•OOK ;: PHYSICS ELECTRICITY AND MAGNETISM ^LOYAL-AV-MORV JOHN LANGTON From the ESTATE OF JOHN LANGTON to tke UNIVERSITY OF TORONTO 1920 A TEXT-BOOK OF PHYSICS ELECTRICITY AND MAGNETISM A TEXT=BOOK OF PHYSICS By J. H. POYNTING, Sc.D., F.R.S. Professor of Physics, Birmingham University And Sir J. J. THOMSON, M.A., F.R.S. Professor of Experimental Physics in the University of Cambridge. In Five Volumes. Large 8vo. Sold Separately. VOLUME I. SIXTH EDITION, Revised, Fully Illustrated. PROPERTIES OF MATTER CONTENTS. — Gravitation. — The Acceleration of Gravity. — Elasticity. — Stresses and Strains. — Torsion. — Bending of Rods. — Spiral Springs. — Collision. — Compressibility of Liquids. — Pres- sures and Volumes of Gases. — Thermal Effects Accompanying Strain. — Capillarity. — Surface Tension.— Laplace's Theory of Capillarity.— Diffusion of Liquids.— Diffusion of Gasses.— Viscosity of Liquids.— INDEX. " We regard this book as quite indispensable not merely to teachers but to physicists of every grade above the lowest." — University Correspondent, VOLUME II. FIFTH EDITION, Revised, Fully Illustrated. SOUND CONTENTS.— The Nature of Sound and its chief Characteristics.— The Velocity of Sound in Air and other Media.— Reflection and Refraction of Sound.— Frequency and Pitch of Notes.— Resonance and Forced Oscillations.— Analysis of Vibrations.— The Transverse Vibrations of Stretched Strings or Wires.— Pipes and other Air Cavities.— Rods.— Plates.— Membranes.— Vibrations maintained by Heat. — Sensitive Flames and Jets. — Musical Sand. — The Superposition of Waves. — Index. " The work . . . may be recommended to anyone desirous of possessing an EASY i DATE STANDARD TREATISE on Acoustics." — Literature. VOLUME III. FOURTH EDITION, Revised, Fully Illustrated. MEAT CONTENTS. — Temperature. — Expansion of Solids. — Liquids.— Gases. — Circulation and Con- vection.— Quantity of Heat ; Specific Heat. — Conductivity. — Forms of Energy; Conservation ; Mechanical Equivalent of Heat. — The Kinetic Theory. — Change of State ; Liquid, Vapour. — Critical Points.— Solids and Liquids.— Atmospheric Conditions. — Radiation.— Theory of Ex- changes.— Radiation and Temperature.— Thermodynamics.— Isothermal and Adiabatic Changes. —Thermodynamics of Changes of State, and Solutions. — Thermodynamics of Radiation. — Index. " Well up-to-date, and extremely clear and exact throughout. ... As clear as it would be possible to make such a text-book."— Nature. VOLUME IV. IN THREE PARTS. PARTS I & II, pp. i-xiv + 246. PART III at press. ELECTRICITY & MAGNETISM IN PREPARATION. VOLUME V LIGHT LONDON: CHARLES GRIFFIN & CO. LTD., EXETER STREET, STRAND. PHILADELPHIA : J. B. LIPPINCOTT COMPANY A TEXT-BOOK OF PHYSICS BY J. H. POYNTING, Sc.D., F.R.S. HON. Sc.D. VICTORIA UNIVERSITY LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE ; MASON PROFESSOR OF PHYSICS IN THE UNIVERSITY OF BIRMINGHAM AND SIR J. J. THOMSON, O.M., M.A., F.R.S. FELLOW OF TRINITY COLLEGE, CAMBRIDGE ; CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE ; PROFESSOR OF NATURAL PHILOSOPHY AT THE ROYAL INSTITUTION ELECTRICITY AND MAGNETISM PARTS I AND II STATIC ELECTRICITY AND MAGNETISM WITH ILLUSTRATIONS LONDON CHARLES GRIFFIN AND COMPANY LIMITED PHILADELPHIA : J. B. LIPPINCOTT COMPANY 1914 PREFACE IT was intended that the subject of Magnetism and Electricity should be a volume complete in itself, of the Series forming a Text- Book on Pnysics, but it has been found necessary to divide it into two. The present volume contains an account of the chief pheno- mena of electric and magnetic svstems when they are respectively charged and magnetised. The effects of changes in the systems are only considered >taticallv, after the changes are effected, and the sv-teniN have become steady again. The phenomena accompanying the progiv.>s of change' belong lo electric current or electro- magneti>m ami will be treated in another volume. The view of electric action taken by Faraday and largely deve- loped in mathematical form bv Maxwell still holds good. Faraday showed that we must regard the material medium between electrified bodie> a> in an altered condition. In this volume the quantity which is taken a> measuring the alteration is termed "electric strain." The term is adopted in preference to "electric displacement" or "electric polarisation," in that it does not <>r imply any special hypothesis. When the- term is first found useful, it is not ncce>sarv to form a hypothesis as to the nature of the alteration, though the somewhat vague representation of it as the beginning of chemical separation, the drawing apart of two equally and oppositely charged portions of atoms or molecules, which Faradav appears to adopt, is helpful. The electron theory i^ substantially, a much more highly developed and dynamical form of Faraday's theory. But the electron theory was suggested by the phenomena of electric discharge, and is so largely a current theory that it is more appropriately considered in the companion volume. In this volume it is only introduced occasionally in magnetism. vi PREFACE The experiments are selected chiefly for their value in establishing the fundamental principles, and no attempt is made to give a com- plete account of experimental methods. For fuller details the reader is referred to Wiedemann's Die Lehre von der Elektricitat , Graetz's Handbuch der Elcldricit'dt und des Magnetismus, Winkel- mann's Handbuch der Physik, or the Encyclopaedia Uriianmcu^ llth edition. The mathematical development is only carried so far as is needed for the account of the experiments described. The aim is to build firmly the foundation on which the mathematical theory may be raised. The companion volume is now «-it press and will be issued shortly. J. II. I1. CONTENTS PART I: STATIC ELECTRICITY CHAFrER I GENERAL ACCOUNT OF COMMON PHENOMENA Electrification by friction — Actions between bodies both electrified by friction — Two kinds of electrification— Law of electric mechanical action — Electric actions not magnetic — Conduction and insulation — Gold-loaf electroscope — Electrification by induction — Detection of kind of charge by the electro- scope—Nature of induced electrification — Only the outer surface of a con- ductor is charged — The two electrifications always occur together, inducing each other — Discharge of electrification — Frictional machines — Ease of dis- charge from points — I. i^htningconductors— Electrophorus— Bennett's doubler — Belli's machine — Holtz's machine — Wimshurst's machine — The Leydi-nj.tr —Franklin's jar— Residual charge Pp. 1-23 CHAPTER II QUANTITY OF ELECTRIFICATION Use of the electroscope to indicate equality of charge — The two electrifications always appear or di>api»-ar in equal amounts whether the electrification is by friction, by conduction, by induction, or by supply from current — Electrification resides only on the outside surface of a conductor unless it contain-, insulated charged bodies— An insulated charged conductor inside a hollow conductor induces an equal and opposite charge on the inside and an equal like charge on the outside — Imagined construction of multiples and Bubraultiples of an arbitrary unit of charge— Imagined method of measuring any charge in terms of this unit Pp. 24-30 CHAFIER III PROPOSITIONS APPLYING TO 44 INVERSE SQUARE " SYSTEMS The inverse square law — The field — Unit quantity — Intensity— Force between quantities 7/4 and 7«2— Lines of force and tubes of force— Gauss's theorem — If a tube of force starts from a given charge it either continues indefinitely vii viii CONTENTS or if it ends it ends on an equal and opposite charge — The product intensity X cross-section is constant along a tube of force which contains no charge — If a tube of force passes through a charge and q is the charge within the tube, the product intensity x cross-section changes by 4^ — The intensity outside a conductor is 47r: CHAFrER IV THE FIELD CONSIDERED WITH REGARD TO THE INDUCTION OR ELECTRIC STRAIN PRODUCED IN IT The two kinds of charge always present and facing each other — Electric action in the medium — Electric strain — No strain in conductors win n the charges are at rest — Diiection of electric strain — The electric strain ju>t outside a conductor is normal to the surface of the conductor — Magnitude of electric strain — Variation of electric strain with distance from the charged bodies : The inverse square law— Lines and tubes of strain — Unit tube— Results deduced from the inverse square law — The transference of tubes of strain from one charge to another when the charged bodies move — Molecular hypo- thesis of electric strain l'j>. 47-60 CHAPTER V THE FORCE ON A SMALL CHARGED BODY IN THE FIELD, AND THE PULL OUT- WARDS PER UNIT AREA ON A CHARGED CONDUCTING SURFACE The inverse square law for the force deduced from that for electric strain — Coulumb's direct measurements — General statement of the inverse square law — Electric intensity — Unit quantity — Relation between electric intensity and CONTENTS ix electric strain — Intensity just outside the surface of a conductor — Lines and tubes of force — Outward pull per unit area on a charged conducting surface — Note on the method of investigating the field in chapters III-V Pp. 61-71 CHAPTER VI ELECTRICAL, LEVEL OR POTENTIAL. THE ENERGY IN ELECTRIFIED SYSTEMS Electrical level — Potential at a point — Equipotential surfaces — Intensity ex- pressed in terms of rate of change of potential — General nature of level surfaces — Energy of an electrified >ystem in terms of charges and potentials —Unit cells — Number of unit cells in a system double the number of units of energy — Distribution of energy in a system Pp. 72-79 CHAPTER VII POTENTIAL AND CAPACITY IN CERTAIN ELECTRIFIED SYSTEMS. SOME METHODS OF MEASURING POTENTIAL AND CAPACITY Definition of capacity— Sphere in the middle of large room — Two concentric spheres — Two parallel plane conducting plates — Two long co-axial cylinders — Two long thin equal and parallel cylinders — Long thin cylinder parallel to a conducting plane— Instruments to measure potential difference— Quadrant electrometer— Attracted disc electrometer — Practical methods of measuring potential— Reduction of the potential of a conductor to that of a given point in the air— Simple methods of measuring capacity— Capacity of a Leyden jar— Capacity of a gold-leaf electroscope Pp. 80-98 CHAPTER VIII THE DIELECTRIC. SPECIFIC INDUCTIVE CAPACITY. RESIDUAL EFFECTS Specific inductive capacity — Faraday's experiments— Kffect of f-pecific inductive capacity on the relations between electric quantities— Conditions to be tied where tubes of strain pass from one dielectric to another — Law of refraction — Capacity of a condenser with a plate of dielectric inserted— The effect of placing a dielectric sphere in a previously uniform field — inl charge and discharge — Mechanical model 1'j.. H9-1 19 x CONTENTS CHAPTER IX RELATION OF SPECIFIC INDUCTIVE CAPACITY TO REFRACTIVE INDEX. THE MEASURE- MENT OF SPECIFIC INDUCTIVE CAPACITY The relation between specific inductive capacity and refractive index in the electro-magnetic theory of light — Determinations of specific inductive capacity — Bo'tzmanYs condenser method for solids — His experiments with crystalline sulphur — Hopkinson's experiments — Boltzman-i's experiments on gases — Specific inductive capacity of water, alcohol, and other electrolytes — Experiments of Cohn and Arons, Rosa, Heerwagen, and Nernst— Experi- ments of Dewar and Fleming at low temperatures — Drude's experiments with electric waves 1'p. 120-133 CHARIER X STRESSES IN THE DIELECTRIC Tension along the lines of strain — Pressure transverse to the lines of strain- Value of the pressure in a simple case — This value will maintain equilibrium in any case— There maybe other solutions of the problem— These stresses will not produce equilibrium if K is not uniform — Quincke's experiments — General expression for the force on a surface due to the electric tension and pressure — The electric stress system is not an elastic stress system and is not accompanied by ordinary elastic strains Pp. 134-141 CHAPTER XI ALTERATIONS OBSERVED IN THE DIELEC- TRIC WHEN IT IS SUBJECTED TO ELECTRIC STRAIN Electric expansion in glass — Maxwell's electric stresses do not explain the effect —Electric expansion of liquids— Electric double refraction : The Kerr effect Pp. 142-147 CHAPTER XII PYROELECTRICITY AND PIEZOELECTRICITY Pyroelectricifcy — Historical notes— Analogous and antilogous poles — Some methods of investigating pyroelectricity — Gaugain's researches— Lord Kelvin's theory of pyroelectricity — Voigt's experiment on a broken crystal— Piezoelectricity — The discovery by the brothers Curie — The piezoelectric electrometer — CONTENTS xi Voigt's theory connecting pyroelectricity and piezoelectricity — Electric deformation of crystals — Lippmann's theory — Verification by the Curies — Dielectric subjected to uniform pressure — Change of temperature of a pyro- electric crystal on changing the potential Pp. 148-163 PART II: MAGNETISM CHAPTER XIII GENERAL ACCOUNT OF MAGNETIC ACTIONS Natural and artificial magnets — Poles — Mechanical action of poles on each other —The two poles have opposite mechanical actions — In a well-magnetised bar the poles are near the ends — The two poles always accompany each other in a magnet — The two poles are equal in strength — The magnetic action takes place equally well through most media — Induced magnetisation — Magnetisa- tion can take place without contact — Retentivity and permanent magnetism — The earth is a great magnet — Magnetisation induced by the earth — Methods of magnetisation — Single touch — Divided touch — Double touch — The electro-magnetic method — Electro-magnets — Ball-ended magnets — Horseshoe magnets — Compound magnets — Distribution of magnetisation — Consequent poles — Magnetisation chiefly near the surface — Saturation and supersaturation — Preservation of magnetisation — Armatures — Portative force — Nickel and cobalt magnets — The magnetic field — Lines of force mapped by iron filings Pp, 165-180 CHAPTER XIV GENERAL ACCOUNT OF MAGNETIC ACTIONS— continued Relation between magnetisation and the magnetising force producing it — The hysteresis loop^— Susceptibility and permeability— Magnetisation and tem- perature— Permeability and temperature— Change of length on magnetisation — Magnetisation and strain Pp. 181-191 CHAPTER XV MOLECULAR HYPOTHESIS OF THE CONSTITUTION OF MAGNETS Molecular hypothesis — Ewing's theory and model — Dissipation of energy in a hysteresis cycle— Temperature and magnetisation— Attempts to explain the constitution of molecular magnets Pp. 192-202 xii CONTENTS CHAPTER XVI GENERAL ACCOUNT OP MAGNETIC QUALITIES OF SUBSTANCES OTHER THAN IRON AND STEEL Faraday's classification of all bodies as either paramagnetics or diamagnetics — General law — Explanation of the action of the medium by an extension of Archimedes' principle — Ferromagnetics, pararaagnetics, and diamagnetics Pp. 203-207 CHAPTER XVII THE INVERSE SQUARE LAW Variation of the force due to a pole as the distance from the pole is varied — The inverse square law— Magnetic measurements founded on the law — Coulomb's experiments — Unit magnetic pole — Strength of pole ;// — Magnetic intensity — Geometrical construction for the direction of the intensity in the field of a bar magnet — Gauss's proof of the inverse square law — Deflection method of verification — Moment of a magnet — Comparison of moments — Some conse- quences of the inverse square law — Application of Gauss's theorem— Flux of force— Unit tubes — A pole m sends out 4iw unit tubes — Potential Pp. 208-218 CHAPTER XVIII SOME MAGNETIC FIELDS Small magnets — Magnetic shells— Uniform sphere Pp. 219-228 CHAPTER XIX INDUCED MAGNETISATION Magnetic induction — Induction and intensity in air — Lines and tubes of induction in air — Induction within a magnetised body — Hydraulic illustration — Equa- tions expressing continuity of potential and continuity of induction tubes — Representation of induced magnetisation by an imaginary distribution of magnetic poles acting according to t he inverse square law — Magnetic suscepti- bility—Imagined investigation of B and H within iron — Permeability and the molecular theory — Induction in a sphere of uniform permeability /u placed in a field in air uniform before its introduction — Long straight wire placed in a field in air uniform before its introduction — Circular iron wire in a circular field — Energy per cubic centimetre in a magnetised body with con- stant permeability — The energy supplied during a cycle when the permeability CONTENTS xiii varies — Calculation of induced magnetisation is only practicable when the permeability is constant — Diamagnetic bodies — Magnetic induction in a body when the surrounding medium has permeability differing from unity Pp. 229-252 CHAPTER XX FORCES ON MAGNETISED BODIES. STRESSES IN THE MEDIUM The forces on a body of constant permeability /* placed in a magnetic field in air — Illustrations — The forces on a body of permeability /i.2 placed in a field in a medium of permeability /^ — Forces on an elongated bar in a uniform field in air when its permeability differs very slightly from 1 — The magnetic moment of a small paramagnetic or diamagnetic body placed in a magnetic field — Time of vibration of a small needle with /*-! very small suspended in the centre of the field between the poles of a magnet — Stresses in the medium which will account for the forces on magnetic bodies — The stresses on an element of surface separating media of permeabilities /*j and ^ Pp. 2:>3-265 ( IIAl'TKR XXI THE MEASUREMENT OF PERMEABILITY AND THE ALLIED QUALITIES IN FERRO- MAGNETIC BODIES Magnetometer method — Ballistic method — Magnetisation in very weak fields — Magnetisation in very strong fields— Kwing's isthmus method — Nickel and cobalt — The hysteresis loop and the energy dissipated in a cycle — Mechanical model to illustrate hysteresis Pp. 266-281 CHAPTER XXII MEASUREMENTS OF SUSCEPTIBILITY AND PERMEABILITY OF PARAMAGNETIC AND DIAMAGNETIC SUBSTANCES Faraday's experiments — Rowland's experiment — Experiments of v. Ettingshausen — Curie's experiments — Curie's law — Wills's experiment— Townsend's experi- ment—Pascal's experiments— The electron theory—The rpagneton Pp. 282-300 xiv CONTENTS CHAPTER XXIII TERRESTRIAL MAGNETISM The direction of the earth's lines of force at a given place — To find the declination — To find the dip or inclination — The earth inductor — The intensity of the field — To determine the horizontal intensity H — The vibration experiment — The deflection experiment — Recording instruments — Results — Sketch of Gauss's theory of terrestrial magnetism Pp. 301-319 CHAPTER XXIV MAGNETISM AND LIGHT The Faraday rotation — Faraday's successors — Verdet's constant and its variation with wave h ngth — Effect of rise of temperatnre — Absolute values of Verdet's constant — Rotation by gases — Rotation by films of iron, nickel, and cobalt — Representation of the rotation by two equal circularly polarised ra\s with opposite rotations travelling with different velocities — An electron theory of the rotation — Confirmations of the theory — Magnetic double refraction when the light ray is perpendicular to the lines of force — Voigt's theory — Magnetic double refraction in colloids — Magnetic double refractiou in pure liquids — The Kerr magnetic effect — The Zeeman effect — Lorentz's theory — Zeeman's veiification Pp. 320-340 INDEX Pp. 341-345 PART I : STATIC ELECTRICITY CHAPTER I GENERAL ACCOUNT OF COMMON PHENOMENA Electrification by friction — Actions between bodies both electrified by friction — Two kinds of electrification — Law of electric mechanical action — Electric actions not magnetic — Conduction and insulation — Gold-leaf electroscope — Electrification by induction — Detection of kind of charge by the electro.^ <>[" — Nature of induced electrification — Only the outer .-urface of a conductor is charged — The two electrifications always occur together, iiidm -iiiir ••a«:h oth»T — Discharge of electrification — Frictional machines — Ease of discharge from points — Lightning conductors — Electruphnnis — Bennett's doubler — Belli's machine — Holt/, s machine — Wim>hur>t'.> inachiiif — Th»j Leydt-n jar — Franklin's jar — Residual charge. Electrification by friction. When a stick of sealing-wax has been rubbed with anv clrv woollen material it is found to attract dust, small pieces of paper, bran, or other very light par- ticle*. A drv glas> rod after being rubbed with silk acts in a .similar way, and the property i> Chared by many other substances. When showing this peculiar action the surface of the rubbed rod is said to be electrified, to have undergone electrification, or to po-c— electricity, this term being derived from the Greek iiXetcrpov — amber, a substance \\liicli was. known to the ancients to possess the>e electrical properties. Manv vcrv simple and interesting experiments may be made on electric actions. We may, for instance, attract egg-shells, paper rings, wood laths .suspended by silk threads, ^c., by electrified rods aling-wax, ivsin, ebonite, and glass. Perhaps the most striking experiment is the electrification of dry paper. If several sheets of paper are thoroughly dried before the fire or by ironing with a hot iron, and are, while still quite hot, laid in a pile and stroked sharply with the finger-nails, they adhere very closely together. It .separated, they for a short time show electrification to a very high degree, and attract small pieces of paper with great violence, adhere to the wall if brought near to it, and so on. In making these elementary experiments, we find that for different substances certain "rubbers" are the most efficient. 1 A 2 STATIC ELECTRICITY Thus, resinous materials, such as sealing-wax or vulcanite, should be rubbed with wool or fur, and glass with silk. Actions between bodies both electrified by friction. We may study electric actions between bodies in more detail by means of some such apparatus as the following, which we may term a disc electroscope. Two discs, «, b (Fig. 1), are attached to the ends of a vulcanite or varnished glass rod, pivoted and free to turn on an upright support ; a is glass and b is vulcanite ; c, d are two other discs, of glass and vulcanite respectively, provided with vulcanite handles. On rubbing a and c with silk, and b and d with wool or fur, all the surfaces are electrified. On presenting the glass disc c to the glass disc a, we find that it repels it. We find also that the vulcanite disc d repels the disc i, or that the similar bodies repel FIG. 1. each other when electrified. On the other hand, c attracts l>, and d attracts a. Two kinds of electrification : vitreous or positive, resinous or negative. We have, then, two distinct kinds of electrification — that of the rubbed glass and that of the rubbed vulcanite. Further experiments show that there arc only t! two kinds, and they are often termed, from their simplest modes of production, vitreous and resinous respectively. But it is to he observed that when acting on a given electrified body they exert opposite forces, which tend to neutralise each other. Thus the detached discs c and d may be so electrified that if held close together in front of a or of b they will produce no motion, their two actions being equal and opposite. Hence, in regard to their mechanical actions, we may apply to the two electrifications the terms positive and negative. Vitreous or glass electrification is always taken as positive, and therefore resinous electrification is negative. We may now make the following statement : Law of electric mechanical action. Bodies if similarly electrified repel each other, and if oppositely electrified attract each other. Two bodies rubbed together are oppositely electrified. This may easily be proved by means of the sus- pended glass and vulcanite discs (Fig. 1), electrifying these by GENERAL ACCOUNT OF COMMON PHENOMENA 3 their appropriate rubbers. Each rubber, if very dry and held at the end of a vulcanite handle, will be found to repel the disc rubbed by the other, while it attracts its own disc. Or if two other discs which electrify each other be rubbed together and presented to the glass disc, one attracts and the other repels it. It must not ;be supposed that a given substance is always electrified in the same way. The nature of its electrification depends on its physical condition and upon the rubber used. It is found that substances can be arranged approximately in order, thus + Cat's fur, Glass, Wool, Feathers, Wood, Paper, Ebonite, Silk, — Shellac. So that if any one of these substances is rubbed by one higher in the list it is negatively electrified, while if rubbed by one lower in the list it is positively electrified. The order in such a list must not be taken as unite fixed, since change in physical condition may alter the position of a substance in the list. A polished surface, for instance, appears to increase the tendency to positive electrifi- cation, so that while polished glass is near the head of the list, roughened glass may below down. Electric actions not magnetic. We may at onco dis- tinguish electric from magnetic action by the fact that iron, nickel, and cobalt are, with all the other metals, absent from the above list. Conduction and insulation. At one time it was supposed that the metals and other sub.stancos not showing electric attrac- tion after friction were non-electric, differing fundamentally from >ub>tances such as shellac and glass, which were alone regarded as being capable of electrification. But afterwards it was shown that the real distinction is in the degree to which a substance will allow electrification to spread over or through it. Shellac and dry glass keep the electrification for a longer or shorter time on the surface where it is first developed, and are therefore termed insulators. Metals allow it to spread over their whole surfaces or to be com- municated to other bodies which they touch, and are therefore termed conductors. If a metal rod is held by an insulating handle it may be electrified easily by friction with fur, but if it is touched by the hand, a conductor, it instantly loses its electrification. There i> no doubt a fundamental difference between conductors and perfect insulators. But we term many bodies insulators which arc really very slow conductors, and we may have every intervening 4 STATIC ELECTRICITY stage between moderate insulation, such as that of wool or that of baked wood, and good conduction, such as that of the metals. Sometimes a body may be in itself an insulator, but may have a tendency to condense water from the air on its surface, and t of water will form a conductor. This explains the difficulty experienced in keeping electrification on glass or the still greater difficulty in keeping it on paper. Sometimes the surface of tin- insulator decomposes. This often happens with vulcanite and glass, and the products of decomposition form a conducting layer. The insulation of glass may be improved by immersion for a short time in boiling water. Dust may also prevent insulation by form- ing a continuous coating of conducting matter on the surface of a ng body. yrkivliich .slitlc.9 un down, the rcci. t'ilnT j . \ulfilttirn The hygroscopic tendency of glass greatly impairs its efficiency as an insulator except in a very dry atmosphere, It is. then-fore. necessary in electrical experiments with glass either to replace the glass surface by another less hygroscopic or to take special precautions to keep the surface dry. Often the g»88 Ifl co\» n fihre moistened \\ith strong sul- phuric- acid. Sliding on the rod is a cork which may be brought down to close the jar when not in use. The sulphuric acid dries the nir of the- jar thoroughly, and the glass is therefore also dry. When in use the cork is slightly raised so that there is no connection between the- upper and possibly damp part of the rod and the jar except through the rod itself, which is insulating. Gold-leaf electroscope. The two electrified discs //. I) (Fig. 1) will serve as indicators of charge on another body if that charge is considerable. Fora charged body will repel one or other of the discs. But a much more delicate test of the existence of electrification is given by the gold-leaf electroscope, which in its original form consists essentially of two strips of gold leaf hanging from the lower end of an insulated metal rod in n glass case, the upper end of the rod projecting above the case. Fig. 3 is a common type of instrument. g,g are two long, narrow strips of gold leaf hanging from the end of the brass rod ?•;-, which is fixed in nn insulating bar of vulcanite rv stretching across the case. The rod passes freely through a large hole in the top of the c GENERAL ACCOUNT OF COMMON which can bo closed by the cork c when the instrument is not in use, in order to prevent the entrance of dust. p,p are two brass side plates which can be arranged at any distance on each side of the gold leaves, the sensitiveness of the instrument being greater the nearer the plates are to the leaves. If the wood of the case is non-conducting the side plates should be connected by wire to earth. The reason for this will be seen when we have considered the phenomena of induction. The top of the rod may conveniently end in a metal table /. If this is touched by an electrified body some of the electrification is communicated through it to the rod and leaves, and the leaves being similarly electrified, repel each other and sta:id out as in the figure. An important improvement in the construction of the gold-leaf electroscope which is now usual consists in prolonging the central rod by a thin metal strip, and having only one gold leaf which is attached to the strip near its upper end. The fixed vertical strip then takes the place of the second gold leaf. On electrification the gold leaf stands out from the strip. Fig. 4 represent* a form of the instrument suitable for lantern projection. It i> convenient to have a glass scale engraved on one of the glass sides. Electrification by induction. We have described above the charging of the electroscope by contact with an electrified body. Hut contact is not necessary. On bringing an electrified fi STATIC ELECTRICITY body near the electroscope the leaves diverge, or the single leaf stands out from the strip, showing that they have undergone electrification. The electrification thus produced at a distance without contact is said to be induced. Detection of kind of charge by the gold-leaf electro- scope. If the electroscope be slightly charged, say by rubbing the table with flannel or fur, it will indicate the kind of charge on any body brought near to the table. If, for example, an excited ebonite rod held over the table makes the leaf or leaves stand out further, an excited glass rod makes them fall nearer together, and vice versa. We shall see later how these actions may be explained. Here we take them as the results of direct ex- periment and shall use them to test the nature of induced electrification. Nature of induced electrification. It may be shown easily that whenever an electrified body is brought near an insulated conductor it induces electrification of the kind opposite to its own in the nearer part, and of the kind .similar to its own in the further part of the insulated body, as in l-'i^. ">. in which A is a body on an insulating support, and is electrified. say, positively, while B is a conductor standing on an insulating support. We may verify this by mean* of a /;;-oo/'y;/////<- (1'iu which is merely a small metal disc at the end of an insulating rod. Fio. 4. o Fro. 5. FIG. 0. When the metal disc is laid on an electrified surface it virtually forms, if small enough, part of the surface, and is electrified in the same way as the surface round it. When lifted from the surface it preserves its electrification, and we may directly test the kind by means of the gold-leaf electroscope. Testing the electrification at each end of B (Fig. 5), we find that on the end nearer A to be negative, and that on the end farther from it to be positive. If we now withdraw A, B returns to its ordinary state, the ends showing no electrification. Thus positive and negative have come together and have neutralised each other. GENERAL ACCOUNT OF COMMON PHENOMENA 7 This gives us an additional reason for ascribing the algebraic signs + and — to the two kinds of electrification. Not only do they show opposite mechanical actions, but when they come together in proper proportion the resultant effect is non-electrification. We may describe induction, perhaps, more easily by materialising our conception of electrification, regarding it as something which we call electricity, in addition to and possessed by the matter, rather than of as a condition of the matter. A neutral body we may regard as possessing amounts of the two electrifications practically unlimited and so mixed together as to neutralise each other. Each electricity must be endowed with an action of repulsion on its own kind and of attraction on the opposite kind. This, we must note, is not a repetition of the statement on p. 2 of the mechanical action between electrified bodies. One gives the nature of the force between portions of electrified matter tending to move matter ; the other gives the actions between electrifications tending to move electricity even though the matter is kept at rest. Using this new conception, we may say that the + electricity on A (Fig. 5) decomposes the neutral mixture on B, drawing the opposite — nearer to itself and repelling the like -f? which we may term complementary, away to the other end. This mode of description is merely provisional. We shall see later that we must suppose the medium round the electrified bodies to take part in the phenomena, and when we try to assign to the medium iU share in the action the above conception of electrifica- tion ceases to be adequate. But if we use it, not as a hypothesis but rather as an illustration to aid us for a time in arranging the ascertained facts of electric induction, it will be at least quite harmless. If we touch B with the finger or with any other conductor, so breaking down its insulation, B becomes part of the general conducting system, consisting of the table, floor, walls, and so on, and then either the positive electricity can get further away from A and so leaves B, or A draws up towards itself more negative from the surrounding conductors on to B. Whichever process takes place the effect is the same, viz. B loses all its positive charge and is only negatively electrified. If now we break the communication with the earth, B retains this electricity even if A is removed ; on the removal of A the electricity on B is redistributed, and negative will be found on either end of it. We may now explain the action of an electrified body in the fundamental experiment of the attraction of light bodies, such as bits of paper or bran. The electrified body acts by induction on these light bodies. electrifying their nearer surfaces with the kind opposite to its own, the complementary charge of the same kind going cither to the further side or passing away to the table or whatever the particles rest upon. The bodies thus oppositely electrified attract each other and the particles fly to the electrifiecl 8 STATIC ELECTRICITY surface. Frequently they only adhere for a moment and are then repelled. This is especially noticeable with pith balls suspended by insulating silk threads. They rush to the electrified surface and then rush away, standing out as in Fig. 7. The explana- tion of this repulsion is that on touching the electrified surface, they yield up their opposite electrifi- cation, being slight conductors, and gather from the surface some of its own electrification ; being now elec- trified in the same sense as the surface, they are repelled from it. The gold-leaf electroscope may be used to illustrate some of the pheno- mena of induction. Eet us use the form with two leaves. If a negatively electrified rod, say of vulcanite, be held above the table of the elect m- scope as in Fig. 8, the table is electrified positively and the leaves negatively. The negative in the leaves induces positive in the side plates, their negative going off through t he- conducting communication to the table or floor, or, let us my generally, to the earth symbolised by the plate E. To be MIIV that we get rid of this negative charge it is advisable to connect the side plates to the gas- or water-pipes In wires. FIG. 7. : A : The gold leaves stand out from each other. We described this before as due to repulsion of the like electrifications; but we may now replace this provisional description by sayino- that each leaf is pulled by the plate near it which is opposite! v e'ectrified, and there is still an action of this kind, even if, as in (iKVERAL ACCOUNT OF COMMON PHENOMENA 9 inanv electroscopes, theiv are no special side plates. In that case the surface of the glass may be dusty or clamp and so become oppositely electrified. If the glass is perfectly insulating the opposite electrification is induced in the nearest conducting surfaces and the leaves are still pulled out by the attractions of those surfaces rather than pushed apart by mutual repulsion. But the nearer the oppositely electrified surfaces the stronger its outward pull. Hence the advantage of having adjustable side plates, which may be brought very near the leaves when the charge is small. If while the electrified rod is over the table of the electroscope we either touch the table or allow the leaves to diverge so far that they touch the side plates, the negative from the leaves either passes away to " the earth," i.e. to the surrounding conductors, or is neutralised by a further supply of positive brought up from the earth, and. ceasing to be electrified, the leaves fall together. On breaking the earth communication the positive on the table above remains, and on removing the vulcanite this positive spreads over the rod and leaves, and induces negative in the side plates. These again attract the leaves, which are once more drawn apart and remain apart as long as the insulation is preserved. \Ye can now see how an electroscope, left charged with positive in this way, may be used as a detector of the kind of charge possessed bv another bodv. If a positively electrified body is brought near its table, then the table is negatively electrified by induction and positive is sent down into the already positively electrified leaves, and they are forced still further apart. If a negatively electrified bodv is brought near the table, it is positively electrified and either this positive is drawn up to it from the leaves or negative is sent down to the leaves, and in either case they are less strongly positively electrified and fall together somewhat. If the body is strongly charged with negative and is brought nearer, a point may be reached when the positive in the leaves is just neutralised and they come together. If the body approaches still nearer than this point, then there is a balance of negative in the leaves over and above that neutralising the positive pre\ iouslv there, and the leaves diverge once more. It will easily be seen how the indications arc modified if the initial charge of the electroscope is negative. Only the outer surface of a conductor is electrified unless there are insulated charged bodies within it. This may be proved as follows: A narrow, deep, tin can, is placed on a block of paraffin to serve as insulator and is charged with electricity either bv contact or induction. If we place a proof plane in contact with the inside of the can and then remove the plane to an electroscope no charge is detected, however strongly the outside may be charged. In fact, the (in acta as ;i screen protecting the inside space from outside electrification, and the more nearly it is closed the more complete is the protection. The tendency of electrification to the 10 STATIC ELECTRICITY + -f 4- 4- 4- outside of a conductor may be shown in another way. Placing the can on the paraffin, uncharged, we may introduce within it a charged conductor, such as a proof plane or a metal ball with an insulating handle. On lowering the charged body well within the can and allowing it to touch the inside of the can it becomes part of the inside surface. Its electrification at once leaves it, and on taking it out of the can it is found to be entirely discharged. By aid of the proof plane we can show that there is no electrification on the inside of the can. But before contact the inside of the can was -electrified by induction, the state of affairs being represented by Fig. 9 ; the + on the insulated body within the can induces — on the inner surface and the complementary + is drawn to the outside. But we may still say that the can acts as a complete screen between the inside and the outside. For the + charge on the outside may be shown to remain in the same position wherever the + body is placed within the conductor so long as it is some di>tanee below the top. We may, for example, conduct the experiment with the can on the table of a gold-leaf electroscope so that the charge on the outside extends to the leaves. Their divergence remains the same, how- ever the bodv within the can may be moved about. When it is well Mow the top, the + on the outside is really related now to a — cha which it induces on the surface of the nearest conductors, the table, wall, &c. The two electrifications always occur together, in- ducing each other ; or electrification is always in- ductive. The analysis of these simple cases of induction prepares us for the general statement that all electrification is accompanied by induction, i.e. that whenever we have one kind of electrification we have somewhere, facing it, the other kind, and the bodies whose sur- faces are subjected to these related charges are being pulled towards each other. In the next chapter we shall describe experiments which verify this, and shall show also that the two related electri- fications are always equal in quantity, so that if any conducting connection be made between the oppositely electrified surfaces the electrifications coming together exactly neutralise each other. Thus in the experiment represented in Fig. 9, the + °n the inside body is equal to the — on the inside of the can. The + on the outside of the can induces equal — on the inside of the walls of the room. If the body touches the can the first + and — come together and only the outside + is left inducing the — on the walls. Now, connecting the walls to the can by a wire or by the body, FIG. 9. GENERAL ACCOUNT OF COMMON PHENOMENA 11 these two charges come together and also neutralise each other ami no electrification of either kind remains. When electrification is produced by friction the two electrifica- tions are at first one on the rubber and the other on the rubbed bodies. When the two bodies arc widely separated they may no longer act only on each other, but on the walls. Thus a charged ebonite rod which after being rubbed with wool is held up in the middle of a room has its opposite + on the arm of the holder, on the table, or on the walls. If two rods both negatively electrified are held up near each other they appear to repel each other, but our further study will show that we ought rather to regard them as drawn apart by the opposite walls, which are positively electrified by induction. Discharge of electrification. When an insulated electrified body is touched by the finger or by a conductor connected with the ground the electrification passes away and the body is said to be discharged. We have described this discharge as if the charge merely went to the earth. We can now see more of the true nature of the process. The electrification of the body, when insulated, is accompanied by opposite electrification of the walls, floor, and surrounding conductors, and we may regard the two charges as tending to come together, but unable to pass through the separating insulating medium. When a conducting bridge is made the two charges spread along or through it and unite to neutralise each other. Frequently they are able to break down the insulation of the medium if the electrification is con- siderable, and a spark occurs. In all cases of electric discharge the energy of the electrified system is dissipated in the discharge, either as heat in the conductors, or as light, heat, and sound, and perhaps chemical energy in the spark. Methods of producing electrification in large quantity. Frictional machines. Many different forms of machine have been devised for producing electrification con- tinuously by friction, and for a long time they were in common INC. As they are now superseded for most purposes by induction machines, it will suffice if we describe one form — Winter's plate machine. A full account of the mode of constructing and using tins and other frictional machines will be found in Harris's Frictional Electricity. In Fig. lO/? is a circular glass plate mounted on a horizontal axis, which can bi- turned by the handle h in the direction of the arrow-head. The plate passes at the lowest point between two pads of silk or leather, stuffed with wool, and these serve as rubbers. Their power of exciting electrification is greatly incmiM'd if an amalgam, usually of mercury, /inc, and tin, is mixed with lard and smeared over the surfaces. The pads are backed with wood and connected to earth, either through the woodwork of tin- machine or, if this is not sufficient, directly by a wire to the 12 STATIC ELECTRICITY gas or water pipes. At the level of the hori/ontal radius I he plate passes between two metal combs c — the collectors. These combs have sharp teeth pointing to the plate and separated from it by a small interval to allow of free running. Two silk flaps .9 attached by their lower left-hand edges to the rubbers extend nearly to the combs and are held so as loosely to cover the section of the plate between rubber and collector. " The combs are connected to FIG. 10. a metal body PC, called the prime conductor, which is supported by an insulating pillar and is terminated by a small knob /.. rr represents a wooden ring with a metal wire as core. This ring \V;ls introduced by Winter and is named after him Winter's ring. If the plate is turned the glass is electrified positively and the rubbers negatively. The + on the glass surface is carried round, the silk flaps appearing to protect it in some way till it comes under the combs. There it acts inductively, and draws — into. the points while the complementary + goes into the rest of the prime conductor. Now electrification is discharged through the air from points with great ease, so that the — passes from the points on to the glass plate, which is thus neutralised, leaving the prime con- ductor charged with +. Sparks may then be drawn from the knob A*, their length and brilliancy increasing rapidly with the >i/e GENERAL ACCOUNT OF COMMON PHENOMENA 13 of the plate, the excellence of its surface and of the surface of the rubbers. The advantage of the particular structure chosen for the ring is not very clear. It certainly makes the sparks brighter, and this is probably explained by the fact that it increases the capacity of the prime conductor so that more electricity is stored in it before a discharge occurs, and more, therefore, passes in the discharge. The ease of discharge from points. If a very sharp metal point is placed on the prime conductor it is found that dis- clrirge takes place from it continuously, the air or dust in the air acting to convey the charge to the surrounding and oppositely charged conductors. We need not here discuss the mode of dis- charge. It is sufficient at this stage to say that the electrification on any conductor tends to accumulate at any projecting point on the conductor and to a greater amount per unit area the sharper the point. It appears then that even with comparatively weak charges the crowding of the electrification on to a point may be so great that the air is unable to insulate, and discharge occurs through it. The point appears in the dark to glow. If the hand be held over the point the impact of the electrified air streaming from it is dis- tinctly felt. A sort of Barker"* mill may easily be made to work by means of thi* point discharge. A light k* whirl " represented in Fig. 11, con*i*ting of a number of spokes with ends pointed and bent tangentially, is pivoted on a vertical ^ metal *upport on the prime conductor of a machine. When the machine is turned the uhirl rotate* uith the points directed back- ward*. \Yc may explain the motion thus: Were the point* covered with some efficient insulator the whirl would be in equilibrium, the pull on the strongly elect iificli;ir^c> and tin- u.-r of lightning conductors^ see Sir Oliver Lodge's Li'jhtuii.y Cuitductorv and Liyktniiiy Cf nurds. GENERAL ACCOUNT OF COMMON PHENOMENA 15 a few points. The charged ebonite therefore acts inductively across the intervening layer of air on the cover, and some positive is induced on its under surface and the complementary negative on its upper surface. The cover is now touched with the finger and so earth-connected. The positive charge on the sole can now get very much nearer to the upper surface of the ebonite and we may regard it as passing round through the earth and the body of the operator to the cover, which receives nearly all the charge previously in the sole. The negative charge in the cover we may regard either as neutralised by the addition of positive or as }>;i>sing away through the operator. Now removing the finger, the cover is insulated, and on lifting it by the insulating handle it contains a positive charge which is quite large enough to be useful for many purposes. The charge on the ebonite still remains practically undiminished and at once induces another charge on the sole. By discharging the cover and replacing it on the ebonite the process may be repeated, the original charge of the ebonite sufficing to give a great number of charges to the cover. Though the original charge is thus the exciter of all the succeeding charges, it must not be supposed that it is also the source of their energy. The cover when placed on the ebonite and charged oppositely is attracted by it, and in lifting the cover up again more work inu*t be done than that required merely to raise its weight, and it is this excess of work which gives to the charge on the cover the energy it possesses. The operator therefore supplies the energy of each charge in the act of raising the cover. As illustrating the fact that the charge on the ebonite acts induc- tively through a thin layer of air on the ordinary cover, it is interesting to note that if a layer of mercury be poured on to the cake and touched to earth the cake is discharged. Evidently the mercury fits so closely on to the electrified surface that the two electrifications are able to unite. It is important to notice that the charge taken away on raising the cover is nearly all derived from the sole. When the cover is fir>t placed on the cake a small electrification of the cover, -f below, —above, no doubt occurs. But the greater part of the charge on the cake is still occupied in inducing the charge on the sole. When the cover is touched the sole charge rushes up into the cover, and really forms the charge carried away when the cover is raised. This may easily be proved by placing the electro- phonis when excited on an insulating pillar. If the cover is put on the cake and touched, the sole charge cannot now pass into the cover and a very small result is obtained. The electrophone is a convenient instrument only when a Mnall quantity of electricity is required. For though any number of iH.-.-irly ( (jual charges can be obtained on the cover in succession from the same electrification of the resinous cake, and may be communicated to a suitably arranged conductor, it would be both 16 STATIC ELECTRICITY troublesome and slow to gather together any large quantity in this way. 3A Bennett's Doubler. — Very soon after the invention of the electrophorus means were devised for increasing the output, the earliest being Bennett's Doubler, invented in 1786 for the purpose of increasing very weak charges. It consisted of three metal plates A B C, Fig. 13, varnished as indicated by the black lines, A on the upper side, B on both sides, and C on the under .side, the varnish acting as an insulator. B and C were provided with insulating handles. Let us suppose the following operation to be gone through : A is placed on an insulator, which we may suppose to be tin electroscope as that will ser\e as an indicator of increasing charge ; B is placed on A. If a conducting body charged, say, positively touches A, while B is touched to earth, almost all the charge will galli- A, for there it can be quite do>e to the opposite negative charge induced on the under side of B. This nega- tive charge is nearly equal to that on A. Now insulate B and remove the body; let B IK- raided and brought again>t C; touch C to earth. A positive charge nearly equal to the negative on B or the positive on A gathers on ( '. In>ulute C and replace B on A; touch B to earth; touch (' to A, uhen nearly the whole of its charge will pass to A. being added to and thus almost doubling the original charge. Repeating the process after each series of operations, the charge on A is nearlv doubled and may soon be made a very large multiple of its original value, as the electroscope will show. The disadvantage of this form of apparatus is that it requires considerable manipulation, and it \\a> soon followed by instruments in which the process was carried out mechanically by the turning of a handle. Probably they were not very efficient, for they seem to have been considered merely as curiosities. One machine, invented by Belli in 1831, should have secured more notice. It appears, however, to have been neglected, and it was not until the independent invention of machines bv Varley, Thomson, Toepler, and Holt/, about the years 1860-65, that general attention was directed to the subject. Since that time various forms of machines, all dependent on induction or " in- fluence "and not on friction, have been devised. These are termed Influence Machines, and have now practically displaced the old frictional machines as sources of electrification. As, perhaps, the simplest in principle we shall first describe Belli's machine. FIG. 13. GENERAL ACCOUNT OF COMMON PHENOMENA 17 Belli1 s machine.* In the form here described there are two metal plates, the field plates A B,bent, as in Fig. 14, like the covers of a book and fixed on insulating pillars. The movable part consists of two circular metal plates, the "carriers'" CD, at the extremities of a glass arm. By turning a handle these can be revolved on a horizontal axis so that they pass through the spaces partially en- closed by the field plates. Projecting from the inside of each field plate is a spring ,9, which each carrier touches as soon as it is completely within the field plates. After contacts with the springs Fro. 14. are broken the carriers come in contact with the two ends of a wire // //, the "neutralising rod " of later machines, which puts them in electric communication while still within the field plates. To understand the action of the machine let us suppose that to begin with there is a positive charge on A and that the carrier plates start from the position of contact with the wire n H, C being within A and D within B. Then negative electricity is induced on C and positive is sent to D. These charges are carried round until C touches the spring s within B,andD touches that within A. Since the carriers are almost enclosed by the field plates and are in contact with them, their charges, tending to get to the outside, pass almost entirely to the field plates, increasing the + charge on A and giving a — charge to B. The carriers then move on to con- tact with the ends of the wire n n, when D will be negatively and C positively electrified by induction. Further rotation brings C again within A and D within B, and on contact with ss the charges go to increase those on the field plates. Thus the charges on the field plates rapidly increase. * I)i VTUI iin>,r,i 1,,'iniera di mvcchinsi cllctrica, Ann. Sci. Lomb. Veneto, I. (1831), p. 111. 18 STATIC ELECTRICITY It is not. always necessary to electrify a field plate to set the- machine in action. There is frequently some charge to begin with, perhaps due to friction of a carrier against a spring, a the two plates are not electrified equally the machine will the larger charge and soon make the smaller charge of the oppo sign to it, if it was not so initially. . In 1860 C F. Varley* invented a machine similar in prn to that of Belli, and later Lord Kelvinf devised a modified « replenisher " used to charge the jar in his Quadrant El* Fir,. 15. Other important developments were made in Toeplers machine and Holies machines.! We shall only describe the machine usually employed in this country and introduced by Wimdnmt 1883, a modification of the previous machines somewhat moiv certain and convenient in its working. The Wimshurst machine. The common form c machine is represented in Fig. 15. It consists of two exactly similar glass plates which are rotated close to each other with equal * Specification of Patent, Jan. 27, 1860, No. 200. f Pavers on Electrostatics and Magnetism, p. 2/0. J A full account of Toepler's and Holtz's machines Electrical Influence Machine \ GENERAL ACCOUNT OF COMMON PHENOMENA '19 velocities in opposite directions on the same axes. On the outer faces of the plates are equal numbers of thin brass sectors arranged radially at equal intervals. On the horizontal diameter are the collecting combs connected with the discharging knobs which are placed above the plates. Two small Leyden jars have their inside coatings connected one to each of the collecting combs, while their outside coatings are connected with each other. At 45° or there- abouts to the horizontal, the best position being found by trial, are two neutralising rods, one opposite each face, and nearly at right angles to each other. These terminate in wire brushes which touch the sectors as they revolve. To start the machine most easily, the discharging knobs are placed nearly in contact and the handle is turned. After a few revolutions of the plates, if the FIG. 16. machine is fairly dry and dust-free, sparks will pass. The knobs may now be separated gradually, and the sparks will continue to pass till the limiting length is passed. Even when the machine will not excite itself it may readily be set iii action by holding an electrified body near one of the brushes on the far side of the plates from it, and then rotating the plates. The Wimshurst is one of the most easily worked and most regular of the influence machines. The explanation of its working can be given most readily by aid of a diagram in which, as represented in Fig. 16, the plates are re- placed by co-axial cylinders. We may then distinguish them as inner and outer. Let us first leave out of account the discharging circuit and suppose only that we have the two neutralising rods m^mv ?t1;?2. I,et the plates revolve as in the figure, and let us suppose that in some way, say by friction as it passes the brush n19 a sector at the highest part of the inner plate is slightly positively electrified. As this comes opposite ml it induces — on the sectors of the outer plate under mv and sends -+- on to the 20 STATIC ELECTRICITY sectors of the outer plate under w2. The smaller the sectors and the more numerous they are, the greater will be the quantity of - induced through m^ by the + on the inner plate, and it may be much more in total amount than the original + sufficient number of sectors pass m1 while the inducing charge near that brush. Now the -- passing round on the outer plate arrives opposite wx when the + on the inner arrives at n.2, and, meanwhile, the + given to the outer plate through m2 arrives opposite w2, and under these charges not only is the + on the inner plate under wa neutralised, but a — charge is induced on it there while a + charge is given to the sectors under wr These are carried on, and when under mlm2 they induce - and + on the outer plate, while when under n^i2 they are reversed. FIG. 17. The charges on the different parts of the plates are therefore as in the figure, and if the sectors are sufficiently numerous the charges will mount up until the increase is balanced by leakage. To understand the necessity of a number of sectors it is sufficient to consider the case represented diagrammatically in Fig. 17, where there are on each plate only two sectors at the opposite ends of a diameter and so arranged that they meet under the brushes. Then if one inner sector receives a charge + Q the most it can do is to induce — Q on the outer sector when the latter is under a brush, the opposite sector being charged with + Q- These + and - charges cannot do more than induce — Q and + Q when in turn the inner plate sectors are under the brushes, and so on, so that the charges do not increase however long the rotation is continued. The addition of the discharging circuit does not alter the nature of the action ; it only leads to an earlier reversal of the charges on the sectors. Thus the + on the inner plate and the + on the outer plate running in opposite directions past the right-hand GENERAL ACCOUNT OF COMMON PHENOMENA 21 combs will clearly tend to draw out — and leave + on the right- hand discharging rod. On the left-hand rod, — will be left, and if the knobs are not far apart the charges may soon be sufficient to spark across. But if the knobs are so distant that no sparks pass, the charges on them may soon reach a steady state, and no more electrification being taken from the combs, the machine works just as if the discharging circuit were removed, and increase is balanced by leakage. Storage of electrification. The Leyden jar. The Leyden jar is a device for collecting and storing large quantities of electrification. It usually consists of a glass jar, as shown in Fig. 18, lined inside and out with tinfoil to perhaps three-quarters of its height ; the neck of the jar is closed by a cork, or preferably FIG. FIG. 19. some insulating material, through which passes a metal rod termi- nated above in a knob and below in a chain, which puts it in communication with the inner coating of tinfoil. If the j.ar is held in the hand, the outer coat is connected to earth, and if the knob is applied to the prime conductor of a machine, the positive charge from the prime conductor passes into the jar and collects on the outside of the inner coating. There it induces a negative charge on the inside of the outer coating. If the knob is held at some distance from the prime conductor the jar is charged by a scries of sparks, and the successive rushes of negative elec- tricity to the outer coating through the arm may frequently be felt. If the source gives large charges in each spark, the rushes through the arm would be painful, and the arm must be replaced by an earth-connected wire. To discharge the jar it is only necessary to connect the two coatings, or to connect the inside to earth. As it is disagree- able to the operator to form part of the connecting circuit it is uMial to employ a discharger, consisting of metal tongs provided 22 STATIC ELECTRICITY with an insulating handle (Fig. 19). The knobs being separated a proper distance, one is first brought against the outer coating, and the other is then brought near the knob of the jar ; the discharge takes place by means of a spark more or less brilliant, according to the charge in the jar. If the tongs are brought first against the knob, discharge sometimes occurs through the operator if there is faulty insulation in the handle. It is therefore advisable to observe the order above described. With a given machine as source of electrification the maximum charge collecting in a jar, which we may take as indicating its electrical capacity, is pro- portional, as we shall prove later, to the area of the coating, and inversely as the thickness. It depends also on the nature of the insulating material of which the jar is made. It would, for example, be greater for an ebonite jar than for a paraffin jar of the same dimensions. Sometimes, instead of a jar, a fiat pane of glass is used, coated on it> two >i. \Ve may evidently regard thi" a» a jar opened out. 'Franklin's jar. If the coating is not ab>o- lutely in contact with thegla^s the charge' resides on the surface of the glass, passing, no doubt, through the intervening thin layer of air. Thi> \\a^ pm\«l by Franklin by means of a jar with movable coat- ings. Fig. 20 represent > the separated part" of such a jar. The lowest i" a tin cup to form the outer coating. The middle is a glass beaker which fits into the cup, and the upper is a tin cylinder to form the inner coating when dropped into the beaker. After charging the- jar we mav remove first the outer coating, and then the inner by an insulating hook so as to prevent earth connection before the coating i> removed from the jar. The two coatings will now he found free from cli After building up the jar again, however, a dis- charge may be obtained on connecting the coating", showing that \\hen the coating" wen- removed the two surfaces of the glass still retained the ch This may seem at liist sight inconsistent with the behaviour of the electrophorus, in which the charge on the cover does not pass acro» the intervening air to the surface of the ebonite. But it must be remembered that in the jar the charge is in general far greater than in the elect rophorus, and its tendency to get over the intervening air space is thuefore greater. Residual charge. If a heyden jar is charged, left for a short time, and then discharged, it appear." to be entirelv fret- from electrification. Hut if left again for a short time another charge of the same kind as the previous one i.s found to have FIG. 20. GENERAL ACCOUNT OF COMMON PHENOMENA 23 ppeared, but much less in quantity, and another spark may be obtained. Leaving it again for a time after discharge, another but still smaller charge gathers and a third spark may be obtained, and so on. These successive charges are termed residual charges, and the successive sparks residual discharges. We shall give the probable explanation of the phenomenon later. CHAFFER II QUANTITY OF ELECTRIFICATION Use of the electroscope to indicate equality of charge — The two electriii (ions always appear or disappear in equal amounts whether the electritir i tion is by friction, by conduction, by induction, or by supply from current — Electrification resides only on the outside surface of a conductor unless it contains insulated charged bodies— An inflated charged conductor inside a hollow conductor induces an equal and « harge on tli- inside and an equal like charge on the outbid- •— Imagined construction of multiples and submultiples of an arbitrary unit of charge — Imagined method of measuring any charge in terms of this unit. WE shall now describe a series of experiments, some illustrating the general laws of the production and distribution of elect ri; tion, and others showing that we may make definite and eon>i>tent measurements of quantity of electrification. Use of the electroscope to indicate equality of charge. If a deep and Harrow metal ves.sel, such as an ordinary round tin canister, is placed on the upper table of a gold-leaf electro* scope, it may be used to detect a very .slight charge of electri- fication on a body, by the introduction of the botlv \\ithin the can. It is not necessary to communicate any of the charge to the can, for it will act inductively, calling up an opposite- charge on the inside of the can and sending a like charge to the gold leaves.* As the body is being lowered within the can the divergen the leaves rapidly increases at first, but soon it approaches a limit, which it does not exceed, and the leavo remain practically !i however the body be moved about within the can, so long a> it does not come near the open top. If the can be closed by a lid with only a small hole for the insulating holder of the body to pass through, the body may IHJ brought quite close to the top, without change in the divergence of the leaves. Thus a change in the position of a charged bodv within a closed can does not affect the indications of an electro- scope connected with the outside of the can. This result, which is * In tin- experiments described in thi> rhapti-r tin- gold-leaf electron o|.»- may be replinvd by tin- Quadrant Klectrom.-ter du.M/ribed <>u p. 87, the ran i insulated and connected with one pair of quadrants. Much i:r«-:it«T M-n.-itiveness is thus obtained, but the gold-leaf elect roM-ope may be made Hittirieiitly sensitive for purposes of demonstration bv briniriiii: tin- >id.