suspension through 0.
At any angular displacement 0, the restoring couple is t0 ; to
twist through a further angle dd, the work done is t0 . dd ; i.e.
the total work done in deflecting the coil is
f.
-cd.de =^ (15)
2
Thus we have
t0 o 2 _ 10*
2 ~~ 2
I0 2 = T0 O * (l6)
Squaring equation (14) and dividing by (16) we have
I = ^H- (17)
Again the time of swing of the coil is given by
T = 2* -yjl-
or * = 7T
Substituting this value in 17
T 2 t = G 2 Q2
4* 2 ~" t0 o 2
Q -f £-t (l8)
or if a is the observed first swing, and x is the logarithmic decre-
ment,
o-H-K'+i) ;•■«*■»
Suppose now we send a steady current of known magnitude, c,
through the galvanometer and observe the steady deflection, <p,
Gc ■-« T9»
AMMETERS, VOLTMETERS AND GALVANOMETERS 479
or _ = -
Hence Q=l.t.*( I+ ?) (i 9 )
76 -K <p 2\ 2/ v
It should be noted that the preceding paragraph gives a very
ready method of finding the ballistic reduction factor for the
galvanometer. We see that
^ , , , T x steady current
Q = ka where k — — J „ — -, —
2re X steady deflection
Subsequent deflections may be converted to quantity by multi-
plying these deflections by k.
See also page 481, equation (23a) for the moving magnet type.
To Correct the observed first swing (a) for damping in either form
of galvanometer.
The observed first deflection, a, is reduced by damping forces
which are proportional to the angular velocity.
The equation of motion, therefore, must be modified from the
simple form :
IB + F0 = o,
where F may be t or MH, according to the type of instrument,
to include a term proportional to the angular velocity. Let this
term be K0, then the equation of motion becomes :
10 + K0 + F0 =0,
an equation similar to that already dealt with on page 27.
_K
The amplitude of the system is therefore O e 3l ', using the
notation of page 161.
_K T
Thus the first observed swing, <x x = 6 Q e *i 4.
K T '
Logarithmic decrement, x = -^ • - ,
2X 2
i.e.
a x = O e~\,
or
O = a i<^
etc
)
In a ballistic galvanometer K is small and although T may be
KT
rge, I is also large, and -^ is small, i.e. X
may be neglected in comparison with unity,
KT
large, I is also large, and -^ is small, i.e. X 2 and higher terms
4 1
4 8o ADVANCED PRACTICAL PHYSICS FOR STUDENTS
i.e. . ».-«(i+i)-
x may be obtained by one of the methods of pages 162 to 164,
and the correcting factor fi -f - ) obtained. The galvanometer
must be in the circuit of the experiment when x is obtained, so that
the damping is the same.
Another way of correcting for damping in the moving part
does not involve an independent measurement of x.
Suppose that in addition to noting the first swing, a lr we
observe the next swing on the same side of the zero, a 3 , we have :
K T _K 3T _k 5T
ai = o e~*i'*; cc 2 =d e *"*; a 3 =0 e *i"4,-etc;
or & = ^ = e + § T (20)
a 2 a 3
K T ( + * T )-*
i.e. a. x =0 e-* 4 = © |^ aI ;
or by (ao) a ' = 9 »t-^'
ie - *•— (£)' (3I)
Thus, suppose there is an error of one per cent in the deter-
mination of an angle, f — J is liable to two per cent error or
(— ) may be in error by one-half per cent.
The correction for damping obtained in this way will be
sufficiently correct for many experiments.
Of course a x , a 2 , and <x 3 are not measured as angles. The
corresponding scale deflections, <5 X , <5 a , and <5 3 , are measured.
Now — = tan 2(Xl for small deflections,
d 3 tan 2<x 3
i.e. since under these circumstances the value of tan 2a is
approximately the same as 2a,
_£i __ «i.
<5 3 a 3
So that (19) for the moving-coil galvanometer becomes
Q_t.i.«*(«!y (22)
* re <p 2 \a 3 y
and (i2«)
AMMETERS, VOLTMETERS AND GALVANOMETERS 481
o-Sff-f-®'! <*3)
In the above method the damping under the conditions of the
experiment is obtained for the correction factor.
Equation (23) may be modified to conform with (22). For
if a steady current, c, produces a deflection q>,
cG = H tan y,
i.e.
Q=7 t w s +Cl)l (23fl)
To Reduce the Excessive Damping in a Moving-coil Ballistic Galvano-
meter used in a Low-resistance Circuit
It was shown in the discussion of damping that the moving-
coil galvanometer has a large amount of electromagnetic damping
when in a closed circuit of low resistances. This would make
the use of a moving-coil ballistic galvanometer inadmissible in
many experiments unless some means were taken to reduce the
damping.
The quantity of electricity passing through the instrument
is, in general, due to some current change. If, therefore, the
galvanometer is in circuit whilst such change takes place, and is
then allowed to swing freely when it has received the impulse
due to that cause, the damping is almost eliminated.
'////A i=i , o
'///A ^ B
WA S
JZZL
E
Fig. 298
By use of a compound key, such as that shown in fig. 298,
this may be brought about. The three brass strips mounted on
an ebonite block (shaded) are connected to separate terminals,
as is the stop S. The battery circuit is connected to A and S,
and the galvanometer to C and B. When the key is depressed,'
C and B, A and S are connected, but B and A remain insulated
by the ebonite stops shown (shaded). When the key is released
A and S are broken ; impulse, due to induction or whatever
cause is operative, is given to the galvanometer, and then C and B
are disconnected by the upward move of the key. If the time
interval between the break of the battery and galvanometer
circuits is small compared with the period, T, of the coil of the
galvanometer, the latter will swing an amount which is almost
independent of electromagnetic damping due to the low
resistance circuit.
3*
482 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If such a key is used for a moving-coil instrument the damping
correction used should be obtained from observations of the coil
when swinging freely in open circuit.
The Grassot Fluxmeter
The Grassot Fluxmeter is an instrument which performs the
same function as a ballistic galvanometer, as for example in
experiments on pages 426, 557, 582, 357, etc. However, the
instrument is specially designed for measuring magnetic field
strengths directly.
Fig. 299
It is a suspended coil instrument which depends entirely on
the electromagnetic damping for control. The coil, D, is
supported by a single cocoon silk fibre, which has a negligible
torsional control, from a flat spiral, E (fig. 299), to eliminate the
effect of shocks. The current enters and leaves the coil by the
silver strip coils, H, as shown in the figure. The usual iron
core, B, and permanent magnet pole pieces, N, S, supply a constant
magnetic field in which the coil swings.
The light frame, FF, carries a pointer and a concave mirror,
not shown in the figure. The points, T T ', correspond to terminals
on the case of the instrument.
In general use a search coil, C, of known mean area and number
of turns is connected to T,T'.
The search coil is placed into the magnetic field to be measured
and the induced electromotive force causes a current to flow
in the closed circuit and so produces a deflection which may be
measured either by a lamp and scale arrangement, using the
Fie, 300
Page 483
AMMETERS, VOLTMETERS AND GALVANOMETERS 483
concave mirror, or, if sufficiently large, by direct reading of the
pointer over a scale.
The instrument therefore produces a deflection which is
proportional to the total quantity of electricity which passes
through it.
Fig. 300 shows the general appearance of the instrument in
use to measure the magnetic distribution along a magnet.
The method of calibration of the graduated scale and general
possibilities of the instrument will be apparent from a con-
sideration of the theory of the instrument.
Let R = the resistance of the circuit, i.e. of the search
coil, C, and leads, and the suspended coil, D,
L = the self-induction of the whole circuit,
I = the moment of inertia of the coil, D, about the
axis of suspension,
E = the E.M.F. induced in C ]
c = the current in the circuit I at any instant,
v> = the angular velocity of the coil J
K = induced E.M.F. set up in D for unit angular velocity,
G — the galvanometer constant, i.e.. the couple exerted
on the coil due to unit current passing through it,
A = the couple due to air resistance for unit angular
velocity (see damping, page 473.)
Expressing Ohm's Law for the circuit, we have :
cR = E - IT- - Ko>, .(24)
i.e.
-i(*- L s-*->
The equation of motion of the coil is :
t d * e r A , ^
1 dtf =Gc ~ A< * -to)
Substituting the above value for c and remembering that
dd
Ci>= —
dt'
T ^6> _ GE _ LG dc _ /GK \
dt R R dt \R +AJ G> -
Now L, G, R, K, and A are constants ; further, if the coil
starts from its rest position when no current passes, and becomes
484 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
deflected through an angle, a, by the quantity discharged through
it, the coil is at rest at the end of the swing when no current
again passes through it.
Therefore, integrating the last equation over the whole swing
with respect to t, we have
■[•I-S/"-¥['T.-<?«)[-]:-
/M,?(!f + A)..
fEdt = (^ + K) a (26)
Thus it appears from (26) above that the value of the deflection,
<x; is determined by / JLdt, since the other terms are constant.
For example, if the search coil is placed in a magnetic field,
thereby setting up an induced E.M.F. in the circuit, the total
deflection is independent of the speed of insertion or withdrawal
of the search coil.
AR
Now in general the value of A is small and the term -^~ may
be neglected in comparison with K. And hence we have :
Edt= Koc,
i.e.
or
/
approximately.
With this approximation the deflection a is independent of
the value of R, i.e. the search coil may be replaced by another,
provided that R does not become very large for then the approxi-
mation is not justifiable.
For the purpose of this account the noteworthy feature is
that the deflection is, as for the ballistic galvanometer, proportional
/■p
g • dt, which passes through the
coils, i.e. a is proportional to the quantity.
The standard form of fluxmeter has a scale, graduated in
maxwells ; this is graduated experimentally. The deflections
corresponding to known flux change define points on the scale,
and since a is proportional to f Edt, which itself is proportional
to B, the number of lines cut, the subdivision of the scale between
such fixed points is a matter of dividing angles into equal parts.
The experimental arrangements for the instrument in measur-
ing magnetic flux and quantity of electricity are described in the
chapters where such measurements find a place.
AMMETERS, VOLTMETERS AND GALVANOMETERS 485
For many of the experiments for which the instrument is used
the graduated scale is too coarse. Use is then made of the
mirror, using a lamp and scale arrangement as in the ordinary
galvanometer. The deflection on the scale is proportional to
the quantity of electricity discharged through the instrument.
A full scale deflection is produced in the instrument by a definite
flux change in a search coil ; the position of the reflected spot
of light is noted, as is the movement of the pointer on the grad-
uated scale. By repetition, the corresponding values are
obtained. From such observation the value of the flux change,
corresponding to 1 mm. scale deflection (at, say, 1 metre), is
deduced from the graduated scale deflection.
MEASUREMENT OF VARYING CURRENT
The preceding account has included the method of measuring
(1) steady direct current, (2) quantity of electricity.
We now consider briefly methods available for the measure-
ment of varying currents. Such currents may be subdivided
into (a) currents of short duration at regular or irregular intervals,
(b) alternating currents.
(a) To detect currents of this kind, and also to measure the
time interval between such short duration currents use may be
made of
The Einthoven Galvanometer
The principle of this instrument may be understood from
fig. 301. A 'string/ CC, of fine platinum or tungsten wire is
supported vertically between the poles of an electromagnet.
If a current be sent down the string, the latter will be deflected
m a direction parallel to the face of the pole pieces. In the
diagram, the direction of movement is shown by the arrow, a,
for a field in the direction, SN.
To observe the movement, the pole pieces of the magnet are
drilled, as shown by the broken lines, in a direction parallel to
that of the magnetic field. Light from a bright point source is
concentrated by a condenser, CF, placed in one hole, and the
movement of the illuminated string is magnified by a telescope,
DE, placed in the other.
The " string ' is usually attached to the ends of two small
springs which keep it stretched. The tension may be altered to
any desired amount by a micrometer adjustment at one end.
The string and tension-varying device are supported on a frame
which may be removed bodily from the gap between the pole
pieces. *
486 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The deflection produced for a given current depends upon
the tension on the string and the strength of the magnetic field.
The former may be varied as indicated above, whilst the latter
may be adjusted by regulation of the current flowing through
Fig. 301
the coils of the electromagnet. The sensitivity of the instrument
may be therefore varied over a wide range quite simply. At a
fixed sensitivity the deflection is proportional to the current.
The usual method of detecting deflections is to use a photo-
graphic arrangement. The beam of light emerging from ED if
allowed to fall on a screen forms a shadow image of the string.
This is reflected on to a cylindrical lens which forms an image
of the central portion of the line, i.e. if the string is still and a
photographic plate is moved vertically at the focus of the lens
a straight white line is produced on the plate when developed.
If a current passes, the string moves, and a shift of the shadow
image results. The point focus of the cylindrical lens is thereby
deflected. This results in a lateral displacement of the lme
image on the photographic plate. The magnitude of the displace-
ment gives a measure of the current strength. The natural
period of the string is small and it rapidly returns to the rest
position when the current ceases to flow (not more than a few
hundredths of a second is required). Thus if a succession of
small currents pass in the circuit, the instrument detects them,
even when but a few hundredths of a second interval occurs
between successive currents, whereas an ordinary galvanometer
would not distinguish the break between them.
To measure the interval of time between successive impulses,
or the duration of one of them, a time scale is imprinted on the
record of the current by means of a 'time marker' as shown in
fig. 302.' This consists of a device for intercepting the light at
Fig. 302
Page 4&G
AMMETERS, VOLTMETERS AND GALVANOMETERS 487
regular intervals and thereby making transverse white lines
across the photographic plate on which the current is recorded.
A small motor is made, as shown in the figure with a soft iron
armature of, say, 10 teeth. Intermittent current is supplied
to the electromagnets by connecting them in series with the
circuit of an electrically maintained tuning fork, the impulses
given to the motor are therefore regular. Suppose the fork
vibrates 50 times per second, the synchronous motor will rotate
5 times per second. This drives a circular disc provided with
projecting arms as shown in the figure. These arms are usually
allowed to move across the beam of light illuminating the
apparatus. The figure shows five such spokes, the fifth being
broader than the others. A spoke intercepts the light 25 times
per second and makes a time scale on the photographic plate of
■£ g second.
By making 20 spokes to the wheel T hs second graduations
may be obtained. The width of a division may be varied by
varying the speed of movement of the plate. For a continuous
record a cinematograph film may be used.
Of course a number of strings may be used, and the image
focussed simultaneously on the same film by placing a 45 right-
angled prism in the path of each image. The whole is therefore
concentrated on to the slit in the camera box. The arrangement
of the parts is seen in fig. 303 which shows a plan of the apparatus.
prisms ~_
Fig. 303
(b) Alternating Current
Two types of instrument may be used for the measurement
of alternating current :
(1) The oscillograph which gives the wave form of the current.
(2) Ammeters or dynamometers which give the effective current.
The Duddell Oscillograph
Fig. 304 shows the essential features of this form of instrument.
A thin phosphor-bronze strip, ss, is supported over a small ivory
bobbin, P, and fastened at the lower ends. The tension on the
strip may be adjusted by regulation of the tension on the spring
suspension of P.
488 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
■ If a current is passed through the loop, ss, the two strips will
suffer a deflection in opposite direction, and consequently rotate
a mirror, M, which is attached to both. If the current direction
is reversed, the direction of rotation is also reversed.
Fig. 304
Thus, for alternating current the mirror would rotate back-
wards and forwards, provided that the natural period of the loop
is small compared with that of the alternating supply.
The image of a source of light reflected by the mirror on to
a scale or screen by M would therefore be drawn out into a line
for such an alternating current.
If the image were focussed on to a photographic plate in a
camera, and the plate were moved at constant speed in a direction
normal to the beam of light and the direction of vibration of
the image, the trace on the plate would be approximately a sine
curve. The form of the current time curve may then be investi-
gated from the record.
To see the wave form on a screen the photographic arrange-
ment could be dispensed with, and a mirror made to rotate and
reflect the first beam on to a screen. The linear patch of light
is again converted to wave form by thus adding a constant
velocity normal to that produced by M.
The amplitude of the curve gives an indication of the maximum
current strength, which is approximately proportional to it.
This instrument can also be employed for many of the purposes
to which the Einthoven galvanometer may be used.
'For a description of the other forms of alternating current
measurers, attracted iron ammeters and dynamometers, the
student is referred to any text-book of electrical engineering ;
e.g., T. F. Wall : " Electrical Engineering." Methuen.
CHAPTER XIX
RESISTANCE MEASUREMENTS
The Wheatstone Bridge
The student will be familiar with the Wheatstone net as shown
in Fig. 305. When the bridge is balanced the relation
PR
Q = s W
holds. For maximum sensitivity, using a fixed galvanometer
and battery and measuring a resistance R, it has been shown*
that P, Q and S should be chosen so that
Q 2 = BG,
P 2 = RG
R + B
S 2 = RB
R + G
R + G R + B
If choice of galvanometer is practicable, it should have a
resistance comparable with the other arms. When P ^= Q =cr R
^= S maximum sensitivity is obtained when G — P.
Measurement of the Resistance of a Galvanometer
This may be done in several ways, of which one or two are
given below.
Fig. 305
Kelvin's Method
In this method the galvanometer acts as its own detector of
balance in a Wheatstone net. The galvanometer is placed in
the arm, DC (fig. 305), and the galvanometer of that figure is
replaced by a single-way key, so that B and D may be connected
together when the key is depressed.
When the battery circuit is completed, a steady current flows
through the galvanometer. P, Q and R are adjusted until on joining
* See Gray : "Absolute Measurements," Vol. I, p. 331 [1
489
490 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
B to D throughthe key, no change is produced in that deflection,
P R
when ^- = -p, where G is the resistance of the galvanometer.
W **
The usual difficulty with a sensitive galvanometer is that the
steady current is too large. The galvanometer may not be
shunted in this experiment, but the E.M.F. applied may be
reduced, e.g. the cell may be connected through a high resistance,
and leads from the end of a small fraction of the resistance may
be taken to A and C instead of the battery directly applied.
This, however, decreases the sensitivity of the method.
For a moving-magnet instrument it is better to apply the cell
directly, and to reduce the steady deflection to zero by adjustment
of the control magnet, or, if that is not sufficiently strong, by
the adjustment of an external bar magnet. The sensitivity is
thereby retained.
For a moving-coil instrument no such adjustment is available,
and the simplest course to follow is to clamp the coil of the instru-
ment and find its resistance using another galvanometer as
detector in the usual way.
See also page 643.
The Carey Foster Bridge
The Carey Foster Bridge, shown in fig. 306, is a modification
of the metre bridge. *lt is provided, as seen, with four gaps,
PR R 1 Q
« 4 D D l E E 1 F F 1
nafe 6 6i g— 3r-arer
FlG. 306
CC 1 , DD 1 , EE 1 and FF 1 , which may be closed by the insertion
of resistances.
E
4
Suppose the gaps be closed with resistance Y in CC 1 , R in DD 1 ,
R 1 in EE 1 , Z in FF 1 , as shown in fig. 307. The battery, E, and gal-
vanometer, G, are arranged at points, A C and B D, which corre-
spond to the same points in the theoretical net diagram, fig. 305.
RESISTANCE MEASUREMENTS 491
If the point D is chosen such that no current passes through the
galvanometer, we have, from (1) :
R = Y+^+SxP , ,
R 1 Z + r t + (ioo-*,)p' - K)
where
P is the resistance per cm. of bridge wire,
x x the length, SD,
DT = (100 — x x ) if ST is one metre,
r x is the value of the resistance at the soldered junction, S,
r 2 the resistance at T.
If the simple metre bridge were used to compare R and R 1 ,
i.e. Y = Z = o, we should have a balance at, say, l x cms., such
that, neglecting r x and r 2 for the moment,
?L — h?
R 1 (100 -l x ) P '
Now suppose Y = vp and Z = zp, equation (2) becomes :
?L= (y + *i)p
R 1 {{z + (ioo-x l )}?'
if r x and r 2 be neglected.
It is obvious from a comparison of the two results that the
Carey Foster bridge functions as though the length of the wire
were increased, i.e. the same error in obtaining a balance point
corresponds therefore to a less percentage error in the Carey
Foster bridge determination.
Comparison of the British Association Ohm and the Legal Ohm
The comparison of two resistances, very nearly equal, serves
to show a common use for this bridge. In the following method
it will be seen that the end resistance is eliminated.
The resistances R and R 1 are made approximately equal to
the values of Y and Z. Connecting as in the fig. 307 with, say,
Z, a B.A. ohm, and Y, a legal ohm, whilst R and R 1 are each,
say, 1 legal ohm, we should obtain a balance at a point, D,
x x cms. from S, so that :
5. = Y + '1 + *iP ( o\
1 Ri Z + r 2 + (100 - x x ) p' u;
If now Y and Z are interchanged, Z being connected in the
gap occupied in fig. 307 by Y, and Y replacing Z, a balance for
such an arrangement could be obtained at a distance x 2 from S,
from which we have :
J* _ Z + r x + x iP ()
R 1 Y + r 2 + (100 -*.)?' ■•" w
492 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Equation (3) may be rewritten :
R _ Y +?!+*!!
R + R 1 Y + Z + r x + r % + ioop'
and (4) similarly becomes :
R _ Z + r x + x 2 p
R + R 1 Y + Z + r x + r 2 + loop '
Equating numerators of these equations we have :
Y -f r x + #jP = Z + r x + # 2 P>
Y-Z = (*.-*,)p (5)
i.e. the difference between Y and Z is equal to the resistance
of the bridge wire between the two points of balance. It is
independent of the value of r x and r 2 , and of the total length of
the bridge wire.
To obtain the value of the B.A. ohm in terms of the legal
ohm, we have, Y being the legal ohm :
B.A. ohm = {1 — (x t — *j)p} legal ohm.
The Value of the resistance of the bridge wire between the
points of balance may be obtained by calibrating the bridge
wire (see page 495), or, if the wire is uniform, the following
simple method will serve.
Resistance of Unit Length of Bridge Wire (p)
(1) The bridge connexions of the main experiment above re-
main as before . R and R x are approximately equal, and may very
well be the same as above. Y and Z are replaced respectively
by a fraction of an ohm, say, Y 1 , and a stout strip of copper
of negligible resistance, say Z 1 = o.
Following the same procedure as before, a balance is obtained
at a distance x x x from S, when Y 1 and Z 1 are in the positions of
Y and Z in fig. 307. When Y 1 and Z 1 are interchanged the
balance will move to another point, x 2 x from S 1 .
Then by equation (5) :
Y*-Z 1 = (V-*i 1 )p.
or since Z 1 = o, and Y is known,
Xn X-i
'2 "*1
Hence, using this value of p, the difference between Y and Z in
the first case may be evaluated.
This method may obviously be applied to any similar case,
and a comparison between two nearly equal resistances obtained.
RESISTANCE MEASUREMENTS
493
Experimental Details
The resistances and Y, Z, Ri and R 2 are connected by means
of stout copper strips. These will have practically zero resistance,
and the small difference to be measured will be truly the difference
in the resistance of the coils. It is also essential, of course, that
all connexions be very tightly screwed, for the same reason.
In performing the second part of the experiment, i.e. to find p,
two or three sets cf observations should be made ; the exact
number will depend on the total resistance of the bridge wise.
For example, Y 1 should be made -i, *2, -3, -4 ohm successively,
and p calculated in each case, from which a mean value is obtained.
It may be found that, -4 ohm acting as Y 1 , no balance is obtain-
able. In that case the total resistance of the bridge wire is less
than «4 ohm.
The following set of observations shows the order of the
result obtained in such an experiment.
Using Z = 1 B.A. ohm, Y = 1 legal ohm,
x 2 = 51-2 cms., x x = 48-2 cms.
The wire was not calibrated.
EXPERIMENTS TO DETERMINE P
Y r in ohms.
•1
•2
•3
•4
x x x in cms.
36-35
2376
11*2
no balance
x 2 x in cms.
6375
76-34
88-9
p ohms per cm.
_ Y 1
•00365
•0038
•00386
x% x^
Mean value of p = -00377.
