ELECTRICITY AND MAGNETISM BY ERIC GERARD Director of Institut Electrotechnique Montefiore^ University of Liege, Belgium TRANSLATED FROM THE FOURTH FRENCH EDITION UNDER THE SUPERVISION OF DR. LOUIS DUNCAN, BY R. C. DUNCAN NEW YORK McGRAW PUBLISHING COMPANY 114 LIBERTY STREET 1897 Engineering Library Copyright, 1897, BY THE W. J. JOHNSTON COMPANY. PUBLISHERS' NOTE. IT is scarcely five years since the original work of Pro- fessor Gerard appeared, but in that time it has reached the fourth edition in the French, and has been translated into the German. The work has evidently become a classic in Europe ; and in view of this fact, and of the further consider- ation that it occupies a place by itself in electrical literature, the publishers have deemed it wise to bring out an Amer- ican edition. The original intention of the author was to produce a work which, while avoiding on the one hand the shortcom- ings of the more elementary works, would not, on the other hand, be so difficult to read as to be only intelligible to the favored few. How well he has succeeded in this intention may best be judged by the favorable reception of the book abroad. The present work is a translation of the fourth French edition by Mr. J. P. Duncan, under the supervision of Dr. Louis Duncan of the Johns Hopkins University and pres- ident of the American Institute of Electrical Engineers. All parts of the work relating to the general subject of elec- tricity have been retained, but the chapters on special sub- jects, such as storage batteries, transformers, and other electrical machinery, have been omitted for two reasons : first, because the information contained in them is easily acces- sible in other well-known works (which is not the case with 257732 IV PUBLISHERS' NOTE. the parts that have been retained) ; and second, because the descriptions of particular machinery and apparatus refer almost exclusively to European practice, which is in some cases quite different from American practice. Again, the saving of space made by these omissions allows the publish- ers to add much valuable new matter. There is a chapter on hysteresis and molecular magnetic friction by Mr. Charles P. Steinmetz, a well known authority on the subject. The short section of the original work on units and dimensions is replaced by a chapter written by Dr. Gary T. Hutchin- son, which gives a comprehensive view of the theory. A chapter on impedance by Dr. A. E. Kennelly, is another valuable addition to the work, as it makes plain those points in the theory of alternating currents which give the average student the most trouble. CONTENTS. CHAPTER I. Introduction. PAGE SECTION 1 2 Fundamental units. 2 2 Derived units. 2 3 Example of a derived unit. 3 4 Dimensions of a derived unit. 3 5 Mechanical derived units. 5 6 Principle of the conservation of energy. 7 7 Multiples and submultiples of the units. 7 8 Application of the dimension of units. GENERAL THEOREMS RELATIVE TO CENTRAL FORCES. 8 Q Definitions. 9 10 Elementary law governing the Newtonian forces. 10 ii Field of force. 11 12 Potential. 14 13 Equipotential surfaces. 14 14 Case of a single mass. 15 15 Uniform field. 15 16 Case of two acting masses. 17 17 Tubes of force. 17 18 Flux of force. 18 19 Theorem of flux of force in a tube of infinitely small section. 18 20 Gauss' theorem. 19 21 Corollary I. 20 22 Corollary II. 20 23 Corollary III. 21 24 Unit tube. Number of lines of force. 21 25 Potential energy of masses subjected to Newtonian forces. APPLICATIONS. 23 26 I. An infinitely thin homogeneous spherical shell exercises no action upon a mass within it. v VI CONTENTS. PAGE SECTION 25 27 II. The action of a homogeneous spherical shell on an ex- ternal point is the same as if the whole mass were con centrated at the centre of the sphere. 26 28 Action of a homogeneous sphere upon an external point, 27 29 Action of a homogeneous sphere upon an internal point, 27 30 Surface pressure. 29 31 Potential due to an infinitely thin disk uniformly charged. CHAPTER II. Magnetism. 31 32 Definitions. 31 33 Action of the earth on a magnet. 31 34 Law of magnetic attractions. 32 35 Unit pole. 33 36 Definitions. 34 37 Action of a uniform field on a magnet. 35 38 Terrestrial magnetic field. 36 39 Weber's hypothesis. 38 40 Elementary magnets. Intensity of magnetizatiou. 40 41 Magnetic or solenoidal filament. 41 42 Uniform magnets. 43 43 Magnetic shells. 44 44 Corollary. 45 j.t; Energy of a shell in a field. 46 46 Relative energy of two shells. 46 47 Artificial magnets. 50 48 Determination of the magnetic moment of i magnet Magnetometer. 53 49 Remarks. 53 50 Measurement of angles. INDUCED MAGNETIZATION. 56 51 Magnetic and diamagnetic bodies. 56 52 Coefficient of Magnetization or Magnetic Susceptibility. 57 53 Cases of a sphere and a disc. 59 54 Case of a ring. 59 55 Case of a cylinder of indefinite extent. 60 56 Portative power of a magnet. 61 57 Variations of intensity of magnetization with the magnet**- ing force. Hysteresis. 66 58 Frolich's formula. 66 59 Formula of Miiller, von Waltenhofen, and Kapp. 67 60 Another way of looking at induced magnetization, magnetic induction, and permeability. CONTENTS. Vll PAGE SECTION 71 5i Work spent in 'magnetizing. 74 62 Numerical results. 78 63 Effect of temperature on. magnetism. Recalescence. 80 64 Ewing's addition to Weber's hypothesis. 84 65 Equilibrium of a body ii&a magnetic field. CHAPTER III. Hysteresis and Molecular Magnetic Friction. 87 66 Hysteresis. 93 67 Hysteretic loop. 95 68 Molecular magnetic friction. 97 69 Determination of hysteresis and molecular magnetic fric- tion. 101 70 Loss of energy. 104 71 Coefficient of hysteresis. 107 72 Eddy currents. 109 73 Effect of molecular magnetic friction and eddy currents, no 74 Equivalent sine curves. 115 75 Hysteretic losses. CHAPTER IV. Electricity. 117 76 Phenomenon of electrification. 118 77 Conductors and insulators. 119 78 Electrification by influence. 120 79 Quadrant electrometer. 121 80 Experiments. 123 81 Distribution on a conductor. 124 82 Law of electric actions. 125 83 Definitions. Electric field. Electric potential. 125 84 Potential of a conductor in equilibrium. 126 85 Potential of the earth. 127 86 Coulomb's theorem. 128 87 Electrostatic pressure. 128 88 Corresponding elements. 129 89 Power of points. 129 90 Electric screen. 130 91 Lightning-rods. CONDENSERS DIELECTRICS. 130 92 Capacity of conductors. 131 93 Condensers. Spherical condenser. 132 94 Plate condenser. r 33 95 Guard-ring condenser. '34 96 Absolute electrometer. Vlll CONTENTS. PAGE SECTION 135 97 Cylindrical condenser, one plate connected to earth. 136 98 Cylindrical condenser disconnected from earth. 136 99 Leyden jar. 137 100 Energy of electrified conductors. 138 101 Theory of the quadrant electrometer, 139 102 Quadrant electrometer formula?. 140 103 Specific inductive capacity of dielectrics. 141 104 Nature of the coefficient k in Coimmo's law. 142 105 Role of the dielectric. Displacements. 145 106 Residual charge of a condenser. 146 107 Electromotive force of contact. Distinction between elec- tromotive force and difference of potential. ELECTRIC DISCHARGES AND CURRENTS. 148 108 Convective discharge. 149 109 Conductive discharge. Electric current. 151 no Disruptive discharge. Electric spark and brush ; their effects. LAWS OF THE ELECTRIC CURRENT. 