- plat»-s nearer the leaves, ami it is much more easily used than the Quadrant Electrometer. L'4 QUANTITY OF ELECTRIFICATION 25 of great importance, may be verified with great exactness by connecting the can to earth until the charged body is inserted. The induced like electrification on the outside of the can is discharged and the leaves remain together. They are then in the most sensitive position, and the sensitiveness may be still further increased bv approach of the side plates. If there were any change in the effect on the leaves due to change of position of the inside body it would now be noted most easily. Here we have a definite effect due to a charged body, which may be easily observed, and one which is consistent. For on successive withdrawals and insertions the same divergence of the leaves is obtained, and the exact position of the charge within the can is unimportant. We may, therefore, use this effect to compare two different electrifications as to quantity. If two charged bodies inserted successively, produce the same divergence, they are to be regarded as having equal charges. If the divergence is not the same the greater divergence corresponds to the greater charge. Again, if two oppositely electrified bodies are inserted in the can at the same time, they tend to send opposite electrifications into the leaves and so to neutralise each the effect of the other. If the leaves do not diverge at all, the charges are to be regarded as equal in amount though opposite in kind. We have thus defined what we mean by equal charges on different bodies, whether like or opposite in kind, and \\e can now state the fundamental law of the production and disappearance of electricity, viz. : The t:co electrification* ti/icat/Jt r.r/V hi equal amounts. If any amount of one u/i/H'tirx or di^ippetir.^ an equal amount of the- other appear* or d'tMi/tjtetir.f at the same tune. The truth of this law may be tested by the following simple experiments with the gold-leaf electroscope and can. Electrification by friction. If a rod of sealing-wax and a rod of glax> are rubbed together within the can the leaves do not diverge. They are both electrified, however, for on withdrawing OIK- of them the effect of the other is at once evident. But since, before the withdrawal of one, the leaves did not diverge, it follows that the two kinds of electrification are produced in equal amounts, one on the sealing-wax and the other on the glass. \V uiav u^e other pairs of substances, and in every case we shall find that the two bodies rubbed together are electrified oppositely with equal amount^. Electrification by conduction. If we first lower an insulated charged conductor into the space inside the can, touching the can to earth meanwhile so that the leaves remain together and in the portion mo>t sensitive to variation of charge, and if we then lower a >< (oiid iiiMilated uncharged conductor into the can till it toiichc.s tin- fir-t. ire know that conduction takes place from the first to the MTond. Hut the electroscope #ives no indication; then the total amount of electrification remains the same, though 26 STATIC ELECTRICITY now it is on two bodies. Again, charging the second body oppositely to the first before inserting it into the can, and keeping the can untouched to earth until both conductors are within it but not in contact, if we now bring them together the smaller charge will be entirely neutralised by a part of the larger and will disappear, while the rest of the larger charge will spread over both bodies. But the electroscope remains unaffected l)y this process, or the algebraic sum is still the same. The two kinds have, therefore, disappeared in equal amounts. Electrification by induction. Returning to the first of the two preceding experiments, while the second uncharged body was being lowered into the can it became electrified bv induction on approaching the charged body. This we know from our general experiments on induction, described in Chapter I. Since the electroscope gave no indication of change of amount, it is evident that equal and opposite amounts must have been induced on the body. The unlike was near the first body, the like mimic from it. On contact the unlike was neutralised bv some of the charge on the first body. So that it is again evident that the t\\o kinds disappeared in equal amounts. Electric current. We shall see hereafter thai UK • clec •trie- current as produced by voltaic cells or dynamos may he regarded as a conduction of electrification along the win- in which it occurs. The quantities conducted even in a short time arc- usually enormously great compared with the quantities with which \\e deal in experiments with the gold-leaf electroscope on cha: produced by friction or induction. Making a small voltaic cell, and connecting the plates together by a \\ire M> thai the current flows, let us insert it within the electroscope can. No \<-ti'm • of charge is shown. Hence the two kinds are being produced and are disappearing in equal quantiti The distribution of electrification on conductors. Electrification /v.v/Vr.v only vn the out side surface of conductor*. unless, being hollow, they contain insulated charged bodiix. If the deep can of the previous experiments, or preferably a hollow insulated metal globe with a small opening to the inside, i- charged in any way, a proof plane inserted inside through the opening by an insulating holder and brought into contact with Ihe inside shows no trace of electrification on withdrawal. Or, if a charged conductor is brought within the hollow conductor and touched to it, it is entirely discharged. On contact it forms parl of the inside of the hollow conductor and can no longer keep its electrification. Faraday (E.i'j). KM. vol. i. §§ 1173-4) made experiments on a large scale to show that the charge of a conductor resides on the outside. A cube with each edge 12 ft. Ion;/, consisting of a light wooden framework covered in with paper, was made thoroughlv conducting by copper wire and bands of tinfoil. It was insulated QUANTITY OF ELECTRIFICATION 27 and charged by a powerful machine till " large sparks and brushes were darting off from every part of its outer surface.11 Faraday " went into the cube and lived in it, and using lighted candles, electrometers, and all other tests of electrical states,11 he was unable to detect any charge within the cube. Faraday devised a very simple and interesting method of showing that charge resides only on the outer surface of a con- ductor by an experiment known as the " butterfly-net experiment.11 A conical muslin bag (Fig. 20A) which may be 10 or 12 inches long is fitted on to a ring about 4 inches in diameter, placed on the top of an insulating pillar with a base fixed to the table. The muslin is preferably slightly starched. After it is starched two FIG. -20 A. insulating -silk threads air attached to the apex, one inside, one out, so that the bag may be easily turned inside out merely by pulling one of the threads. A charge is then communicated to the bag, and it can be detected on a proof plane which has been applied to the outer surface while no charge is thus detected on the inner surface. The bag is then polled inside out by one of the silk threads, and the outside, which was previously the inside, is found to possess the whole charge. When (in in xulntcd charged conductor is brought within a hollow conductor It induces an equal and opposite charge on the inside of the hollow conductor, while an equal like charge goes to the outside. Let Fig. 21 represent a deep can on the table of an electro- scope and let a positively charged insulated conductor be inserted within the can. Outside the can and in the leaves there is a positive charge, as may be shown by the collapse of the gold leaves on bringing near the can a negatively electrified body, say an ebonite rod. In>ide the can there is a negative charge, for if we touch the can to earth to discharge the outside positive electrifi- cation and then withdraw the charged body from within, the 28 STATIC ELECTRICITY outside of the can and the gold leaves become negatively electrified. We may at once prove this by again bringing up the ebonite rod, when the leaves will diverge still further. Now, discharging the electroscope and again inserting in the can the positively charged body, we know from our experiments on induction that the quan- tities on the inside and the outside surfares of the can are equal, though opposite. Touching the charged body to the inside of the can, it is entirely discharged, and at the same time the inside charge of the can dis- appears, while the outside charge is un- affected. Hence the inside charge must liuvr been exactly equal in amount, though opposite in kind, to the charge on the inserted Ixxlv. Faraday used the cube already described to show that a charge within a conductor induces an equal and opposite charge on the inner surface of the conductor. For this purpose he passed a long glass tube through the wall of the cube so far that it reached well within it. A wire from a large electric machine passed through the lube, and bv discharge from the end of the wire the air within the cube could be highly electrified. If while the machine was being worked the cube remained insulated, the outside became strongly charged. If it was put to earth this charge was conducted away. If at the moment that the machine was stopped the cube was insulated again, no charge gathered on the outside, showing that the electrification of the air within \\a> exactly neutralised by the charge which it induced on the inner surface of the cube. Imagined construction of multiples and sub- multiples of an arbitrary unit of charge. 1 fa fore pro- ceeding further with the account of electric induction it may assist us if we consider a method of making quant itathc measure- ments of charge which depends upon the foregoing experiments. The method is impracticable owing to the impossibility of perfect insulation, but. as a conception, it is a legitimate deduction from the principles to which we have been led, and it KTrefl to show that we can attach a definite meaning to the term '• quantity of electricity." Let us suppose that perfect insulation is possible and that we have a conductor on a perfectly insulating stand. Let us impart to A, Fig. 22, some arbitrary positive charge. With a perfect insulation this charge will remain on A for ever, and so being definite and consistent we may chooie it as the unit of charge. FIG. 21. QUANTITY OF ELECTRIFICATION 29 Let a small can B have a lid furnished with an insulating handle h and let the under side of the lid have some arrangement by which the base of A can be attached when A is inverted. Let H be an insulating handle by which the can may be lifted up. Let an insulating base plate be fixed to the can. Let C and D be two other cans similar in general plan to B, but large enough to con- tain it. Let a metal piece S be fastened to the lid of each of the larger cans so as just to touch the smaller can when inside. We shall now show that we may give to C a charge equal to FIG. 22. any positive multiple of the unit, and to D a charge equal to any negative multiple. To charge C with a positive multiple, place the unit within B, place B within C, and put the lid on C, taking care not to touch the conductors in any case, but moving them always by their insulating handles. Then following out the process by the aid of the upper row of diagrams in Fig. 22, it is easy to see that when B is taken out of C, C will have a positive unit on the outside. On taking A out of B, B may be inserted in D, as represented in the lower row, and its negative charge is imparted to I). On taking B out of D, B is free from charge. As A has its original charge we may repeat the process any number of times, each repetition adding a positive unit to C and a negative unit to D. We may imagine the preparation in this way of a series of multiples of the unit, both positive and negative, by having a number of cans like C and D. By making a number^of exactly similar cans, charging one with a unit and sharing this unit 30 STATIC ELECTRICITY with another, then sharing one of the half units with another, and so on, we may suppose sub-multiples prepared on the scale i,J,i,&c. Imagined method of measuring any charge in terms of the unit; We may now use these quantities some- what as we use the weights of a balance. Suppose that we wish to measure the quantity of electricity on a given body, let us place it insulated within an insulated can connected to an electro- scope. Then let us place within the can and also insulated from it quantities of the opposite kind of electricity until the leaves of the electroscope fall together. Then the two kinds are equal in amount. It is evident that we may keep the can to earth while we are inserting the charged body and the opposite quantities and then test their equality by lifting them all out at the same time. If the opposite charges are unequal the electroscope will at once show a charge. CHAFrER III PROPOSITIONS APPLYING TO 44 INVERSE SQUARE" SYSTEMS The inverse square law — The field — Unit quantity — Intensity — Force between quantities m-^ and mz — Lines of force and tubes of force — Gauss's theorem — If a tube of force starts from a given charge it either continues indefinitely, or if it ends it ends on an equal and opposite charge — The product intensity x cross-section is constant along a tube of force Avhich contains no charge— If a tube of force passes through a charge and q is the charge within the tube, the product intensity x cross-section changes by lirq — The intensity outside a conductor is 4ir2. Experiments which are described in Chapter V show that a small body charged with ///t acts upon a small body charged with m2 and distant d from it with a force proportional to m± Wg/rf2, whatever be the unit in terms of which we measure the charges. Unit quantity. The inverse square law enables us to fix on the following convenient unit: If two small bodies charged with equal quantities would act on each other with a force of 1 dyne if placed 1 cm. apart in air, each body has the unit charge on it. This unit is termed the Electrostatic or E.S. unit. If two bodies have charges mt and m2 in E.S. units and are d cm. apart, the inverse square law shows that the force between them is mi m2/d'2 degrees. Intensity of field. The intensity at a point is the force which would act on a small body placed at the point if the body carried a unit charge. The intensity due to a charge m at a distance d is m/d?. Lines of force and tubes of force. If a line is drawn in t In- field so that the tangent to it at any point is in the direction of the intensity at that point, the line is termed a line of force. A bundle of lines of force is termed a tube of fom •. < )r we may think of a tube of force as enclosed by the surface obtained by drawing tin- lines of force through every point of a small closed curve. We shall now prove a theorem due to Gauss which enables u> to obtain the intensities in certain cases very simply, and which also 'shows us that lines and tubes of force indie ate for us the magnitude as well as the direction of the intensity at every point in their length. Gauss's theorem. If we take any closed surface S, and if N is the resolved part of the intensity normal to the surface at the element ^/S. positive when outwards, negative when inwards, then f^d* = 4--^, where (J is the quantity of charge within the surface S. Any charge without the surface makes on the whole no contribution to JNdS. To prove this let us consider an element FIG. 28. of charge q, situated at a point O within a closed surface, such as is indicated by S in Fig. 23. Let an elementary cone of solid angle dw be drawn from O, intercepting an element of area dS of the surface S. Let this cone "INVERSE SQUARE" SYSTEMS 83 be represented on a larger scale in Fig. 24-, in which AHBK repre- sents dS. The intensity at dS may be taken as q/OA2 outwards along the axis of the cone, and if 0 is the angle which the axis makes with the normal to AHBK, then N = ? and OA2 But if we draw ALCM, a section of the cone dw, through A B FIG. -Jl. and perpendicular to the a\i>, the angle between AHBK and A I. CM is fJ, and since die i^ ><> -mall that BC is practically parallel to AO, ALCM = dScos0. Hence NWS ~ If we now >um up for every element of the surface, the contribution of fj i> J SdS =f qd(t) = qj du = 4nrq since the total solid angle round q is 4?r. The normal intensity at any point of S due to any number of elements of charge is equal to the sum of the normal intensities due to the separate elements. Hence for the whole charge Q within S we shall have to add up all the quantities such as 4^^, and so Now let (] be outside the surface at O, Fig. 25, and let an elemen- tary com- (Its cut. S in r/Sj at A, and JS2 at B. Let the intensities at A and B normal to the surface be ^ N2 respectively. Then, as we have just proved, NjCfSi^^w, while at B, N being in wards and negative, — — f/dw. The two, therefore, neutralise each other. Tins 34 STATIC ELECTRICITY will hold good for every elementary cone drawn from () to cut S- and therefore an element of charge g without S on the whole makes no contribution to /NdS, and only the charge within S counts. We have taken the simple case in which each cone drawn from a point inside cuts the surface once, and each cone from the out- Fio. 25. side cuts it twice, but it is easy to sec that the ioult> arc true if the surface is re-entrant, so that it is cut any odd number of times in the first case and any even number of times in tl it- case. We shall now apply Gauss's theorem to tubes of force. If a tube of force starts from a given charge it either continues indefinitely, or if it ends it does so on an equal and opposite charge. Let a tube start from a (juantity 7 on an element of surface Sr Fig. 26'. In dcM-ribing the t ; art ing from q we imply that within SA the intensity is /no. Suppose that the tube continues some distance from SA and FIG. 26. that it ends at P, as represented in the figure, without reaching any other charge. Prolong the tube within Sr make another closed end, and to the closed surface thus formed applv Gauart theorem. Over the whole of this surface N = 0, for alon^ the " INVERSE SQUARE " SYSTEMS 35 side.^ of the tube the intensity is parallel to the surface of the tube and has no component perpendicular to it, and therefore J NWS = 0. But J NWS = 47T dotted within Sx and S, and thu> make- U<> ciuk Apply (iati^V theorem to the closed surface thus formed. Then J NWS = 0, since N = 0 at every point, and l-('/i+'72) = 0. Hence q^= —qr The product intensity x cross-section is constant along a tube of force which contains no charge. Let SjSg, ed ion- of >uch a tube, and let I^ be the inten- at those section.-. Let us apply Gauss's theorem to the closed Fio. 28. MII face formed by the tube between Sjand S2and by the ends SjSg. N ha> no value over the tides of the tube since the intensity is along the tube, Over the end S2, N(/S = o . rtheendS^ NWS a I1S1 — IjSr ne^ali\. I inuard-. Then J Nf/S = 1^ — I2S2. But >iii( e there i«, no charge within the tube between Sx and N 3 ' 'iid thi-refon- 1^!— I2S2. 36 STATIC ELECTRICITY If a tube of force passes through a charge and 7 is the charge within the tube, the product intensity x cross-section changes by 4^. Let SXS2, I'm- ::'.). t>< MU-II a tube having charge q within it between Sx and S2. Applying Fio. 29. Gauss's theorem to the tube and its ends SjSr it is evident that TO _ T Gl JThe2 intensity just outside a conducting surface is normal to the surface. Let «, Fiic- '50, represent the section of a small circular area of a conducting surfa. « (harmed with a p» T unit area, and let a be so small thai it ma\ In • regard « -d M pi the surface density of itfl charge as uniform. Let it becircul.. l^Pjj be two points Oil the a\is of ,, respectively at e.|iial ({Malices just outside and just inside the surface and M> near that \\ 1', is indefinitely small compared with the radius of „. Now the intensity at ////// point may )»< -1 as tin i> Miltant of the intensity due" to «, and the intensity due to tin tin- surface and to oth> in the system which \\ r ma\ di ii- tixely hv S. At 1', the inteiisitx is • point within I ductor, so that the intensity thue due to S is eijual and oppu- that due to «<. 1-Vom that due to K is normal. So also must that due to S be normal. At l\ the intensity due t«»> is theaame as at Pr since the distain < the points is negliffiUe comjMired FIG. 30. with the distance of ilie nearest part* of S. The intensitx at \\ dut is equal and opposite to its value at P2 and is tli 1 to and in the same direction as the intensity due to S. and hoth are normal to the surface. The intensity just outside a conductor is 47r! = 4x, is clo^e to the surface. bv the last proposition it is parallel to the surface, and also has unit area, and therefore Ij = 4— - Representation of intensity by the number of lines of force through unit area perpendicular to the lines. Along a tube of force intensity X cross-section is everywhere constant if there i- no charge within the tu!x>. Draw a cross-section Sj where the intensity is jp and imagine that through each unit area at S1% ll lines of tone pass along the tube. There will be in all IjSj lines in the tul>e. Let S, be a section further on where the intensity is It. All the IjSj lines pass through S2, and the number through unit area of it will Ix* c"^^ '2' or W}^ ^ equal to the 1 ^ intensit \ - Hence if ue draw lines of force from any surface at the rate of I lines p< r unit area (the ana being perpendicular to the the mnnlxT passing through unit area at any point in their coins,- (the area being perjK-ndicular to the lines) will l>e equal e intensity at that area. If the area of the surface from which we draw the lines is S. the total nuinlxT through it, f I}>« n out the field is \uaker. Number of lines of force starting from unit quantity. If a charge q is at a point, the intensit \ OY< r the surface of a sphere vn with the point as centre is l>\ [-mall, m and positive electric-it v are replaced by sources, and South -seeking magnetism and negative electricity are replaced 1>\ sinks. The quantities of fluid introduced or withdrawn are proportional to the quantities of matter, magnetism, or electricity which they represent. Lines and tubes of flow in the fluid system will then follow the same course as lines and tubes of force in the correspond}!. ijTSteOM. Drawing a displacement tube in the fluid system, since no fluid passes out through its walls, equal quantities must pass across every section in any part of the tube not containing sources or sinks. If, then, dl d2 be the displacements at two sections St Sr r/xSj = r/2S2, or displacement X cross-section i^ constant. Replacing displacements by the corresponding and proportional intensities, we obtain for a tube of force the corresponding property that intensity X cross-section is constant. We may also prove easily the other properties of tubes of force, but these ire le.-ive to the reader. \\ shall now find the value of the intensity in some simple Spherical shell uniformly charged. Intensity out- side the shell. Ix.-t S be a sphere, l«'ig. 33, over which a e radial, too. at a distance from this plane small compared with the distance from either end, if the distance from the lao small compared \\ith the distance from the ends. f 11. I 'ids. and ( ' is the central point, take a F CD E' point I*, not (jiiite in the plane through C. Draw PD perpen- dicular to the axis and make DF---DE'. The intensity due to 11. bei identlv radial. That due to EF is in comparison negligible, since EF is )>\ supposition not large, and it is very distant from 1' -.mpared with the part of the cylinder immediately under P. II«-nce tli<- intensity at P is radial. Further, if the position of P ( -liaiiiM -s bv a small amount parallel to the axis the intensity is oulv changed by a removal of a small length from one end of the cvlindcr to the other with negligible effect. Draw a cylinder radius ; and length PQ = 1. I • Jfi, co-axial with the charged c\linder, and applv (iatis^s theorem to the surface thus formed. The intensity I, as we have just seen, is normal over the circular ice and e\ ci \ u liei <• the > ime, and it has no normal component the flat ends of the cylinder I\). The (juantity of charge within is 2-7ro and so on. When we add up, evidently all the terms but the first and l.-ist we ur twice and with opposite signs, and the total work done = *\SF OAA The \\ork done from A to 1' is, therefore, the same, whatever the path from A to P. I \ is at an infinite distance the work done to P is q/OP. That is, it is equal to the potential at P due to Q. The work done from A to P is equal to the difference of the potentials due to q at P and A. In any extended system we may divide the charges into elements so small that each may be regarded as at a definite point. The intensity at any point in the field is the resultant of the intensities due to ti it. dements, and the resolute of the intensity in din-rtion is t he algebraic sum of then -solutes of all theseparate intensities in that direction. The work done in moving unit itity along any path is, therefore, the sum of the works done against th< forces due to the separate element-.. !!• nee the work done in moving the unit from an infinite mcc to the point P in the field is ^-i^or is equal to the poten- tial at the point. The work done in moving from 1* to Q is evidently -(-£r " ~fju'or is (M|IIa' t() the difference of potentials at Q and P. The resolute of the intensity in any direction in terms of potential variation. Ix-t IV Fig. .'38, be a given direc- tion, and let X be the resolute of the intensity in that direction. P Q X Fio. 38. \ \ .!• i te the potentials at P and Q, two neighbour- ing I The work* done against X in ^()jnir from P to (^ I'M. >in« we h;m- supposed X to act from P to Q. Then \ Vr=- IV \ If we denote !*<») by gl is ft/AB. If instead of unit charge we bring up charge qr tin- work done is g1g2/A'R. Let us suppose that we have a system consisting of el. charges ft at A, q^ at B, q3 at ( . and so on, and let us put AH AC = r13, BC = 7-23, and so on. Imagine that initially q} on! in position, and that all the other eh t .it inlin away and apart from each other. Bring q« up to B. The uork done is-ii^ ^ow i)rjng g^ Up j.o Q an(j tju> ^^ (j0||r js ri2 Ms + Ml. Continue this process till the system is built up, and e\ ident shall have the product of each pair of charges divid.-d b\ their distance apart occurring once and once only. The total \\ork at an infinite distance. The work done in bringing ition is 23 when- V2 istlu- potential at H dm- to the rot of the system. ng thus with each element in turn, the "total work is '/i^i + f/i^t + 7»^3 + *>'<-'• = 2>/V. Hut in the work thus clone each product such as *-^lz occurs twice, and twice only, once when qm ' | :. i> brought up, and once when tjn is brought up, and therefore the total work done ^ twice the potential energy of the .system. We have tin Potential energy of the system =-^-2/yV, where e.-n-h i-Iement of charge is multiplied by it.s potential. \Yc may illiistiat. tln-r i. Milts bv considering The potential due to a uniformly charged sphere at points without and within its surface. Ix;t a charge Q be distributed imiformU uHlM O.\ «, l«'ig. J5J). \Ye ha\e shown 'p. ;>(.)) that the intensity due to the charge at any I ; i if the charge wen concentrated at O. II done iii bringing up unit charge from an infinite H i> the sune M if <»> were concentrated at O and \ l\ (l OH. Within the sphere, as we have shown, 90 that the potential is constant, 46 STATIC ELECTRICITY since no work is done in carrying the unit from point to point. It is, therefore, the same as the value at O. As all UK- di is at the same distance from O, r OA 7 The same result is obtained by putting the potential of the spl equal to the value at A, which is obtained by supposing H to n up to A, whence The. energy of the charge, since it is all at the same- poU-ntinl, i- ± CHAPTER IV THE FIELD CONSIDERED WITH REGARD TO THE INDUCTION OR ELECTRIC STRAIN PRODUCED IN IT The two kinds of charge always present and facing each other— Electric action in the medium — Electric strain — No strain in conductors when the charges are at rest — Direction of electric strain — The electric strain just outside a conductor is normal to the surface of the conductor — Magnitude of electric strain — Variation of electric strain with distance from the charged bodies : The inverse square law — Lines and tubes of strain — Unit tube — Results deduced from the inverse square law — The transference of tabes of strain from one charge to another when the charged bodies move — Molecular hypothesis of electric strain. The two kinds of charge always present and facing each Other. l'.\p« riments of tin- kind desc-rilx-d in Chapter II show that tin- two kinds ot electrification an- always produced together in et tip v ili^ipp. .11 in i-.jn d quantities. Sometime i, in experimenting, we te all our attention to a charge of one kind alone and speak as if it :«-nllv ; hut it isoftl. - 1 .importance to mlier that tin- opposite or complementary charge is really in U*nce, perhaps on the neighbouring surface of a non-conductor usecl •••lop tin- ell 'ion \\hich we an- studying, or perhaps on the 'in^ table, Hoor, or walls of the room, so that not only is it in existence, hut is in the presence of the inducing c har^« . If we have a charge in one room we cannot have the iry charge in another. If we attempt thus to separate the two kii i will at once induce its own opposite surface of the surrounding conducting walls. Again, when we put a charged conductor " to earth " we sometimes speak of the disappearance of the charge as if it merely spread away to infinite conducting earth. lint in fact we are making a con- due* ILJI- hv means of which the charge and its opposite may md neutralise each other. The two kinds of ///<•;• in the electric field, each, a» it were, inducing the other. A I lay* put it, "Uodics • Erp. to*, vol. I. § 1178. 48 STATIC ELECTRICITY cannot be charged absolutely," i.e. with one kind of electrification alone in existence, " but only relatively " to other bodies some- where in their presence which take up the opposite charge " and by a principle which is the same with that of induction. All charge is sustained by induction.*" We may arrange our experimental knowledge of induction more clearly by the aid of a hypothesis, originally due to Faraday and subsequently developed and extended by Maxwell. UV shall here give a gene/al account of this hypothesis, filling in the details as they become necessary for the explanation of the various points in the theory of electric action. Electric action in the medium. Let us suppose that a charged insulated body is placed within a hollow conductor. We. may think, for instance, of a charged brass sphere hanging bv a -ilk thread in the middle of a room. Then there will be an opposite charge on the walls. In describing the two opposing surfaces as electrified the most important experimental fact which we connote is that the two surfaces are being pulled each towards the other. This action was formerly supposed to be direct, each little bit of wall surface, for instance, pulling at each bit of brass surface at a distance, .the intervening matter playing no part in the action and being unaffected by the existence of the two electrified M between which it lay. But Faraday succeeded in showing that the action varies with the nature of the intcrxciiiiig iiiMilator. and we may describe the variation by saving that the pull on the electrified surfaces with given charge- i : with insulators than with others. This fact is entirely unexplained, unless we adopt the view taken by Faraday that the insulator plays an essential part in the action. He supposed that it i- altered in some way during the process of electrification, so that it exerts pulls on the conducting surfaces with which it i- in contact. In fact, he regarded electrical actions as similar to the drawing towards each other of two masse- connected by a stretched india- rubber cord. In the case of a cord the pulls on the mussel depend on the nature, quality, and strain of the cord, and can be expressed in terms of the strain. The energy which appears in the motion of the masses is regarded as having been previously stored up as strain energy in the cord during its stretching. So the electrical pulls are supposed to be accompanied by a condition in the in- sulating medium analogous to strain and to be expressible in terms of this condition, which we may call electrical strain. Th< trical energy is supposed to be stored in the medium during the electrification and to pass out of it again on the motion of the charged bodies towards each other or on the neutralisation of their charges. The electrification of a conducting surface may therefore be regarded as a surface manifestation of an alteration in the state of the contiguous insulator, somewhat as the pressure on th< of a hydraulic press may be regarded as a surface manifestation of THE INDUCTION OR ELECTRIC STRAIN 49 the compressed state of the water which connects it with the plunger. When Faraday first gave this account of electrical action it was on I v supported hv the variation of the effect of a given charge with variation of the medium, and no independent evidence of the state of k< electrical strain" was in existence. But it has been dis- covered that some insulating media when between charged surfaces are affected in regard to the transmission of light. Unless we make the further and very far-fetched hypothesis that the light itself is acted on at a distance by the charged surfaces, Faraday ""s supposition becomes for these media a proved fact. We have also gooa evidence of electrical strain in the electro-magnetic waves dis- covered by Hertz and used now in wireless telegraphy. We know only one way of explaining the phenomena discovered by Hertz, vi/. hv the supposition that the disturbances are waves travelling through the air with a definite velocity. This implies the existence of both electric and magnetic energy in the air. The electric energy implies the existence of that which we have called electric strain. It i> aKo called electric polarisation, electric displacement M -\well) and electric induction (Faraday). No strain within conductors when the charges are at rest. When the charges and the charged bodies in a system are iiis condition of electric strain is confined to the air or other insulator bet \\ccn the charged surfaces and does not occur within the substance of the conductors. This is evident if we the experiments already described with closed conductors. These show that a hollow closed conductor may be regarded as cntirelv MT. cuini; the -pace within from the space without. No electrification is induced on bodies within by external charges ; no mutual null is exerted on an electrified surface inside by an electrified surface outride. Interpreting these facts by Faraday's hypothcsk we must Mippose that the substance of the conductor is not its, If in ;l state of electric strain. If it can acquire that iiti >n it rapidly loses if, he in or incapable of permanently n J, ^2* defined. For instance, if two proof \ be held together in the position shoun in Fig. 40, the arrow-head denoting the d tion of strain, the one is entirely posit the other entirely negatively electrified. FIG. 40. on separating the two while in this portion the charges may be shown to be equal opposite by the electroscope. Further, the charge on each plane is a maximum when so held. For on holding the planes at some other angle to the strain, and then separating and testing the charges, these will be found appreciably less than when held as in Fig. 40. When the planes are as in Fig. 41 the electrification is as indi- cated, and on separating and testing the planes they are found to be unelectrified. From this effect of induction, combined with the tendency of a positively electrified body to move with, and a nega- tively electrified body to move against, the strain, we may obtain a Till- INDUCTION OR ELECTRIC STRAIN 51 very easy mode of indicating the direction of electric strain. Suspending a small elongated conductor, say a piece of a match, a straight cotton fibre, or a short piece of wire, by a cocoon fibre passing as nearly as possible through its centre of gravity, the two ends will be oppositely electrified unless the con- ductor is at right angles to the direction of strain. When so electrified the ends will be acted on by forces constituting a couple, which will set the con- ductor with its axis along the direction of strain as a position of stable equilibrium. The position at right angles is evidently unstable. Such a conductor may be termed an "electric pointer." Finally, we may indicate the direction of strain by the direction of the force on a very small posi- tively charged body, say a pith ball hung by a silk Its charge should IK- so small that surround- ing charges are not sensibly affected by its presence. Direct experiments in the field round a charged Fio. 41. globe or round the knob of n charged Leyden jar show that tin- three methods — (1) by two proof planes, (2) by an electric pointer, (3) by a charged pith ball — give the same direction for the strain. We may use any of tin -se tests for the direction to prove that The electric strain just outside a conductor is normal to the surface of the conductor. Perhaps the simplest proof of this is attained by suspending a small electric pointer from tin-end of a thin ebonite or shellac rod and bringing it near the sur- face of an insulated conductor, when it will always take the direction IK- normal. A very striking verification is obtained by fixing the pointer at some little distance from the conductor and then introducing another conductor, say the hand, into the field and gradually bringing it up to the pointer. When the second con- ductor is brought quite close to the pointer it will be normal to this second surface. If pith balls be fixed by short silk fibres to the sides of a conductor, when the conductor receives a charge they will all hang in positions agreeing with the supposition that the strain is normal to the su i >r if two proof planes be used, they give the maximum charge on separation after being held parallel to the surface and close to it, and no charge when they have been held perpendicular to the surface. This last method enables us also to show that the direction of the strain is from a positively electrified surface and towards one negatively electrified. The law thus proved may be regarded as a case of the more general law that positive electrification t< ncls to move in the direction of strain and negative electrification against it. Had the strain a component parallel to the surface of the conductor, the electrification would move either with or against that component. Such motion of the charge does actually occur \\hile a body i* receiving its electrification. The 52 STATIC ELECTRICITY strain is not then altogether normal to the surface, and the elect riti- cation moves about until a distribution is attained such that the strain is normal to the surface. Then there is equilibrium, for, the conductor being surrounded by an insulator, the electrification can move no further. In the case of a conducting circuit ram a current produced, we may suppose, by a voltaic cell, equilibrium does not exist, and the strain has a component along the surface of the wire from the copper terminal towards the /.IMC terminal, and therefore either positive electrification moves with it or negati \ c electrification moves against it, this motion of electrification being one of the chief features of the current. While the motion goes on there is also strain in the conductor itself, but such strain can only be maintained by perpetual renewal, the energy for the renewal being supplied by the cell. The magnitude of electric strain just outside a con- ductor. Usually we are practically concerned with the magnitude of the strain only where it ceases at the MII We shall take the amount of charge gathering per unit urea the conductor as the measure of the strain t li- on a given conductor there is at one time twin- the charge per unit area that there is at another time, the strain jiM out-id«- tin- conductor in the former case is twice as much as in the latti r.* Definition of surface density. The charge pn unit area on a conducting surface is termed the surface density. Tin- surface density, then, may be taken as iiiea^urin^ t M in the insulator just outside the conductor. If we wish to men strain at a point not close to a conduct MIL bring a small conductor there and note the charge gathering per unit area on one side or the other. But in general the introduction of a conductor into a system alters the strain in the part of tin • immediately around it, so that the charge }>• r unit area docs not in general measure the original strain, though it is probably | portionate to it. But there is one case in which the introdm -tion of a conductor does not appreciably affect the strain in it- bourhood, viz. when the conductor is a small exceedingly thin conducting plane held perpendicular to the direction of strain, and this case may be more or less nearly approached in practice l>\ the proof plane. By using a pair of exceedingly thin equal proof planes of known area we may measure, at least in imagination, tin- strain at a point by holding the planes in contact at the point and normal to the strain, then separating them and measuring tin- charge of either. The strain equals the charge per unit area. U shall therefore take as a definition of the magnitude of strain tin- following : Measurement of strain at any point. We measure the * In taking this definition, it may be noted that the analojry with elastic strain fails, for elastic strain has zero dimensions, while electric »traio has dimensions charge /length*. THK INDUCTION OR ELECTRIC STRAIN 53 electric strain at any point in a medium by the charge per unit area gathering on a conducting surface which is introduced so as to pass through the point, but which does not alter the strain in the region immediately outside itself. \Ve shall usually denote the strain by D. The quantity which is here termed electric strain is, as already stated, that which Maxwell termed "electric displace- ment." We have thought the former term preferable as having no meaning beyond that which we put into it. The latter term almost irresistibly carries with it the idea that electric strain is a displacement • »inethin«: in tin- direction of in- duction, an idea \\hich may prove a hindrance rather than a help. The variation of electric strain with distance from the charged surfaces or bodies. The inverse square law. It a conducting spin i \ 1 iJ.uitHa Fio. 42. charm- -f V 'N huiii: bv an insulating thread within a hollow spin : ity in a conductor B, the two sphrrir.d surface* bt'inij concent i ic, — <^) will be induced on the i micr surface of B. If the insulating medium is homogeneous, then . s\miii<-tr\ each ch.-r uniformly so that if A and B are the radii and tr;«in uhicli ue should obtain 64 STATIC ELECTRICITY by calculating it on the supposition that it is due to the din-ct action of the charged surfaces, each little bit of charge ij acting directly at its distance d without regard to the nature, conduct or insulating, of the intervening matter, and producing sti ? always from or towards itself according as it is posit h ^_,x72 * negative, all the strains due to the various elements being com- pounded as vectors. For with this law of action, as we proved in Chapter III, the charge on B will have on the whole no effect on a point within its surface, while the charge on A will 1 the same effect as if collected at its centre, giving therefore ^^3 as the resultant. Within A and without B there ix no xtrnin, a result again agreeing with that obtained on the supposition <>t direct action. For within A the point is within both -ph« i surfaces, and neither will on the whole have any effect . Without B each charge will have the same effect as if collected at the centre of the sphere on which it lies. Being equal and 0] tin- two charges will evidently produce equal and opp> neutralising each other. These resiil t s suggest tin 1 a w t hat— The strain at anif point of any f/ntrinil .w/v d homo- geneous insulator may be calculated In f ntppoitaftkot it i\t/n- n. \nltant of the strains due to w/>arutf clumnt* <>f fin' chargf. according to the law charge -f 4-U//.v///mv)-, these elemcn being compounded according to the irriur /< Assuming this, we may pass to the general case in which \u have any assigned arrangement of conducting elect ri lit «1 lx>dics of known shape possessing Known charges, a: it.d 1>\ a homo- geneous insulator. Whatever the distribution, we know (1) t there is no strain within the conductors, and (2) that th- just outside each surface is at every point perpendicular to tin- surface. Assuming that the strain is e\er\ where, both in conductors and in the medium, the same as if each element a separately according to the inverse square law, it is possible to fit: general mathematical expression for a distribution which will satisfy the conditions stated above, and it may be shown that there is in each case only one possible distribution. The mathe- matical expression can be interpreted numerically onlv in a few- cases. Among these cases are an ellipsoid with a given charge, a sphere acted on inductively by a charged point, and two charged spheres in contact. In some cases the results of calculation 1. been verified by experiment, and we may therefore consider that the employment of the inverse square law i> justified.* * A proof of the inverse square law is frequently jrivL-n which n the fact that there is no strain within a i •••Hnlm-tin. It is usually assigned to Cavendish, but it appears to have been given firv Priestley in his " Electricity," 1st ed. 1 7r>7. l>. 7 II be found in the Report of the Hritish A**uctatiuH. l^ THE INDUCTION OR ELECTRIC STRAIN 55 Lanes and tubes of strain. A line, straight or curved, drawn in the electric field so that at every point of its course the tangent to it coincides in direction with the strain at the point is termed a line of strain. We may suppose such a line traced out by an " electric pointer " moved always in the direction of its length, and with its positive end forward. Imagining the lines drawn closely together and in all parts of the field, they map out the field for us in regard to the direction of strain. There is no method like that of the use of iron filings in magnetism which will show the whole course of the lines, though sawdust scattered on a thin ebonite plate gives some indication of the electric field. We may, however, have a number of pointers in the field to show the direction at different points. It is interesting to study in this way the field due to the knobs of two or more charged Leyden jars. \\ may show, for instance, that the field due to two similar jars charged equally and oppositely from the terminals of an influence machine resemble the magnetic field of a bar magnet. A bundle of lines of strain forms a tube of strain. Results deduced from the inverse square law. We may at once applv to electric strain the results proved in Chapter III for systems in uhirli the inverse square la\s holds. It i> important to notice that tl»? quantity whial v.«- ha\e there called "intensity" must be di\i'led by 4ir to ^i\e what uc hen- call strain, for the intensitv just outside a conductor is I-T, while the strain is &. product strain X cross-see t ion i> constant for every >\\ of a given tulie. Calling strain x cross-section the total strain at the section, we may say that the total strain is the same throughout the tube. , Unit tube. If a tube starts from +1 of electrification it is ied n unit tube. Jf the area on which that + 1 is spread is a, and the surface d- T. then ide the surface, D = (r = -. Then just outside the surface a Da = l,nnd this product is constant along the tube. The total strain in a unit tube is, therefore, equal to unity. A tube starting from a given quantity of positive electri- fication either continues indefinitely or ends on an equal quantity of negative electrification. I '>r example, each unit lx-<;ins on +1 and ends on — 1. Then the statement in Chapter II, p. i^.j, that equal quantities of the two itions always induce each other is true not only of the whoN- charge, but also of the separate elements. We may connect each element of the one with an equal element of the other by a tulx in. Appl\: '. ,\ closed surface and remembering that we ha\e to di\idc intensity by I-TT to give strain, 56 STATIC ELECTRICITY The total normal strain over any closed surface draun in the medium is equal to the total quantity of the elect tion within. The total quantity is, of course, the algebraic- sum of tlu- potttivc and negative electrification >. If the surface is unclosed w» \ It the surtace is ttOdOMa we J ^? have a useful expression for the total 1 i • " i normal strain over it. FIG- 43. If Fig. 43 represents a unit tulx- with cross-section a and strain 1). Da = l. Let S be a cross-section making any angle 0 with ,». Then S = a/cos 0. If DN be the component of strain normal to S,DN = Dcos0. Then DNS = Dcos0 x— ^ = Da =1. Or tlu COS (7 total normal strain over any section of a unit tube is units. Now suppose that we have any surface through which n unit tubes pass. The total normal .strain OUT their « mis cut oil' by tin- surface will be n. Or The total normal .strain over anv Mirtacr is equal to the number of unit tubes passing through the surface. The transference of tubes of strain from one charge to another when the charged bodies move. We have seen that it follows from tin- inverse square law that in an electric system occupving a finite region each 1 1» positive charge is connected with an rqnal ilrnu-nt of nr^atix*- charge by a tube of strain. Hut if the- charged bodies mou-. tin- pairs of elements connected may ch. 1 \M- « general way how this may occur. As an illustration the case of a charged body originally near a condiu-ting tabh , Fig. 44, with nearly all its tul- ,g down to the surface. Suppose that then an insulated conductor \*>. \ brought between A and the table. If B were an insulator the tubes would pass through it. But the parts of the tubes within H \\ill disappear, since a conductor cannot maintain any electric strain ; and \u- 1 the arrangement more or less like that in Fig. l~>. uhcrc HH Tin: iNDi ( noN on ELECTRIC STRAIN 57 j>ositiu> elements of charge on A are now joined up to negative elements of charge on H, and positive elements on B are joined up to negative elements on the table. 1 A and H are connected by an insulated conductor, most of Fio. 45. the tubes from A to H will move sideways into that conductor and disappear. Others of the tubes from A to the table will move inward**, and the up|x.T part* will melt awav in the connecting piece, so that their po>itivc < n will lie on H, and we shall have Fio. 47. u^«- on H cMimrrtrd uith the table, as in Fig. 46. It \ ut ^hall lia\r something like I''ig. 47. 58 STATIC ELECTRICITY As another illustration, suppose we have two inflated spheres charged with equal + 5 and - 5 respectively and near together, so that nearly all the lines of strain go from one to the other. Let us further suppose that we have two plates charged with H and - 5, and with five lines of strain between them, as in Fig. 4 Let us suppose that in the spheres the -f charge is below am in the plates the + charge is above. Then the lines of strain a \x Fio. 48. in opposite directions, as indicated by the arrow >. Now mo\e the two spheres towards and finally into tin ip the plate*. When the nearest lines of strain of the two systems come togvtlu i the strains are in opposite directions, and where ti. will neutralise each other and disappear, and ur ma\ trier tin- successive steps in what happens to OIK- line of -train some* hat as in Fig, 49. Repeating thi> process, we may imagine that in this way, say three out of the five line- of -train between the spheret FIG. 49. are neutralised one after the other, and that ultimately \\e 1 the arrangement shown in Fig. 50. We still have five lines from each sphere, but three come to the upper from the top plate, and three go from the lower to the bottom plate, and only two go between the spheres. Molecular hypothesis of electric strain. 1 1 may be u >e f 1 1 1 to give here a hypothetical representation of the condition of a dielectric when it is the seat of electric strain. We IM^UI 1»\ assuming that the molecules of the dielectric are really made up THE INDUCTION OR ELECTRIC STRAIN 59 each of two portions, one positively, the other negatively electri- fied, and that the two portions are connected by tubes, or, as more easily drawn, lines, of strain, and that they are pulled towards each other by the forces accompanying the strain. In making this sup- position we are really identifying chemical with electric attraction. Fio. 50. \\ nm>t al>o have motion of the two parts of the molecule round each other, otherwise the tuo uould lx? pulled into contact and the tuU-s would disappear. But \\r may for our present purpose leave the motion out of account and con vcntionally represent a molecule as in Fig. 51, where we suppose, for dcliniteness, that there are •iinccting the two parts. If the dielectric is in a neutral condition we nm>t suppose that the molecular axes are distributed tonally in nil dhfebons, so that there are equal iiiiinhfrs of and negative ends of axes facing in any fion, and Unit is no resultant electric action outside, and tore no external indication of the molecular charge-*. : Fio. 52. Let us now suppose that the medium is between two conducting i ~M, and that these become charged. We may suppose that, in tli« < li irging, -f and — arc moving along the surfaces, each pair' iii< < i. (1 }>\ ;i tube of strain aa PQ, and \^- think ot MM ii ,i tube as picking out, as it \\eiv. the moli-cuh - 60 STATIC ELECTRICITV suited for its purpose— those which will help it to stretch from plate to plate. Let a ft y be three such molecules. The line of st rain PQ coming against these will neutralise one of the lines of strain of each in the way followed out in the illustration last considered, and we shall get the arrangement of Fig. 53 changing to that of Fig. 54, where now one line of strain goes across the spaces between Fio. 63. the molecules in the row forming <1«1 link^ in flu- chain. The strain between the two i>arts of each BolecOk i only lessened, but the + of each is alto mm connected OJ - of the next, and so they will tend to move apart. thus separation of the CODftltlWntl of the im.WuIr*, tin- MgllH uingof chemical di»m-iution, and this iK-ginning constitute tin- electric strain in tlu> medium on the h Jpothedf COMidmd. If a second tube moves into the >anu- i-liain of moliriilcii it will leave three tubes between each portion, and then* *ill now be two outside, and the separation i> still greater. If a third tube move* in there will be only two tubes between each portion of the original molecules, and three tulu > ronmrting eacn with the opposite portion of the next, and still greater separation. If five tubes move in, then, there will be a total neutralisation of the strain between the parts of the original molecules, and each will be con- nected with the opposite of the adjacent molecule. There will be re-pairing all along the line and no continuity of M; The electrification will, in fact, have been destroyed. We may imagine that something of this kind occurs when a spark discharge begins. CHAPTKK V THE FORCE ON A SMALL CHARGED BODY IN THE FIELD, AND THE PULL OUT- WARDS PER UNIT AREA ON A CHARGED CONDUCTING SURFACE The inverse square law for the force deduced from that for electric strain— Coulomb'* direct measurements — General statement of the inverse square law— Electric intensity— Unit quantity— Relation be- tween electric intensity and electric strain— Intensity just outside the surface of a conductor— Lines and tubes of force— Outward pull per unit area on a charged conducting surface— Note on the method of inrwtigatiitfthe field in Chapters II I- V The force on a small body in the field. Electric intensity. In the fore^oin^ inNe-ti^.ition- of the mode in which • barges are di-tri: nature of electric- induc- tion, we have not consider* <1 tin- magnitude of the forces acting on v PI the resultant according to the vector law of the two strains repre- sented by PC and PI), and in order that the resultant <>t I I -Mould also be in the direction PK we must h FI PC or the force on a given small charged body due to an cl< i another charged body is proportional to charge -f (distance)1. Further, assuming that action and reaction are equal and opposite, the force is proportional to the charge on the body at P. For if that charge be Q the force exerted on the body at A by the body at P is proportional to -, /.<•. is proportional to Q. II- the reaction of the body at A on that at P which U ecjual and opposite is also proportional to Q. Then the force exerted by the body at A on the body at P is proportional to Coulomb's direct measurement of the forces between electrified bodies. Having regard to the importance of the law just stated, it is worth while to consider the direct method of obtaining it employed by its discoverer. Coulomb. To measure the force, he used the torsion balance of which the general arrangement is represented in Fig. 56. be represents a " tor- rod" of shellac with a pith ball b at one extremity, c and d being small loads to keep the rod horizontal. It is suspended by a fine metal wire from the torsion head /. This can be rotated and the angle through which the upper end of the wire is thus turned can be read by means of a pointer p moving over a scale * THE FORCE ON A SMALL CHARGED BODY 63 lixed at the top of the tube through which the suspending wire passes. The position of the torsion rod is read by means of a scale running round the side of the glass case containing it. A ball a is at the end of a thin vertical shellac- rod which can be passed through a hole in the top of the case and fixed so that the ball is in the horizontal plane containing the torsion rod and at the same distance from the centre a^ the ball b. The method of using the balance will be understood from the following account of one of Coulomb's experiments. Both a and 6 being without charge, the torsion head was turned no that b was just touching a when the wire was free from twist. A charge was then imparted to a ; it was at once shared with />, which \\a^ then repelled through 36°, so that the torsion on the wire was 36°. The torsion head was then moved round till the angle between the upper igh 1S?. The total torsion was fore only 126° + 18° = 144°. Tlu- head \UIN then turned round until the angle between a and /; was 8J°; the additional turn of the torsion head was 441°, giving a total torsion of 441° 4- 126° + 8£° = 575£°. Assuming as a first approximation that up to 36° the repulsive force acts always at the same arm and that the distance between a and b is proportional to the angle between them, then, if the first observation is correct at a distance 36, the force is proportional to the torsion due 0 to 36°. According to the invi-r-e - tl,< true x of the arm at which the force on the ball acts. L- AOB = a, the angle between a and b. Ixit 0 be the t< M0 the couple due to it. Let F be the force on b due to a. let ON be perpendicular to AB. Then F.<)N = M0 F.ON and 6 = - But ON-OA< and assuming the invn>e Mjuare law, P« ' 1 in. \ Then to verify the law we ought to find that 6* Cl.s '' Or that 0 tan | sin | is constant. The values of this product for the experiments we have 3-568 :3 1 The numbers in the third column are fairly near together. The torsion balance may be used also for the case in which the charges on a and b are opposite in sign, though the experin is not so easy when the repulsion is replaced by attraction. within a certain distance equilibrium is unstable, a small di^- THE FORCE OiN A SMALL CHARGED BODY 65 placement making a greater change in the attraction than in the torsion. Coulomb also tested the inverse square law by a method which possesses interest since it corresponds to the vibrating-needle method which he employed in his magnetic experiments. An insulating shellac needle was suspended horizontally by a cocoon fibre, a small charged disc being fixed at one end of it. It was placed in the field of an insulated globe charged in the opposite way, and the time of oscillation was observed to vary directly as the distance from the centre of the globe, the result which we should expect, assuming the inverse square law. Coulomb showed in the following manner that the force on a given charged body is proportional to the charge acting. Sup- posing a and 6 in the torsion balance to be exactly equal spheres, on contact they will be equally charged and the torsion rod will be repelled through a certain angle. If a is now withdrawn and its charge shared with a third equal sphere, only half the charge remains on it. Replacing it in the balance, it is found that only half the torsion is needed to keep the ball b at its original ince from a. Hence it follows that if two small bodies a distance d apart contain charges Q and Q' the force on either is proportional to QQ'/^1' Two other researches of historical intm ->t made by Coulomb may be here referred to — that on the distribution of electrification on conductors of various forms by means of the proof plane which he invented for the purpose, and that on the rate of loss of charge and on the best methods of insulation. He showed that with a givi-n conductor insulated in a given way the rate of loss was proportional to the total charge, though the rate varied on different days, being, in accordance with common experience, more considerable on damp than on dry days. Coulomb ascribed the loss partly to the air and partly to the supports, supposing that the air was a conductor when containing much water-vapour. We know that his supposition as to water- vapour was a mistake for it, like air, is an excellent insulator. It is true that if air is nearly saturated with water-vapour the rate of loss of charge is icntly great, but this is almost certainly due to the condensa- tion of a thin layer of water on the surface of the supports, which latter may be hygroscopic and may so condense the vapour into water before normal saturation is reached. We now know that neither dry air nor water-vapour is a perfect insulator, for there is a loss of charge due to the presence of electrified "ions" in the air, and the electrified surface draws to itself the ions with opposite charges. But this loss is exceedingly slow under ordinary condition-. inverse square law may be illustrated by the following expeiim Two metre rules are fixed horizontally, one over the other (Fig. 58). A conducting sphere A is mounted on a pillar, 66 STATIC ELECTRICITY which must be a very good insulator, with a base having an index mark which slides along the lower rule. On the- upj. is a slider with an index mark, and from the slider depends a pith ball B hung by a cocoon fibre, the ball being on a bid "it centre of A. A telescope sights B when it is uncharged, and position of the slider is read. Then A is brought into the middle of the field of view, and the position of its base is read. A 11 charged, B touches it, is charged and repelled. A is then removed say, 10 cm. to the left, and the slider is moved to the left till the middle of the telescope field. If the slider has ben m the force on B is for small angles proportional to d. Now n to, say, 20 cm. from the central position and move the slider till is in the middle of the field ; let its displacement be «f. It ii ' ' ' ' j ' ' ' ' J" J I i I i I i FlQ. 58. found that with fair accuracy d : d = 4 : 1. If A be now touched to another equal insulated sphere, is charge is halved and the forces at the same distances are huht n l.ition ho Id ing for each element of the electrification to which irld is due, it must hold for tin- resultant of all the elements. I > to be noted that it is only true when the medium is air, for then i^ the intensity Q/cP for each element. As a particular case we have another proof of the proposition already proved in Chapt. i III. \i/. : The intensity just outside the surface of a con- ductor is 4?r x surface density. Tor the strain is by defini- tion measured by a-. Hence the inten>itv N ITTO-. Lines and" tubes of force. The lines and tubes of strain may also be regarded in in»n-crv>talline media as lines and tubes of force or inteiiMty, and the various strain properties of such tubes may be stated in terms of intensity. The most important of /. that area x strain is constant and equal to unity along a unit tube, becomes: area of cross-section X intensity = 4?7r along a unit tube. The relation between strain and intensity is not to be regarded merely as a numerical definition of one in terms of the other. It expresses a phyncftl f «ct, viz. that whichever method we adopt to explore the field, the force on a small electrified body or the charge gathering on one side of a properly held proof plane, the field varies in the same proportion from point to point by either 68 STATIC ELECTRICITY system of measurement. The numerical relation obtained depends on the units employed. Whatever be the unit of charge, if \u- define strain as on p. 52, the strain is ql^-jrd2, but the intensity is only q/d? when a particular unit is chosen. We shall see here- after that with this particular unit, but with a medium other than air, the relation is KE = 4?rD, where K is a constant for each medium. When we discuss the relation between intensity and strain in media other than air we shall be able to show that \\ e may interpret the intensity as analogous to elastic stress and the strain as analogous to elastic strain. The outward pull per unit area on a charged con- ducting surface. Since the intensity is always outwards from a positively electrified surface and in- wards to a negatively electrified one, it is evident that in each case an actual element of the surface, if lc>« would tend to move outward- \\ I shall now find the amount of the pull outwards per unit area. NNV 1 seen in Chapter III that the intensity at a point Pp Fig. 59, just outside a conductor is normal to the surface and is equal to 47nr. Also we found that it may be regarded as made up FIG. 69. of two equal parts, one due to the charge on the area a i tn media t< 1\ under it, and the other due to the rest of the charges of the system. Each is therefore 2x0-. Now consider an element da of a just under P,. The outwards on this element is not due to the re.st of , B fa practically all in the same plane as da. It mu>t thei due to S alone, and its value is : intensity due to S x charge on da = %7rar X ordu The pull outwards per unit area is therefore F = 27TC71. If, then, we know the distribution of charge on a condiu body, we know all the forces acting upon it, and the resultant force is the resultant of all the elementary pulls outward-. Note on the method of investigating the field in Chapters III-V. The mode of analysing the total strain at a point into the actions of the separate elements each supposed to act directly, and without regard to the intervening medium, is that THE FORCE ON A SMALL CHARGED BODY 69 suggested by, and most appropriate to, the old "action at a distance " view of electric action, and it might appear out of place with our new conception of action through the medium. But if, as in the text, we only say that the strain is the same, however produced, as if the direct action occurred, the method is perfectly justifiable for purposes of calculation, and it is the best in the first treatment of the subject owing to the ease with which it admits of mathematical representation. As an illustration of this method, consider the somewhat similar case of the equilibrium of a body pulled by a number of strings, with given forces applied at the free ends. We consider these forces as applied directly at the points of attachment of the strings, and do not trouble to consider how the strings behave or to take them into account, so long as we are dealing only with the equilibrium of the body. For experi- ment has shown us that if a force is applied at one end of a string an equal force is exerted by the string at the other, and we there- fore are justified in a kind of " direct action" treatment, though obviously the string has an all-important part to play, and must be considered if we are to have a complete investigation of all the phenomena accompanying the equilibrium of the body. There is a method of treating electrical actions in which the mathematical representation fits in perfectly with the supposition of action by and through the medium. It is fully set forth in Maxwell's Electricity mul Afn^n, ti\tn. vol. i, chap, iv, and we shall here only point out verv briefly the principle of the method for the sake of advanced readers who may wish to compare it with the easier, though much less general, method which starts from the inverse square law. Let us suppose that we \\i-li to find how electrification will be distributed under given conditions on any given system of conductors in air. Starting with the idea of the electric field as a space in which electric .strain and intensity may be mani- fested, we must deduce from experiment that in air the strain has v where the same direction as, and is proportional to, the intensity, and that we have chosen units so that E = 4<7rD. Then introducing the idea of the energy possessed by the system, we must show from our experiments that its amount depends solely on the configuration. From this it follows that a potential V exists, and the intensity at any point may be expressed in NTIIIS of V. The total energy of the system may be shown to be where T is UK- §ui nsity on the element of surface dS, the integral being taken over all the conducting surfaces. 