Hence since
Y - Z = (# 2 - x x ) P ,
Legal ohm — B.A. ohm = (51-2 — 48-2) (-00377),
B.A. ohm = 1 — 3 x -00377 = -989 legal ohm.
(2) Another equally simple method for finding the resistance
per cm. of bridge wire may be used. Suppose Y, Z, R and R 1
(fig. 307) are all 1 ohm coils (of the same kind, e.g. B.A. ohms).
R and R 1 are set up as before. To introduce a small difference
between Y and Z, one of them is shunted with, say, 10 ohms,
i.e. the net result is ££ ohm. If now the process described above
is carried out, interchanging the 1 ohm and effective ££ ohm, we
have two balance positions, l x and l 2 cms., say, and
Y -Z = {l x -l 2 ) 9 ,
i.e.
Hence p is determined.
5- <*i-^ P.
494 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
To Construct a Resistance Coil of Known Magnitude
To construct, for example, a i ohm coil, a wire with small
temperature coefficient between io° and 20° C. is selected, say
manganin, and the resistance per cm. of the specimen available
is obtained by finding the resistance of about ioo cms.
The length of wire required to have a resistance not less than
1*1 ohms is calculated, and cut off. The insulation covering is
removed from the ends and, using a non-corrosive flux (say,
resin or ' fluxite '), the two ends are soldered to two stout copper
wires which are soldered to flat copper forks, A and B (fig. 308).
Fig. 308
A and B and the rods are then fastened to opposite sides of a
wooden bobbin by means of terminals as shown, and the wire
wrapped, as in the diagram, in a non-inductive manner round
the bobbin. The middle of the wire T is freed from the silk
cover. The resistance between A and B is obtained by the
Carey Foster method. The middle, T, is twisted with pliers,
cutting out the end loop until, compared with a standard 1 ohm
coil by the method of page 491, the balance is in the centre of
the bridge. T is then soldered in position, and the value of this
copy of the standard ohm when completed is compared with
the true standard as already described.
CALIBRATION OF A BRIDGE WIRE
The simple metre bridge, briefly referred to on page 489, also
the Carey Foster Bridge, the potentiometer and similar instru-
ments, depend upon measurement to a point of balance on a
stretched wire. For simplicity it is often assumed that the wire
is of uniform cross-section, and that its resistance per cm. is
consequently constant throughout the length. Further, the
soldered end and the thick copper connecting strip is assumed to
be of zero resistance. '
In a practical measurement it is better to make no such
assumptions, but to determine the variation due to these causes
by a preliminary calibration of the bridge.
RESISTANCE MEASUREMENTS 495
To Determine the End Correction of the Bridge Wire
For this determination the outer gaps of the bridge (CC 1 , FF 1
in fig. 306) are closed by short clean thick copper strips of
negligible resistance, and in the inner gaps are two unequal
resistances, say a 10 and a 1 ohm coil (R and R 1 ).
A balance is obtained at, say, x x cms. from A. The 10 ohm
and 1 ohm coils are interchanged, and a second balance is obtained
at x 2 cms. Now, if the ' end resistance ' at A is equivalent to
l x cms. of the bridge wire, and if the end resistance at B is equiva-
lent to l 2 cms. of bridge wire, we have :
R. _ x x + h
R 1 100 — x x + /, '
and R1 = *■ + '»
R 100 — x 2 + 1%
from which l t and l 2 may be calculated.
Calibration of the Wire
This may be done in many ways, of which we will consider two.
The object of these experiments is to find, at different points
along the wire, lengths having the same resistance, usually equal
to that of a gauge employed. Knowing the total resistance, S,
of the wire, the mean value of the resistance of such a length is
readily calculated, and hence the correction to be applied at
each segment taken, to reduce to the mean value, may be deter-
mined. Alternatively, having obtained the lengths which have
the fixed resistance, the method of analysis used in the calibration
of a tube may be applied (page 43). Suppose we wish to test
every 5 cms. of the wire, and this will be quite sufficient for most
cases, the following methods may be used.
(1) Carey Foster's Method
^ We saw on page 492, that if two resistances, R and R 1 , are
fixed in the inner gaps of the Carey Foster Bridge, and resistances,
Y and Z, are balanced, and then interchanged and once more
balanced, that :
Y — Z = (% 2 — *i)p>
where x t — x x is the difference in the balance points, and p the
mean resistance per cm. between these balance points, i.e.
the difference between the resistance of Y and Z is equal
to the resistance of the wire included between the balance points,
the end resistances being eliminated by this method.
If now we make Y-rZ equal to the approximate resistance of
5 cms. of the bridge wire, and so arrange R and R 1 that the
496 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
values of x 2 and x t are made in turn to pass along the whole
wire, the calibration is completed. This is done in the following
manner.
The value of S, the total resistance of the bridge wire, is found
either by using another measuring device or the same bridge,*
Knowing S, the mean value of 5 cms. of the bridge wire may
be found. A length of wire, preferably of the same material as
the bridge, is then taken ; its resistance is measured and a
length cut off 2 cms. in excess of that required to be of the same
resistance as 5 cms. of the bridge wire.
This is then soldered to two stout copper lugs as shown in
fig- 309- The two cms. excess being soldered to the lugs. This
gauge is used as resistance Y. Z is composed of a thick copper
connecting strip of practically zero resistance (fig. 310).
B 1 1
Q)
^B
«r—
Fig. 309 Fig. 310
Carrying out the process of balancing Y and Z, inter-
changing, etc., we have, if x 1 and x 2 are the two values of SD,
as before,
Y — Z = resistance of wire between x 2 and x v
By altering R and R 1 the part of the wire compared with
Y — Z may be varied. This process is best done by using
another bridge wire as (R + R 1 ) seen in fig. 310. PQ is the
second bridge wire connected with thick copper strips to the
bridge as shown. The galvanometer, G, is joined to B and D
by means of variable contacts : the gauge is placed in the left-
hand side and the zero resistance in the right, the point D is
chosen near the end of the wire and B adjusted until the galvano-
meter when closed in the circuit shows no deflection. Y and Z
are interchanged and, leaving B fixed (i.e. R and R 1 are fixed),
D is adjusted to some position, D', where a balance is again
obtained. , The resistance of the length, DD ', is then equal to
the resistance of the gauge. Keeping the contact at P 1 fixed,
* The value of S may be found using the same bridge. Referring to'fig. 307,
the gap, Z, is closed with a stout copper strip of negligible resistance. The
galvanometer is permanently corrected between points S (x=o) and B. R, R 1
and Y are adjusted for balance, the whole bridge wire making the fourth arm in
R 1
a Wheatstone net. When balanced we have S «■ ^r * Y. This gives S with-
out allowing for the end corrections.
RESISTANCE MEASUREMENTS
497
the gauge and the strip are returned to their original positions
and B is adjusted to some position, B 1 , where balance is obtained
(i.e. a new ratio R : R 1 is obtained). Y and Z are interchanged,
and the contact D' is moved until again a balance is obtained,
i.e. it is moved a further distance having a resistance equal to
that of the gauge.
This method is carried out until the whole wire, SP, is covered
in approximate 5 cm. steps. The length of wire (x 2 — #1) for
the different steps are noted for each part of the scale, and
tabulated as under.
REGION OF
LENGTH OF WIRE
DIFFERENCE
CORRECTIONS TO
WIRE ON METRE
OF SAME RESIST-
FROM
REDUCE TO
SCALE
ANCE AS GAUGE
MEAN
LENGTH OF MEAN p
0- 5
4-8
+ •15
+0-15
5-10
5*1
— 15
0-00
10-15
5'2
—25
—0-25
15-20
4'9
+ •05
— 0-20
20-25
5-o
—05
-0-25
etc.
etc.
etc.
etc.
90-95
- -05
95-100
o*o
Mean length = 4*95, say.
The correction is obtained by finding the mean of column (2),
and finding what must be added or subtracted from the first
value in column (2) to give the mean value
For the 5-10 cm. range the correction is the algebraic sum
of the first 0-5 cm. range, and the 5-10 cm., and so on.
A correction curve is drawn which will convert any length
of the wire to the equivalent length of a uniform wire of the
same length and resistance.
Thus, if in use in an experiment the bridge balances at 65 cms.,
and from the correction curve the correction, —0*2 say, is obtained,
if the wire were uniform and of same length and resistance,
64*8 cms. would be the balance point, and the resistance ratios
of the two segments of the wire is 648 : 35 '2.
It will be seen that in this method the interchanges which are
frequently made must be assumed to cause no variation in
resistance. The gauge and the straight copper connecting strip
are thoroughly cleaned and always screwed tightly in the gaps
to avoid, as far as possible, such changes. ' ,
This method may be used to calibrate wire, PQ, since the
contact, B, is moved a distance corresponding to equal increments
32
498 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
of resistance. It would therefore serve well for the calibration
of a potentiometer wire.
Direct Calibration by a Potentiometer Method
The wire, ST, of the bridge is connected, as shown in fig. 311,
in series with an adjustable resistance, R 2 , and a 2 volt accumu-
lator which is in good condition and well charged. A similar
accumulator is connected in series with a second adjustable
resistance, R x , and a length of, say, 20 cms. of stretched wire
of the same material and approximate cross-section as the
bridge wire. G is a high-resistance galvanometer.
If
A B
3 IE
13
"n2k
C D
{Elr
1^— ^
K
Fig. 311
C is soldered to the wire, M?and with A and B in contact with
ST, say, between the o and 6 cm. readings, and about 5 cms.
apart, D is adjusted until no deflection is given in the galvano-
meter. D is then soldered to M.
It should be noted that convenient resistances for R x and R 2
are of the order 20 to 50 ohms. The adjustment of these resis-
tances is made until the currents are about equal.
Leaving B in contact with ST, reverse the current in M by
means of the commutator, K, and move A to the other side of
B; adjust till again no deflection is produced. Repeat this
process over the length of ST.
If the latter is uniform, then of course the length AB will
be the same. Variations of the cross-section will be made
apparent by the different values of AB.
Tabulate the length of AB as, before, and deduce the correction
curve for the wire.
This method may also be used for any stretched wire, e.g.
potentiometer, etc.
* Alternatively, the potentiometer may be made up of two resistance boxes,
Rj and R/, in place of the wire M and box R x .
RESISTANCE MEASUREMENTS 499
DETERMINATIONS OF THE VALUE OF A HIGH OR LOW
RESISTANCE
Using the ordinary Post Office Box method of finding resistance
is a very ready method, but for resistances of a million (10 6 ) ohms
or more it is unreliable. The ordinary Post Office Box enables
a magnification of 1000 to 10 only to be obtained, and in the
adjustable arm 10000 ohms is the maximum resistance available,
so, for a resistance greater than 10 e ohms a special method is
required.
Equal limitations for very small resistances make it necessary
to employ special methods in this case.
Some of these special methods are described under.
Low Resistance
(1) The Direct Deflection Method
To find the resistance, r, of a low resistance wire, AB, the
following simple method gives a fair approximation. AB is
joined in series with a known low resistance, R, and an accumu-
lator (2 volt), and an adjustable, fairly large resistance, S.
Fig. 312
Having by means of S adjusted a small current, c, through
the circuit, there will be a drop of potential, cr, across AB, and
cR between the ends of R. If a high-resistance galvanometer is
used, the current in the main circuit is least disturbed, and the
resulting current in the galvanometer is proportional to the
potential cr or cR applied to it. If this causes a small deflection,
d x cms., using the usual lamp and scale method, cr oc 0, approxi-
mately, and 0j oc d x approximately.
The same galvanometer will have deflection i a cms. (angle 0^
when connected to the ends of R. Again cR oc 2 and 0„ oc d 2
approximately
i.e.
B x d x cr
~e t ~~ d 2 ~ cR
r
~R
r = ^-R
d %
Hence r = -=*.- R „. (6)
5oo ADVANCED PRACTICAL PHYSICS FOR STUDENTS
This method of observing deflections may be used to find the
specific resistance of copper.
A length of copper wire is soldered to two terminals about
i metre apart on a wooden base, and is connected in series with
•i or «oi ohm, and a constant source of potential such as a steady
2-volt accumulator, and a resistance box of o to ioo ohms. (It
is not advisable to have less than 20 to 30 ohms in the circuit.)
The ends of the wire AB (r) and of the ^ ohm (R) are con-
nected to a double pole two-way switch (indicated by the broken
lines in fig. 312), which is connected to a high-resistance galvano-
meter, G, the deflections of which may be observed by the usual
lamp and scale method.
The deflection given when the potential difference between
the ends of r and R are applied is measured by taking the reading
of the deflected spot of light in each case. The battery is then
reversed and the reading is obtained on the other side; half
the difference in readings giving d x and d 2 .
S is adjusted and the experiment repeated for 2 or 3 values
of the current, and the mean value of r is obtained.
Hence, putting R = t& or y^r in (6), r is evaluated.
(2) Potentiometer Method
A steady lead accumulator E x is set up in the potentoimeter
circuit AB.
Fig. 312a,
The two small resistances to be compared, R and r, are
RESISTANCE MEASUREMENTS
5oi
joined in series with a third variable resistance S, which is
adjusted so that the current is the maximum compatible with the
capacity of the lead accumulator E 2 .
A galvanometer G, is connected as shown, and a double pole
two-way switch is used. This should be a mercury cu|> key,
using well amalgamated copper connecting strips.
A balance is obtained at H when cups 1 and 4, 2 and 5 are
joined together. A second balance is obtained at H' when
2 and 4, 5 and 3, are joined together.
The P.D. between C and D is cR where c is the steady current
in the circuit CFS ; in the same way the P.D. between D and F
is cr. So that if AH = l x and AH' = l 2 ,
cR l x R l x
— = -i or — = -1
cr l 2 r l 2
The length l x is obtained, then l 2 , and finally l x is checked ;
if any difference is found the mean of l x and l x ' is compared with
1 2, in the usual way.
(3) The Kelvin Bridge
To obtain a more accurate comparison of two small resistances
than may be obtained by the foregoing experiments, the Kelvin
Bridge method is used. Fig. 313 shows the general arrangement
of this network.
r x and r 2 are equal resistances, as are r 8 and r 4 . These are
arranged as shown, so that the total resistance, D to N = that
from N to G and resistances in the arm EK = resistance in KF.
The two resistances to be compared, if in the form of wires,
are arranged as at AB and BC ; B being a mercury cup or other
low-resistance junction.
The two resistances, r x and r s , are connected to AB at points
D and E, and the second low resistance is connected to r 2 and r x
at F and G, the latter being an adjustable point.
502 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The two pairs of equal resistances (about i or 2 ohms each)
should be made of wire of the same material, or having similar
temperature coefficient, and should be placed in close proximity
to ensure a common temperature throughout the experiment.
Further, when the current from the battery is sent through the
circuits, its duration should be short to eliminate excessive
heating in these resistances, and in the specimen AB and in BC ;
the latter, presumably, will have different coefficients, and the
comparison might therefore be upset if such temperature variation
are introduced.
The galvanometer between K and N should be of low resistance.
The points E and F should be near B and the point of contact,
G, is adjusted until on closing the battery and galvanometer key
no deflection is produced in the latter; then if the resistance
DE = r, FG = R, we have
1 = r -l = *J.
R r t r 4
Thus it is apparent that r v r* r z , and r 4 are not of necessity
equal, but the relation
r 1 = ^ w
r 2 *4
must be satisfied. This condition is satisfied in the construction
of the bridge which, like the Post Office Box, may be used with
equal ratio arms or with factors 1 : 10, 1 : 100, 10 : 1, 100 : 1, in the
usual form.
To establish the relation,^-= - = -, let us put c = the current
■K ^2 ?3
in DE (f), Cj the current in EKF (when in the balanced condition
no current passes through the galvanometer), c t the current through
DNG.
Then, since the current c — c t passes along EBF, the current
in FG (R) is c.
The above is the balanced condition of the net of conductors,
i.e. when the potential at K is the same as the potential at N,
i.e. when
cr + c x r x = ctf* _ r _z;
c x r 2 +cR c % r A r 4
i.e. r A {cr + c^) = r z {c x Yi 4- cR),
or c(*r 4 - r 3 R) = c x {r# 3 - r x u).
But the condition satisfied by the bridge, by construction, is that
£ = £2 or r x u = r 2 r 9 ,
r t r A
i.e. c(rr A — r s R) = o ;
and since c is not equal to o :
rr k = y,R;
RESISTANCE MEASUREMENTS 503
or 4 = ~ = - fr om (7) (8)
With such an arrangement of resistances as already described
the comparison of two nearly equal low resistances such as two
wires, can be made ; then if a, I, and s, represent the radius, length
and specific resistance of r, and similar terms apply to R, we have :
7 o /» 2 y o
/ 1 2 2 = -* ,whence — may be calculated.
To obtain the specific resistance of a metal in the form of a
thick wire or rod, we may carry out the above process using a
copper wire as a second low resistance, and assume s for copper
(s for drawn copper at i8°C. =178 x io- fl ).
Alternatively the copper rod may be replaced by *oi ohm (r),
and the unknown rod placed in BC (R). The length, FG, to
balance may be found as above ; whence s for the material :
•01 r t
sl/a* r %
When a definite length of wire, or fixed small resistance, is to
be measured it becomes advisable to have a variable value for r.
This may be a variable length of copper wire of known cross-
section for very small resistances (R), or a series of »oi ohm coils,
and a calibrated wire in series with the coils for larger resistances.
This latter method is usually used in the ready-made Kelvin
Bridge now on the market.
Of course it must be realised that when the values of R, the
unknown resistance, is very small, the current passing through
it must be correspondingly large if any appreciable potential is
to be attained between the ends, and if the measurement is to
be at all satisfactory, the reason for a sensitive /oze/-resistance
galvanometer will also be apparent.
As examples of the types of Kelvin Bridge on the market,
consider figs. 314 and 315.
For measuring the specific resistance of a wire of low resistance,
the special clamping contact arrangement shown at the back
of fig. 314 is used. For other forms of low resistance, the ter-
minals X of that figure and fig. 315 are used directly.
Such an apparatus has a wider range and is more adjustable
than the simpler form already described but is, in principle,
identical with the bridge illustrated in fig. 313. To emphasize
this, fig. 315 has been drawn showing the internal arrangements
of fig. 314, lettered to agree with fig. 313.
504 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The ' ratio arms ' may be adjusted as indicated to give various
ratios, viz. -oi, «i, i, 10, ioo. The coils which make up
?xt r* ?z, r if are accurately adjusted to ensure the condition of
(7) above.
As shown in the figure, r is the unknown resistance, and R
may be one or more of the four coils each of -02 ohms, together
with a part, F'G', of the graduated and calibrated wire, G'G".
The length of the wire is about 450 mms. and each mm. has a
resistance of '00005 ohm, i.e. 40 cms. of the wire add -02 ohms
to the value taken from CG' (-04 in the diagram = GG'). The
wire is graduated in fractions of an ohm.
A comparison of figs. 315 and 313 will show the general arrange-
ments.
For example, suppose an unknown resistance r were balanced
w j 1 en — was made — «oi, by -04 ohm from the coils, and 290
units of the slide wire, as shown approximately in fig. 315,
t — R f — 1 = (-04 -f 2qo x -00005) * 01
\ r, /
= (-04 + -0145) -oi
= "000545 ohms.
This instrument will measure resistances of 10 to -ooooi ohm.
Fig. 317 shows a second form of Kelvin Bridge made from the
potentiometer illustrated in fig. 338, and a ' double ratio ' box
(fig. 316). This enables a range of measurement from 1-5 ohms
to -ooooi ohm, about.
The ' double ratio * box contains resistances (fig 317), LN, NN',
N'N" and N"M, which are respectively \, ^, tttt and
1
TTJI
Of
the total resistance of LM. A similar arrangement holds for
I'm. 314
Fig. 3 r6
Pw 504
RESISTANCE MEASUREMENTS
505
the parts of HI, so that the resistances LN : NM = HK : KI
(1 : 1) or LN' : N'M' = HK' : K'l (= 10 : 1), etc. The scheme
of connexions is lettered to conform with the letters of fig. 313.
A comparison with that figure shows the reason for this scheme :
R, the unknown resistance, is equal to the resistance between
E and D when the galvanometer is connected to NK as in the
case taken, or is ^ ED when the galvanometer is connected
at N'K', etc.
The coils to which D is tapped are each «i ohm and the resistance
of the slide wire is *ooi ohm per small division. Thus, using
N* K* as galvanometer terminals, a resistance of -ooooi may be
measured.
<2X
r.
» 6 vd
C.3 4-C-* T* VO I* 1-4
B«
Fig. 317
High-resistance Measurement
As already stated, a special method is required to determine
the resistance of greater value than 1 megohm (i.e. 10 6 ohms).
One of the simplest ways of evaluating resistances of this order
is the method of substitution. Fig. 291, page 463, shows the
scheme of connexions.
A steady accumulator, E, is connected in series with the
unknown resistance, R, and connected to a commutator, C,
whence the current may be sent in either direction through a
galvanometer, G, which is provided with a variable shunt, S.
Let the steady deflection due to the current be d cms. on a
scale one metre away (corresponding to a movement of the
suspended system of 0°). A known adjustable resistance is
substituted in place of R, and if on adjusting the known resistance
a steady deflection d cms. is obtained again, the known resistance
is of the same value as the unknown. If no variable known
resistances of the order of the unknown resistance are available,
the galvanometer is shunted so that one-thousandth part of the
current is allowed to pass through it. When a large known
resistance (e.g. 10000 to 20000 ohms from two P.O. boxes in series)
is placed in series with it and the steady accumulator, a deflection
of the same order as that given when the unknown resistance is
placed in the circuit may be obtained.
506 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Under these circumstances, suppose that d x is the deflection
caused by the battery when in series with the unknown resistance,
R ohms, and d 2 when in series with r, the known resistance.
Then if E is the E.M.F. of the accumulator, B the resistance
of the battery, and G the resistance of the galvanometer (if
the latter is a moving-coil galvanometer with a radial field), we
have, where k is the galvanometer constant,
Mx = B + G + R'
kd* =
E
S .
(■» SG , I
S + G'
. *i |fr + B)(S+G) + SG} S + G
'•<*■ {(S + G)(B+G + R)} S
M S + G
(,+B)(S+G)+SG W s + ^ _ ()
S(B + G + R) G + R w
when, as is obviously the case, B may be neglected in comparison
with r or R.
If further, G is negligible compared with r or R : *
m
dx
d % R »
i.e., if Trnrry part of the current goes through the shunted galvano-
* S + G
meter, — ^ — = iooo,
d x _ iooor
1,e - T 2 -~R
The experiment may be carried out, using an adjustable
known resistance, and shunting the galvanometer with a known
shunt. The known resistance is adjusted until d 2 = d v i.e.
equal deflections are obtained. Then, if G is small compared
with R,
R = iooor, (10)
if the shunt is -^.
Alternatively, if the known resistance is not sufficiently
adjustable to cause equal deflections, the values of d x and d 2 ,
both of the same order, are noted, then, neglecting G in com-
parison with R, we have :
rf, S, + G
* But according to p. 473 G should be as high as possible to make G as near
equal to R as possible for maximum sensitivity when the experiment is per-
formed without the shunt S.
RESISTANCE MEASUREMENTS 507
Again, if the shunt takes ^^ of the current
R =-r • iooor.
d x
In both these cases, if G is not negligible compared with R,
then the corresponding formulae for the two cases is given by
equation (9) above.
Determine by the above methods the value of the resistance
of a leaky condenser. Another suitable high resistance to be
measured by these methods is made by taking a sheet of ebonite,
say 20 cms. by 5 cms. Two holes are drilled about 15 cms.
apart, and the surface of the ebonite blackened, round the holes,
with a black lead pencil. Two terminals are screwed down in
the holes, and a fine black lead pencil line is ruled between them.
The sheet is covered with a thin protecting second sheet of
ebonite to prevent any accidental change in the dimensions of
the line.
The apparatus should be set up as in fig. 291, and the values
of R obtained.