154 in General considerations. 155 112 Law of successive contacts. 155 113 Thermal and chemical electromotive forces. Means of keeping up a constant difference of potential in a con- ductor. 156 114 Ohm's law. 157 115 Case of a conductor of constant section. 158 116 Graphic representation of Ohm's law. 159 117 Variable period of the current. 160 118 Application of Ohm's law to the variable period of the cur- rent in but slightly conductive bodies. 161 119 Application of Ohm's law to the case of a heterogeneous circuit. 162 120 Graphic representation. 163 121 Kirchhoff s laws. 165 122 Application to derived circuits. 166 123 Wheatstone's bridge or parallelogram. ENERGY OF THE ELECTRIC CURRENT. 167 124 General expression. 168 125 Application to the case of a homogeneous conductor. Joule's effect. 168 126 Case of heterogeneous conductors. Peltier effect. CONTENTS. IX JAGH SECTION 169 127 Chemical effect of the current. Faraday's and Becquerel's laws. 170 128 Grothiiss' hypothesis. 171 129 Application of the conservation of energy to electrolysis: Voltaic cell, THERMO-ELECTRIC COUPLES. 174 130 Seebeck and Peltier effects. 176 131 Kelvin effect. 178 132 Laws of thermo-electric action. 178 133 Thermo-electric powers. 183 134 Thermo-electric pile. CHAPTER V. Electromagnetism. MAGNETIC PHENOMENA DUE TO CURRENTS. 184 135 Oersted's discovery. 185 136 Magnetic field due to an indefinite rectilinear current. 186 137 Laplace's law. 189 138 Action of a magnetic field on an element of current. 190 139 Work due to the displacement of an element of current under the action of a pole. 191 140 Work due to the displacement of a circuit under the action of a pole. 192 141 Magnetic potential due to a circuit. Unit of current. Am- pere's hypothesis on the nature of magnetism. 196 142 Energy of a current in a magnetic field. Maxwell's rule. 197 143 Relative energy of two currents. 197 144 Intrinsic energy of a current. 198 145 Faraday's rule. APPLICATIONS RELATING TO THE MAGNETIC POTENTIAL OF THE CURREN 1 / 200 146 Case of an indefinite rectilinear current. 202 147 Case of a circular current. Tangent-galvanometer. 205 148 Thomson galvanometers. 207 149 Shunt. 208 150 Measurement of an instantaneous discharge. 210 151 Solenoid. Cylindrical bobbin. 214 152 Electrodynamometer. 215 153 Case of a ring-shaped bobbin or solenoid. ELECTROMAGNETIC ROTATIONS AND DISPLACEMENTS. 218 154 General statements. 218 155 Rotation of a current by a magnet. X CONTENTS, PAGE SECTION 219 156 Rotation of a brush-discharge. 220 157 Barlow's wheel. 221 158 Rotation produced by reversing a current. 222 159 Mutual action of currents. 222 160 Reaction produced in a circuit traversed by a current. 223 161 Explanation of electromagnetic displacements based on the properties of lines of force. ELECTROMAGNET.;. 226 162 Description and definitions. 227 163 Energy expended in electromagnets. General definition of the coefficient of self-induction of a circuit. 230 164 Magnetic circuit. Magnetomotive force. Magnetic resist- ance or reluctance. 234 165 Forms and construction of electromagnets. 240 166 Magnetization of a conductor. 240 167 Modifications in the properties of bodies in a magnetic field. 241 168 Hall effect. CHAPTER VI. Units and Dimensions. 244 169 Units. 245 170 Dimensions. 247 171 C. G. S. system of units. ELECTRICAL AND MAGNETIC UNITS. 248 172 General considerations. 249 173 Systems in terms of K and jit. 252 174 Proposed nomenclature. 254 175 Dimensions of K and ju. 256 176 Value of ratio " z>." 257 177 Practical system. 259 178 Nomenclature of practical units. 260 179 "Rational" system. 263 180 Electrical standards of measure. 265 . 181 Recommendations of Congress in 1893. CHAPTER VII. Electromagnetic Induction. 266 182 Induced currents. 268 183 Lenz's law. 269 184 General law of induction. 270 185 Maxwell's rule. 271 186 Faraday's rule. 272 187 Seat of the electromotive force of induction. CONTENTS. XI PAGE SECTION 273 188 Flux of force- producing induction. 277 189 Quantity of induced electricity. APPLICATIONS OF THE LAWS OF INDUCTION. 278 190 Movable conductor in iv a uniform field. 278 191 Faraday's disc. 279 192 Measurement of the intensity of the magnetic field by the quantity of electricity induced. 280 193 Expression for the work absorbed in magnetization. Loss due to hysteresis. 283 194 Self-induction in a circuit composed of linear conductors. Case of a constant electromotive force. Time constant. 287 195 Work accomplished during the variable period. 289 196 Application to the case of derived currents. 290 197 Discharge of a condenser into a galvanometer with shunt. 292 198 Self-induction in a circuit of linear conductors where there is a periodic or undulatory electromotive force. 298 199 Graphic representations. 301 200 Mean current and effective current. Measurement by a dynamometer. 305 201 Mutual induction of two circuits. 305 202 Mutual induction of two fixed circuits. 306 203 Quantity of induced electricity. 307 204 Expression for mutual inductance. 308 205 Induction in metallic masses. 309 206 Foucault currents. 310 207 Cores of electromagnets traversed by variable currents. Calculation of the power lost in Foucault currents. 312 208 Self-induction in the mass of a cylindrical conductor. Ex- pression for the coefficient of self-induction in such a conductor. ROTATIONS UNDER THE ACTION OF INDUCED CURRENTS. 317 209 General rule. 318 210 Ferraris' arrangement. 320 211 Shallenberger's arrangement. 320 212 Repulsion exercised by an inducing current upon an induced current. CHAPTER VIII. Impedance. 324 213 Inductance. 327 214 Inductance and capacitance. 328 215 Inductance and reactance. 331 216 Joint impedance. Xll CONTENTS. CHAPTER IX. The Propagation of Currents. PAGE SECTION 333 217 Phenomena which accompany the propagation of the current in a conductor. 337 218 Special characteristics shown by alternating currents. 339 219 Comparative effects of the self-induction and capacity of a circuit. 340 220 Effect of a capacity in a circuit traversed by alternating currents. 342 221 Combined effects of a capacity and a self-induction in a cir- cuit traversed by alternating currents. Ferranti effect. 344 222 Oscillating discharge. 350 223 Transmission of electric waves in the surrounding medium. 357 224 Present views on the propagation of electric energy. CHAPTER X. Electrical Measurements. 361 225 Case of a continuous current. 361 226 Siemens Wattmeter. 362 227 Case of a periodic current. 362 228 Non-inductive conductors. . . 364 229 Conductors having self-induction. MEASUREMENT OF THE INTENSITY OF A MAGNETIC FIELD. 367 230 Method by oscillation. 367 231 Electromagnetic method. 368 232 Method based on induction. MEASUREMENT OF MAGNETIC PERMEABILITY. 369 233 Method based on induction. 371 234 Another similar method, 372 235 Remarks on these methods. 373 236 Magnetometric method. 374 237 Method by portative power. MEASUREMENT OF COEFFICIENTS OF INDUCTION. 375 238 Maxwell and Rayleigh's method for measuring a coefficient of self-induction in terms of a resistance. 377 239 Maxwell's method, modified by Pirani, for measuring a self- induction in terms of a capacity. 