70 STATIC ELECTRICITY But contains energy which may appear in the kinetic form. \NY must suppose that before its appearance as kinetic energy, it was stored in some way as electric energy,and in that form was the equivalent of the work done in charging the system, Regarding the s\- from this " work " point of view, we are led to the idea of elert i level or potential, which affords meet valuable aid in dex-rihing the system in respect to the forces exerted and to the energy stored. We have already discussed the potential in invi TM- Mjtiare systems in Chapter III, but we shall now approach the subject from another point of view. Electrical level. If a positively charged conduct* suspended in the middle of a room a small body brought again-t it will receive a little of the charge and will thru tend to move to the wall, on which is the negative charge. Let us imagine that the small carrier has no weight so that we need not think of any but the electric forces. At the wall we may think of it as going into a hole and touching the side of the hole, when it will be entirely discharged. It can then be moved back to the large body, where it will receive a little more charge, go again to the wall, and so on. On each journey from the body to the wall it can be made to do work, and the system will gradually yield up its energy. Now compare this with what happens in a system containing gravitative energy, such as a reservoir of water on a high level. We may let the water run down to a lower level and gradually transform the potential energy into some other form in the process. Or, to make the cases more alike, we may let the water down by a bucket which corresponds to the electrical carrier. The water gives up its energy by falling in level. Using this analogy, we may describe the electrified body in the 72 ELECTRICAL LEVEL OR POTENTIAL 73 middle of the room as being at a higher electrical level than the \\alls and think of the level as gradually falling from the body to the walls in every direction. The conductor is all at the same level, for the charged carrier will not move along its surface, but will be pulled straight out from it at every point. The walls, if conducting, are also at one level, since the intensity is everywhere perpendicular to the surface, and the carrier will not tend to move along the surface at all. The gravitative analogy, if followed out, also suggests a definite method of measuring electrical level. We usually measure difference of level above the earth's surface in vertical feet or metres. But this measurement will not give us consistent results in extended n stems. For think of two canals at different levels reaching from the latitude of London to the Equator. Suppose the surface of the upper one is 978 cm. above that of the lower at London. At the Equator the difference in level will be, not 978, but 981 cm. For the work done in letting a given mass of water down from one canal to the otluT must be the same at each end. Otherwise it would be possible to get an endless store of energy out of the water by allowing it to trickle down at the end where the most work was i out. and t<> do work as it fell. We might use some of this work to raise an equal mass <>f water up again at the other end and we illicit transform or store the balance. This is contrary to all experience, so that we are certain that the work done in lifting a gramme of water from one level to the other is the same at each end and indc.-d at every intermediate point. Since then g at Ion : g nt the Kqimtor = 981 : 978, if one canal is 978 cm. higher than the other at London, it mu>t be 981 em. higher at the Equator. It is worth noting that if we were concerned with exact measurements of work done in ing mosses, our Ordnance : >s .should be marked, not in feet above sea- level, but in ergs >t-poundals. Now, turning to an electrical system, let us imagine a surface Sj, Fig. 60, drawn in the dielec- tric all at one level, drawn, that is, so that at every point of the surface the force on a small FIG. 60. charged carrier is perpendicular to the surface, and no work is done in moving the carrier along tht1 . Imagine another sin face S2 in the dielectric, also . but at a higher level than the first. Then the work done in taking a small charged carrier from the first surface to the second is the same by all paths, l-'or imagine that it is greater by one path. Ah, than by another path, CD, from the first surface, Sr to the second surface, Sr Taking the carrier round 74 STATIC ELECTRICITY the circuit BACDB, we shall get a supply of energy each time we go round. This could only be accounted for by a gradual discharge of the system due to the motion, and experiment shows that no such discharge occurs. Hence the work along AB equals the work along CD. As a particular case the work from A to B is the same by all paths, such as AEB or AFB. If then we choose some convenient surface as having zero-level, corresponding to sea-level with gravity, the level of any other point, measured by the work done in carrying the unit positive charge from zero- level to that point, is definite, whatever be the path pursued. Level as thus estimated is termed Potential, and we have the following definition : The potential at a point is the work done in carrying a small body with unit positive charge from zero-level to the point. The potential at a point is usually denoted by V. It is evident that the work done in carrying unit charge from a point A at potential VA to a point B at potential VBis VB— VA. For if O is a point in the zero-level, VA is the work done in going from O to A, VB that in going from O to B, and we may go to H through A. Hence the work along AB is V« — VA. Alteration of surface chosen as zero -level or zero- potential. If we alter our starting-point or zero-level to one at potential U referred to the old starting-point, the potential at every other point is evidently decreased by U. Equipotential surfaces. The surfaces which we have hitherto termed level, surfaces everywhere cutting the lines of force at right angles, are also equipotential surfaces. Drawing a series of such surfaces corresponding to potentials 0, 1, 2, 3, &c., we may map out the variations of level in the system. A conducting surface is an equipotential surface, and any closed conductor containing no electrification within it is all at one potential, for there is no component of the intensity along the surface, and if there is no charge within there is no intensity within, and no work is done in taking unit charge from one point to another. The intensity at a point in any direction expressed in terms of the rate of change of potential. Let A be the given point and B a point very near it such that AB is the direction in which the component of the intensity is to be expressed. Then the work done in carrying the unit charge from A to B is VB — VA ; but if X is the component of the intensity along AB, the component along BA is — X, and this is the force against which the unit charge is carried. Then - X.AB = V - VA or, using the notation of the differential calculus and putting AB = dx and VA - VB = dV, - X.cLr = d\ ELECTRICAL LEVEL OR POTENTIAL and ultimately X = — -y- 76 or the intensity in any direction is the rate of fall of potential per unit length in that direction. General nature of level surfaces. If the surfaces are drawn at intervals such that the successive potentials differ by unitv, it is evident that the surfaces crowd most closely together where the intensity is greatest, for there the shortest distance will be traversed with the unit charge in doing unit work. On the other hand, where the intensity is less the surfaces open out. Indeed, we can at once obtain a relation between the distance from surface to surface and the cross-section of a unit tube of force. If E be the intensity at a point and d the distance between two consecutive level surfaces there. .'. Ed = l. But if a be the area of the cross-section of the unit tube at the point, Ea = 4-Tr (p. 67), d or the distance d is proportional to the cross-section a of the tube. When a conducting surface is charged entirely with one kind of electricity, the intensity is everywhere outwards if the charge be 61. Charged sphere in the middle of a cubical room. FlG. 62. Charged sphere inducing opposite charge on a neighbouring insulated sphere in the middle of a cubical room. positive, and everywhere inwards if it be negative. Then in the former ea-e the potential falls and in the latter rises as we go outwards from the surface in all directions. In other words, 76 STATIC ELECTRICITY the conductor is entirely surrounded by level surfaces lower in the former case, higher in the latter, than the surface itself. If the surface is partly positively and partly negatively char the surfaces are lower near the one part and higher near the other part. We may illustrate these statements diagrammatic-ally by Figs. 61 and 62, which are sections through systems showing iu each case lines of force and level lines in the section. There is no attempt to draw tu scale in these figures. Energy of an electrified system in terms of charges and potentials. Let the charges be Qr Qr Q3, \r.. ropirti vi-ly on conductors at potentials Vr V2, V8, £c., the /em-level bring any conveniently chosen conductor. If all the charges be altered in any given ratio the same for all, say to XQX, XQf, XQ8, «\ dently the intensity at every point will still remain the vun< in direction, but will have X times the original magnitude. li- the original level surfaces will still be leu -1, ;m perhaps suflicieiiily evident that the quantity of energy obtained must be independent of the arbitrary zero of potential choM-ii. but we may also deduce this from the expression for the total. For i t \\ e had chosen another zero at potential U relatively to tin liiM, then the potentials would have been V, — U, V2 — U,£c., and the total energy would have been expressed by for Q1 + Q, + . . . = 0. I ;> to be noted that if any charge is on the original zero-level surface, it must be taken into account in expressing the energy in terms of the levels referred to the new zero. 78 STATIC ELECTRICITY Unit cells. Drawing all the unit tubes of force and all the level surfaces at unit differences of potential, all the space between the conductors of a system is divided up into cells. Maxwell has termed these unit cells. The number of unit cells in a system is 2QV. In the case of one conductor at potential V entirely surrounded by another at zero-potential, it is evident that each tube is cut into V cells, and as there are Q tubes there are in all QV cells. When there are several conductors each tube does not necessarily pass from the highest to the lowest level, but some may begin or end at intermediate levels. Take any one tube passing from a conductor at level VM to one at level VN. Evidently it contains VM' — VN cells, or, since it has + 1 of electricity at one end and — 1 at the other, we may write this 1 xVM + (-l) XVN. Doing this for every tube in the system, we have Total number of cells = 2{1 X VM +(- 1)VXJ = 2 each element of charge X its potential. If on any conductor at potential V there be both positive and negative charges, say QP and QN, both charges will enter into this expression and will contribute to the result. But if QP - QN = Q, this becomes QV. Hence the total number of cells = 2QV. Distribution of the energy in the system. From the preceding result we see that the number of unit cells is always double the number of units of energy in the system. We have already seen that we must suppose the energy to be in the dielectric, accompanying the strain there. Bearing in mind that where the cells are larger the electric strain is less, so that t lie- energy is presumably less densely stored, this relation between the quantity of energy and the number of cells suggests that the energy is distributed at the rate of half a unit to each cell. Adopting the suggestion, let us find the energy per unit volume. The area of the cross-section of a tube at any point being a, and the distance between the two consecutive level surfaces being d, we have volume of cell = ad but Ea = 47r (p. 67) and Ed = 1 (p. 75) 4-7T whence ad = . ELECTRICAL LEVEL OR POTENTIAL 79 If this volume contains half a unit of energy, the quantity per unit volume J_ E2 47T STT E2 or, substituting for E its equivalent 47rD, Now this result is of the kind which we should expect from Faraday's view of the nature of electric action. For if electric strain be analogous to elastic strain, we should expect that the energy stored up per unit volume would be proportional to the square of the electric strain, just as the energy stored by an elastic strain for which Hooke's law holds is proportional to the square of the strain. Since then the distribution at the rate of half a unit E2 to each cell or of — = 2?rD2 to each unit volume just accounts for OTT the energy of the system, and also carries further the analogy between electric and elastic strain, it is accepted as the true law of distribution. E2 It is noteworthy that the energy per unit volume £- may be O7T expressed in the form -^-. This supports the suggestion on p. 101 that the intensity E is analogous to elastic stress. For we know that in elastic strain stress x strain energy per unit volume = - ^ - . CHAPTER VII POTENTIAL AND CAPACITY IN CERTAIN ELECTRIFIED SYSTEMS. SOME METHODS OF MEASURING POTENTIAL AND CAPACITY Definition of capacity— Sphere in the middle of large room— Two con- centric spheres — Two parallel plane conducting plates — Two long co-axial cylinders — Two long thin equal and parallel cylinders — Long tl cylinder parallel to a conducting plane — Instrument* to measure poten- tial difference — Quadrant electrometer — Attracted disc electrometer — Practical methods of measuring potential — Reduction of the potential of a conductor to that of a given point in the air — Simple methods of measuring capacity— Capacity of a Leyden jar— Capacity of a gold-leaf electroscope. WE have shown in Chapter III that if we define tin- potential at a point as the sum of each element of electrification divided by its distance from the point, or 2*, it is the work done in bringing unit charge from a zero-level at a distance so great that the system has no appreciable action there. We may frequent!? bring the zero-level quite close to the system. Thus, if a room has conduct- ing walls, an electrified system within it has no intcn>itv outside, and the walls may be regarded as having zcro-le\el. We shall now consider some special systems which will illustrate the use of the formula for the potential. Infinitely thin small metal plate held normally to the lines of force or strain. The plate is to be so small that the lines of force may be regarded as straight and parallel in the region just about the point at which it is introduced before that introduc- tion takes place. Then, when it is introduced, since it is perpen- dicular to the previous course of the lines, the positive charge on one side will evidently be equal, element by element, to the negath e charge on the other side. The potential due to the plate at any point outside it will be made up of equal and opposite terms, since for every element of positive charge there is an equal element of negative charge practically at the same distance from the point, as the plate is infinitely thin. Then the plate does not affect the potential in its neighbourhood, and therefore does not alter 80 POTENTIAL AND CAPACITY 81 the course of the lines of force. This justifies the use of such a plate in the measurement of strain at a point. In practice no plate is infinitely thin. A real proof plane will therefore produce some slight modifica- tion of the surrounding field, especially near the edges, where the lines must turn inwards, somewhat as in Fig. 64, to enable some of them to meet the edge normally. Capacity of a conductor. If the potential of one conductor of a in is raised while all the rest are kept at zero- potential, as the charge rises in value the induced opposite charges rise in proportion. Hence £*, or the potential at any point, rises in the same proportion. Thus the ratio of the charge on the conductor to its potential is constant and is termed the capacity of the conductor. We have then this definition : If a conductor receives a charge Q and is raised thereby to potential V, while the surrounding conductors are at zero-potential, ~ is constant, and is termed the capacity of the conductor. It is usually denoted by C. Sphere in the middle of a large room. Suppose a sphere radius a to receive a charge -f- Q. Let the sphere be placed insulated in a room and let its distance from the walls be great compared with its radium The potential of the sphere will be uniform throughout, and therefore we may find it by calculating the FIG. 64. potential at the centre. In the formula V the positive elements are all at the same distance from the centre, and therefore contribute Q/a. The negative elements arc all at a great distance, and the sum of the terms 2 is negligible in comparison with Q/a. We have then The capacity is C = * = a. Since the potential at the surface of the sphere has everywhere the same value as that at the centre, viz. : — , and is due to the dis- tribution on the surface, the wall charges having negligible effect, p 82 STATIC ELECTRICITY that distribution must be uniform. Hence the potential n distance r from the centre outside the sphere is, by Chapter III, provided that r is small compared with tin- distance t<» tin- Two concentric spheres with equal and opposite charges Q. Let the radii bo a and b. Tho potential of the innorsphoro is uniform and equal to that at tin- centre. >imv tin- whole of the positive charge N distant // from the md the .whole of the negative charge is distant I>. \\e \\ V = 2<- = -y r a b Between the spheres, at r from the centre, tin- potential due to the positive charge is - (Chapter III), while the potential due to the negative charge is still y. Then " ~r ~~ 1}' The outer sphere has potential v = F - ? = a The capacity of the inner sphere U .,_ Q ab = it v is 4frr(T and the distance apart is rf, the work done in carrying unit charge from one plate to the other is l-To-d. II 'he difference of potential V = 4t7r the area of either charged face, the total charge on the positive plate is approximately ACT- The capacity then is Q AT _A_ ' V 4W 47rcT Two long co-axial circular cylinders. The cylinders are to l>e supposed of length great compared with the radius of either. Tin- inner cylinder is charged to potential V above the outer. The radii are a and h. The lines of force will evidently be radial at a distance from the ends, and the tubes of force will be wedge-shaped as in Fig. , the apex of each wedge lying along the axis of the cylinders. Th«-n the area of the cross-section of a tube at distance r from the axis 84 STATIC ELECTRICITY is proportional to r, and since cross-section X intensity is constant, the intensity is inversely proportional to r, and may be written as \/r. Now if (T is the surface density on the inner cylinder the intensity just outside it is 47r<7, and as this is distant a from the *-0 FIG. 65A. G5B. axis we have 47nr = \/a or A = ±ir7raince the quadrants are insulated from one another by narrow air spaces. The opposite quadrants are connected in pairs by very fine wires. Under the quadrants and between the supporting pillars is an open Leyden jar lined outside with tinfoil and con- taining inside strong sulphuric acid, the surface of contact of the acid with the glass serving as the inner coating of the jar. The acid also serves to dry the air within the case of the instrument and so maintain^ tin- insulation of the various parts. Within the hollow space made by the quadrants is a " needle," really a" figure- of-eight'1 shaped piece of sheet aluminium, Fig. 66 (6), the sectors being each about a quadrant. The needle is supported from the top of the case by an insulating suspension which also introduces torsion wlim the needle is displaced from its " zero " position. The torsion may be obtained from a bifilar suspension, a quartz fibre suspension, or from a small magnet fastened on to the needle system. The position of the needle is indicated by a beam of light reflected on to a scale from a mirror above the quadrants and rigidly attached to the needle. From two adjacent quadrants two brass rods pass vertically * In Lord Kelvin's Papers on Electrostatics and Electro-magnetism a form of the instrument salted for exact work is described in the Report on Electrometers. 88 STATIC ELECTRICITY downwards through holes in the brass plate so large that the rods do not touch the sides. The rods turn outwards, ending in binding screws. A third vertical brass rod passes through an insulating block in the brass plate, and can turn about its own axis so as to bring a horizontal wire at its upper end in contact with a platinum wire which is attached to the needle, and passes vertically down FIG. 66. into the acid in the jar. When the jar is to be charged the contact is made and the jar may be charged by two or three sparks from an electrophorus cover. The contact is then broken and the horizontal wire turned well out of the way so that the needk free to move. The instrument is contained in a glass case resting on the brass plate, which is itself supported on levelling screws. The glass should be lined with tinfoil except where the indicating beam of light passes, and the case should be put to earth by a wire going to the gas or water pipes. The instrument is thus completely secured from external electrifications. The general principle of its action is simple. To begin with, the needle should be adjusted in its zero position so that it- median line is parallel to one of the lines of separation of the quadrants. If now one opposite pair of quadrants is connected by the outside terminal to earth, and the other pair to a body of a potential differing from that of the earth, the needle, being at the potential of the jar, which we may suppose positive, is itself positively charged and tends to move so as to carry its charge to the region of lowest potential. If the outside body POTENTIAL AND CAPACITY 89 is at positive potential the needle will tend to set under the earth-connected pair of quadrants, while if the outside body is at negative potential the needle will tend to set under the pair of quadrants connected to it. If the torsion couple introduced by the displacement is so great that the displacement is always small, then the angle of displacement is, as we shall show below, nearly proportional to the difference of potential between the pairs of quadrants. It is also, for a certain range, nearly proportional to the potential of the needle, so that the sensitiveness of the instru- ment may be adjusted by altering the charge in the jar. It is usual to calibrate the scale of the electrometer by putting on known differences of potential from the terminals of cells of known voltage. Dolazalek electrometer. A modification of the instrument introduced by Dolazalek, which is more sensitive than the original type, is much used. In this the quadrants are smaller and are mounted often on amber pillars. The needle is made of silvered paper and so is lighter, ana it is preferably suspended by a quartz fibre made conducting or by a very fine metallic wire. The Leyden jar can then be dispensed with and the needle can be charged directly from a battery of, say, 100 volts. There is a potential of maximum sensitiveness, however, which can be found by trial for the particular instrument used. The instrument is contained in a brass case with a window for the beam of light to pass to and from the mirror. Elementary theory of the quadrant electrometer. We shall take the case in which the two pairs of quadrants, which we denote respectively by 1, 1' and 2, 2', are maintained at constant potentials \\ V2. The needle is connected to the jar, which has very great capacity compared with its own. The potential of both jar and needle therefore remains sensibly constant even if the charge on the needle varies. We denote this potential by Vn. If V4 is greater than Vj the needle tends to move towards 1, 1', and it will only be in equilibrium when the electrical couple is balanced by the torsional couple. Now had the quadrants and needle, after charging, been insulated, the motion of the needle would have diminished the electrical energy of the system, the differences of potential of the fixed charges decreasing, and the position of equilibrium would have been that in which the decrease in electrical energy on small displacement would have just supplied the work necessary for the increase in torsion. But the potentials remaining constant, there is an actual increase in the energy of the system. The area of the needle under 1,1', where the difference of potential i> greatest, increases, and therefore its charge increases and consequently the electrical energy increases. The energy is -upplied by the sources which maintain the constant potentials, the supply being sufficient both for the increase in electrical energy and for that used in ining the torsion of the wire. We shall make use of the following important theorem : 90 STATIC ELECTRICITY When a system of conductors is maintained each at a constant potential by connection with a source of electricity, the total energy supplied by the source in any displacement of the system is double the increase in the electrical energy of the system. This may be proved as follows : Let Qi,Q2, Q3 be the charges at potentials Vr V2, V8 : then the total energy is + . . Let a displacement be given to the system so that the charges become Qx + qv Q2 + qv Q3 + fc, &c- : the ener^ is now whence the increase in energy is But the quantities qv qv qy &c., have been drawn from sources of energy at constant potentials V^ V2, £c., and therefore the energy given up by these sources is which proves the theorem. Applying this to the quadrant electrometer, the sources supply double the energy required for the electrical system. At first, then, while the torsion couple is small, an excess of energy is supplied. But the torsion couple increases, and ultimately a position of equilibrium is attained when on further small displace- ment the increase of torsional energy equals the increase in elec- trical energy. To calculate the latter we assume that the motion does not affect the distribution on the outer edges of the needle and under the line of separation of the quadrants ; this implies that the linear displacement of an edge is small compared with its distance from a quadrant other than that under which it is moving. Hence the motion through a small angle dO may be regarded as merely transferring a part of the needle distant from both the edge and the median line from the 2, % pair to the 1, 1' pair. The capacity of such a part may be taken as proportional to the angle it subtends. Let it be represented by CdO. Hence the charge under 1,1' is increased by CdO(Vn — V1), while that under 2, 2' is decreased by CdO(Vn — V2). The potential differences of these charges from that on the needle are Vn — Vl and Vn — V, respectively. Then the total gain of energy is POTENTIAL AND CAPACITY 91 The increase of torsional energy in the same displacement is \6d0 where 0 is the total angle of displacement from zero and \0 is the corresponding couple. Then for equilibrium we may equate these and t - vt) (vn - If Vn be great compared with V1 and V2 we may take as an approximation If the needle and one pair of quadrants be both connected to the same source at potential Vn and the other pair be earthed, V, = Vw Vt = 0 and This is independent of the sign of Vn, and the electrometer may then be used to measure an alternating potential, giving the mean square of Vn. By taking into account the increase or decrease of charge induced on the two pairs of quadrants as well as the increase on the needle it is easy to verify directly for this particular case the fact that the sources supply double the energy added in the field within the quadrants.* The attracted disc, or the trap -door electrometer. In Thomson's trap-door electrometer two parallel plates are arranged as a condenser, one being connected to the source of which the potential is required, the other being, say, to earth. We might measure the pull of one plate on the whole of the other, and in the first instrument of the kind — Harrises attracted disc electro- meter — this was actually done. If we could express the potential * The formula found above is quite sufficiently exact for many of the ex- periments in which the quadrant electrometer is used. The explanation of the voltage for maximum sensitiveness will be found in Thomson's Elements of Electricity and Magnetism, § 61. 92 STATIC ELECTRICITY difference in terms of this pull the instrument would be a satisfac- tory electrometer. But the density in a condenser is not uniform near the edges, and when the plates are some distance apart the edge effect may extend some distance inwards. We cannot there- fore apply the results of p. 83 to the whole plate. In the central part of each plate, however, the density is practically uniform ; the lines of strain go straight from plate to plate and these results apply. Taking tha surface density as cr, the difference of potential as V, and the distance of the plates apart as d, V = Since the pull per square centimetre is ZTTO*, the total pull P on a central area A is AV* =_ (i). In practice the pull is measured on a " trap-door," i.e. on a movable plate nearly filling a hole in one of the plates ot t he- condenser, but with free edges. The surrounding plate is tern ml the guard-ring, its function being to guard the density on the trap- door from variation. The trap-door and guard-ring are electrically connected and the force measured is that which is required to keep the two in the same plane against the electrical pull of the opposite plate. The effective area A is approximately (Maxwell, Electricity and Magnetism, 3rd ed. vol. i. p. 333) the mean between the aperture of the guard-ring and the area of tlu •disc or trap-door. Fig. 67 is one form of the instrument,* showing only the essential parts. The guard-ring is fixed, while the movable disc or trap- door is hung by metal wires from the end of a metal lever ha\ a counterpoise at the other. The lever is supported by a wire stretched horizontally between two insulated metal pillars connect id to the guard-ring so that disc and ring are in connection. The position of the trap-door is indicated by a hair stretched across the forked end of the lever. This is viewed by a lens, and is so adjusted that when it is midway between two black dots on a white upright passing up through the fork behind the hair, the trap-door and guard-ring are coplanar. Suppose that a known weight P is placed on the disc and that a movable rider is adjusted on the lever until the hair is in the central position when there is no charge on the plates. Now remove P and connect the plates to bodies at different potentials, adjusting the distance d of the lower plate until the hair is again central. Evidently the electrical pull is equal to P, and if V is the difference of potential we have from (1) * Thomson's Papers on Electrostatics and Magnetitm, p. 281. POTENTIAL AND CAPACITY 93 V = d (2). It is very difficult to measure d accurately, and it is better to make the results depend on the differences of the distance in two successive experiments, that is, on the distance the lower plate is moved. To do this let the lower plate be connected to a source of uparvd metfrscrtw. FIG. 67. potential V, the upper plate being connected to earth ; then equation (2) holds. Now connect the upper plate to a source at potential V', the lower plate being connected as before. If the distance, after ad- justment, is now d\ V- V" (3). Subtracting (3) from (2), V =(<*-<*') which gives V in terms of measurable quantities. It may be noted that in obtaining (4) we suppose V either less than V or opposite in >ign. I another form of the instrument the attracted disc is hung from one arm of a delicate balance and counterpoised so that when there is no electrification it hangs exactly in the plane of the guard-ring. A stop is then arranged so that the disc cannot move upwards and an extra weight P is put on the other arm. The subsequent working is as with the arrangement just described. 9* STATIC ELECTRICITY Practical methods of measuring potential. If the centre of a conducting sphere is brought to the point at which the potential V is required and is then connected to the zero-level surface, say the earth, by a wire, its potential becomes zero. Let Q be the charge induced upon it. The potential at the centre of the sphere being zero, we have 22 = 0. r But, if the wire is exceedingly fine and if the sphere is sufficiently small, this may be split up into the terms due to the original electrification practically undisturbed by the introduction of the sphere and giving value V, and the terms due to the charge gathering on the sphere and giving value — . Then On breaking the earth connection and taking the sphere away, it we measure its charge, we obtain a number proportional to tin- potential at the point, though opposite in .sign. A very obvious weakness in this method lies in the neglect of the charge on the connecting wire. It is not easy to make the capacity of the win- negligible in comparison with that of the sphere. Reduction of the potential of a conductor to that of a given point in air. Suppose, for example, that we wi>h to equalise the potential of a pair of the quadrants of a quadrant electrometer to the potential of a given point P, Fig. 68, in the air. Suppose, to begin with, that they are at higher potential. Take an insulated wire from the electrometer to the point 1* : the end of the wire is evidently at higher |p\ potential than its surroundings. Thru \ it is positively electrified, while the quad- . - - ) rants are negatively electrified. Imagine V J now that the outer layer of the wire is ^JL^ loose, so that it can be drawn of}' into the FIG. 68. air with the charge on it. The poten- tial of the wire at P is thereby lowered more nearly to that of the air near it. A new positive charge will now gather at P and more negative will go into the quadrat If the new surface is also loose, the new charge will also be drawn oft' into the air, and if we so imagine successive layers to be removed they will carry with them positive charges until the end POTENTIAL AND CAPACITY 95 of the wire P is brought to the same level as its surroundings, or at least to the mean level so that each layer takes with it equal and opposite quantities. If the electrometer is at lower potential to begin with than the neighbourhood of P, it is evident that negative will go off, while the electrometer will be positively electrified. This process has been realised practically by various methods. In the method first used a burning match was fixed at one end of an insulated conducting rod, which was brought to the given point, the other end being connected to the electrometer. The hot mass of the flame is a conductor, and as it is continually being thrown off it carries electrification with it as long as there is any difference of potential between the end of the rod and the surrounding air. In a second method, still sometimes used, an insulated can of water discharges through a fine nozzle drop by drop. In Lord Kelvin's original description of this instrument as applied to find the potential outside a window he says : " With only about ten inches head of water and a discharge so slow as to give no trouble in re- plenishing the can of water, the atmospheric effect is collected so quickly that any difference of potentials between the insulated conductor and the air at the place where the stream from the nozzle breaks into drops is done away with at the rate of 5 per cent, per half second, or even faster. Hence a very moderate degree of insulation is sensibly as good as perfect so far as observing the atmospheric effects is concerned." (Electrostatics and Magnetism, p. 200). It is usual now to employ a wire tipped with radium. The air is ionised by the radium, and the charge on the end of the wire is neutralised by the ions of opposite sign. Atmospheric electricity. Either of these instruments may be employed to determine the potential at any point in the air with regard to the earth. It is found in general that the potential rises upwards from the surface, especially in clear weather. This of course implies negative electrification of the earth's surface, the corresponding positive electrification being scattered through the air above the surface. When clouds are formed they act as conducting masses and become electrified on their surfaces. If the weather is not stormy the potential in the air is still usually positive with regard to the earth, the under surface of a cloud being probably positively charged and its upper surface negatively charged. But in stormy weather it frequently happens that the earth is positively charged under a cloud. This may possibly be explained on the supposition that the cloud was electrified by induction bv the ground under it and that the positive charge on the cloud nas been removed by the rain falling from it; the negative then spreading over the whole might produce a negative potential in the air under it. We cannot here go into details of the very puzzling subject of Atmospheric Electricity, a subject in 96 STATIC ELECTRICITY which the facts are only beginning to fall into order. We refer the reader to the Encyclopaedia Britannica, llth ed. vol. ii. p. 860, or to Gockel Die Luftelektrizitat. Simple methods of measuring capacity. We shall describe two methods of measuring capacity merely in order to make the idea of capacity more definite. Other methods will be described later or may be found in laboratory manuals. The definition of capacity shows that it is in dimension a length. Thus the capacity of a sphere, far from surrounding conductors, is equal to its radius, and the capacity of two parallel plates in air is equal to the area of either /4?r distance apart, when the distance apart is so small that the edge effect may be neglected. We may then determine the capacity in either of these cases by actual measurement of the dimensions of the system. But when the system is of such form or with such a dielectric that we cannot determine the capacity by direct measurement of dimensions, then we may use a method of comparison. Capacity of a Leyden jar by comparison with a sliding condenser with parallel plates. Let AB, Fig. 69, A B ITI'IT'I'I'I'ITI'ITI'I FIG. 69. represent the sections of two round plates mounted on insulating supports with bases sliding along a divided scale, so that the plates are always parallel to each other and so that their distance apart can be observed. The capacity of this "sliding condenser'1 can therefore be varied at will. The method consists in charging this sliding condenser and the system of which the capacity is required with electrification at the same difference of potential, and then varying the distance apart of the plates of the sliding condenser till the charges on the two are equal. Suppose AB, Fig. 70, are the plates of the sliding condenser, and let J be the jar on an insulating stand. Let A and the outside coating of the jar be connected to earth. Let B and the inside coating of the jar be connected to the positive terminal of a battery whose negative terminal is earthed. Then disconnect the battery from both and insulate A and the outside coating of the POTENTIAL AND CAPACITY 97 jar. Connect A to the inside coating of J, and B to the outside coating of J. If the charges are equal these connections result in complete discharge, and if A is now connected to a quadrant electrometer no charge is shown. But if J has the larger capacity some positive charge is left unneutralised and is indicated by the electrometer. If J is the smaller, some negative charge is indicated. The distance apart of the two plates A B must be varied till no charge is indicated and the capacities are then equal. The A B FIG. 70. difficulty in such measurement arises from "residual charge," which will be discussed in Chapter VIII. This method was used by Cavendish (Electrical Researches, p. 144?). Capacity of a gold-leaf electroscope by comparison "With a sphere. We shall describe the method used by C. T. R. Wilson (P.R.S. Ixviii. p. 157). The electroscope was charged by a battery of small accumulators to different potentials so that the relative values of the deflections of the gold leaf were known. A brass sphere, radius 2*13 cm., was suspended by a silk thread at a distance great compared with its radius from all other conductors except two fine wires, one earth-connected with which it rested in contact, and another leading from the electroscope and with its free end near the sphere. The electroscope was charged to some potential V given by the deflection of the gold leaf. Then the charge on it was CV, where C was its capacity. The sphere was then drawn aside by a silk thread so that it momentarily broke contact with the earth and came into contact with the wire from the electroscope. Suppose the new value of the potential indicated by the electroscope was V. The charge on it was now CV. But the charge lost, viz. CV — CV7, was given up to the sphere with capacity 2*13, and raised its potential to V so that its amount was 2-13 V. Equating C(V - Y) = 2-13 V' G 98 STATIC ELECTRICITY V' or C = 2-13 V - V By this method Wilson found the capacity of an electroscope which he was using to be 1*1 cm., and this was practically constant for different positions of the gold leaf. In carrying out this experiment the wires, in order to reduce their capacity, should be made as fine as possible and not longer than is necessary to keep the sphere sufficiently distant from the electroscope. CHAPTER VIII THE DIELECTRIC. SPECIFIC INDUCTIVE CAPACITY. RESIDUAL EFFECTS Specific inductive capacity — Faraday's experiments — Effect of specific inductive capacity on the relations between electric quantities — Condi- tions to be satisfied where tubes of strain pass from one dielectric to another— Law of refraction— Capacity of a condenser with a plate of dielectric inserted — The effect of placing a dielectric sphere in a pre- viously uniform field — Residual charge and discharge — Mechanical model. FARADAY, as we have seen in Chapter IV, abandoned the old method of regarding electric forces as due to direct action of the charges at a distance, and sought to explain electric induction by the action of contiguous particles on each other in the dielectric, an action which he suppo>ed to be "the first step in the process of elect rolysution." Taking this view of electric induction, " there seemed reason to expect some particular relation of it to the different kinds of matter through which it would be exerted, or something, equivalent to a specific electric induction for different bodies, which, if it existed, would unequivocally prove the dependence of induction on the particles/* He was thus led to the great discovery that the Quantity of electricity which a condenser will receive when charged to a given potential — that is, -ipacity — depends on the nature of the dielectric. This implies that the force which a given electric charge will exert depends on the medium through and by which it acts. The nature of Faraday's discovery may be illustrated by supposing that we have two exactly equal condensers of the same dimensions, the plates in one being separated by air and those in the other bv, say, ebonite. If each is charged to the same potential difference, that with ebonite will have about two and a li ilf times as great a charge as that with air as the dielectric, and the ebonite is said to have two and a half times the specific induc- tive capacity of air. A very simple experiment with a gold-leaf electroscope suffices t<> -how the greater inductive capacity of ebonite. Let the electroscope have a table on the top of the rod to which the gold leaves are attached, and let there be a cover the size of the table provided with an in-ulating handle. Put a thin plate of ebonite * Exp. Re*. Ser. XI. i. p. 373 (November 1837). 100 STATIC ELECTRICITY on the table and put the cover on it, the cover and table thus forming the plates of a condenser. Then connect the table with the negative terminal, and the cover with the positive terminal, of a voltaic battery of several cells, when charges will gather pro- portional to the potential difference of the terminals. Disconnect the wires, first from the table, and then from the cover, and lift the cover. The charge on the table is then shared with the leaves, which diverge by an amount to be noted. Now remove the ebonite plate and put three very small pieces of ebonite of the same thickness as the ebonite plate on to the table merely to serve as spacing pieces, and place the cover on them. We now have an air condenser with the plates the same distance apart as before, for the spacing pieces occupy a very small fraction of the volume and may be neglected in a rough experiment. On con- necting and then disconnecting the terminals of the battery as before and lifting the cover we note that the leaves diverge much less than before, or the table has received a much smaller charge when air is the dielectric than it received when ebonite was the dielectric. The experiment is not suitable for exact measurement, for the capacity of the table when the cover is removed will not be quite the same in the two cases, so that the gold leaves will not get quite the same fraction of the charge. If a plate of india- rubber or a plate of sulphur be used, similar effects are noticed; the induced charge is always greater than with air, or, as Faraday expressed it, the specific inductive capacity is greater. We may give exact signification to this term in the following definition : Specific inductive capacity. Let two condensers A and H have exactly equal dimensions, and let the dielectric in A be air, while in B it is some other substance. Then the ratio capacity of B capacity of A is termed the specific inductive capacity of the dielectric in B. It is usually denoted by K. The specific inductive capacity of a given specimen is probably constant over a wide range of electric intensity, and assuming this constancy, it is frequently termed the dielectric constant. It is also termed the electric or electrostatic capacity of the material. If we consider the ease or difficulty of producing electric strain in the dielectric, we obtain an analogy with elastic strain which has some value. Suppose two equal condensers, A with air as dielectric, B with a dielectric with constant K. If we charge A to potential difference V, producing surface density cr and strain D = t B. With a flint-glass cup in place of the shellac he found K = 1'7(>, and with sulphur K = 2'24. When liquids were introduced into the condenser no certain measurements could be made owing to conduction, and \\hcn different gases replaced air no difference could he detected, for tin- apparatus used was not sufficiently sen-iti\e. The difficulties <>t experiment with liquids and gases ha\e only been OUTCUIMC ^ince Faraday's time. Faraday also uaed two cowlenierB consisting of three parallel equal circular plates, the middle plate formi: fkdenter With each side plate.* He charged the middle plate, and then slum ed that by introducing a slab of shellac or sulphur between the middle and one side plate the capacity on that side was increased. He saw that this arrangement might be used for exact measuren as indeed it has been used later in a modified form. Though our knowledge of the existence of specific inductive capacity is entirely due to Faraday, it is not a little remarkable that it was discovered by Cavendish some time between 1771 and 1781. But he communicated his results to no one, and t remained buried in his MS. notes till these were edited anil published by Maxwell in 1879 as The Electrical Researches of the Hon. Henry Cavendish. His discovery was thus entirely without influence on the progress of electrical knowledge. For many years after the publication of Faraday's original paper little experimental work was done on the subject, but the publication of Maxwell's Electro-magnetic Theory of Light,| * Ex$. Res. i. p. 413. t Phil Trans. , 1865, p. 459 ; Electricity and Magnetism, vol. ii. chap. xx. (1873). THE DIELECTRIC 103 according to which the dielectric constant should be equal to the square of the refractive index, led to a renewal of interest in the matter, and this was no doubt increased on the practical side by the necessity of knowing something about the electric capacity of the insulating material of telegraph cables. Since then a great amount of work has been done, and the specific inductive capacities of a great number of solid, liquid, and gaseous dielectrics have been determined in a great variety of ways. The phenomena termed residual charge and discharge, which so much complicate the measurements, have also been investigated. Before proceeding to an account of the work we shall examine the effect of the existence of specific inductive capacity on the various electrical relations, assuming that for a given material' it has a constant value. The effect of specific inductive capacity on the relations between electrical quantities. Let us suppose that we have two condensers (Fig. 71), A with air as the dielectric Air FIG. 71. and K = 1, H with a substance as dielectric which has constant K, and let the two be charged to the same surface density &. Electric strain. Since the electric strain is measured by the quantity induced per unit area on a conductor bounding the im-dimn. and this quantity is & in both A and B, the strain D is iint- in cadi. Difference of potential. To produce the same difference of potential we should require Ko- in B. Then usually an imaginary and impracticable way of regard- ing intensity in B, \\v must now define it as the slope of potential, and again, of course, we get it as «r of the value in air. If then in A we put E = 4-7nr= 4-TrD, in B we must put ,„ _ -K : Energy of charge. If Q is the charge in each, and if V is the V potential in A, and V = r? is that in B, the energy of charge will 104 STATIC ELECTRICITY QV QV still be J charge X potential, or -g- in A and -£- in B. For the work done in raising element q through potential v will still be qv whatever the medium between the conductors. Energy per cubic centimetre. Let us consider a tube of unit cross-section going from plate to plate and let d be the distance between the plates. The energy to be assigned to this . tube in B is |ei«rn being given to ABCD because it is work done on the system. Equating the two expressions 106 STATIC ELECTRICITY dO j %7T(TZ "K j£ dQ*-TFm*'TP"3SM and ^Q : JV IV U< I/ 0 dK or the heat energy added per unit volume is + ^ -yg of the energy added by the work done in charging the system. In some experiments by Cassie* the following values of -^ -^ and « ^ were obtained at about 30° C., say 300° A A a(7 !_ j/K 0 dl\ K ,/tf K do Glass Mica ()•()()() I- 0 \<> Ebonite 00007 Ml We see that the heat supplied to keep the temperature constant when glass of the kind used by Cassie is electrically strained is 0*6 of the energy supplied by the work done. If this heat is not supplied and the charging takes place under adiabatic conditions the temperature falls. K is dec T« MM <1. and the medium becomes as it were electrically stronger, and \\e ha\e the analogue to the increased elasticity of solids under adiabatic « 1 strain. But it can be shown that the adiabatic capacity of a condenser bears to its isothermal capacity a ratio which differs from unity by a quantity quite negligible in practical meaMiivnu-nK even though the excess over unity is proportional, as investigatlOD shows, to the square of the potential diffi rence.f Conditions to be satisfied where tubes of strain pass from one dielectric to another. Law of refraction. When tubes of strain pass from one medium to another with different dielectric constant, no charge being on the surface, they change their direction unless they are normal to the separating surface, and they are said to be refracted. There are two conditions to be satisfied. We may describe the first as (1) continuity of potential on passing through the surface. This continuity of potential implies that the potential is the same at two points indefinitely near to each other, one on each side of the separating surface ; and this implies that the intensity in any direction parallel to the surface, and close to it, is the same in each * Thomson's Applications of Dynamics to Physics and Chemistry, p. 102. | If C , and C0 are the two capacities, ^0 = ! V_2_l_rfK_*_ is the density, and continuous. We may describe the second condition as (2) continuity of strain. The equality of the opposite charges in an electrified system is not affected by the presence of dielectrics of different capacities, and we have every reason to suppose that if we draw unit tubes, each starting from +1, they will end each on —1, whatever dielectrics thi-y P.-INN through. If then ABCD, Fig. 74, represents a unit K, FIG. 74. tube in a medium with dielectric constant Kr continued as CDEF in another ini-diiim with constant K2, CDEF is also a unit tube. If the area of the surface cut out by each tube is a, and if the two tubes make angles Ot 02 with the normal, the cross-section of ABCD i- a CM ffj, while the cross-section of CDEF is a cos 0r If the strains on the two sides of CD are Dx and D2, the equality of the total strain in the two tubes gives 108 or STATIC ELECTRICITY Dx a cos 01 = D2 a cos 6t Dj cos Oi = D2 cos 02 or the normal component of the strain is the same in each medium. If we are considering non-crystalline media, in which the directions of strain and intensity coincide, let the plane of Fig. 74 pass through AC and the normal; the tube in the second medium must also be in that plane. For the components of the intensities at C in any direction parallel to the surface are the same in both media. The intensity in the first medium has zero component perpendicular to the plane of the figure. Therefore the intensity in the second medium has zero component in that direction. ThU implies that CE lies in the plane of the figure or the plane through AC and the normal. Then the incident and refracted tubes are in the same plane with the normal. The continuity of potential gives us E1sin ^! = K2 sm #r The continuity of strain gives us whence or, since x cos 01 = D2 cos We shall now consider some effects of the presence of a dicloc trie- other than air, which will be useful when we come to dcM rilx- methods of measuring specific inductive capacity. Capacity of a condenser with a slab or plate of dielectric inserted. Let us suppose that the distance bet" FIG. 75. the charged conducting plates is d, and that a slab of dielectric with constant K occupies a thickness t of the space between the plates and that its surfaces are parallel to the plates as in Fig. 75. Let the plates and slab extend indefinitely. Let the surface density THE DIELECTRIC 109 of charge be or. We must suppose that the tubes of strain go normally from plate to plate, and that the strain D = o- is the same in air and in the slab. The intensity in the air is 4-Tro-, while that in the dielectric is -^. Then the potential difference A between the plates is The capacity per unit area is a- 1 or is equal to that of an air condenser in which the plates are nearer j£ _ 1 together by — ^- 1. If such a slab of dielectric is inserted between the plates of an attracted disc electrometer maintained at a given potential, a layer of air intervening between the slab and the attracted disc, the charge a- is increased in proportion to the capacity. If then P is the pull per unit area when air alone is between the plates, it becomes with the insertion of the slab The effect of placing a dielectric sphere in a pre- viously uniform field deduced from the effect of placing a conducting sphere in the same field. If a sphere of dielectric constant K is placed in a field in air, previously uniform, the lines of strain crowd in upon the sphere, for it is more easily strained than the air. The effect of a sphere with K about 1 '35 is shown in Fig. 76, where it will be seen that the strain within the sphere is uniform and parallel to the original direction. \Ve shall not give a strict proof that this is the distribution, but shall merely show that it will satisfy the conditions required at the surface of separation. I-'irst, let us suppose that a conducting sphere occupies the position. Then on one hemisphere a + charge is induced, and on the other a— charge. These two charges must be so distributed tlmt they will produce a field within the sphere, just neutralising the field E, which previously existed, for there is no field within the conductor. We may find this distribution by the following device. 110 STATIC ELECTRICITY Let electrification, density p, be distributed uniformly within the sphere centre O, radius a, Fig. 77. At any point P within the sphere the intensity due to this is along OP and since the shell FIG. 76. external to P has no effect and the sphere within P may be regarded 4 ( )lu 4 as collected at its centre it is equal to ^ irp Tjp^a71"/0 ®^' *""' let electrification density — p be uniformly distributed within FIG. 77. a sphere, radius a, centre O' close to O, O'O being in the dinrtion of E. The intensity at P due to this second sphere is | -jrp PO' along PO', since the electrification is negative. The two distribu- tions superposed will have resultant | irpOO' through P parallel to OO'. The result of this superposition is zero density, or no charge m the overlapping part of the spheres, a + layer on the hemisphere towards A, and a - layer on the hemisphere towards B. rv^A* • CKneSS °f the layer at any Point Q in a Direction 0 with OA is GO cos 0, and therefore the surface density is p OO' < Now adjust p and OO' so that THE DIELECTRIC 111 I vp 00' = E. o Then the two layers will give a uniform intensity — E in the overlapping region, which will neutralise the external field E in that region, and the surface density of the layers is 3 E cos 0 4 7T Externally, and only externally, to the spherical space the two distributions will act as if each were collected at its centre, and 4 4 therefore as if we had electrification ^7r/oa3 at O and — B-rr/oa3 at O'. These two constitute what is termed an electric doublet, and they correspond to the two poles of a magnet. We define their moment 4 to be M = £ 7r/oa3OO/ = Ea3. Just as with a magnet, the in- ten-ity at distance d making 0 with OO' is 2Ea3 cos S d* along d, and — -^ perpendicular to d (see Magnetism),* and this field is superposed on the uniform field E. It may be noted that just Fio. 78. outside the sphere at the ends of the diameter parallel to E where d = a and 0 = 0, the total field is E + 2E = 3E. The field in the neighbourhood of the sphere is shown in Fig. 78. Since the field without is modified by the presence of the * These components may be obtained very easily by noting that the two nearly equal forces -rpaa/rfa and £jrpa3/(d + 5)2 acting at an angle differing from r by the small angle have resolutes along the two bisectors of the angle respectively equal to the difference grpa.3& W3 and to ^irpa3(f>/d2. Here 5 is O'O cos 0 and 0Js OOr sin 0 >l. 112 STATIC ELECTRICITY conductor as if it were a doublet, the forces on the external charges producing the field will be the same as those of the doublet. We have just seen that two almost overlapping spheres with equal radii a and equal and opposite densities of charge, will change the internal intensity from E to 0, i.e. by an amount E if they are equivalent externally to a doublet of moment Ea8. Then in order that the internal intensity may be changed from E to E', i.e. by an amount E — E', the pair of spheres must be equivalent externally to a doublet (E — E')a3. Let us assume that when a dielectric sphere, radius a, and with dielectric constant K is substituted, the field is uniform and equal to E' within, and without is E with the field due to the doublet (E - E')a8 superposed on it. In order that this may be the actual arrangement of intensity it has to satisfy the two conditions: — (1) equality of intensity in the two media tang«-nti:il to and close to the surface of the sphere; and (2) equality of strain in the t\\o media normal to and close to the surface. The tangential intensities at a point on the surf'.. -nt // in the field assumed are : — just outside, E sin 9 — (E — I •'.') >in 0 = E' sin 0 ; just inside, E' sin 6. So that any uniform value of E' will sati-fx tlii> condition. The normal intensities at the same point are: — just ouNidc, 2(E-E') cos 0+ E cos 0 =(,'3K-^K') cos th. ju>t ins: Since the strains are respectively intensity /4-Tr in air and K x intensity /4?r in the sphere, we" must have for equalitx normal strain 3E - 2E' = KE' whence E' The moment of the doublet equivalent for the outside is (E - E') a» and the values of the internal and external fields thus obtained satisfy the conditions, and so constitute a solution. We shall assume that it is the only solution. Since the field without is modified as if the dielectric \\i-re replaced by the doublet, the action on any external charge will be TjT _ I the same as that of the doublet, or will be -^ - -^ times the action JV + * of the equal conducting sphere. Similarly, the reaction of the tr _ 1 charge on the dielectric will be ^ - « times the action on the equal conducting sphere. We shall see how Boltzmann used this result to obtain K. THE DIELECTRIC 113 RESIDUAL CHARGE AND DISCHARGE The investigation of specific inductive capacity is very much complicated by what are termed residual effects. If a Ley den jar is charged and, after standing for a short time, is discharged by a spark in the ordinary way, it appears to be completely discharged and the two coatings are at the same potential. But if the jar is allowed to stand for a short time, with the inner coating insulated, a new charge gathers of the same sign as the original charge, and a second much smaller spark may be obtained on discharging it. This process may be repeated, and with some jars three, four, five, or more visible sparks may be obtained in succession, the jar being allowed to rest insulated after each discharge. If, immediately after the first discharge, the inner coating is connected to a gold- leaf electroscope, the charge can be seen to gather, for the leaves diverge till they touch the side plates and so discharge the jar. They will diverge and discharge many times in succession. The electroscope may be used to show that all these charges are of the same sign as the original charge. They are known as Residual Charges, and the discharges as Residual Discharges. If a jar which shows very conspicuous residual effect is charged to some measured potential and left insulated, it is found that the potential gradually falls, and the discharge obtained on connecting the coatings for a moment will be less than the original charge. This fact suggests that the phenomenon is in some way connected with conduction, and Faraday sought to explain it by supposing that the two charges left the plates to some extent and penetrated the dielectric towards each other, some of the + charge on AB, Fig. 79, for instance, reaching ab, while some of the — charge on CD reached cd, ab and cd being probably further in Fio. 79. the longer the time.. On discharging AB and CD the charges ab and cd would no longer be pushed from behind by charges on the plates, and some part of them would return to AB and CD and be ready for a second discharge. But when we come to examine the process of conduction we shall see that this account, though probably containing a good deal of truth, hardly gives the correct view as it stands.