For example, using a thin black lead pencil line,
&x = i«95 cms., R = ? no shunt ;
^2 = 9*3° cms., t = 10000 ohms, shunt fa,
Hence :
R = -2-2- x 10000 x 1000 = 47-7 x 10 « ohms.
1.95
Using a thicker line on ebonite, i.e. smaller resistance, the
first method (equation (10)) :
deflection 10 cms. when R was in circuit, no shunt,
deflection 10 cms. when 3640 ohms replaced R, shunt fa.
Thus, R = 3640 x 1000 = 3-64 x 10 6 ohms.
THE VARIATION OF RESISTANCE WITH TEMPERATURE
The value of a resistance of a specimen of wire, in most cases,
increases with temperature.
The relation between R , the resistance at o° C, and R f , the
resistance at another temperature, t° C, is :
R, = R (i +at + &*), (11)
where a and p are constants.
The constant, p, is small, and therefore over small ranges of
temperature the resistance is practically a linear function of
the temperature, i.e.
R t = R (i + «t) (12)
expresses the relation, for small temperature ranges, between
508 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
resistance and temperature, (t should not exceed ioo° C. for (12)
to be valid).
From (12) we have :
^a ^ R * ~ R °,
R t
and a may be called the ' coefficient of increase of resistance with
temperature ' for the limited range taken.
To measure such resistance changes and determine the value
of a, which is practically constant for all pure metals, we may
make use of the Carey Foster bridge, since these resistance
changes are small. This method is specially advantageous for
such a determination of small differences, as has already been
shown.
The theory of the method is almost identical with that already
given, so that we may deal with the experiment as such.
Determination of the Resistance of a Wire at Temperature from 0° C.
to 100° C.
A length of thin platinum wire of about 1 ohm resistance is
mounted on a mica frame in the non-inductive manner shown in
fig. 318. The ends of this wire are soldered to two thick copper
wires whose resistance is negligible compared with that of the
platinum. This arrangement is mounted rigidly in a glass tube,
with mica or rubber supports as shown at M. The free ends of
the copper wires are soldered to two long leads, and two identical
leads, cut from the same wire, are joined (D) together. The
four free ends of the leads are soldered to copper connecting
strips, shown in fig. 322, P, P 1 , C, and C 1 . These ensure good
contacts with terminals under which they are fixed.
The pair, P, P 1 , are the ends of the leads from the platinum
wire ; C, C 1 are the compensating leads.
- The inner gaps in the Carey Foster bridge are occupied by
two equal resistances of the same order as that of the platinum
wire ; in the case taken these resistances, R t and R 2 , are both
made 1 ohm (fig. 319).
P and P 1 are connected in one of the outer gaps and the com-
pensating leads in series with a resistance box, S, fill the fourth.
It must be remembered that R 1} R 2 and S should be connected
to the bridge by means of copper strips, and every source of
uncertain resistance, such as a bad or dirty connexion, must be
eliminated.
All the resistances should be ' non-inductive/ for in this
method it will be seen that the galvanometer is permanently
connected in the circuit, and the battery takes the position in
the sliding contact.
RESISTANCE MEASUREMENTS
'509
This is essential, for we must balance the resistance of the
platinum at the temperature which is fixed by the surroundings,
so that the current should not pass for any appreciable time
and cause a heating in the spiral.
fr
Ur y
sn»«r
Copper"
Leads
Wame
Platinum
Wire
Fig. 318 Fig. 319
S is a resistance box in which are all values from *i to :eo or
more ohms.
The function of the compensating leads will be apparent.
They are in opposite arms to the bridge to PP 1 , and so, since
the material and length of the leads are identical with those of
the leads to the platinum, the value of the resistance of the last-
named leads is eliminated ; and since the two pairs of leads are,
in the main, side by side, variation in the resistance of this part
of the circuit is also balanced.
In obtaining a balance at a point, A, x cms. from the end of
the wire, the value of the resistance, S, is so adjusted that A is
near the middle of the wire. Also the balance is obtained by
having no immediate deflection of the galvanometer when contact
is made with the bridge wire. If the current continues to flow,
a change in the resistance of the platinum wire, due to the
heating by the current, will cause a deflection in the galvanometer.
Suppose the glass tube containing the platinum wire specimen
is immersed in a constant temperature bath and a balance is
obtained at x cms. from the end of the bridge, we have, if R, is
the resistance of the platinum at this temperature t°,
Rj ^ R« + r + x ?
R, r -f S + (100 — #)V
510 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
where r is the resistance of either pair of leads, p is the resistance
of i cm. of bridge wire.
Since R x = R 2 = i ohm,
R t -f r + x 9 = S +r + (ioo -*)p,
R t = S + (ioo — 2X) P (13)
Hence, if p be known, R f may be calculated.
The value of R is obtained by surrounding the platinum, etc.,
with melting ice. After twenty minutes or half an hour the
whole of the immersed tube will have attained the temperature
of o°C. ; the balance is obtained near the centre of the bridge by
making S have a value S . A is x cms. from the end. If
after five minutes the balance is still at x , we may safely assume
that the resistance is at o° C.
The value of P may be obtained at this stage, by varying
S to S ' and finding the new balance at x '. S ' is again ad-
justed to a third value, S ", and the balance is now at x ".
The platinum wire is meanwhile at o° C. and therefore its
resistance remains R„,
i.e. R = S + (100 - 2*0) p,
Ro = S ' + (IOO "2* ')p,
R = S " + (I00-2« *) P .
These taken in pairs yield three values of p, and further give
by any one, preferably the value of S which makes the balance
in the neighbourhood of the centre of the wire, a value of R .
The glass tube and contents are then placed in a hypsometer,
and a balance is obtained once more, when the temperature of
the wire has acquired that of the steam which is made to pass
round it in the hypsometer.
The value of the resistance at, say, 40 C, 6o° C, 8o° C. about,
is obtained in the same way by immersing the container in a
water bath maintained at steady temperatures near these points.
In each case time is allowed for the platinum to acquire the
temperature of the bath.
The values of R obtained are plotted against temperature.
The result will be approximately a straight line. From the slope
of the straight line drawn through the observed points, calculate
a as defined above.
In the theoretical account of the balance given above it was
assumed that there were no end corrections to the bridge wire.
If l x and l 2 are the equivalent lengths obtained in the bridge
calibration (see page 495), equation (13) would become
R t + r + x? + h? = S + r + (100 - x) 9 + l 2 ?,
' R t = S + (100 - 2*)p + (h - h)?-
Since l x and l t are known the effect produced may be allowed
in this way throughout the determinations.
RESISTANCE MEASUREMENTS
5ii
r\
The values of x and (100 — x) should be corrected from a
calibration curve as obtained in the manner given on pages
495-8.
The Platinum Resistance Thermometer. Callendar-Grifflths Bridge
The platinum resistance thermometer is a temperature measur-
ing instrument which depends for its action on the variation of
the resistance of a wire with temperature, as investigated in the
last experiment.
The temperature of any enclosure is calculated from the
observed value of the resistance of a calibrated specimen of
platinum wire.
The instrument consists of a length of fine platinum wire,
wound in a non-inductive manner on a mica frame. The ends
of this thin wire are soldered to two thick leads which are con-
nected at the other ends to two terminals on the cap of the
porcelain tube which contains them.
Arranged parallel to these thick leads are a second pair, of
identical material and size, which are likewise soldered to two
more terminals on the cap. The other ends of this second pair
are joined together as shown in fig. 320.
The terminals are usually marked in a distinc-
tive manner, P, P, C, C ; C, C being the ends of
the compensating leads.
We have already seen (page 507) that the re-
sistance, R f , of a given specimen of wire, at a
temperature t° C, may be expressed in terms of
the resistance at o° C, R , and two constants a
and in the following manner :
R, = R (i + <*t + ftf 2 ) (n)
Thus, if the resistance of the platinum wire
were measured at o° and two other known
temperatures, we should have two equations
from which a and p could be calculated, and
the wire would be standardized, so that if R t were
measured at an unknown temperature the latter
could be calculated from equation (11) above ;
or having R , a, and p, a calibration curve could
be drawn showing the relation between R and t,
and hence the temperature corresponding to any
resistance could be obtained.
Alternatively we may define a new scale of
temperature which expresses the relationship
rm. 3*u between temperature so defined and the value of
the resistance in a very simple manner.
512 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Let R and R 100 be the resistance of the platinum wire at
o° C. and ioo° C. ; R 100 — R being the increase in resistance for
ioo° rise in temperature, and is called the ' fundamental interval.'
The platinum scale makes the size of the degree such that
each degree rise in temperature on this scale corresponds to an
equal increase of resistance of the specimen of wire and is equal to
Rioo Rq
ioo
The platinum scale so defined coincides therefore with the
gas scale at o° and ioo°, but will differ at other points
since equation (n) above expresses the true relation between
R and t on the gas scale.
Let R be the resistance of the wire at any temperature, t° C,
or t P on the platinum scale ; by definition :
tp ==ioo p ~ p • ( x 4)
■•Moo ss -o
The difference between t p and t has been found to be given by
t
t - U = k-
(14a)
100 j 100
where k = 1-5 for pure platinum, but its value for any particular
specimen may be obtained by obtaining the resistance R , R 100 ,
and R t at o° C, ioo° C. and a third standard temperature, say,
the boiling point of sulphur corresponding to a known t° C.
The values of R , R 100 , etc., may most conveniently be deter-
mined by using the Callendar-Grimths Bridge.
The Callendar-Grifflths Bridge
The Callendar-Grimths bridge is a compact form of the Carey
Foster arrangement described previously in considering the
variations of resistance with temperature. Fig. 321 shows the
general appearance of a modern example of this type of bridge.
Reference to fig. 322 may make the principle of construction
clear.
R x and R 2 are equal resistances. EF an adjustable resistance
capable of giving 5, 10, 20, 40, 80, 160, 320, 640, 1280 arbitrary
units of resistance. ML is a straight stretched wire chosen by
reason of its uniformity ; T a second parallel wire of the same
material, which may be connected to ML by means of a short
length of the same material wire. Thus, the possibility of
thermo-electrical effect when the galvanometer is connected to
the wire is eliminated.
Fig. 321
Pag* 512
RESISTANCE MEASUREMENTS
513
Suppose the thermometer be balanced at a temperature t p ,
against S units from EF, when the contact is made at O, the
centre of ML, then using the same notation as before (page 509).
R t + r + Resistance LO = S -f r + Resistance OM.
or R, = S.
If now at a second slightly different temperature t\ the
Lcadi
Fig. 322
thermometer has resistance R v and is balanced by moving the
contact 1 unit of length to the right,
Rj+rH- resistance of (OL— 1) =S+r-f resistance of (OM+i),
i-e. R 2 =S -f-2 resistance of 1 cm. of LM.
If the resistance per cm. of ML is half an arbitrary unit, we
have :
Rx = S + 1 arbitrary units.
The scale on which measurements of MO, etc., are made is
usually inscribed in arbitrary units equivalent to the movement
of the contact position.
If the balance is to the right of the mid-point of LM, the
number of units must be added to S ; if to the left the number is
taken from S.
The length of the wire, LM, is usually sufficient to allow about
15 units each side of the centre.
To arrange this simple relation between the length of the
wire and the arbitrary units, a coil, s, of suitable resistance is
shunted across the bridge wire. The coil is adjusted to give the
desired value of resistance per cm. to the bridge wire.
The arbitrary unit of resistance often chosen is deduced from
the value of the fundamental interval of the thermometer (i.e.
the change in resistance for a change in temperature from o°
to ioo° C).
33
514 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The platinum thermometer is constructed with such a resistance
that its fundamental interval is I ohm and one hundredth part
of this, i.e. *oi ohm, is taken as the unit. The coils in S have,
therefore, resistances of -05, *i, etc.
The resistance per cm. of bridge wire in the example taken
would therefore be £(-oi) or -005, the effect of 1 cm. change in
balance being 2 x -005 = •01, i.e. a change of balance of 1 cm.
corresponds to a change in resistance of the platinum wire of
one unit.
The form of Callendar-Grimths bridge illustrated in fig. 321,
is provided with a scale for the slide wire bridge which is gradu-
ated from o to 15 units, i.e. the readings are continuous from
one end of the wire to the other. The balance points give directly
the number of units to be added to S to give the equivalent
resistance of the thermometer. The half, MO, of the wire is
obviously in this case within the instrument in the form of a coil.
A more recent feature of the bridge is the use of mercury cup
contacts instead of the usual plug contacts in the adjustable
arm, S. Fig. 323 shows an enlarged view of one of the contacts.
Fig. 323
The coil is connected to two mercury cups inside a tightly
fitting cover.
A plug, D, when inserted in the hole, G, corresponding to this
resistance, strikes a thin circular sheet, E, which is covered with
baize and is ordinarily held tightly against the hole by means of
the spring, A, and a spiral spring, F, thus keeping dirt and dust
from the mercury. When D is allowed to depress this arrange-
ment, the copper connecting strip, B, which is amalgamated at
the ends, dips into the cups and the resistance is thereby cut out.
RESISTANCE MEASUREMENTS
515
The total value of the resistance, S, is therefore the sum of the
numbers opposite the holes without plugs.
The makers suggest that when not in use all plugs should be
in to maintain the amalgamation of the connecting strip, B.
The balance point on the bridge wire may be maintained by
a rough movement of the slider, followed by a fine adjustment
by means of the small lever attachment. The position of the
oalance is read on a vernier which enables one-tenth of the
small scale division to be measured, i.e. if graduated in lengths
corresponding to a unit and subdivided into tenths, one may
read to y^ of the unit.
Reference to fig. 324 may make clear the internal wiring of
the bridge. The lettering in this diagram corresponds to that
of fig. 322. A two-volt accumulator or Daniel cell is connected
to BB 1 and, say, a Broca galvanometer to GG 1 . PP 1 and CC 1
are gaps for the thermometer and compensating leads. It will
be noticed that Rj and R a are contained in the bridge.
Fig. 324
The point A in fig. 324 is capable of slight adjustment in many
forms of the bridge ; so that, if in error, Rj may be made
equal to R 2 .
Before using the bridge with the thermometer it is advisable
to check some of the points of construction which have already
been described.
(a) See that the zero of the scale is truly the mid-point ot the
bridge. This is readily done by inserting equal resistances, e.g.
two thick copper strips, in the gap provided for the thermo-
meter and compensating leads. With S = o, find the balance
point on the wire. This is the mid-point and should coincide
with the zero graduation ; if this is not so, then probably the
values of R and R 1 may not be truly the same ; this may be
further checked by using two practically equal resistances of
5i6 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
5 or 10 ohms in the gaps PP 1 and CC 1 . A balance is obtained.
The coils are then interchanged and a second balance obtained.
The mean of the two readings should be the zero of the scale.
This method cannot be very well used in the second form of
bridge wire described above.
However, for the purpose of comparing resistances the slight
error in equality will not affect the comparison.
(b) Verify the relation between arbitrary units of resistance in
S and the resistance per unit length of bridge wire, and see that
the resistances in S are consistent within themselves.
The gap CC 1 is closed by means of a copper connecting strip,
and an external resistance or resistances are connected by copper
strips to the gap PP 1 .
The coil 1280 is unplugged and balanced against an external
resistance of, say, i2«9 ohms. Suppose x x be the reading on
the bridge at balance.
Replace the 1280 coil by the rest of the coils, i.e. 640 +
320 + .... 10 + 5, and again balance these coils (nominal
value 1275 units) at a point, x x x , using the same external re-
sistance. Let l x be the resistance corresponding to a change in
balance x x x — x x .
Then coil 1280 — sum of the rest =l x (15)
Carry out this test using the 640 coil : balance against an
external resistance making the point of balance at x 2 , say. Cut
out the 640 coil and, again leaving the external resistance the
same, balance the rest of the coils (320 -f 160 + 80 + 40 + 20 +
10 + 5) at a second point x 2 x cms. Let the resistance corres-
ponding to the change x 2 x — x 2 = h>
then :
Resistance (coil 640 — the rest) — l 2 (16)
Carry out this test with each coil in turn.
Finally :
Resistance (coil 5 — o) = / 9 .
Nine equations are obtained in this way for the bridge described.
The difference between the first and second equations, where
each term represents the corresponding resistance is :
coil 1280 — coil (640 + 320 ... +5) — {coil 640 —
coil (320 + . . . +5)} =li—h>
i.e. coil 1280 — 2 coil 640 = l x — 1 2
.. coil 1280 , l 2 — l x . v
or coil 640 = + 2 (17)
In the same way, taking the third equation from the second :
.. coil 640 , l z — 1 2 ,_™
coil 320 = — — — + ± — — » (1-0)
RESISTANCE MEASUREMENTS
517
and so on. Substituting in (18) the value of coil 640 in terms
of 1280 from equation (17) :
coil 1280 . l z — h
coil 320 = • + -
l z — 1 2
(19)
442
This process is carried on throughout the range. We may
express each coil in terms of the largest one. Equation (17)
and (19) give the corrections to be applied to each coil to make
them consistent with the largest.
For a perfect set of coils l x = / 2 = • • • = h = h = tne
length of wire having 5 units of resistance.
It is important to note that the unit often taken is -oi ohm,
EXTERNAL
RESISTANCE
COIL
BRIDGE WIRE
READING
x units re-
sistance
DIFFERENCE
I
CORREC-
TION
1280
640 +320 +...+5
64O
320 +160 +...+5
320
160+80+...+5
160
80+40+. ..+5
80 (+1280)
40+20+10+5
(+1280)
40 (+1286)
20+IO+5(+I28o)
20 ( + I28o)
10+5 (+1280)
10 (+1280)
5 (+1280)
5 (+1280)
(+1280)
1
so that for safety the 1280 coil should be out as a permanent
addition when testing the 80, 40, 20, 10 and 5 unit coils.
The results may be tabulated as shown above, the last
column giving the correction to each coil.
The above calibration gives the mean value per unit length
of bridge wire.
518 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
For some purposes, and especially when the bridge has had
considerable use, it may be necessary to calibrate the bridge
wire. This may be done by a slight modification of the method
given on page 498. But a new bridge wire should be uniform
to within 0-3 per cent, and therefore will give readings correct
to the second decimal place in degrees on the platinum scale.
Calibration and Use of the Platinum Thermometer
Having tested the bridge and calibrated the coil and wire as
indicated above, the thermometer is placed in melting ice and
allowed to remain there until the resistance remains constant.
The value of this resistance is noted.
The thermometer is next placed in a hypsometer, in which
water is boiled, and again the steady resistance, at the tempera-
ture of the steam, is obtained (after being in the steam for, say,
20 to 30 minutes).
The temperature of the steam corresponding to the atmospheric
pressure is determined from tables (= b° C. say) then, if R ft is
the resistance obtained in the steam, and R in ice at o° C,
R& — Rp
b
is the change in resistance per degree centigrade.
The thermometer is next immersed in the vapour from boiling
sulphur (see below for method), and from the balance of the
bridge, when steady, the value of R, the resistance at the tempera-
ture of the sulphur vapour, is obtained.
The boiling point of the sulphur in degrees on the platinum
scale is from (14) :
_ b(R - R )
tp ~ K 6 - R *
Now the value of the boiling point of the sulphur at 76 cms.
of mercury pressure is 4447 C, and the value at any other
pressure, j^mms., is :
2=4447 + -0904(/> — 760) — «oooo52(£ — 760) 2 .
The value of t, the boiling point at the pressure which obtains
during the experiment is calculated and hence, substituting
t p and t in (14a), the value of k may be calculated.
It will be found to be very near 1-5 in most cases. The thermo-
meter is now fully calibrated and may be used, for example,
to find the B.P. of aniline first in degrees platinum from which, by
(14a), the corresponding value in degrees centigrade may be
found.
To Obtain the Boiling Point of Sulphur
The apparatus of fig. 325 is convenient. This is readily made
by taking a length of iron tube about two inches diameter, and
RESISTANCE MEASUREMENTS
519
about four or five inches longer than the thermometer. The
lower end of the tube is closed by brazing on a circular disc of
iron.
The lower end of the tube is covered completely with asbestos
paper and then wound with nichrome wire of suitable gauge and
length to produce sufficient heating. For example, the one
Fig. 325
shown in the figure was made by winding with nichrome wire
of -092 cm. diameter and six metres in length. This was
satisfactory for use with a series resistance on 100-volt mains.
The length of the same gauge wire may be varied to suit the
potential of the mains available. It might be found advisable,
in addition to the thick layer of asbestos around the nichrome
winding, to surround the boiler with dry sand. The sulphur
vapour condenses on the upper walls of the tube and runs back.
The boiling point under the pressure which obtains during
the experiment, may be obtained from the formula given on
page 518.
The Variation of the Resistance 0! a Bismuth Spiral in a Magnetic
Field
In this experiment a bismuth spiral such as that shown in
fig. 326 is placed in a variable magnetic field and the resistance
Fig. 326
is determined for each value of the field strength. The measure-
ments involved are therefore (1) the strength of the field, (2) the
resistance of the spiral.
The magnetic field should be of as large a range as possible, and
may be obtained by using a large du Bois magnet as shown in
520 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
fig. 327. Current from the electricity mains is passed through
a series of frame resistances, R, to the magnet, and an incandes-
cent lamp, L, in parallel. The current passing through the
windings of the magnet may be measured on a calibrated ammeter,
A, and may be reversed by the commutator, C. The ammeter
is placed at as large a distance from the magnet as is convenient.
SHIHS
Fig. 327
By varying the frame resistance, the current passing may be
of values from, say, to 5 amperes.
The bismuth spiral is placed centrally between the poles of
the magnet with the plane of the spiral normal to the magnetic
field of the magnet and coinciding with the meridian. The
resistance of the spiral is measured either by a Post Office Box,
or by the Carey Foster method. In the latter case the scheme
of connexions shown in fig. 319 may be used, where the gap,
PP 1 , is closed by the spiral and the resistance, S, may be of any
value from o to 50 ohms. R x and R 2 may be given convenient
values ; for a spiral of resistance 15 to 20 ohms make R!=R 2 =20
ohms, say.
In either case the determination of the resistance, R 1} say, of
the spiral will present no difficulties.
Its magnitude is determined for values of the field corresponding
to currents in the electromagnet windings ranging from o to 5
amperes, by, say, '3 ampere steps.
The initial value of the resistance will depend upon the past
history of the specimen, but will increase with increasing magnetic
fields. (The order of this increase will be 10 per cent of
the original value, depending upon the field change which a
current variation of o to 5 amperes creates.)
RESISTANCE MEASUREMENTS 521
The values of the current and resistance are tabulated.
The next part of the experiment is an estimation of the mag-
netic field corresponding to the currents used.
This may be carried out in either of the following ways :
(1) Fluxmeter Determination of the Field Strength (H)
The fluxmeter is set up, away from the electromagnet, and a
search coil of the same area as the bismuth spiral is made.
This is connected to the fluxmeter and introduced between the
poles of the magnet. The deflection of the suspended system
is noted, either directly or on a scale one metre away (for the
weaker field this method is essential). The scale readings are
converted to maxwells\>y comparison with the graduated scale
on the instrument as described on page 485.
If a is the area of the search coil (per turn) and there are
n turns, the total flux recorded in maxwells (m maxwells, say) is
m = nail, or H = — gausses.
na °
As explained in dealing with the theory of the fluxmeter
(page 482) the total deflection is independent of the speed of
insertion of the search coil.
The field is found in this way for each of the current values
used in the resistance determination.
(2) Ballistic Galvanometer Method of Finding H
In this method a similar search coil is made and connected
to a ballistic galvanometer. This forms a low-resistance circuit,
so that a moving-needle type would be least damped by such a
low-resistance circuit. This, however, should be removed to a
very great distance from the magnet and ammeter for obvious
reasons.
It will generally be better to use a moving-coil instrument,
also removed from the magnet, though not necessarily as far
away as the needle type. In such a case a key such as described
on page 481 should be used so that the circuit may be broken
immediately after the passage of the discharge.
Under such circumstances, when the current is passed through
the windings of the electromagnet, the search coil being in the
gap between the pole pieces of the magnet, replacing the bismuth
coil, a transient E.M.F. is set up in the galvanometer circuit
through which a quantity of electricity, Q, will be discharged.