378 240 Ayrton and Perry's method for comparing a coil's self- induction with that of a standard coil. 380 241 Mutual induction Carey-Foster method. THERMOELECTRIC COUPLES. INTRODUCTION. UNITS OF MEASUREMENT. A phenomenon is well known only when it is possi- ble to express it in numbers. (KELVIN.) I. Fundamental Units. All electrical actions are re- ferred to forces, and are consequently expressed by the aid of the three fundamental quantities, length, mass, and time. To measure these quantities electricians have chosen the centimetre, the gram, and the second as units. The centimetre is approximately the billionth part of the terrestrial quadrant ; rigorously speaking, it is the hundredth part of the standard metre measured by Delambre and Borda, and kept at the international conservatory at Sevres. The gram represents about the mass of a cubic centi- metre of distilled water at its maximum density. There is also a standard kilogram at Sevres. The second is the 86,4OOth part of the mean solar day. These units, called fundamental, are represented by the symbols [], [J/], [T]. The numerical value of a quantity is expressed by its ratio to the unit chosen. A length measured by a number / will have a concrete value equal to /[]. If we adopt another unit [Z/], there will be a numerical value /', such that /'[//] = /[], whence l jt = . 2 IN TROD UCTION. We see, therefore, that the numerical value of a quantity is in inverse ratio to the magnitude of the unit chosen. 2. Derived Units. In order to express the various physical quantities, arbitrary units might be chosen quite independent of each other. This method, which was long followed, presents no inconvenience when the measures are relative, that is, when they are directly compared with their units. But more often quantities are measured by units of other kinds, making use of the inter-relations between the different quantities. Such a system of measurement is called absolute. For example, to measure a surface we do not com- pare it directly with a standard area, but determine its linear elements, by aid of the unit of length, and then apply the relation existing between an area and its linear dimensions. For a square s, with a side /, the relation is s kP. If /= i, j = k. The arbitrary factor k, which represents the area of a square having unit side, may be put equal to i. The unit of area thus determined is the square whose sides are equal to one centimetre ; it is connected with one of the funda- mental units, and for this reason is called the derived unit of area. In the same way the derived unit of volume is the cube having sides of one centimetre. We can thus define derived units for all the physical mag- nitudes, getting rid of the arbitrary coefficients in the rela- tions which unite these magnitudes together. The system of units determined in this way is called by the initials of the fundamental units chosen. The C. G. S. system of units, adopted by electricians, has as its basis the centimetre, the gram, and the second. 3. Example of a Derived Unit. The velocity of a mov- UNI TS OF ME A S U RE MEN T. 3 ing body, traversing a path / in a time ^, is given by the equation ' k expresses the velocity of % moving body traversing unit length in unit time. This velocity is chosen as unity, thus eliminating a factor inconvenient in calculation. This unit is the velocity of one centimetre per second', it may be ex- pressed in symbolic form 4. Dimensions of a Derived Unit. Such an expression, which shows the dependence of the derived unit on the fundamental units, exhibits the dimensions of the derived unit. It enables us to follow the variation of the derived unit when the fundamental units are changed. If, for ex- ample, we measure the time in hours and the length in metres, the derived unit of velocity will be \L'T' ~ '] = 100 X 3600 ^[ZjT- 1 ], or the thirty-sixth part of the unit defined above. Every relation between physical quantities is independent of the units chosen to measure it, therefore this relation must be homogeneous with regard to the fundamental units. Thus the equation v al, in which v represents a veloc- ity, / a length, and a an abstract number, is inconsistent, for only the first member would vary with the unit of time. 5. Mechanical Derived Units. Following the line of reasoning used above, it is readily seen that the unit of an- gular velocity is the velocity of a moving body which passes over unit angle in unit time. As the unit angle or radian (arc equal to the radius) is defined by a simple numerical ratio, the dimensions of angular velocity reduce to \T~ ']. 4 IN TROD UCTION. The unit of acceleration is the acceleration by which the velocity is increased by one unit per second. Dimensions [LT-1. The unit of quantity of movement (momentum) is the mo- mentum of unit mass moving with unit velocity. Dimen- sions \LMT " ']. The unit of force, which has received the name dyne, is the force which, applied to unit mass, impresses upon it unit acceleration. Dimensions \LMT~*\ The ordinary unit of force is the weight of the gram, that is, the force capable, in our latitude,* of impressing on unit mass an acceleration approximately equal to 981 cm. per second. The gram is consequently equal to 981 dynes. The unit of work, called erg, is the work done by unit force on a body moving in the direction of its action over unit length. Dimensions \_UMT-*]. The ordinary unit of work is the kilogrammetre, which is equal to 981 X IO 5 ergs, in our latitude.* The unit of po^ver, or erg per second, is the power de- veloped when unit work is done in unit time. Dimensions [/, 2 MT ~ ']. The ordinary units of power are the cheval-vapeur (French horse-power), which is equal to 75 X 981 X IO 5 = 736 X io 7 ergs per second, and the poncelet, defined by the Congress in Mechanics, 1889, as equivalent to 100 kilogrammetres per second, or 981 X io 7 ergs per second. The unit of density is the density of a body which con- tains unit mass in unit volume. Dimensions [L~ 3 M], The unit of modulus of elasticity is the modulus of a body which, supporting unit force per unit of section, receives * Li&ge, lat. 50 45' approx. UNITS OF MEASUREMENT. $ an elongation equal to its original length. Dimensions 6. Principle of the Conservation of Energy. Work ap- plied to a system is capable^of various effects. It may be used : (i) To increase the active energy of the masses, or, to use Rankine's expression, to develop kinetic energy, repre- sented by the product of half the sum of the masses into the square of their velocity. (2) To overcome the friction of the system ; it was long believed that this effect represented a loss of energy, but thermodynamics has shown that in such a case there is generated an amount of heat equivalent to the work expended. (3) To overcome molecular forces, such as elasticity, chemical affinity ; or to overcome natural forces, such as gravitation, magnetic attraction, etc. In this case the work is stored up in the system in the form of po- tential energy, which is again transformed into kinetic energy or heat, when the system is abandoned to the reaction of the forces concerned. Let us suppose, for example, that we raise a weight or stretch a spring which sets a clockwork in motion. The potential energy given to the weight or spring is trans- formed into kinetic energy when the mechanism is allowed to operate, and