* R. Kohlrauschf showed that if a given jar is charged to a certain * Exp. Res., vol. i. § 124:,. f ?W> Ann. xci. (1854). n 114 STATIC ELECTRICITY potential and is then insulated, the fall of potential in a given time is proportional to the initial potential, and, further, that if at any instant the jar is discharged, the instantaneous discharge is pro- portional to the potential just before the discharge was made. Kohlrausch formed a theory of the action going on which apparently involved the idea of conduction, though he did not express it in that form. But he pointed out the resemblance of the residual phenomena to those of "elastic after-action" in strained wires, which gives a very valuable analogy. If a wire is twisted — for instance, if a glass fibre is fixed vertically in a clam}) at its upper end, and if the lower end is twisted and held, it will return on release towards its original position, but not the whole way. If it is now held in its new position for a time and is then released, it will return another portion towards its original position, and so on. We may explain this after-action, as it is termed, by supposing that some parts of the glass retain strain energy as long as the strain is retained, while in other parts, though the sh remains, the energy is dissipated, or the diminishes. Imagine, for instance, that the outer shell of the fibre is " tnu " in its elasticity, i>e. that the stress is always proportion*] to tin- strain, but that the inner core gradually lo>e> its "train ^'ix-Tgy, even though its strain is maintained. Now twist the compound fibre. If it is instantly released before the energy in the core has had time to become dissipated it will return to the original position. But if the fibre is held twisted for a time the stress in the core gradually decreases and the effective strain <1< \\lnn tin fibre is released the effective strain in the core will be entirely removed before the fibre is entirely untwisted. But some stress still remains in the outer shell when this point is reached, and the outer shell still tends to untwist, and will continue to untwist till the core is strained in the opposite direction so much that the negative stress in it just balances the positm itNH -till mi mining in the shell On again holding the fibre this negative stress decreases, and on release the outer shell will be able to impart some more negative strain to the core, and so on. Gradually the fibre will return to its original position if the outer shell is perfectly "true" in its elasticity. But if it too exhibits some dissipation of energy the return will not be complete. The most probable explanation of residual effects is analc to this, and was given by Maxwell* somewhat in the following form. It takes account of effects which must certainly exist, and so far it must be a true explanation, though it may not be com])! In it the dielectric is regarded as heterogeneous, of which \>; are slightly conducting, while other parts remain completely insu- lating. We may imagine, as in Chapter IV, that the strain" in the * Electricity and Magnetism, vol. i. chap. x. An account of Maxwell's theory on somewhat simpler lines is given in the P/til. Mag., Series V. vol. xxi. (1886), p. 419. THE DIELECTRIC 115 dielectric consists in the formation of chains of molecules. If each molecule consists of a positively charged part a and a negatively charged part 6, in the entirely unelectrified condition these molecules ab will be turned indifferently in all directions. When the dielectric begins to be strained we may suppose that some, at any rate, arrange themselves in chains between AB and EF, and if AB is positively charged we shall have the negative elements towards AB thus : AB | ab ab ab ab \ EF. As the strain increases we may suppose the links of the chain, as it were, stretched out thus: AB \ a b a b a b a 6 | EF and conduction implies a breakdown of the chain, the first a going to AB and neutralising part of its positive charge, while its b unites with the a of the second molecule, and so on along the line, the last b going to EF and neutralising part of the negative charge there, thus : AB | a ba ba ba b\ EF We may suppose that, in the slightly conducting dielectric, a very small fraction of the whole number of chains breaks down per second, the fraction being proportional to the slope of potential. To illustrate Maxwell's theory, let us imagine a condenser, ABCD, Fig. 80, with a dielectric of specific inductive capacity K throughout. Let the upper half above EF be a slight conductor, while A B the lower half is a perfect insulator. Let the condenser be charged till the strain is, say, 32 throughout, c 0 and then let the upper plate be F10- 80. insulated. Then strain 32 remains in the lower half, EFCD. But conduction in the upper half means that the effective strain in it is gradually diminishing. Now let us suppose that the condenser remains insulated till strain 16 only remains in the upper half. There will then only be charge + 16 on AB. On EF there will be charge 32 - 16 = -f 16, and on CD there will be — 32. If we now discharge the condenser by connecting AB and CD we shall get in the first place + 16 of AB neutralising- 16 of CD. AB is now discharged. But there is still — 16 remaining on CD, which will be equally shared between AB and CD. This implies a transfer of — 8 from CD to AB, which is equivalent to a further discharge of + 8 from A B, or a discharge of 24 in all. The potential of AB is made equal to that of CD, and the strain is — 8 in ABEF and -f 8 in EFCI), while there is a charge + 16 on EF. If AB is now insulated, the negative strain — 8 between EF and AB gradually breaks down, while the positive strain between EF 116 STATIC ELECTRICITY and CD remains, and the potential difference between AB and CD becomes positive again. Let us suppose that AB remains insulated till the strain between it and EF has decreased from — 8 to — 4. There will then be charge + 12 on EF. On connecting AB and CD the two charges —4 and —8 will be equally shared, which implies a discharge of — 2 from CD to AB, equivalent to a discharge of + 2 from AB. There will then be— 6 on each of AB and CD, with the corresponding -f-12 on EF. Again insulate AB till the strain — 6 between EF and AB has fallen to — 3. On connecting AB and CD there will be a passage of — 1'5 from CD equivalent to +1*5 from AB, and 9 remains on EF. If we suppose the strain between AB and EF to be halved in this way each time, the successive discharges from the first will be 24, 2, 1-5, 1*125, &c. Evidently discharges will be obtained till the space bet\\e»-n EF and CD is completely free from strain — that is, until all the charge on CD is gone, Hence, if we have a completely inflating layer, the sum of all the discharges must equal the original charge. If, however, the layer between EF and CD is not completely insulating, but is a worse conductor than the upper layer, though we shall have residual discharges their sum will bo less than Un- original charge, owing to the decay of strain terminating on CD. It is obvious that if the dielectric is homqfleneoua and .slightly conducting throughout there will be no residual phenomena accord- ing to this theory. For the strain will break down .simultaneously from plate to plate, and the first discharge will be complete, as there will be no charges left within the medium as we have supposed those left on EF in the above explanation. We may note here how this explanation differs from that of Faraday. He appears to have thought of the charge from AB as gradually moving towards EF, the first links of the chain, as it were, breaking down first. Then after discharge he thought of the charge as gradually moving back again towards AB. It is evident, however, that Faraday had the essential features of the prt- theory. If a jar is charged, allowed to rest, discharged, and then charged in the opposite direction to a less extent than at first, it is found that it may ultimately show a residual charge of the same sign as the first charge. The theory gives an explanation of this phenomenon. For suppose that after the first discharge in the case we have considered, where — 8 is the strain in the upper half and + 8 is that in the lower, we give a negative charge — 4 on AB and + 4 on CD, we shall begin with strain — 12 in the upper half and + 4 in the lower, or with AB at lower potential than CD. But if we leave the jar long enough the strain in the upper half will decay to less than — 4, and the potential of AB will then THE DIELECTRIC 117 rise above that of CD, or we shall again be able to get a positive discharge from AB. It is easy to work out a more general theory wHere the dielectric consists of any number of layers of different thicknesses with different dielectric constants and different conductivities or rates of decay of strain, if we assume that Ohm's law holds — that is, that the rate of decay is proportional to the intensity. If D is the strain and E the intensity, we have _dD _ _ 47rcD dt TT _4irrt whence D = Y>0e ~^~ where D0 is the initial strain and c is the specfic conductivity of the material. If then at any instant the strains are Dx D2 . . . in layers having conductivities ctct . . . dielectric constants Kx K2 . . . and thicknesses d^ dz . . . at any future time t they will be K, &c., and the potential will alter from , But this general investigation has little value, for in the first place the dielectric heterogeneity does not consist in a parallel arrangement of layers, each homogeneous, but much more probably in an irregular granular arrangement. That it is complex was shown by Hopkinson (Original Papers, ii. p. 2). He found that the poten- tial of a jar charged and then insulated could not be expressed as a function of the time by two exponentials only. If it could be expressed by a series of the above form, certainly more than two terms would be required, or the heterogeneity is more than twofold. In the second place, even if Ohm's law holds, we cannot assume that K and c are constant for each element of the structure while it is breaking down. If the breakdown is, as we have supposed, electrolytic, the products of decomposition may alter the values of K and c. It is even possible that they may alter the values of d if the heterogeneous portions are of molecular dimensions. Hopkinson (loc. cit. ii. pp. 10-43) investigated the rate of fall of potential of a Leyden jar, and though he could not obtain a mathematical expression for the rate, he found that it was not /\ very different in some cases from — , where t is the time from 118 STATIC ELECTRICITY insulation. Boltzmann had found that the strain in a twisted win- decays at this rate. But though we cannot as vet give a full quantitative explana- tion of residual phenomena, there can be no doubt that they arc. at any rate, largely due to conductivity in parts, and that Max well's theory contains a large element of truth. On the one hand exceedingly good insulators, such as air and other how no trace of residual charge, while on the other hand poor insulators, with structure probably heterogeneous, show residual phenomena in a marked degree. Rowland and Nichols* showed that a plate of Iceland >j>ar exhibited no residual effect whatever, as might be expected it lit -tt n>- geneity of structure is a necessary condition for its existence. Mechanical model illustrating the theory.f Tin- model consists of a trough of semicircular < T<>^--r< -t KMI ( I'i^. iSl), say 24 in. long, 6 in. diameter, and divided into eight equal compart- A A ir Fio. 81. ments by a middle partition along the axis and three crow- partitions. It is supported at the two ends, so that it can rock about its axis, and a pointer attached to one end moves in front of a scale. Four pipes, with taps, connect the opposite compartments when the taps are turned on. The trough is balanced In weights on an upright, so that when empty it is in neutral equilibrium. On turn- ing the taps off, and on pouring in water to the same depth in all the compartments, the equilibrium at once becomes stable, and the trough, if displaced round the axis, stores up energy. It may be considered as analogous to a tube of strain connecting charges ± q on the surfaces of two opposite conductors, the angle of displace- ment representing the charge at either end, or the strain along the * Rowland's Physical Papers, p. 204. f Proc. Birmingham Phil. Soc.. vol. vi. p. 314. THE DIELECTRIC 119 tube. As long as the taps are turned off, the trough represents a perfect insulator, for the energy is undissipated. Release of the trough corresponds to discharge, and we have oscillations corre- sponding to the electric oscillations in the discharge of a condenser. If the taps are all turned on equally the trough represents a leaky dielectric of the same conductivity throughout, and holding it in the displaced position for a time and then releasing it, it returns to and remains, after the oscillations have ceased, in a position short of the original position. But if the taps are turned on unequally — if, sav, the two end taps are turned off and the two middle ones are turned on — it is easily seen that the phenomena of residual discharge are exactly imitated. For on turning the trough through a given angle and holding it there, the energy in the middle compartments decreases, and on release the trough only moves part way back, going to the point at which the mean level is the same on the two sides. There is now a negative difference of level in the middle compartments if the original difference is called p<»itivr. If the trough is held in its new position for a short time the negative difference is reduced, and on release the trough returns by another amount towards its original position, and this may be repeated several times until finally the original position is practically regained. CHAPTER IX RELATION OF SPECIFIC INDUCTIVE CAPACITY TO REFRACTIVE INDEX. THE MEASURE- MENT OF SPECIFIC INDUCTIVE CAPACITY The relation between specific inductive capacity and n-fr iciiv.- index in the electro-magnetic theory of light — Determinations of specific induc- tive capacity — Boltzmann'e condenser method for solids — His exp- ments with crystalline sulphur — Hopkinson's exprriim-tit^ — Boltzmann's experiments on gases — Specific inductive capacity of water, alcohol, ami other electrolytes — Experiments of Colin nnd Arons,Rosa, Hcerwa^r and Nernst — Experiments of Dewar and Fleming at low temperatures — Drude's experiments with electric waves. The relation between specific inductive capacity and refractive index in the electro -magnetic theory of light. Maxwell's electro-magnetic theory of light MippoM> that light consists of waves of electric strain transverse to the direction of propagation accompanied hy magnetic induction perpendicular to the electric strain, and also transverse to the direction of propaga- tion. If, for instance, the light is plane polarised, we mu-t -nppose the electric strain always in one plane, say that of the paper as represented by the vertical lines in Fig. 82, and alternately up and down in successive half wave-lengths. The accompanying magu Fio. 82. induction will be in a plane perpendicular to that of the paper, and alternately in and out in successive half wave-lengths. An analogy with sound waves will suggest to us the ratio of the veloci ties of pro- pagation in different media. The full investigation belong to Optics. In sound waves there are the two types of energy, elastic strain and kinetic, and the velocity of propagation is given 1>\ y _ /modulus of bulk elasticity density The numerator is proportional to the energy stored per unit 120 SPECIFIC INDUCTIVE CAPACITY strain, and the denominator is proportional to the energy possessed per unit velocity. If we have two media with different elasticities Ex E2, but with equal densities, the ratio of the velocities of propagation or the refractive index of sound waves from one to the other will be Guided by this analogy, we may regard the energy of electric strain as corresponding to the energy of elastic strain, and the energy of magnetic induction as corresponding to kinetic energy. Since in all transparent media the magnetic permeability is prac- ticallv the same, the energy due to unit induction in the two media is the same, or the media for electric waves correspond to media of equal density for sound waves. The electric modulus is, as we have 4?r already seen, ^, so that if for two media the dielectric constants are Kx and Kr the analogy suggests that the refractive index should be given by the If one of the media is air, for which K, = 1, and K is dielectric constant ot the other medium with respect to air, Absorbing region ' Fio. 83. But the analogy is obviously incomplete. In sound waves in gases the elasticity is definite and independent of the periodicity of the waves. A 1 1 waves travel with the same velocity, and the refractive index from one to another is a definite constant. But in light the < itv varies with the periodicity, and we have the phenomena of'diftpenion. The refractive index y. for transparent substances (hcK.iM > in general as the wave-length increases. If, however, a sub- re al)M>ri» ;i particular wave-length or a group of wave-lengths, tin- K fr;i( live index in that neighbourhood varies in a manner which is termed anomalous, and the general nature of the connection between /x, the refractive index, and A, the wave-length in air, is ««hown in Fig. 83. STATIC ELECTRICITY In ordinary methods of determining K by experiments on capa- city, we charge and discharge a condenser in times enormously long compared with the period of a visible light vibration, and we should only, therefore, expect to find v/K = yu if the value of M is that for very long waves with period long, and if that value is unaffected by absorption— that is, if the substance is transparent to very long waves. We shall now give an account of some of the determinations of the dielectric constant for solids, liquids, and ga^-. selecting typical methods, and not attempting to give any complete account of the subject.* At the time when Maxwell published his treatise on Electricity and Magnetism the only substance of which the dielectric constant was at all accurately known was paraffin. (iib-on and Kuvlay (Phil. Trans., 1871,* p. 573) had shortly betore found K to he 1'975. From the refractive index of melted paraffin for the A. 1 ). and H lines Maxwell calculated the refractive index for light of infinite wave-length to be T420., whence /z2 = '' Boltzmann's condenser method for solids.f Holt/m ann used an air condenser (Fig. 84) with parallel plates so arranged FIG. 84. that a slab of the dielectric to be experimented on could be inserted. He compared the capacities of the condenser with and without the slab by charging it in each case to a definite potential, and then sharing the charge with a quadrant electrometer with a small air condenser added in order to increase the capacitv. This was done so that any change in the capacity of the electrometer due to the motion of the needle should be negligible. The fall of potential din- to the division of the charge gave the ratio of the capacity of the * An account of the earlier work on specific inductive capacity will be found in Gordon's Electricity and Magnetum, vol. i. chap. xi. ; or Gray's Ab* sinrmcnti*, vo1. i. Later work is described in Winklemann's Handbnch, vol. iv. f Carl's Repcrtorinin. x. p. 1<>9. See Gordon's Electricity, vo'. i. SPECIFIC INDUCTIVE CAPACITY 123 large condenser to that of the electrometer + the small condenser. In order to eliminate the distance of the plates apart, a quantity not easily measured, Boltzmann moved one plate and took observa- tions at different distances, so that the distance the one plate was moved alone came into consideration. The principle of the experi- ment may be represented as follows : Let CE be the capacity of the electrometer + small condenser c, and let V be the potential of the battery. When the key k is open and t is connected to a, the electrometer will indicate V. Now let k be connected to e, the electrometer being thus dis- charged. Let t be connected to 6, the large condenser C being thus charged to V. Let its capacity be Cl when the plates are distance- dl apnrt. Let t be disconnected from a and 6, and let k be connected toy. Thus the charge on C is shared with CE, and we have the potential falling to Vr where whence Cx = CE_ (1) Since the two readings of the electrometer give us V1/V, we have C'j in terms of CE. Repeat these operations when the distance of the plates in C is altered to C* effect of N in disturbing the charge on M. 1 M on N will be equal to this. Now replace N by a dielectric sphere of the same siie with constant K. The dielectric sphere is equivalent to a doublet j£ _ I y , a °f that representing the conducting sphere, and the pull on K — 1 the dielectric due to M will only be .. of that on the , ducting sphere. Boltzmann arranged a small conducting sphere suspended bv silk threads from one end of a torsion arm, and mcaMin d the deflection when an elect riti< ^ in its neighbourhood. The "ondiu-ting sphere was then repla. ing for exact measurement a parallel plate con- dcn>er in which the distance between the plates could be varied till the capacity was equal to that of another condenser. The principle of the method may be gathered from the diagrammatic repre- sentation in Fig. 85. Let B be a battery of cells earthed at its middle point, so that the potential of one terminal is as far above that of the earth as the K f/i Fio. 85. potential of the other terminal is below it. Let C be the variable condenser with parallel adjustable plates, and let S be the con- -er with which it mu*t be equalised, in practice a sliding of two co-axial cylinders. Q is a quadrant electrometer. Ix-t the lower plate of C and the outer cylinder of S be • I the upper plate of C and the inner cylinder of S r« -pectively with the -f and — terminals of tin li.-itt. iv hy means of the k( -\N K, Kt. Then move Kx and K2 •he tuo condenser! are connected together and to the met r. If the capacities are equal the two charges are equal and opposite, and will neutralise each other, but if .-in excess of one kind of charge indicated by the electro- • Original Papert, vol. ii. p. 54. 126 STATIC ELECTRICITY meter the condenser with that kind has the greater capacity. C must be adjusted till the electrometer shows no charge and t In- equality is obtained. Now let a layer of dielectric be interposed in (', of thickness t. If the distance of the plates apart is 1 1 Density K Very light flint (i-fil Light „ M 8-7« Dense „ tf-66 7-87 Extra 4-5 9-90 am 1-710 It is evident that /x2 is very much less than K. Hopkinson also showed that the result was the same whether C and S were connected to each other, and to the electrometer, for a time comparable with a second or for a minute fraction of a second. Residual phenomena, then, did not come appreciably into play. With plate glass, however, it was necessary to have only an instantaneous connection, for with long connection the method failed owing to residual charge. To show that K is really constant, i.e. independent of the potential difference, for a given specimen of glass, a flask of extra dense flint glass containing sulphuric acid was placed in water, the contacts of the liquids with the glass forming the coatings. The capacity of this flask was found to be the same. comparing it with the sliding condenser S, when charged \\ith 20 elements as when charged with 1800 elements of a chloride of silver battery. To find the dielectric constant of liquids a fluid condenser SPECIFIC INDUCTIVE CAPACITY 127 was used, consisting of a double cylinder in which an insulated cylinder could hang, the vertical cross-section being as represented in Fig 86. The capacity of this condenser was com- pared with that of a sliding condenser, first with air and second with the liquid to be tested as the dielectric. The sliding con- denser was brought to equality in each case. The sliding condenser was graduated, and its capacity for so many divisions of the inner cylinder within the outer was known. The hydrocarbon oils agree in giving K = /z2 nearly, while their values for vegetable and animal* oils differ widely. The Table below is extracted from Hopkinson^s paper (Original Papers, ii. p. 85). The square of the refractive index for infinitely long waves, which is given as well as K, is calculated from the dispersion in the \i>ible spectrum by the formula p = a + b/\, using the sodium and hydrogen lines. Fia. 86. K IVtroleum spirit 1'92 Petroleum oil (Field's) 2'07. Turpentine 2'23 Castor oil 4'78 Sperm oil 3'02 Olive oil 3-16 1-922 2-075 2-128 2-153 2-135 2-131 Boltzmann's experiments on gases. Boltzmann made determination! of K for different gases by a simple method, using a condenser within a closed metallic earth-connected vessel which could be exhausted or filled with any gas. The connections with the plates of the condcn^ r were made by wires passing through the tlit \ »—>(•!, but hermetically sealed in, as represented diagram mat ically in Fig. 87. The vc^el uas first exhausted, and one plate, A, was connected to a battery of about 300 Danielfs cells, the other plate, H, being connected to an electrometer and to earth. After A was thus raised to a potential which we will call Vj it wa> insulated, B and the connected pair of quadrants of the electrometer \\cre insulated, and the gas to be experimented on was Admitted The potential of B >f ill remained /ero, for, as there was TIO leakage from A, all the electrification on B was connected with that on A, and then wen then-fore no lines of force between B and the case, and no change of its potential. The potential of A fell by the admi-ion of the gas to, say, V2, where Vx = KV2, K being the dielectric constant of the gas, that of vacuum being 1. 128 STATIC ELECTRICITY Connecting A again to the batteiy, its potential was once more raised to Vr But B being insulated, its potential was raised by an amount proportional to the additional charge on A, for this additional charge induced on B an equal and opposite charge con- HI FIG. 87. nected with itself and an equal like charge connected with the si of the vessel. We may therefore put the potential of B as equal to - Va) where m is a constant, or Suppose the charge observed in the electrometer to be dri. bv 8. The number of cells was now increased from n to n + 1, and the potential of A was consequently -raised to Vx+ -1, and V that of B was raised by m — *. If the change observed in the electrometer was 3'9 we have whence K The following Table gives BoltzmaniTs results at 0° C. and 760 mm., with the values of ^ as determined by Dulong for white light. As the difference between this and p er and determined the capacity by a Wheatstone bridge method, using * Phil. Mag., xxxi. (1891), p. 188. t Wild. Ann. (1893), xlviii. p. 35, and xlix. p. L7i(. t Zeit. Phys. CAem. (1894), vol. xiv. p. «22. SPECIFIC INDUCTIVE CAPACITY 131 an alternating current. For water at 17° C. he obtained K = 80 '00, and other observers * by various methods have found nearly the same value. For alcohol he found K to be about 26. Experiments of Dewar and Fleming at low tem- peratures.f Dewar and Fleming made experiments on the dielectric constant of ice and other substances from a temperature of —200° C. upwards, using a condenser consisting of two co-axial brass cones, about 15 cm. long, the outer tapering from an inside diameter of 5'1 cm. to 2*6 cm., and the space between the two being 3 mm. The condenser was charged and discharged about 120 times per second by an interrupting tuning-fork and the circuit was arranged so that either the charging or the discharging current alone should go through a galvanometer. The equality of the two was taken to show that conduction was not coming into play. The space between the cones was filled with the substance to be examined, and cooled to the temperature of liquid air, and the galvanometer deflection was observed on charging at given potential or on discharging. Then the substance was melted out and replaced by gaseous air at the same temperature, and the galvanometer deflection was again observed. The ratio of the deflections, after certain corrections, gave the specific inductive capacity of the substance. Observations were also made at higher temperatures by allowing the temperature of the condenser to rise gradually. The dielectric constant of pure ice at —200° C. was 2 '43, rising with rise of temperature to 70'8 at — 7'5° C., though here conduction had set in and the measurement was not so trust- worthy. A large number of solutions and compounds were thus examined, and the general result was that at —200° C. the dielectric constant was not much greater than the square of the refractive index for exceedingly long waves, as calculated from dispersion formulae. For castor oil, olive oil, and bisulphide of carbon they were nearly coincident. Drude's experiments with electric waves. As a type of an entirely different method of research, in which electric- waves are used, we shall take Drude^s experiments on water and on other liquids.^ If Uj is the velocity of electric waves in air and U2 is their velocity in another medium, the refractive index for the waves is /x = U1/U2. But if A! X2 be the lengths in the two media of waves starting from a source of the same frequency X1/X2= Ui/lT2» so that /x = X1/X2. According to the electro-magnetic theory K = /x2 = X12/X22. To find the ratio X^X^ Drude used a Lecher system in which waves were transmitted between two parallel wires, first through air and then through a trough containing the * For a summary see Dewar and Fleming, Pt'oc. Rot/, Soc., Ixi. i\ 2. t Proc. Roy. Foe.. Ixi. (several papers). % Ann. der plnjgf, 1896, lyiij. p. 1, lix. p, 17, STATIC ELECTRICITY liquid to be examined. The principle of the method may be gathered from Fig. 88. J was an induction coil connected by wires A A to the terminals of the Blondlot exciter E E, which consisted of two semicircle with diameter in some cases 5 cm. and in other cases 15 cm. The brass terminals of E E were 5 mm. diameter, and the spark gap between them could be varied by a micrometer. Hound E E was a circular wire continued by the parallel wires D D about 2 cm. apart. These might be continued, when desired, into a trough containing the liquid to be tested and from 150 cm. to 60cm. long. When sparks took place across the gap in E V. ua\< - were propagated in the space between D D with the velocity <>t free waves. U1 H2 were holders to keep the wires adjusted and Bj and B2 were two wire bridges. Between these, stationary waves were formed. Bx was adjusted to be at the first node; the second node was where the wires entered the trough of liquid, FIG. 88. J induction coil ; A A wires from its terminals to the exciting circuit E E, round which is a circuit prolonged by the two wires D D ; fy Ba bridge*, B! fixed, B2 in the liquid trough movable ; Z Xehnder vacuum MM Hffctin| up at a loop. and B2 was moved about till it was at a third or further n- In the original arrangement Z was a vacuum tube with terminals connected by a wire s of such length that the electrical period of the tube coincided with that of the exciter, but a neon tube laid across the wires suffices. When B, was at a node and Z at a loop. / lighted up with maximum brilliance. Several successive nodal positions were observed by means of this maximum brilliance, and these were, of course, at intervals of Xt/SJ. Then the trough detached and replaced by a continuation of the wires in air. Nodal positions of B2 were again observed, and thus Xx/2 in air was found. Thence yu = X1/X2 was known. With this apparatus Drude was able to show the existence of dispersion — that is, a difference of velocity with difference of frequency. He used two frequencies, respectively 150x10* and 400 xlO6. The velocity for water was practically the same for both frequencies, but for glycerine /x2 = 39*l for the former and /x2 = 25*4 for the latter, and for other liquids there were consider- able differences. He found that for water at t° C. the value SPECIFIC INDUCTIVE CAPACITY 133 sufficiently expressed the dependence on temperature, though there were slight variations with variation of frequency. With solutions of small conductivity the refractive index was nearly the same as for water, but as the conductivity increased, the value of the refractive index decidedly decreased. For details the reader may consult the original papers. CHAPTER X STRESSES IN THE DIELECTRIC Tension along the lines of strain — Pressure transverse to the lines of strain— Value of the pressure in a simple case — This value will maintain equilibrium in any case — There may be other solutions of the problem — These stresses will not produce equilibrium if K is not uniform — Quincke's experiments — General expression for the force on a surface due to the electric tension and pressure — The electric stress system is not an elasti< stress system and is not accompanied by ordinary elastic strains. WE have shown that a charged conducting surface in air is pulled out by a normal force 2-Tnr2 per unit area, and that in a dirleetric of specific inductive capacity K the pull become* ^Tro^/K per unit m Tension along the lines of strain. In accordance \\ith the dielectric theory of electric action we must suppose that this pull is exerted on the conductor by the insulating medium in contact with it. Further, assuming that reaction is (<|ii.il and opposite to action, the surface is pulling on the medium with an equal and opposite force. Let AB, Fig. 89, be a small area a of a charged surface; AC BD the tube of strain or force starting normally from it. The conductor is pulling on the medium in the tube with force , since D = \anishingly .small compared with the area of either end. Then for equilibrium the part of the medium above A'B' must pull on AA'B'B with a foivr „ „ equal and opposite to that across AB, \i/. 27rD2a/K. There is therefore a tension FIG. 89. 27rD2/K across A'B'. Now imagine the charged surface to be removed to A"B" some distance back, but so adjusted as to position and charge that strain in the neighbourhood of A'B' remains the same in direction and in magnitude. We can hardly suppose that the stress across A'B' is altered, and so we obtain the result that a 134 STRESSES IN THE DIELECTRIC 135 any point in the medium there is a stress along the lines of force, a tension 2-7rD2/K, where D is the electric strain. Since E = 4-TrD/K, we may put the tension equal to KE2/8?r. Pressure transverse to the lines of strain. A portion of the medium, say the portion between two cross-sections a± a2 of a tube of strain, could not be in equilibrium under these forces unless the lines of force were parallel and the field uniform. Equilibrium under the end pulls is obviously impossible unless aiis parallel to az. To show that the field must also be uniform, let D! and D2 be the strains at at and o2 ; the difference of the pulls on the two areas is and since D^ — D2a2, this may be put which only vanishes if D1 = D2 or al = u2, and this is the condition for a uniform field. When the field is not uniform there must be forces across the sides of the tube to make equilibrium possible. Value of the pressure in a simple case. A value for the side forces is suggested by considering a special case. Let a particle charged with Q be placed at O, Fig. 90, and let the DA o FIG. 90. C F opposite charge be so far away that the lines of force radiate straight away from O. Consider the equilibrium of the shell between two hemispheres, ABC radius r and DEF radius r+dr, drawn with O as centre. We have a tension normal to the surface O2 O inwards across ABC equal to 27rD2/K = — , since D = Now a closed surface is in equilibrium under a uniform tension or pressure, so that if we put an equal tension, Q2/87rK?*4, on the diametral plane AC we have equilibrium for the hemisphere OABCO. Or the tensions over the curved surface ABC have a 02^7-2 Q2 resultant inwards equal to the total tension 5 ^ ,= srr-o across the diametral plane. 136 STATIC ELECTRICITY Dealing in the same way with the tensions across DEF, they Q2 have a resultant outwards equal to g. When dr is very small, the resultant of these two is a pull inwards 8K _1 \ _ (?dr (r+drf) 4Kr3' This inward pull can only be neutralised by forces applied round the rim of the shell. Let us assume at the rim a uniform pressure P upwards, i.e. perpendicular to the lines of force. The total area of the rim is 2irrdr9 so that for equilibrium and p_ _ Q2 _ or a pressure normal to the lines of force equal to the tension along the lines of force would maintain equilibrium. These equal values will maintain equilibrium in any case. It is easy to show that when we have a field in which the le\el surfaces have double curvature tlu-i equal values of the tension along the lines and of the pressure tran-vi r^r to the lines will suffice for equilibrium. Let ABCD, Fig. 91, be a small rectangle on a level surface with its sides in the planes of principal curvature. Let Oj O2 be the centres of curva- ture and let OjAsRj, O2B = R2. Let AO^ = <^ and BO2C = r The area of ABCD is l^R^^. The normals through "ABCD form the tube of strain. Now draw a section of the tube nearer to the centres of curva- ture by <$, where S is very small, and let A'B'C'D' be the comers of this section. Its area is - 4 - if), neglecting -*_. H] IV ^Ili2 If DXD2 are the strains over the two surfaces ABCD, A'B'C'D respectively, or STRESSES IN THE DIELECTRIC 137 The tensions being 2-TrD^/K and ^7rD22/K, the resultant pull inwards is And substituting from the previous equation the total pull inwards is But now suppose there is a transverse pressure P normal to the surfaces of the rim of the lamina between ABCD and A'B'C'D'. The area of ABB' A' is R^c. That of BCC'B' is R22S. Resolving P along the normal to ABCD and perpendicular to it, the latter gives resolutes neutralising each other in pairs. The former gives so that there is equilibrium if „ = There may be other solutions of the problem. Faraday was led to the idea of longitudinal tension and lateral pressure by considering the nature of electric induction. He says : * " The attractive force which exists amongst the particles of the dielectric in the direction of the induction is accompanied by a repulsive or a diverging force in the transverse direction." Clerk Maxwell showed that equilibrium would be maintained if the tension were equal to the pressure. But it is to be remembered that the system thus obtained is only a possible solution of the problem of the stresses in the medium. It is a solution, but there may be others. Meanwhile its simplicity recommends it as worthy of trial, and we shall assume henceforth that it is the solution. These stresses will not produce equilibrium if K is not uniform. It is important to observe that if we assume the existence of these stresses the medium is not necessarily in equili- brium unless K is constant. Thus a solid dielectric body suspended in air behaves like a magnetic body in a magnetic field, and Boltzmann's method of determining K by the force on a small sphere hung up in a field radiating from a centre depends upon this fact. Even in a system in equilibrium stresses other than the electrical stresses must intervene if there is a change of value of K. * E*p. Re*., i. p. 409, § 1297. 138 STATIC ELECTRICITY If, for example, the lines of force pass normally from air into a dielectric of specific inductive capacity K the electrical tension x in air is ^TrD2, while that in the other medium is , so that TT _ 1 for equilibrium a force 2?rD2 — ^— must be supplied to keep the surface layer in equilibrium. The medium is no doubt elastically strained, and the corresponding stress supplies the required force. If the lines of force are parallel to the surface separating one dielectric from another, the level surfaces are normal to that surface, and the electric intensity is the same in each medium. If the electric KE2 pressure in air is E2/87r, that in the other medium is — — . Hence E2 (K — 1)— is the pressure needed on the surface of the other medium in addition to the electric pressure to maintain equilibrium. Quincke's experiments. Quincke made a scries <>i experi- ments of historical interest to test the existence of these electrical tensions and pressures in dielectric liquids. He used a condenser * with horizontal plates, the upper being hung insulated from a balance beam, while the lower was insulated and so arranged that it could be connected to a battery of Leyden jars charged to given potential. The condenser being charged, the pull G, on the upper plate was first determined in air. Then, the condenser being sur- rounded with the liquid to be experimented on, the pull G2 was determined, when the potential difference, and, therefore, the elect He intensity, was the same as before, the plates being the same distance apart. Taking the area as A and neglecting the edge effect, we have by the foregoing theory : Gf_AKE» / AK2_ ~= or the pulls are in the ratio of the specific inductive capacity. In each case the condenser was discharged through a ballistic galvanometer. The quantity flowing through the galvanometer and indicated by the kick-off' was proportional to the capacity of the condenser, and, therefore, to the specific inductive capacity of the medium. This, of course, gave another determination of K. At first the values by the two methods were somewhat widely apart, owing to the neglect of the capacity of the connecting wire and key. When this was taken into account the determinations were fairly in accord. f The existence of the pressure at right angles to the lines of force was verified as follows : The upper suspended plate was replaced by a fixed plate with a short vertical tube passing up * Phil. Mag., vol. xvi. (1883). f Xalure, vol. xxxv. (1887), p. 334. STRESSES IN THE DIELECTRIC 139 from a hole in the centre. This was connected to an india-rubber tube provided with a stop-cock and to a bisulphide of carbon manometer. The plates being immersed in the liquid near together, and to begin with to earth, air was blown into the space between them through the india-rubber tube until a large central space extending from plate to plate was cleared of liquid. The cock was then turned off* and the manometer read. The lower plate was then brought to the same potential as in the experiment above described, and at once the manometer showed an increase of pressure, this increase being needed to balance the difference between the electrical pressures at right angles to the lines of force in the air and in the liquid. If h is the increase of height in the mano- meter and (T the specific gravity of the liquid used, we have, since E is the same in the liquid and in the air, If the difference of potential and distance apart are the same as in the previous experiment we may eliminate E by the result of that experiment. If we write Kp for the value of K as involved in the pressure at right angles, and KT for its value as involved in the tension along the lines of force, the two experiments give us respectively : <\c i\ E2 7 (KP - A) -5 — = g'1^ and whence A' Kp = The values obtained for Kp and KT were nearly equal to each other and to the value obtained by the ballistic galvanometer, which we may denote by K. The following Table shows the results in a few cases obtained in two series of experiments, the earlier comparing Kp and KT, the later comparing KT and K. EARLIER. LATER. Kp KT KT K Sulphuric ether 4-672 4-851 4-394 4-211 Carbon di sulphide . 2-743 2-669 2-623 2-508 Benzene . 2370 2-389 2-360 2-359 Petroleum 2-149 2-138 2-073 2-025 ', 140 STATIC ELECTRICITY The difference in the values for KT in the earlier and later experiments is probably to be ascribed to slight differences in the constitution of the specimens. General expression for the resultant force on a surface due to the electric tension and pressure. Let a rectangular tube of strain meet the surface in AB, Fig. 92, making 6 with the normal, and let the area of AB be a. Consider the equilibrium of the wedge with section ABC. 2^1)2 Perpendicular to BC we have a pull .,. a cos #, since the area of BC is a cos 0. Perpendicular to AC we have a push — ^ — a sin 0. The forces on the sides parallel to the paper neutralise each other. Resolving along the normal and tangent, we have Along the normal outwards a (cos2 0— sin2 0)= a cos .. -Tr— K K Tangential along AB, 2-TrD2 2 1 ); -^ a (cos 6 sin 0 + sin 0 cos 0)=" .. « >in20. The force on AB must be equal to the resultant of these, sincv P FIG. 92. the wedge considered is in equilibrium. wards per unit area, and at an angle p FIG. 93. Thus there is a force out- with the normal. The system of forces thus made up should give as resultant the force actually observed in any case. The electric stress system is not an elastic stress system, and is not accompanied by ordinary elastic strains. The electric stress system applies equally whether the STRESSES IN THE DIELECTRIC 141 dielectric be solid or fluid. Consider a small cube with four vertical edges parallel to the lines of force, Fig 93. The system of stresses is obviously equivalent to a shear stress in the plane of the figure and to pressures on the two faces parallel to that plane. Now, in a fluid in equilibrium ordinary elastic shear stresses cannot exist and can- not contribute to maintenance of equilibrium. Indeed, hydrostatic equilibrium is only possible in a fluid when the pressures about a point are equal in all directions. Further, even in a solid the elastic system of a shear stress and a perpendicular pressure would produce on the whole a decrease in volume. For the shear would not affect the volume, while it is easy to show that the pressure P 1 — 2 cr would decrease it bv P — — , where or is Poisson's ratio and Y is Young's modulus. But experiments to be described in the next chapter show that, at any rate in the case of glass, the presence of electric strain is accompanied by a uniform dilatation. We must therefore suppose that the electric stresses are not elastic forces accompanied by elastic changes of shape, but that they are called into play in some other way. Where the medium is material we may probably account for the stresses by molecular arrangement, on the supposition that the molecules are electric doublets. We may illustrate the idea by considering the corresponding case of a magnetic system. Suppose that we have a number of little magnets pivoted on points arranged in rows and columns, as in Fig. 94. If the magnetic axes are equally distributed in all directions there will be no continuous lines of force going in any one direction. But if the axes are arranged as in the figure the opposite poles in the columns will attract each other, forming tensions along the lines of axes. The like poles in the neighbouring columns will repel each other, forming pressures perpendicular to the lines of axes. Some such arrangement of electric doublets in a material medium may account for the electric stresses accompanying electric strain. We cannot say, u />riori, what effect such a rearrangement should YLU. 94. have on the dimensions of the system. We only know by experiment that it appears to lead to a uniform dilatation. When the matter is exceedingly attenuated it may be doubted whether we can account for the stresses by the molecules present, and we may have to imagine some structure in the ether before we can frame a hypothesis to supply the forces. CHAPTER XI ALTERATIONS OBSERVED IN THE DIELEC- TRIC WHEN IT IS SUBJECTED TO ELECTRIC STRAIN Electric expansion in glass — Maxwell's electric stresses do not explain the effect — Electric expansion of liquids — Electric double refraction : The Kerr effect. WHEV a dielectric is subjected to electric strain it shows in MHIH* cases a change in volume and in other cases it becomes doubly retracting. Electric expansion in glass. The dielectric in a glass condenser generally expands when the condenser is charged, and the dielectric becomes the seat of electric strain. This expansion was known to Volta and has since been studied by Go\i, Duter, Righi, and especially by Quincke.* It may be conveniently observed and measured by using a common thermometer as a Leyden jar. The bulb is immersed in melting ice and water, tin- water serving as the outer coating, while the liquid contained in the bulb serves as the inner coating. On charging the inside by connecting it with a large Leyden jar the level is observed to fall. Quincke found that if Ai> is the change in the internal volunx the bulb, if e is the thickness of the glass and V is the potential difference, then - is proportional to — -2 nearly. .With thicknesses between O'l mm. and 0'5 mm. and with a potential difference sparking at 2 mm. between knobs '2 an. in diameter, Quincke found that the change in volume lav between 1 in 105 and 1 in 107. In general, when it rose above 1 in 10* sparking occurred and the glass was perforated. The intensity and strain are proportional to V /e so that the change in volume is proportional to the square of the electric strain in the glass. The expansion occurs equally in all directions, for .simultaneous measurements of increase in length / and of volume v * Phil. May. [5], vol. 3, j>. 30, where references to earlier work will be found. ALTERATIONS IN THE DIELECTRIC 143 of a condenser like a thermometer with a long cylindrical bulb showed that Maxwell's electric stresses do not explain the effect. The result just given is important as showing at once that no such system of stresses as that considered in the last chapter will account for the effect. That system would give extension parallel to the lines of force, and compression perpendicular to them. But it fails to account for the effect in that it would give on the whole a contraction in volume. If, as in the last chapter (Fig. 93), we take a cube of the dielectric, with its vertical edges parallel to the lines of force, it is acted on by its surroundings with the set of forces represented in the figure. The vertical tensions and one pair of side forces will give a shear with no change of volume, while the other pair of pressures will produce a decrease in unit volume (1 — 2 cm. in diameter. We may perhaps take this as a potential difference about equal to 25 E.S. units, and it gives O ^v intensity within the glass E = ^- X 1C3. For flint glass K is K ]•';- probably about 7, .so that P = — - = 106 nearly. \\ C may take 1 - 2train. Electric expansion of liquids. Quincke also experimented with liquids. A conden>er was formed in the liquid by a pair of vertical platinum plates. The containing vessel was entirely filled with the liquid and closed except that a capillary tube rose from it to show any change in ve>lume of the liquid. On charging the plates as a concfenser the level in the tube was observed to alter. The results indicated that the change in volume is probably proportional to the sejuare of the intensity. In general there xpansion, but with fatty oils, such as colza and almond oils, there is a elccre.-a.se in volume. 144 STATIC ELECTRICITY Quincke also observed a decrease in the torsipnal rigidity ot glass, mica, and india-rubber when electrically strained. Electric double refraction: The Kerr effect. Km discovered* that a liquid dielectric is in general doubly refracting when it is the seat of electric strain. It behaves as a uniaxial crystal with the optic axis along the lines of force. There is there- fore a difference in the velocity of rays which traverse the liquid perpendicular to the lines of force, according as their plane of polarisation is parallel to or perpendicular to the line- of force. The effect may be shown by the arrangement of which I-'ig. D~> is a plan. A ray of light is sent through a polariser P set so that its plain of polarisation is at 45° to the vertical. It then passes into ft trough T with glass ends containing the liquid, winch may be carbon bisulphide very pure and dry. In the trough arc t\\o bra— ex Under- 3 in. or 4 in. long, with their axes parallel and in the -mm- hoi i/ontal plane. These are connected by rods rising out of the trough to the terminals of a Wimshurst machine. After emerging from tin- trough the ray passes through the analyser A, set at 90° to the polariser, so that when the liquid is not electrically strained there is complete extinction. The ray then passes on to the eye of an observer E, or by using lenses we may project an image of the end - of the conductors in the trough on to a screen. On working the Wimshurst the field betxveen the two conductors traiiMiiit- the light to some extent, and the more the stronger the field. The vibration of the incident beam may be resolved into equal components vibrating respectively vertically and horizontally, and therefore respectively along and perpendicular to the lines of force. Ti travel at different speeds through the strained medium so that when they emerge from the electric field the phases differ, and instead of uniting to form a plane polarised beam, with pi parallel to that of the original, they unite to form an elliptically polarised beam which cannot be entirely extinguished by the analyser. By inserting a Jamin compensator C, Kcrr determined * Phil. Mag. [4], 1875, vol. 1. pp. 337 and 446 ; [5], 1879, vol. viii. pp. 85 and 229. ALTERATIONS IN THE DIELECTRIC 145 the delay in phase of one ray relative to the other and showed * that the delay is proportional to the square of the strength of field as measured by a Thomson electrometer. Kerr investigated the effect for a large number of liquids f and found that some, which he termed positive liquids, behave like quartz with its axis parallel to the lines of force, or like glass extended in a direction parallel to the lines of force. Of the liquids in which he was able to measure the effect, carbon disulphide shows it in the greatest degree. Paraffin oil, toluene, and benzene are other positive liquids. Others, which Kerr termed negative liquids, behave like Iceland spar with its axis parallel to the lines of force, or like glass compressed in the direction of the lines of force. Of these colza oil gives a strong effect ; others are olive oil and seal oil. The effect may be expressed conveniently by the number of wave-lengths difference between the two components in a length I of the dielectric when the transverse electric field has intensity E. Experiment shows that if S is the difference of path pursued in the same time, and if X is the wave-length of the light used, the number of wave lengths difference or S/\ is given by S/X = B/K- ulu-re B is a constant for a given substance and wave-length — the Kerr electric constant. The difference of phase is STT^/X, and 3 and X must both be measured in the liquid or both in air. It is easilv shown that if //0 and //, are the refractive indices for the two rays and X is the wave-length in air Mo - yu. = BX/E2. QuinekeJ was the first to obtain a value of B, working with carbon disulphide. Since then various measurements for this liquid have been made § and the value of B for sodium light appears to be very near to 3-1 x 10~7. But the v&lue changes with the wave-length, increasing as X diminishes. This was observed by Kerr, and he thought that /ULO — /mt varied inversely as the wave-length. Subsequent observers || have not verified the simplicity of the relation though confirming the general nature of Kerr's observation. Cotton and Mouton have investigated the effect of change of temperature and have found that B ; lilarkwell. /'/• \rad.ofArl -res, XLI, 1906, p. 147, McComb, /'"-. "'. Cotton and Mouton, Jmimnf *\, />/,,,.<;,<•, 5th. ser., vol. I, 1911, p. 5. I 146 STATIC ELECTRICITY sufficiently good insulator to allow a persistent field to be estab- lished in it, yet on making a spark-gap in one of the conncct- ing wires, at each spark the field lighted up, due no doubt to the oscillations producing a large oscillating field between the conductors. Schmidt* succeeded in measuring the effect in nitrobenzene, He used a compensation method in which the ray of light pas-cd through the liquid in one cell and then through another cell con- taining a liquid for which B was known, and arranged so as to have an effect of the opposite kind on the phase relation of the two components. It was adjustable so that the effect of the two cells exactly neutralised each other when the same rapidly alter- nating potential difference was applied to each. He found that the Kerr constant is sixty times as great for nitrobenzene as for carbon disulphide, and that it is about the same for the latter and for water. McComb found that the effect is large for other aromatic compounds.! Cotton and Mouton (loc. cit.) discou-ml I hat BlllobeBieM exhibits magnetic double refraction, that is that a polarised ray transmitted perpendicularly to a magnetic field established in the liquid travels at different speeds according as its plant- of polarisa- tion is parallel or perpendicular to the lines of force. They found that the effect could be expressed by a formula exactly corre- sponding to that for the Kerr electric effect, vi/. : S/\ = C/H* where H is the magnetic intensity and C is the constant of magnet ic double refraction for the substance for the wave-length u-t-d. I'm nitrobenzene they found that C/B is nearly the same for different wave-lengths. This has been confirmed for several otlu i aromatic compounds by McComb and Sk inner, t In Cotton and Mou ton's paper there is some reference to theories of the subject by Voigt, Havelock, and themselves. These, how- ever, belong to optics rather than to electricity. We may state the Kerr effect in terms of the electro- magnetic theory of light by saying that K is slightly different in t la- electric field. Since the change in velocity is given by dv _ldK v 2 K and Dr. Kerr's observations show that — is proportional to K2. 7|»- where E is the intensity, then -^ is proportional to E1, and * Ann. derPhys. [4], 7, 1902, p. 142. f A number of values will be found in Tables Annucttci Intcrrjationole* de C nstantes I. J PJiys. Rev., 29, pp. 525 and 541. ALTERATIONS IN THE DIELECTRIC 147 K is not constant, but decreases as E increases, and by an amount proportional to E2. This is on the supposition that the speed of the rav with its electric vibrations along the lines of force is alone affected. Some experiments by Kerr * appeared to establish this, but some experiments by Aeckerlein f showed that in nitrobenzene the ray with its electric vibrations along the lines of force was retarded and that with its vibrations perpendicular was advanced and about in the ratio 2:1. It is to be noted that the change in K is exceedingly minute, the change in phase being only a fraction of a period in a path of several centimetres through a field of perhaps 100,000 volts per centimetre. The Kerr effect in glass is exceed- ingly small, so small that it cannot always be ceitainly distinguished from the effect due to the elastic stresses present. * Phii. May. [5], 37, 1894, p. 380. t Phys Zeit., VII, 190fi, p. 594. CHAPTER XII PYROELECTRICITY AND PIEZOELECTRICITY Pyroelectricity — Historical notes — Analogous and antilogous pole? — Some methods of investigating pyroelectricity — Gaugain's researches — Lord Kelvin's theory of pyroelectricity — Voigt's experiment on a broken crystal — Piezoelectricity — The discovery by the brothers Curie — The piezoelectric electrometer — Voigt's theory connecting pyro- electricity and piezoelectricity — Electric deformation of crystals — Lippmann's theory — Verification by the Curies — Dielectric subjected to uniform pressure — Change of temperature of a pyroelectric crystal on changing the potential. CERTAIN crystals initially showing no electrification develop, if heated uniformly, opposite surface electrifications on opposite surfaces. If they are cooled from a neutral condition the polarity is reversed. The phenomenon is termed Pyroelectricity, and it \\.-is observed first in the eighteenth century. If these crystals, \\ ithout being heated, are subjected to pressure along the axis of electrifica- tion which was observed when the temperature was changed, then opposite electrifications develop at the ends of the axis. If they are subjected to tensions the polarity is reversed. The phenomenon is termed Piezoelectricity. It was discovered by J. and P. Curie in 1880. They found that the electrification under pressure was the same in sign as that due to cooling, while that under tension was the same in sign as that due to heating. The two phenomena are evidently related to each other. The crystals in which they are observed are those known as " hemihedral with in- clined faces." They have axes with the faces at the two ends unlike, so that they are unsymmetrical with regard to a plane at right angles to the axis through its middle point. As an example we may take tourma- line, of which Fig. 96 represents one form, the axis of electrification being as shown by the arrow. We shall describe first the phenomenon of pyroelectricity. 148 FIG. PYROELECTRICITY AND PIEZOELECTRICITY 149 Pyroelectricity : Historical notes. It was discovered about 1700 that a tourmaline crystal placed in hot ashes attracted the ashes. In 1756 J£pinus showed that the effect was electrical and that the charges were opposite in sign at the two ends of the crystal. Soon after, Canton* showed that the charges were equal as well as opposite. He connected an insulated tin cup filled with boiling water to a pith-ball electroscope. A tourmaline crystal was dropped into the water and the pith ball showed no trace of electrification, either then or during the subsequent cooling of the water. Canton found also that the development of electrification depended only on change of temperature from the neutral condition and not on the absolute temperature. For if the crystal was in a neutral condition at any temperature it showed polarity of one kind if raised above that temperature, and polarity of the opposite kind if cooled below that temperature. He broke a tourmaline prism into three pieces and found that each piece exhibited the same kind of action along the same axis as the whole crystal. He found that Brazilian topaz acted like tourmaline, and his con- temporary Watson discovered other pyroelectric gems. Haiiy about 1800 discovered that only crystals hemihedral with I inclined faces developed pyroelectricity, and that the end forming the most acute angles with the axis was positively electrified on.' cooling. Other crystals may show electrification when not heatedj uniformly, but this is due to local strains set up by the non-uniform heating and a consequent local alteration, as it were, of thq crystalline form. Analogous and antilogous poles. Riess termed the pole on which + electricity appears on heating the analogous pole, and the other the antilogous pole. A crystal may have as many as four pyroelectric axes. Quartz has three axes, bisecting the three angles of 120° between the faces of the prism. The axis of the prism is not a pyroelectric axis. Some methods of investigating pyroelectricity. The earliest method consisted in merely heating a crystal and then bringing one part of its surface near an electroscope. Hankel heated the crystal in copper filings, passed it through a spirit flame by which it was entirely reduced to the neutral condition, and then allowed it to cool, when its electrification could be tested by an electroscope. Kundt t heated the crystal in an air bath and after neutralisation allowed it to cool. He then blew on to it, through muslin, a powder consisting of a mixture of red lead and sulphur. The red lead was positively electrified by the friction with the muslin and the sulphur was negatively electrified. The positive region of the crystal attracted the sulphur and the negative the red lead, and the two regions became differently coloured. If a tourmaline prism '^Priestley on Electricity, 4th. ed. p. 298. t Wied. Ann., 20 (1883), p. 692. 150 STATIC ELECTRICITY is immersed in liquid air and is then lifted out and dusted with red lead and sulphur before it has time to condense ice on its surface, it shows the colours excellently. Gaugain' s researches. Gaugain* employed an air bath in which a tourmaline crystal was suspended. One end of the crystal was earthed and the other end was connected to a self-discharging gold- leaf electroscope, and the total charge developed was measured by the number of discharges of the gold leaf. He found that 1. The charge is proportional to the change of temperature from neutrality. 2. With different tourmalines of the same kind the charge is independent of the length and is proportional to the cross- section. Rlecke's researches. Rieckef made important contribu- tions to our knowledge of the subject. He investigated the loss of charge through leakage and showed that it was very much diminished if the crystal was suspended in a dry vacuum. He showed that if $ is the change of temperature from neutrality, the charge is not exactly proportional to $, but may be represented more nearly by E = «S + b&. He worked in a manner similar to that of Gaugain, and graduating the electrometer by known charges, he was able to determine the charges developed in electrostatic units. Thus for a certain tourmaline which gave the largest effect he found that 180 E.S. units per square centimetre cross-section were developed per 100° C. rise of temperature. Lord Kelvin's theory of pyroelectricity. To Lord Kelvin J we owe the theory of the subject which is generally accepted. Let us take the simple case of a crystal like tourmaline with one pyroelectric axis. We suppose that each molecule of the crystal is an electric doublet with lines of force or strain passing between the pair and into and out of the ends. The molecules set in the crystal with their electric axis parallel to a certain direction, and there will therefore be a definite amount of strain passing from each positive element of a doublet to the negative element of the doublet next to it. The crystal as a whole will be the seat of electric strain along the direction of the axis. The constitution corresponds to that of a saturated permanent magnet on the molecular theory of magnetism. The electric strain will manifest itself almost entirely at the ends of the crystal if the doublets are near enough together, and the crystal will produce a field outside equal to that which would be produced by charges respectively * Ann. de Chim. [3], 57, p. 5 (1859). f Wied. Ann., 28, p. 43 (1886) ; 31, p. 889 (1887) ; id. p. 902 ; 40, p. 264 (1890) ; id. p. 305. J Nicol's Cyclopaedia, 2nd ed. (I860), or Math, and Phys. Papers, vol. i. p. 315. The fullest account is in the Baltimore Lectures, p. 559. PYROELECTRICITY AND PIEZOELECTRICITY 151 positive and negative at the two ends, and giving out the same number of tubes of strain as those actually issuing. But through the positive and negative ions which always exist to some small extent in the air, through conducting films on the crystal and sometimes by conduction in the body of the crystal itself, charges will in time gather on the surfaces at the ends of the axis, negative on the end from which tubes of strain issue, and positive on the other end into which they enter, entirely masking the existence of the strain so that there is no external field. This is the ordinary neutral condition of a pyroelectric crystal. We have now to suppose that the electric strain passing forward from molecule to molecule depends on the temperature and that it. alters nearly in proportion to the change of temperature. Then when the temperature changes there will be an unbalanced strain proportional to 'the change, producing the same effect as a positive charge at one end and a negative charge at the other until con- duction again does its work in bringing up masking charges. The apparent charges are proportional evidently to the cross-section and independent of the length. We may picture a molecular model which might give the effect, and though it suffers from the common defect of such models in being statical, whereas in reality everything must be in motion, it will at least serve as a working hypothesis. We suppose the molecules to be doublets set with their axes all parallel to a given direction and with the positive part of each to the right, as in Fig. 97, and we suppose that there is some connection between the positive of each molecule and the negative in the next. For instance, in Fig. 97 there are five internal lines of force in each FIG. 97. molecule, and three external lines connecting it to the next on each side, and these lines run in opposite directions from the same charge. The external lines to the extreme molecule on the left begin on the positive masking charge, and those from the extreme molecule on the right end on the negative masking charge. Let us further suppose that through a rise in temperature the molecules are more separated. Then the external lines of force will bulge out, as it were, and this bulging out may take place to such an extent that one between each pair of molecules may touch one between each of the next pairs on either side, as represented in Fig. 98, where only the external lines concerned are drawn. At the points of contact P and Q the strains are in opposite directions in the two coalescing lines and neutralise each other. 152 STATIC ELECTRICITY Then more or less straightening occurs and we get something of the arrangement represented in Fig. 99, which shows that one of the external connecting lines disappears, there is one more internal connecting line, and there is one extruded line passing from end to end of the crystal, not through the chains of molecules, and not FIG. even necessarily through the crystal. At each end of this line there will be an element of what was previously the masking charge, but is now what we may term free charge, and the con- dition is represented in Fig. 100. This model would explain pyro- electricity as due merely to molecular separations, and we ought FIG. 99. IS produced K's found. to get a similar effect if the molecular separation by tension, and, as we shall see, this is just what the Curie: But we may imagine another molecular model in which pyro- electric effects should occur if rise of temperature produces separation of the constituent charges in each molecule without extension as a whole. It will easily be seen that then the internal lines of force bulge out and finally extrude a line of force running in the opposite -t- FIG. 100. direction to that in Fig. 100, and finally we have one more external line and one less internal line, as represented in Fig. 101. In this case the charges developed at the ends will be opposite to those developed in the last case. It is possible that both effects occur. We may describe the electrical condition within the crystal on this molecular chain theory by the number of tubes of strain issuing per square centimetre from the end of an axis. We may term this the " Intensity of Electrification " along that axis, and it PYROELECTRICITY AND PIEZOELECTRICITY 153 corresponds to the intensity of magnetisation of a magnet. We should expect, on the theory, that if a pyroelectric crystal is broken across an electric axis, then the two broken axes would show apparent electrifications equal and opposite. Canton is sometimes supposed to have shown this,* but he does not appear to have gone further than a demonstration that each fragment •4- FIG. 101. behaved with change of temperature like the whole crystal. The experiment was first successfully made by Voigt.f Voigt's experiment on a broken crystal. He fastened the two ends of a small tourmaline rod to two brass rods which were connected electrically. He then broke the tourmaline in the middle and dipped the two broken surfaces into mercury cups connected to a Nernst-Dolaxalek electrometer which showed charges on the ends. He found as a mean result that there was an intensity of electrification at 24° C. of —33*4 E.S. units, or the electrical moment of 1 c.c. at 24° C. was — 33'4, the sign being negative because the electrification was opposite to that which would have appeared on the same surface with rise of temperature. Now for this kind of crystal the electrification gathering per 1 °C. rise of temperature was found to be about -f 1'2 E.S. units per 33*4° square centimetre. Hence at 24° H — =-^- = 5%° C., the electrical moment of the crystal should be zero. Or, if we return to the molecular models, it may be that both effects, dilatation and inter- molecular separation, exist, and that at 52° C. they balance each other. Piezoelectricity: The discovery by the brothers Curie. The discovery of piezoelectricity was made by the brothers Curie by an experiment represented diagrammatically in Fig. 102. C is a crystal hemihedral with inclined faces cut into the form of a rectangular block. It was placed between the jaws J J J J of a vice. Between the jaws and the crystal were ebonite plates ee and tinfoil 1 1, the tinfoil being against the ends of the crystal. The tinfoil was connected to the quadrants of an electro- meter, and one tinfoil and the connected quadrants could be earthed. * Kelvin's Hultimore Lectures, p. 560. Cf. Priestley on Electricity, 4th ed. p. 300. t Wied. Ann., 60, p. 368 (1896). 