The quantity, Q, as shown earlier, may be expressed by equation
(19), page 479.
* n <p 2 \a 3 /
522 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
where
T is the undamped periodic time (when the key described
is used),
a x is the first deflection,
a 3 is the second deflection on the same side,
c is a small steady current,
<P is the corresponding steady deflection.
The correction for damping (—V will only be applicable
for the moving-coil instrument when the circuit is broken, as
described, as soon as the impulse is given to the galvanometer.
The value of <p corresponding to a current, c, may be obtained
by passing a current from a 2 volt accumulator through one
megohm, and the galvanometer shunted by a small resistance.
The deflection is observed and the current, c, calculated, as on
page 465. T, a lf a 3 are observed in the usual way.
Now suppose that the field strength to be measured is H
gausses, and let c 1 be the current passing through the coil at any
instant, t, after the circuit is made. During a small interval of
time, it, let ^N be the number of lines threading the circuit, then
cZN
dE = -=- (numerically),
, rfN
or re 1 = -jt,
at
where r is the total resistance of the galvanometer circuit (galvano-
meter, leads and search coil),
i.e.
i.e.
fredt = fdN,
when a is the area of one turn (= itR 2 ), and n is the number of
turns, in the search coil,
~ Han
i.e. Q = — *
it (Tc
i.e. H
;N(-) 4 -
[jc tp 2an)
The bracket term is constant during the experiment and may
be evaluated and used as a reduction factor throughout.
The observations described above under distinct headings may
be carried out successively. The search coil is placed in the gap
between the pole pieces ; a lf a 3 are observed. The bismuth
* Alternatively, if the search coil is placed in the gap between the magnets
2 anil
and the current reversed in the windings, Q = — — •
RESISTANCE MEASUREMENTS 523
spiral is then introduced and its resistance measured. The
search coil is replaced and the current cut on. The values,
"fci 1 * as 1 * are again noted. A new value of the current strength
is obtained by an adjustment of the frame resistance, and the
current switched on. The throws are observed and then the
bismuth spiral resistance is again measured. The check value
for the throws is again obtained when the current is switched off,
and so on.
With either method of working compile a table showing the
relation between the magnetic field strength and the resistance,
and plot a curve showing this relation.
The form of the curve will suggest that the relation between
R and H is of the form :
R H = R + d¥L + 6H 8 ,
where R H is the resistance in a field of strength, H,
R the resistance in zero field,
a, b are constants.
By substituting values for two points, find a and b.
Calculate the expected value of R for another magnetic field,
in the range of the experiment, as far removed from the first
two points as convenient. The observed value will be found in
close agreement with this value.
Determination of the Absolute Resistance of a Metal Rod. (Lorentz's
Method)
This method of measuring resistance in absolute units is one
in which a steady drop of potential at the ends of a rod (as
determined by Ohm's Law) is balanced against one set up by
induction. The measurements are reduced to those of length
and time.
Fig. 328 shows the details ; the sketch shows a copper disc,
mounted on a horizontal axis about which it may be rotated,
either by hand or by a motor. A metal brush, C, makes contact
at the rim of the disc, and a second contact to the disc is made
at the axle.
At an adjustable distance, d, from the plane of the disc is a
coil, D, of N turns of wire, of about the same radius as that of
the disc (a cms).
By regulating the position of the movable base, S, the distance,
d, may be made any length within the limits of the length of
the base. Fig. 328 also shows, in a diagrammatic form, the
connexions used in the experiment. AB is the metal rod, say,
of brass. A single accumulator is connected in series with the
coil, D, and the resistance, AB.
524 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If R is the resistance of the rod in absolute E.M. units, the drop
in potential at the ends of AB is cR, when c is the current in like
units.
The rod is also connected, by copper wires, in series with the
disc and a galvanometer.
Fig. 328
When the current from the accumulator, E, circulates through
the coil, a definite number of lines of magnetic force cut the
disc. If M is the coefficient of mutual inductance of the disc
and the coil, there is a flux Mc lines through the disc. If, there-
fore, the latter is made to rotate, an E.M.F. will be set up in the
disc circuit, which will depend on the direction of rotation of the
disc for its direction.
The direction of rotation is chosen such that the E.M.F. set
up by induction opposes the E.M.F., Re, due to the steady
current of the coil circuit. By adjusting the speed of rotation,
or by altering the distance, d (usually by both these methods),
we may balance the potential due to the two causes. Suppose
there are n revolutions per second when this balance occurs, as
shown by no deflection in the sensitive galvanometer G.
In the disc circuit we have an E.M.F. due to the induction
equal to
(Mc)n,
since Mc lines are cut by any radius for one revolution,
there are n revolutions per second. ~M.cn is the number of lines
cut per second, and opposing this is an E.M.F., Re. For balance
Men = Re,
or R = nM.
The resistance, R, is therefore determined if M, the coefficient
of mutual induction, is known, and n the number of revolutions
per second is counted.
Maxwell's formula for the coefficient of mutual inductance in
the case of two circuits of one turn each, of radius, a, and separated
by a distance, d, is :
= 4*«(log.^( I+i ^)-( 2+ -g 5 )
m
RESISTANCE MEASUREMENTS 525
The coefficient, M, above is thus Nw,
1 - e - R = wwN,
or R= 4 ^N« J 2-303 -10^.(1+^) _( 2 ■ ^
16a 2 , ,
The resistance in ohms, where the ohm is defined as io 9 E.M.U.
is, of course, R x 10-'.
Experimental Details
To avoid the complication of the Earth's field in the induction
of the E.M.F. in the disc, the plane of latter is turned so that
it is in the magnetic meridian. *"
The Broca, or other sensitive moving-magnet galvanometer,
works very well in this experiment.
If no motor is available, a fair result is obtained by rotating
the disc by hand. The speed of rotation which may be best
maintained steady is found by a trial experiment, and the coil
and disc placed at a convenient distance, d. When all is ready
the disc is rotated, until the spot of light from the galvanometer
is brought back to the zero reading. At this stage the rotation
is maintained constant. The number of revolutions made is
counted and timed, with a stop-clock, over as long a period as
the light spot may be kept at zero.
From this a knowledge of n may be obtained.
d and a are measured in cms., and the value of R calculated.
If the length and cross-section of the rod are measurable, the
value of the specific resistance may be also calculated.
The following is an experimental result for a determination
made with such an apparatus.
400 revolutions of the disc in 172-5 seconds,
.*. n — 2-325 revolutions per second,
a = n-8 cms.,
d = 5-0 cms.,
N = 100 cms.,
R = 100 x 2-325 x 148-1 1 2-303 x 1-2761 ( 1 -j- — ^~ )
( \ 2227-8 /
V ^ 2227-8/ (
= 35372 absolute E.M. units.
Specific Resistance :
Length of rod = 167 cms., area of cross-section = ^(.64)2,
Specific Resistance = 35372* 1-20,
16-7
= 2732 • E.M. units
(or -000002732 ohm).
CHAPTER XX
RESISTANCE OF ELECTROLYTES
Resistance of a Battery. (Mance's Method)
The resistance of a battery may be found by a modification of
the Wheatstone net as shown in fig. 329. The resistances,
P,Q,R, and the battery of resistance, B, form the network.
The connecting wire and key, K, take the place of the battery
in the usual form of the net.
K
Fig. 329
It will be seen that for all values of P, Q, R, a current will
pass through the galvanometer, G (i.e. in circuit, DCB).
The resistances, P, Q and R are adjusted until, on depressing K,
no change is produced in the steady deflection ; under such
circumstances we have :
P R „ OR
— = — or B = —
Q B P
The main difficulty in such a determination is that the steady
current in the galvanometer causes a deflection which is too
large to keep the reflected beam of light on the scale ; or, if on
the scale, the galvanometer is insensitive in detecting change in
deflections, as seen by the fact that over a large range of resistance
in the arms, R or Q, the above condition seems equally well
fulfilled.
526
RESISTANCE OF ELECTROLYTES 527
This is best overcome by placing a condenser of, say, one-third
microfarad in series with the galvanometer (shown in broken
lines in the figure).
The steady current in the galvanometer is therefore eliminated.
When the key, K, is closed and the bridge is unbalanced, a
throw of the galvanometer will be recorded. When balanced,
no kick will be given by the galvanometer when K is closed.
Another method which may be employed to overcome the
steady deflection, using a moving magnet galvanometer, is to
alter the control field in such a way as to reduce the deflection
to zero, and proceed as first described. This method, if possible
with the control magnet on the instrument, gives good results
as the galvanometer is sensitive to the variations produced by
closing K, when in the zero position.
If the control magnet is too weak to restore the moving magnet
to the original zero position, it may be restored by an external
magnet.
Using a Kelvin (Thompson) galvanometer, the same result
may be brought about by sending the current through one pair
of coils only. The second pair of coils are connected to an
independent circuit composed of an adjustable resistance and
cell, and the current sent in such a direction as to produce a
field in the opposite direction. The magnitude of this current
is adjusted until the spot of light is at the zero position. The
effect of tapping the key, K, will then be seen in the resulting
deflection. The control magnet, in such a case, may be adjusted
to give good sensitivity to the instrument.
Conductivity of Salt Solutions (Electrolytes)
When a direct current is passed through an electrolyte, the
resulting polarization causes an increase in the resistance of the
electrolyte. Further, for continued passage of current through
such a liquid, the resulting decomposition also causes a change
in the resistance, due to alteration of the concentration of the
solution.
The usual special methods adopted to measure such resistances
are designed to overcome these difficulties, and are either a
potentiometer method, or one making use of an alternating
current.
The second is the one most generally employed. If the
alternating current is small, and the area of the electrodes large,
the polarization effect is reduced to a negligible amount. The
more completely is this brought about when very rapidly alter-
nating current is used.
Thus, whereas it is impossible to obtain reliable values for
528 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the resistance of salt solutions by the Wheatstone bridge method
in the ordinary way, by using alternating current in conjunction
with a Wheatstone net, the value of the resistance of the electro-
lyte may be found. Of course in such an arrangement an ordinary
galvanometer is useless as detector ; a telephone replaces it in
the usual modification.
It is not advisable to introduce any inductance or capacity
into the net, so for this reason the wire bridge is preferable to
the post office box type, as the one or ten metres of wire have
less self-inductance than the coils of the post office box.
The scheme of connexions is shown in fig. 330.
AB is a stretched wire of one or ten metres in length,
R an adjustable resistance,
S a vessel containing the solution,
T a telephone,
C a small induction coil. \
Fig. 330
The alternating current is supplied by a small induction coil
which will give but small potential. The induction coil is
driven by a cell and the secondary is connected to the ends, A
and B, of the bridge. If possible an induction coil without an
iron core should be used ; the coil with the greater number of
turns should be used as primary, and the current supplied to it
should be reversed by a rotating commutator driven by a small
motor.
The telephone, T, should be a head-piece receiver.
J The vessel, S, shown in fig. 330, should be surrounded by a
water bath, and be provided with platinum electrodes. The
temperature of the water jacket is maintained constant,
as the resistance of the solution varies rapidly with temperature.
The platinum electrodes are coated with finely divided platinum
to increase their effective area, and to decrease the back electro-
motive force due to polarization. If this has not already been
RESISTANCE OF ELECTROLYTES 529
done, the electrode should be immersed in a solution of
platinum chloride* and a smaller current passed first in one
direction, and then the reverse. To prevent an absorption of
salt from the solution, the platinized electrodes are then raised
to dull red heat.
The solution of known concentration is placed in S, and R is
adjusted so that a point P is obtained near the centre of the
bridge, such that the sound in the telephone is entirely cut out
or, as more often happens, until the sound is reduced to a
minimum, when the usual Wheatstone result may be applied,
RAP
S ~PB'
It is usual to express the results in terms of the specific con-
ductivity of the solution, i.e. the reciprocal of the resistance of
one centimetre, one square cm. in cross-section.
If r is the resistance in ohms between the electrodes,
s the specific conductivity,
then sec— ,
r
A
or s = — ,
r
where A is a constant depending on the dimensions of the vessel
used
A may be obtained by finding r for a liquid of known specific
conductivity. We may take potassium chloride as a standardizing
solution, using ihe data in the table shown on page 530,
where the specific equivalent conductivity is the quotient of
the specific conductivity and the number of gramme molecules
of the salt per litre.
It will be seen that for dilute solution this quantity is nearly
independent of the concentration.
Make up a solution of KG containing a definite number of
gramme molecules per litre, and find the resistance, r, of the
solution when filling the vessel, S. Dilute this solution so that
the solution contains, say, half this number of gramme molecules.
The value of the specific conductivity for these two strengths
may be taken from the above table, and the mean value of A
calculated.
Then, using this calibrated vessel, find the resistance of several
solutions of NaCl from a concentration of, say, 29*25 grammes
* The Platinum chloride solution to use is made by taking
1 part platinum chloride,
30 parts water,
'008 part lead acetate.
34
530 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
per litre P^\ to -2925 grammes per litre, by diluting the concen-
trated solution first made.
Plot a curve showing the relation between specific conductivity
and concentration, and specific equivalent conductivity and
concentration.
The variation of the resistance of one of the solutions with
temperature may also be investigated by heating the water baths
surrounding the cell, S, and the value of the temperature co-
efficient may be calculated.
Using a container for the solution, of more measurable dimen-
PARTS BY WEIGHT
GRAMME
SPECIFIC
SPECIFIC
OF KCI IN IOO
MOL. PER
CONDUCTIVITY
EQUIVALENT
PARTS OF SOLUTION
LITRE
AT l8°C.
CONDUCTIVITY
•OO746
•001
0-125 x 10- 3
125 xio- 8
•O746
•01
1-206 „
120-6
743
•I
11-300 „
113-0
I-48
•2
22-00 „
IIO-O
3^4
*5
50-60 „
ioi-6
7'43
1-0
99-10 „
99-1
137
2*0
188-8
94-4
197
3'0
274-2
91-4
sions, the specific conductivity, and hence equivalent conductivity
may be calculated in the way usually employed in solids. For
example, in fig. 331, a uniform tube connects two bottles in
dh
X j
Fig. 331
each pf which is immersed an electrode ; the column of liquid
conveying the current is practically one coinciding with the
tube, i.e. of length I cms. and cross-section equal to that of
the internal cross-section of the tube. The resistance is
measured as above ; the specific resistance is then directly
calculated in the usual way, and the experiment proceeds as
already described.
CHAPTER XXr
MEASUREMENT OF POTENTIAL
Standard Cells
'the two standard cells in common use are the Weston or
Cadmium cell and the Clark cell ; of these, the former is better for
general service.
The two cells have each a constant electromotive force at one
temperature, and also a definite temperature coefficient.
The Weston cell is shown in diagram form in fig. 332.
Two tubes are arranged as shown, each being provided with
an external lead which is in contact with the bottom layers.
These layers consist of pure mercury, M, and an amalgam of
pure mercury and cadmium, A, respectively. Above the pure
mercury is a layer of a paste of mercurous sulphate, P, made as
described later (shown by horizontal shading). Above this and
the cadmium amalgam is a layer in each tube of pure cadmium
sulphate crystals, CC. Finally, a layer of a saturated solution
of pure cadmium sulphate occupies the upper parts of the tubes.
To make a Cadmium Cell
The following is a method of construction for the Weston cell ;
for a more permanent and exact cell the specification given in
the Report of the British Association Meeting, 1905, on page 98,
by F. E. Smith, should be consulted.
531
532 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
(a) Mercury is at first obtained in as pure condition as possible
commercially. It is then passed through a dilute solution of
nitric acid, drop by drop. To bring this about take a tube of
about three-quarters to one inch in diameter, and about 60 to 100
cms." long ; draw out the end to a smaller diameter and bend
this smaller tube so as to leave a short U-tube at the end ; the
shorter end is bent over as seen in fig. 333.
Fig. 333
A second length of the wide tube is drawn out at one end to
a fine capillary of such diameter, that when the tube is filled
with mercury the latter will just emerge as very small drops.
This tube is inserted at the upper end of the first, which is filled
with dilute nitric acid. The mercury is collected in the U-tube
and passes over into a collecting vessel.
The process is repeated and the partially purified mercury
is next distilled under reduced pressure, air being bubbled
through it during this process. To carry out this distillation
Fig. 334
the mercury is placed in a round-bottomed flask, provided with
a side tube which leads through a condenser to a second round-
bottomed flask, itself connected to a good water filter pump,
as shown in fig. 334.
MEASUREMENT OF POTENTIAL 533
Into the first flask, through a tightly fitting cork, a narrow-
glass tube passes under the mercury. A clip regulates the
inflow of air. When the pump has reduced the pressure inside
the system, the mercury is heated on a sand bath, and the
condensed mercury is collected in the second flask.
The air is allowed to enter through the mercury as a slow
succession of bubbles. This oxidizes such impurities as zinc
and minimizes the risk of their distillation with the mercury.
(b) The amalgam is made by dissolving pure cadmium in pure
inercury, to make a 12 to 13 per cent cadmium amalgam (i.e. about
in the proportion of 1 gramme of cadmium to 7 grammes of
pure mercury).
(c) The paste is made in a mortar by grinding together pure
mercurous sulphate, cadmium sulphate and purified mercury in
the proportion of 8:4:1. The mixture is made into a thin
paste by the addition of a solution of cadmium sulphate.
Before filling the cell the platinum wires must be amalgamated.
In the form of tube shown in fig. 332, this is done by the electro-
lysis of mercuric nitrate in the glass tubes, using the wires as
electrodes and reversing the current.
Another form of glass container for the cell consists of a
double tube with open upper ends. Corks are selected to fit
the open ends tightly, and through holes in the corks a narrow
glass tube may be inserted in each side tube, to carry the wire
through the cell content to the mercury or amalgam.
Such platinum wires could well be amalgamated by heating
to red heat and dipping in mercury.
Having amalgamated the platinum wires, the tubes are next
carefully filled as shown in fig. 332. If the open-tube type is
used, the corks should be finally coated with marine glue or a
mixture of beeswax and resin.
The E.M.F. of the cell so formed will be found constant with
constant temperature. On no account must a current of any
appreciable magnitude be taken from the cell.
The International Conference on Electrical Units and Standards,
1908, adopted the following formula as giving most accurately
the E.M.F. of the cell :
E t = 1-0184 — 4'o6 x 10- 5 (t — 20) — 9-5 x 10- ' (t — 20) 2
+ I0- 8 (tf-20) 3 VOlt, . . « M ... (I)
where t is expressed in degrees centigrade.
The temperature coefficient is therefore small.
The Clark Cell
This is made in a manner identical with that described above,
With the exception that cadmium is replaced in this case
534 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
by zinc ; cadmium sulphate by zinc sulphate, etc. Proceeding
as above, again using pure salts and mercury, the standard cell
so constructed has an electromotive force expressed by the
formula :
E t = 1-4328 - 1-19 x 10- 3 (t - 15) — 7 x 10- 6 (*-i5) 2 . .(2)
Thus this cell has a larger temperature coefficient than the
Weston, a fact which explains the more general use of the former.
When using either form of cell in potentiometer work the
device of fig. 337 (i.e. a large series resistance) is a useful one as a
safeguard against damage to the cell when the ' balance point '
is not approximately found.
Comparison of Electromotive Force
The potentiometer method of comparing two electromotive
forces is the most satisfactory one. It is assumed that the reader
is familiar with the direct comparison of two E.M.F.'s, using a
stretched wire potentiometer.
B
In that method a steady accumulator is arranged as at E in
fig- 335 > and E x and E 2 whose E.M.F.'s are to be compared
are in turn placed in series with the galvanometer, and a point
of balance obtained at l x and Z,*cms. respectively ; then, if the
wire is of uniform resistance per cm.,
E 2 l 2
Using one of the methods given on page 495, tt seq., the wire,
AB, could be calibrated and then, more accurately, if l x x and Z 2 *
are the corrected lengths corresponding to l x and l a
For an absolute value of the E.M.F. of either cell a third
balance could be obtained at, say, l 3 for a cadmium cell whose
E.M.F. is known at the temperature of the experiment and is
* l t is the mean of, say, three observations, and l 2 the mean of two observations,
taken alternately ; this eliminates the effect of the variation of E.M.F. of E.
MEASUREMENT OF POTENTIAL
535
approximately 1-0184 volts. Whence E x and E 2 may be obtained
in volts.
For making such a comparison, the accuracy of the deter-
mination depends on the accuracy of obtaining the balance
poiftt. If instead of using a one-metre potentiometer, a wire of
ten metres (i.e. ten wires in series, each one metre long),
be used, then each cm. of wire has a potential drop equal to
one-tenth the drop in the simpler potentiometer, i.e. a movement
of 1 mm. in the single wire bridge would correspond to 1 cm.
movement in the ten-wire instrument ; hence by using the
ten-meter potentiometer the true balance point may be more
nearly estimated.
It is often convenient to make such a potentiometer direct
reading. To do this we arrange that between the ends of a
definite length there is a fixed potential difference, say, 10- s volt.
Using a ten-metre potentiometer the most convenient length
to employ to correspond to io _a volt is 5 mms. This is brought
about as follows :
A standard cadmium cell (E.M.F. = 1-0184, assuming tempera-
ture is 20 C.) is connected to A,* and the jockey makes contact
with the wire at 1018-4 units of length from the end.
We have chosen the unit for this purpose as 5 mms. i.e. P is
fixed at 509-2 cms. from A as in fig. 336. R is now adjusted
E 1
. — I— 1 — B
S.C. G
Fig. 336
until the galvanometer gives no deflection, i.e. the current in
the wire AB, due to E, is such as to cause io -3 volt drop per
5 mm. of the wire (assumed uniform). Leaving R fixed, any
other E.M.F. may be found by balancing on the potentiometer
at, say, / cms. from A or 2/ x 5 mms., i.e. 2.1 x 10- 3 volts is the
value of the balanced E.M.F. The above assumes that the
accumulator E remains steady. This should be checked at
intervals, if a series of comparisons are to be made, by reinserting
the standard cell, SC, and adjusting R to bring a balance at
509-2 cms.
* A resistance of about io 4 ohms should be in series with S.C. until balance is
almost complete. See fig. 337 and page 537.
.536 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
For many purposes it is convenient to replace the wire by-
variable standard resistances when comparing potentials. A
suitable arrangement of apparatus using such a method is seen
in fig. 337. R 2 and R 2 are two resistance boxes, each having a
E x l |U? K 2
Fig. 337
resistance up to, say, 10,000 ohms (two post office boxes do
very well).
E is a steady accumulator which is connected in series with
R x and R 2 . The cells to be compared are connected to A and P
through a galvanometer, G.
For direct comparison the resistance (Rj + R 2 ) is kept constant,
say 10,000 ohms, and R x and R 2 are varied until no deflection
is obtained when K 2 is closed. The value of R x is noted.
The process is repeated with the second cell, say, a cadmium
cell, a balance being obtained for a resistance R^ in the box
between AP.
Then, as before ~r = ^j>
Ji 2 Kx 1
E R
for the drop in potential between AP due to E is = ^ — 5 i
R-i + J^2 + -fc>
where B is the resistance of the cell E. When a balance is
obtained
similarly
and
Hence
E. =
R x + R 2 + B
R1 1
E,
R 2 i + R, 1 + B
Rj 4. R 2 == Rji -j- R 2 i = 10,000 ohms.
5s
E.
5l
Rx 1 '
Another way of using the above form of potentiometer is
similar to the direct reading method of using the wire potentio-
meter.
MEASUREMENT OF POTENTIAL 537
One of the cells to be balanced against E is a cadmium cell.
If the temperature of the experiment is not far removed from
20 C, the electromotive force of such a cell is 1-0184.
R x is given the value 1018-4 ohms (a fraction ohm box may
be included in series with R a ) . Knowing the approximate electro-
motive force of the cell, E, the value of R 2 may be estimated
such that the potential drop in AP ^ 1-0184 volts. The standard
cell is placed in series with the galvanometer G ; K 3 being open,
r, of about 10 4 ohms is in series with SC, to avoid damaging it
during the preliminary balancing. R 2 is now adjusted until no
deflection is noted in G. K s is closed and the final balance
verified. The total value of R x + R 2 under these conditions is
noted and maintained constant throughout the comparison.