this kinetic energy is itself reduced to heat by the friction of the wheelwork. The tendency of modern science is to refer these diverse varieties of energy to a single one, kinetic energy ; calorific, luminous, or electrical radiations, for example, which to us seem potential forms of energy, might be reduced to special modes of motion of the ether. The study of physical phenomena has given us a natural law of the highest importance. The energy of a system is a quantity which cannot be either increased or diminished by any mutual action between the bodies which compose the system* IN TROD UCTION. This law of the conservation of energy, together with that of the conservation of matter, rules supreme in physical science. From this principle it results that a system cannot of it- self produce more than a limited quantity of external work, whence the impossibility of perpetual motion. The persistence and the indestructibility of energy make it as much a physical entity as matter is, and give it a leading place among the magnitudes considered in me- chanics. Energy assumes indifferently a mechanical, elec- trical, thermic, or chemical form. Experiment shows that the two former are capable of being entirely transformed into one of the two latter, but that only a portion of ther- mic or chemical energy can be made to assume a mechan- ical or electrical form. In whatever form energy may be, it possesses a mechani- cal equivalent ; it is therefore homogeneous with work \UMT~*~\) and maybe measured in mechanical units. It follows that the C. G. S. unit of heat equals the erg. One gram-degree, or lesser calorie (caloriegram), rep- resents 4.2 X io 7 C. G. S. units of heat. The electrician has constantly occasion to apply the principle of the conservation of energy, which we have just defined, for the essential role of electricity is to serve as agent for the transformation of energy. The energy of the electric current is produced from the work done by chemi- cal affinity in batteries, by using up heat in thermo-electric couples, or by an absorption of mechanical power in dynamos. The energy of the current is, in its turn, transformed into heat and light in the conductors and electric lamps ; it is capable of decomposing an electrolyte or of overcoming the resistance offered to the motion of an electromotor. The marvellous facility with which electricity lends itself to the transmission and transformation of energy, and which UNITS OF MEASUREMENT. 7 justifies the increasing number of applications of this agent, leads the electrician to compare phenomena of very diverse kinds, the measurement of which demands such a system as the C. G. S., embracing all. physical magnitudes. 7. Multiples and Submultiples of the Units. The use of the units just described leads sometimes to very large or very small numerical values. By way of abbreviation we make use of multiples or submultiples designated by such prefixes as kilo-, mega- (one million), milli-, micro- (one millionth). Thus, one megadyne = 10" dynes ; one microdyne = io~ 6 dynes. 8. Application of the Dimensions of Units. The di- mensions of the units are of use not only in verifying the homogeneity of formulas, but they allow us, as Bertrand has shown, to predict the form of a function when the physical quantities which enter into it are known. Suppose, for ex- ample, that experiment has shown that the velocity of prop- agation of an undulatory movement in a medium depends on the modulus of elasticity and the density of the medium. Then the velocity v is a function of the elasticity e, and the density d\ v = (e, d). If we consider the dimensions of the quantities which enter into this equation, we have vLT~ l = (eL- l MT-\ dL~*M). As Mis wanting in the first member, the homogeneity of the function requires it to be eliminated from the second, which is obtained by adopting the form 8 IN TROD UCTION. To bring L and T to the same degree in both members, it is clear that the function must be a radical of the second degree. From what precedes we conclude that v is a linear function of And in fact experiment shows that the relation sought is v GENERAL THEOREMS RELATIVE TO CENTRAL FORCES. 9. Definitions. Forces are called central whose direction passes through definite points, called centres of force, and whose intensity is a function of the distance between those points. The Newtonian central forces, such as gravitation, electric and magnetic attraction, are inversely proportional to the square of the distance between the acting centres. In studying the effects of these forces, it is a matter of indifference whether they emanate from the centres them- selves or have their seat in the medium which separates these centres. Thus, to account for the universal attrac- tion of matter, the simplest way is to assume that the attractive force is a property of all ponderable bodies, which act upon each other at a distance. This hypothesis has the advantage of lending itself readily to calculation. It has sufficed as a basis for celestial mechanics. Nevertheless it does not satisfy the intellect. The ordi- nary methods used in the transmission of forces show us the necessity of an intermediary, such as a tense cord, air or water under pressure, and this permits us at least to limit to GENERAL THEOREMS RELATIVE TO CENTRAL FORCES. 9 intermolecular space the idea of action at a distance. Again, the direct action of one body on another takes for granted an instantaneous effect. Now physical phenomena, even the most rapid, have p. finite time of propagation. To account for observed phenomena, physicists have been led to suppose the 'universe filled with an ocean of ether, whose waves, representing heat, light, and electrical energy, are propagated with a velocity of 3 X IO 10 centimetres per second, so that they take about eight minutes to reach us from the sun. However, for simplicity of treatment, we shall admit pro- visionally that central forces are due to the bodies from which they seem to emanate, or to an agent diffused through these bodies. In the case of gravity the observed actions are attributed to the mass of the body. In the case of the electrical phenomena that are mani- fested between bodies that have been rubbed, we shall say that an agent, called electricity, has been developed on these bodies, and, without making any supposition as to its nature, we shall speak of quantity, mass, or charge of the agent, these terms expressing merely a factor proportional to the effects produced. Thus we shall say that two bodies possess equal quantities of the agent when they produce equal effects on a third body. The quantities of the agent will be doubled, or tripled, when the forces developed are double or triple. The quantity of agent per unit of area or per unit of volume is called, the surface density or volume density. 