154 STATIC ELECTRICITY When the vice was screwed up, a charge was shown in the electrometer. On discharging this and then releasing the crystal from pressure the polarity was reversed. A number of crystals were thus tested, and it was found that always compression gave the same sign of charge at one end as cooling, and extension the same sign as heating.* Working with tourmaline and applying pressure by a loaded lever by which the force could be measured, it was found that 1. The positive and negative charges were equal. 2. The change in charge de developed was proportional to the change in the total force dF and changed sign with dF. 3. The charge was independent of the length. 4. The charge was independent of the cro*s-section for a given total force. Hence de -LJU 2? FIG. 103. gave \>tem to be charged. The charge is proportional to the total load and if the load be increased the additional charge is proportional to the additional load. The instrument may be graduated by the use of an electrometer and condenser in the manner already explained. Voigt's theory connecting pyroelectricity and piezoelectricity.* The Curies found that pressure produces electrification at the ends of an axis the same in sign as that produced by cooling, and tension the same in sign as heating. This suggests that the change in dimensions of the crystal may alone be concerned in the two classes of phenomena, and Voigt has investigated the question. The following is a mere indication of the method. * Abhand. Gesell. d. Wins. Gottingen, 36 (1890). PYROELECTRICITY AND PIEZOELECTRICITY 157 The deformation of an element of a body is completely expressed in terms of six quantities * which we may denote by ABC abc. If the intensities of electrification due to the strains are El E2 E3 in three directions at right angles, we may suppose that each term in the strain contributes to the electrifica- tion, and so we get A2B A3C with similar values for E2 and E3, and for the most general case with different values for the coefficients A. Thus for this most general case — a hemihedral crystal with inclined faces in the triclinic system — we shall have eighteen independent coefficients of the type A. In all but this general case there are relations holding between the coefficients owing to the symmetry of the crystal about certain planes or directions, and their number may be greatly reduced. Thus we get definite values of the intensities of electrification in terms of the change of shape whether it is produced by heat or by stress. Riecke and Voigt f have tested the theory for tourmaline and quartz. Knowing the piezoelectricity produced by given deformations, they were able to determine the coefficients. Then, measuring the dilatations in different directions with rise of temperature, they were able to determine the charges which should be produced merely by the change of shape on heating. Voigt } found that in a tourmaline crystal about four-fifths of the pyroelectric charge could be accounted for by dilatation. Pockels § found a fairly close agreement with Voigt's theory in the case of sodium chlorate. Electric deformation of crystals : Lippmann's theory. Soon after the discovery of piezoelectricity Lippmann \\ pointed out that there should be a converse effect, viz. a change in the dimensions of a piezoelectric crystal when it was subjected to electric strain. We may present Lippmann's theory as follows : * Thomson and Tait, Natural Philosophy, vol. ii. p. 461. t Wied. Ann. 45, p. 523 (1892). £ Wied. Ann. 66, p. 1030 (189s). § Winkelmann, Ifandbuch der Physik, 2nd. ed. iv. p. 783. Ann. >i<. ''/,/,„ ,/ pert Integrating from x = 0 to # = #, the rise of temperature is where E is the electric intensity. In practice this is Mn.-ill. If, for instance, E is 100, 0 is 300, and pa- is of the order 107, S> of the order 0-005° C. Let us now apply the formulae to find the temperature change on charging an ordinary condenser. We are to find -*- when is constant. d0 h We have where h = and b = Now rp, neglecting the change in t PVROELECTRICITY AND PIEZOELECTRICITY 163 and 0\dOJx 0- Then fd0\ x dK _#_ dO po-f If we integrate from x = 0 to x = x the total change in 6 is small, so that we may put 0 constant on the right hand and the change is *» . dK . 0 ~ dO o- = _ _ HTT K dO pa- where E is the intensity. This is much smaller than the value just found for a pyro- electric crystal, unless E is very much greater than 100. It is easily found that the isentropic capacity dm\ A.^ = cl + and ,- is found to be be which 'is quite negligibly small.* * Pockels has generalised the theory of Lippraann. The reader is referred to Winkelmann's Ifandbuch, vol. iv. , where a full account of pyro- and piezoelectricity will be found, with references to the original memoirs. KE2,M dK\*_0_ 47r \ K dO/ pa- PART II : MAGNETISM CHAPTER XIII GENERAL ACCOUNT OF MAGNETIC ACTIONS Natural and artificial magnets — Poles—Mechanical action of poles on each other — The two poles have opposite mechanical actions — In a well- magnetised bar the poles are near the ends — The two poles always ac- company each other in a magnet — The two poles are equal in strength — The magnetic action takes place equally well through most media — In- duced magnetisation — Magnetisation can take place without contact — Retentivity and permanent magnetism — The earth is a great magnet — Magnetisation induced by the earth — Methods of magnetisation — Single touch — Divided touch — Double touch — The electro-magnetic method — Electro-magnets — Ball-ended magnets — Horseshoe magnets — Compound magnets — Distribution of magnetisation — Consequent poles — Magnetisa- tion chiefly near the surface — Saturation and supersaturation — Preserva- tion of magnetisation — Armatures — Portative force — Nickel and cobalt magnets — The magnetic field — Lines of force — Mapped by iron filings. Natural and Artificial magnets. Magnetite, a mineral found in various parts of the world, having the composition Fe3O4, has been known for ages to possess the property of attracting small pieces of iron. It was found by the ancients near Magnesia, in Lydia, and from the locality, probably, the Greeks gave it the name yuayyr??, whence we derive the name magnet. A piece of the mineral exhibiting the attractive property is now called a " natural magnet,11 to distinguish it from the steel magnets which are made by processes described hereafter. If a natural magnet is dipped in iron filings it is usually found that the filings cling in tufts, chiefly about two points at opposite ends of the specimen, and if the magnet is suspended so that the line joining these two points is horizontal, this line will in this part of the world set nearly North and South, and from this directive property the mineral acquired the name "lode-stone," that is, way-stone. If a bar of steel is stroked in one direction with one end of a natural magnet, the bar itself becomes a magnet, acquiring the property of attracting iron filings, and the tendency to set North 165 166 MAGNETISM and South when freely suspended horizontally. Such a bar was formerly termed an artificial magnet. In describing the fundamental phenomena we shall suppose that we are dealing with steel magnets. Natural magnets are now mere curiosities and have not baen usefully employed since the eighteenth century. The prefix "artificial" has long ceased to be applied to steel magnets. Poles. The two ends of a magnetised bar round which the attractive properties are exhibited are termed its poles. The end setting towards the North is termed the North -seek ing pole, the other the South-seeking pole, or more shortly the North and the South poles respectively. They may be designated by NSP and SSP, or more shortly by N and S. The NSP is frequently marked on the magnet either by a cut across it, or by N cut in the metal. Mechanical actions of poles on each other. li ire suspend one magnet so that it is free to move round on a pivot, forming in fact a compass needle, and then pre>ent to its poles in succession each of the poles of a second magnet, \\e find that two unlike poles attract each other, while two like poles repel each other, or : NSP attracts SSP and repels NSP and SSP „ NSP „ „ SSP. The two poles have opposite mechanical action f. \Yhere one at t iaeU the other repels, and the actions of the two poles lend to neutralise each other. We may therefore call them respectively positive and negative, and if we consider a NSP to be positive, \\e musf consider a SSP to be negative. Position of poles. In a well-magnetised bar the poles are near the ends. For, on dipping the bar into iron filings, we find S N =* 1= FIG. 108. that the filings cluster chiefly about the ends, and hardly adhere at all near the middle. But though we speak of the poles of a magnet we must remember that they are not definite points, but rather regions, in general near the ends of the magnet. The two opposite poles always accompany each other on a magnet. If we break a magnetised knitting needle, for instance, we do not isolate the two poles, for two new poles are at once developed at the broken ends, as in Fig. 108, so that each fragment becomes a complete magnet with its opposite poles. All experi- GENERAL ACCOUNT OF MAGNETIC ACTIONS 167 ments go to show that it is impossible to obtain one kind of pole alone on a magnet. The two poles are equal in strength — that is, they exert at equal distances equal forces respectively of attraction and repulsion. This may be proved by bringing up a magnet to a compass needle, the magnet being held vertically with its centre on a level with the needle, Fig. 109, and its poles N and S equidistant from the pole n. The two forces then have a vertical resultant, as will be seen from Fig. 110, and the compass needle is not deflected in the horizontal plane, but only tends to dip. FIG. 109. FIG. 110. Another proof of this equality is given by the fact that a bar of steel weighs the same before and after magnetisation. In the latter case it is acted on by the earth's magnetisation, but with forces at the two ends equal and opposite, and therefore at most only forming a directive couple. The forces being equal, we conclude that the poles are equal. The magnetic action takes place equally well through most media within the limits of our powers of observation. If, for instance, a magnetised needle is suspended so as to set North and South and a magnet is brought near it as in Fig. Ill, the needle is LJ FIG. 111. deflected with the North-seeking pole towards the East. The deflec- tion is still the same if a screen of wood or cardboard or brass be 168 MAGNETISM interposed. There is, however, one striking exception. If the screen is iron and is of sufficient extent the deflection diminishes, the action of the magnet being screened off by the iron from the space beyond. Induced Magnetisation. A piece of iron brought into contact with a magnet becomes itself a magnet by induction, the iron being magnetised in such a manner that the end of it in contact with the magnet is a pole of the kind opposite to that with which it is in contact, while the other end of the iron is a pole of the same kind. The adherence of the iron to the magnet is explained by the formation of the unlike pole, while the existence of the like pole at the further end may easily be tested by means of a suspended compass needle. Several pieces of iron may thus be hung one below the other from a magnet pole, each mag- netising the next below it, unlike poles being together and the lasi pole being like the inducing pole of the magnet, as in I-'ig. 112. The NSP of the last piece of iron ivpeK the NSP of a needle. 77//.V magnetisation ant take />/t the earth, and therefore they will he acted on by equal and opposite forces by that pole. The resultant action of the two poles of tin- earth thus being a couple is diivctive and does not tend to move the centre of gravity of a magnet. Magnetisation induced by the earth. Not only is the magnetisation of the earth sufficient to direct magnetised but it produces quite considerable induced magnetisation. If in this part of the world a poker be held upright and struck sharply the lower end becomes a NSP and the upper a SSP, the polarity acting strongly on a compass needle. If the poker be re\vi M d and again struck, the magnetisation is quickly reversed, and what is now the lower end becomes a NSP. Any iron lying more or in the general direction of the dipping needle is thus magnetised by induction, as, for instance, the front plates of iron fire-grates or iron gas-pendants. Iron ships are magnetised by the earth and their compasses are affected seriously by this magnetisation. The deviation of the compass thus produced must be allowed for. Methods of magnetisation. Various methods have been devised to magnetise bars of steel by stroking them with magnets. GENERAL ACCOUNT OF MAGNETIC ACTIONS 171 They are all modifications of three, called respectively the methods of single, double, and divided touch. In our description we shall assume that the object is to make flat bar magnets, and it appears that good dimensions for such magnets are on the scale — length 10 to 15, breadth 1, depth \. Single touch. The bar to be magnetised is rubbed several times in the same direction by one pole of the magnetising magnet, which is moved parallel to itself either at right angles to the bar or inclined as shown in Fig. 114, the course of the magnet being \ FIG. 11 1. indicated by the solid line, the dotted line indicating the return journey at such a distance from the bar that the magnet has no appreciable influence on it. The probable action of the process is that when the N pole of the magnetising magnet is put on the end of the bar to be mag- netised it induces a SSP at a immediately under it and North- seeking polarity chiefly on the other side, b, but also at the more distant parts of the bar and at the other end, c. But as N moves along the bar it carries along with it the SSP originally at a, while the NSPati moves round to occupy the whole of the end N', and the SSP carried to the other end neutralises the weak North-seeking polarity already there. Repetition with the same magnet somewhat in- creases the effect, but if a magnet weaker than that employed at first be substituted in the repetition it partially demagnetises the bar. The process is unsatisfactory, inasmuch as it does not lead to strong magnetisation or to symmetrical distribution. It may be somewhat improved by beginning with, say, the N magnetising pole in the middle of the bar and drawing it along to the end. The magnetising magnet is then turned over and the S magnetising pole is drawn from the middle to the other end. It is best in this last process to slide the one over the other as indicated in Fig. 115. But practically the same process may be carried out much more effectually by the method of Divided touch. This method was devised by Dr. Knight about the middle of the eighteenth century, and he made by it 172 MAGNETISM small magnets of very great power. One, for example, weighing 1 oz. troy, when armed with iron at the two ends so that the weight was still under 2 oz., lifted forty-four times its own weight. The method is especially applicahle to compass needles or short hars. Two magnets are used and are placed with their unlike poles initially as close together as possible, and the needle or bar is laid IN. N' FIG. 115. under them as in Fig. 116, and fixed by a button. The two ma-ix -Is are then gradually drawn apart till their ends slide off the bar, leaving the bar magnetised as indicated in the figure. The t\\o magnetising poles are again brought together or mar each other, and then brought down on the middle of the in ague I. and the process is repeated three or four times. Tin- bar is then turned over and the same number of operations rep-ated on the other face. The explanation of the magnetisation is obvious. The N / S FIG. 116. me ...cthod of divided touch requires in general moiv poueri'ul in than the following method, known as that of Double touch. In this method the magnet i.sing s\.|( ,,, consists of two unlike poles. NS, \\hich are kept a constant distance apart. A horseshoe magnet may he u-cd.or its equivalent, t\\o bar magnets with a piece of wood between their lower poles, their upper poles bfin» united bv M>ft iron, Fig. H7. If one bar is' to be magnetised, the magnetising poles are brought down on to the middle of it and moved to and fro several times, each half being traversed the same number of times. They are then withdrawn from the centre. If several bars are to be magnetised they may be laid end to end and treated as one bar. After several journeys of the magnetising poles the end bars are brought into the middle and the process is continued until all the bars have thus been in the middle. w S S N Y//J.,.;:A 1 1 FIG. 117. GENERAL ACCOUNT OF MAGNETIC ACTIONS 173 These methods of making magnets give interesting illustrations of magnetisation by induction, bub they are now little used except sometimes for compass needles and for the needles used in the determination of the dip of the earth's magnetic force, in which case the divided touch is employed.* They are for general purposes entirely superseded by The electro-magnetic method. In this method a coil of wire is wrapped round a hollow cylinder, forming what is termed a solenoid, the bar to be magnetised is inserted in the cylinder, and a strong electric current is passed through the coil. When the current is broken and the bar is withdrawn it is found to be magnetised. Electro -magnets. The electro-magnetic method is not only useful for thus making permanent magnets by the inser- tion of bars and their withdrawal when magnetised. If a coil of wire carrying a current surrounds an iron core that core is a powerful magnet while the current is running, though on the stoppage of the current the magnetisation in general falls off very greatly. By far the strongest magnets are made by some such arrangement as that represented in Fig. 118, where the two limbs U^k rm rm A^J FIG. 118. consist of soft iron surrounded by coils through which a current may be passed, the limbs being joined by a soft iron cross-piece as the base. With a strong current passing round the two limbs from inside to outside in both limbs as seen from the front, or in the opposite direction in both, the ends of the soft iron become exceedingly strong poles, far stronger than can be obtained in permanent magnets of steel. It is to be observed, however, that the magnetism only maintains its great strength while the current is passing. * A full account of the methods used before the introduction of the electro- magnetic method will be found in the Library of Useful Knoulcdgt, arUc.te Magnetism, chap. v. (1832). 174 MAGNETISM Horseshoe magnets. For purposes in which the two poles are required close together, steel magnets of the horseshoe form are often convenient. Bars of this shape may have their length greater in proportion to breadth and thickness than straight bars, and they may be conveniently magnetised either by the electro- magnetic method, a coil being placed on each limb, or by the method of double touch after connecting the two ends by a soft iron cross-piece. Compound magnets. Very powerful " compound " magnets are often formed by building together a number of single bars or horseshoes magnetised separately. This method is not so much used now as formerly, when it was difficult to get thick bars of homogeneous steel. Distribution of magnetisation. If a straight bar magnet is very carefully magnetised the magnetic action appears to radiate from very near the ends, and at some distance from the bar the resultant action is very nearly the same as if the poles were actual points within the bar and near the ends. The thinner the bar in comparison to its length, the nearer are the poles to the ends. If the bar has dimensions, say, 10 x 1 X J, then the poles may perhaps be considered as an eighth or a tenth of the length from the ends. This is, however, only an approximation. If we imagine an ideal magnet as consisting of a very fine wire or fibre, with its poles actually at the two ends, then any real n induct may be regarded as a bundle of such fibres of unequal lengths. If the magnet is well made the fibres will have their middle points at the middle of the bar so that the magnetisation is symmetrically distributed, and the greater number will be of nearly the same length as the bar. Ball -ended magnets. Balls of steel or soft iron are bored and are put one on each end of a steel rod which is magnetised. The force is directed very much more nearly to a definite pole at the centre of the ball than in other magnets.* Consequent poles. If there is any irregularity in the stroking of a bar in the process of magnetisation — as, for example, magnetising it first in one direction and then in the other — consequent poles may be developed, i.e. poles of the same kind may be formed together within the bar. We may easily obtain a knitting needle magnetised as in Fig. 119, each end being North- s s FIG. 119. seeking, two consequent South-seeking poles occurring somewhere between them. Magnetisation chiefly near the surface. The per- * Searle, Camb. Phil. Soc. Proc., 12, p. 27, 1903. Ball-ended magnets were devised by Kobison in the eighteenth century, but had been entirely forgotten till they were re-invented by Searle. GENERAL ACCOUNT OF MAGNETIC ACTIONS 175 manent magnetisation is usually near the surface of a bar, for if the surface layers be dissolved away by nitric acid the inner core will be found to be nearly unmagnetised. Or if a steel tube with a steel core be magnetised the core on removal is only slightly magnetised. Saturation and supersaturation. It is found that there is a limit to the magnetisation of a given bar, depending on its composition, temper, and shape. When the limit is reached the bar is said to be saturated. Just after being magnetised the bar is frequently found to have attained a degree of magnetisation greater than it can permanently retain, and it is then said to be supersaturated. The supersaturation disappears in time if the bar is left to itself, and rapidly if it is shaken. Preservation of magnetisation. Armatures. A single magnet falls below even the normal saturation in course of time, especially if subjected to rough usage. To preserve the magnetisa- tion it is usual to keep bar magnets in pairs with their like poles turned in opposite ways, cross-pieces of soft iron termed armatures connecting the unlike poles. Horseshoe magnets are provided with a single armature connecting the poles. The armatures have poles developed by induction as marked in Fig. 120, and each pole FIG. 120. Pole oF soFt pieces iron. being faced by an unlike pole, they are supposed to tend to strengthen rather than to weaken each other. Mr. Hookham * has found that if a bundle of bar magnets has large soft iron " pole pieces," Fig. 121, fixed on the ends, the surfaces of the pole pieces being separated only by a very small air space, the horseshoe magnet which is thus virtually formed retains its magnetisation indefinitely without an armature. Long thin bars retain their mag- netisation much better than short thick ones. Portative force. Experiments have been made to find a general formula for the weight which a horseshoe magnet can sustain, or rather the force which is required to pull away its armature. Hacker gives * I'hil. M«:/.,[5], vol. xxvii, p. 186. > a r ri a q n e c s FIG. 121. 176 MAGNETISM W = awi', where m is the weight of the magnet W, the weight just sufficient to detach the armature, and a a constant for a given quality of steel, the magnet in each case being in the state of normal saturation. Inasmuch as a differs very widely with the quality of steel employed, the formula has little general value, but it may be pointed out that for magnets of similar form and of the same quality it is equivalent to W oc Z3X* oc /2, where / is a linear dimension, say the breadth ; in other words. \V is proportional to the cross-section. Nickel and cobalt magnets. While iron is pre-eminent in its magnetic properties, all bodies, as we .shall sec in Chapter 1'i, give signs of polarity under magnetic induction, though in general so slight that the polarity can only be observed easily in exceedingly strong fields such as are produced by powerful electro-magnets. There are, however, two other metals, nickel and cobalt, which, though far less magnetic than iron, still show very appreciable polarity under induction. Pieces of tln^e metals will fly to, and be held on by, a strong pole. They may be very appreciably magnetised by insertion within a magnetising coil and they exhibit some retentivity, nickel, especially when hardened, more than cobalt. It has been found possible to make a compass needle of hardened nickel. The magnetic field. When we consider the space round a magnet in connection with the magnetic actions which may be manifested in it, we term the space the " field **" of the magnet. If we place a magnet on a table, a small compass needle also placed on the table tends to set in a definite direction in the field. Its NSP is acted on by a force which we may regard as the resultant of the repulsion of the NSP and the attraction of the SSP of the magnet. We shall suppose that the needle is very small, M small that its poles may be taken as equidistant from the magnet poles ; then its SSP is acted on by the resultant of the attraction of the NSP and the repulsion of the SSP of the magnet, and these two forces are respectively equal and opposite to those on the NSP. Then the two resultants are equal and opposite, and a couple acts on the needle. It therefore sets with its axis in the direction of the resultant forces on the poles. If it is disturbed from this equilibrium position the couple tends to restore it, and if left to itself it will vibrate more or less rapidly according to the magnitude of the force on either pole. In fact, the needle is a kind of double magnetic pendulum with opposite pulls on the two ends, and just as the time of vibration of a given gravity pendulum is inversely as the square root of the force of GENERAL ACCOUNT OF MAGNETIC ACTIONS 177 gravity, so the time of vibration of the compass needle is inversely proportional to the square root of the force due to the magnet acting on either pole. Hence by moving the needle about we may investigate the direction of the force on a pole at any point by the direction of the axis, and we may compare the value of the force at different points by observing the time of vibration. Lines of force. If a curve is drawn in the magnetic field so that at every point of the curve its direction coincides with that of the magnetic force on a North-seeking pole placed at the point, the curve is called a " line of force." It is to be regarded as having for its positive direction that in which a North-seeking pole would tend to move, and the opposite direction is negative. If, for instance, the line of force is in a horizontal plane and a small compass needle is laid at a point in it, the needle will be a tangent to the curve, and we must think of the line of force as coming in to the SSP of the needle and going out of the NSP. We may trace a line of force in a horizontal plane by laying a magnet on a table and then moving a small compass needle in its field so that its centre is always travelling in the direction of its length. What the compass needle shows in the horizontal plane would be shown all round the magnet in other planes if we could suspend a needle so that it was free to move in any direction round its centre of gravity. Lines of force mapped by iron filings. There is a very simple and beautiful method of exhibiting these lines of force in the horizontal plane through the magnet by means of iron filings. The magnet is laid on or under a sheet of smooth cardboard, and the card is sprinkled with iron filings. The filings are elongated fragments of iron, and owing to the inductive action of the magnet each filing becomes a little magnet and tends to set with its longer axis in a line of force. The friction, however, prevents immediate setting. If the card is gently tapped the filings are thrown up a little from the card, are momentarily relieved from friction, and so are pulled into the lines of force. They thus map out the direction of the lines. Fig. 122 shows the general course of the lines of force round a single bar magnet. It is important to notice how the lines of force starting from the NSP curve round and enter the SSP. They start mostly from near the end, and where they are most crowded the frequency of vibration of a compass needle shows that the force there is strongest. Near the middle of the magnet few lines issue or enter, and there the force is less. As the lines issuing from the pole pass outwards they diverge, and a needle shows by its diminishing frequency that the force diminishes as the lines diverge. Fig. 123 shows the field due to two magnets arranged parallel, with their unlike poles near each other. It should be noticed in 178 MAGNETISM this case how many of the lines issuing from the NSP of one magnet enter the SSP of the other. Fig. 124 shows the field due to two bar magnets arranged in one FIG. 122. line with unlike poles towards each other, and it may be noted that the mutual induction pulls the two neighbouring poles towards each other. Fig. 125 is the field due to two bar magnets arranged in one line with like poles towards each other, and here we notice that FIG. 123. the lines of force from one NSP do not enter another NSP, but as it were the two sets repel each other and travel out sideways, each set bending round and entering the other end of its own magnet. We notice, too, that there is a point midway between the adjacent 3- FIG. L i i GENERAL ACCOUNT OF MAGNETIC ACTIONS 179 poles where there are no lines, and a compass needle at that point shows no directive force. If two bar magnets are placed in line with unlike poles in con- tact, then they act very nearly as one magnet with poles near the two distant ends, and very little polarity is shown at the junction. But if they are separated by a small gap a strong field is exhibited in the gap. When the course of the lines of force in a field has been traced, as in any of the above figures, a compass needle shows us by its \ "K ///// '"• / .\^i///; FIG. 125. frequency of vibration that, where the lines crowd together, there the action is stronger ; where the lines open out from each other, there the action is weaker. We shall see later that the lines may be drawn according to a plan such that the number passing through a unit area, held so that they pass through it perpen- dicularly, serves as a measure of the action. We regard the lines of force as symbolising some change which has taken place in the field, some action which is going on there, an action which is indicated by the setting of a needle and by the magnitude of the pulls on its poles. Now we have seen that we always have the two poles on the same bar and that the lines of force leaving one end of the bar in general bend round and enter the other, and above all we have seen that if we break a magnet in two and slightly separate the broken ends, or if, as equivalent to this, we slightly separate two magnets as in Fig. 124, then there is a strong field between the two ends. These facts lead to the irresistible conclusion that the lines of force pass through the magnet, that the inside of the magnet is as it were the origin of the action, and that the polarity over its ends is merely the manifestation at the surface of what is going on within ; or, putting the same idea in another form, the polarity is due to the passage of the lines offeree from the steel into the air. The lines of force * are to be regarded as continuous closed curves issuing from * In a later chapter we shall see that it is convenient to introduce a term to denote the alteration due to magnetic action taking place in the field, and the term used is "induction." The lines which are continuous, and which form closed curves, are really lines of induction. But in attempting to get a general idea of the subject it is sufficient to use the idea of the lines of force which are exhibited so directly by experiment. 180 MAGNETISM the NSP, bending round through the air, entering the SSP, travel- ling through the steel body of the magnet, and joining on at the point of issue. The condition symbolised by the lines of force is somehow maintained by what goes on in the body of the magnet. We may roughly illustrate the idea of continuity by imagining a number of steel flexible rings of different sizes a, &, c, d, e, f, bound together along NS as in Fig. 126. The part where they are bound together represents the magnet, the region where they bend round repre- sents the field in air. Magnetisation by induc- tion. Permeability. We may use the filing method to show some- thing of what is going on in magnetisation by induction. If a pure of soft iron is placed in a field — for instance, in the field between two unlike poles as in Fig. 127 (a) — and the field thus produced is com- pared with that when the soft iron is removed, Fig. 127 (/;), it is seen that the lines from N bend round and enter the iron a^ if tU-y FIG. 126. FIG. 127. found it easier to go through the iron than through the air. Where they enter the soft iron the passage from air to iron is manifested by a SSP ; where they leave the iron at the other end the passage into air is manifested by a NSP. But the idea to be dwelt upon is that the iron forms an easier passage for the magnetic action — is, as Faraday put it, a better conductor — than the air, or, as we now say, using terms introduced by Lord Kelvin, is more " permeable," or has greater " permeability." CHAPTER XIV GENERAL, ACCOUNT OF MAGNETIC ACTIONS— continued Relation between magnetisation and the magnetising force producing it — The hysteresis loop — Susceptibility and permeability — Magnetisa- tion and temperature — Permeability and temperature — Change of length on magnetisation — Magnetisation and strain. IN this chapter we shall give some account of certain less obvious phenomena of magnetism, which for their complete discussion require a knowledge of the methods of magnetic measurement. But without quantitative details a general idea of the phenomena may be obtained, and this general idea will be valuable when we come to the more exact investigations of magnetic quantities in later chapters. Relation between magnetisation and the mag- netising force producing it. Without entering into detail we shall now give a short account of the relation between the ooooooooooooooo ^ /Dpooooooooooooooooas.o< FIG. 128. magnetisation in iron and the magnetic force by which the magnetisation is produced, and we shall suppose that we use the electro-magnetic coil method of producing it. We have already seen that a coil carrying a current is itself a magnet. If Fig. 128 represents the section of such a coil by a plane through its axis, the course of the lines of force may be investigated by inserting a card with iron filings on it, and they are somewhat as represented in the figure. The lines of force are practically parallel within the coil except near the ends. They diverge as they pass out from one end and converge as they come in at the other. If we turn a screw round so as to travel along the lines of force inside the coil 181 182 MAGNETISM in their positive direction, the direction of flow of the current in the coil must be that in which the head is turned round. If the current be reversed, the magnetisation is reversed. The force on a pole inside the coil is proportional to the current. Now let us imagine that two equal coils, or solenoids, A and B, are arranged as in Fig. 129 so that the same current flows in opposite directions round them. Let a small needle be placed mid- way between the near ends. It will be at a neutral point — that is, a point where the two coil magnets neutralise each other — and if B !••]<;. Hi1.'. the coils are placed East and West the needle will set N and S under the earth's force which is perpendicular to the common a\i-. Now let a bar of iron be placed within A. When the current is put on, the bar at once becomes a magnet and there is nothing on the other side, B, to counteract its magnetic action on the needle, which will therefore be deflected and by an amount which will indicate the magnetisation of the bar. We shall here assume that the magnetisation is proportional to the tangent of the angle of deflection, and that the magnetising force producing it is due to the current in A and is proportional to that current, B being so far away as to have no appreciable action. Imagine that we make a series of measurements of magnetising force as given by the current and of the accompanying magnetisa- tion as given by the deflection of the needle, gradually increasing the current from zero to a large value, and let the magnetisation be plotted against magnetising force on a diagram. The general nature of the result is indicated in Fig. 130. The curve of magnetisation, though varying greatly with different kinds of iron, shows in general three characteristic parts : a part O A for very small magnetising forces, at a very small slope, a part A B sloping very steeply, and a part B C tending to become a nearly horizontal line for very great forces. These three parts change from one to another more or less gradually. If, instead of merely increasing the magnetising force indefinitely, we take it up to a certain point and then gradually decrease and GENERAL ACCOUNT OF MAGNETIC ACTIONS 183 ultimately reverse it, we find that the magnetisation will in general decrease in different ways according to the position of the point P, FIG. 130. Fig. 131, on the curve representing the condition arrived at before decrease. If P is in the straight initial part O A, the magnetisation will decrease in proportion to the decrease in the magnetising force. But if P is on the upper part of the curve when we begin FIG. 131. to diminish the magnetising force, the magnetisation for a given current on the return journey is greater than it was on the outgoing journey, and the magnetisation curve P M now lies above O A P. The magnetic condition tends then to persist, and this tendency for a preceding magnetic condition to persist is termed magnetic hysteresis, a term introduced by Ewing. If the current is brought down to zero so that there is no magnetising force, there is still 184 MAGNETISM magnetisation retained, represented by O M in the figure, and this retained magnetisation is the residual magnetism. It measures the retentivity of the specimen. If the current is now gradually increased from zero in the opposite direction the magnetisation decreases and will be zero for a certain magnetising force O N in the direction opposite to that of the original force. This is termed the coercive force. On still further increasing the reversed current the magnetisation is reversed. If we cease to increase it at Q and then begin to diminish again, hysteresis is shown, and when the current is again zero there will be residual magnetisation O M' now in the opposite direction to O M. Increasing the current from zero in the original direction, we shall travel to a point near P, but a little below it, when the original maximum current is reached, and repeating the cycle we shall get another curve. After a number of cycles the curves obtained for each practically coincide, as we have supposed in the figure. The area P M Q M ' P is termed ///<• hysteresis loop. The hysteresis curve varies for different specimens of iron and steel. Thus for magnet steel O M is not very great, but O N is considerable; while for soft iron O M may be very great and ON may be small. Susceptibility and permeability. The greater the poles developed in a bar of iron or steel under a given magnetising force, the greater is the snwrplibilitii of the bar to magnetisation. When Fit;. 182, the magnetising force and the magnetisation are Miitably measured the ratio magnetisation /magnetising force is termed the nm^m'tic susceptibility (see p. 240). It is usually denoted bv K. Without at- tending to the scale of measurement we can see that if O A B C, Fig. 152, represents the curve of magnetisation, then the eurve repre- senting the susceptibility will have a form somewhat like that of the dotted line. In the first straight line stage O A, the susceptibility will be constant. Then in the early part of A B as the curve beads round it will rise rapidly, increasing to somewhere near the second bend about B and then falling off' again. GENERAL ACCOUNT OF MAGNETIC ACTIONS 185 In the last chapter we introduced the idea of magnetic permea- bility as describing the power of conducting the lines of magnetic force possessed by a magnetic body. Though permeability is connected with susceptibility, it implies a somewhat different way of looking at magnetic actions and is not measured in the same way. We may realise the difference, perhaps, by considering the case of a bar inserted in a magnetising solenoid A B, Fig. 133, and placed some distance from a needle N, which it deflects. Before the bar is inserted the space A B contains air, and the lines of magnetic force due to the electric current in the coil are passing along this space and ultimately out into the field, and are deflecting the needle N. Let us suppose that N is brought back to its original \ M FIG. 133. position by a magnet M placed to the right. Now let the bar be inserted. Many more lines of force pass through it than through the air which it displaces, and we say that poles are developed on the bar. The needle N is again deflected, and we measure the strength of the poles or the magnetisation of the bar by the in- crease in the lines of force, that is, by the deflection which is due to the substitution of iron for air. We do not consider the magnetisation of the space A B before the bar was inserted, but only the increase when the iron displaces the air. If we inserted an equal bar of wood or of glass instead of the iron, the lines of force would pass through it practically as through the air and the needle would show no appreciable change in deflection or development of polarity in the wood or glass. Hence, though they are the seat of magnetic action, yet as they only have the same action as air, or as a vacuous space, we say that they have no susceptibility — their /c = 0. We shall see later that this is not strictly true, for K has some small value for so-called non-magnetic substance, but so small that we could not detect the poles developed in the experiment which we are imagining. Susceptibility then depends on the excess of lines of force over and above those passing through air. Permeability depends on the total number passing through, not on the excess. Before the iron is inserted the air carries or conducts lines of force, and its permeability is taken as the standard and as having value unity. When the iron is inserted many more lines of force pass through, and when the number is measured in a way to be described later, the permeability, which is denoted by yu, is measured by number of lines through iron/number of lines through air, when the magnetising current is the same. The susceptibility 186 MAGNETISM is measured by the excess and the permeability by the multiplica- tion of lines of force. We shall see later that it is convenient to adopt such units of permeability /j. and of susceptibility K that yU=l+4 TTK. We see then that the permeability curve is like the suscepti- bility curve, but we must multiply the ordinates of the latter by 4 TT and raise the whole curve by 1 to give the former. "Permeability" is the more fundamental idea, for it takes account of the whole magnetic action going on. " Susceptibility " is somewhat artificial, but it is convenient to use the term, as K is more directly measured in certain methods of experiment. Magnetisation and temperature. If we heat a magnet to a bright red heat and then cool it, we find that the magnetisa- tion has entirely disappeared. If we place a bar of iron or steel at a bright red heat within a magnetising coil it shows no suscepti- bility. But as it cools to a dull red usually a little below 800° (', its susceptibility is almost suddenly regained. If an iron ball heated to a bright red is hung by a spiral spring over a strong pole it is not attracted, but as it cools a point is reach rd when it is suddenly pulled down to the pole. Hopkinson* found that with a certain specimen of iron the permeability increased with rise of temperature up to 775° C., when it was many times the value at ordinary temperatures. A slight further rise to 786° sufficed to render it non-magnetic. He called the temperature at which the magnetic permeability ceases to be more than that of air the critical temperature. The critical tempera- ture varies, with different specimens of iron and steel, from 690° to 870° C. It may be taken usually as not very far from 750° C. This sudden change in property is related to other phenomena. Gore j" observed that if an iron wire was heated to redness and then allowed to cool, at a certain point a slight elongation occurred. Barrett { observed that if a piece of iron, or, better, hard steel, is heated to bright redness and then allowed to cool, at a certain point it suddenly glows more brightly. This can easily be observed in a dark room. The phenomenon was termed by Barrett " recalescence," as implying a sudden rise of temperature. Hopkinson § measured this rise. He found that a certain piece of hard steel, after cooling to 680°, suddenly rose to 712°C. and then began to cool again, and he showed that the magnetic condition was assumed at the same time. No doubt the lengthening observed by Gore occurs at the same point. There is then some sudden change in the molecular condition of the iron when it passes from the magnetic to the non-magnetic condition, or rather when its sus- ceptibility disappears and its permeability falls to that of air. Permanent magnets and temperature change. If a permanent magnet has already been subjected to M-\eral vnria- * Phil. Trails. (1889), A, p. 443. f Proc. /?..*., xvii. (1869), p. 260. J Phil. Mag., Jan. 1874. § Proc. R.S., xlv. (1889), p. 4 GENERAL ACCOUNT OF MAGNETIC ACTIONS 187 tions of temperature, to get rid of a sort of temperature hysteresis which occurs at first, then subsequent rise of temperature within ordinary range results in diminution of magnetisation, but on the fall of temperature to the original value the magnetisation increases to its original value. This may be easily verified by an experiment represented in plan in Fig. 134. A needle, N S, with an attached mirror is suspended so as to throw an image on a scale. Two lJi o Sccda IN JU IS NJ -W- IN si ""' y Cool S I HJ. 134. magnets are brought up, one on either side, at such distances that t lit-y neutralise each other's action on the needle. They are fixed in position. Then one magnet is warmed, say, by bringing up from below a beaker of warm water to surround it. At once its magnetisation is diminished, and the needle shows by its deflec- tion that the cool magnet is the stronger. When the temperature is reduced once more to its original value the needle returns to its original position. By this arrangement the magnet whose tempera- ture is varied is brought quite near to the needle, so that a small change in its magnetisation may produce a considerable deflection. K wing* found that if a bar is only very weakly magnetised, then rise of temperature produces increase of magnetisation. As the bar is more strongly magnetised this increase falls off and ultimately changes sign. In ordinary steel bar magnets the effect is always decrease in strength with rise of temperature. Permeability and temperature. A large number of experiments have been made with different qualities of iron and steel to investigate the change of permeability with change of temperature.")" We shall here only describe the results of some experiments of Hopkinson on a specimen of Whitworth mild steel. * .!/./'//"//'• Ii«l <«-t ;,,ii, in Iron and other Metals, 3rd ed. p. 181. t Kwing, loc. cit. chap, viii, gives an account of the fundamental work on the effects of change of temperature. A paper by S. P. Thompson on The Magnetism of Pennant ut .)/"7l07. 500 1OOO Mag net ts inp force FIG. 137. into an observable quantity will be understood from the figure. The results obtained are given in Fig. 137, taken from the paper in the Philosophical Transactions, loc. cit. p. 228. * Proc. R.S., xl. (1836), p. 109 ; Phil. Trans. A. (1888), p. 205. 190 MAGNETISM The results for cobalt and nickel rods are also given. It will be observed that the greatest lengthening in iron is about two in a million, while the ultimate shortening approaches seven in a million. The behaviour of cobalt is just the opposite to that of iron, while nickel contracts always and is much more affected than either of the other metals. In Rapports du Congres International de Physique, vol. ii. p. 536, there is an account of the work done on this subject. It has been found that there is an increase in volume on magnetisation of the order of 1 in 106 for a field of 1000 in iron and steel, and an increase somewhat less in nickel. Stress and magnetisation. When a body either pre- viously magnetised or under the action of a magnetising force is subjected to stress, the magnetisation is affected by the stress. We shall here only give a short account of two cases, referring the reader for a fuller treatment of this very complicated subject to Ewing's Magnetic Induction in Iron and other Metals, chap, ix.* Matteucci discovered that if a bar of magnetised iron "is pulled lengthwise " its magnetisation was increased. Villari found that the nature of this effect depends upon the degree of magnetisa- tion. If the bar is weakly magnetised, then the effect of a pull is to increase the magnetisation as Matteucci observed, but if the bar is strongly magnetised the pull decreases the magnetisation. That is, after a certain point in the magnetisation the effect is reversed. This is known as the Villari reversal. Comparing this with the effect of magnetisation on change of length, we note that there is a reciprocity. Weak magnetisation lengthens a bar. Lengthening a weakly magnetised bar increases the magnetisation. Strong magnetisation shortens a bar. Lengthening a strongly magnetised bar decreases, while shortening increases, the magnetisation. It may be noted here that with nickel and cobalt bars the reciprocal effects also hold. Thus nickel shortens, whatever the value of the magnetising force, and reciprocally a longitudinal pull leads to diminution of magnetisation. Weakly magnetised cooalt shortens, and a pull diminishes the weak magnetisation. Strongly magnetised cobalt lengthens, and a pull increases the strong magnetisation. These reciprocal effects are illustrations of recipro- cities which are continually met with in physical phenomena. They are dealt with in J. J. Thomson's Applications of Dynamics to Physics and Chemistry. Lord Kelvin,f who was the first to investigate these effects of stress with requisite exactness, pointed out that we may regard a pull on a bar in a weak magnetic field as increasing its permeability, and a push as decreasing its permeability. In a strong field the change of permeability is reversed. * For earlier work see Wiedemann's Galvanism.ii*, epitomised in Encyc. Erit., 9th ed., Magnetism, p. 253. Later work and references will be found in Ewing (loc. cit.), who has himself largely contributed to tho exact study of the subject, f Math, and Phys. Papers, vol. ii. p. 332. GENERAL ACCOUNT OF MAGNETIC ACTIONS 191 The second case which we shall consider is one discovered by G. Wiedemann. If an iron wire is carrying an electric current, then the lines of force are circles round the axis of the wire and the wire is circularly magnetised, but as the lines of force are complete within the wire, the iron does not produce an external field. But Wiedemann found that if the wire is twisted it becomes longitu- dinally magnetised. If the current is flowing down the wire and the lower end is twisted so that the front part moves from right to left, that lower end becomes a NSP. Suppose that the square A B C D represents the unsheared form of a piece of the metal in the front part of the wire, A D being parallel to the axis of the wire and the direction of the current. The magnetising force H due to the current will be perpendicular as to A D. Now let the wire be twisted so that C D moves to the left relatively to A B. This shear may be represented by a lengthening of the diagonal A C and of all lines parallel to it, and a shortening of the diagonal B D and all lines parallel to it. With weak magnetisation the permeability is increased along AC, and is decreased along B D. The magnetising force due to the current is still H. Resolve it into Hj parallel to A C, and H2 parallel to D B. These are each inclined at 45° t'o H, and are equal. But Hr acting in a direction of greater permeability, produces stronger magnetisation along A C than H2 produces along D B. The resultant magnetisation, there- fore, has a component downwards. For if M1 M2 represent these magnetisations, their resultant M slants down. Hence the lower end of the wire has a NSP. Again we have a reciprocal effect. If the wire be magnetised longitudinally and a current be passed along it, then the lower end twists round. We might expect that with strong magnetising current these effects would be reversed owing to the Villari reversal, but so far they have always been found to be in the direction described, perhaps because the intensity of magnetisation has not reached t he critical point. CHAPTER XV MOLECULAR HYPOTHESIS OF THE CONSTITUTION OF MAGNETS Molecular hypothesis — Ewing's theory and model — Dissipation of energy in a hysteresis cycle — Temperature and magnetisation — Attempts to explain the constitution of molecular magnets. Molecular hypothesis of the constitution of magnets. The fundamental phenomena of magnetisation receive an explana- tion on the hypothesis that the molecules of iron and steel are themselves small permanent magnets, capable of being turned round their centres. This hypothoi^ \\a> lir>t propounded in definite form by Weber.* It has .since been developed by G. Wiedemann, Hughes, Maxwell, and above all by Ewing.f In the eighteenth century Aepinus supposed that magnetism consisted of two fluids with opposite properties, uniformly mixed in an unmagnetised bar, and that one ua^ pulled to one end and the other pushed to the other end under a magnetising force. Coulomb and others supposed that the separation \\ent on in each molecule separately, and this was the first step in the molecular hypothesis. Poisson developed it in this form and in\< I iS mathematical theory. But Weber gave it a new form, now univer- sally adopted, in supposing that each molecule of iron or -ted, or at least each molecule which contributes to the magnetic condition, is itself a permanent magnet, with definite polarity, and capable rotation about its centre. No attempt was made to account for the magnetisation of the molecules, so that the hypothesis attempts to give an account of the structure of a magnet as built up of smaller magnets in a particular way, and does not in any way explain magnetism. According to Weber, the axes of these molecular magnets in an unmagnetised bar are pointing equally in all directions, as represented in Fig. 139, so that in any space small compared with the whole magnet, but large compared with that occupied by a single molecule, an equal fraction of the whole number of molecules will point in every given direction, and so neutralise each other's external * Pogg, Ann. Ixxxvii. (1852), p. 145. t Ewing's paper describing his magnetic model is in Proc. X.S., xlviii. (1890), p. 342. He gives an account of the molecular hypothesis in chap. xi. of Magnetic Induction in Iron and other Metals. 192 MOLECULAR HYPOTHESIS 193 field. When a magnetising force acts, it is supposed to turn the magnetic molecules round with their axes more or less in the direction of the force, though there is some constraint (not accounted for by Weber) tending to restore them to their original position. But the stronger the magnetising force the more the constraint is overcome, and in a very strongly magnetised bar we FIG. 139. may suppose that all the molecular axes are turned nearly into line, with their north ends towards one end of the bar, and their south ends towards the other end, forming molecular magnetic chains as in Fig. 140. The poles are to be supposed to be arranged on all the chains as shown on the central chain. The unlike poles of successive members of a chain are close together so as to neutralise each other, but obviously each chain will have an unneutralised TV J Tl 3 TV .? TV 7V STVSTV3TLSTV FIG. 140. pole at each of its ends so that there will be a number of un- neutralised NSP's on the surface near one end of the bar, and an equal number of unneutralised SSP's near the other end. And the molecules all having equal poles, the total polarity is the same at the two ends. Further, if the bar be cut through at A B, on the new surface to the left a row of unneutralised SSP's will be < \posed, while on the new surface to the right a row of NSP's will be exposed. Obviously, as we do not cut through the molecules we cannot obtain polarity of one kind alone. On this hypothesis there is a limit to the magnetisation which ;\ given bar will undergo, reached when all the molecular magnets 194 MAGNETISM are turned round so as to be parallel to the magnetising force. The existence of such a limit is borne out by experiment. The curve BC in Fig. 130, p. 183, tends to become nearly parallel to the axis when the magnetising force is increased to a very great value. An interesting experiment by Beetz * also shows that there is probably a limit. A varnished silver wire with a very narrow scratch through the varnish was made the cathode in an electrolytic cell containing a salt of iron in solution. This was arranged in a weak magnetic field parallel to the scratch, and the iron molecules as they were laid down on the silver one by one were free to turn in the direction of the field. The magnetisation of the iron deposited was very great for so small a mass, and was found to be only very slightly increased when the wire was placed in a very powerful field. When a bar undergoing magnetisation is vibrated, say, by hammering, it is magnetised more strongly than if quiet, and we may perhaps assume that the vibration frees the moleculei from their constraint, so that they yield more easily to the magnet is ing force. A modification of the theory was made by Wiedemann + and by Hughes.J If a number of bars of iron are made up into a bundle, magnetised along their lengths, and then partially demagnetised by, say, hammering, on taking the bundle apart it is found that some of the bars have their polarity reversed. The ap- parent demagnetisation is partly (llR, to the completion of the mag- netic circuit of the stronger bars through the weaker ones, which S they reverse as represented in Fig. 141. Carrying this idea down S to molecular regions, Wiedemann and Hughes suggested that in an FIG. 141. un magnetised bar the molecular magnets are arranged in neutral- ising circuits of two or more molecules, as in Fig. 142. As originally described, Weber's theory gives no account of hysteresis. Wiedemann sought to explain it by supposing that there is some sort of friction to be overcome in turning the molecules round, friction which has to be overcome again when the molecules seek to return after the magnetising force is removed. Maxwell § introduced the idea that the constraining force on each molecule is analogous to an elastic force, that for small displace- * Pogg. Ann. cxi. (1860), p. 107. t Cfalvani*mu9, 2nd ed. vol. ii. [1], p. 373. $ Proo. A.S., xxxv. (18*3), p. 178. § Electricity and Maynffitm, 3rd ed. vot. H. p. 85. MOLECULAR HYPOTHESIS 195 meats it is proportional to the displacement, and on removal of the magnetising force there is complete return. With large dis- placements, however, there is something analogous to permanent n & \ \ 6' 71 FIG. H2. set and therefore not complete return. He did not attempt to account for the permanent set, but simply assumed it. Ewing's theory and model. Ewing brought a great simplification into the hypothesis by showing that the mutual action of the molecular magnets would account for the controlling or constraining forces, tending to restore the molecules exactly after small displacements, and for the permanent set, that is, the residual magnetisation, after large displacements. Thus no sort of friction need be assumed, and the hypothesis in its simple form accounts for the facts. To illustrate the ideas he constructed what is known as Ewing s model of a magnet, consisting of a number of compass needles placed together as in Fig. 143 (in plan), each needle representing a molecule. Such a model, if inserted within a FIG. 143. magnetising solenoid in place of the bar in Fig. 129 shows all the phenomena of magnetic induction and gives a curve of magnetisa- tion repeating all the features of the hysteresis curve in Fig. 131.* To understand how these mutual molecular actions account for the phenomena let us follow Ewing in supposing that we have four needles arranged at the corners of a square. With no external magnetising force and under their own mutual actions only, they will set as in Fig. 144, where the arrow-heads are north-seeking * Induction in Iron and other Metals, 3rd ed. p. 351. 196 MAGNETISM poles. The magnetic circuit is completed within the group and there is practically no external field exhibited at a distance. If a small magnetising force is put on, represented by H, each magnet will be turned round a little, its N pole tending to move in the direction of H, as in Fig. 145. But the displacement of a FIG. 1-H. Fi<;. 1 !'-. magnet from a position N S to a position N'S', Fig. 146, may be represented by the superposition of two other magnets with equal poles, for superpose on NS, Fig. 146, a magnet SjN'and a magnet NjS', then N and Sx neutralise each other, and Sand N, neutralise each other, and we have N'S' left. Thus the effect of the rotation of the four magnets in Fig. 145 may be represented by superposing eight little magnets as in Fig. 147, and these will have a field generally in tin- direction of H. .1- N *N — - % FIG. 146. FIG. 1 17. While the magnetising force H is small, these deflections, and therefore the lengths of the small magnets in Fig. 147, will be proportional to the magnetising force, and the net magnetisation will also be proportional to it. But when a certain point is reached in the value of the magnetising force the configuration becomes un- stable and the needles swing round into some such position as that in Fig. 148, and their external field suddenly and greatly increases. MOLECULAR HYPOTHESIS 197 On still further increasing H the magnets tend to set more and more nearly parallel to H, as in Fig. 149, and then the limit of magnetisation is practically reached. The curve of magnetisation is represented in Fig. 150, where O A represents the initial stage with no hysteresis, A B the sudden ^^^x^, turning round, and BC the subsequent approach to par- allelism with H. If H be removed the return will be from Fig. 149 to Fig. 148 — that is, a very large amount of residual magnetism H remains in the direction of H. Reversal of H will ultimately FIG. 148. send the magnets round in the opposite direction and we shall have a sudden reversal of mag- netisation. The behaviour of such a group may be studied by placing four compass needles in a square. The earth's field must be as nearly as possible neutral- • ) ised by means of a large magnet, placed so that its field is equal • ) and opposite to that of the H ^ earth. On gradually approach- . ing a second magnet towards the group the phenomena will v ) be observed. If there are many such square groups near together, and with the sides of the squares at different inclinations to H, instability will set in at different values of H for the different squares, and "the transition from one stage Fu;. 100. 198 MAGNETISM to another will be gradual in the aggregate, for it will happen at different values of H in different groups. Hence the curve will assume a rounded outline in place of the sharp corners " of Fig. 150. As each group separately shows hysteresis the aggregate will also show it. Figs. 151, 152, 153* show the imagined behaviour of a large 1U1 1 1 i 1 } I i I l 1 \ H=0 FlQ. 151. FIG. 152. FIG. 153. group of square set molecules respectively under no torn.1, ;i force which has pulled them nearly parallel, and a very large forcv. From E wing's model we see how retentiveness is accounted for by the molecular poles holding each other more or less in their displaced positions. We see, too, that as H increases in the later stage the mutual action tending to hold the magnets in the nearly parallel position increases. Hence if H is increased to a large value and then decreased to a smaller value the magnets, being under a larger con- trolling force due to their mutual action, do not return to the portion * Ewmj.', /inlucti'iii in Iron lk< r JA {«/.•>•. ord ud. j>. il.M, MOLECULAR HYPOTHESIS 199 which they had under that smaller value when the value of H was rising through it. Fig. 154, taken from Ewing (loc. cit. p. 351), shows the magne- tisation curve of a group of twenty-four pivoted magnets when the magnetising force is taken through a complete cycle. Dissipation of energy in a hysteresis cycle. A study of the behaviour of the members of a model at once suggests that there is dissipation of energy when a piece of iron or steel is taken through a cycle. For work is done on the magnets in putting FIG. 154. them into line, and when on the diminution and reversal of the magnetising force a point of instability is reached, the magnets fly round and vibrate about their new position through large ampli- tude. This means that some of the work done appears as energy of vibration. In the molecular magnets we should expect similar behaviour, and the molecular vibration would imply the develop- ment of heat and a rise in temperature of the iron. In fact, it can be shown that the area of the hysteresis loop is proportional to the energy dissipated in a cycle. The cycle may be performed in another way than that which we have supposed, viz. by the rotation of the specimen in a con- stant field. This is what takes place in a dynamo armature when the core is a cylinder of iron rotated between strong poles. Besides changing the magnetisation, the rotation will tend to produce electric currents in the iron and to prevent these currents the iron is laminated transversely to the direction in which the currents would flow, so that they are practically destroyed. The work done in a revolution is then due to the hysteresis. The molecular groups as they move round from a or b in Fig. 155 tend to keep in their original line, given to them by the magnetising force, so that at c or d they should be perpendicular to the magnetising force ; but 200 MAGNETISM before they reach these points they become unstable and swing round, thus dissipating energy. A remarkable conclusion was predicted by Swinburne as a consequence of Ewing's theory, and was subsequently verified by experiment, viz. that when the field in such a case as that repre- sented in Fig. L5."