Now 1018-4 ohms have a drop of potential of 1-0184 volts,
i.e. each ohm corresponds to a potential drop of io~ 3 volt.
Therefore, when a second cell (E 1 ) is introduced, if R t has a
new value, R x * ohms at balance (R^ + R 2 X being equal to
R, and R 2 as obtained in the first test), E 1 is R x x x io~ 3 volts.
The above methods of making the potentiometer direct reading
are only suggested for those cases where several comparisons
are to be made, for under such circumstances subsequent
calculation is eliminated.
The experimental arrangements described above are most
sensitive for comparison of electromotive forces of the order of
one volt. If now a small difference of potential is to be deter-
mined it will be apparent that these arrangements are not
sufficiently sensitive. In the case of a stretched wire, the
sensitivity increases with increase in length ; therefore, to
measure a potential difference of the order of, say, io~* volts
with fair accuracy the potentiometer wire would require an
extension of several metres of wire, or, what is more convenient,
the inclusion in the circuit of a resistance several times that of
the wire (r). For example, if a potentiometer wire of one metre
were of one ohm resistance, and the accumulator E had an
E.M.F. of 2 volts, a potential difference of the order of 10 milli-
volts would balance at a distance of 5 mm. from the end of the
wire. This could very easily be estimated at 4 mm., or 20 per
cent in error. Further, under such circumstances the calibration
of the wire would be of greater importance.
If on the other hand 99 ohms were placed in series with the
wire, the drop of potential in the wire would be 1/100 of 2 volts,
and the- true balance point would therefore be at 50 cms. An
error in estimating the balance point of 1 mm. would only be
•2 per cent.
An example of the application of this method is seen in fig.
343, page 542.
538 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Direct Reading Potentiometers
■> As described above, the potentiometer may be made ' direct
reading.' Several forms of potentiometer are now on the market
which are constructed on the above principles and are calibrated
directly in potential.
Of the simpler forms of compact manufactured instruments,
which measures potential of the order of a volt to one millivolt,
we will describe the form illustrated in fig. 33,8. The internal
wiring is indicated by white lines drawn on the case, and the
instrument is an application of the form shown in fig. 336. The
2 volt accumulator which is connected to the terminals EF,
sends a current through an adjustable rheostat, a series of coils
provided with tappings to the studs shown, and through the slide
wire. This constitutes the main circuit.
When the current is adjusted to the correct amount, the fall
in potential along each of the resistances, connected to the studs,
is *i volt, and along the slide wire, «I2 volt.
The wire is divided into 120 parts, each corresponding in
adjustment to a fall of potential of 10- * volts.
To adjust the current to this strength a standard cell is inserted
in the gap marked ' potential,' and a galvanometer of low resis-
tance in the gap marked ' galv.' The two adjustable contacts
are set at points which correspond to the true E.M.F. of the
cell at the temperature of the room, e.g. if the E.M.F. of the cell
is 1-018, the sliding contact at the back of the instrument is set
at i«o and the contact on the wire made at •018.
The rheostat is then adjusted so that when the key is closed
no current is indicated in the galvanometer.
If the rheostat is not sufficiently large to bring this about
the battery should be connected to E and G, not EF. This
introduces more resistance in the circuit as indicated in the
figure, and, using a normal 2-volt accumulator, the balance for
the standard cell will be attained.
To obtain the value of an unknown electromotive force, it
is connected to replace the standard cell. Leaving the rheostat
in the balanced position, the sliding contacts are adjusted until
a balance is obtained, as indicated by no current in the galvano-
meter. The value of the electromotive force is then directly
obtained on the calibrated scales.
The width of the small divisions is sufficiently large to allow
eye estimation to £ of a division, but is not of suitable range
for the measurement of the corresponding £ of a millivolt. Thus,
whereas the instrument would measure a potential of 1-0182
with a good degree of accuracy, it cannot be used to measure
•0002 volt with any certainty, nor is it reasonable to expect
PtlRfi 5 3 B
Fig. 339
Pg£B 5J9
MEASUREMENT OF POTENTIAL 539
such measurement with an instrument having a range o to 1*5
volts.*
For such measurements, as, say, for a thermo-junction, an
instrument having a range of to 50 millivolts is more suitable.
Potentiometers of such a range are manufactured by many
firms, e.g. Nalder, Cambridge and Paul, Crompton, Gambrell, etc.
Figs. 339 and 340 show the general appearance and internal
arrangements of such an instrument.
The current to main circuit is supplied by a 2 volt accumulator,
B. As seen, this circuit consists of adjustable resistances, Rj
and R 2 , fixed resistance E and F, MVC, D x and D 2 , and the two
stretched wires SS and W.
The range of the instrument is 30 millivolts (o to 30, or 30 to 60,
or 60 to 90) . The resistance MVC is made up of 29 similar coils of
such a resistance that when the current is adjusted as described
below, the potential difference between the ends of each coil is 1
millivolt. Also the wire, VV, is of a resistance such that for this
adjustment the drop of potential along its length is 1*2 milli-
volts. By subdivision, a value of '005 millivolts may be
obtained.
Further, the value of the resistance, F, and the slide wire, SS,
is such that the graduation along SS gives the potential between
M and N when adjusted.
To standardize the potentiometer, the standard cell, SC, is
connected to the galvanometer, G, by means of the double pole
switch. The point, N, is chosen equal to the potential of the
standard cell, R x and R 2 are then adjusted until the low resistance
galvanometer shows no deflection, i.e. the drop of potential along
MN corresponds to the graduation value. Under such circum-
stances the potential difference per coil of MVC is one millivolt,
etc.
To measure an unknown electromotive force between .0 to 30
millivolts, plugs are inserted at the points shown in fig. 340, and
the unknown potential connected to X. The commutator is
thrown over so that XX are connected to the galvanometer,
and a balance is obtained by varying the point of contact, P,
on MVC and Q on VV. If D 2 is of zero resistance the unknown
potential corresponds to that between P and Q and is therefore
obtained directly from the scale.
Suppose a bigger potential, say, 30 to 60 is to be measured, D a
is now made of such resistance that there is a potential drop
equal to 30 millivolts along its length, and therefore for the
* A modification of this form of instrument, having a range suitable for
thermo-junction work, is now to be obtained.
540 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
balanced position the potential is 30 + the readings of MVC
and VV.
Similarly for 60 to 90 millivolts. D 2 is increased so that for the
steady current in the main circuit the potential difference between
the ends of D 9 = 60 millivolts.
r-0
Fig. 340
To maintain the current in the main circuit at a fixed value,
the resistance of D x must be decreased by the same amount as
the increase in resistance of D 2 . This is done in the manner
shown in fig. 340.
Thermo-Electrieity— Thermo-Junetions
When a circuit is composed of two dissimilar metals and the
junctions of these metals are maintained at different temperatures
an E.M.F. is set up in the circuit. This electromotive force
varies with the difference in temperature between the junctions,
and when one junction is maintained at o° C. is given by
E f = at + U\
where a and b are constants and t expresses the temperature of
the hot junction in degrees centigrade.
The direction of the electromotive force depends on the metals.
It is customary to express E t for any metal with respect to a
standard metal which is taken as one of the pair. The usual
choice of standard metal is lead.
In drawing the curve giving the relation between the E.M.F.
and temperature, the E.M.F. is taken as positive when the
current tends to flow from lead to the metal at the hot junction.
MEASUREMENT OF POTENTIAL
54i
Thus fig. 341 shows the form of these curves for Pb/Fe and Pb/Cu.
At a temperature t° C. AB represents the E.M.F. developed in
a Pb/Cu junction and, according to the above rule, the electro-
motive force is from the lead to the metal at the hot junction.
Similarly, AC is the magnitude of the electromotive force
developed from lead to iron at the hot junction.
+
c
y
1 ^r
EMF
f B
A?
O
L^ A
Temperature
—
Pb
Fig. 341
The law of intermediate metals may now be applied to deter-
mine the value of the E.M.F. developed at a copper-iron junction
at a temperature t° C, the other junction being maintained at
o°C, for, according to this law the E.M.F. developed at the
temperature t°, for a Cu/Fe junction, is expressed by
E.M.F. Cu/Fe = E.M.F. (Cu/Pb + Pb/Fe)
= - AB + AC
. , = + BC,
i.e. there is an electro-motive force of magnitude BC from copper
to iron at the hot junction since the BC is positive.
Reference to fig. 342 will show this from another point of view.
If junctions, X and Y, are maintained at temperature, t°, Z being
maintained at o° C, the arrows show the direction of the E.M.F. 's
542 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
of magnitude AC at X counter-clockwise and AB at Y clock-
wise — a net result BC counter-clockwise. This, according to
the Law of Intermediate Metals, is the value of the E.M.F. if
XY are brought together, the lead being removed. The result
is an E.M.F., BC, in counter-clockwise direction, i.e. from the
copper to the iron at the hot junction.
Experimental Determination of the Thermo-Electromotive Force —
Temperature Diagram
The magnitude of the thermo-electric E.M.F. is of the order
of a few millivolts. It is best investigated by means of a potentio-
meter of the form described on page 537. An instructive result
is obtained using the three metals, copper, iron and lead. As
will be seen from the above, it is only necessary to obtain the
E.M.F. for two pairs of the metals, the third pair may be estimated,
making use of the law of intermediate metals.
A uniform wire potentiometer of, say, 10 metres is employed
(one of one or two metres, if the other form is not available). If
the best results are to be obtained a preliminary calibration of
the wire is advised.
A steady accumulator, C, is joined in series with R lf R 2 , and
the potentiometer wire (R x and R 2 should contain resistances
up to 10,000 ohms). A preliminary experiment gives the value
of r the resistance of the ten metres of wire.
Her ice
Fig. 343
A thermo-junction is constructed, using copper and iron
wire. Care must, of course, be taken that the wires are in
contact at the junction only. For this purpose a suitable form
of junction is seen in fig. 344. The one wire, B, passes down
MEASUREMENT OF POTENTIAL 543
a thin glass tube, G, the other, A, is joined to it at J. To
ensure good contact, J is dipped into mercury at the bottom of
the test tube, T. This does not affect the E.M.F. If one or both
of the metals are affected chemically by mercury, it must be dis-
pensed with, and special care paid to the welding of the junctions.
j . Three such junctions are made. One, the hot
/ / A junction, using iron and copper wires. The free
ends of the copper and iron wires are each joined
to connecting leads as illustrated. These two junc-
tions are maintained at the same temperature, o° C.
in the experiment described. The use of such junc-
tions is to eliminate thermal electromotive force
between the metal of the connecting leads and the
metals of the thermo-junction proper. In the par-
ticular case taken, the copper to copper connecting
lead is not essential generally, but is a safeguard
against the possibility of impurity in one wire, and
is necessary for the Pb/Fe junction.
If the junctions to the connecting wires are main-
tained at o° C. by surrounding with ice, and the
' hot junction ' is placed in a water bath at a tempera-
Fig. 344 ture t o c ^ a;a £jip_ yrfft be developed in the direction
A to L via the junctions. C is therefore connected with the
positive pole to B.
The maximum E.M.F. developed in the above couple is of
the order of 1500 micro-volts : the rise in potential along AB
should therefore be arranged not very much in excess of this.
A voltmeter gives the approximate potential available from
C, and r being known, the value of R x + R a to cause such a drop
may be calculated,
potential difference in AB r
E.M.F. of C r + Ri + R»'
the accumulator resistance being negligibly small.
Having fixed R x and R 2 , the point, L, is made to coincide with
A, and the standard cell, say a cadmium cell, is put in series with
the galvanometer ; R 2 is adjusted, keeping R x + R 2 at the
value determined above, until no deflection is given in G when
contact is made at A.
If the room temperature is approximately 20 C. the E.M.F. of
the cadmium cell is 1-0184. Hence the drop of potential along
AB is :
£- x 1-0184,
whence the drop per cm. of the wire is :
^- x 1-0184 X I0 " 3 volt • — — — ~ (3)
544 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The thermo-junction is now placed in circuit and the hot
junction is raised to about 95 C. in a water bath : the length
of wire required for balance is obtained. The water is allowed
to cool and balance points obtained for intervals of temperature
of about 5 C.
The hot junction is placed in a hypsometer and again the
E.M.F. is balanced : when the junction is at the temperature
of the steam a steady balance point is obtained and noted.
The hot junction test tube is next placed in a boiling tube
containing mercury. This is heated slowly. Balance points are
obtained at intervals and the temperature noted on a special
mercury thermometer which reads to 360 C. The process is
carried on until the mercury boils.
The results are tabulated — temperature and lengths for balance.
The lengths are converted to E.M.F. by multiplying by the
reduction factor given in (3) above.
The process is repeated, using a lead wire in place of the iron,
and the results are plotted as in fig. 341, E.M.F. in micro-volts
(io - 6 volt) against the temperature of the hot junction.
From these results obtain the corresponding curve for lead — iron.
It will be found that at a temperature of about 240 ° C. the
Fe-Cu junction will give a maximum E.M.F. This tempera-
ture is called the neutral point.
The same results could be obtained directly, using a direct
reading potentiometer. With a ten-metre instrument r would
probably be just greater than 10 ohms. If a shunt, S, were
placed between A and B (fig. 343) and S given a suitable value
which can be calculated from a knowledge of r, the wire and
shunt may be made to have exactly 10 ohms resistance.
As before, suppose the E.M.F. of the cadmium cell were
1-0184 at tne temperature of the experiment (any other value
can be treated in the same manner). Then to obtain a drop of
10- 6 volt per cm. of bridge wire R 2 is given the value (1018*4 — 10)
1008-4 ohms. The cadmium cell is placed in series with the
galvanometer, and the sliding contact, L, is made at B (fig. 343).
Ri is adjusted until a balance is obtained. Then the drop in
potential from A to B = 5 — of 1-0184 or I0 ~ 8 volt, hence
1010-4
the drop per cm. is io -6 volt.
The lengths for balance in the thermo-couple experiment now
give the potential in micro-volts directly.
The same process applies in the case of a single-metre poten-
tiometer. The value of r is found, and the shunt value, S, to
make the value for the two in parallel 1 ohm, is calculated and
R 2 made (1000 E — 1). In that case each millimetre of the
wire corresponds to a potential drop of io~ 6 volt.
MEASUREMENT OF POTENTIAL 545
If no standard cell is available the value of the potential per
cm. of wire may be calculated from a knowledge of r, R x and
R 2 and E, the E.M.F. of the cell, C, as determined by means of
a high-resistance voltmeter.
Consider the curves obtained. If we take any two fixed
temperatures and determine from the curve the value of the
E.M.F. developed, in micro-volts, we shall be provided with two
equations having two unknown constants, for we have seen that
the relation between the E.M.F. developed E, and the temperature
t° C. is :
E = at + bt*.
So by choosing two such pairs of temperatures and finding E
on the Fb-Cu and Fb-Fe curves, the values of a and b for
iron and copper may be obtained.
dF
Hence since ' -=- = a-\-2bt,
at
we know the value of the thermo-electric power, -=--, at any
dt
temperature.
The thermo-electric power lines for Cu and Fe against the
standard, lead, may thus be drawn and the point of intersection,
which gives the neutral temperature for iron-copper, may be
ascertained. This will be found, of course, to agree with the neutral
temperature as found directly from the E.M.F. temperature curve
for the iron-copper junction.
The Use of a Thermo-Eleetrie Couple as a Thermometer
It will be seen from the curves obtained (fig. 341), that a couple
such as copper-iron is not a suitable one to use as a thermometer.
The neutral point is too near o° C. for such use.
The ideal couple for such purpose is one with a neutral point
well removed from o 6 C, and which will therefore give an E.M.F.
approximately proportional to the temperature difference. In
many experiments in this book such a couple is required. A
couple of copper and eureka or constantan serves well for
this purpose.
When the thermo-junction is used in this way the form of
potentiometer described on page ^ 539, fig. 339 is a great
convenience, but the form of direct reading potentiometer of
ten metres of wire (page 544) when once adjusted serves quite
well.
In either case the electromotive force-temperature curve of
the junction should be obtained as described up to, say, 240 C.
The value of the temperature corresponding to any other electro-
motive force may then be obtained from the standardizing curve
so obtained.
35
546 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Lippmann's Capillary Electrometer
The essential feature of the Lippmann Capillary Electrometer
is a very fine capillary tube drawn from a thin glass tube, mounted
so that a column of mercury of variable height may be placed
over the meniscus of a liquid which rises in the capillary.
Fig. 345
Fi g- 345 shows how this is usually arranged : C is the capillary
attached to a vertical glass tube, AB, by means of a short length
of pressure tube at B. The side tube is connected by pressure
tubing to an adjustable reservoir, R, full of mercury. A scale
in millimetres is placed behind either AB or the reservoir R,
so that changes in level of the mercury may be measured.
The capillary tube, which dips into sulphuric acid, should be
of sufficiently small diameter to prevent the acid from being
driven from the tube by the mercury above it.
At the bottom of the beaker, E, which contains the acid, is a
layer of mercury, into which a wire, passing down the centre of
a glass tube, D, may be placed.
A second copper wire dips into the mercury above the meniscus,
either by means of a platinum wire lead fused into the tube AB,
or by inserting a wire into the mercury in AB or R. The two
leads are connected to a potentiometer which consists of two
Post Office boxes in series, R x and R 2 , so that, maintaining R t +
R 2 at 10,000 ohms, and adjusting R x and R 2 , any fraction of the
potential of the cell, E, may be applied to the junction of the
upper mercury and the acid.
MEASUREMENT OF POTENTIAL
547
If the meniscus is focussed and made to coincide with the
horizontal cross-hair in the focal plane of a high-power microscope,
it will be found that when a potential is applied in one direction,
the meniscus descends, and when applied in the other direction
the meniscus ascends. The effective value of the surface tension
of the acid mercury surface is altered by the applied voltage.
Suppose the reservoir, R, is lowered so that only a small
pressure is applied to the surface, and the meniscus is focussed
on the cross-hair of the observing microscope (which should be
of high magnifying power), the observed level does not corres-
pond to the true surface tension level.
At the junction of two liquids there is a contact difference
of potential set up, and the value of the effective surface tension
depends on this contact potential. In whichever direction this
contact potential acts, the result is a decrease in the effect
surface tension value. For two given liquids, i.e. sulphuric acid
and mercury, the value is fixed in direction and magnitude.
We may regard the surface of separation of the liquids as a
double layer functioning like a condenser, and having an energy
per unit area of £cV 2 , where c is the capacity and V the value
of the contact potential.
Thus, if T is the effective surface tension, and T the value of
the surface tension if no potential exists, we may write :
T=T -£cV2 (4)
so that whatever sign be given to V the value of T < T .
If a potential be applied to the surface of separation in the
same direction as the contact potential, then V increases and T
becomes less, causing the column of acid to descend to a new
equilibrium position, whereas, if the applied potential be of the
opposite sign to the contact potential, we obtain a larger value
for T, and consequently the acid rises in the tube.
S.-4A
- Fig. 346
The capillary when drawn out from the glass tube is always
slightly conical, and the state of things showing two positions
is seen in fig. 346.
If the applied E.M.F. opposing the contact potential increases
548 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
T becomes larger, until, when the applied potential is further
increased, we have the double layer effect again coming into
play, due to a net potential difference of the opposite sign.
For each value of the applied opposing E.M.F. the meniscus
will take up a definite position, and it is obvious that when the
contact potential is just neutralized, the meniscus will be at the
highest point.
The conical shape of the capillary tube, with resultant change
of diameter and of focus, renders unreliable the observations of
the level of the meniscus. To be sure that the diameter of the
tube at which the surface of separation lies is the same, the
meniscus is always observed at one point.
For example, as the value of T increases the level is brought
back to the previous one by raising the reservoir R and increasing
the pressure on the surface. The microscope used for observing
the level of the meniscus should be high-power and the reservoir
readily adjustable, so that the level of the mercury in R or AB
may be read for each value of the applied potential.
This process should be repeated for all values between o and
2 volts, or until electrolysis interferes with observation.
The values of the applied E.M.F. should be plotted as abscissae,
and the pressure in cms. of mercury as ordinates.
The form of the curve as obtained from the results under, is
seen in fig. 347. The maximum of the applied pressure^corre-
Fig. 347
sponds to a maximum, T, or if the value of T may be represented
in some such form as (4) the maximum T is T when the term
depending on the contact potential is zero., i.e. it corresponds
to an applied potential equal to the contact potential. From a
MEASUREMENT OF POTENTIAL
549
knowledge of the direction of the applied potential we may say
at once which of the liquid is electro-positive and which electro-
negative to the other. » t .
In performing the experiment the first thing to do, having
assembled the rest of the apparatus, is to draw out a suitable
capillary. After one or two attempts a suitable one will be
obtained, which is of sufficiently small dimensions to support
the pressure.
It should be noted that the capillary tube should be drawn
from a clean tube. This may be obtained by boiling the tube
in nitric acid and rinsing in tap water, or by leaving the tubes
to be drawn in a solution of potassium bichromate and sulphuric
acid for 12 to 24 hours and then rinsing in tap water.
The mercury also should be cleaned, preferably redistilled.
The following is a record of an experiment where the above
precautions were observed. ?
Rj + R 2 = 10000 ohms.
Ri
APPLIED POTENTIAL
APPLIED PRESSURE
ohms
volts
cms. Hg
o-o
227
1000
•2
25'3
2000
*4
26-8
3000
4000
•6
•8
27-2
267
5000
1*0
25'9
25-8
6000
1-2
7000
i-4
25'4
8000
i-6
24-9
9000
1-8
247
Contact potential from curve = -57 (fig. 347)- The negative
terminal was connected to the upper mercury, i.e. in the experi-
ment the mercury was electro-positive to the sulphuric acid.
CHAPTER XXII
MEASUREMENT OF CAPACITY AND INDUCTANCE
COMPARISON OF CAPACITIES OF CONDENSERS
(1) Deflection Method
Suppose Ci and C 2 are the condensers, of capacity K x and K 2
and that to each is imparted a difference of potential equal to
E volts. Let £>! and Q 2 be the respective charges on the plates.
I hen we have :
K —Sdl If _ Q*.
Kl - E ' K » ~ "E '
or El „ 2l / T ,
In the experiment described below the quantities, Q x and Q 2 ,
are measured by discharging the condensers in turn through a
moving-coil ballistic galvanometer. In such a case, if B x and 2
are the angles of the first throw of the ballistic galvanometer,
corresponding to movement of the spot of light, d x and d g cms.,
from the zero on the scale, we have (see page 479) :
q, - 1 . 1- ?tf 1 + *Y
Wl 7T G 2\ 2/
Q,_J .i.i»( I+ i\
*" 7T G 2 \ 2/
or $i = -i,
If the deflections are small :
e t _ 2 tan At _ tan 20 t _ rfi
2 ~ 2 tan 2 ~ tan 20 2 d a '
whence Ml = -i ^ _ - 1 , from (1)
Q 2 *2 K,
KT^iT- — (2)
550
MEASUREMENT OF CAPACITY AND INDUCTANCE 551
If the deflections are not small this approximation cannot be
used. The values of X and 2 may easily be obtained, for
tan 20! == y 1 ' where L is the distance from the scale to the mirror
of the galvanometer ; hence 20 x and B v
It often happens that with a steady source of potential, say,
a steady 2 volt accumulator, that the throw 6 X and a is too
large, i.e. E is too great. Under such circumstances, instead of
using E directly we may take any fraction of E by a potentiometer
method.
Connect E in series with two resistance boxes, R x and R 2 ,
making R t -f R a = 10000 ohms. From the ends of R t connect
up the leads which in the diagram (fig. 348) are shown directly
connected with E.
By adjusting R lf making R x + R 2 = 10000, we may obtain
a suitable fraction of E, for the potential drop through the
resistance, R 1( is
Ri
E. These resistances are then
Ri + R t
kept constant and the throws are obtained.