10. Elementary Law Governing the Newtonian Forces. The preceding definitions amount to saying that the force exerted between two quantities of the agent is proportional to the product of these quantities, since it is proportional I O IN TROD UCTION. to each one of them. It is also a function of the distance between the masses concerned. In the case of Newtonian forces it is inversely proportional to the square of the dis- tance. If, then, we express by m, m' two quantities of the agent, and by / their distance, the force . The action exerted on one of the masses considered as unity would be expressed by H = k j- In the case of electric and magnetic actions, masses of the same nature repel each other, contrary to what holds in case of gravitation. In a logical system of units, the constant k is not a simple numerical factor. Consider the attraction of heavy bodies and replace force, mass, and distance by their dimensions : then the condition of homogeneity demands that k have dimensions \DM-*T-*]. II. Field of Force. Let us suppose that the quantities m, m', m" of the agent are concentrated in physical points, occupying given positions in space. If we bring into their vicinity a mass of the agent equal to unity, it is acted upon by the forces emanating from m, m', m", which form a result- ant having a definite direction and intensity. By changing the position of the point charged with unit mass of the agent we can obtain the intensity and sign of the resultant force for every point in space. The space in which such forces are manifested is called a field of force, and the resultant force, just defined, is the intensity of the field at the point where the unit mass is GENERAL THEOREMS RELA TIVE TO CENTRAL FORCES. 1 1 placed. The direction of the resultant is called indirection of the field. The value of the intensity of a field is directly deduced by the application of the elem-entary law ; but the necessary calculations by this method , become extremely complicated in the case of a number of acting masses, since the elements to be combined are vectors, that is, quantities having given magnitudes and directions, and combining according to the parallelogram of forces. The procedure is especially in- volved when the analytical method is employed. The solution of the problem is reduced to a simple alge- braic addition followed by a differentiation, by taking into consideration a new function defined by Laplace and inves- tigated by Gauss and Green under the nz.mz potential. 12. Potential. Let us suppose that unit mass is dis- FlG. I. placed in the field by an infinitesimal distance, under the action of the forces acting upon it. Let oo' = dr be the displacement. The force due to the mass m is -rr, and the work done under the action of this force is km , km dr cos a = 1 2 IN TROD UCTION. Likewise the work done by m' is km' j f by ", ~^r. These single expressions for the work done are to be added, since they are taken in the same direction ; the total work is therefore expressed by ,mdt This sum is the differential of the function m k*2~r -\- const. The expression -^-k'S, whose differential, taken with the contrary sign, represents the elementary work of the forces of the field, has been given the name oi potential by Gauss. We shall designate it by the letter U: = . For a point in space, therefore, the potential is proportional to the sum of the ratios of the acting masses to their distances from the point. The potential permits us readily to define the work ac- complished by the forces of the field. Thus, if we integrate the expression between two positions (9,, <9 2 , occupied by the unit mass, we get - o, l GENERA L THEOREMS RELA TIVE TO CENTRAL FORCES. 1 3 The work done by the field on unit mass displaced from the point O l to <9 2 is equal to the difference of the values of the function U at the two points. The work depends solely, upon the position of the initial and final points, and not on the path followed by the unit mass between these points. If the unit mass should pass from the point O 1 to an in- finite distance from the acting masses, we would have r A/ r* / -7T = / -dU = U,. J o l J o l Hence we see that the potential at any point is measured by the work done by the field in displacing unit mass from the given point to an infinite distance from the acting masses, that is, to the limit of the field. The potential function furnishes a simple expression for the intensity of the field. Let H be the component of intensity in a direction /. The elementary work //d/ is likewise expressed by the dif- ferential, taken with contrary sign, of the potential in this direction: ffdl= - whence The component of field intensity in a given direction is ex- pressed by the derivative, taken with contrary sign, of the po- tential in that direction. The force is directed towards the points where the poten- tial diminishes. 14 IN TROD UCTION. 13. Equipotential Surfaces. Put U = (#, y, z) = constant, x, y, and z representing the points of the field by rectangular co-ordinates. This equation represents a surface at every point of which the potential has the same value. Consequently the forces of the field have a zero resultant along this surface, the normal to which represents the direction of the field at each point. The surfaces thus defined are called equipotential surfaces or level surfaces, by analogy with the free surface of a liquid, everywhere normal to the force of gravity. Designating by n a direction normal to the equipotential surface, the field intensity in a point of the surface is ex- pressed by We can get a representation of the distribution of the forces of the field by imagining in the field a series of similar surfaces sufficiently near to each other and corre- sponding to potentials which increase in arithmetical pro- gression. A mass free to move in the field will follow a path cutting the equipotential surfaces perpendicularly. This curve, whose tangent represents in each point the direction of the field, has been named by Faraday a line of force. The field intensity is obviously in inverse ratio to the seg- ment of line of force comprised between two consecutive equipotential surfaces. 14. Case of a Single Mass. The case of a single acting GENERAL THEOREMS RELA TIVE TO CENTRAL FORCES. I 5 mass gives an example of a field easily defined. The equi- potential surfaces are concentric spheres, whose radii repre- sent the lines of force. Let us suppose a mass m, such that km = 6, concentrated in a point A. The concentric circles represent the intersection by a plane passing through the mass m of the equipotential surfaces I, 2, 3, 4, 5, 6. The radii of these circumferences are respec- tively 6/1, 6/2, 6/3, 6/4, 6/5, 6/6. 15. Uniform Field. We see that as the potential de- creases the equipotential surfaces are successively further and further apart. At a sufficiently great distance from the FIG. 2. centre the lines of force drawn through a region of small ex- tent are practically parallel, and the equipotential surfaces are comparable to planes in this region. In the case of gravitation, for example, no appreciable error is caused by taking, in the space occupied by a laboratory, the verticals as parallel. A field represented in this manner by equipotential planes and lines of force perpendicular to them, whose intensity is constant in magnitude and direction, is called a uniform field. 