> is exceed- ingly strong, hysteresis should disappear. If the field i- -o strong that the molecular magnets are held all the time in the line of the magnetising FIG. 155. force and rotate round their centres, as it were, as the iron travels round, there is no instability, no conversion of magnetic energy into energy of vibration, and no hysteresis. Temperature and magnetisation. We ha\e seen that in a permanent magnet sgiall rise of temperature is accompanied by diminution of magnetisation, the magnetisation returning to its initial value when the temperature falls again. We mav perhaps explain this by supposing that rise of temperature is accompanied by increased vibration of the molecular magnet - about their centres, so that they do not point so entirclv in the direction of the magnetising force-. For hisl-mcc. lei \ >. Fig, \~>('^ be the equilibrium position of a molecule under a force H and let it vibrate In •! \\een «! S1 and 7f2 .v.,. When at t^ .s-j, we may represent the effect of the displacement of N by superposing a small magnet ,y nl ; when at 1I2 *.» by superposing a small magnet s nv * n« has more effect in destroying the magnetisation in the direc- tion of H than ft nl has in increasing i(. since it is more nearly parallel to it. There is another possibility which mav be men- tioned. The permeability of iron increuM - with rise of temperature. The magnetic circuit of a molecule or of a group of molecules is partly completed through the iron, and it is only part which issues into the air. As the permeability rises with rise of temperature the groups may find it easier to complete more of the circuit within the iron, and some of the lines of force mav be withdrawn from the air. The increase, of . permeability with rise of temperature \\hich occurs in weak fields may probably be accounted for by supposing that initially the iron is in the first stage of magnetisation, where the molecular magnets are only slightly deflected, as in Fig. I IV But as the temperature rises molecular vibration increases and MOLECULAR HYPOTHESIS 201 some of the groups become unstable, and for those groups we pass to the second stage, represented in Fig. 148. As the temperature rises, more and more groups pass to this stage, and between 700° and 800° all the groups have become unstable and have then come more or less into line with the magnetising force. This is repre- sented by the great rise in permeability. The sudden disappearance of magnetic quality at the critical temperature, about 785°, may perhaps be explained by supposing that after that temperature the vibration has become so violent as to change into molecular rotation. With strong fields the iron is already in the second or third stage. Vibration due to heating has two opposite and neutralising effects. On the one hand it eases the breaking up of groups and alignment with the force and so tends to increase the magnetisa- tion, while on the other it tends to reduce the magnetisation by the movements out of the line of the force in the way illustrated in Fig. 156. We may suppose that these opposite effects keep the curve of permeability nearly level, till at a temperature near 800° the diminution due to vibration overpowers the increase due to facility of group breaking and the permeability decreases. The change in length of a bar on magnetisation is yet to be explained on the molecular hypothesis. It would obviously require some further assumptions as to molecular grouping or as to molecular dimensions in different directions. Attempts to explain the constitution of the molecular magnets. \Ve have already pointed out that the molecular hypothesis only carries down the magnetisation from the visible mass of the bar as a whole to its molecular constituents, and does not account for their magnetisation. Attempts have been made to explain the magnetisation of the molecules in terms of electricity. The earliest was Ampere's hypothesis. He started from the- experimental fact that a current flowing in a circular wire forms a magnet of which the axis is perpendicular to the plane of the wire. He supposed that there was a channel round a magnetic molecule through which an electric current could flow, and he further supposed that this channel was perfectly conducting, so that the energy of the current was not dissipated, and therefore, if once started, the current should persist unless an equal and opposite current were in any way superposed. But a continuous closed current, such as that flowing in a circular wire, is essentially a locus into which electric lines of force are continually converging and are there dissipated. As these lines sweep through the surrounding space they are accompanied by magnetic action, and >o in a sense produce the magnetic field. We know of no such current as that which Ampere supposed. Hut if we could imagine a bundle of lines of magnetic force in the form of those of a small magnet, bound round as it were with a perfectly conducting channel, then no lines of force could be 202 MAGNETISM added to, or subtracted from, the bundle. For an addition of lines of force through the channel would mean an inverse current in the channel which would just suffice to neutralise the lines added. Thus Ampere's hypothesis really gives no explanation in terms of known facts, but when analysed is a hypothetical device for securing permanent magnets. Another hypothesis much more in touch with experimental knowledge has lately gained acceptance. It is founded on the discovery that a moving charge of electricity produces a magnetic field. If, adopting astronomical language, we suppose that a molecule consists of a large positively electrified primary and a small negatively electrified satellite revolving round a centre of mass much nearer the large primary, then it can be shown that the system would be magnetically equivalent to a small magnet with axis through tin- centre of mass perpendicular to the plane of the orbit.* Thus the magnetism of iron and steel would be explained in terms of the magnetism which accompanies moving electricity. Still, that magnetism remains to be explained. We shall refer to the molecular hypotheses and to an attempt to determine the strength of the molecular magnets in Chapter XXII. * This "electron" hypothesis has been most fully developed by Lar.gevin, Ann. dc Chim. et dc Phys., 8th series [5], 1905, p. 70. A brief account is given at p. 297 infra. CHAPTER XVI GENERAL ACCOUNT OF MAGNETIC QUALITIES OF SUBSTANCES OTHER THAN IRON / Jix^ AND STEEL Faraday's classification of all bodies as either paramagnetics or diamagnetics — General law — Explanation of the action of the medium •by an extension of Archimedes' principle — Ferromagnetics, para- magnetics, and diamagnetics. Paramagnetics and diamagnetics. — We have in the previous chapters only described the magnetisation of iron and steel and have mentioned that nickel and cobalt show sensible magnetisation by induction, though in a far less degree than iron. They attract and are attracted by the poles of a magnet and possess retentivity in some conditions. Biot found that a certain needle of carefully purified nickel — which stands next to iron in its capacity for mag- netisation— retained its magnetism and possessed about one-third the directive force towards the North of that of a certain steel needle of exactly the same dimensions. Faraday experimented with cobalt, which could easily be made to lift more than its own weight, though losing all its magnetism on withdrawal of the inducing magnet. Magnetisation was qfteii apparently detected by early experi- menters in other metals and alloys, but we may probably ascribe the effects they observed to the presence of small quantities of iron. Towards the end of the eighteenth century it was observed that antimony and bismuth were repelled from the pole of a strong magnet. On the one hand, then, iron, nickel, and cobalt are attracted, and on the other hand bismuth and antimony are repelled, by a strong magnetic pole, and it was at one time supposed that other substances possessed no magnetic properties at all. Faraday, how- ever, showed that these two groups are really only at the two extremes and that all bodies are either attracted or repelled, though the forces are usually so slight that they can only be detected with very powerful magnets. The mode of experiment by which he discovered this consisted in suspending a small bar of the substance to be tested between the poles of a strong electro-magnet by a fine thread. If 204 MAGNETISM 1.1. the substance resembled iron, the bar set itself "uxially, with its longer axis along the line NS, Fig. 157, joining the two poles, and he classed all such substances together as paramagnetic*: If it resembled bismuth or antimony the bar set itself "equa- r FIG. 107. torially," i.e. along the line crat right angles to N S, and all >uch substances he designated diamagneiict. Liquids. By enclosing liquids in gla>^ vessels Farada\ \\a^ enabled to determine to which cla>s thcv belonged. Thus hi- found that solutions of salts of iron are paramagnetic, while alcohol and water are diamagnetic. Gases. He also succeeded in showing the magnetic character of gases in several ways. For instance, if a stream of gas rendeird evident by traces of ammonia and hydrochloric acid was allowed to ascend between the poles, if dinmagnetk it divided into h\<» streams pushed out from the central region. Bubbles filled \\ith the various gases were attracted into or repelled from the central space between the poles. He found from hi> cxpcrimi-nt> that oxygen in air is strongly magnetic, while nitrogen and hulrogrn are diamagnetic. The following Table gives a feu of tin- mon important substances, arranged under the two heads, air being the medium surrounding the poles (A'./y;. Res. vol. iii. p. 71): PARAMAGNETIC Iron Nickel Cobalt Mangam-M- Crown G lav- Platinum DIAMAGNETIC Araenic Ether Alcohol Gold Water .Mercury Flint (ila>- I It -a\ v (ila» Antimonv Bismuth SUBSTANCES OTHER THAN IRON AND STEEL 205 The arrows show the direction in which the magnetic action increases. Faraday was able to state the following General Law : Para- magnetic substances tend to move from positions where the magnetic action is weaker to .where it is stronger, i.e. from weaker to stronger parts of the field ; while diamagnetic substances tend to move from stronger to weaker parts of the field. The axial setting of paramagnetic bars and the equatorial setting of diamagnetic bars at once come under this general law. The field in Fig. 157 is strongest along and close around the axis N S, and a paramagnetic body tends to stay there. A bar is in the strongest part of the field when it lies along the axis. The field rapidly weakens as we travel along the line er either way from the axis. A diamagnetic bar therefore gets, on the whole, into a weaker part of the field when it sets equatorially. The law is further illustrated by an experiment devised by Pliicker, which shows whether a liquid is paramagnetic or diamag- netic. A watch glass containing some liquid is placed on the poles ; the liquid if paramagnetic is attracted to, and if diamagnetic is repelled from, points where the intensity of the field is greatest and so the level of the surface is disturbed, its shape depending on the nature and distance apart of the poles. Thus when the poles are very close together, say y1^ inch apart, a paramagnetic liquid has the section shown in Fig. 158 («), while a diamagnetic liquid has the section shown in Fig. 158 (6), the liquid retreating from the central strongest field. When the poles are further apart the strongest parts of the field are near to the pole pieces and the centre is weaker and the effects are nearly reversed. Effect of the medium. Faraday also made experiments on the effect of the medium, and he showed that a paramagnetic solution enclosed in a glass tube behaves as a paramagnetic if Mispended in a weaker solution of the same kind, and as a diamag- netic if suspended in a stronger solution. For example (Eocp. Res. vol. iii. p. 58), a clear solution of protosulphate of iron was prepared containing 74 grains of the hydrated crystals to 1 ounce of water. Another weaker solution was prepared by adding to one volume of this three volumes of water. Suspended in air in tubes both were paramagnetic, the former the more strongly so. If the tube containing the first solution was suspended between the poles in a vessel containing the same solution no directive force acted upon 2 ()« MAGNETISM it. Suspended in the second weaker solution it behaved again as a paramagnetic. If a tube containing the weaker solution was suspended in a vessel containing the stronger, it pointed equatorially, behaving as a diamagnetic. Surrounded by a solu- tion of its own strength it was neutral, and in a still weaker solution it was paramagnetic. He also prepared a solution of protosulphate of iron in water which was neither paramagnetic nor diamagnetic, the iron salt being neutralised by the water. Explanation of the action of the medium by an ex- tension of Archimedes' principle. In Chapter XX we shall investigate the action of the medium in detail. For the present we may regard the action as being in accordance with an extension of Archimedes'* principle, the resultant force on a body in a magnetic field being equal to that on the body minus that on the medium which it displaces. We may put this extension of the principle in the following form : Let N, Fig. 159, be a pole with a FIG. 159. magnetic field round it. We may regard all parts of the medium as acted on by the pole, and as coming, under that action, into a state of equilibrium in which the medium is in a condition of strain. We may resolve the total action on any part of the medium A into the magnetic action of the pole on it, and tin- actions of its surroundings due to their state of strain, tin- resultant of the latter being equal and opposite to the former. If now A be removed and replaced by some other body, A', of exactly the same shape, the magnetic action on A' differs from that on A, while, unless A' is so strongly magnetic as appreciably to alter the field, the magnetic action on the rest of the medium remains un- changed and its strain is the same as before. Since the action of the medium on A' depends on this strain it is unchanged. Hence there is no longer equilibrium, and A' will be attracted to or repelled from the pole according as it is more or less paramagnetic than the medium. A difficulty in making this explanation per- fectly general so as to account for all diamagnetic actions is that we have to suppose a vacuum magnetic, for the diamagnetics in the list given above are diamagnetic in a vacuum, and we should have to suppose that their repulsion from a pole is due to the stronger attraction on the surrounding vacuum. This is only another way of saying that what we term a vacuum is not empty space, but contains something capable of acting on and being acted on by material bodies, a supposition to which we are forced by the transmission of light through it. In a beam of light there SUBSTANCES OTHER THAN IRON AND STEEL 207 is energy and there is momentum, whether it is passing through ordinary matter or through the highest vacuum of which we have experience. Ferromagnetics, paramagnetics, and diamagnetics. Though Faraday only used the two classes, pararnagnetics and diamagnetics, and to one or other of these referred all hodies, it is usual now to separate out the three metals iron, nickel, and cobalt, and to these may probably be added manganese, and to class them as Ferromagnetics. Their magnetisation does not increase in linear fashion with magnetising force and they show hysteresis and retentivitv. The term paramagnetic is reserved for the bodies with far feebler magnetic qualities. In these the magnetisation is proportional to the magnetising force. Curie* showed that with given magnetising force the magnetisation for paramagnetics is inversely as the absolute temperature over a very wide range, thus extending a result which had been found to hold in certain cases over a smaller range by G. Wiedemann. This is now known as Curie's Law. Ferromagnetics may probably be regarded as passing into paramagnetics when above their critical temperature. Diamagnetics, if we describe them in terms of polarity, may be regarded as having a like, instead of an opposite, pole formed by an inducing pole, and Curie found that their magnetisation, except in the case of bismuth, is very nearly independent of temperature. Thediamagnetisation of bismuth decreases with rise of temperature between —182° and the point of fusion very nearly linearly, and by about j^Vo for 1° rise-t This has led to the suggestion that diamagnetism is not to be regarded merely as a negative paramagnetism. The temperature effect on paramagnetics would appear to show that paramagnetism is concerned with molecular structure, which changes with change of temperature, while the absence or smallness of the effect of change of temperature on diamagnetics suggests that diamagnetism is concerned rather with the atomic structure, which is, we may suppose, independent of temperature. We shall return to the forces on paramagnetic and diamagnetic bodies in Chapters XX and XXII. * Ann. de Chim. et de Phys., 7e serie, t. V. (1895). An account of the subject is given by H. du Bois in the Congres International de Physique, vol. ii. p. 460. f Conyres International, loc. cit. p. 503. CHAPTER XVII THE INVERSE SQUARE LAW Variation of the force due to a pole as the distance from the pole is varied — The inverse square law — Magnetic measurements founded on the law — Coulomb's experiments — Unit magnetic pole — Strength of pole m — Magnetic intensity — Geometrical construction for the direction of the intensity in the field of a har magnet — Gauss's proof of the inverse square law — Deflection method of verification — Moment of a magnet — Comparison of moments — Some consequences of the inverse square law — Application of Gauss's theorem — Flux of force — Unit tube? — A pole m sends out 4ir m unit tubes — Potential. Variation of the force due to a pole as the distance from the pole is varied. The inverse square law. If a NSP is placed at any point in the field of a magnet, the force on it is due to the action of the magnet as a whole. We cannot isolate one of the two polarities. Always the two come into play and the action is the resultant of the North-seeking repulsion and the South- seeking attraction. The best approach we can make to isolation of one polarity consists in so arranging the acting magnet that one pole has practically very little action as compared with the other. Coulomb's experiments. This method was used by Coulomb, to whom we owe the decisive experimental proofs which finally established the inverse square law.* In one experiment he used a steel wire magnet 25 inches long. He first found that the poles might be considered an inch from each end. Then, placing the wire vertically, he noted the time of vibration of a needle opposite the lower pole and at different small distances from it in the plane of the magnetic meridian. The upper pole was not onlv at a much greater distance, but its action was nearly vertical, so that for two reasons the horizontal component of the force due to it was sufficiently small to be negligible. It was necessary to take into account the action of the earth which was added to that of the magnet in all cases. Let us suppose that the horizontal force on each pole of the needle due to the earth is Hr and that the force due to the pole of the magnet at distance d± is Fx and at distance d2 is F2, and let the times of * For a history of the experiments made by Coulomb's predecessors, sec Brit. 9th ed., Magnetism, p. 236. 208 THE INVERSE SQUARE LAW 209 vibration be t when the earth alone acts, and ^ and t2 when the pole is at the distances dt and d2 from the needle. Regarding the needle as a sort of double pendulum with equal and opposite forces acting at the two ends, it is seen that the times of vibration are inversely as the square roots of the forces acting : H : H + Fx : H + F2 =^2 : 1 : i. *1 r2 whence the forces due to the magnet alone are given by F - F 1 • 1 - -1 - l 2 t* t* ' t* I2' By observing the times of vibration Coulomb found that the forces were approximately proportional to the inverse square of the distances c^ and 6?2, or that the force due to a given pole is inversely as the square of the distance from that pole. He also verified the law by means of the torsion balance (see ante, p. 62), using two long wire magnets about the same length as the one described above, and about J inch in diameter, one being suspended horizontally in the torsion balance, the other being fixed vertically so that if the suspended one had remained in the magnetic meridian, their two like poles, taken as being one inch from the end of each, would have been in contact. The suspended needle was repelled from the magnetic meridian, and the torsions were then measured which were required to bring the two poles to observed distances from each other. In each case the action of the earth aided the torsion, and a preliminary experiment was necessary to determine the torsion equivalent to the earth's action. Thus in one experiment, before the vertical magnet was put in position, two turns of the torsion head, i.e. 720°, pulled the magnet round 20°, or 700° torsion were required to balance the earth's couple due to 20° deflection from the meridian, giving 35° of torsion per 1° of deflection. Putting the vertical magnet in position and bringing the torsion head back to its original position, the suspended magnet was deflected 24°, the torsion being therefore 24°. To this we must add 24 x 35 = 840° for the torsion equivalent to the effect of the earth, making a total of 864° at 24°. To halve the deflection, the torsion head had to be turned eight times round, giving a torsion of 8 X 360 + 12 = 2892°. To this must be added 12 X 35 = 420 for the torsion equivalent of the earth's effect, making a total of 3312° at 12°. According to the inverse square law, with this value at 12°, taking distances as 3312 equal to arcs, we ought to have at 24° a torsion of ~ = 828°, which is not very far from the observed value 864°. Magnetic measurements founded on the law. We shall describe below a still more accurate method of verifying 210 MAGNETISM the inverse square law, but it is convenient here to show how the law enables us to arrange a definite and practical system of measurement for the strength of magnetic poles. Were it possible to obtain perfectly constant long magnets, it might In- convenient to use one pole of such a magnet as the standard, and copies of this might be used just as we use copies of the standard unit of mass. But no magnet retains its magnetisation un- changed, and we are obliged to fix on a standard depending on the magnetic force exerted. We therefore take the following definition : A unit magnetic pole is that which at a distance of 1 cm. would exert a force of 1 dyne on another equal pole in air. In practice it would not be advisable to bring two poles so near as 1 cm. At so small a distance they might weaken each other appre- ciably by induction. Also in actual magnets we cannot fix on any definite point as the pole. The region of North- or South-seek ing polarity might have dimensions quite com parable with 1cm., but the inverse square law at once enables us to get over this difficulty. Arranging the two poles at a distance d apart, \\hcre (/ i^ so great that these objections do not hold, the force will be -% dvm^. Having thus a definite standard, if magnetic poles ^«u constant in strength we might measure the strength of any other pole by finding the number of unit poles which, placed together, would produce at an equal distance an effect equal to that pro- duced by the pole to be measured. Hut it i.s easv to show that if wu put a number of equal poles together the total effect is not the sum of the separate effects, since thev \\cakcn each other. If one of two exactly similar equal bar magnets is placed \\ith it> axis East and \Yest. and a compass needle i- placid at some distance from it at a point in the axial line, and the deflection noted, if the second magnet is now superposed on the first uith like poles together the deflection is \)\ no means doubled. But \\e may still measure the strength of a pole in terms of the unit by the ratio which its action bears to that of the unit at the same distance. We say that The strength of a magnetic pole is m if it exert- on a unit pole at a distance d a force of j, dyne-. We have now to investigate the force with which a pole ;// will act on a pole m' at a distance n m,' I f <* -> 4 *' > ACs FIG. 100. adjusted that the compass remains in the meridian, any other compass needle placed in the same position will also lie in the meridian. If then we place a pole or strength m at C it acts on 1 at B with a force -y,2 an(l is reacted on by an equal force which balances the action of m upon it. Hence the force exerted by won in = -y- / . 1 _ in \ _ mm Y\ * result may he verified by the method of deflection. A small needle ?/,?, Fig. 163, is suspended so that it lies in the .v I i i i i i i FIG. 163. magnetic meridian. A magnet NS is then brought up, say, to the west of it in the line through its centre perpendicular to the meridian. The magnet is said to be in the u end-on position." Considering the horizontal action on the pole n of the needle, if its strength is /x it will be acted on bv a force ull in the meridian. where H is the horizontal intensity of the earth's action ; it will be acted on by a force // -- at right angles to the meridian, and t he- needle will set with its axis in the line of the resultant of these two, since there are equal and opposite forces on the other pole, 8. From the parallelogram of forces on the right hand of Fig. it will be seen that if 9 be the angle of deflection, THE INVERSE SQUARE LAW 215 Zml ^~~ tan 0 = or the deflection is the same whatever the strength of the suspended needle. Now move the magnet NS till it is north or south of the suspended needle, with its centre in the meridian and at the same distance d away, and with its axis perpendicular to the meridian : the magnet is now said to be in the " broadside-on position." The deflection 0' will be given by /v = .*. tan 0 = 2 tan 0'. If the angles are small we shall have e = 20'. Gauss* verified the law in this manner with very great accuracy. He used a more general formula for the action, carrying the approximation a step further than in the investigation just given. Moment of a magnet. It will be observed that in the above values for the intensity the strength of pole does not occur alone, but in the product, strength of pole x distance between the poles. When we may consider the poles as points, this product is termed the " moment of the magnet," and we may denote it by M. In actual magnets we cannot consider that the poles are two points at a definite distance apart, but we may still give a meaning to the term moment. Whatever the nature of the magnets there are always equal amounts of the opposite polarities. Let us treat each kind of polarity as if it were mass, and find the centre N of North-seeking polarity at one end, and the centre S of South-seek- ing polarity at the other, just as we should find the centres of mass. I^et those two points be denoted by N and S. If the total quantity of either kind of magnetisation is m, then the moment of the magnet is given by M = mx NS. Axis and centre. The line NS is termed the axis of the magnet, and the point bisecting NS is the centre of the magnet. \Y" may give a physical interpretation to the moment thus: It the magnet is placed in a field in which the lines of force are parallel and the intensity everywhere of the same value, H, i.e. in a uniform * Enr,yc. Erit., 9th ed., Magnetism, p. 237. 216 MAGNETISM field, and if the magnet is turned round so that its axis is perpen- dicular to the lines of force, the couple acting on it is MH. Though we thus imagine two "centres of polarity/' it is to be remembered that a distribution of matter, or of electricity, or of magnetism, cannot be replaced by a concentration at a single point in calculating the field which is produced. Comparison of moments. We may compare the moments of two magnets of small dimensions to a first approximation by using the value for the field at a distance d from the cent. IT along the axis, 2 ml 2 M For this purpose we may bring up the two magnets, one east and the other west of a suspended needle, and placed so that they tend to deflect the needle in opposite directions ; then adjusting them a! distances dl and d2 so that the needle is not deflect. ?•-$ - Or we may use each in succession to deflect the needle at the same distance. If the deflections be 01 and 02 rcsp,-cti\ely. the earth's horizontal intensity l>eing H, tan «t tan 0f = Some consequences of the inverse square law. Application of Gauss's theorem. Flux of force. As all experiments go to show that the intensity at any point <>t I magnetic field in air or in any other non-magnetic medium may be regarded as due to each element of polarity (hn acting according to the law rfw/;-2. we may apply (iaiiss\ theorem (p. JW) ' magnetic system. If we define the flux of force through an element of surface dS as NJS, where N is the component of the intensity perpendicular to r/S, the theorem tells us that the total flux offeree through any closed surface, which is entirely in air or other non magnetic medium, is /ero. For the total polarity within the surface, even if it encloses magnets, is /ero. THE INVERSE SQUARE LAW 217 Tube of force. If a small closed curve is drawn at any place in a field, and if a line offeree is drawn through every point of it, the lines so drawn will enclose a tubular space termed a tube of force. Let us apply Gauss's theorem to the closed surface in air or other non-magnetised medium, formed by a portion of a tube of force, and any two cross-sections of it at right angles to the lines of force. Thus, let Sx, S2, Fig. 164, be the two cross-sections, and let Hx, H2 be the intensities at S15 S2. The normal intensity over the curved sides of the tube is everywhere zero. At Sj it is Hx and at S2 it is — H2, negative because H2 is inwards. We have, therefore, H! x Sx — HL X S2 = 4-7T X magnetisation within the tube = O FIG. 164. That is, the product H x S, or the flux of force, is constant throughout the tube so long as it does not contain magnetisation. Unit tubes. If we draw tubes so that for each, intensity X cross-section = 1, that is, so that there is unit flux of force along it, each is termed a unit tul>e. A pole m sends out l^m unit tubes, for if we draw a sphere of radius r round m as centre the intensity on the surface of the sphere is m/r*9 and the total flux of force through the sphere is 4xr2w/r2 = 4-7TW. There are, therefore, 4>7rm tubes passing through the surface. If we imagine one line of force along each unit tube the number of lines of force passing through any area will be equal to the total flux of force through the area. The number of unit tubes or of lines of force passing through unit area perpendicular to them is equal to 1 /cross-section of unit tube or is equal to the intensity of the field. Potential. If we define the potential at any point in the field as the work done in bringing a unit positive pole to the point from a distance so great that the field is negligible, we can show (sec Electricity, p. 42) that where every element of magnetisation dm is to be divided by its distance r from the point. If the unit pole is moved a distance dx in any direction x to a point where the potential is V + T- dx, the work done is -^ dr. So that the force on the unit pole opposing the motion is 218 MAGNETISM -p. Then the force on the unit pole, in the direction of motion dx or the component of the intensity in the direction x is H ^V- **x — " T~ d.r Level or equipotential surfaces can l>e drawn in magnetic as in electric systems. CHAPTER XVIII SOME MAGNETIC FIELDS Small magnets — Magnetic shells — Uniform sphere. The potential of a small magnet at any point in its field. Let ACB represent a small magnet of length I, poles ± m, moment M = ml. Let P be a point at a distance r from its centre C in a direction CP making 0 with CA. The potential at P is m m ~ AP~BP' Draw MCN perpendicular to CP, and draw PA M, PNB. We 'P FIG. 165. suppose CP so large compared with / that AM and BN may be taken as perpendicular to MN, and PM = PN = PC. Then PA = PM - AM = PC - AM = r - -' cos 0 and PB = PN + NB = PC + NB = r + J cos 0 ;// m ml cos 0 r-jcosfl r+^cos0 M cos 0 .I2 •% — when we neglect -3 219 -|2cos20 4 220 MAGNETISM M If 0 = 0 the point P is in the axis of the magnet and V = — . The intensity along the axis is dV 2M If 0 = 90°, i.e. if P is in a line through C perpendicular to the axis, the intensity at P parallel to the axis is 1 <*V Msinfl = M r dO " 7-3 ~ r3 d\ and is in the direction in which 0 increases. The component ^ along r vanishes, since cos 0 = 0. These are the values which we obtained in the last chapter for the end-on and broadside-on positions. the values on which (iaiiss founded his proof of the inverse square law. Vector resolution of a small magnet. It M be re- garded as a vector, drawn in the direction of the axis the potential -^— at a point P is the potential of the resolute of M in the direction of r, and if M be resolved like a force into any number of components, Mr M2, &c., making 0r 02 &c -. with ;•. the potential of the components will be M! cos Q1 + M2 cos fl, 4- . . . + M2cos^2-f . . . _ since the resolved part of M equals the smn of the resolved parts of its components. Hence the potential of M equals the sum of the potentials of'iK vector components. This result follows also from the consideration that a magnet AB (Fig. !()()) may be replaced by two magnets AC, CH. \\itli equal pcles. For the two magnets HC. CA have equal and opposite poles at C, neutralising each other, and are equivalent therefore to m at A and — ;// at B. The direction and magnitude of the intensity at any point in the field of a small magnet. Let AB(Fig. Hi7) be tin- magnet with moment M ; resolve it into 1)K along CP with moment M cos 0, and FG perpendicular to CP with moment M sjn 0. The intensity due to I)K i^ along CP and i " ACP = 0 and CP = r. SOME MAGNETIC FIELDS „,,, . . ., , „_ . ,. , , . Msin0 Ihe intensity due to rG is perpendicular to /• and is § — • The resultant is ^-/y/4 cos2 0 + sin2 0. The direction makes an angle with CP given by 'sin 0 1 tan 0 = ^ ^ = r, tan 0. 2 cos 0 2 FIG. 100. FIG. 167. The following construction gives the direction of the intensity. Draw PN, Fig. 108, perpendicular to CP, meeting the axis produced FIG. 168. in N: complete the rectangle CPNQ. Bisect CQ in T. Then TP is the direction of the intensity at P. For 222 MAGNETISM CT 1 CQ 1 NP 1 . tan ( = CP = 2CF=2CP = 2bl'ia Either from an easy geometrical construction or from the trigo- nometry of the two triangles CPM CPN it can be seen that CM = oCN. Hence the following somewhat simpler construction: Draw PN perpendicular to CP, meeting the axis in N ; trisect CN in M. Then MP is the direction of the intensity at P. We may easily work out the force on one small magnet due, to another, for particular cases. For these we refer the reader to J. J. Thomson's Elements of Electricity and Mu<*neti.fm, chap, vi, as we shall not need them in what follows. Uniform magnetic shell. Imagine a very thin steel or iron plate of uniform thickness bent into a surface of any form. The plate will of course be bounded by a closed curve. Let the plate be magnetised uniformly at each point and normal to the surface. This is equivalent to the supposition that it is built up of equal small magnets, placed side by side, the axis of each perpendicular to the surface of the plate ; or it is equivalent to the supposition that a layer of North magnetism of uniform densitv i> spread over one face of the plate and that a lavcr of South magnetism of equal uniform density is spread over tin- other fare. This system is termed a uniform magnetic shell. If & is the uniform surface density and t the uniform thickness, a-t is the moment of the plate per unit ana. The product rrt is termed the "strength "of the shell We shall denote this strength by 0. Though we ne\er meet with a uniform magnetic shell in practice, the conception is very useful, since the magnetic field, for \\hicli \\r can find a simple- explosion, is. the IM(I- lb<), represent a cylindrical element of the shell with cm --Action «. [U poles are ± cur, and its moment is ao-t = a(f). The potential at P of the element is : art cos 0 _ „ cos 0 ~^~ =*— If dw is the solid angle subtended by a at P, du = \. projection of a 011 a plane perpendicular to AP = q °^S ^. Then the potential SOME MAGNETIC FIELDS due to the element is , where w is the solid angle subtended at P by the rim of the shell. At an infinite distance the angle subtended is zero and the potential is therefore zero. If we bring a unit NSP from an infinite distance to the point on the positive side of the shell where the angle subtended is w, then we do more work against the near side than is done for us by the more remote side and the potential is -f <£w. If we approach from an infinite distance to a point on the negative side where — w is subtended the potential is evidently — <£w. The potential depends only on the strength and the angle subtended by the rim. Hence all shells of the same strength and the same contour have the same external magnetic field. Difference of potential of two neighbouring points, one on each side of the shell. Let PQ, Fig. 170«, be two . 170«. Fio. 1706. neighbouring points : P on the + side, Q on the — side. Let us sup- pose them so near that they subtend the same solid angle w at the rim. In reckoning the potential, if P subtends o>, Q must be con- sidered to subtend 4?r — w. For take a point o, Fig. 1706, and draw a sphere of unit radius with o as centre. Let a line from P5 Fig. ITOrt, trace out the rim A BCD of the shell and let a parallel line from o, Fig. 1706, trace out the curve abed on the sphere. The area abode represents w. And as the pole moves from an infinite distance on the 4- side to P the angle increases from zero to w and the potential at P is rf)tt). Now let the pole move from an infinite distance on the — side to P. The angle subtended increases from zero to abcdf = whole area of sphere — abcde : = 4>7T — W and the potential at Q is — ^ (4-Tr — a>), Hence potential at P — potential at Q = 4-7T0 MAGNETISM or the difference is the .same at all pairs of points one on each side of the shell. Potential of pole m due to a shell. Since the potential at any point of the unit pole is w, the potential of pole m is (/>mw. We have seen that pole m sends out 4>7rm lines of force or unit tubes through the whole sol id angle 4?r surrounding it ; therefore through solid angle w it sends out 0?/iw, or its potential is x number of lines of force it sends through the closed curve forming the rim of the shell. Conversely the potential of the shell with respect to the pole is (j) x number of lines from the pole passing through the shell. It is easy to see that if lines of force which enter the shell on the positive side are reckoned positive, and if those which enter on the negative side are reckoned negative, the number will be tlu- same. whatever the form of the shell, so long as it has the same bounding curve. For FIG. 171. let ABCDE and A IT, ( Fig. 1 71 ) repre- sent two forms of the shell. A line ( 'H which passes through the first shell at C in one direction and at It in the other direction will not count in the sum ; while a line 1)F which passes only once through the first shell miM pass through the second shell also, and will count equally for each. Potential^of a shell in any field/ If we regard the field as made by a distribution of polarity, the potential of a shell will be equal to X ^ lines sent through, summed up for e\cr\ element of polarity, or 0xN, where N is the total number of lino sent through. Force on a shell in any direction. Let a shell l>c displaced parallel to itself through small distance < ./ in the gi\en direction. Let the rate at which N. the total number of lino of force through the shell, increases in the given direction be Then the number pa»ing through in the displaced position N d^ N -f y-.cM' and the potential has been increased bv This is the work which must be done in the displacement. There is therefore a force resisting the displacement equal to 0 ~ or the force tends to decrease N, and in the positive direction of oV is SOME MAGNETIC FIELDS The shell, therefore, tends to move into the position in which as many lines of force as possible pass through in the negative direction. This can easily be remembered if it is considered that it will tend to set with its negative face towards a positive pole. The resultant force on the shell evidently acts in the direction in which a given motion produces the maximum decrease in N. Representation of the force by elementary forces each acting on an element of the rim of the shell. The value of N can alter only by lines moving in or out across the boundary curve of the shell as it is displaced. This is seen at once if we imagine the lines of force to be materialised into wires. The number can only be altered by the rim cutting through the wires. We may expect then that the force on the whole shell can be represented by forces acting at the rim where the cutting takes place. Let us suppose that the force on an element $s of the boundary in direction &r is — 0?*, where n is the number of lines cut by Ss per unit displacement in the direction &x. Then the total force is — 0Zra = — 0 -j-, since 2n must be equal to -^—. Then we may regard the total force — -=- as the resultant of forces such as (uC — 0 n on the elements of the boundary. The direction in which the force on each element acts is that in which a given motion will make n the greatest. If H, Fig. 172, represents the field at an element Ss, it is evident that n is the greatest when &r is perpendicular to H and to Ss. Then H * F FIG. 172. the direction of the resultant force F on the element is along the normal to the plane through Ss and H. The number of lines of force cut per unit motion of Ss along F is H sin 9Ss, where 6 is the angle between H and Ss. Then F = 0H sin BSs. We shall see when we study the magnetic field of a current that the current equivalent to a shell circulates counter-clockwise round the rim when seen from the side towards which the positive layer faces. Hence in the figure we may assign to ^considered as an element of the current, the direction inwards through the paper wheii the shell lies above Ss with the positive layer to the right. The force MAGNETISM on Ss tends to lessen the positive lines of force through the shell, that is, Ss tends to move upwards when H is as drawn. If we represent current by C, field by H, then force is in the direction in which a screw would move if its head were turned round from C to H, where C and H both act towards or both form the point of intersection. The rule may be remembered bv noting the alphabetical order of CtoH. Field due to a uniformly magnetised sphere. Let us suppose that a sphere consists of small parallel bars or fibres of equal cross-section a and with equal poles ± la. Then it is said to be uniformly magnetised with intensity I in the direction of the fibres. Let PQ, Fig. 173, represent the end of one of the fibres, and let the radius OP make 0 with the direction of magnetisation. The polarity at PQ is Ia. The area of the end of the fibre on the sphere is ^— , and if the surface density on that area is ^ <>-M ^ cm: But this difficulty has to be fa.ced in considering the phenomena <»f radiation. Radiant energy passes at very nearly the same speed through air and through the highest vacuum \\hich we can create, and it passes through the vacuum, as far as we yet know, without any loss of energy. There is something, then, in the vacuum which can take up and propagate radiant energy and its accompanying momentum, and nearly in the same \\ay as if air were present. When we speak of a field in air, we must not, then, imply that tin- air is the seat of all the action, the storer of all the cncrgv. Doubtless it takes some share in the action, but the medium which pervades a so-called vacuum is no doubt present, and in all proba- bility takes its share in the action. Induction and intensity in air. When magnetisation of iron takes place by induction, we symbolise it by supposing that lines of induction pass through the iron, and we say that induction goes on within it. We may think of the induction within as implying a condition in the iron similar to that which exists in the neighbouring air outside, but in general the action, whatever it is, will be greater. The term induction is therefore extended to denote the alteration existing in the air, as well as that in iron. In the air, or in any non-magnetic medium, we take the intensity H, the force on a unit pole, to represent the induction, both in magnitude and direction. In doing this we make no hypothesis as to the nature of the induction. All that we imply is that the inductions at two points P and Q in air will differ from each other in magnitude and direction in the same way as the intensities at P and Q, and we so choose the unit of induction INDUCED MAGNETISATION 231 that if the intensity at a point in the non-magnetic medium is H and the induction at the same point is B, then B = H. Since the induction and the intensity are thus made to coincide in air, the only medium in which we directly measure H, the employment of the two terms appears at first sight needless. But we shall see directly that we are led to assign to them different values in magnetic media, and in permanent magnets and in crystalline media they may even have different directions. They imply, too, even in air, different modes of viewing the action. "Intensity" is the result of experiment. We suppose that we place a unit test pole at the point considered and measure the mechanical force upon it. It is the measure of the action of the medium on something, as it were, outside itself, viz. the test pole. It corresponds in fact to stress in ordinary elasticity. " Induction " implies that there is an alteration in the medium and corresponds to strain in elasticity. In elasticity we have stress = modulus x strain, and when we know the modulus we can determine the strain by measuring the stress. The magnetic equation B — H implies units so arranged that the magnetic quantity corresponding to the modulus is unity in air. We may here follow the analogy with elasticity one step further. The energy per unit volume in a strained body is proportional to stress x strain. In the magnetic field, then, we might expect it to be proportional to the product HB, or, since we make B = H, proportional to H2. We shall see later in the chapter that there is some reason to suppose that it is H2/8?r. Lines and tubes of induction in air. Lines and tubes of force in air may now be described as lines and tubes of induction, and along a given tube induction x cross-section = constant. The total flux of induction through any closed surface, the surface being entirely in air, will be zero whether it encloses magnetised matter or not. For if it does enclose it the two polarities are equal and opposite, and the total polarity within the surface is zero. Induction within a magnetised body. Let us suppose that an iron sphere without permanent magnetism is placed in a field in air which before its introduction was uniform, i.e. was such that the lines of induction were straight and parallel. We take the case of a sphere, since on a certain supposition, as we shall see later (p. 242), the course of the lines of force within the sphere can be determined, but the general ideas we gain from the sphere will apply for any form. The course of the lines of induction in the air will be altered by the introduction of the iron as indicated in MAGNETISM Fig. 177, the lines being drawn in towards the sphere. The figure represents those on a diametrical plane parallel to the original direction of the field. The total polarity over one hemisphere is equal and opposite to the total polarity over the other hemisphere. FIG. 177. so that as many lines of induction converge on to one hemisphere as diverge from the other. Now imagine that the >pheiv is cut through in a plane perpen- dicular to the central line of induction and that the two halves are drawn a very small distance apart. Then we shall have tubes of induction crossing the air gap thus formed, and as many tubes will cross the gap as enter the curved surface of one hemisphere and leave that of the other. For we can draw a closed surface entirely in air closely surrounding either hemi>phere, and the total flux through such a surface is, as we have seen, /.em, or the INDUCED MAGNETISATION 233 flux across the gap must be equal and opposite to the flux over the curved hemispherical surface. All this may be regarded as experimental. We now make the fundamental supposition that the lines and tubes of induction are continued through the iron, that the tubes crossing the air gap are the continuations of those in the air outside, and that along each tube, induction x cross-section is constant, and has the same value FIG. 179. as the same product for the same tube outside the iron. We shall see later that on a certain supposition the tubes will go straight through the iron, or that the induction within it is uniform. We may now close up the gap, having only used it to emphasise the idea that the induction runs through the iron, and we have the lines of induction as in Fig. 179, and along any tube induction x cross- section = constant whether it is in iron or air. Let us now, Fig. 180, FIG. 180. 234 MAGNETISM draw two level surfaces, or surfaces cutting the lines of induction at right angles, one through Pclose to the point at which the central line enters the iron, and the other through Q close to the point at which it leaves the iron. It is evident that these surfaces must bend, the one downwards, the other upwards, as indicated in the figure, coining nearer together far out at KS, where the field is practically un- affected by the iron, than they are at PQ, close to the iron. Now, we have no way of experimentally measuring the intensity H within the iron. But we make the supposition that it is such that in passing from one level surface to another, intensity x path is constant for all paths whether in air or iron ; or, if H is not uniform, then if ds is an element of path, fHds is the same for all paths from one level surface to another. Since, then, the level surfaces are further apart at PQ than they are in distant parts of the field, H is less within the iron than in those distant parts, i.e. less than it was before the iron was introduced. But just outside the sphere at P or Q the tubes of force or induction have narrowed, in converging on to the iron, and so H is greater there than it was before the iron was introduced. Then H in air at P is greater than H in iron close to P. But B in air at P is equal to B in iron close to P, and since B in air is equal to H in air, B in iron close to P is greater than H in iron at the same point. Let us put in the iron ju. is termed the permeability of the iron. If we put H = - B and compare with the elasticity relation .xtre>s — modulus x strain we see that 1/yu corresponds to the modulus of elasticity. It represents as it were the dif lie-nit v, while /m represents the ra^e. of magnetisation. To sum up. We have in air B = H, where H can be experi- mentally determined, and therefore B can be determined. In iron we can suppose that B is determinate from the continuity of the induction flux along a tube as it passes from air into iron. We can also suppose that H is determinate from the constancy of potential difference as we pass from one level surface to another, whether through air or through iron. We have seen that in iron B must be greater than H, and its ratio to H is termed the permea- bility of the iron. We have considered only the case in which the magnetisation is produced by induction, and have disregarded the possibilitv of permanent magnetism in the iron. If there is permanent magnetism we can no longer assume that fHds between two level surfaces is the same through air as through iron. INDUCED MAGNETISATION 235 Hydraulic illustration. A hydraulic illustration may help us to realise the meaning of magnetic permeability. Imagine that ABCD, Fig. 181, is the cross-section of a slab of spongy or porous material indefinitely extended to right and left, and that a liquid which entirely fills the pores is being forced through it by a pressure across AB in excess of that across CD. Then the liquid will flow with a velocity from AB to CD such that the viscous resistance balances the difference of pressure. On the average the flow will be straight across from AB to CD and the pressure will fall uniformly from one face to the other. This corresponds to a uniform magnetic field. The pressure corresponds to the potential, and its slope, or fall per centimetre, to the intensity. The velocity will be O FIG. 181. proportional to the pressure-slope and will correspond to induction. Evidently, if we draw a tube of flow, velocity x cross-section, or total flow, will be the same throughout its length. Now imagine a portion of the slab — represented as spherical in the figure — made more porous, and therefore less resisting, than the rest. Evidently the liquid will flow more rapidly through this portion. The lines of flow will converge on to the end nearer AB, and diverge from that nearer CD. The surfaces of equal pressure will be deformed just as are the magnetic level surfaces in Fig. 180. The total flow along a tube will be the same whether it is within or without the larger-pored space occupied by the sphere, and the pressure-slope will be less within than without that space. We could replace the sphere with its larger pores by one having pores equal to those outside it, if we supposed that where the tubes of flow enter the sphere, there are "sinks" or points at which fluid is removed somehow from the system, the amount removed being the excess of fluid coming up over that which the given pressure- slope will drive through the sphere on the supposition of pores equal in size to those outside. On the surface of the other hemisphere, where the fluid emerges, we must suppose that there are " sources " or points at which fluid is somehow introduced into the system to make up the excess of outward flow over the flow through the sphere. The " sources " must introduce just as much as the " sinks " remove. This mode of representation corresponds to the representation of the magnetic induction by polarity distributed over the surface and it brings to the front the artificial nature of that representation. Though we are not here considering permanent magnetism, we 236 MAGNETISM may point out what would correspond to a permanent magnet in the fluid illustration. A steel magnet in air would correspond to a larger pored legion, and in the pores we should have to imagine small turbines fixed, driving the liquid in a given direction. The turbines would represent the magnetic molecules, and each turbine would have to produce a flow proportional to the strength of the molecular pole. As much fluid would be drawn in at one end of the region as was driven out at the other, and the fluid would circulate back through the surrounding space, the lines of flow rc-pn-s<-nting the lines of force. Equations expressing continuity of potential and continuity of induction tubes. We may conveniently term the condition that the difference of potential between two points U the same, whether we pass from one to the other through air or through iron, the condition of continuity of potential, for it implies that the potential is the same at two points quite close together, one in air, the other in iron, and we may put the condition into the following form : I^et PQ be two neighbouring points on the surface of the iron. Let Hj be the intensity in the air close to PQ and let its din (lion make 0j with the normal to the surface at 1\>. linn II, sin t\ is the component of the intensity parallel to the surface in tin- air. Let H2 be the intensity in the iron close to PQ and let its direction make #2 with the normal. Then Ht sin 6t is the component of the intensity parallel to the surface in the iron. In order that the difference of Fie. 182. potential between Pand <<) shall be the same in air and in iron, these com- ponents must be the same in direction and equal in magnitude. The first condition implies that Hl and Ht are in one plane with the normal, while the second requires that II, sin flj = II2sin 0r We may conveniently term the condition that the flux of induction across a tube is constant, the condition of continuity of induction tubes, and we may put it in the following form : Let Fig. 183 represent the section of a tube of induction enter- ing the iron at PQ by the plane through the normal QM and the line of induction in the air through Q. It passes through the line of induction in the iron also, if we assume that the directions of induction and intensity in the iron coincide, as they do in air. Let #1 02 be the angles which the direction of the induction makes with the normal in the two media. Let QR PS represent cross- sections of the tube in the two media, the breadth of the tube perpendicular to the paper being the same in each. The con- INDUCED MAGNETISATION 237 tinuity of induction gives Bx . QR = B2 . PS, where Bx is the induction in air, B2 that in iron ; or ^ . PQ cos 01 = B2 . PQ cos 02 whence Bx cos 0I = B2 cos 02. Thus while the components of intensity parallel to the surface are M equal, the components of induction perpendicular to the surface are equal. If instead of air and iron we have any two media with permeabilities ^ and yit2, intensities Hj^ H2, and inductions Bx B2, we still have H! sin 0j_ = H2 sin 6% Bj cos 01 = B2 cos 62. Putting Bj = /XjH^ and B2 = /x2H2, the equation of continuity of induction becomes j cos 01 = //2H2 cos 02 and dividing the two sides of the equation of continuity of potential by these we get tan tan 0. 2. We may term this the law of refraction of magnetic tubes. It applies of course only to cay* in which the conditions of continuity hold. 238 MAGNETISM When we compare the foregoing investigations with the corresponding investigations in electricity we see that magnetic intensity corresponds to electric intensity, and magnetic induction corresponds to electric strain. The magnetic equation corresponds to the electric equation so that /UL corresponds to K/4-7T. The introduction of 4?r in one of these is very unfortunate, for it destroys the exact correspondence of formulae in the two CUM ->. It arises from the fact that the units are so chosen that unit quantity of electricity sends out one tube of strain, while unit quantity of magnetism sends out 4?r tubes of induction. Ileavisidr* has introduced a system of "rational units " in which unit magnetic pole, like unit quantity of electricity, sends out one tube. But tilt- difficulty in the way of change in the practical units has not yet been faced. Representation of induced magnetism by an imaginary distribution of magnetic poles acting ac- cording to the inverse square law. We have seen that the intensity changes when we pass from air into iron, or more generally when we pass across a surface at which the permeability changes. For purposes of calculation we can imagine the ch, of field to be produced by a layer of magnetism, a layer of poles distributed over any surface where /u changes from one con- stant value to another, or distributed through the volume if the change in /UL is gradual. These poles are supposed to act according to the inverse square law and as if the whole space \\ere of unit permeability. Let us consider a change of per- meability at a surface PQ, Fig. 184, from 1 to /z, and let the figure represent the section of a tube of induction by a plane containing Hx and H2, and let the tube stamp out an area a on the surface, represented in section by \\). FlG- 184- Let ^ section of the bar,./ its length. The force pulling the bar down is J 0 0 The surface force Sira* per area 1 may be neglected since K is small. As an illustration of a case in which this surface force is not FIG. 200. negligible, let us suppose that two iron bars of constant permeability are magnetised by coils near their neighbouring ends and placed as in Fig. 200, with unlike poles near each other. Let the bars be so FORCES ON MAGNETISED BODIES 257 long that the intensities at the remote ends are negligible. We shall assume that the intensity near the neighbouring ends is parallel to the axis. As above, the volume force gives us a pull -/cH22 where H2is the intensity just inside the iron. The surface pull 2-7r<72 gives us 27nc2H./. The total pull is therefore 8-7T/X where Hj is the intensity in the air in the gap. The assumptions of constant permeability and of field parallel to the axis in the bar near the facing ends are only very rough approximations to the truth at the best, so that the result obtained has no real value except as an illustration of the formulae. Forces on a body of permeability /x2 placed in a field in a medium of permeability ^r The field is altered by the introduction of the body just as it would be by the introduction of a layer of magnetism of surface density and acting everywhere according to the law m/far2 (p. 251). The forces due to the body, and therefore the reactions on the body, can be calculated by replacing the body by this layer and finding the forces upon it. The forces on the surface layer a- may be obtained as on p. 254, remembering that now the intensity due to cr is 27r0-/Mi> and that this is now the value of DH or HC in Fig. 197. The intensity H acting on the surface layer may be regarded as the resultant of 27nr/yu1 along the normal and of H2. The former gives a tension along the normal 27rc72//x1 = 27T(*2 - *i)2H22 cos2 0a//*r The latter may be treated as on p. 255, remembering that now I = (/c2 — A^H. The volume force is then found to be 1 2("2 ~ Kl)~3Z hich may be represented as due to a pressure system In para- and diamagnetic bodies in which AC is very small the tension on the surface is negligible. Faraday's experimental 258 MAGNETISM observations that a paramagnetic body in a more paramagnetic medium behaves as if it were diamagnetic is at once explained by 7TT2 the expression for the volume force, -2— — -=— . For suppose that the body is first in air : the force on unit volume in direction x is ^- —T—> where -=— may be determined from the value of the field before the introduction of the body, since when K is small the intensity of the field is not sensibly altered by its introduction. Now surround the body by a medium with susceptibility K^ and the force changes to AT2 - KI dR* 2 dx and if ^ is greater than KZ the force changes sign. Forces on an elongated bar in a uniform field in air when its permeability differs very slightly from 1. If we neglect the outward tension proportional to /c2 the other term in the surface force is H2 tend to set with their longer axes along the lines of force. This result is important, for it implies that the equatorial setting of small diamagnetic bars placed midway between the poles of a magnet i> not due to a tendency to set transverse to the lines of force, but to a tendency of the material to get into a weaker field. By setting transversely, the two ends of the bar get into the weakest fields available, for the field diminishes in the equatorial plane as we move out from the centre. To understand the longitudinal setting in a uniform field let us imagine a bar of square section to be made of separate cubes placed end to end. First let the body be paramagnetic and let it be held longitudinally in the field. Each cube will tend to increase the magnetisation of its neigh bours, just as each of a series of magnets tends to increase the magnetisation when they are placed end to end. Thus the polarity at the ends is increased. Next let the bar be placed transversely to the field ; then each cube will tend to weaken the magnetisation of its neighbours, just as each of a mumber of magnets set side by side tends to weaken its neighbours. * Papers on Electrostatics and Magnetism, Sect. 691. FORCES ON MAGNETISED BODIES 259 FIG. 201. Now let the bar be inclined at any angle to the field, as in Fig. 201. The mutual action of the cubes tends to increase the end magnetisation and to weaken the side magnetisation. Neglecting this mutual action, that is, neglecting the effect of the surface distribution on the intensity, the bar, as we have seen, will be in equilibrium — that is, the "centre of polarity11 of the end A and of the side B must be at c in the line of force OH through the centre O, Fig. 201 . But taking into account the increase of polarity at A and the decrease at B, the mag- netic axis will be thrown on to the A side of oc to cv say, and the force, acting through c will tend to pull the bar into the direction of the field. Now take a diamagnetic bar made up in the same way of cubes. If it is held longitudinally in the field, we represent its polarity by supposing that a NSP induces a NSP, while a SSP induces a SSP. At the plane of contact of two cubes where there are opposite polarities each therefore tends to weaken the other, and the net result is that the end magnetisation is de- creased. Now put the bar transversely. Since like induces like, each cube will tend to strengthen its neigh- bours and the side magnetisation is increased. If the bar is inclined at an angle to the field the mutual action of the cubes tends to increase the side magnetisation and to decrease the end magnetisation, and the centre of polarity is thrown towards B, say to cir This is a SSP, and the force due to the intensity H will push it. The moment round the centre O will tend to make the bar turn round parallel to the lines of force. The magnetic moment of a small paramagnetic or diamagnetic body placed in a magnetic field. If the field intensity is H before the body is introduced, that value is almost unaltered by its introduction when ju. — 1 is exceedingly small. The magnetisation at any point may therefore be taken as I = /cH. Any small cylinder of length / and cross-section a and with axis along H may be regarded as a magnet with poles ± la = /cHa, and with moment Mai = /cH X volume. The total magnetic moment will be the resultant of all such moments, and if the body is so small that H may be regarded as uniform in it, its moment will be /cH X volume. Force on the small body. We showed in Chapter XVIII that the force on a small magnet of moment M in direction x is— -^ — . If its volume is dv. this is equal to 260 MAGNETISM „ dH , 1 . M -j— dv = s K —j — dv. dx % dx If the body is large, the total force in direction x is ,v from which we may again draw the conclusion that if A: is positive the body tends to move towards the position where f}3Pdv is a maximum, and that if AC is negative it tends to move towards the position whereyH2^ is a minimum. Time of vibration of a small needle with M— 1 very small suspended in the centre of a field between the poles of a magnet, the field being symmetrical about the axis and the equatorial plane. Lord Kelvin has shown * that the time of vibration of a needle-shaped body depends only on its density and magnetic susceptibility, and not on the particular shape or size of the needle so long as it is a needle and small. This very remark- able result has led to a method of de- termining susceptibilities. First imagine a particle of volume V9 density p, and susceptibility K to be _ ___ somehow freed from weight and to be a: suspended, under the action of magnetic forces only, by a very fine fibre from the central point C of the field represented in Fig. 202. If paramagnetic, it will set at P with the fibre along CP ; if diamagnetic, at Q, with the fibre along CQ; and if dis- Fio. 202. turbed it will vibrate, forming as it were a magnetic pendulum. Let us take axes as in the figure. We may put the intensity near C in the form H* = Hc* - A,r> + For putting H as a function of x and y and expanding, first powers must disappear to satisfy symmetry. Powers higher than the second may be neglected if we are near enough to C. The signs of the terms with A and B as coefficients are determined by the fact that H increases from C towards P, while it decreases from C towards Q. 1 7H2 The magnetic force on the particle in any direction s is ^ KV —j—t Let us suppose it to be diamagnetic, so that its undisturbed * Electrostatics and Magnetism, Sect. 670, FORCES ON MAGNETISED BODIES position is Q. Let it be disturbed through a small angle 6 from CQ. The force along the arc ,?, the restoring force, is putting ds = ad9, where a is the length CQ of the fibre. But in the disturbed position x = a cos 0 and y = a sin 0, so that H2 = Hc2 - Aa2 cos2 0 + B«2 sin2 0 and - = 2a2(A + B) sin 0 cos 0 = a\A. + B) sin W = 2a2(A + B)0 since 0 is small. The acceleration on the particle is therefore _ pva dO and as we suppose K negative, this is towards Q. The vibration is therefore harmonic of period T — " + B) This is independent of a. If, then, a thin bar or needle is pivoted at C so that its weight need not be considered, every particle of it tends to vibrate in the same time, and it vibrates as a whole with the above period. If the bar is paramagnetic, it is easily seen that the sign of the denominator must be changed, and that the time is 2lVK(A + B) The experiments made by Rowland and Jacques, using this formula, are referred to in Chapter XXII. Stresses in the medium which will account for the forces on magnetic bodies. It is not so easy to work out a system of stresses in the surrounding medium to account for the observed forces on magnetised bodies as it is to find a system which will account for the forces on electrified bodies or for 262 MAGNETISM dielectrics in the electric field. In the electric field, for instance, between two parallel metal plates charged with ± cr the tension along the lines of force must be ^Trcr2 in order to account for the work done in separating the plates. But there is no exactly corresponding magnetic system, nothing like a charged conductor. Since the magnetic induction tubes are continuous, there are magnetic stresses on each side of the surface separating two media, and energy is distributed throughout the tube. We cannot make a real magnetic experiment like that with an attracted disc electrometer, where we weigh the pull by the dielectric on the surface. Even the electric system is indeterminate, for we can hardly argue conclusively from the pulls in a uniform field to those in a field in which the level surfaces are curved. The tensions and the pressures keeping an element in equilibrium might depend on the curvature of the lines of force. But we can at least show that the electric stress sy»tem devised by Maxwell holds with a uniform field, and can further show that it gives force -s which agree with the actual forces on bodies. We have no satisfactory starting-point for a magnetic system ofstres8e8,aodit is probably In M to be guided by analogy with the electric field, remembering again that the problem is really indeterminate and that the >y«*trm assumed is only one of an infinite number of possible solutions. Maxwell* proposed a magnetic stress system which in a medium of perme- H2 ability 1 gives a tension — along the lines of force and an e\ >tem of tension and pressure, each equal to H2/8-7r, would keep each element of a homogeneous medium in equilibrium, and it can be shown that it would account for the surface forces on a magnetised bodv in air, that is, for the forces on the surfaces of discontinuity. But the additional tension /cH2 within the body would only form a system in equilibrium in a uniform field. We shall consider in place of Maxwell's system one analogous to the electric system, and shall therefore assume that there is in a homogeneous isotropic body a TT2 TJ2 tension ^ — along the lines of force and an equal pressure ^1— perpendicular to them. * Electricity and Magnetism, ii. Sect. 642. The stresses in the medium are discussed at length in Walker's Aberration and the Electro-magnetic Field, and in Heaviside's Electrical Papers, vol. ii. FORCES ON MAGNETISED BODIES 263 This system, like the electric system, will maintain each element of a body in which ^ is constant in equilibrium. Tne forces acting on the two sides of an element of surface where /UL changes will not be in equilibrium, and their resultant ought to be equal to the mechanical force acting on the surface, equal, in fact, to the forces deduced by supposing that the imaginary surface layer is acted on by the field existing about it. We shall find that the two systems are not equivalent, element by element, but that they give the same resultant force on the body as a whole. Perhaps this is as much as could be expected. The tensions and pressures — are in equilibrium on any ele- ment of a medium in which JUL is constant, exactly as the tensions and KE2 pressures -^ — are in equilibrium on an element of dielectric. Any O7T portion of a medium with /* constant is therefore in equilibrium, and if we draw a closed surface S wholly within the medium, the tensions and pressures acting on S from outside form a system in equilibrium. Equally the reactions from inside on the medium outside S are in equilibrium. If we imagine the medium within S to have the same distribution of induction, but some other constant value of ft, we can superpose the stresses due to this on the actual system without destroying equilibrium. Obviously we need not even have the distribution of induction the same in the superposed system so long as it is a possible distribution obeying the tube law, but we shall only require the more limited case. The stresses on an element of surface separating media of permeabilities ^ and /z2. Let the plane of Fig. 203 be that through the normal to the surface and the axis of a tube in the /^ medium, meeting the surface in an element d$ represented by its trace AB. Let the tube make 01 with the normal. Let the tube be rectangular in section with two sides parallel to the plane of the figure, and consider the equilibrium of the wedge ABC. The forces across AC and BC must give the force across AB. Let N be the normal force outwards across AB and T the tangential force in the direction AB. Since BC = dS cos 0 and AC = dS sin 0, we have, on resolving the assumed tension and pressure, ^i2 • dS cos2 0! - OTT or N = co,2 0 - sin" 0) = l cos 264 MAGNETISM Similarl T = iL cos 6, sin 01 + sin ^ cos Ol 07T sin The resultant of N and T is a tension on unit area of the surface in a direction making an angle Wl with the normal. Tin- resultant of the system N and T over the bodv >hould gi\i- tin- actual force. But we may add to this system the system due to the internal stresses, for these form a system in equilibrium. \Ve may take these internal stresses either of their actual value, — — , or, as N oTT tt II 2 more convenient, of value —g-*~* '•<• Ut> >ii|><-r|><>^- on the actua . - //2H22 II system -g-- a system (MI-MJ)^-' When the internal system i^ superposed \\ • N = ^i-2 (cos2 01 - sin* 0,) - &&L(c**t 0, - sin2 f)t) O7T O7T ,2 COS2 0* - MlH.