The scheme of connexions is shown in fig. 348. K x and K a
are condenser keys. K is a single-way tapping key, useful in bring-
ing the galvanometer to rest after observing the swings. BG
a ballistic galvanometer of the moving-coil type.
-\
<L_S
E
Fig. 348
K x is depressed and the condenser, C lt is charged. Kj is now
raised. The charge, Q lt is thus sent through BG, and X noted.
As soon as the galvanometer is brought to rest, the key K 2 is
depressed for the same short time and then raised, a being
noted. - During the interval of the two depressions of the keys,
the potential difference between the poles of the accumulator
will remain sensibly constant.
Of course the experiment could be done using one condenser
key, and placing C x and C 2 in turn in the single circuit.
The observations are repeated, say, four times with Cj and
three times with C 2 , and the mean values taken.
552 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
(2) Null Method, (de Sauty's Method)
For this method the two condensers, C t and C 2 , are arranged
as two arms in a Wheatstone net ; R x and R 2 , two adjustable
high resistances making the bridge complete. Fig. 349 gives
the scheme of connexions. G is a high-resistance galvanometer
and E a battery of cells having an E.M.F. of several volts (say,
4 or 6 Leclanche cells).
FIG. 349
The key, K, may very well be an ordinary condenser key, so
that when depressed the condensers are charged, and when
raised the condensers are discharged. The values of R t and
R 2 are so arranged, that when the key K is moved up and
down, charging and discharging the condensers, there is no move-
ment of the galvanometer coil, i.e. in both processes the potential
at B is the same as at D.
Let E 1 be the potential at A, and
v x be the potential at B at a time, t,
v 2 be the potential at D at a time, t.
The drop in potential along R x is (E 1 — v x ) and along R 2 is
{E 1 - v 2 ). Thus, the currents in R x and R 2 are ^-^ and ^"^
K t K 2
In a small interval of time, dt, the quantity of electricity
passing along R t is ( ~ Vl \ dt, and along R 2 is ( E * ~ ^2
In charging the condensers the quantity, / III!- 1 • dt, flows to C,
J Ri
and f ~ v * -dt to C r
dt.
MEASUREMENT OF CAPACITY AND INDUCTANCE 553
Now when the condensers are charged, since the potential
difference is E (that of the cells) we have :
/jri v
T5 L * &»
and similarly
K 9 E
/
Ri
E 1 - v,
R<
dt,
where K x and K 2 are the capacities of the two condensers ;
or K^E = /(Ei - v x )dt,
K 2 R 2 E = /(E 1 - v 2 )dt,
when no flow occurs through the galvanometer, i.e. v x =
throughout the charging,
i.e. /(E 1 - vjdt = /(Ei - v z )dt ;
or, KiRiE = K 2 R 2 E,
ie Ki-^1
K 2 -R x
In performing the experiment R x and R 2 may be obtained
sufficiently high by using a Post Office Box for each, and following
the scheme of connexions shown.
(3)
Fig. 350
(3) Method of Mixtures
Comparison of the capacity of two condensers may be made
in one or two ways using the 'mixture' method. In this particu-
lar method the condensers are arranged to have equal charges
*and the potential required to do this is measured or compared
by a potentiometer method. The theory of the method presents
no special points which require separate treatment.
554 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
A battery of cells (two accumulators) is connected as in
fig- 350 to send a current, c, through two high resistances, Ri
and R,, which are adjustable and connected together at a point, B.
The potentials between A and B, B and C, are proportional to
Ri and R 2 .
B is connected, as shown, to the two condensers whose other
plates are connected to the central cups of a cleaned ebonite
Pohl commutator from which the cross-connexions have been
removed (or a slab of clean paraffin wax with six holes full of
mercury will serve).
A galvanometer, G, is connected to the cups, X, X 1 , of the
commutator.
A and C are joined to Z and Z 1 .
When connexion is made (by the rocker of the Pohl commutator)
between Y and Z, Y 1 and Z\ the condensers, C t and C 2 , are
charged to the potential differences of AB and BC, respectively.
If Qi and Q % are the charges, and K t and K 2 the capacities,
we have :
K «=fir K - = fc- -(4)
c being the current through R x and R 2 .
When the switch is thrown over so that Y and X, Y 1 and X 1
are connected, the condensers are discharged through G.
Suppose the current flow from A to B to C in the potentiometer
circuit, then the inner plate of C x is at a lower potential than
the outer, when charged, whereas for C a the inner plate is at
a higher potential than the outer. So when the condensers
discharge through G, C 2 will send its charge in opposition to C v
The above process is repeated, altering Rj and R 2 between
observations, until finally, a value of R x and R 2 is obtained,
such that on discharging, no current passes through the galvano-
meter. Under such circumstances, Q t = Q 2 .
From equations (4) above we have :
Q 1 = K 1 R 1 c, Q 2 = K 2 R 2 c.
In the adjusted position, since Q x = Q 2 ,
KjR^ = K^,
K, R 2
or — - = — =•
K 2 Rj
Determination of the Absolute Capacity of a Condenser
The value of the capacity of a condenser may be determined
in any system of units if a measured potential, E, in these units,
MEASUREMENT OF CAPACITY AND INDUCTANCE 555
applied to the condenser imparts Q units of charge in the same
units ; for the capacity, K, of the condenser is denned by Q =K-E,
or K=2 (5)
For example, if Q is expressed in coulombs and E in volts, K»
the capacity, obtained in the equation above is expressed in
farads. This can be transformed to micro-farads since this unit
is 10- 6 of the farad.
The usual method of finding K experimentally is to apply a
known potential (in volts) to the condenser and measure the
charge, Q, by discharging the condenser through a ballistic
galvanometer.
The moving-coil ballistic galvanometer is the best form to
use, as in most capacity experiments, whence if a! be the first
observed throw of the instrument, due to the discharge, we
have (see page 479) :
or, if the method of page 480 is used, the value of x is not obtained
so fully. If ax and a 8 are the first and second displacements of
the spot of light on the same side of the zero, we have :
it 9 2 \a 3 /
when <p is the angular deflection produced by a steady current c.
L®J
K
ISP
K:
' ^ ■
K*
£5
»1
R<
E
Fig. 351
For small displacements of <5, d lt and d 3 cms. on the scale corres-
ponding to <p, a, lf and a 3 , we have :
O = - .-.^±( dl
djf
dj
(6)
556 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Fig- 35 1 shows the scheme of connexions for such an experiment.
K is the condenser whose capacity is to be determined ; R x and
R 2 are resistance boxes, introducing a high resistance to the
circuit ; AB a resistance box of decimals of an ohm, r say ;
K 1 a single-way switch ; K 3 a tapping key, whereby the galvano-
meter may be brought to rest ; K 2 a condenser key ; and BG a
ballistic galvanometer of the moving-coil type. c-
The circuit, R 1} R 2 , and the battery, E, serve in the first part
of the experiment as a potentiometer.* R x and R 2 have a
constant sum of about, say, ioooo ohms. By adjusting these
two values, keeping (R x + R 2 ) constant the potential difference
between B and C may be made any desired fraction of the
potential, E, of the battery. This adjustment is carried out,
and the deflection produced in the galvanometer when jthe
condenser is afterwards discharged through it is noted.
Since R t + R a is high and r is never greater than I ohm, the
potential between BC = = — -~-*'E=y volts, say.
Ri + K 2
When K 2 is depressed, the condenser is charged, the potential
applied being V. The charge, Q, will flow through the galvano-
meter causing a deflection, d x cms., when K 2 is raised.
This process is repeated, say three times, and a mean value
of & x obtained for the fixed potential, V.
For each discharge, the reading corresponding to the second
deflection, d 3 , on the same side as d x , is measured. This
provides the data for the damping correction for the
galvanometer under the identical conditions under which oc 2
is measured.
The time of swing, T, is obtained by timing, say, 20 swings
of the needle with a stop-watch. It is most convenient to use
the position of rest of the spot of light as a reference point of
such counting and timing of swings.
To standardize the galvanometer a steady current of known
magnitude must now be sent through it and the scale deflection,
8, measured.
The key, K lt is now closed. The current from the battery E
(the same 2 volt accumulator throughout), now passes through
R 2 and R x , and then through the galvanometer, shunted by the
small resistance, r.
The resistance of the galvanometer and shunt is :
rG
G +r
where G is the resistance of the galvanometer, t
t G may be obtained by including a resistance between B and BG, and adjust-
ing till the deflection is reduced to half the value, when G is obviously equal to
this resistance.
MEASUREMENT OF CAPACITY AND INDUCTANCE 557
Hence, if E is the potential difference in the circuit, the current
in the main circuit, *, is :
E
R 1+ R,+B+Jg;
where B is the resistance of the battery. This may be neglected
when the latter is an accumulator, and — ^is > 1 ohm, and is
also negligible compared with R + R 2 -
Of this current, the value of the part, c, through the galvano-
meter is :
Er
=
(r + G)(R 1 +R 2 )
We have also seen that the potential applied to the capacity is
V = 5i — • E.
R x + R 2
Hence, substituting this value of V and Q from (6),
K=2
V
E-r
2* (r + G) (R, + R 2 )
d JL(<h\*
Ri
Ri + R 2
^ILftA* ( r V
27T d \dj XRfi)
if r is negligible, cf. G. E is assumed constant throughout the
experiment.
Z
Fig. 352
Measurement of Capacity, using a Fluxmeter
A simple method of estimating the capacity of a condenser
is illustrated in fig. 352.
558 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The condenser, C, of capacity, K x , is connected to two condenser
keys, Si and S 2 . The lower studs of the keys are connected to
the mains of the electricity supply (i.e. to 100-200 volts). When
the keys have been depressed the condenser is charged to that
potential, V. On releasing the keys the charge on the condenser
is discharged through a low resistance, r, which is shunted across
the fluxmeter, F (the position shown in the diagram).
r is made small, about ^ to ^ ohm is suitable, and the shunt
may therefore be regarded as taking the whole of the discharge,
say, Q units.
The potential difference at the ends of r varies from V, to zero.
i.e. the current through r at any instant when the potential is E
E
is — and the total quantity of electricity passing through r is
therefore : Q = f— ■
Now, if the resistance of the fluxmeter, usually about 30 ohms,
is large compared with r, the instrument gives a deflection which
is a measure of /E • dt (see page 484). Let this deflection be
x divisions or x x 10 4 maxwells,
o — x x iq4
W — f
If now r is expressed in ohms, the units must be made consistent,
x x 10 *
i.e. a; xio* maxwells should now be written ^ — practical
units to obtain Q in coulombs,
i.e. Q (in coulombs) =
If the potential applied to the condenser is measured by
means of a voltmeter, the capacity in farads may be readily
obtained.
The Effect of Inductance and Capacity in a Circuit Conveying an
Alternating Current
In many of the subsequent methods of finding the self -induct-
ance, mutual inductance, or capacity in a circuit, alternating
current is employed and a system of non-inductive resistances,
capacities, inductances are arranged to produce a balanced
Wheatstone net, the usual galvanometer being replaced by a
telephone as detector. In such circumstances it is not a difficult
matter to find the relation between the values of the resistance,
etc., when such a balance is obtained. The majority of cases
may be solved by precisely the same method.
MEASUREMENT OF CAPACITY AND INDUCTANCE 559
Consider an alternating potential E cos pt to be applied to a
circuit, of resistance R, self-inductance L, and containing a
capacity K (fig) 353), (p is, of course, given by
27m = pt
where n is the frequency of the alternation).
'Ofl'OWOO N — —
I
B
At/^
E co* yt
Fig. 353
Now E^ = E (cos pt -h i sin pt),
where i = V^i a* 10 ^ tne rea ^ ^ axt °* E^*' expresses, therefore,
the alternating potential applied.
If c is the resulting current at any instant the net applied
potential = Re,
Lie Q
i.e. Re =E cos^--^--^,
when Q is the charge on the condenser at any time, t. Further,
we have :
c ~ dt'
dc
It
£2.
dt*
** dt* + dt + K y
Hence : L ^ + R 3? + ^« = real of E^*',
If Q = A-e*< , i.e. § = #Ae*< , ^ - - p*A& ,
we have, where A is a constant,
- L£ 2 Ae*' + R# Ae*< + g Ae* = E e*';
or
and
A =
E (
Rip-Lp*+±
Q
Epg^
R#-I4> a +^
560 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
or
i.e.
c =
dQ ipE^ 1
dt Rip -Lp* + ±
E^
c =
R + i( L p-^)
Ea e»<\R-i(Lp-±)}
(7)
R8 + ( L ^- ^)
. E e» f
R
if
i R2 +(^-^) a r^i Ra +(^-i) 2 }
R
1 7 FTm* = cos *
: R ' + ( L *-^)I
R 2 +
(*-b)
Fjl = sm a,
we have
c = -
E n e*'
The real part of this is]
^ E cos (pt — a)
"> ,+ ( i *-i5) , r
(8)
i.e. the current lags behind the applied E.M.F. unless a = o,
i.e. cos a = I,
MEASUREMENT OF CAPACITY AND INDUCTANCE 561
i.e. unless R = \ R 2 +
(*-*)'}'
or Lp = =jr-, or p =- .
F Kp' r VK-L
Thus we see that in the general case taken the method gives
the value of the current at any time.
From equation (7) we see that the maximum current C is equal
to the applied electromotive force divided by ( R + Lip — — Y By
analogy with the simple Ohm's Law equation ( R + Lip — J
may be termed the ' quasi-resistance ' of the circuit, and be used
in the same way as ordinary non-inductance resistance values,
when alternating electromotive force is applied to a circuit.
For example, consider a Wheatstone net as illustrated in
fig. 360. If a direct current were applied at AC and the net
balanced we have :
Resistance AB _ Resistance AD
Resistance BC ~~ Resistance DC
Now for alternating current applied using the quasi-resistances
in the corresponding arms :
i
r s + Lip R _ *
K^
which gives on rearranging and equating real and imaginary
quantities, thus :
(real) r x R = M or L = K^R,
Kjp
(imaginary) ]0 = 1T^ or K i r i = K * r »
a result which agrees with the fuller treatment given on page 575.
The above method of investigation assumes that a true sine
E.M.F. is applied. Using a ' hummer ' or the secondary of a
small induction coil does not produce such a simple form of
E.M.F., but would be the resultant of a series of such simple
wave forms.
Such an E.M.F. would be satisfactory in those cases where
the end result, as in the case taken above, is independent of the
frequency or p.
36
562 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
In cases where p is involved in the result, a timed telephone,
which responds to one frequency only should be used. As an
alternative, a vibration galvanometer such as that designed by
A. Campbell (" Physical Society of London Proceedings," 1907,
page 626). As such methods are not very much used, they are
not included in this account.
MEASUREMENT OF SELF-INDUCTION
The measurement of the coefficient of self-induction of a coil
may be carried out in several ways, of which the following
methods are representative.
Some of these methods are very tedious as they involve a
double balance of a bridge, for steady and variable currents.
Perhaps the most unsatisfactory methods from this point of
view are (1) the direct comparison with a standard self -inductance,
(2) Maxwell's method. Rimington's method, too, is apt to
be tedious. The methods which have been found to be most
satisfactory in the laboratory are Rayleigh's, Anderson's, and
Owen's. These methods are therefore used generally when a
choice of methods is available.
Fig. 354
(1) Rayleigh's Method
In this method the inductance, L, is placed in series with a
small variable resistance, r, and arranged in the arm, AB, of
the Wheatstone net, as shown in fig. 354. The remaining arms
are of the same order of magnitude as the resistance, AB. G is a
ballistic galvanometer preferably of the same order of resistance ;
K x and K a are keys. The resistance, r, is reduced to zero and'
the network is balanced for steady currents by closing the key,
K 2 , before making the galvanometer circuit through K x .
MEASUREMENT OF CAPACITY AND INDUCTANCE 563
To obtain an accurate balance, it may be necessary, when the
arms are equal, to introduce a smaller resistance than is available
in the Post Office Box which provides the three resistances, r 8 ,
r z and r A . This may be effected by having a length of platinoid
wire in series with L, and adjusting the length until an accurate
balance is obtained.
If now a small E.M.F., E, is introduced in one arm, a current
which depends on E will pass through each of the other resistances
in the network.
Thus, if the battery key is closed, K a being meanwhile closed,
dc
an E.M.F. of magnitude L-y is established in the arm, AB,
resulting in a current in each of the arms in the net. Let the
current in the galvanometer branch, BD, be k • (Lt), where k is-
a constant which depends on the value of the resistances.
Under such circumstances the total quantity of electricity
which passes through the galvanometer due to this cause is :
/* dc
kL -j t dt = kLc , „ (9)
where c is the maximum steady current flowing through AB.
This quantity, Q, may be calculated from the observed throw
in the ballistic galvanometer, and thus L is obtained in terms
of the constants of the galvanometer, k and c .
Assuming that a moving-needle galvanometer is employed
we have, page 477,
Q -v i • «i (*+!)- «*■ ™ (I0)
where x is the logarithmic decrement.
To eliminate k and c , a measurable small potential charge
is introduced into the arm AB. This is brought about by
adding a small resistance, r, to AB. (r should be not greater
than g£o r 1 , usually j^ ohm does very well.)
Assuming that the current, c , will not be materially affected
by this small charge, the potential introduced in the arm AB
is c& : this causes a current, kc^/m. the galvanometer, producing
a steady deflection, r
Then G being, as above, the galvanometer constant, i.e. the
field strength at the centre of the coil due to unit current, we
have :
kc rG = H tan Q lt
, H tan 0« .
or kc = ~x • * •
G r
564 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Substituting this value in equation (10) above, we have :
0/ X^
rp sin -j
L = r
T sin M I+ ^
n tan0j
If r is expressed in ohms and T in seconds, L is in henrys
(io 9 C.G.S. E.M. units).
If a moving coil type of ballistic galvanometer is used and
r ? and r 4 or r x and r 3 are small, the galvanometer may give but
a small deflection as it is shunted by these resistances. This
is the case when the resistance of L is small. Measurable and
reliable results can be obtained if the galvanometer circuit is
broken the moment the discharge has passed through it. This
condition is most conveniently brought about by using a single
key as shown in fig. 298 on page 481, for both battery and
galvanometer circuits.
The three brass strips, A, B, and C, are insulated and connected
to separate terminals. When C is depressed contact is made
between C and B, then A and S, but not between B and A which
are separated by an ebonite stop. S and A replace K 2 ; C and B
replace K 2 , when C is pressed down and a steady current flows
in the circuit, the galvanometer which is in the circuit shows
no deflection for a balanced bridge. On releasing C, the bottom
contact is first broken (i.e. the battery circuit), and a very short
interval of time afterwards C and B are separated. This interval
is sufficient to allow the impulse to be given to the galvanometer,
coil which, due to the separation of B and C, swings without
excessive damping due to induced currents.
If the moving-coil instrument is used in this way, the end
result is slightly modified, for in this case (see page 479)
where is the corrected value of the first throw for no damping,
i.e. is the observed throw X f J.
Now for a steady current equal to kc^ the couple on the coil
is GkCff, so that
Gkc r = T0J,
t being, as above, the restoring couple in the suspension, per
unit angular displacement. Combining the last two equations
we see that
n 2 * 0i "
MEASUREMENT OF CAPACITY AND INDUCTANCE 565
(2) Comparison with a Standard Inductance
To effect a direct comparison of an unknown inductance, L^
with a standard inductance, L 2 , the arrangement of fig. 355
may be employed. /
In series with L x is a non-inductive resistance of a variable
Fig- 355
magnitude, say a length of manganin wire, which is adjusted
so that the total resistance included between A and B is r x ohms.
By means of a key, S^ the galvanometer, G, is included in the
circuit to which a direct current from a 2 volt cell is supplied via
key, S 2 , and a balance is obtained for steady currents, i.e. the
balance is obtained by closing the battery circuit first, and
when the current is established, the equality of potential at
B and D is tested by closing the galvanometer circuit.
Having balanced the bridge for steady currents the balance
is now tested for variable current. The first steady balance will
not be one which also is in adjustment for variable currents.
The resistance in series with L x is given another value and the
steady current balance obtained once more for the net, which
is once more tested with variable current. This process, which
is often a lengthy one, is repeated until the resistances, r v r 2 , r t , / 4
are of such magnitude that the points, B and D, are at the same
potential both with steady and varying currents.
It will be shown that under such circumstances
Li = ri = U,,
The variable current may be obtained by breaking and making
the battery circuit; or by switches, S x and S 8 , a telephone ear-piece
566 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
may be made to replace the galvanometer, and an alternating
current from a buzzer supplied instead of the direct current used
in the steady balance experiment.
Assuming an alternating E.M.F. be applied to AC and the
network adjusted so that the potential at B is the same as that
at D, at all times we have, using in the Wheatstone relation
the ' quasi-resistances ' of the arms :
Ljp + r x = Tj
L/p + r z r 4
Equating the real parts we have :
»V« = r 2 r 9 , or £ =~ 3 , (n)
a condition which was fulfilled by first balancing for steady
currents.
Equating imaginary parts of the equation, we find
T y *»
or _-L=— 9 — _J i, from (n).
(3) Maxwell's Method
The arrangement of resistances and the inductance to be
measured is shown in fig. 356. The method involves a comparison
of L, with a capacity of known value. In the present method
the resistances r lt r t , and r 8 , together with L, are made into a
balanced Wheatstone net, using a cell and galvanometer in the
usual way. In parallel with r x is placed a condenser whose
capacity, K, is known. During the process of balancing the
resistances for steady currents, care is, of course, taken to ensure
that the current is established before the galvanometer key is
depressed to test the balance. The experiment may now be
completed either with the aid of the galvanometer across BD
and the cell connected through a switch to AC, or by means
of a telephone ear-piece as detector and an alternating E.M.F.
applied from a hummer.*
If the galvanometer is retained, the current is made and broken
whilst the galvanometer is permanently in the circuit, and
r % and r 8 are adjusted until no deflection is obtained in the
galvanometer.
If the alternating E.M.F. from a hummer is used, the resistances
are adjusted until no noise is heard in the telephone ear-piece.
In either case the steady current balance will, in general, be
upset. This must be again obtained and then the test repeated,
until finally the balance is equally good for steady and varying
currents.
* It should be noted that the galvanometer and steady E.M.F. may be dis-
pensed with in experiments (2) (3) and (5). Alternating current may be used
alone. The first adjustment is to reduce the noise in the phones to a minimum,
MEASUREMENT OF CAPACITY AND INDUCTANCE 567
When the network is balanced for steady and alternating
currents the relation between the four ' quasi-resistances ' may
be applied.
K
B
Fig. 356
Since the arm, AB, consists of — -r=~ and r x in parallel, the
effective ' quasi-resistance ' is
1 -f- ipKr x '
whence :
"\
r,(i + ipKrJ r 4 + Lip '
where r 4 is the total ohmic resistance in the arm, DC.
Equating real quantities in the above equation :
r*r« == y,r^ or -^ =— *,
2'3
(12)
a condition which was experimentally established in the steady
current balance.
Equating imaginary terms :
r % r z pKr x = r t Lp,
or L = r 2 r 8 K,
or from (12) above = r jftK,
from which L may be calculated.
(4) Rimington's Method
A method due to Rimington enables a determination of L to
be made in terms of a known capacity, making use of a modifica-"
tion of the Maxwell arrangement. The self-inductance is placed
and the second to reduce this minimum to zero. N.B. — The value of the resistance
for intermittent current is only the same as that for alternating current when the
frequency of the latter is small.
568 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
in the arm, DC (fig 357). The capacity, K, is placed in parallel
with a variable portion, r, of the resistance r x (AB). The network
is balanced for steady currents, and then the point, E, is chosen
such that when the battery circuit is made and broken no
deflection is produced in the galvanometer. This method has
the advantage that in the second balance the adjustment of the
resistances, r x , r s , r s and r 4 , is not disturbed.
When the above conditions are fulfilled we have the following
relation between the physical quantities involved.
L = K — - henrys.