16. Case of Two Acting Masses. Let us consider the i6 INTRODUCTION. case of two acting masses, such that for one of them km = 20, and for the other km' = 5. To determine the intersection of the equipotential sur- faces due to the two centres by a plane passing through these centres, commence by tracing the circular equipoten- tial lines due to each centre considered separately. Let n t , n^, n z , n t ... be the circles drawn around the first, and /, #', #,' . . . those enveloping the second. FIG. 3. The equipotential line of the order 5 will evidently pass through the intersections of the circumferences n t , / ; ,, *,'; ,, ,'; i, / The equipotential line of the order 4 will pass through GENERAL THEOREMS RELA TIVE TO CENTRAL FORCES. I/ the intersections of the circumferences n a , n/ ; n. lt / ; ,, ,', and so on for the other orders. The lines of force will be curves normal to the equipoten- tial lines obtained. > In Fig. 3, taken from Maxwell's Electricity and Magnetism, the two masses above mentioned are concentrated in the points A and B. The full lines are equipotential ; the lines of force are shown by dotted lines. 17. Tubes of Force. Trace any closed curve in a field, and imagine that a line of force passes through each point of this curve. All these lines taken together form a tubular surface, called a tube of force. In the case of a single centre of force the tubes of force are conical. In a uniform field they are cylindrical. 18. Flux of Force. The intensity of any field is con- stant over an infinitely small surface dj. The product of this surface into the component of the intensity normal to the surface is called \\\z flux of force across the surface. Let a be the angle of the direction of the field with the normal, the flux of force will be represented by dN = H cos a ds. The flux of force across a finite surface is given by N j H cos ads, the integration being extended to every element of the sur- face under consideration. In the case of a closed surface the flux is said to be issuing when the lines of force are directed towards the exterior of the surface, and entering in the opposite case. By considering the angle a to be made by the direction of the field with the normal exterior to the surface, the change of sign of cos a allows us to distinguish the issuing from the entering flux. 1 8 IN TROD UCTION. 19. Theorem. The flux of force which traverses a tube of infinitely small section is independent of the inclination of the section to the axis of the tube. Thus, &H = H cos ads = #dcr, dcr representing the section of the tube normal to the axis, and H the intensity of the field at this point. 20. Gauss' Theorem. Between the masses of a field and the flux traversing a surface which envelops these masses there exists a simple relation, very frequently used, as fol- lows: The flux of force traversing a closed surface in a field is equal to ^.nk times the sum of the masses enveloped by this sur- face. I. Let us first consider a single mass m, concentrated in FIG. 4. a point P, within a surface which, to make our treatment more general, has a re-entrant portion (Fig. 4). From the point P as apex draw the elements of a cone corresponding to a solid angle do?, which is measured by the surface intercepted by the cone on a sphere of centre P and radius equal to unity. Call ds, d.y', d/' the areas bounded by the intersections of the cone with the surface described; do? represents the apparent surface of these intersections as seen from the point P. Call /, /', I" the distances of the intersected ele- GENERAL THEOREMS RELA TIVE TO CENTRAL FORCES, ig ments from the point P\ and a, a', a" the angles of the axis of the cone with the normals to the elements. The flux of force traversing these elements are respectively i km , km ,, , . km , ,, + cos ads, + -^ cosg ds , + j^ cos ads But ds cos a ds' cos a' ds" cos a" since these expressions, by definition, measure the solid angle doo. The flux reduces then to kmdao, whatever be the number of intersections, provided that the number is uneven. The total flux across the surface is given by the sum of all the elementary cones that can be drawn about P\ thus V r. 47T represents the total surface cut by the cones on a sphere of unit radius. II. If we had considered a mass m, outside of the closed surface, the elementary cones can traverse this surface only an even number of times, and would thus give a zero re- sultant. III. Finally, if we suppose in the field masses, m l , m y , m 3 , some of which are in the interior and others on the exterior of the surface, the total flux through the surface will be the sum of the flux due to the masses within, 2m : I Hds cos a = 21. Corollary I. Suppose that the closed surface be bounded by the lateral walls of a tube of force and by two sections of this tube, s and s' '. 20 IN TROD UCTION. As the walls of the tube cut no lines of force, the flux of force traversing the closed surface is limited to the flux / Hds I H'ds', ds and ds f being normal equipotential sections of the tube. Consequently, If there are no masses within this closed region, The flux entering by one base issues from the other, that is to say that the flux is constant in a tube of force, as long as the tube does not encounter acting masses. This property, comparable with that of fluid circuits in which the flow remains constant as long as no outflowing sources are met, justifies the namey?&;r, given to the mathe- matical expression which we have been considering. We shall see that this property, known by the name of continuity of flux, plays an important part in electric and magnetic phenomena. 22. Corollary II. If the tube of force were infinitely thin, we should have ffds = H f ds f = dN\ whence ILL *L H ' d/' In such a tube the intensity of the field is in inverse ratio to the section normal to the axis. In a uniform field the tubes of force are necessarily cylindrical. 23. Corollary III. The expression H = shows that the intensity of afield is the flux per unit equipotential sur- face at the point under consideration. GENERAL THEOREMS RE LA T2VE TO CENTRAL FORCES. 21 24. Unit Tube. Number of Lines of Force. A tube chosen so that the expression / Hds = i is assumed a unit tube. , ' Following a convention du^to Faraday and admitted by many authors, the number of lines of force of a field, which in reality is indefinite, is limited to the number of unit tubes of which they form the axes. In accordance with this convention, Gauss' theorem is enunciated as follows: The number of unit tubes or lines of force traversing a closed surface in a field is equal to ^.nk times the sum of the quantities of the agent enveloped by this surface. 25. Potential Energy of Masses Subjected to New- tonian Forces. In consequence of the repulsion exerted between masses of the same kind, a certain amount of work must be done to bring the masses m, m', m" to the neighboring points o, o', o" . This work is stored up in the system in the state of potential energy, and is restored when the masses, being set free, separate indefinitely from one another under the effect of their mutual actions. To determine the expression for the work done, suppose that the masses are formed by means of elementary masses brought up successively to the given points 0, being the solid angle subtended by the element ds at the point P'\ therefore r> dH = kcrdoo X - OP* The action of the whole shell will be This action is the same as if the entire mass were concen- trated in the point O. Corollary. For a point infinitely near to the surface the action of the shell would be 28. Action of a Homogeneous Sphere upon an Exter- nal Point. If the sphere were composed of a number of similar shells superposed, this conclusion would still hold. We can therefore say that a homogeneous sphere, or sphere composed of homogeneous shells, acts upon an external point as if the mass were concentrated at the centre of the sphere. Calling in this case # the mass per unit volume, we have OP APPLICA TIONS. 