* COS1 St . * - * ^l * - ", since IIj sin H1 = !!„ sin vn OTT _ Ht cos ei + Ht cos 0, Ht cos Ql - H, cos 8t 2 '^ " 47T H -4- 1*1 = Normal component of - i-^ — * X surface density f the highest importance at A guide to dynamo construction. In this chapter we shall uri\r an account of the principles upon which are based *onu- of tin- methods used to investigate the relation between H and II for iron and steel, and we shall indicate the general nature of tlic rcsulK* It \\ill be necessary to assume some knowledge of electro-magnet i-m. Magnetometer method. In tin- m near a magnetic needle —the magnetometer — and from the deflection of this needle the magnetisation of the rod is calculated. A verv convenient arrangement, due to Ewing, is represented in Fig. 204. * The reader will liml th.» details <>f tin- methods and th*-ir ndapta: t«-cluiir:il nrtMls ill Kwinp's M,i>in,ii,- Indurtnn, in Iron and t>(! • MEASUREMENT OF PERMEABILITY 267 A represents the solenoid placed vertical. B is the magneto- meter nearly level with the upper end of A, and at a distance from it so short that the action on the needle is almost entirely due to the upper pole. The rod must be adjusted to the height at which the action is a maximum. The magnetising current through the solenoid is supplied by the cells on the right, and on its way to the amperemeter G and the commutator F it passes through the adjustable liquid resistance H, which consists of a tall glass jar with three discs of zinc immersed in a dilute solution of zinc sulphate. It will be seen from the figure that the circuit from one terminal of the battery passes through the top disc on its way to G and the solenoid, while the other terminal is connected by an insulated wire to the bottom disc. The other end of the solenoid is connected to the middle disc, which is movable up and down. If the middle disc touches the top one, the current flows through the jar and none goes through the solenoid. As the middle disc is lowered the current through the solenoid increases and is a maximum when the middle disc touches the bottom one. On its way to the solenoid the current passes through the " compensating coil " E, which is adjusted in such a position that when there is no rod within the solenoid the action of E on the magnetometer is just equal and opposite to that of the solenoid. The magnetic action of the solenoid itself may therefore be left out of account. In order that the earth's field may be eliminated a second solenoid is wound round the first and a constant current from a cell C is passed through it and through a resistance D so adjusted that the field due to the constant current within this second solenoid is equal and opposite to the earth's vertical field. When a specimen is to be tested it is necessary in the first place to demagnetise it. This is effected by the " method of reversals." The rod is placed within the solenoid and the full current is put on by lowering the middle plate in the liquid resistance to contact with that at the bottom. The commutator F is then rapidly revolved, so as to alternate the field within the solenoid. While the rapid alternation is going on the middle plate is slowly raised through the liquid to the top so that the amplitude of the alternating current slowly decreases from its largest value to zero, and the rod is found to be completely demagnetised by the process. This method serves also to adjust the current through the outer solenoid to the value required to neutralise the earth's field, for the reversals only completely demagnetise the rod when the field is finally reduced to zero. It is useful to employ for this adjustment a special rod of annealed soft iron. The specimen having been demagnetised, let a current of A amperes be established in the solenoid. If there are n turns per centimetre the field due to it is H' = 4>TrnA/lO. 268 MAGNETISM If I is the intensity of magnetisation of the rod and TTQ* is its cross-section, its pole strength is -rra2!. Let QQ', Fig. 205, repre-« -n t the two poles of the rod,o being the magnetometer, the hori/ontai component of the field in which it is placed being perpendicular to the plane of the figure, and being of inten-itv !•'. If 0 is the deflection of the needle, then -i J) and l = d* F tan 0/ira* (l - |^ This gives I in terms of measurable quantities. Since B = /zH = (l+4™)H = H + 4x1, if ire know H and I we find B and /x. The value of H' due to the mag- netising current i> not, however. the actual value of the field, for the poles tunned in the xprcinien produce Fio. 20t. an opposing field, and SOUR- assumption is needed to allow for the reduction due to this. If the, specimen is very thin with diameter, say, 1/400 of the length, the reduction -3-2-101*3 to a 12 a i+ u 16 n it 19 •: FIG. 206. is very small. Swing supposes that such a specimen may be treated as an elongated ellipsoid and finds that the actual field is H = H'-0 00045 I. MEASUREMENT OF PERMEABILITY 269 The following Table gives particulars of a test made by Ewing on an annealed wrought-iron wire of diameter 0-077 cm. and length 30'5 cm. or 400 diameters. The current was increased step by step from 0 to a value giving a field of 22*27 gausses and was then reduced by steps through 0 to —2*87 gausses. The continuous curves in Fig. 206 (taken from Ewing) show the values of I as ordinates plotted against H' as abscissae, the lower curve for increas- ing current and the upper one for decreasing current. Field doe to mag- netising current H'=4W»A/10 Intensity of magnetisation I Corrected field H=H'- 451/108 Magnetic BUS ceptibility K Magnetic induction B = H + 4,rI Magnetic permeabilit j^B/H o-o 0 o-o _ 0 0-32 8 0-32 9 40 120 0-85 19 0-84 15 170 200 1-38 33 1-37 24 420 310 2-18 93 2-14 43 1,170 550 2-80 295 2-67 110 3,710 1390 3-50 581 3-24 179 7,300 2250 4-21 793 3-89 204 9,970 2560 4-92 926 4-50 206 11,640 2590 1009 5-17 195 12,680 2450 6-69 1086 6-20 175 13,6^0 2200 8-46 1155 7-94 145 14,510 1830 10-23 1192 9-79 122 14,980 1530 12-11 1212 11-57 105 15,230 1320 16*61 1238 15-06 82 15,570 1030 1255 19-76 64 15,780 800 22-27 1262 21-70 58 15,870 730 16-42 1258 — — — 8-46 124.'» — — — 4-92 1235 — — — 3-15 1225 — — 1-38 1205 — — — o-o 1162 — — — - 0-41 1186 — — - 0-81 1092 — — - 1-10 1056 — — . - I-}:. 979 — — — — - 1-80 840 - 2-20 551 — — — - 2-51 232 - 2-87 -40 — — — — The figure shows how the correction from H' to H may be made graphically. Since the correction is —0-00045 I, draw a >tr,-iitrc^c> on magnetisation, .since the wire tested can be subjected to end pulls or to Uists while within the solenoid. It serves well, too, to give a general idea of the results obtained on subjecting a .specimen to a magnetising force and on carrying the magnetising force through a cycle. ina>- much as it is easy to work and the theory i- simple and L fairly good results when the .specimen trxted i> UTV thin compared with its length. By it \\<- ma\ obtain the curve alnad\ Ini described in Chapter \I\. p. 1S:J, and represented in I _ W7, Fi<;. 207. It is interesting to note that in the ti st represented b\ the residual magnetisation when the current is reduced to < still 1162, or 92 per cent, of the value it attained \\ith the increasing current. AVhen the current is reversed the magnetisa- tion does not disappear till the field H= —2*75 gausses, and this is the value of the "coercive force" for the specimen. It is to l>e remembered that though the external eflict of the magnetisation disappears under this coercive force, the wire is not in the de- magnetised condition from which it started. It is still affected by MEASUREMENT OF PERMEABILITY the process through which it has been carried. For if the current in the reverse direction is further increased the curve does not descend as it originally ascended from O, but is much steeper. The Ballistic method. This was first devised by Weber and has since been used in many important researches. We shall describe its application to a ring-shaped specimen of the metal to be tested. Round the ring is wrapped a coil of wire A, Fig. 208, and in the circuit is a battery, an amperemeter Gx, an adjustable resistance B1? and a commutator K. If r is the mean FIG. 208. radius of the ring and if the number of turns of the coil is N per cm. length of the circle of radius r, a current A in the coil gives a field of intensity H = 4?rN A /1 0 at the distance?-. We shall suppose the ring so thin that H is constant across a section, and of this value. Round one part of the ring a secondary coil I of n turns is wrapped. In the circuit of this coil is a ballistic galvanometer G2, an adjustable resistance B2, an " earth coil" E, and a small coil D, which can be used to reduce the swings of the needle G2. The earth coil is used to find the total quantity of electricity passing through G2 for a given throw. SuppoM.1 that E is laid on a horizontal table. If the sum of the areas of its turns is S and the intensity of the earth's vertical field is V, the total fiux of induction through it is VS. If E is suddenly turned over through 180°, the change in induction is X \ S, and if R is the total resistance of the circuit the total flow 2 V S of electricity is • coulombs. 10 R MAGNETISM If the throw due to this is 6 any other throw 0 will indicate flow of 10 KG coulombs. In carrying out a test on a ring it is first demagnetised by reversals, as already described in the magnetometer method. Thru a small current A1 is established in the coil A,prodocing within it afield H1 = 47rNA1/10. If this creates induction \\1 through tin- secondary coil the flow of electricity in the ballistic galvanometer will be -rjr-i and if the throw is Ol It /iB, TT whence . Unless the primary coil A is very close in every turn to the iron a correction must fee subtracted from Bj for the induction through the space between wire and iron. If the mean area of the coil i> S'and the cross-section of the iron i>S. the amount to lu-siiht ra< -ted from BA to give the induction in the iron is (>' — S) IIr But the correction is very small. The current in A is then inerea>ed by an addition of \ amperes. This increases the field by H2, and the induction bv B., measured by the new throw of the ballistic galvanometer. The MEASUREMENT OF PERMEABILITY 273 total field will be Hx + H2 and the total induction Bx + B2. To find the induction per sq. cm. in the iron, we must divide the total induction by the cross-section S. The current may be increased step by step to any desired amount and then taken round a cycle in any desired way, the values of H and B being determinate at every point. Fig. 209 represents a test carried out by Ewing (loc. cit. pp. 70-72) on a wrought-iron ring, mean radius 5 cm. and cross-section 0-0483 sq. cm. The magnetising coil had 474 turns and the secondary had 167 turns. H was gradually raised from 0 to 9'14, then taken back to 0 and again raised to 9'] 4. The induction per sq. cm. in the iron rose to 12,440 at the maximum. The maximum value of /* = B/H was 1740 when the value of H was 495.* Magnetisation in very weak fields. Baur,f experi- menting by the ballistic method on a ring of soft iron with values of H rising to 0'384, found that I and B could be represented by the parabolic formulae I = 14'5H+HOH2 B = 183H+1382H2 whence K = 14'5+110H and M = 183+1382H. These results suggest that for extremely small values of H, K and IUL are constant. This point was investigated by Lord Rayleigh J and the suggestion was confirmed. He used the magnetometer method with a second compensating coil through which the magnetising current passed, adjusted in such a position that it neutralised the effect of the iron wire tested (Swedish iron un- annealed) on the magnetometer needle when the intensity of the magnetic field within the solenoid did not exceed 0'04. When the intensity of the field was still further reduced the compensation remained perfect and the reduction was carried down to an intensity about one two-thousandth of the initial value. It might therefore be concluded that, between H = 0 and H = 0'04, K and JUL are constant and that within this range there is no retentivity. The value of IUL was about 100. Above 0'04 the constancy no longer held. Lord Rayleigh also found that if a specimen was already magnetised by a moderate force, and there was then made a small change in the magnetising force opposite in direction to that * For other arrangements in which the ballistic method is used, especially adapted for technical requirements, the reader should consult Ewing on Magnetic Induction in Iron and other Afetals, chap. xii. t Wied. Ann., xi. p. 399 (1880). j Scientific Papers, vol. ii. p. 579. 274 MAGNETISM already employed the susceptibility for that change was nearly tin- same as when the specimen was initially unmagnetised. Thus if GAP, Fig. 2 10, represents the magnetisation curve with an increasing current, OA is the part under small forces with constant JUL and K and no retentivity. If the magnetisation is carried up to N M H Fio. 210. P and then a small decrease MN is made in II, PQ parallel to OA is the magnetisation curve. If H is again increased 1>\ VM \\r arrive again at P. But when the initial magnetisation is much increased and the region of saturation is approached. 1\) has a less slope, or (In- susceptibility for the small change, which we may denote by TTT> decreases. With annealed iron there- is a lime lag in the magnetisation. This effect has been examined by Ewing, and we refer tin- reader to his work, p. 129, for an account of the experiments. Magnetisation in very strong fields. Ewing' s isthmus method. In Fig. Jll NS an the pole pieces of a very powerful electro-magnet, the tips being bored out cylindrically about an axis through o perpendicular to the plane of the figure. A bobbin B just fits in between the poles. It has a coil of out- turn wrapped close about the central neck and with its ends connected to a ballistic galvanometer. It can be rotated through 180° about the axis through o so that the induction through it is reversed. The throw of the galvanometer on this rotation «: the value of the induction through the neck. Since there is no free magnetism on the surface of the iron about the neck the field MEASUREMENT OF PERMEABILITY 275 is continuous within and without the iron there, and Ewing calculated that if the pole pieces had an angle of 78° 28' the field would be nearly uniform as well as continuous. In order to measure the intensity H of the field a second coil of somewhat larger area was wound round the first, the difference in areas being FIG. 211. known. The difference in the galvanometer throw when the second coil was used and when the first coil was used gave the value of H in the air, and therefore in the iron. The following Table (Ewing, loc. cit. p. 150) gives the results obtained with a specimen of Swedish iron, and will serve as a type : H 13 I V- 1,490 22,f).->0 1680 15-20 3,600 24,650 1680 6-85 6,070 27,130 1680 4-47 8,600 30,27<) 1720 3-52 1 N310 38,960 1640 2-13 19,450 40,820 1700 2-10 19,880 40,140 1700 2-07 All the specimens examined showed the same tendency towards a limiting value of I, that is, towards saturation, and a consequent tendency in /j. to diminish towards unity. These are merely two aspects of the same fact, for H H H If I has a finite limiting value and H is indefinitely increased IUL must ultimately equal 1. These results have been confirmed by du Bois, using an entirely different method depending on the Kerr effect of rotation of the plane of polarisation of light reflected from a magnet pole.* * Phil. Mag., vol. xxix. (1890), pp. 263 and 293. 276 MAGNETISM The relations between permeability and magnetising force vary greatly with the quality of the iron or steel tested. The softer the iron the greater the maximum permeability. With steel the permeability is in general lower the harder tin- steel. Alloys of iron with manganese, tk manganese steel," have been made with a permeability less than 1'5, and Hopkinson found that a certain specimen of nickel steel containing °.."> per cent, of nickel had practically a constant permeability of 1*5. Nickel and cobalt. The methods which we have described for iron may be used also for nickel and cobalt, and similar results are obtained, though the magnetisation and permeability for a given magnetising force are much less than for iron. For an annealed nickel wire Ewing found that K attained a maximum value of 23'5 when the field was 9'5. This corresponds top = 2i)(>. In general the saturation value of I for nickel is a third or a fourth the value for wrought iron. Cobalt has less susceptibility than nickel in weak fields, but greater in strong fields. The hysteresis loop and the energy dissipated in a cycle. We have seen that if we start with demagnetised iron and apply a magnetising force gradually increasing to a maximum of considerable value + Hm, and if then we carry H through a complete cycle + IIm, O, — Hm, O,-f Hm, the magnetisa- tion I and the induction 13 return to very nearly the same val and the magnetisation curve forms a very nearly closed loop. Probably I and B do not return to exactly the same values. But successive repetitions of the cycle give nines \u\ nearly overlying the first, and the longer the cycle is repeated the more nearly do the sucoettive curves coincide. \\Y shall suppose that the stage is reached in which \\c may take the magnetisation curve as a closed loop — the hv>tcresi«, loop. In each cycle a certain amount of energy is dissipated and appears as heat in the iron. We may calculate the energy dis- sipated in a way somewhat different from that already given in Chapter XIX. We shall suppose that we are using the ring method and that the magnetising coil is wound so closely on the iron that the induction may be considered as all within the iron. Let E be the E.M.F. put on to the magnetising coil from outside. Let B be the number of induction tubes through unit area of the iron, and let a be the area of its cross-section. Then aB is the total induction. If there are n turns of the coil the virtual number of tubes through it is ;iaB. If C is the current through the coil and R its resistance the current equation is or Multiplying by C, CEdt = naCdB MEASUREMENT OF PERMEABILITY 277 The left-hand side, CEefa, is the energy supplied to the coil by the external source of current in time dt. C*Rdt is the energy dissipated as heat in the wire according to Joule^s law. Then ?iaCdB is the energy supplied to the iron in increasing the induction by dB. If / is the mean length of the iron its volume is /a, so that the ,. , . 7iCdB energy supplied to unit volume is — - — . If H is the intensity of the field within the coil, H/ = 47rnC, ., . nC H so that — = - . / 4-7T Hence the energy supplied per unit volume in the increase of magnetic induction dB is HdB 47T " Let CD, Fig. 212, represent the hysteresis loop when the curve is drawn with abscissa representing H and ordinate repre- senting B. Let PQ be an element in the ascending branch. The PQNM strip PQNM is H//B, so that the energy supplied is 47T In the descending branch there will be an element Q'P', with both H B s' / Fia. 212. and dB negative, so that HdB will be positive and the energy supplied there is + FQ'NM The total for the two elements is PQQT 47T ' For an element RS higher up the curve on the ascending branch and for the corresponding element R'S' on the descending 278 MAGNETISM branch H is + for both, while dB is + in one, — in the other, and Tms'H the total energy supplied is the difference - — . It is easily seen, then, that in the complete cycle the energy supplied i> rudB = j_ x guin of all such §t . as pQQ/p/ = area of J 4?r 4-7T 4-7T hysteresis loop. If we put B = H + 471-1, then dB = (III + 4WI, and Now 7j— is 0 round a cycle, since H returns to its initial value. Then the energy supplied is f lldl. Or if the h\Mt curve is drawn for II and I, tin- energy supplied i> represented by the area of the loop. Since tin- iron returns to its original con- dition after a complete cycle, and mcanw bile there i>an addition of energy, we may conclude thai thi^ en (i and is tin- equivalent of ihc heat \\hirh appear-. It can be shown by a thennodvnaniie cycle that if u is eon-tant for a given temperature so that the iron can be magnet i^-d l»\ a reversible process, and if yu varit-s with change of temperature, then energy must be supplied to a magnet iM-d body to keep the temperature constant if the magnrf i-at ion i> mcica-ed. The amount thus supplied * is _ ' 47T fl dtf We can onlv applv this formula to n-al iron in the initial investigated by Lord Ha\ K -iglu l-'ig. ^10. and in the small revci from subsequent stage > \\lure the \ ti>ation is practi- cally reversible. We cannot diducc the heat mjuired to keep tlie temperature constant in a non -reversible process, nor the In at appearing in a non-reversible cvclc. It is possible that if we could account for the heat required in the reversible process by some physical explanation, the same explanation would account for the heat dissipated in hysteresis. But the mechanical illustration gi\en (p. 279) would suggest that they are distinct, that the magnetisation consists of two parts, one strictly reversible and with the heating also reversible, another part non-reversible, dissipating energy by something analogous to friction. In connection with this subject some very interesting experi- * Thomson, Applications of Dynamics to Physics and Chemistry, p. 103. MEASUREMENT OF PERMEABILITY 279 ments by Baily are to be noted.* Swinburne had pointed out that if E wing's model, described in Chapter XV. gives a right idea of the physical nature of the magnetisation of iron, then the hysteresis of iron rotating in a constant magnetic field should show a falling off* as saturation is approached. For if all the molecular magnets are parallel to the imposed force they will simply rotate as the field rotates relative to the iron, without dissipating energy. Baily found that up to a value of B in the neighbourhood of 15,000 the hysteresis increased, and for both soft iron and hard steel it is not very different in a rotating field from its value in a field reversed in the ordinary way. If the rotating field is further increased, then hysteresis falls off and apparently would vanish for a value of B about 21,000, though that value was not actually reached. Mechanical model to illustrate hysteresis. It may be interesting to describe a mechanical model which, if it could be constructed, would give relations between force P and displacement d very nearly corresponding to the relations between magnetic intensity H and magnetisation I. Let AB, Fig. 21 3, be a cylinder of air with unit-area cross-section and of length 2D. Let CDEF be a short cylinder moving within it with friction F. Let G be a piston FIG. 213. moving without friction in CDEF, but against springs SS attached to CDEF. Let the piston-rod R pass out through an air-tight hole at B, and let any desired force P be applied at the end of R. Let us suppose that G has area practically equal to the unit-area of cross-section of AB. Let us start with the piston in the centre and with equal pressures of air p on the two sides. It is easily seen that if the piston is displaced d to the right, then, as far as the air pressure alone is concerned, R must be pulled with a force and the curve giving the relation between Pa and d will be somewhat as in Fig. 214. But the springs will also introduce a force P2, which may be considered separately, on the supposition that the ends of A and B are open. At first the spring force, and therefore P2, is pro- portional to the displacement of G, and will continue so till that displacement has come to a certain value dv at which the spring * Phil. Trans., A, '.vol. 77 (1896), p. 715. 280 MAGNETISM force can just overcome the friction F. Thereafter the pull l\ will be constant and equal to F as long as the motion is from right to left. To reverse the motion the force must be reversed from Pa = -f F to P, = — F, i;i \ing a di-^l.u rm< nt of (• Mtant F. The Fio. 215. giving the relation between P, and d will be as in Fig. 215. After the first motion from the centre it is a parallelogram. Now let us add the two forces Pt -f P! = P for a given displacement and we get Fig. 216, a hysteresis loop very much MEASUREMENT OF PERMEABILITY 281 resembling the magnetic hysteresis loop. The reader may easily follow out the close analogies between the two systems. We have only attempted in this chapter to give such an account of the magnetisation of the ferromagnetic metals that the reader may gain a general idea of the methods of experiments and a FIG. 216. general id« a of the results obtained. For details of the methods and results, for effects of temperature and stress, and for modifica- tions of the methods for technical purposes, the reader should consult Ewing's Magnetic Induction in Iron and Other Metals on which this chapter is based. That work is a very clear account by one who has largely contributed to our knowledge of magnetic phenomena. Additional information will be found in Rapports au Congres International, 1900, vol. ii, in articles by H. du Bois and E. Warburg. The latter contains some account of the various changes in iron as the temperature rises, investigated by F. Osmond. Another and a very full account of the present state of the subject will be found in Winkelmann's Handbuch der Physik, vol. v. Further information will be found in the paper by S. P. Thompson referred to on p. 187. CHAPTER XXII MEASUREMENTS OF SUSCEPTIBILITY AND PERMEABILITY OF PARAMAGNETIC AND DIAMAGNETIC SUBSTANCES Faraday's experiments — Rowland's exprriiii>-iit — ExjH'riiiHMits v. Ettingshausen — Curie's experiment* — Curie's law — Wills' experiment — Townsend's experiment — Pascal's experiments — The electron theory — The magneton. THE earliest measurements of the magnetic qualities of stihstances other than ferromagnetic \\civ made, MUMI after I-'aradas's diMo of (liaiuagnetism, by Pliicker,* by K. Hcctjuercl.t and I) iv.J Though excelled in accuracy by later work, HUM- earl Y experiment! arc worthy of attention, since tin- methods UM d an >imple in principle and easily understood. Tliev consisted in su^u-nding a body at a gixen point in the fit-Id between the poles of a strong magnet \ to, but not quite in, the a\is, and in measuring either fa * tioMm balance or by a torsion balance tin- force acting on the body. That force in any direction .r is **~*1 • - per unit \ohnne, \\h- is the susceptibility of the bodv and ^ that of th«- -nrnmnding medium, and H is the intensity of flu- field (p. ^~>^). If bodies of the same \olmnr an- u^d m Mio-e^ion ut the same point in tin- same field, the forces are proportional to Art — KI or to Mt~Mr \Ve shall select for descript ion hen- the t-xpt riinen' ulay. It is to be noted that he ga\e an account of them only in a lecture at the Royal Institution, and the account \vas e\i hung at ///. UMially (V5 in. fn»m tne strongest part of the field c. To measure the force on it, it was hung by a fine glass fibre 5 in. long from one end of a torsion rod parallel to ae, but with the end exactly over ///. A counter- poise was attached to the other arm of the torsion rod. The . Ann., Ixxiv. (1849), p. 321. t Ann. d« Chan, et de Phy*., xxxii. (18:>1), p. 68. EJP[). Ecs., iii. p. 497. 282 PARAMAGNETIC & DIAMAGNETIC SUBSTANCES 283 fibre supporting the torsion rod was attached to a graduated head, and the position of the rod was read by a mirror and scale. Its zero position was determined by removing the cylinder and placing an equal load on the rod itself, where any force due to the field was negligible. This zero position was so arranged that the vertical through the end of the rod passed through the point m, 0'5 in. from c. Then the body was suspended, and it was forced inwards or outwards according as it was more or less paramagnetic than the sur- rounding medium, the torsion rod being pulled round one way or the other. The torsion head was then turned round till the rod was again in the zero position and the angle of torsion measured the force acting. Since the force due to the field was exceedingly small compared with the weight of the body, the suspending fibre was >o near the vertical that the body could be regarded as at the point vertically under the end of the arm. The standard force selected was that on water in air, and was taken as 100. To determine it a cylinder of glass was suspended in air. It was repelled from the zero position, and required 15° of torsion to restore it. It was then suspended in water. Now it was attracted and required 54*5° of torsion to restore it. If /CK, /ca, /cvv are the susceptibilities of glass, air, and water respectively, FIG. 217. KK — K w oc + 54'5 whence KW — /ca oc — 69*5. Then 69'5° of torsion are represented by 100 on the scale chosen. Now. Mipposc that the same cylinder is hung in any other liquid with Mix-rptibility KV and that 6° of torsion are required to nature it. Then KS — K± a: 0. But Ks — ACa OC — 15. Subtracting KI - /ca x - (15 + 6) and this is expressed on the air-water scale by If the susceptibility /cs of a solid was required, taking a cylinder 284 MAGNETISM of it of the same shape and size as the glass, if the restoring force in air was 0, KB — fa QC 100 whence /c. — AT, on the air-water scale was 0 X — — . ( ). / • > Full details of the work are not given, but apparently a bull) was used containing in succession various liquids and ga-ex. and the same bulb was used when exhausted so that the values of * relative to a vacuum could be deduced. Taking a vacuum as having zero susceptibility and K* — KW = 100, Faradav found for K* the value 3-5, and therefore for water KW = - 96'5. The following Table gives some of his measurement^ on this scale. Becquerel's \.-dues for some of the same substances are given, adapted to Faraday's scale. Becquerel took /c» — /or = — 10 and *-, — 0. To compare them his numbers have been multiplied by 10. and have been added to 3*4. The numbers for air and water must ofcour- Oxygen 17'5 '.Ml Air :$•!• Vacuum 0 Carbon dioxide o Hydrogen 01 Zinc -7H; -21fi Alcohol Water <>ll li - %•<> Carbon disulphide - 99'6 Sulphur -lls - in- Bismuth - 1 -21 The closeness of the \ahus in many cases is e\ideme o! tin- excellence of the work of the two experiment! ; Faraday made experiments at different distances and found different results. The value for bismuth, for instance, rose <]uite considerably as the distance increased. This has not been con- firmed by later work, and there was no doubt some undetected error important at the greater distances, but apparently not so important at the half-inch distance, since the result ^ there obtained are in very fair accordance with the best measurements. We shall now describe some of the later experiments, selecting typical methods. Rowland's experiment. Rowland and Jacques * used the method of a vibrating needle, of which the simple theory i^ given on p. 260. According to that theory the time of vibration of a needle-shaped body in the field is * Rowland's Phytical Paper*, pp. 75 and 1 -I. PARAMAGNETIC & DIAMAGNETIC SUBSTANCES 285 where A and B are known if we know the rates of change of field along and perpendicular to the axis. These rates were determined by means of an <; exploring coil " which was connected to a ballistic galvanometer and placed, to begin with, at the centre of the field and with its plane perpendicular to the field. It was then moved short measured distances and the galvanometer throws gave the changes to be measured. The coil was also moved from the centre right out of the field and thus the field at the centre was found. The method was applied to crystals of bismuth and of calc spar, and it was found that the value of AC in each case was different along different axes. The values of 109/c obtained along two axes were BISMUTH CALC SPAR lOVi - 12554 - 38 10V, - 14324 - 40 These appear to be the earliest determinations in absolute measure. Experiments of von Ettingshausen. Four different methods were used by von Ettingshausen * to determine the sus- ceptibility of bismuth. In the first method two primary coils were placed in series and an intermittent current was sent through them. Round the primary coils were wound secondary coils con- nected in series through a galvanometer but in opposition — that is, so that the simultaneous currents induced in them by the primaries went in opposite directions through the galvanometer. There was a commutator in the circuit worked at such a rate that both make and break currents went through the galvanometer in the same direction. First the coils were so adjusted that the secondary currents just neutralised each other. Then a bismuth cylinder was inserted in one of the primaries. Since the per- meability of bismuth is less than unity, the mutual induction between that primary and its secondary was decreased, the current induced in it was less than the current in the other secondary, and the galvanometer was deflected. The bismuth was then withdrawn, and in its place was inserted a solenoid through which a known current was passed, made and broken as frequently as the primary current in the previous experiment. This induced an alternating current in the secondary, rectified as before through the galvanometer. The magnetisation of the bismuth under the original known primary could thus be compared with the mag- netisation — if we may so term it — of the inserted solenoid with its known current, and the susceptibility of the bismuth could be determined. * Wied. Ann., xvii. p. 272 (1882). 286 iMAGNETISM A precaution is necessary in this method. Foueault currents are induced in the bismuth at both make and break in the primary. If each Foucault current has time to he formed and die away before the next change in the primary, it will induce equal and opposite currents in the secondary and may be left out of account. But if the interval between make and break is too short, the Foucault current formed in one interval extends partly over the next interval, where its effect on the secondary is commuted, and so a balance is left over to affect the gaKano- meter. Experiments showed that the frequency must not exceed more than 8 or 9 per second. In a second method a horizontal bismuth cylinder was suspended from one end of a torsion rod so as to be partlv within, partly without a horizontal solenoid through which a known cunvnt passed and the force on the bismuth was meaMired. That force is, as we have seen, I g - II could be calculated for each element of the cylinder and hence K was determinate. A third method resembled that of Rowland and Jacques, and need not be described. In a fourth method a bismuth cylinder was brought bet the poles of a strong magnet. At a distance from tin poles cither 9 or 17cm. was a magnetometer ncere the bismuth was inserted the effect of the magnet on the needle was compen- sated by a second magnet. When the bismuth was inserted it behaved as a feeble reversed magnet and deflected tin i meter needle through where M is the magnetic moment of the bismuth and II is Oil- field at the magnetometer. The deflection was too minute for direct measurement. The bismuth was inserted and withdrawn with a periodicity the same as that of the needle, producing ultim itelv a swing g. If K is the dampii.. .cut of the suin^s the d. deflection A is then given oy The values obtained by the successive methods for — 10*/c were : I, 13-57; II, 14-11 ; III, 15%S; IV, l:i-<>. Ti,e noond method is probably the most exact. Curie's experiments. P. Curie * determined the value of K for a number of substances and investigated the effect of change of temperature, u>ing a method like that of Bt«|uerel and Faraday. The body to be tested was on a torsion arm, and the torsion was measured which was needed to keep it at o in the field * inn. de Chim. et nhy». , 7 ser. V. (1895), p, 289 ; or (Euvre* df P. Curie, p. 232- PARAMAGNETIC & DIAMAGNETIC SUBSTANCES 287 of an electro-magnet EEEE, Fig. 218. The figure represents the horizontal plane through the magnetic axis. The torsion rod was arranged so that the body could move along ox. The force upon JTT 2 it was F = -g — -r- v9 where Hy is the intensity, which from *\ mmetry must be perpendicular to ox, K is the susceptibility, and v is the volume. If p is the density and 772 is the mass, we may put F_l *OT<*Hya 2 p dx dkl. w here \~ ~- Curie investigated the value of ^, which he termed " the specific coefficient of magnetisation." To find Hy a small coil connected to a ballistic galvanometer was placed at o perpendicular to Hy and then turned through 180°. 288 MAGNETISM The galvanometer throw gave Hy. To find —77- use was made of the fact that — = -7— ^> since each is equal to — -, — j-, where V is ax ay • il.nl if the magnetic potential. The coil was placed at o with its plane perpendicular to H,, and was moved a given small distance dy along oy, and the throw gave — r— • dy By these experiments, then, Hy —r1 could be found. Pre- liminary experiments were made to determine the values of dU dH Hy and —^ or -T— *• at various points along ox, and the point where the product was a maximum was fixed as at o. Since, there- abouts, it varied slowly, exact adjustment of the body at o not necessary. Further, the force along ox was practically con- stant for a small displacement along ox. In the experiment! the 219. t torsion fibre, r torsion rod, 6 body, EE electro-magnet, * scale, p pointer, m microscope displacement due to magnetic action was never more than (H5 cm. It was not necessary, then, to restore the body to iN original position, when the deHecting force acted. It was sufficient to read the deflection. This was done by viewing through a microscope a pointer on the other end of the torsion arm which moved over a scale. The arrangement is represented diagram matically in Fig. 219, where certain details for measuring the moment of inertia are PARAMAGNETIC & DIAMAGNETIC SUBSTANCES 289 omitted. The force for a given deflection was obviously deter- minate from the moment of inertia and the time of vibration. The body was contained in a small bulb. When the effect of temperature was to be investigated the bulb could be surrounded by a jacket electrically heated, as represented diagrammatically in Fig. 220, the heating coil being surrounded by porcelain. Outside ^ T FlG. 220. " bulb containing the body, titn- ent. As T increases x approaches — H. Curie also studied the behaviour of iron, magnetite, and nickel when carried to high temperatures. In the case of iron he experimented in succession on three \\iiv-, each about 1 rm. lon^, and of respective diameters 0*002, O'OU, and 0'035 cm. These were enclosed in glass or platinum tube* to protect them from oxidation and arranged on the tor-inn rod nlon.i: Mf, 1 The temperature was raised from 20° C. to 1360° ( .. and tin- field from 25 to 1300. About 750° ('. tlic value of I, the intensity of magnetisation, began to be nearly independent of the field and tell rapidly. «>n- tinuing to fall to 950° C. Then it ;lv con-taut, though falling slightly to 1280° C. Then it rose llightly and ftf fell again. Apparently at a sufficiently high temperate I as for feebly paramagnetic hod I MID the < \JM i munN In obtained the following roults for iron : Tetnperatmv <>:> B I KXXX) 20° 1300 1000° 25 to 1300 Thus at 1000° C. the value of x i- independent of the lield : is nearly the same as for air at 20°.* Similar results were obtained \\ith magnetite and nirkrl. Wills' experiment.! In this i pillar plate of the Mih.stai.i • ^ as liiin_u t'lmn oiu- arm balance. Its lower edge was hori/ontal and \\a- l>. t \\een the poles of an electro-magnet as represcn t ed in filiation and plan in 1'ig. 221. If A is the area of i -ro^-si ction of the plate indicatttl in in (b\ the forcv on the plate in direction ./ upuanU i> F= / A — where the limits for II are the field at the lower t dg«- and that at the upper. Hut the field at the upper edge is negligible in comparison with that at the lower, and we may put the force measured bv the balance where H is the field between the poles. i tMire this a "dummy" plate of plaster of Paris with tinfoil strip round its * For details of the work on iron at high temperature?, Curie's paper (Joe. ett.> should be consulted. t Phil. Mag., xlv. (1898). p. 432. PARAMAGNETIC £ DIAMAGiNETIC SUBSTANCES 293 two sides and its lower edge was suspended in the position of the plate, and a known current C was sent through the tinfoil strip. If the length of the lower edge was /, the vertical force was CZH, since the action on the sides was negligible, being practically horizontal. Hence H was determinate by the balance. The value of K for bismuth was found to be the same within (//) Elevation N 1 (.' Fi ) < s Plan . 221. the limits of error with fields varying from 1600 to 10,000. The following results were obtained : 10«/c Bismuth . . . — 12'25 to —12-55 Antimony . . . —0*714 Sulphur . . . -0-765 Marble . . -0'7 Aluminium . . . +1*88 Townsend's experiment.* The method was devised to measure K for liquids, and was in principle like the first method of v. Ettiiigsiwuseii. A tall jar, which could contain the liquid to be experimented on, was placed within a solenoid which served as a primary coil. Round the middle of this was a secondary coil. There were a compensating primary and a secondary respectively in series with the coils round the jar, and a galvanometer with commutator to rectify the current was in the secondary circuit. The two sets of coils were so arranged and so connected up that when the jar was empty an alternating current of frequency 16 in the primaries produced no current in the secondaries. If there are * Phil. Tnin*.. A. (1896), p. 533. 294 MAGNETISM N turns on the jar-primary and N' turns in the secondary round it, and if the area of cn>»- Action of the primary is A and it* length M. the mutual induction of primary and sccondan If the jar is now filled with liquid of susceptibility *-, and if we suppose its cross-section to be A also, the mutual induction is altered to or increased hy The balance between the pair of secondaries will therefore be destroyed, and a current \vill flow through the gal To measure Ihis increase in mutual indu. ; lir of coils was arranged, one in the prii d the c the secondary circuit, so that the mutual induction, ai i dl, might IK- \aricd from 0 h .lount mca-i dimensions and positions ,,f tin coiN. This third mutual induction MM 10 arranged that it exactly neutralised the current di:> the insertion of the liijiiid, and so the \alue of the nurease 47r/f.47rNN/A/2/ was known, and K was determinate. '1 showed that Foucault currents in tin- jar c<>uld not have an appreciable effect. The value of 10** for water was found to be — 0'77, agreeing closely with Curie's value - The suMcptibilr salt solution iu water arises partly from the water and partly t the salt, so that, if * is the s,,sceptibilit v of the sohr K + 0-77 X 10-* is that of the salt alone I : old iron salts this was found to depend only on the weight of iron in tin- salt and not on the nature of the acid radicle, U-ing proportional to the weight of iron NY per c.c. of solution, thoi for eijual weights in tin I and ii .-trie salts. The values obtained for t em }>eratures about 1" • re: 10* Fe.C'e - Fed, aoc»\\ The values obtained for the dry salts were approximatel same. All the salts showed a fall in *, with rise of temperature of the order of i per cent, per degree C. PARAMAGNETIC & DIAMAGNETIC SUBSTANCES 295 The susceptibilities of solutions of potassium ferrocyanide and potassium ferri cyanide were the same as if water alone were present so that when iron is present in the acid radicle it must be at least 100 times less magnetic than where it takes the part of a metal in a salt. The paper contains an interesting suggestion. Imagine a magnet set spinning about its centre in the earth's field. The N pole will go round the semicircle on the Northern side more rapidly than it will go round that on the Southern side. Thus the end towards the North will be a longer time S than N, and the average effect at a distance will be that of a reversed magnet. If the molecules of a body consisted of small magnets rotating about their centres under the action of a field, the body would appear diamagnetic. We may note here that iron carbonyl, the liquid compound Fe(CO)5 discovered byMond and Quincke in 1891,* is diamagnetic. Experiments of Fleming and Dewar on oxygen at low temperatures. Dewar had already found that liquid jen was strongly paramagnetic.f Fleming and Dewar made investigations on its susceptibility. The first method J which they used was an induction method in which the primary and secondary (oils were immersed in liquid oxygen and then in gaseous oxygen at nearly the same temperature. But this was not so satisfactory >iid method j in which balls of silver, of bismuth, of copper, and of gl.-i^s containing mercury, were weighed above the pole of a strong electro-magnet. When the magnet was excited there was a change in weight of (/c1-Aca)VH-^-, where V is the volume of the ball, /^ is its susceptibility, and /c, is that of the surrounding medium and x is measured upwards. The field H was determined for different values of x and for different exciting currents by means of an exploring coil connected to a ballistic galvanometer. The susceptibility of each ball was first found by weighing it in air with the current off and on, and assuming that 106/c for air is 0'024. To test the method the silver ball and a glass ball were each weighed also in water with the current off and on, and the mean value of 10V for water was 0'74, agreeing fairly with the values obtained by others. The balls were then weighed in liquid oxygen at— 182° C. with the current off and on. The susceptibilities of the balls were estimated at — 182° C.,and thus the susceptibility of liquid oxygen was found. The mean result was 106/c = + 324, whence /x = 1 *004. Curie found that for gaseous oxygen 1ft, ,v* 33700 0«x=— -^-, * /r.ins. Chem >V. , l-'.M, vol. 50. p. 604. t /'.//..V. v..I. I. p. 247. { 76trf., vol. lx. p. 283 (1896). . v.,1. Ixiii. p, 311 (1 *'.»*), 296 MAGNETISM where T is the absolute temperature. If we .i^mne that this holds when the oxygen is liquefied at - 182° C. or + 91° A., and if take the density of liquid oxygen as 1*14, thi-n y i The change from gas to liquid, then, makes no great change in the value of \. Pascal's experiments. Pascal* studied the magnetic quali- ties of a large number of solutions, of some liquids, and liquefied gases by a method due to Quincke. In principle tin- liquid to be experimented on was contained in a U-tubc with limb narrow {(j mm. diameter), the other wide (Gem. di.t narrow limb was between the poles of an electro-magi x • 1>\ whit h a horizontal field could be establishes! ot 'any intensity up to i thousand gausses. The surface of the liquid in this limb was at the centre of the field and it s 1. \t 1 \\ ax <»!,., ( \ , ,| \\ ith a microsi The field at the surface of the wide limb was in cnmp.t: negligible. Hence when field II was put on. if the difl. level in the limbs was //, if K is the Misn-|,t ibiht \ o; the liquid /c0 of whatever is above it, the toice per unit area was as in Wills' experiment : Hut this balances the column of height //. It' the dcnsjt\ of the li(iuid is n, The actual change ob i the microscope is tliat of the 1< in the smaller limb, say f, If 8 be the cross-s<-( tion «.: r row limb and S that of tin w ider, it is easy to see that h = (S-f *>5/S, so that To avoid measurement of H, the experiment was repeated on water with the >amc value of II. 1 )• not i u ptihihtx by *„ its density by f>l% and the change of level in the nai limb by £x, By division * -I"". '/' i'1'ii.i. -M S ser. XVI. (1909). p. 3.';'. nn»l XIX. (1910>, p. 6. PARAMAGNETIC & DIAMAGNETIC SUBSTANCES 297 where A is (riven by the measurements made. Pascal took /ct as — 7'5 x 10~ 7 and KO for air as 0*25 X 10~7. When coal gas or a vacuum was above the liquid KO was taken as zero. For a solution under air it follows that 107/c = 0-25 - 7-75X while for one under coal gas or a vacuum 107 AC = - 7'5A. If the density of the solution was p and it contained/? grammes of salt per c.c. there was p — p of water per c.c. Assuming that salt and solvent each produced its own effect, and taking x as the susceptibility of the salt as dissolved, -7 Kp = px + (p —p)7'5 x 10 The measurement of X, /?, and p therefore gave x. For the modifi- cations of the method for liquefied gases the original papers should be referred to. From his results with solutions of salts of iron, nickel, and cobalt Pascal inferred that as an ion of a ferromagnetic metal |>a--r> into a complex ion or into a colloid its susceptibility diminishes, and it may even become diamagnetic. Hence the magnetic moment of a molecule in a given field is not the sum of the moments of its constituent atoms, but depends on the grouping in the molecule. With the diamagnetic metalloids he found that the atomic Mi-ceptibihty KO. (a being the atomic weight) was nearly pro- portional to e a+Pa where a and /3 are constants for the same family. With several diamagnetic compound gases, when liquefied he found that Km (m being the molecular weight) was nearly proportional to 5 -f- n, where n is the number of atoms in the molecule. In diamagnetic organic compounds his results showed that the different atoms preserved, as a rule, each its own diAmagnetic susceptibility , but the presence of oxygen appeared to lead to anomalies. The electron theory. The following is a brief sketch of Langevin's form of the Electron Theory of Magnetism.* Let us, as a preliminary, imagine an atom which consists of a positive charge on a large central mass, and a negative electron circling round it as the moon circles round the earth. The positive charge will hardly move, and the electric lines of force will sweep through the space round it, all the lines directed outwards from the centre, and their motion will produce a magnetic field equivalent to that of a current along the orbit but opposite in direction to the motion of the electron. The magnetic moment * Ann. tl <'!,;,„. ,i ,1, Phy*.t s scr. V. (1905), p. 70. 298 MAGNETISM will be given by M = tfA, where e is the elect ionic- charge and A i- the area of the orbit swept out per second. Now suppose that a field II is put on perpendicular to the plane of the orbit, and in the positive diivction, so that if the equivalent current is going round clockwise' in the plane of tin- paper, or the electron is going round counter-clockwise. II is from above downwards. During the increase in II then- U a negatixc. i.e. counter-clockwise, E.M.F., and this, acting on : nc electron, gives a clock \\i-e or retarding toree. \Yhcii II i>establi- the angular velocity in the orbit is less than initial! diminished, and it can be shown that A M/ M = - cllr!*-™ where r is the period of revolution and /// is the mass of tin- electron. Now imagine a more complex atom in which there is a central body with a large number of elect rons circling round it in orl with their aspects indificrentlx (list i United in all direction-. On the whole the moment of each atom, then, i- \\ field II is put on, the effect on each orbit is like ti above, but less as the inclination of II to the perpendicular t«< orbit increases. Those going round one \\-.\\ \\ill ha\e their positive moments decreased, tho-e going round the other will have their negative moments increased, so that on the whole the moment of the atom will become negatixe. SYe max suppose that the ordinary dian atom is of this t\; formula for the change of moment shoxx.s that, in -Mill experience, it is very minute. Further. lineC tl D to siippo-e that temperature describes the agitation of the molecules and atoms as wholes and not the motions x\ ithin the atoms. \\c should not expect diamagnetic susceptibility to depend on temperature. Curie showed that, excluding bismuth and antimony, the susceptibility was constant through a MIX wide rang* temperature. To explain paramagnetism. imagine a bodx to consist of atoms in each of which the aspects of the electronic orbits are not indifferently distributed, but that tl .,u»uped more or about a particular axis. Each atom, then, will have a magnetic moment, like the molecular magnets in \Y« U-r's theory* \\hen there is no external field and the body is unmagnctist d. the ax« the atoms and the molecules into which they are grouped will be distributed indifferentlv in all directions. When a field II is put on, it acts in txvo opposite ways, In the first place it tends to decrease each atomic magnetic moment in the way already explained for diamagnetic atoms. In the second place it tends to pull the magnetic axes into its own line and so to give a positive moment to a mass of molecules. This second effect is vastly PARAMAGNETIC & DIAMAGNETIC SUBSTANCES 299 greater than the first and quite masks it. If there were no interrnolecular and interatomic agitation — that is, if the temperature of the body were absolute zero — the molecules might set under sufficient force all with their axes in the direction of H, like the magnets in E wing's model. The intensity of magnetisation would then be a maximum and would be independent of the field when this had reached the value needed to secure parallelism. But molecular and atomic agitation — that is, heat — with the resulting collisions, produces disturbance of the axes. Langevin examined specially the case of a paramagnetic gas such as oxygen. Let us imagine that we are dealing with a constant mass kept at constant unit volume. Let its moment in field H be M. Let the moment be increased to M + dM.. The work done from outside on the gas is therefore HdM. But work done on a gas at constant volume goes to increase its temperature. If we keep the temperature constant we must take away heat dQ = II<7M. But if T is the temperature, -^ is a perfect differential, TT for it is equal to the entropy removed, d(f>. Then 7^. dM. is aUo a perfect differential. Now the condition of the gas, in- cluding the condition M, is a function of H and T only. It follows easily that M must be a function of ^ -/l¥> But as far as experiment has yet gone, when T is constant MocH. M — a r^ Therefore or M H C T' But M/H is proportional to the susceptibility, so that this result gives Curie's law. This cannot hold, however, down to the absolute zero. There we shall have a moment M0 due to parallel alignment of all the molecules. But of course the gas law assumed ceases to hold before we get to zero temperature. By a treatment analogous to that by which the mean square of the velocity is found, Langevin found that the magnetic suscepti- M 2 bility should be znr£,, where R is the gas constant. The magneton.* P. Weiss, in conjunction with Kammerlingh- * Weiss, Journal de Physique, 5 ser. I. (1911), pp. 900 and 965. 300 MAGNETISM Onnes,* made measurements at the temperature of liquid hydi. of the intensity of magnetisation of iron, nickel, and m .and by certain inferences of that of cobalt. At this temperature the bodies might be regarded as saturated, \\ith all the molecular parallel. Weiss found that the magnetic moment | .me- molecule was very near in each case to a small integral multiple of 1123*5. This value he termed the " nwgneton-^ramnn-." II- puts forth the theory that there is in each atom of these nil at least one elementary magnet of constant moment and the for all of them, and he terms this the ••magneton." Assuming Perrin's value 68*5 x 1022 as the number of molecules in the gramme-molecule, and assigning one magneton with moment ;// to each molecule in the magneton-gramme, \\e h 68-5 X W2m = IK' whence w = 16'4 x 10-". The magnetisations of iron, cobalt, magnetite, and nickel respectively agree \\ith the possession per molecule of 11, 9, '. magnetons ivsp»-rtivcly. I"- '. tain extensions, and assuming that it is applicable to solids as \s gases, certain results obtained by himself and 1'oex 4- on b<>< above the Curie point, or the point where ti ability : almost to 1, appeal- to fall in line with the magneton hvpof! on the supposition that the numb ;nctons changes. Weiss also uses IV show that in | solutions the magneli-ation can be expressed in terms of the magneton. i. p. 3 (1910). Jnur,,. :. ser. 1. (1911), pp. 274 ai CHAPTER XXIII TERRESTRIAL MAGNETISM The direction of the earth's lines of force at a given place — To find the declination — To find the dip or inclination — The earth inductor — The intensity of the field— To determine the horizontal intensity H— The vibration experiment — The deflection experiment — Recording instruments — Results — Sketch of Gauss's theory of terrestrial magnetism. Is* many magnetic and electric measurements it is necessary to know the intensity and direction of the earth's magnetic field at the place of experiment. It is also of the highest scientific interest to determine the field of the earth as a whole, in the hope that we may answer the questions how and where is the earth magnetised and what is the origin of the magnetisation ? We shall give here a brief account of the methods usually adopted to obtain the field at a given place, a summary of the results of observations made at the various observatories distributed over the earth's surface, and of the progress so far made in the investigation as to where the magnetisation resides and how the magnetic field is disturbed. As to the origin we have at present no theory of certain value.* The direction of the earth's lines offeree at a given place. It is convenient to describe the direction of the earth's field at a given place by two angles : (1) The angle which the vertical plane through the line of force — the plane of the magnetic meridian —makes with the vertical plane through the geographical North arid South — the plane of the geographical meridian. This angle is called the Declination. (2) The angle between the line of force and the horizontal plane. This angle is called the Dip or Inclination. To find the declination. Let us suppose that we have a compass needle with axis of figure coinciding with the magnetic axis balanced without friction on the point C, Fig. 222. If the poles are ±w, and if the earth's total intensity is I, we have two equal and opposite forces ml, one acting at each end. These may be resolved into horizontal components ±mH, and into vertical components ±mV. The latter will give a couple r«V/, * A full account of the Kew instruments will be found in Gordon's Electricity inn! MM where / is the distance between the p -1 «iil tend to make the needle dip in the direction of I. But the m-rdle max In- horizontal at a place in the Northern magnetic hemisj putting c towards the N end of it a dMaiuv t'rom (i Mich thai is the weight of the needle zr.fG = wA7. The needle will tlun rnH rn rn w Fie remain horizontal even \\hen displaced from thi for the only other forces acting on it are ±mll IMH i/,,nt.il When so displan d i hoe forces will form a c-onp iit(dk- into the ina^netir meridian, ami it \\ill ultimately M-ttK- tiirn-. Kvcn it c i> not acvurahK j»la. . needle will still settlr in tin- plain- <>t th< magnetic uu-ridian. I^et us Mippo.se that it m» i^radua of which c is the ccntiv. and that \\t- kn«»w tin- .U'-o^ra; \.»rth point on this circle ; «il»\ioiisly the dtu-lination circle between this point and the N end ol 111 practice the magnet ic axis cannot 1 •• with the axis of figure, \N • ra H axis from the position^ of the a\ ;f the in tii>t suspended with one face npuanU, and is then turned over and suspended with the other face upwards. Let (a), Fig. 223, be a plan of the needle in the first case. We read the direction of AB TERRESTRIAL MAGNETISM 303 on the circle. Now turn the needle over about NS as axis and Mi-pend it with the other face up, and let (6), Fig. 223, be the plan in the new position. NS is still in the same direction, but AB is thrown as much on one side of NS as it was previously on the other, and the mean of the two readings for the direction of AB gives the direction of NS. The form of the apparatus adopted in this country is known as the Kew magnetometer. The needle is a hollow magnetised steel tube about 4 in. long and J in. diameter, Fig. 224, with a very fine FIG. 224. horizontal scale on glass at one end «y, and a lens I with s as its focus at the other. It is provided with two points of suspension, pl and p.,. The needle is hung up in a torsion box by a silk fibre, as represented diagram matically in the figure. This torsion box is mounted so that it can be turned about a vertical axis, easily adjusted vertically by levelling screws and a level not shown. A telescope focused for a long distance away moves round the same ;i\i^. and i> provided with verniers on arms moving on a horizontal divided circle c. First, it is necessary to eliminate the torsion of the fibre. This is done by placing the torsion box as nearly as possible in the magnetic meridian, and then hanging a non-magnetic plumb bob of the same weight as the magnet and in its place. When the bob has come to rest the magnet is hung up by pv and the telescope i*. moved round so that the centre of the scale *, viewed through the window zc^ and lighted through the window w2, is seen on the rrovs wire of the telescope. The axis of figure is the line joining the centre of the scale s to the centre of the lens /. The readings Of the telescope verniers are then taken. Say that the mean reading 304 MAdM.TIvM is l)r The needle is now turned oxer and hung : telesc-ope is moved round >o that again tip of 8 appeal- on the cross wire, and the telescope xernicrs are again i say, D.2. The magnetic axis bi>. ct> the two directions given bx the telescope readings, and corresponds to a reading J ( l)j -f l)t). For simplicity we shall suppose that by prcxion- (ions on the sun >omc object on the hori/on at a d: on as the geographical South point. The t< is turned so that this point is on it* CTottirirt and thevern MKa mean reading S, say. ANY have then the declinati. 1 0 — - — S. The instrument is proxidid x\ith apparatus h\ which a sun ohservat ion ni.ix I) taken and the reading of the South point on the circle r may be directly determined, l>nt into tin- ue shall not enter. The determination of the declination bx diffi instruments* may differ bx -< \c ial minutes. To find the dip or'inclination. The ideal is for the- determination of the dip \\ould consist of a needle with axis of figure- coinciding with the magnetic a\ freelv round an axis through its , ,\it\. this .