Fig- 357 shows the arrangement of apparatus. For this
experiment it is not possible to use a telephone and alternating
current, as will be seen by working out the case by the general
method given ; the conditions required for balance with such
currents reduces the method to Maxwell's, i.e. r x — r = o.
To establish the formula given above for the case of a balance
with intermittent direct current, let c x be the steady current
in AB (resistance, r x ) when the key is closed ; c 4 the steady current
in DC. Then when the circuit is broken, the quantity of elec-
Lc . /*° hdc
tricity which passes through the inductance is -~- i.e./ ^-,
where R is the resistance between the points D and C (r 4 ) together
with the resistance of the rest of the net, i.e.
R = * 4 + r 2 +
G(r 3 + r x )
where G is the resistance of the galvanometer.
MEASUREMENT OF CAPACITY AND INDUCTANCE 569
The fraction of this quantity which passes through the galvano-
y i +r 3 of LC4
r x + r 3 + G R
i.e. is
which reduces to
LC4K + r z )
G(r 3 + r t )
G + r 3 + r x ,
Lc 4 r x
('i + r z + G)
(13)
'i('i + '3) + Gfo + r 2 )
Also when the circuit is broken, the condenser discharges a
quantity of electricity through the galvanometer, in the opposite
direction to that due to the inductance.
Now the quantity of electricity on the condenser due to the
steady current, c v in AB is Kfrcj).
The part of this quantity passing through the resistances
other than r is :
Krc 1 . _ . -
ri + f » + LG+r, + r;
Of this amount, that passing through the galvanometer is :
Kr 2 ci x Jr 2 + rj
r + r -I G ( y 2 + y 4) (G + r. + ftf-
1 8 G+r 2 +y 4
This reduces to
Kr 2 Cjr 2
(14)
' 2 ( r t + r 3 ) +G(r! + r 2 ) '
when substitutions are made using the fact that r#i — r 2 r z .
When the quantities represented by (13) and (14) are equal
and opposite, the galvanometer will be unaffected by making
or breaking the circuit by means of the key shown in the figure,
i.e. K> 2 CjT 2 = Lc^j,
or
We may further express — in terms of the resistance for when
the currents have acquired the steady values, c x and c 4 ,we have :
£1 = *z + U .
c 4 ?i+r 2 '
'1 r 2 r x + r 2 c A r 2
Hence L = ^-^
57© ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Experimental Details
The chief disadvantage to this experiment is that much time
may very easily be spent in a fruitless effort to determine the
value of an inductance, unless the order of suitable resistances,
etc., to balance the network, is known. If, however, the approxi-
mate magnitude of L is known we may, as in the following
example, obtain this information.
Example
An inductance (of 98*15 ohms resistance) is to be evaluated,
using a capacity, one micro-farad.
Now from above we have :
L = K -^,
and the inductance is known to be of the order of one henry,
i.e. y 4 is of the order 10 2 ohms.,
K is of the order 10- 6 farad ;
10- 6
r 1 = —
so
x io 2 r 2 = io _4 r a «
i.e.
r is a fraction of r v or r = ar lf where a is not greater than 1,
Zj__- -rn-4
2*
10*
a 2 r,
The arm, AB, of the figure could be therefore two Post Office
boxes in series. To obtain a subdivision as at E, however, it
will be convenient in this case to use three Post Office boxes, or
boxes having a similar range of resistances, for this arm of the
net, as in fig. 358. The condenser, K, is placed in parallel with
boxes 1 and 2 as shown.
MEASUREMENT OF CAPACITY AND INDUCTANCE 571
The balance for steady currents having been obtained, the
value of r is varied by taking out plugs from 1 and 2, and inserting
the same value in 3, thus keeping r x constant and of the value
required fpr a steady current balance, until r is finally obtained,
such that the potential drop along AE is such as to cause a
current through the galvanometer equal and opposite to that
due to the inductance. Values of r too small will be shown by a
movement of the galvanometer to the one side due to the induct-
ance effect predominating, whereas when c is too large the
capacity effect will be large and cause a deflection in the opposite
direction.
This process was employed to find the value of L for the coil
mentioned above, and the following values obtained,
r x = 20000 r z = 1963, r = 14500
r % = 1000, r t = 98*15, K = 1 micro-farad,
K-lxiHS'xio-xgiis _ x . 03 henrys-
whence L =
20000
The self -inductance was a nominal 1 henry.
(5) Anderson's Method
The scheme of connexions shown in fig. 359 illustrates a very
convenient method of finding the value of an inductance in
terms of capacity and resistance.
Fig- 359
The Wheatstone net, r t , r 2 , r 3 , r 4 (r 4 being the total resistance
of the arm, DC, including the self-inductance) is arranged as
shown. The condenser, K, is connected from A to a variable
point, E, in a resistance r (BE).
572 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The network is first balanced for steady currents using a
battery between AC and a galvanometer, G, between ED. For
such steady currents K and r do not have any effect. When a
balance is obtained, by first depressing the battery key and then
the galvanometer key, we have :
^i = - 3 .
The network must now be balanced for varying currents.
This may be done in one or two ways : either by using the
galvanometer and an intermittent direct current, or an alternating
current and telephone receiver. If the galvanometer is used
as a detector, a tapping key is inserted in the battery circuit
and r is so adjusted for the particular value of K, that on making
and breaking the circuit no deflection of the galvanometer is
produced.
When this condition holds, it may be shown, by the application
of Kirchoff's Laws to the circuits, ABED A, ABCDA, that
L = K{fy, + r(r,+r«).}
The more convenient way of proceeding with the experiment,
however, is to replace the galvanometer by a telephone and
the battery by a source of alternating current. In the general case,
when these changes have been carried out to conform with fig.
359 a note will be heard in the ear-piece when AE is closed.
By adjusting r (the resistance BE) this note may be reduced
in intensity to zero.
We may readily find the relation which exists between the
various resistances, L and K, under such circumstances. The
condition satisfied is that the potential drop between A and E
is the same at all times as the potential drop between A and D.
Let the currents in the arm, AB be c- l ; in AE, c 2 ; and in
AC, c. Then, when no current passes in either direction along
ED, the magnitude of the current along BC is c x + c 2 . Consider
the circuit ADC, and let the applied potential be taken as the
real part of E e*', The potential difference between A and
D is r z c, where c is the solution of
dc
E e*' = r z c + Lj t +r 9 c.
Putting c = he*** in the above equation we have :
E
U) +
E e^
A = E o ;
whence c = -. ; — » , T . . »
and potential difference between A and D is therefore
E e**
r a +r] + Lip' r * ••' ( I5 >
MEASUREMENT OF CAPACITY AND INDUCTANCE 573
Now let Q be the charge on the condenser, K, at any instant.
From a consideration of the circuit, ABC, we have :
E^*' = r x c x + r 2 (c x + c 2 ) (16)
But the potential drop along AB is the same as the drop
along AEB, or
Q / *
ClTl = K + c *' " * ' ^
Substituting for c x (17) in (16) and putting c 2 = -J^, we have :
whence as before :
Q *£L
from which the potential difference between A and E ( ^ J is
p Eof^ (18)
■( I+ &)+K*(, i+ a + r )
The condition specified above for no current through the
telephone is that (15) and (18) are the same for all values of t.
Equating these expressions we have :
('. + U + Lip) E«** = f ,|(i + £j) + ^(fif .+Wi+frJ JE^';
whence equating the real parts we have :
r. + ,.-,.(*±r,).
or — = — ,
a condition which is assumed by the initial balance for steady
currents.
Equating the imaginary quantities we have :
Lp = ^Kfor, + rr x + rr 2 ),
or L = K{r z r s + r(r s + u)} . (19)
This result could have been obtained using the quasi-resistance
method already outlined. For we may consider the circuit taken
574 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
above and write down the complete ' Ohm's Law Equation ' and
find c,c x and c 2 and hence the potential drops, AE and AD, and equate.
Thus, for circuit ADC :
E/»<
'* + y 4 + L #
ial drop, AD =
Circuit ABC
and
Potential drop, AD = r * B ° e *\ (20)
r z + r * + ^P
E e** = e 1 r 1 + (c s + cjr s (21)
By Kirchoff' s Law since the E.M.F. in circuit, AEBA, is zero,
Substituting in (21) above,
whence
and since ^f=*c Xi we have, integrating, Q = ^ and the potential between
AE, which is g., is
XV.
EI**'
.(22)
Equate to the value given in equation (20), we have, equating the
real quantities :
'* u
and equating the imaginaries,
I* = K{r s r t + r(r t + r 4 )}.
(6) Owen's Method
The following method of finding the value of the self -inductance
of a coil was described by Dr. D. Owen in the " Proceedings of
the London Physical Society," 1914-15, vol. xxvii.
The coil whose inductance, L, is to be found is placed in one
arm of a Wheatstone network, in series with a variable non-
inductive resistance, so that the total resistance is r 2 ohms.
The scheme of connexions is shown in the diagram (fig. 360).
K x is a standard capacity, K a a second capacity, and R an
adjustable non-inductive resistance.
MEASUREMENT OF CAPACITY AND INDUCTANCE 575
It will be noticed that in this method there is only alternating
current supplied to the bridge. In the first case the resistance
R is made equal to zero, and the resistance r 2 is adjusted so that
a minimum intensity is heard in the telephone. R is then
gradually increased until this minimum is reduced to silence.
Under these circumstances the relation
L = K^R^K^R
is found to hold.
Fig. 360
Now it was shown on page 561, as an example of the ' quasi-
resistance ' method of solving such problems, that when such a
balance is obtained
L = K^R = K> 2 R.
As a contrast with the method given on that page, we may
show that /the above expression gives the relation between the
capacity and resistances used, from a fuller consideration. For
let the applied E.M.F. be E^*'; and, since no current flows
through the galvanometer in the balanced state, let c x be the
current in ABC.
576 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
For the circuit ABC we have :
E^ = Cl (r 1+ r 2 )+L^;
hence, putting o x = Ae*' and substituting to find A, we have :
Ep^
'* ^ + r 2 + Lip'
The potential difference between A and B is r x c x or
/jE^
r% + U + L#
Now, if c 2 is the current in ADC, we have for that circuit :
E^ = Rc 2 + Q(^+^-),
dQ
it
(23)
or, since -jr= c lt
E ^= R f+Gt + r>
gives the following
Hence putting Q = Be* 1 and -^ = B#d?*' we find B, which
Q **
The potential difference, at any time, t, between A and D, is
therefore ~-
_x_ E g»*
The condition that there is no sound in the telephone is that
at all times, t, the potential at B = potential at D, i.e. for all
values of t :
i.e. r&^Rip + (± + i) | = (r x + r t ) + Up.
Equating real quantities :
' iK ' it + r.) -'• + '•
MEASUREMENT OF CAPACITY AND INDUCTANCE 577
Ki + K 2 _ r^jfyg
K 2 r x *
K 2 tr
or rj&i = rz& 2
Equating imaginary quantities :
or L = /iKjR or r 2 K 2 R
MUTUAL INDUCTANCE
Direct Measurement with a Ballistic Galvanometer
The coefficient of mutual induction of two coils may be denned
as the number of lines of magnetic force which pass through one
coil when unit current circulates through the other.
Thus, if a current of maximum strength, c , passes through
one of the coils, whose mutual induction is M, Mc lines of mag-
netic force thread the second ; and whilst the current grows
in the primary, the number of lines of magnetic force threading
the secondary is changing. Therefore an induced E.M.F. is set
up in the secondary during the time of growth of the primary
current. This E.M.F. is numerically equal to the rate of change
of the number of lines of magnetic force in the secondary, i.e.
= — (Mc)* where c is the instantaneous current in the primary
at
during the growth of that current.
If L is the coefficient of self-inductance of the secondary coil,
and c 1 is the current in the secondary corresponding to c in the
primary, we have a further E.M.F. in the secondary due to the
hdc 1
self-inductance numerically equal to -tj—> i- e - if R is the total
resistance of the secondary coil circuit, neglecting signs,
Rcl = L Tt + M Tf
Now Q, the quantity of electricity passing through the second-
ary, is / cHt where the integration is carried out over the whole
time during which c rises to the steady value c .
Q =jcHi -=f~dc^ +f~dc.
* N.B. — This only applies to coils with non-magnetic cores. For if there is
an iron core the value of the flux is not proportional to the current.
37
578 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
A
= and
rfc-"*
The value of c 1 at the commencement and the end of this
integration is zero, hence :
!*•■
If, therefore, the second coil is connected to a ballistic galvano-
meter and the throw is 0, due to the passage of this quantity of
electricity, M may be calculated in terms of R, c , the constants
of the galvanometer, and 0.
In this case it will be well to calibrate the galvanometer in
the circuit since R includes the resistance of the galvanometer
and the second coil.
G
<2>
Fig 361.
Fig. 361 shows a convenient disposition of apparatus to carry
out a direct measurement of M on these lines.
The current in the primary coil may be regulated to a suitable
value by means of the resistance R x . In series with R x is a small
resistance r(A»). C is a four-segment commutator, or may be a
switch consisting of four mercury filled holes in a block of paraffin
wax.
K 1 and 2 are connected together the ballistic galvanometer
is in direct circuit with the secondary coil, and when K is depressed
a deflection, 0, is obtained for the establishing of the steady
current, c , in the primary. If a moving-needle ballistic galva-
nometer is used, we have
T
H . 0„ Mc
G 2
Tf-
(25)
where d is the value of the first deflection corrected for damping,
i.e. O = f i + - 1 where x is the logarithmic decrement and
is the observed throw.*
* If a moving-coil instrument is used, the form of double key, described on
page 481, should be used ; K is replaced by the lower pair of contacts in such a
key and the upper pair act as a key in the galvanometer circuit.
MEASUREMENT OF CAPACITY AND INDUCTANCE 579
If now C is arranged so that connexion is made between
1 and 3, 2 and 4 only, and a steady current, c , is passed in the
primary circuit, the potential drop established at the ends of r
is C&, i.e. the current through the galvanometer is -^ » smce r is
very small compared with the resistance of the galvanometer.
If this causes a steady deflection, lt we have :
G. c ^=Htane 1 , (26)
as shown on page 481.
Combining (25) and (26), we obtain the following value for M :
it T Smi
If a moving-coil galvanometer must be used, since Q =-• ^ • —
the value for M becomes :
M =
J . 1 . 1°.
w 2 2
It will be noted that the value of R is eliminated : r may well
be a standard ^ or jf ohm. T is obtained in the usual
way by timing 20 or 30 swings, under the conditions of damping
which o.btain during the observations above ; O = 6 1 1 + - j
The value of x may be obtained by one of the methods given
on page 480.
Comparisons of Mutual and Self-Inductance. (Maxwell's Method)
The one coil of self-inductance, L, is arranged in one arm of
a Wheatstone network ; the other coil of the mutual inductance
is connected, as shown in fig. 362, in series with the battery.
For this method the resistance, r (AC), is not present. The coils
are arranged by trial so that the self and mutual induction effects
in the arm AB are opposed.
The resistances, r lt r t , r z , r 4 , are balanced for steady currents.
The battery circuit is then made and broken, the galvanometer
circuit being closed. r z , r % , r 4 are adjusted until the minimum
effect is produced in the galvanometer for such current changes.
The steady balance is again tested. By repetition the values
r v r z , r 4 , may be finally arranged such that for steady and varying
currents the galvanometer remains unaffected.
58o ADVANCED PRACTICAL PHYSICS FOR STUDENTS
The steady current balance gives in the usual way the relations
between the resistances,
i.e.
During the fall of the current at break, the self-inductance
dc
effect is an E.M.F. L^- 1 , where c t is the current in the arm ABC.
B
s
In the opposite direction an E.M.F., M ^-, is set up in the same
arm due to the mutual inductance. If these two E.M.F.'s are
equal and opposite,
L-^ = M jt (numerically),
where c = c x + Cj,
Lc x = M(c x + c 2 ),
L c x +c a
M c x '
-* + '?
'4
It is apparent from the above equation that the experiment
may only be performed if L is larger than M.
MEASUREMENT OF CAPACITY AND INDUCTANCE 581
This method is open to the same criticism as the Maxwell and
comparison methods of finding L, and is therefore seldom used.
A modification of this method is obtained by the insertion
of the arm, AC, of a variable resistance, r, as shown in fig. 362
With K 2 open a steady current balance is first obtained.
This having been accomplished, the resistances, r v r 2) r z , r 4 , are
fixed ; K 2 is now closed and r adjusted until the galvanometer
is unaffected by intermittent current in the network, e.g. when
the current is made and broken or reversed.
As before, no resultant E.M.F. is set up in AB during such
changes of current, otherwise a current proportional to such
E.M.F. would pass through the galvanometer. Thus we have
again :
I^i =M^ (numerically),
at ctt
where c is the current passing through the coil M, i.e. is
{c x + c 2 + c 3 ).
Integrating as before, we have :
LCi = Mc,
L _£_
1,e# M c/
Now if R is the resistance of the whole net between A and C,
we have :
cR = c 1 {r 1 + r 2 )
L r x + r 9
M~ R '
= r(r t + r t ) + r{r 3 + u) + {r x + r t ) (r 3 + r 4 )
r{r 9 + rj
or
and
^3
+ (rx + rj
M r
L = r{r 1 + r 9 ) + (^i + r 2 ) r s
M rr z
This arrangement of resistance eliminates the trouble-
some double balancing of the last method. The steady balance
is obtained with a cell and galvanometer in the usual way.
The battery circuit is then made and broken or reversed ; r is
582 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
adjusted until the galvanometer shows no deflection during that
process. As before, care must be taken that the coils are so
connected that the effect of the self and mutual induction oppose
each other.
Alternatively a hummer may be used as a source of alternating
E.M.F., instead of making and breaking the battery circuit,
and a telephone used as detector.
If the self-inductance is less than the mutual inductance a
modification of the method above may be used. In this case
the battery and the resistance, r, are interchanged, i.e. M is in
series with r.
For balance, the effect of the induced E.M.F. in the arm, AB,
due to L and M, must neutralize each other. As before, equating
the numerical values of such E.M.F.'s we have, if c 8 is the current
through r and M,
T dc i _m^ 3
Lc t = Mc 3 , and c 1 {r l + r 2 ) = c 3 (r -J- x),
M c, r + x
i.e. — - = - 1 = — ^— ~,
where x is the resistance of the coil, M. This method, however,
cannot be performed using alternating current and telephone.
Determination of Mutual Inductance by the Fluxmeter
The mutual inductance of two coils may be very readily
obtained, using a fluxmeter (which is described on page 482).
AJ } R
— Mttl'l '
E
Fig. 363
One coil, M, is connected directly to the fluxmeter, F, fig. 363.
The other coil, L, is connected through a commutator, C, to a
circuit consisting of an accumulator, E, an ammeter, A, and a
1 variable resistance, R.
The current in L is adjusted to some convenient amount,
c amperes, and then reversed, causing a deflection of x divisions
in the fluxmeter.
MEASUREMENT OF CAPACITY AND INDUCTANCE 583
If each division of the fluxmeter (as usual) corresponds to 10000
maxwells, the change in the number of maxwells in the secondary,
M, due to a reversal of c amperes in L is ioooo* , i.e. for — E.M.
units of current in L there are 5000* lines of magnetic force
threading M
Hence, since the coefficient of mutual inductance is denned
as the number of lines in the one coil for unit current in the
other, it has a value,
50000* .
c
for the coils used.
The experiment is repeated using various values of c, and the
mean value of - is obtained, and hence the coemcient of mutual
c
inductance in E.M. units may be calculated.
The mutual inductance in henrys is io~ 9 times the above
value, for two coils are said to have a coemcient of mutual
inductance of 1 henry when a current change of 1 ampere per
second in the primary causes an E.M.F. of 1 volt in the secondary,
i.e. a flux of io 8 lines for a current of 10- 1 E.M.U., i.e. the henry
is io 9 E.M. units.
CHAPTER XXIII
THE QUADRANT ELECTROMETER
A Modern development of the Kelvin Quadrant Electrometer
is seen in fig. 364, and is due to Dr. F. Dolezalek. The four
quadrants, QQ, etc. are supported on ambroid pillars, AA.
As shown in the figure, two quadrants are mounted on a pivot,
and may be swung aside to allow of the introduction of the
' needle.' Alternate quadrants are joined together to terminals
Fig. 364
under the base plate of the instrument, care being taken that
such terminals are very well insulated from the case.
The needle may be either a light paper frame (fig. 365), coated
with a metal to make it conducting, or as in fig. 366, a thin
584
THE QUADRANT ELECTROMETER 585
mica sheet spluttered with silver. In some of the more recent
instruments a very thin aluminium needle of the shape of the
second form is used. The needle is attached to a light rod
which carries a small mirror, M, and is supported by means of
a thin quartz or phosphor-bronze strip from the torsion head, R.
Fig. 365 Fig. 366
The method of observing the deflection produced by a difference
of potential on the quadrant consists of the use of the usual lamp
and scale, as with galvanometers.
A beam of light from a lamp is directed on the mirror, M,
which reflects the light on to a scale placed one metre away.
If the mirror is concave, and of the correct focal length, a clearly
denned image of the source will be obtained on the scale. If
M is a plane mirror, a lens is required to produce a sharp image.
Such a lens is placed in the path of the incident beam from a
source of light below the scale ; the reflected beam does not,
therefore, pass through the lens.
The Suspension
(1) Quartz Fibre
If a quartz fibre is used as a suspension, the needle may
be charged by means of the charging device shown at K.
K is connected to the source of potential and the metal rod
turned until contact is made between it and the rod which
supports the mirror on the needle. K is then turned to its
original position leaving the needle charged. Due to the very
good insulating property of the quartz, this charge will be main-
tained. However, there is a danger of breaking the quartz
suspension during this process.
For most purposes it is more convenient to avoid using the
charging device To make this possible the suspension is coated
lightly with a calcium chloride solution. The hygroscopic
properties of the latter ensures a conductivity which is sufficiently
good to maintain the needle at the potential of the source which
is permanently connected to R.
An alternative method is to splutter the quartz with silver,
but this cannot as a rule be carried out conveniently in the
laboratory.
586 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Suitable quartz fibres are very simply made by the aid of
a coal gas-oxygen flame. A burner for this mixture is shown in
fig. 367. A piece of quartz rod is heated in the flame until
thoroughly soft ; when at this stage the ends are drawn apart.
If this is done rapidly a fine quartz thread will result. If a still
finer fibre is required draw out the quartz rod to about 1 mm.
diameter, and then reintroduce into the flame. When the
quartz becomes very soft the pressure in the flame itself will
blow the quartz outwards into very fine fibres. It is advisable
to have a black velvet cloth on the bench to receive these threads.
Coed
Gas
T
Oxygen
Fig. 367
Small hooks may be fastened to the ends of the fibre, when cut
to the correct length, by means of a small globule of shellac
(solution in methylated spirits) ; a hot iron is held over the
globule, which is placed on the hook over the end of the quartz.
The spirit evaporates and the fibre adheres.
Another method of fastening the hook to the fibre is to use
indian ink. The end of the hook is dipped into indian ink
which is allowed to become ' thick ' by evaporating. The fibre
is then placed on this plastic drop ; when dry the two will be
found to be very firmly held.
(2) Phosphor-bronze Strip
Of the metal suspensions phosphor-bronze strip is most
satisfactory Platinum and tungsten wires may be obtained
with a smaller diameter, but usually have the disadvantage
that the zero of the needle is not stable when very fine suspen-
sions of these metals are used.
THE QUADRANT ELECTROMETER 587
For the experiments described in this chapter, the finer
phosphor-bronze strip obtainable commercially gives a sensitivity
which satisfies the needs of the experiment. If, however, the
phosphor-bronze strip available is too thick for the purpose, it
may be treated as described below ; which method enables a
fair sensitivity to be attained.
A solution of one part nitric acid to four parts of water is
taken, and the Suspension immersed in it. The action of the
acid causes an evolution of gas which adheres to the suspension,
causing it in time to rise to the. surface. The chemical action
should be slow. If violent action takes place the solution should
be further diluted. Of course, a preliminary test should be
made with a small sample of the phosphor-bronze to be used in
order to avoid undue waste.