27 This property justifies the hypothesis of the concentration in physical points of masses which in reality occupy definite volumes around these points. As a particular case, if th^e point is at the surface of the sphere, the preceding expression reduces to ff=> 3 29. Action of a Homogeneous Sphere upon an Inter- nal Point. If the point were inside of a homogeneous sphere, this latter could be divided into two parts, separated by a concentric sphere passing through the given point. The action of the external portion is null; the action of the sphere internal to the point is equal to that of an equal mass concentrated at the centre. Denoting by / the dis- tance of the point P from the centre, H = *7tkld. 3 30. Surface Pressure. In the case of a homogeneous FIG. 7. spherical shell the component due to the element ds de- pends only on the solid angle subtended by it at the point 28 INTRODUCTION. P '. It is therefore equal to that of the element ds', corre- sponding to ds. The same holds for all the elements of the segment adb, taken in pairs with those of the segment acb. The plane projected on ab divides the sphere into two zones exercising the same actions upon P, equal to If the point Pis removed indefinitely from the sphere, the two segments tend to become equal. If, on the other hand, the point P approaches indefinitely to the spherical shell, one of the segments has as its limit the entire sphere, while the other tends towards zero. In this last case the preceding expression becomes 2ttk(j. Now we have just seen that the whole shell exerts upon unit mass, situated infinitely near to its surface, an action equal to ^nk The work done by unit ppsitive mass, in passing from a point on the surface of a shgll to a point infinitely near situated on the other^side, is equal to 4?r multiplied by the strength of the shell. This work is, moreover, independent of the path followed between these two points ( 12). 45. Energy of a Shell in a Field. Let us consider a field of force due to a pole m situated at a point O (Fig. 12)\ the work expended to bring the shell to its present position represents the relative energy of the shell and the field. It is equal to the work required to bring the mass m to the point O whose potential is U, or Now m/3 is the flux of force from the pole across the solid angle limited by the contour of the shell. We will call this flux $, considering it as positive when it enters by the negative face of the shell, and as negative when it enters by the positive face: W= - JF,$. If the field is produced by several poles m, m', m n ', the total energy will become The relative energy of a shell and a field is therefore equal to the product of the strength of the shell by the flux included within the contour of the shell. If the flux penetrates, as in Fig. 12, by the positive face, $ is negative, and the product takes the plus sign. When a shell is free to move in afield, it tends to move so that the expression for the potential energy becomes a minimum ; 46 MAGNETISM. that is, the flux entering by the negative face tends towards a maximum. It will easily be shown that this condition is satisfied in the case of a shell and a positive pole when the latter touches the negative face. A plane shell situated in a uniform field will take up a position normal to the direc- tion of the field, so that the lines of force penetrate it by the S-face. 46. Relative Energy of Two Shells. Let us consider two neighboring shells A, A', of strengths SF, and IF/. Let $' be the flux of force from ^['across the section of A entering by its negative face. The energy of the shell A is, as we have just seen, expressed by Now the flux $' may be represented by the product of SF' into a factor L m ; whence W=-SS'L m ........ (i) This expression must evidently represent the energy of the shell A', for the same work is expended to bring the shell A' up to A as to bring A up to A ' . But as the energy of A ' is also given by the product of SF/ into th-e flux # passing from A to A', we see that $ = &,L m , just as we found $' = & s 'L m \ whence we obtain The factor L mj called coefficient of mutual induction of the two shells, represents, as seen above, the ratio of the flux across one of the shells to the strength of the neighboring shell. Equation (i) shows that the dimensions of L m reduce to [L]. 47. Artificial Magnets. Instead of presenting, like uni- form magnets, a surface distribution of magnetism, magnet- PROPERTIES OF MAGNETS. 47 ized bars possess free magnetic masses internally. We can even superpose opposite magnetizations in a steel bar by submitting it successively to magnetizing forces of opposite directions. When such a bar is dissolved in an acid, there appear progressively layers rrjagnetized in opposite direc- tions. This experiment proves that the magnetization affects at first the superficial layers of the bar, which are moreover tempered harder than the inner layers. Hence the utility of employing thin plates of steel, separately magnetized, in order to obtain powerful magnets. We can show in a striking way the form of the magnetic FIG. 14. field due to a bar magnet by placing over it a sheet of paper covered with iron filings. The particles of iron be- come magnetized by induction, orient themselves along the directions of the field, and arrange themselves in continuous rows, representing the lines of force. The image thus produced may be fixed if sensitized pho- tographic paper be used and then exposed to the actinic ac- 48 MAGNETISM. tion of light while covered with the filings. The develop- ment of the image shows the shadows produced by the filings. Fig. 14 thus represents the magnetic field of two adjacent bar magnets. By observing the distribution of iron filings in the field of a single or of several magnets, and the curious patterns re- produced by the particles, Faraday was led to the idea that the seat of the magnetic forces is in the medium which separates the acting poles. According to him, lines of force are not a mere mathematical conception, but have a real existence corresponding to a particular state of the space around the poles. Faraday imagined this medium as being strained along the lines of force, and he readily substituted mentally for these lines elastic threads having a tendency to contract and thus cause neighboring poles to approach. To explain the curvature of the lines of force, Faraday assumed that they repel each other when they proceed in the same direction, so that each of them takes a curved form whose tendency to return to a rectilinear form balances the repulsion of the neighboring lines. Although experiment shows that a magnetized bar has free magnetic masses within it, it is possible to imagine a surface distribution of magnetism producing the same exter- nal field as the actual distribution. Suppose the magnet, for example, to be formed of longitudinal magnetic filaments some of which end on the end faces of the magnet and others on its lateral surfaces. Then the poles of the fila- ments constitute the surface charges of the magnet, as we have seen in the case of a magnetized sphere ( 42), and the curves taken by the iron filings (Fig. 14), may be considered as the prolongation of the axes of the filaments. To determine the imaginary distribution of magnetism giving the same external results as the real distribution, we measure the variation of the field intensity around the mag- PROPERTIES OF MAGNETS. 