-n .supported so as to be hori/ontal and perpmdicular to the plai the magnetic meridian. A \ertical d would be placed in the magnetic meridian with its rent re in tin 'ion of the mcdle and its /ero leading in the hori/ontal thm •Kis, the ends of the needle King just in trout of th< The reading of cither end of tl, would then gixc the dip. In practice, of com il conditions ar« more or less nearly approach. d. Th, 1\ of a vertical circle contained in a rectangular box which is gla, and back, and it is mount, d -<> as to turn round an axis which be adjusted to be vertical by a level and lexelliug screws. There is a hori/ontal dix idcd circle round the axis \s Inch gives the a/imuth of the vertical circle. The \ ne'e, shown without the box in Fig. !W(>, i> divided from 0°at the ends ot th KIO. 225. to 90° at the ends of the vertical diam. g. 225, is a flat plate of steel from 3 in. to (> in. long, with sharply poiir and an axis is ti\ed as nearly through the centre nfgrax it\ as p<>- and perpendicular to the plate. Tin- thin cyhndiical steel rod projecting on both sides of the plate, and is a> truly circular in section as it can be made. Almost level with the centre of the vertical circle are two hori/ontal and parallel agate knife edges, on which the axis of the needle itself can roll, the central line of that * Thorpe and Rttcker, Phil Tro**., A, vol. clxxxriii. (1896), p. 1«. TERRESTRIAL MAGNETISM 305 axis being level with the centre of the scale. The needle when in position moves in a plane midway between the agate knife edges. A cross arm provided with verniers and two microscopes having cross wires in their fields of view can be turned round till, say, the upper end of the needle is on the cross wire of one microscope and the upper vernier is read. The cross wire in the FIG. 226. Diagrammatic representation of the Dip Circle. other microscope is then brought to coincide with the lower end and the lower vernier is read. The mean reading gives the position of the needle on the scale. When an observation is to be made, the axis round which the dip circle rotates is made vertical and the needle is put on to V- bearings, which are then lowered so as to leave it on the knife edges. We shall suppose that the dip circle is already adjusted in the plane of the meridian, and that the face of the instrument is East. The position of each end of the needle is then read. The mean of these eliminates error due to a small error in the centering of the needle. For if NS, Fig. 227, is the line through the ends of the needle and it does not pass through the axis of the circle, draw N'S parallel to NS, and HOH' the horizontal through (). Then SON' = SOS', and HON + H'OS = HON' -- NON' + H'OS' -f- >()>' = I ION' + H'OS', or the mean of the readings for N and S gives us the mean of the readings for N' and S'. The vertical circle may not be correctly set — that is, the points H and 1 1' may not be at 0° on the scale. The circle with the needle on it is therefore turned through 180° round the vertical so as to face West, and the positions of the ends of the needle are again read. u MAGNETISM Considering this error to exist alone, if Z, Fig. 228, i> t lu / the first position, we read NX instead of MI. In the second ZZ' is brought to the position Z'"Z" and we read NZ". Hut X is as Fio. 227. much below H as Z is above it, so that NZ + NZ" = 2NH, and the mean of the readings eliminates the zero error. The a\i- of figure of the needle does not in practice coincide exactly with the magnetic axis. To eliminate the error Fio. 228. produced the needle is taken up and its face reversed. The positions of the ends are again read. Then, as with the declination needle, it is easy to see that the mean of the readings, with tin face of the needle West and with its face East, will give the position of the magnetic axis. At the same time part of tin error due to the centre of gravity lying out of the axis of rotation will be very nearly corrected. For if G, Fig. 229, is the cc i of gravity and O the axis of rotation, drawing the rectangle ( )AGB, the moment of the weight of the needle acting through G will be equal to the moments of two equal weights at A and B. By turning the needle over, the moment of the weight at A is reversed, and tin TERRESTRIAL MAGNETISM 307 error due to the component of the displacement of G perpendicular to the axis is eliminated on taking the mean of the readings before and after turning over. The face of the instrument is then turned round into its original position facing East, and the positions of the ends of the needle are again read. Denoting the aspects of the face of the in- strument by E and W and those of the face of the needle by e and a>, we have to read each end in each of FIG. 229. the four arrangements Ee, Wo>, We, Ew, and the mean of the eight readings taken eliminates the errors so far considered. There is still outstanding the error due to OB, Fig. 229, the component of the displacement of the centre of gravity parallel to the axis. To eliminate this the needle is taken out and its magnetisation is reversed by the method of divided touch. It is assumed that the magnetisation is exactly reversed — an assumption quite certainly not fulfilled, but it at least gives us a method of pro- cedure which tends to diminish the error. B is now thrown as much above O as it was previously below it, and going through the four cases again the mean of the readings with the magnetisation direct and reversed gives as good a value of the dip as can be obtained with a single needle. It is usual to find the value with two different needles and to take the mean in the hope that it is nearer the true value than that given by either separately. In this country, with a value near 70°, skilled observers may obtain determinations with different needles differing by one or two minutes, and different instruments may give results differing by quantities of the same order. We assumed that the observations began with the circle in the magnetic meridian. To make this adjustment it is usual to set the microscope verniers at 90°, and turn the circle facing S till the ends of the needle are on the cross wires. The position on the horizontal circle is then read. The needle is then turned round and the dip circle is turned round the vertical axis till the microscopes again sight the needle-ends on the cross wires, and another reading on the horizontal circle is taken. The dip circle is then turned through 180° to face North, and the operations are repeated. The mean of the readings on the horizontal circle then gives the position of the circle in which the magnetic axis of the needle is vertical and, there fore, that in which the plane of the circle is perpen- dicular to the magnetic meridian, for in that plane the vertical force alone can move the needle. The dip circle is then turned through 90° and the needle is assumed to be in the magnetic meridian. Sometimes the dip is taken by another method which does not require this adjustment. 308 MAGNETISM Suppose that thedip circle is in a plane making an unknown aii^lc 0 with the magnetic meridian. If H is tin- hori/ontal intensity in the meridian, the effective horizontal intensity in the piano con- sidered is H cos 69 and if Sx is the dip observed when the needle is free to move in this plane, and if D is the true dip, The circle is now turned through 90° about the vertical axis and the dip (5a in the new position is observed. Then cot 3, = S cos (0-90) = cot D sin 0 and cot2^1+cot*^1 = cot8D, whence D is obtained from observation of Sl and Sr The earth inductor. There is a totally different method of obtaining the dip — the earth-inductor method. It is cmpl. at some observatories and appears likely to come into use. The principle may be given thus: Suppose that a coil in the form of a large ring has total area A, counting the areas of all the turns. Let it be laid Hat on a hoi and be connected to a ballistic galvanometer. The total flux of induction through it is AV, where V is th« \crtieal ii Now let it be turned over suddenly so that the oilier face of the coil is on the table. The change in induction is °.AV. and the throw of the galvanometer will be due to a flow <>: proportional to 2AV. Say that the i meter ^JM-S thi- Mow as Qi coulombs. Now let the coil be raised into the \« rtical plane perpendicular to the magnetic meridian. The total flux thr« it is now All, where II is the hori/ontal intensity. Now let turned suddenly round a vertical axis through 180°. Tip in induction is 2AH. Say that the galvan ..... » t« r indicates a of Q2 coulombs. Then 2AV V The intensity of the field. The total intensity I is determined directly. If H is the horizontal component and 1) is the dip, H = I cos D. The value of H can be found much more easily and much more exactly than I, and if D is known I may In- deduced if needed. But in almost all CHM I ire want to know H. To determine the horizontal intensity H. The horizontal intensity is determined by a combination of two separate experiments. By one, known as the vibration experi- ment, we find the product of H and of M the magnetic moment of the magnet used. By the other, known as the deflection experi- TERRESTRIAL MAGNETISM 309 ment, we find the ratio of these two quantities. The two results combined give us H, and incidentally M. The vibration experiment. If a magnet of moment M suspended so as to be horizontal and is displaced through 0 from the magnetic meridian, it is easily seen that the magnetic couple tending to restore it to the meridian is MH sin 0, or, if 0 is very small, it is MHO, and we shall have a simple harmonic vibration. If K is the moment of inertia of the magnet and if the magnetic couple alone acts, the time of vibration is T = 27TA/— V MH 47T2K or To find K, a non-magnetic bar of calculated moment of inertia B is attached to the magnet and the new time of vibration T' is observed and T' = K+B MH or MH = 4^T/ + B). Equating the two values of MH we obtain K+B_T/2 K T2 and K = The value of K need not be redetermined for each vibration experiment. The vibration experiment is carried out by suspending a needle like that used in the declination experiment in the same torsion box, Fig. 224. The vibration needle, however, is provided with a holder for the bar B. Corrections, into which we shall not enter, are made for the additional torsion couple dueto the thread, for the arc of swing not being negligibly small, and for the change in the magnetic moment M due to the temperature and for its increase due to the magnetisation induced by the earth's field. The deflection experiment. In this experiment the vibration needle used in the previous experiment is set to deflect another needle from the meridian. The general principle of the method is as follows : If the vibration needle ScN, Fig. 230, is set end on, E or W of another needle it will deflect it through 0, given by (p. 215) 310 MAGNETISM where d is the distance of the centre of the deflecting needle from the deflected needle, when we neglect small quantities. If then \\e observe 6 and measure d, we know the ratio of H to M, for ^ M tan 0 Multiplying this by the result of the vibration experiment and taking the square root, 27T./~2K" H Or another procedure may be adopted, and this N the more usual. The deflecting magnet, always in the end on position, i- N FiO. 230. turned round the centre of the deflected magnet until the lathi is in equilibrium and at right angles to the lonm-r. Then from FIG. 231. Fig. 231 we see that the angle 0 through which the deflecting magnet has been turned round is given by whence sin S T 2M _ sin 9' TERRESTRIAL MAGNETISM 311 In using the last method, a graduated horizontal cross bar about a metre long is fixed at right angles to, and just under, the torsion box. A carriage slides along this bar to carry the deflect- ing magnet, and the distance d is read on the scale of the bar. To measure 0 a short magnet provided with a mirror is suspended in the torsion box of Fig. 224 to serve as the deflected magnet. A telescope with a scale attached replaces the telescope used in the declination experiment to fix the position of this deflected magnet. The deflected magnet is first suspended and its position is observed when the deflecting magnet is not acting. The reading of the horizontal circle at the base of the instrument is also to be taken. The deflecting magnet is then put in position at a distance indicated as d on the bar scale and the instrument, with the telescope, is turned round until the deflected magnet appears in the same position in the field of the telescope. The horizontal scale at the base is again read, and the difference in readings gives the deflection 0. If the observed d were the actual d this would suffice in so far as the simple formula holds. But the centre of the magnet may not coincide with the index mark on the carriage. The deflected magnet is therefore turned end for end. The deflec- tion is now reversed and the mean value eliminates error of centering of the deflecting magnet. The graduation of the bar may not date from a point exactly under the centre of the deflected magnet. The deflecting magnet is therefore moved on its carriage to the other side of the bar to the distance marked as d, and the deflections are again taken with the magnet in its last position on the carriage and when it is turned end for end. We have then four values of the deflection, and the mean eliminates the errors considered. But the formula for the deflecting force — - is only approxi- mate. A nearer approximation is given by * where P depends on the distribution of magnetisation along the magnet. We may eliminate P by taking another value of the deflection 0' at another distance d' and it is easily found that H 2 (d* - M == d5 sin 0 - d6 sin & ' TT The error in the determination of ^ is a minimum if the value 7/ of -r is about 1 *3. a * Maxwell, Electricity and Magnetism, vol. ii., 3rd ed., p. 106. 312 MAGNETISM Values frequently chosen are d = 30 cm. and d' — 40 cm. Corrections for temperature and for magnetism induced bv tin- earth in the deflecting magnet must be made. Dr. Chree has shown* that the approximation represented by the factor 1 + P/f* is not justifiable with the magnets usually employed and that another term, Q/r4, should be taken into account. The reader is referred to Dr. Chree's papers for a discussion of this point. The value for H in this country is in the neighbourhood of 0'18, and different instruments may give values di fieri ng by as much as two or three units in the fourth place, say by as much as 1 in 1000 of the whole. In the regions near the magnetic pole-, i.e. the two points where the force is vertical, the hori/ontal intensity is only a small fraction of the total intensity and too small to be nun- accurately. Special instruments which we need not describe f have been devised to measure both dip and total intensity in these i Recording instruments. H.M.I,., the instruments \\hich give the absolute values of the magnetic elements, it is usual to equip magnetic observatories with self-recording instruments which register continuously the small changes which are alwav> occurring in the direction and intensity of the force. For this purpose t magnets are used, one to record changes in declination, another those in horizontal intensity, and a third those in vertical intensity. A satisfactory self-recording ; dip instrument ha> not been de\i-. d. Each magnet is suspended in a dark box, and is furnished \\ith a mirror which reflects a small beam of light from a lamp on to a volvingdrum covered with sensitixd paper.J Each magnet is sur- rounded with a thick copper ring to damp it- vibrations. The declination magnet is suspended so that it is five to set in the direction of the hori/ontal component. The hori/ontal force magnet has a bifilar suspension, and the torsion head is tiirnii the area of the sunspots increases.* In addition to these changes there is a secular cha: secular change in dip and declination is well represented 1 method due to Bauer. Fig. J^JI. givei tin- rhan«;e> in the declina- tion at London since 1580.| These secular variations make it Decenary to take a particular epoch when giving the value of the element- at a place. FIG. 235. Figs. 235 and 236 show the lines of horizontal force (Duperrey's * For an account of the inequalities see Chree, Studies in Terrestrial Magnetism On the connection with sunspots and auroras, see Ency. lint., I.e. f A Table of the mean annual changes in the elements at different positions in these islands is given by Thorpe and Ru'cker, Phil. Trans., A, clxxxi. (1890), p. 325. TERRESTRIAL MAGNETISM 315 lines) and the lines of equal dip for 1876. It will be seen that there are two poles or points of 90° clip. That in the Northern Hemisphere in the neighbourhood of lat. 70° 5' N. and long. FIG. 236. 96° 46' W. was reached in 1831 by Captain James Ross. In 1903 the region was again visited by Amundsen, who made a stay of two years, carrying out an extensive magnetic survey there. The Vest Variation FIG. 237. South magnetic pole was reached in 1909. It is about 72° S. lat. and 155° E. long. Fig. 237 shows lines of equal declination or isogonic lines, and The 316 MAGNETISM Fig. 238 those for equal dip or isoclinic lines for 1876. figures on the right of Fig. 238 arc the tangents of the dip. Several careful magnetic surveys of limited areas on the earth's surface have been recently made. Thus, Thor, Kiuker made a survey over Great Britain and Ireland in 1891-92,* finding various local deviations due, apparently, to the presence of magnetised rocks. Some of these deviations could be 140 160 180 100 140 120 IOO BO GO 4O 2O O SO 4O «O »O IOO «?O I4O «0 Fio. to the known presence of such rocks, and from others the Mih- terranean existence of magnetised matter ua> intern d. Sketch of Gauss's theory of terrestrial magnetism. In 1839 Gauss published a paper on the (n in nil Theory of Terrestrial J/^^v/<7/-v?//,f in which lie investigate! a formula \\hich should represent the earth's field at every point of its surface. We shall give only a brief account of the principles ot the th< The original memoir is well worthy of stiulv. not onlv tor the subject-matter, but for the admirable introduction which it gi- to the use of spherical harmonics. Let us assume that the earth is a sphere and that its \\\u±- field is due in part to magnetisation within the surface, in part to magnetisation without the surface, and in part to electric currents which lie wholly within or wholly without the surface. These electric currents may be replaced by magnetic shells (p. ~ - so that the field may be regarded as due to a definite distribution of magnetism. This distribution will have a potential given by V= / — , where dm is an element of magnetism and r its distance from the point considered, and V will have a single definite value * Phil. Trans., A., vol. clxxxviii. (1896). f Werlie, v. p. 119 ; Taylor's Scientific Memoire, ii. p. n, p. 223, a brief abstract is given. 184. In Lloyd's TERRESTRIAL MAGNETISM 317 at each point on the surface. This would not be true if currents existed passing through the surface. For in order that the potential shall be definite the work done in carrying unit pole from one point, A, to another point, B, must be the same by all paths, or, what is equivalent, the work done round any closed path ABA must be zero. But if a current C passes through the area enclosed by ABA, the work done round the path is 4?rC, and this work will be different for different paths enclosing different amounts of current. Hence the work from A to B is indefinite and in this case there is not a potential.* Let us first consider the distribution of magnetism within the surface. Let R be the radius of the earth, r the distance from the earth^s centre of a point P outside the earth, I the latitude, and X the longitude of the point of intersection of r with the surface. Then it can be shown (Gauss, I.e.) that the potential at P may be put in the form of a converging series : v U v= Bor where B0, Br B2, &c., are certain functions of / and X known as spherical harmonics. The general form of each of these functions is known. f If the order is designated by the suffix, that of the nth order Bn contains %n + 1 arbitrary constants. At a very great distance from the earth the potential tends to a value V = — ^ — , where M is the earth's magnetic moment and 0 is the angle its magnetic axis makes with r. The first term, B R ;-, must, therefore, vanish, or B0 = 0 ; and it is easily shown, too, that in the second term Bx = a sin I + b cos / sin X + c cos /cos X, since B^2 must be equal to M cos 0 ; a, &, and c are constants depending on the position of the magnetic axis. Next let us consider the distribution without the surface. It can be proved that the potential within the surface may be put in the form of a converging series : where again A0, Av &c., are spherical harmonics, but with arbi- trary constants other than those in Br B2, &c. V0 is the potential at the earth's centre, since it is the value of V when r = 0. Being * It is now known that there are minute vertical currents in the air, but too minute to be considered in a theory which does not pretend to give an exact account. f Thomson and Tait's Natural Philosophy, vol. i. p. 187. 318 MAGNETISM constant, it may be omitted, for in determining the forces we differentiate V. If we suppose a thin shell round the Mirfnrr of the earth to be free from magnetism, within that shell the potential will be the of the potentials due to external and internal magn«-ti>m. Then The intensity in any direction .v is -- T-' It I is drawn in tin- geographical meridian towards the North, f/.v = }{dl. component of the hori/ontnl intensity in that direction 1>< \ It ,9 is drawn towards tli phical West, <2f K I/t tin- component in that direction In- V. It .v i-. drawn \erticall\ down- wards, ds = — dr. Let the vertical intensity = /. _ \ l dV ~ IT 3T v ~ 7 -dV -777' Gauss began by assuming that the earth's field is entin 1\ due to inside magnetisation, and that the series converges so rapidly that the first four terms are sufficient to express V. So that YD H R3 H « '* = Bj— Bj contains three coiManK H2 ti\e. H3 >. ven. H4 nine, giving twenty-four in all. It mu>t he noted that though these constants depend on the internal distrihution. they tell us nothing as to that distribution, for a given distribution of i-xtcriml potential may be produced by an infinite number of different arrangements of mag- netism. To determine the constants it would be sufficient to • eight stations at which X. Y, and Z were known, if these were known exactly. But, of course, there are errors in the values, not only of observation, but also of reduction to the same instant of time. Gauss took twelve stations in -i \cn different latitudes, and computed the values of the twenty-four constants best satisfying the values oi \ . Y . and Z at the stations. He then calculated the values oi \. \ . /, for a larger number of known stations all over the earth where X. ^ . / had been observed. The differences between observation and TERRESTRIAL MAGNETISM 319 calculation were not much greater than the errors of observa- tion. In Taylor's Scientific Memoirs, vol. ii., will be found maps giving the lines of equal dip, equal declination, and equal total intensity according to Gauss's formula. Considering that the observations could not all be correctly reduced to the same epoch, these lines show a remarkable agreement with the observed lines and justify the preliminary assumption that the earth's field is, at any rate, mostly due to internal magnetism. Subsequent recalcula- tion of the constants * has tended to confirm this assumption. Gauss himself suggested that the disturbances of the magnetic elements might be treated in a similar way, and that it might be shown whether they arose from internal or external changes in magnetism. Schuster f undertook such an investigation on the diurnal variation for the year 1870. His work led to the con- clusion that the variation was chiefly " due to causes outside the earth's surface, and probably to electric currents in our atmosphere," and that "currents are induced in the earth by the diurnal varia- tion which produce a sensible effect, chiefly in reducing the amplitude of the vertical components and increasing the amplitude of the horizontal components." * Km-y. Itrit., 10th ed., xxx. p. -462 ; llth ed., xvii. p. 881, t Phil. Trans., A, vol. clxxx. (1889), p. 467, CHAPTER XXI\ MAGNETISM AND LIGHT The Faraday rotation — Faraday*g successors — V.-s-.l.-i's constant and its variation with wave length— Effect of rise of temperature — Absolute values of Verdet's constant — Rotation by jfanes— Rotation by films of iron, nickel, and cobalt— Representation of the rotation by two equal circularly polarised rays with opposite r..t:it iini; with different velocities — An electron theory of the rotation -('•-•. I tlMOfJ — M;;:riirtic double refraction wh'-n : ray is perj^ndicula- liii«-n of force — V, -i-_'t'> tlj.-oi- - Magnetic doul liquids — The Kerr magnetic effect — l!i, Keen -Lorents'a theory— Ze*man's Teriflcnt The Faraday rotation, l opera hi* paper describ- ing his givat di »f an action of magnetism on li^lit by saying: *• I have lon^ held an opinion, ai: viction, in common I In-lit \c \\itli many oth» r 1 natural knowledge, that the vari. I under \\ hidi the forces of ma' are made manifest lm\e one common origin : <>r. in other words, are so directly related and mutually dependent that tlit-v are coin tible, as it were, one into another, and possess equivalents of p<> in their action.""* He then states that this persuasion had led him formerly to make many unsuccessful attempts to obtain ; D of light and electricity. Recently, bo i-dcd in " magnetising and eL ^>it. (in through long the lines of force in the field of A permanent magnet or in that of a solenoid through which i* .rrent of electricity tlie j)lane of polarisation gradually rotates a> the ra\ on. The first substance which exhibit i M TV heavy glass, a silicated borate of lead which he had ma ra earlier. A plate of this glass two inches .square and half an inch thick was polished on its narrow sides and placed between the poles of an electro-magnet in sucb a position that when the current was on, lines of force went, as nearly as might be, straight through the inches of glass from one side to the other. A ray ot lignl Iron Argand burner was made to fall on a glass plate at the polari'sino; /•;./•;•. Re*. III. p. 1 (N 320 MAGNETISM AND LIGHT angle and then passed plane-polarised in thd horixontal direction through the glass close to the poles and in the direction of the lines of force. It was then received through a Nicol prism into the observer's eye. The prism could be rotated about the ray as . and the rotation could be measured. Before the current was put on, the Nicol was turned into the position of extinction. On making the current the field of view lighted up. If the Nicol was then rotated, a new position of extinction was found showing that the light was still plane- polarised, but that the plane was turned round into a new position. He found that there was no visible effect if the magnet were so turned that the lines of force were perpendicular to the ray. The (ields produced by a permanent magnet, and by a helical coil or solenoid in which an electric current flowed, were effective, and as far as he could estimate he found that the rotation with a given length of glass was proportional to the strength of field, and with different lengths of glass proportional to the length. He examined a large number of substances besides the glass, and found that m. my exhibited the effect, such as flint and crown glass, water and alcohol. The solenoid arrangement was useful for the liquids, which could be enclosed in tubes with glass ends, placed along the axis of the solenoid. It was this solenoid method, in which an electric current was effective, which led him to say that he had electrified a ray of light. With optically active bodies such as turpentine or sugar he found that the magnetic rotation was merely added to or subtracted from the natural rotation according to its sign. He was unable to detect any rotation in air. In every case which Faraday observed, the rotation was in the direction of the current which would produce the lines of force, if we imagine them due to a solenoid surrounding them. Thus if the lines of force are supposed to come out from the paper to the reader's eye, the creating current would circulate counter-clock wise, and that would be the direction of rotation of the plane of polari- sation. The rotation would be reversed if the lines of force went from the eye to the paper. Subsequent researches have shown that all substances which light traverses exhibit the Faraday effect, and for the most part with the rotation in the direction which he observed and which is described as positive. But salts of iron and some few others, though not salts of nickel and cobalt, exhibit a negative rotation. The distinction in no way corresponds to the distinction of paramagnetic and dia- magnetic, for iron nickel and cobalt, though enormously para- magnetic, coincide with the great majority of diamagnetics in giving positive rotation, while the diamagnetic chloride of titanium gives a negative rotation. The magnetic rotation differs in one very important respect from the rotation by a naturally active substance such as sugar solution. In the sugar solution the rotation is always "right- 322 MAGNETISM handed11 or clockwise in a rav coining to the ol»ei -ver. It is therefore counter-clockwise to him in a ray going from him. So that if a ray is sent along a solution from the observing end and then reflected back to the observer the rotations in the two journeys are opposite and cancel each other, and the ray emerges polarised parallel to its original plane. But with a " magnetised " ray the rotation is always in the direction of the current which would produce the lines of force, so that for each journey along a line of force the rotation is increased by the same amount. Thus Faraday * found that the rotation for one passage through a piece of glass was 12°, but on reflecting it to and fro so that it made five passages the rotation was increased to 60°. Faraday's successors. Faraday does not appear to have made further experiments after the publication of his paper, his attention being called off' by his investigations into paramagnetism and diamagnetism ; but the subject was pursued by many work E. Becquerel was the first to observe that different colours were rotated by different amounts, red least, violet most — a phenomenon known as rotary dispersion. By making the magnetic rotation "left-handed" and then passing the rotated light through a sugar tube of suitable length to neutralise the rotation for one colour. it was neutralised for all as nearly as he could judge, or the magnetic rotation was proportional to the natural rotation for all colours. G. Wiedemann used different parts of a solar spectrum as the source of light, a solenoid to produce the field, and a tangent galvanometer to measure the current. He was thus able to show that the rotation was proportional to the current, and therefore to the field intensity within the coil. He found that the magnetic rotary dispersion of carbon bisulphide, for instance, was not very far from being proportional to the natural rotary dispersion in a sir solution but yet sensibly different, and it was not far from being inversely as the square of the wave length, though again quite sensibly different. Verdet's constant, and its variation with wave length. Verdetf made a long series of experiments on various substances with an accuracy probably superior to that attained up to his time, and he proved conclusively that the rotation per unit length of a substance traversed is proportional to the component of the intensity of field in the direction of the ray. If, then, ds is an element of the path of the ray in a substance, and the intensity H makes an angle 0 with ds, the rotation R of the plane of polarisation between 8 = a and s = b, usually expressed in minutes of arc, is * Exp, Res. III. p. 453. f CEurres, tome i. ; or Gordon, Electricity and Magnetism, vol. ii. MAGNETISM AND LIGHT 323 where C is a constant for the substance for the particular wave length used. It is known as Verdet's constant, though it might more appropriately be named after Faraday. If V", V6, be the magnetic potentials at the beginning and end of the path con- sidered we may put R in minutes = C (V6— Va), a concise form of expression, though the integral form is the one practically useful. Verdet worked before the value of absolute units was realised, and he was content to express the constant of a substance in terms of the constant for water. Taking the rotation in water of light at the E line in the spectrum as 1, he found the rotations of the light of the various lines as follows. Underneath are the rotation, which would have been observed if they had been inversely as the squares of the wave length. WATER Lines in spectrum C D E F G Observed rotations 0'63 0'79 I'OO 1'213 1'605 Calculated from 1/X2 0'64 0'80 I'OO 1'08 1'50 For carbon bisulphide, creosote, and various other liquids the dispersion is about as near to that calculated from l/\2 as in the case of water, departing as notably from it as the wave length diminishes. If the substance examined absorbs the light of one particular wave length there are peculiarities in the rotation, which were first predicted from theory, and which will be more appropriately described when we discuss the theory now accepted. Effect of rise of temperature. Bichat was the first to make a careful examination of the effect of change of temperature of the Verdet constant, and he found that in general the constant lessens as the temperature rises, though his numerical values have not been confirmed by subsequent workers. Absolute values of Verdet 's constant. The first determination of the absolute value of a Verdet constant was made by Gordon* for sodium light in carbon bisulphide. A few years later another determination of the same constant was made by Lord Rayleigh. f He used a coil with a known number of turns surrounding a tube of known length containing the bisulphide. The current was measured by a potentiometer method, assuming the E.M.F. of a Clark cell. If the tube had ex- tended a great distance beyond the coil at each end the difference of magnetic potential between the ends would have been 4x X number of turns of coil X current. But as the ends were at points where the field was still appreciable a correction was needed. To measure the rotation Lord Rayleigh used a special " half- * Gordon, El. and Mag,, vol. ii. chap, xlviii. f Phil. Traitx. 176, p. 343 (1885) ; or Scientific Papers, vol. ii. p. 360. 324 MAGNETISM shadow"" method,* on the general principle now alwav- adopted in polarimctry, in which the field is divided into n gion- with their planes of polarisation slightly inclimd to that one half of tin- 1>« am passed through it, while the other half passed bv its , 1 so through an extra thickness ot-ugar. solution. This half of the h- was therefore a little more rotated than the half passing thru the glass, and when the beam emerged from the cell its two h > were polarised in plain-- making a small angle with each n* Then the beam pa--, d through the tube and in' -ing Nieol in front of the obsi I i cither fonissed his eye directly on to the the glass plate in the -ugar cell, or viewed tl, G -pc. To un tin- method let us stippo-e that, to begin with, there : t in the coil. Let OK OL, Fig. °.M, r« pic-nit the c<|iial amplitudes and N O M 239. the directions of vibration in the two hai he field , The angle K()L is .neatly exag :i tl,e li.: :;,. I more than <> or I* degrees OSUIuTy, 1 .. t the analvsing NIC.., turned so that it extinguish :i0,,s in the r the bisector of KOL. If M< ).\ il ,„-, pendicular to (}('. and KM LN are perpendicular to it, the . when placed M it would extinguish vibration- < < d vibrations OM ON from the two halve- ot the field. If it be now turn.-d through a small angle so that the vibrations admit) • \1 ON - Fig. 240, it is obvious that there will be a great difference in the illumination of the two halves. Hence the position of the analj for equality of illumination can be fixed with accuracy. When this has been done let the current be put on so that the magnetic field is established. The planes nf OK * Foynting, Phil. Mag. [5], x. MAGNETISM AND LIGHT 325 and OL will be equally rotated, and the analysing Nicol must be turned through the same angle to secure equality of the two halves of the field once more. The angle is read on a divided circle with which the analyser is provided. The result which Lord Rayleigh obtained for the Verdet constant for sodium light for bisulphide of carbon at 18° measured in minutes per unit difference of potential is 0-04202. Hodger and Watson* made determinations of the Verdet con- stants for sodium light in carbon bisulphide and in water over a range from 0° C. to 40° C. for the former, and over a range from 0° C. to 100° C. for the latter. Their values for the constants were: for carbon bisulphide 0*04347-737rf/107, and for water 0-01311 -4 //107-4 t*/\0*. Their formula for bisulphide of carbon would give the value of the constant at 18° as 0*4214, which only differs from Lord Rayleigh's value by 1 in 350. Constants for many other materials were determined in terms of that of carbon bisulphide by H. Becquerel, and these may be turned into absolute values by using the constant just given for that liquid. The following values are extracted from a table of Becquerers results uriven by Gordon, I.e. : carbon bisulphide, 1; water, 0'308 ; benzene, 0'636 ; rock-salt, 0*843 ; crown glass, 0*481; bichloride of titanium, - 0*358. f Rotation by gases. In 1879 both H. Becquerel and Kundt and BdotgenJ succeeded in measuring the rotation by gases. H»v.jiu-!vl pa-x-d a ray several times to and fro along a copper tube with gla>s ends surrounded by a coil. Kundt and Rontgen aKo ti>ed a copper tube with glass ends surrounded by a coil. Into this the gas to be dealt with was compressed to 250 atmo- spheres. The glass ends were so seriously strained by the internal jirr^iire that they became doubly refracting. It was therefore nece>sarv to put the polariser and analyser within the tube, each being fixed to the tube at the end at which it was placed. The polariser end was clamped in a fixed position. The analyser end c.mld be twisted round, subjecting the tube to torsion, and the angle of twist could be measured. It was found that the rotation was proportional to the density of the gas, so that it could be reduced to normal density at 0° C. and 760 mm. In all cases i mined the rotation was positive. The following rotations at N'lT in terms of that of bisulphide of carbon were obtained : CS2 O N II C02 1 0-000109 0000127 0*000132 0-000232 * /V//7. 7/v///«., A., 180, Part II. p. 621 (1895). f A nti'iiber of values by different workers will be found in Wiukelmann's IfnndhtK'h, vol. v. * Som<- in/count of both researches is given by Gordon, I.e. 326 MAGNETISM Rotation by films of iron, nickel, and cobalt. In 1884 Kundt * succeeded in depositing on glass films of iron, nickel, and cobalt Jess than a wave length in thickness which easily transmitted light. When one of these was placed in a magnetic field normal to the lines of force the plane of a polarised ray passing through was rotated in the positive direction — i.e. the direction of the magnetising current — and by an amount pro- portional to the thickness of the film. The rotation increased with increase of the field up to about 17,000° ; but when the field exceeded this the rotation did not increase much. The maximum rotation of a certain red ray in iron was such that passage through 1 cm. of iron would at the same rate give a rotation of 200,000°. The rotation of blue light was less than that of red. The approach of the rotation to a limit suggested that the rotation is proportional, not to the external magnetising force H, but to the intensity of magnetisation I. With H = 17,000 we may suppose the iron to be saturated or that I has reached its limit. In an ordinary non-ferromagnetic substance the intensity of magnetisation is given by I = /cII. wheiv K is a minute constant and I is proportional to H. In a ferromagnetic metal K is not constant and I is not proportional to H. Du Boisf plotted I against H for each of the three metals iron, nickel, and cobalt, using Rowlands'^ values. He then plotted rotation for a definite thickness against H, using the rotations determined by Kundt and by himself, and found that the form of the two sets of curves was the same, or the rotation was proportional to I. Representation of the rotation of the plane of polari- sation by two equal circularly polarised rays -with opposite rotations travelling with different velocities. Let a point 1* move with uniform angular velocity w in a circle radius a. Drop a perpendiculai PN on a iliameh r ACA' It' 1* starts at t = 0 from A, ACT = o>£ and CN = a cos tot. Thru N has a simple harmonic motion with the same period as P. When P starts from A let P' start from the same point with the same angular velocity 01 round the same circle but in the opposite direction, so that P' is always in PN produced. Then the sum FlQ 241> of the displacements of P and P' parallel to AA' = %a cos wt, and the sum of the displacements perpendicular to AA' is /ero, since the displacements are equal and opposite. Hence a simple harmonic vibration may be made up of, or conversely may be resolved into, two circular motions, each of half its amplitude and with equal period. * Phil. Mag. [5], xviii. (1884), p. 308. t Ibid. [5], xxiv. (1887), p. 445. MAGNETISM AND LIGHT 327 A train of waves of circularly polarised light may be regarded as consisting of circular disturbance, represented by a radius of given length travelling round each point in the path and with the same angular velocity for each radius, the ends of the radii being on a revolving spiral. A corkscrew turned round counter-clockwise without advancing exactly represents such a train of waves with counter-clockwise rotation travelling towards its point, and if it be laid lengthwise on a mirror its reflection in the mirror repre- sents the transmission of a train with clockwise rotation. Let (a), Fig. 242, represent the right-handed corkscrew as seen from above, the thick lines representing the upper part. Then if it is turned round (a,) M B1 M FIG. 242. A1 (b) FIG. 243. counter-clockwise as seen from A, a given displacement will travel forward from A to B. If (b) represents its reflection in the mirror MM, a clockwise rotation as seen from A' will send a given displacement forward from A' to B'. If the two displacements in (a) and (b) are combined, they will always give an up-and-down displacement perpendicular to the plane of the paper, or a plane-polarised disturbance. But now suppose that while the speed of rotation of the two is the same, one travels faster forward than the other, that is, has a longer wave length, or is a steeper spiral. .Thus in Fig. 243 let ACDB represent the faster-moving and longer wave, AEFG the slower and shorter. At the instant represented they coincide at A, and as the revolution is at the same rate for both spirals they combine now and always to give at A a disturbance perpendicular to the paper. At this instant they coincide again at G, and combine now and always to give at G a disturbance in a line which is turned round in the direction of the circular motion which has the greatest forward velocity. That is, there is rotation of the plane of polarisation. 328 MA<;M-;TISM We may obtain the actual rotation bv noting tlmt the forward propagation of a circular disturbance with spied D i> represented by • - - where 0 is the angle the radius makes \\illi a fixed direction per- pendicular to the line of propagation, the distance X being measured from a starting-point on the line at which 0= " when t = 0. For at any fixed point the angle 6 gro\\> at tin r.v or at each point there is circular motion, and at di>tance ./. fJ = 0 when x = r/, or the displacement which is at ./ = 0 when / = 0 is propagated forward with velocity r. Now let an equal circular disturbance, but with opposite rotation, — 6», travel forward with a !« -^ \rlocifv ;•'. Then the direction of disturbance at ./ at time / \\ill ta as.siiming that it is 0 at ./• = u when / — 0. It is easily seen that the sum of two displacement-. < ach >i and 0+& in directions 9 0', is in tliv direction — 5— , and is in magnitude £% That is, it is in the direction ^— -), % \ V V / and its magnitiKi- 1 It is, then, a simple harmonic vibration in a constant din at a constant point, but as we travel forward the direction t round in the positive direction of w at a rate per unit dist. i,> 1 1 »,. ;• — !•' i or if n is the frecjueney of revolution. IO that to = %-n. we may put the rotation of the plane of polari-ation r—v — . and in the direction of rotation of the faster-moving constituent. So far this is meiely a geometrical representation of the MAGNETISM AND LIGHT 329 and at first sight it appears somewhat artificial. But it appears more natural if we assume, as we surely must, that there is some sort of motion round the lines of force in a magnetic field. Then a resolution into circular motion would seem to be appropriate. An electron theory of the rotation. We have now to .show that if we suppose that in a circularly polarised ray the electrons are whirled round with the frequency of the waves (i.e. a number of times per second equal to the number of waves passing a point per second), then the velocity will be different for the t\vo directions of whirling if a magnetic field exists parallel to the path of the ray. According to Sellmciers mechanical theory, the velocity of a train of light waves in a medium depends on the nearness of the wave frequency to the natural frequencies of vibration of the molecular or atomic systems constituting the medium. By the natural frequencies we mean those of the different types of motion of each system if it is disturbed and then left to vibrate without auv external action. I jet N1? N2, £c., be the natural periods of the constituents of the medium. The velocity v of a train of waves of frequency n is given bv where K, /3,,/32, &c., are constants for the medium.* For simplicity we shall suppose that there is only one mode of natural vibration N, and that this takes place under an acceleration towards a centre at distance d equal to io2d or 4<7r2N2d. The velocity of a train of waves is given in this case by \Ylicn n — N, the formula makes v — 0. The physical inter- pretation is that there is at this value "resonance," and the waves spend their energy in setting the constituent systems of the medium into 'vibration so that they are absorbed and not trans- mitted. There will be absorption too, in a real system, for some little range on each side of n = N. As we rise from n = 0, i.e. from waves of infinite length towards n = N, l/r>2, which is pro- portional to jUL2 where JUL is the refractive index, gradually increases and becomes great as n nears N. When n has just passed N, l/v2 i> at first negative or /m is unreal. But when l/v2 becomes positive and IUL re.il it increases as n increases. There is, however, a limit K towards which it tends as n goes on increasing. * Sue Schnst.-r's Theory of Optics for an account of the theory of dispersion and of majrni'to-optirs. Voitft, who has very largely contributed to the theory, is the author of a work on M>ii/net<>- and Elclrtro-optilt, Teubner, 1908. 330 .MACiNKTISM We may represent the general nature of the change in l/^orM2by Fig. 244. This theory holds good in the clcctro-magm tit thcorv it we suppose that the medium consists of mitral bodies round uhirh electrons or small charged bodies are revolving, each with a pi-rind or periods of its own. \Ve make the iiMinl supposition that the t I K+/3/N K **7t N charge on each electron has the same n< iluc. NVe repi it numerically by f and the ma>s of the clc< tum by m. Now let a circularlx polarised beam of fr« n travel through the medium. We suppose that it makes t: whirl round the centres of their natural orbit- frequency n, instead of the natural fiequencx N. the «a\e di-- turbance" calling into play a force which, added to m.in^N1^ the natural force of the s\stem, accounts lor the i When a field II is put on, let tix suppo-i coincides with the dirt ct ion in which > tiaxrlling. Il the distance of the electron < tiom it is %-jriid, and as it is moving transxciscly to t II idial :• . to II, equal to girfio»it: lating counter-ilockwisi-. 11 ix inwaids on an electron moving clockx\i>e and outw.-,- clockwise. If m \^ the i ion, th- n at distance d is ±2ir« ra» and a circularly polarised ray in which the rotation as seen from the source is clockwise moves with greater velocity than a ray with counter- clockwise rotation. The rotation therefore has the same sign whetluT n \^ li-vi than or greater than N. If a ray of plane-polarised light is resolved into two co-existing circular components, and we assume that they have different velocities as given by (3), the clockwise component moves the faster and the plane of polarisation rotates clockwise or in the direction of the current round the ray which would produce the field in which it is travelling. If vl and r2 are nearly equal, we may put the rotation pel unit length (p. 328), by aid of (4), in the form MAGNKTISM When the rotation is in the opposite direction, as in titanium chloride and in salts of iron, we mav perhaps account for it bv a negative sign for /3, i.e. a decrease in ,/ at \ i . or perhaps by assigning the opposite sign for r, which implies that positive charges are revolving. Confirmations of the theory. Highi * showed that tin- wave lengths of two oppositely circularly polarised rays \\eie different when thev were made to traverse a magnetic- field along the lines of force, before the electron theorv \\as elaborated to account for the difference of their velocities. A pencil from a source was divided into two pencils which \\ere circularly polar with rotations in opposite directions and then ran alongside each other through a tube containing an active liquid and surrounded by a solenoid. The two pencils then P.-ISM d through a Nicol and inter- fered in the focal plain- of a micrometer e\ epiecc by which the frii could be seen and their positions fixed. Wh<-n the cm put on the position of a fring. id b\ the micrometer \\ iie. Thru the current Wai I6V< r-ed and t he trin^ - xhifh-d, and by an amount sufficiently near to that which woi'ld be e\pc cted from t he oi.-. i \ . d rotation of the- planes of polarisation in the liquid 1 ame current. Hraee4- made a similar experiment with a similar result. - time later, by making the tuo ra\s travel in an an nt of 5890 D, I'M. prisms to and fro along the lines of fosce, 1 edt-d in separating them on emergence. Macahiso and Corbinot >ho\\i(l that the lotation is in the same direction on opposite ridefl -dium \apour absorption band, anil W«)od§ measured the i in the neigh bourhood of both 1) lines, and obtained results of the kind icpn-entcd in Fig. 245. Butintht i solution of praseodidymium chloride. * 1 1 W. JfriMiittir, ii. p. 71."» (1 Phil. Maij. i. p. n;i (inn]). II the inlnxliii-iin- carlior work, :ind tlit-n t d'Hipt. llrud. cxxvii. ]>. .MS; W,,«,,l. ft/,fi,-M. L'n.l re two periods t-xist for a ray travelling perpendicularly to the lines of force in which the motion of the electrons is also perpen- dicular to the lines of force, the equation for the velocity v before the field exists, which is 1 _R+ ft must be replaced by the less simple form -i,=K + ft W-v-n- , neglecting higher powers. The velocity before the field is established is also the velocity of a rav which has its vibrations along the lines of force, and which i> therefore unaffected by the field. Let us put it vp = v. If we take off' the field, v becomes = 0 and vn — vp. This requires that 2ft1 = ft. We have then _ _ v* vp* (N2-/*2)3* It will be noted that this is a small quantity of a higher order than the corresponding difference which accounted for the rotation * P!,U. Miiy. ix. p. 7'2~> (11)0.")). 334 MAGNETISM of the plane of polarisation. Further, it changes sign with N — n ; or when n < N,z^ >Pn, and when n is > N, vp < *>»• The theory of which this is a mere indication led Voigt to the result just obtained, and it was verified by Voigt and Weichert in the case of sodium vapour.* Magnetic double refraction in colloids due to sus- pended particles. In 1901f Kerr observed that if a small quantity of Fe8O4 is chemically precipitated in invisibly fine particles in water, which is thereby rendered slightly hazy, the liquid becomes doubly refracting when it is placed in a magnetic field and is traversed by a polarised ray perpendicularly to the field. The vibrations in which the electric component is along the lines of force, travel more rapidly than the vibrations in which it is at right angles. At the same time the former are more absorbed. This agrees with the supposition that the particles of the oxide arrange themselves in fine filaments along the lines of force, and act like a Hertz's grating of parallel wires. Very shortly after this, MajoranaJ independently discovered the phenomenon in colloidal solutions of iron, and found that it \va^ very notable in long-prepared "Fer BravaU." Cotton and Mouton § also worked at the subject, examining many colloidal solutions. Magnetic double refraction in pure liquids. At first Cotton and Mouton supposed that the colloidal condition was necessary for magnetic double refraction, but they discovered that it is exhibited by nitrobenzene, and it has been observed in other aromatic compounds. The phenomenon is c-losely parallel to the Kerr electric effect. If 8 is the retardation of one ray behind the other in traversing length /of a substance placed in >i field H which is at right angles to the path, and if X is the wave length C is the constant of magnetic double refraction for the substance for the wave length used. Cotton and Mouton found that for nitro- benzene C increases as X diminishes, and that if B is the Ken- electric constant C/B is very nearly the same for different wave lengths. This has been confirmed for other aromatic compounds by McComb and Skinner.|| A theory of magnetic double refraction has been developed by Havelock,H who supposes that the molecules are differently spaced along and perpendicular to the field. Cotton and Mouton prefer * Wiffd. Ann. 67, p. 345 (1899) ; or Wood's Optic*, chap. xvii. t Brit. Asuoc. Report (1901), p. 568. i Science Abstracts, 1902, No. 1844. § Ann. dr. C/iim. ct de Phys. [8], vol. xi. p. 146 and p. 289 for their earlier work; Journal de Physique [6], vol. i. (1911), p. 5, for their later discovery. j| Phys. Rev. xxix. p. 525 and p. 541 (1909). f Prop. Roy. /Sue., A, vol. Ixxvii. p 170, and vol. Ixxx. p. 28. MAGNETISM AND LIGHT 335 to account for the effect by supposing, not that the molecules alter their distances apart, but that they are oriented with axes of some kind along the lines of force. They consider that Voigt's theory will not account for the double refraction which they have discovered, for his theory should apply to all bodies, whereas their double refraction is so far confined to a particular class. The Kerr magnetic effect. Reflection from magnets. The reflection of light at a surface is not merely a return of the waves at an impenetrable surface, for the nature of the reflection depends on the constitution of the medium within the surface. For instance, the polarising angle differs with different reflecting media, showing that the medium near the surface takes part in the process. Con- sidering this, and further considering that, as the Faraday effect in some salts of iron is already great, it might be expected to be vastly greater in the steel of a magnet, Dr. Kerr* was led to expect some kind of rotation in a plane-polarised ray when reflected from the pole of a magnet. To test his expectation he polished a flat as perfect as possible on one pole of an electro-magnet and then brought the other pole near to it, just leaving between them a chink through which a ray of plane-polarised light could be directed on to the surface at an angle of incidence between 60° and 80°. When the plane of polarisation was either parallel to or perpendicular to the plane of incidence and the current was off, the plane was unaltered by reflection. These two planes were selected because, in accordance with the ordinary result of reflection from a metal, the light in any other plane would after reflection be elliptic-ally polarised. After reflection the ray passed through a Nicol to the observer's eye. The Nicol was turned to extinction. On putting the current on, the light reappeared faintly, the plane of polarisation always turning in the direction opposite to that of the magnetising current, and the polarisation was slightly elliptical, as the principle of reversibility would lead us to expect. In further experiments Kerr worked with normal incidence through a perforation in the opposing pole and found again a rotation, though less in value, and again in the opposite direction to the magnetising current. According to the later work of Kundt, the light entering the metal and finally absorbed by it would have its vibration turned round in the direction of the magnetising current so that we might expect the reflected vibration to be turned in the opposite direction. Afterwards Kerr obtained similar effects with reflection at surfaces worked on the equatorial parts of a magnet. He summed up his results in the statement that a new small component was introduced in the reflected ray with vibration perpendicular to that of the ray reflected before the iron was magnetised. The vibration * P/til. Mag. [5], vol. iii. p. 321 (1877), for reflection from a pole ; vol. v. p. 161 (1878), for reflection from the side. 336 MAdNKTlSM in the original direction \\as not changed in pha-c h\ ma tion. The new component differed fro in it in pi -mail ai I)u Bois* measured tin- rotation on reflection at normal incidence from mirrors worked on the cmN of iron, nickel, and cobalt cores of electro-magnets. The cores were in tin- spheroids. He Used this >hapc b. cording to the them induction, if without the metal core the lie-Id is uniform, the magnetisation I, when the metal is put in, is uniform parallel to the axis. Measuring the rotation t and thcint magnetisation I. du Hois found that e = KI, and he termed K the Kerr constant. If the magnetisation was not normal to tin- surface, then I was replaced bv I,., the normal component. K found to be nearly independent of temp- and \\lnn t was expressed in minutes it was found that K For iron — IH> for cobalt — 0-0198, and for nickel -n-oHJO. while for magn. it was +0*0 l!,'. Tin :.Ur tor iron from red to violet, while for Cobalt there \\as a minimum in tb and for nickel a minimum in tin- \ello\\ of the .spectrum. The Zee man effect.4- Tin- ti. light which \\r ha\e a I read \ de-« MisM in the light due to changes in ti \hieh it milled, oral \\hichit is reflected \\hentb -id. The modification^ ma\ be a-mlxd to a!' not to alterations in the Impicm 1 alteration in \\a\e fre.pienc\ \\ In . lati\e n.. sourci- and medium and tin- condition i- stiaed the Dlii. fine and sharp. When the current \\as put on thev \\id- perceptiblv. The red lithium lin similar effect. To show that the effect was directly du. magnetic field and not to anv indin-i -tich as | ti-mperature through the change ot si, apt- of the ll . Mth gUlfl ends \\as put across the magnetic axis in place of the Hame. A piece of sodium was ins^Tted in the tube, \\hich \va-> heated so that sodium vapour \\a> formed. A b-am of white light from an * /'//;/. May. xxix. --0). t '/•• •' Mircht* in Magnfttt-Optict, Macmilla < in NtoNj. The total force l<» the ccnl're it' tl 47r2N2wd + lift* = 47r2NW 4- J^rNJIrr/, putting 2-N^/ 1, The acceleration is (4-Tr2 N2 + 2- m t; \ \ A oounier-clodcwiM motion v^ill 1. : D «»ut- \\aid-, and M> the frccjiu-ncy i^ reduced. I \ n 1>\ V N or approximately There are therefore ^t the original 1 y N stil ^remaining in the vihrations nlnn^ the lines of force, nn«l them are two sets of circular motions in a plane perpcndiculni lines of i <1 \\ith fivijucneies, one rather greater and tin- other rather less than \ Consider the li«:lit _ur«»in.i: out c.juatoi ;lar vibra- tions will only send linear vil. in that M, for it is in their own plane, and the direction of vil> n the waves will be perpendicular to the lin source aloni: the lines of force will send out vibrations parallel to that direction and of t li- the light in a spectroscope, ue mav expect to lia\. a line in tin- position of the original line polarised in the equatorial plane, or, as it is easier to picture, vi! ..arallrl to the lines of force, and this should he Hanked 1>\ Uo lines \\ith \il>i erse to the lines of force. If we consider the light travelling out axially, then the vibration along the lines of force \\ith the orii: frequency sends n<> waves in that direction, but the Uo opposite sets of circular vihrations send opp« ilarly polarised rays, and in a spectroscope we ought to have two lines, one on each side of the original position, and the light in them should be circularly polarised in opposite din Zeeman's verification. Zeeman found that experiment verified these anticipations of theory. But further research with MAGNETISM AND LIGHT 339 stronger fields and better resolving power showed that the effect of establishment of the field was usually considerably more compli- cated than in the simple case which we have treated, though for some lines there is mere tripling, as described above.* For instance, the sodium line D1 shows a quadrupling in the equatorial direction, the polarisation indicating that the central line is doubled. D2 shows six lines, and the polarisation indicates that the two side lines are also doubled. Lorentz f showed that if there are n degrees of freedom in the vibrating source, all coinciding before the establishment of the field, and so giving one line in the spectrum, we may suppose them to be made all slightly different when the field is established, thus giving n lines. Thus in our simple case w = 3, and each of the degrees has the same period if there is no field. When the field is put on there are three periods and three lines in place of one. Since Zeeman made his great discovery a very great deal of work has been done by himself and by others in investigating the details of the effect, but these details belong more to optics than to magnetism. J We shall conclude with a very interesting calculation which Zeeman made, in his first paper, of the value given for e/m by the observed widening of the lines. From the values of Nx and N2 we get NX2- N22 = and where \v Xa are the wave lengths of the two side lines, and V is the velocity of light, ..gTryXg-Xj 1 fir JT~ *x' -luce in the denominator X15 A2 may be put equal to A, the original wave length. Now, as 2~ * was found to be ^QQQQ, for X = 0-000059, when H was 10,000, and as V is 3 X 1010, e/m is about * S.-hu-ter's Tlicnn/ of (Jj>ti<-*, I.e. . . t The Theory of VUctrvH* (1909), i». 112 ; or Congrts International cle Physique (1'JOO), vol. iii. p. 1. J A bibliography is giveii iii Zeemaii's work (see p. 330). MAGNETISM 107. Various later measurement* of the separation of the lim known fields have shown that elm ob; n thU way is a 1; different for lines in dillri ^, but that it is the same for 1 in the same series. Tli- \alue is very near to 1*77 X i which again is very near to tl i»e»t measurements of elm for the kathode rays and for p ra\ I, INDEX ALCOHOL, specific inductive capacity, Ampere, hypothesis as to constitution of magnets, 201 Analogous and antilogous pyroelectric poles, 149 Armatures for magnets, 175 Artificial magnets, 1 Atmospheric electricity, 95 Axis of a magnet, 215 BALL- ENDED magnet > Ballistic method of measurement of permeability, LTI Belli's machine, 17 Bennett's Doubler, 16 Biot, experiments with nickel needle, 203 Bismuth, von Ettingshausen's experi- ments with, 285 Boltxmann, experiments in specific in- ductive capacity for pases, 127 experiments in specific inductive capacity for solids, 122 investigations with crystalline sul- phur, 124 CAPACITY, Measurement of, 96 of a conductor, 81 Centre of a magnet, 215 Charge, Measurement of, 30 Unit of, 28 Charged surfaces, pull per unit area, 104 :iar disc, Intensity due to, 40 Clouds and lightning conductors, 14 Cobalt and nickel, magnetisation and permeability, magnets, Coil as a magnet, 181 Colloids, Magnetic double refraction of, 334 Condenser, potential and capacity, 87 Conduction, 3 Electrification 1 Conductor, Capacity of, 81 Intensity outside surface of, 36 proof that charge on outside only, actors. Distribution of electrifica- tion on, 26 Consequent poles, in a magnet, 1 7 1 841 Coulomb's direct measurement of forces between electrified bodies, 62 experiments to establish inverse square law for magnetic field, 208 Curie, discovery of piezoelectricity, 153 experiments on suscepibility and permeability, 286 law for paramagnetic bodies, 291 Current, Electric, produces equal positive and negative, 26 Cylinder, Intensity due to charged, 41 Cylinders, Potential and capacity of two co-axial, 83 Potential and capacity of two parallel, 84 DECLINATION, 301 Lines of equal, 315 Diamagnetic bodies, Induced magnetisa- tion in, 249 Diamagnetics, 203 tend to move to weaker field, 205 Dielectric, 50, 99 Alterations in, when subject to electric strain, 142 constant of ice, 131 Quincke's experiments, 138 Stresses in, 134 Dip, or inclination, 304 or inclination by dip circle, 305 or inclination by earth inductor, 308 Discharge from points, 13 of electrification, 11 Disturbances in magnetic field of earth, 313 Dolazalek electrometer, 89 Drude's experiments on specific inductive capacity with electric waves, 131 EARTH as a magnet, 169, 301 inductor, 308 Earth's lines of force, 301 magnetic field, disturbances in, 313 magnetic poles, 315 Ebonite, Inductive capacity of, 99 Electrification by conduction, 25 by friction, 1, 25 by heating, 149 by induction, 5, 26 Coulomb's test of distribution of, 05 342 INDIA ElectrificatioD, Discharge of, 11 Distribution on conductors, 28 Occurrence of the two kii.cU together, 10 on outer surface only of conductors, 9 positive and negative, 2, 25 Quantity of, 2-1 Units of, 28, 32 Electrified bodies, action between, 2 Electric action in the medium, 48 current produces equal positive and negative, 28 deformation of crystals, 157 displacement (Maxwell). 49 double refraction (Kerr effect), 144 expansion in glass, 142 expansion of liquids, 143 field, 32 intensity of field, 32 intensity in dielectric, 103 mechanical action, Law < strain and intensity, Relations be- tween, 87 strain, 48 strain, direction of, 60 strain, measurement of, 52 strain, molecular hypothesis of, 58 strain, variation of, with distance, 58 waves, Drnde's experiments OB specific inductive capacity with, 131 Electrical level, or Potent i. Electricity, Atmospheric, 95 Electro-magnetic waves, 49 Electro magnets, 178, 182 Electrometer, Dolasalek, 89 Harris's attracted disc, VI Piezoelectric, 156 Quadrant, 87 Thomson's trap door, VI Electron theory of magnetism, 297 theory of rotation of plane of polari- sation, 328 Electrophone, 14 Electroscope, gold-leaf, 4 gold-leaf, capacity of, 97 gold-leaf, detection of charge by, 6 gold-leaf, use of, to indicate equality of charge, 24 Energy, dissipation of, in a hysteresis cycle, 199 distribution of, in a dielectric 104 distribution of, in an electric system, of charge in a dielectric, 103 of a system, 44, 70 per c.c. in magnetised body with constant permeability, 248 supplied during a cycle when perme ability varies, 248 Equipotentiiil surfaces, -I : Ewiog's model (molecular theory of magnetism), 195 I.,..: ..-..;• : i: . .„•!.• I |.il. meability, 268 FARADAY, dbcovery of specific indi. capacity, 99 experiments with cobalt as a magnet, | , with para- and dia- on action of magnetism on light, 820 Faraday's " butterfly -net " experiment, M Ferro-magnetios, 207 32 electric, force on small body in, 61 electric, intensity of, 82 Fleming and Dews experiments on sosceptibiht r of oxygen, 295 Flux of force in magnetic field. 216 Force on a small body in electric field, 61 Forces between electrified bodies, bodies, 258**,. Kranklin'sjar, 22 Friction*! Bnohine* ,...,., Gauss, proof of in in pyroelcc square law (mag- applied to tube* of force, 84 . . , .Ml. «:. .• Ulaas* gf iaaalalor 4 •ejld in. 142 UuM.|cafclcctr'u»co|*.4 capacity of. V7 detection of change by, 6 HAM MI. ui NO, Effect of, on tion. 194 !• • . • . . • .' .:...-.., city of water, 180 ian waves, 49 HopUnsoo on rate of fall of potent Leydenjar, 117 on specific inductive capacity Hydraulic illustration of magnetic per- Hysteresis cycle, Dissipation of loop, and energy *