The strip should be very well washed and dried, and its sensi-
tivity tested in the instrument. The process is repeated until
the required sensitivity is obtained. When this process has
once been performed with the specimen of strip available, the
time of immersion in the acid required to reduce cross-section
to a suitable value may be very readily estimated, and the time
spent in testing will be reduced, e.g. the strip may be sufficiently
reduced by immersing four times in the acid, allowing it to come
to the surface each time.
In this way a sensitivity of 700 to 1000 cms. per volt may be
obtained with the instrument shown, but for the experiment in
this chapter a sensitivity of about 25 cms. per volt will be found
to be quite sufficient.
The above process is carried out using the wire without hooks.
If the hooks are soldered on to the ends, the action of the acid
will sever them. The small hooks are then soldered to the ends
of the strip, using a very small pointed iron. Care should be
taken that the ends and hooks are clean and the iron hot. Use
soft solder and a trace of fluxite.
Adjustment
To prepare the instrument for use, the following adjustments
must be made. Both pairs of quadrants are connected to earth,
and are therefore at the same potential ; R is also earth connected.
Thus, the only forces acting on the needle are those due to the
torsion of the suspension. The needle is raised or lowered by
means of the screw, T, until it swings about the mid-plane within
the quadrants. By means of the levelling screws, B, the whole
instrument is then levelled, so that the suspension hangs centrally
within the quadrants. To ensure this, the rod which supports
the mirror, M, is sighted along the two diagonal spaces between
588 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
the four quadrants and a slight adjustment made if required to
bring the rod truly central. The torsion head, E, is then turned
until the needle appears symmetrical with respect to the quadrant.
A suitable high potential of, say, ioo volts is applied to the
needle (in the manner described later, fig. 368). If it were
adjusted to be precisely symmetrical no movement should result
on making this change. Any error in the adjustment by eye for
symmetry may now be corrected, by noting the direction of
deflection of the needle. The potential is removed and the
needle earthed. The torsion head is given a very slight turn
in the direction of the movement previously observed. The
needle is again charged and the process repeated, until on
charging and earthing the needle alternately the deflection
produced is of the order of a few cms. only. To reduce this
small deflection to zero a slight adjustment of a levelling screw
will be found sufficient — a portion of a turn of one of the levelling
screws will be found to increase or decrease the deflection produced.
When the potential is applied to the needle, the screw is turned
in the direction required to reduce the deflection until it is
finally eliminated.
To needle
E
nh
>>J
D
-M'H'I'I— -#1
O B 2
Fig. 368
With a fine suspension the above process is apt to be somewhat
tedious until the student becomes familiar with the instrument ;
but it is essential to the success of any observation.
When this adjustment has been satisfactorily made the instru-
ment should be tested for leak. To do this the arrangement of
apparatus shown in fig. 368 is a convenient one, for by means
THE QUADRANT ELECTROMETER 589
of this the adjustment already described may be also performed.
Thus, when the two-way switch* is to the left, the needle is at
zero potential : and if Kj* is open and K 2 * closed, the quadrants
are at zero potential. The needle is charged by closing the
two-way switch to the right.
Bj is a steady 2 volt accumulator ; R a high resistance,
provided with a slide contact, or two resistance boxes in series.
Either arrangement serves as a potentiometer by means of
which any fraction of the E.M.F. of B x may be obtained between
S and P when K x is closed. When K 2 is closed this potential
difference is applied to the quadrants. The needle, being
maintained at a fixed potential, say 100 volts, by the battery, B 2 ,
will be deflected, causing a movement of the spot of light of, say,
d cms. K 2 is then opened ; if the quadrants are fully insulated
no movement will be given by the needle. If on the other hand
a decrease in the deflection occurs, it indicates that the quadrants
are losing charge, i.e. the insulating supports are faulty. The
usual cause for this is dust or grease on the surface of the pillars,
which should therefore be cleaned. When this has been carried
Out as thoroughly as possible the test is again made. In general
one cannot remove the leak entirely, but it may be reduced to
a very small amount. The rate of movement of the needle,
when the one pair Of quadrants is charged and, then insulated, is
measured by timing, with a stop-clock, the movement of the spot
of light. The number of divisions per second is called the
natural leak of the instrument.
Note
The high potential is connected to the needle of the electrometer through a
water resistance. This may be a small U-tube filled with water ; the wire from
the high potential is inserted in one limb, and the lead from the needle into the
other. In the case of an accidental contact between the quadrant and the needle,
the latter would be ruined, in the absence of a water (high) resistance. The poten-
tial on the needle is unaffected by such resistance, but unless measured by an
electrostatic voltmeter there will appear to be a fall in potential. This is due to
the fact that an ordinary voltmeter is not of a very high resistance compared
with the water resistance. If only a moving-coil voltmeter is available to test
this potential it should be applied to the point at which the high potential enters
the water resistance.
A simple theory of the instrument (see, for example, Whetham
" Electricity and Magnetism ") shows that if one pair of quad-
rants is maintained at a potential, V lt and the other at a potential
V 2 , the needle, being maintained at potential V„, will move,
* These switches are best made by boring small holes
in a clean slab of paraffin wax. The holes are filled with
clean mercury and connection is made by using a length
of copper wire, bent twice at right-angles and mounted
on a sealing-wax handle, as in figure 369.
Fig. 309
590 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
through an angle, 0, which depends on the potential difference,
Vj — V 2 , and
e = c(V 1 -V 2 )(v n + ^i^y
where c is a constant.
In general, V t and V 2 are small compared with V w , and we
may write :
e = c(V 1 -V 2 )V n (i)
Such a simple theory therefore leads to the conclusion that
the deflection for a given potential difference on the quadrants
is proportional to the potential of the needle. One would
conclude, therefore, that the sensitivity is proportional to this
potential. This, in fact, is not so, as may be seen from the
following experimental test, using the arrangement of fig. 368.
Adjust the potential difference between P and S to such a
value that the electrometer gives about 20 or 30 cms. deflection
(K ± and K 2 being closed) when a potential of about 100 volts
is applied to the needle.
Note the deflection produced. Throw the switch over to the
left and so earth the needle and reduce the deflection to zero ;
when the spot of light is steady take the zero reading. Move D
to some other point in the battery, and then put the switch back
to the right-hand side ; again read the value of the deflection
Repeat this for a wide range of values of V„. The value of V„
is, of course, taken for each setting, by means of a voltmeter
which is connected to D and earth.
The zero readings between each deflected reading should be
the same. Plot a curve showing the relation between the
deflection produced for the fixed potential difference on the
quadrants, and the potential on the needle. It will be found
that the deflection produced increases withV a , but not indefinitely,
i.e. there is a potential above which there is no gain in sensitivity,
this is usually between 70 and 120 volts for the type of instrument
described.
Maintain the potential of the needle at this value and verify
the fact that the deflection is proportional to (V a — V 2 ) by
applying different values to the quadrants. By adjusting P on
the resistance R, the ratio, — r-r ~~, may be chosen to give,
say, -oi, -05, *i, -15 volt. For these values the deflections will
be found to be as 1 : 5 : 10 : 15.
Thus, for a fixed potential on the needle the deflection is
proportional to the difference of potential of the quadrants.
Thus, the equation (1) holds so long as the needle is maintained
THE QUADRANT ELECTROMETER 591
at fixed potential, and the instrument may be used to compare
potential.
A more complete theoretical treatment of the instrument may
be found in " Phil. Mag." 1903, by G. W. Walker, or later (" Phil.
Mag." 1912, p. 380) by Prof. A. Anderson. Here the variation
of with V„ is more correctly stated. This leads to the expression :
2aV 2 (v n - p) - 2 Y KV» 2
F + KV n *
where F is the torsional couple due to the fibre per unit angular
displacement, a, K and y are constants depending on the
particular instrument for their value.
Sensitivity
By the sensitivity we understand the number of cms. or mms.
deflection obtained on a scale one metre away, when one volt
potential difference is applied to opposite pairs of quadrants.
With the same instrument this factor may have vastly different
values depending upon the dimensions of the suspending fibre.
In most cases, especially in laboratory work, the fibre is con-
ducting and very conveniently made of phosphor-bronze strip.
The type of instrument with a fine phosphor-bronze strip may
give, say, 60 cms. deflection per volt.
This factor may be found by use of the arrangement of fig. 368,
as already set up for the previous experiments.
The needle being raised to the high potential already chosen,
K x and K 2 are closed and P is adjusted so that a deflection of
about 20 or 30 cms. is obtained. The value of the resistance
in SP, r ohms say, is noted as is the total resistance, R, of ST.
A standard cell is then connected to S and P through a galva-
nometer, and the resistance, SP, is adjusted, keeping R the same
as before, until no deflection is given in the galvanometer, i.e.
the usual method of standardizing the potential is followed.
If r x is the resistance, SP, under such circumstances, then the
Y
potential previously applied to the needle is obviously — E, when
r i
E is the E.M.F. of the standard cell.
From this the number of cms. deflection per volt may be
calculated.
As already mentioned, a suitable sensitivity is about 25 to 50
divisions per volt. The suspension should be made of suitable
size to give this value. The method has already been described.
592 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
Comparison of the E.M.F. of Two Cells
The adjusted quadrant electrometer is connected as shown in
fig. 370. The needle is raised to the potential already found to
be most satisfactory. The three-way paraffin wax switch is
made and connected to the batteries B x and B 2 by mercury
cups, 1 and 3. Cup No. 2 is connected to earth as are the negative
poles of Bj and B 2 .
13 § 1|
r 9 -}-*!*
2 T * T
E EBB
Fig. 370
By connecting to cup 2, all the quadrants are at zero potential.
The battery, B x , is then placed with its positive pole in contact with
one pair of quadrants (via 1), the other pair being earthed.
The deflection, d x , is noted. The quadrants are then earthed,
and B 2 is connected (via cup 3) ; the deflection, d 2 , due to this
also being observed.
Then & = i
Ji 2 »2
from the previous investigation, i.e. (Vi — V 2 ) oc 0.
For large deflections d x and d 2 cms. must be replaced by the
corresponding angles, X and 2 ; this may be readily done, for
tan (2^) = A, tan (20 2 ) = -^-,
v 1; 100 v ' 100
whence : ~ = -~.
Verification of Ohm's Law
For this purpose a battery, B, an adjustable resistance, R,
and a tangent galvanometer are connected in series with a
resistance of fixed amount, r ohms, which may very well be a
length of manganin or platinoid wire (fig. 371),
The end, C, is connected through a paraffin wax switch, K,
to one pair of quadrants of an adjusted electrometer, and the
end, D, is connected to earth and the other pair of quadrants.
The current flowing in the wire, CD, may be measured By means
of the tangent galvanometer, for if is the deflection
THE QUADRANT ELECTROMETER 593
produced in this instrument as measured by means of the lamp
and scale method, we have :
c = • tan 0,
or
c oc
tan
n
where n is the number of turns used.
E
-vwwvwvwv »»
Fig. 371
Further, the potential difference between the ends, C, D, is
proportional to <p, the deflection produced in the quadrant electro-
meter.
If <p is small, the corresponding scale deflections, d, may replace
<p below.
For a range of values of c it might become necessary to alter
the number of turns, n, used in the galvanometer. Vary R and
tabulate the result as under.
tan
c oc
n
E oc<p
E <pn
— « ^ — ^~
c tan 6
It will be found, if care has been taken to avoid large currents
which would cause appreciable heating in the wire, that the
value of , -•• is constant,
tan
If the sensitivity of the Q.E. has been found, the value of
the relation, — > may be found absolutely, i.e. for a
current J J
38
594 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
sensitivity of d cms. per volt, deflections, <p x , <p 2 , etc., being small,
the potential drop along r is ~, -~, . . , volts.
c may be calculated in amperes from the usual tangent galvano-
meter formula :
c =
5H0
tan 6,
where a is the mean radius of the coils, H the horizontal com-
ponent of the Earth's magnetic field (about '185 in London).
Whence tabulating absolute values the third column gives
the constant value of the resistance, r , at room temperature in ohms.
This should be tested independently.
Measurement of High Resistance
The following is a method of measuring a high resistance by
finding the rate of leak of a charged condenser through the
resistance.
A quadrant electrometer is set up and adjusted as previously
described, care being taken to reduce the natural leak to a
minimum. One pair of quadrants is connected to earth ; the
other pair, as shown in fig. 372, is connected to a condenser, K,
and through a switch, K x (see footnote, page 589), to a 2 volt
cell, B, the other pole of which is earth connected. Connexion
is also made to a two-way switch, K 2 , by means of which this
pair of quadrants may be connected to earth or to an earth-
connected resistance whose value, R, is to be determined.
QE
S^Ko K
13 21
i£_2n —
B
\[
1
I — to ois
B
» — I
■ 1 ■XT
Fig. 372
With K 2 open, close K t ; in this way the condenser will
be charged to a potential difference equal to that of the
cell, B. Note the deflection, & x , corresponding to an angular
deflection, <p x . This is due to a potential difference of Y x volts,
THE QUADRANT ELECTROMETER 595
say. Now close K 2 by joining I and 3 ; open K x and start a
stop-watch. The charge on the condenser will slowly leak to
earth through R. The potential of the quadrants will in conse-
quence be reduced, i.e. d or <p will gradually decrease. If this
process be allowed to continue for about two minutes or until
the deflection is reduced to about 60 per cent of its original
value (t seconds), the final deflection, d 2 (angular deflection <pj,
will be shown to be such that
R = — '—.
K log 2S.
fa
For let
Q and V be the charge and potential on the quadrants
at any time,
Qj V x the corresponding values at the moment K 2
is opened,
Q 2 V, the values after t seconds,
K the capacity of the condenser,
c, the current at any instant, is equal to the rate of decrease
of the charge on the condenser,
or
Also by Ohm's Law c = = =
i.e.
or
Integrating over the limit indicated below
c =
dQ
dt'
V
= R ~"
Q
KR'
since
V
dQ
dt
Q
KR'
dQ
dt
Q
KR"
f^dQ = r* dt
and since
log|
■p
t
KR'
t
Qi
R
Klog S
V, " 9,*
t
9%
(2)
596 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
If <p x and <p 2 are small we may approximate and write :
~K.lo g (J) = 2 - 3 o 3 Klog 10 (g •- (3)
In making an estimation of R several values of t should be
taken — the range of values will be largely fixed by the value
of R and the condenser used. When several values are obtained,
the mean value of - is taken and substituted in equation
log i
(3) above. When K is expressed in farads, t in seconds, R is
in ohms.
The method outlined above may be very conveniently used
for a determination of the specific resistance of, say, cadmium
iodide solution in xylol. Such a solution is very useful when a
high resistance is required for any purpose. The value of the
specific resistance of solution of different concentrations should
be measured. Pure xylol will be found to be practically
without effect on the leak of the condenser.
A suitable container of liquids is made by selecting a length
of glass tubing of uniform bore (about 2 cms. diameter) and
about 80 cms. long. This is closed at both ends by corks and
the tube clamped vertically (fig. 373). A disc of brass which
just fits the tube rests on the upper surface of the lower cork
and can be connected to an outside circuit by means of a thin
brass rod soldered to the under surface of the disc and passing
through the cork. This disc serves as one electrode.
A second brass disc of the same area is supported
vertically above this by means of a brass rod which
passes through the upper cork, as seen in fig. 373.
Using such a container a definite ' length of liquid/
I, with an area of cross-section practically equal to
that of the area of the discs may be employed ; and
hence the specific resistance may be calculated from the
value of R measured in the experiment, /, the distance
( ^ between the discs, and a, tjie radius of the discs.
Note
This experiment could also be performed using a
ballistic galvanometer. The value of Q x at the com-
mencement could be obtained by discharging the
condenser through the ballistic galvanometer. The
condenser is then recharged, allowed to leak through
the resistance for a measured time, and then dis-
charged through the galvanometer once more, thus
FiG.373 Q t and Q 2 are measured ; and R is obtained from (2).
THE QUADRANT ELECTROMETER
597
Measurement oi the Capacity of the Quadrant Electrometer
Fig. 374 shows a simple arrangement of apparatus which will
enable an estimation of the capacity of the quadrant electro-
meter to be made. .
Fig. 374
Kj is a small capcity of known size. When switch, S X) is closed
from i to 2, and S 8 is closed, the condenser becomes charged,
and the electrometer is deflected an amount, tp x say, corresponding
to a scale deflection of the spot of light oi d x cms. If now
switch, S lf is closed from I to 3, the electrometer becomes dis-
charged, and when the connecting strip is replaced to the position
I to 2, S 2 now being open, the condenser, K 1} shares its charge
with the electrometer. If K x is properly chosen of the same
order of magnitude as the capacity, K, of the electrometer, the
latter will show a marked drop in deflection compared with the
former <p x . Let this new deflection be <p 2 .
The charge lost by the condenser, K x , is equal to that gained
by the electrometer, KV 2 , where V 2 is the final potential corres-
ponding to the deflection <p 2 . This loss is also K^Vx — V 8 )
where V x is the original potential,
KV,- = K x (Vx - V 8 ),
1 =Kl (Xi^-Va) = K r 9 ' 1 -^
i.e.
or
K
<p 2
whence, if K x is known, K may be calculated.
If the known small condenser has a capacity which is large
compared with the capacity of the electrometer, the value of <p 2
will not be very different from q> x , and the value of K, as calculated
from the above equation, will probably be inaccurate.
If no smaller capacity is available, the following slight modifica-
tion of the method is very readily carried out.
The key, S x , is moved from position 1 and 2 to 1 and 3, a definite
number, n, times, i.e. the electrometer is charged and discharged
n times, until the deflection is finally reduced a distinct readable
amount.
598 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
i.e
At the end of the first sharing of charge we saw above that
KV 2 = K 1 (V 1 - V 2 ),
V 2 =
V x .
When the condenser, K x , which is now at potential, V 2 , is again
connected to the discharged electrometer, a new common potential
V,, will be the result where
KV 3 = K x <y 9 - V 3 ),
V a =
Kx
V„
so, after n repetitions
or
-(k. + k) V * ;
Vb+1 = VETlhc) v *
of if 9»j and <p n+1 are the corresponding deflections,
K x
whence
A suitable form of measurable capacity to use in this and the
following experiments, which require a known capacity, is a
circular parallel plate air condenser having a guard ring.
Scale S
M
M
',
G A G ^
Fig. 375
The guard ring, GG (fig. 375), surrounds a central plate, A,
which may be adjusted by a micrometer screw attachment, M,
moving past a fixed scale, S, so that the distance, d, between the
plate, may be measured, provided the zero reading on the scale
THE QUADRANT ELECTROMETER
599
is known, whence, if plate, A, has an area A sq. cms., the capacity,
Kj, in the above experiment is
A
— 5 E.S. units
477*1
A A
or rr — 5 E.M. units = r\ — -; farads.
9 x io 20 4toZ 9 x 10" 4«*
To obtain the zero reading of the instrument, the plate, A, is
connected to a quadrant electrometer, and the system is given a
charge, and then insulated. The plate, A, is moved towards the
lower plate until the electrometer deflection is reduced to zero,
To allow for possible irregularity of plate surface or a slight
inclination of one of the plates, the effective zero of the condenser
may be obtained by a method which consists in finding
the definite value of the deflection of the electrometer (<p) for
each scale reading of the condenser (d). Several values of
d and <p are taken as the plates approach each other. No
observation of q> less than one-tenth of the original deflection
need be observed. Plot q> against d. The point where this
curve produced cuts the axis of d corresponds to the zero of
the scale. *
Another form of condenser suitable for the above experiments,
and which is simply made, is described on page 605. This
condenser has a fixed capacity.
Comparison of Small Capacities
Capacities such as small air condensers may be compared,
using the method indicated in fig. 376. S x and S 2 are keys made
as already described, and B is a 2-volt cell.
7=T
K,
*i
4
-o o-
S 2
E
Fig. 376
With S 2 closed and Sj open, K t and the quadrant electrometer
are charged to a potential, V lf equal to the E.M.F. of B.
If now S 2 is opened and Si closed, the charge on K x and the
Q.E. is shared with K 2 .
Let q> x be the deflection of the Q.E. before S x is closed, and
<P 2 the deflection after the charge is shared, corresponding to a
potential, V,.
600 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
We have K 2 V 2 = charge acquired by K 2 , and (Kj + K)
(Vj— V 2 ) is the charge lost by the Q.E. and K lt where K is the
capacity of the electrometer,
i.e. K 2 V 2 = (K x + K) (Y, - V 2 ),
or V 2 = K, + K = <p 2
Vj — V 2 K 2 <p x — <p z '
or Kl = K 2 (-^-)-K
Whence by observing the deflections before and after the
sharing of the charge, and knowing K, the relation between
Kj and K 2 may be obtained.
When <p x and q> 2 are small the corresponding scale deflections,
d x and d 2 cms., may be used in the equation.
Determination of the Dielectric Constant
The value of k, the dielectric constant of a medium, may be
obtained by measuring the change in capacity of a parallel plate
air condenser, when a parallel plane-faced slab of the medium is
introduced between the condenser plates.
The capacity of the condenser when the plates are entirely
separated by air is
4*d' * U/
where A is the area of the plates and d the distance between
them.
If the slab of thickness, t, and dielectric constant, k, is intro-
duced between the plates of the air condenser, the capacity
becomes
(4)
4B |*-*(i-^)[
Thus the change produced in the capacity is numerically equal
to the change produced when the distance between the plates is
reduced by t ( i — r- J , when air is the dielectric, for equation
(3) would then become identical with (4).
Suppose the capacity be determined when the plates are a
definite distance apart. Then let the slab be introduced ; the
capacity increases. If now the plates are separated until the
capacity is restored to its original value, the distance, D . through
THE QUADRANT ELECTROMETER 601
which the plates are moved is obviously equal to the equivalent
movement produced by the introduction of the slab ; thus
To make a determination of k experimentally a convenient
form of condenser is a circular parallel plate condenser, one
plate of which is surrounded by a guard ring, as described on
page 598, fig. 375.
Such a condenser is connected as shown at K x in fig. 374, the
large plate, B, being earth connected and the central plate, A,
inside the guard ring, connected to switches S x and S 2 . The
quadrant electrometer is adjusted, and the condenser, K lf and
one pair of quadrants are raised to the potential, V lt of the cell,
by closing S x and S 2 , producing a deflection, y x or d t divisions.
The key, S 2 , is then opened, and the slab of dielectric is introduced
between the plates of K x . The slab should be of uniform thick-
ness and have an area not less than the plate, A, of the condenser.
Since the system has a fixed charge, the effect of the increase
in the capacity of the condenser is evidenced by a drop in potential
to V 2 (deflection <p 2 ). The reading of the micrometer adjustment
of the condenser is taken and the movable plate is then with-
drawn until the deflection of the quadrant electrometer is again
<Px (di cms.), i.e. the whole system has once more the original
capacity. The movement of the condenser plate (D cms.) is
known from the micrometer readings.
Hence, if t is measured in the usual way, and a mean value
taken, k may be calculated.
Comparison of a Large Capaeity with a Small
The method of comparing capacities given on page 599 is
applicable when the capacities are of the same order. To
compare a large capacity with a small one (both measured in
electrostatic units) the method of repeated sharing of charges
may be employed. This method was outlined for the case where
the difference in the capacities was not very great on page 597.
Now, when the order of the capacity of the two condensers is
very different, as in the case of, say, £ micro-farad, and a
simple parallel plate air condenser, n, the number of times the
charge is shared, must be a large number to cause an appreciable
change in the potential, or in the deflection, <p. Some mechanical
means must therefore be introduced to carry out rapidly the
n steps. In fig. syy, which shows the arrangement of apparatus
602 ADVANCED PRACTICAL PHYSICS FOR STUDENTS
for this determination, such a mechanical device, which is
described later, is placed at M.
Cj is the condenser of small capacity, K x . A guard ring
parallel plate air condenser as described on page 598 or page 605,
would do very well. The distance between the plates is fixed
and measured (dcms.) ; hence K, = — =.
v ' 4nd
"To earrti
Fig. 377
The large cap