49 net along its axis. To do this, we determine the period of oscillation of a small magnetized needle moving on a pivot, and which is brought successively to the various points where it is desired to know the intensity. This means is not, however, rigorous, because the force is not the same at both poles of the needle, and the latter's magnetism may be altered under the influence of the field that is being investi- gated. By this experiment we find that the field decreases rapidly from the extremity of the magnet towards the mid- dle, unless there be intermediate or consequent poles. In long needles of hard steel the poles are very near the ends, and the neutral line extends over the greatest part of their length. In every case the neighborhood of the edges gives a more intense field than the neighborhood of the plane surfaces. FIG. 15. Fig. 15 shows the curves obtained by marking off on the perpendiculars to the axis of the bar lengths proportional to the components of the field along these lines. The ordinates are inversely proportional to the square of the periods of os- cillation of the magnetized needle opposite various points of the axis and at the same distance from it, and oscillating in a plane normal to the magnet. These ordinates may be con- sidered as proportional to the thickness of the magnetic shell having the same external effect as the real distribution of magnetism. Magnets undergo a slow demagnetization, which may be explained by the repulsion exercised between poles of the same name in neighboring molecules. This loss is retarded by joining the poles by a piece of soft iron called armature? keeper. Opposite poles are developed in this latter which 5O MAGNETISM. retain the magnetization of the magnet, since closed mag- netic filaments are formed through the armature, and the poles of the elements composing these filaments attract and neutralize each other in couples. The best steels for permanent magnets are those capable of acquiring the hardest temper. The addition of 3 per cent, of tungsten increases the coercive force of steel very perceptibly. The tempering may be done in oil, water, or mercury. The bath should be of sufficient volume to prevent great rise in temperature and splashing of the liquid. According to Strouhal and Barus, the best way to obtain a powerful and constant mag- net is to make the steel as hard as possible by tempering, and then to anneal it for 20 to 30 hours in steam at 100 C. It is next magnetized by placing its extremities on the poles of a powerful electromagnet, and finally annealing it again for at least 5 hours in steam. This method secures a magnetization which resists, as far as is possible, both blows and the daily variations of temperature. According to Preece, the intensity of permanent magnet- ization obtainable in prisms of I cm section and 10 cm length, made of good magnet steel bearing the stamp Mar- chal, Clemandot & Allevard, varies from 100 to 225 C. G. S. units. These numbers express the ratio of the permanent moment of the magnets to their volume. 48. Determination of the Magnetic Moment of a Magnet. Magnetometer. When we cause a magnet to oscillate horizontally in the earth's magnetic field, the mag- net being hung by a thread having no sensible torsion-pull, the duration of a complete oscillation, of sufficiently small amplitude, is /~H - 2 *V gft 3C PROPERTIES OF MAGNETS. 5 I K being the moment of inertia of the magnet, 311 its mag- netic moment, and 5C the horizontal component of the earth's field. From this we deduce (i) If 3C be known, the magnetic moment can be determined from this equation. If this is not the case, a second experiment may be made, The magnet being placed in a horizontal plane and normal to the magnetic meridian, a small magnetized needle is hung FIG. 1 6. at a certain distance on the prolongation of the given mag- net's axis (Fig. 16). Let 2/ be the distance between the poles of the magnet, 2/' that between the poles of the needle, L the distance from the centre of the magnet to that of the needle, -f- m and m the poles of the magnet, + m' and m' the poles of the needle. The magnetic moments are respectively 9fTl = 2ml, 3ft' = 2m' I'. The movable needle, being drawn in one direction by the earth's magnetism and in the perpendicular direction by the magnet, takes up a position of equilibrium corresponding to an angle a between its axis and the magnetic meridian. $2 MAGNETISM. On account of the small dimensions of the needle, we may consider that the forces exerted upon its poles by the poles of the magnet are equal and opposite and form a couple. The couple due to the earth's field is expressed by w 3ft ' 3C sin a. The couple due to the magnet is, calling F the force which it exerts on the poles of the needle, w' = 2FI' cos a. Now by Coulomb's law mm' mm' A.LI * "(_/)' The condition of equilibrium w \ gives T c*c")C /y SOI 'X sin a = 2SfnSfn ' '_/)' whence 3ft/ /V a _Z 8 tana; But if / is sufficiently small, we may neglect the powers of ~ higher than the second, and write !(< + 4} =n-- - w PROPERTIES OF MAGNETS. 53 In general, the actual distance / between the poles is un- known. This term may be eliminated by making another experiment on the deviation of the needle at a different dis- tance. * We get an angle of .fJeviatiotf <*', such that 3fft/ / 2 \ L fi tan a' whence by subtracting (3) from (2), after having multiplied (2) by U and (3) by L'\ ' 3TI D tan a L" tan a' . , OC " 2 JC - -7T3 AT3C := -- The susceptibility of iron is always much higher than unity. It follows, then, that the value of 3 is never very TC TP distant from - = - . Consequently the spherical form 7t 3 is not suitable for obtaining high intensities of magnetiza- tion. In the interior of an infinitely thin disk, transversely magnetized in such a way that the magnetic density of its faces, equal to the intensity of magnetization, be 3, the com- ponent of the force due to these faces is ( 31) -f 2;r3 ( 2?r3) = Consequently, the intensity of magnetization becomes 3 = /c(3C 47T3), whence 3 -f- ^TTK a value which tends towards . The transverse magnetization of a disk of iron is therefore always very feeble. The same can be shown with regard to the magnetization of an iron cylinder in a direction normal to its axis. INDUCED MAGNETIZATION. 59 54. Case of a Ring. A ring subjected to magnetizing forces, constant in magnitude and directed in every point of the ring along the tangent to the parallel circle passing through that point, will assume a constant magnetization without free poles, since the^rows of magnetic molecules will form closed circular chains. The original field will not, therefore, have its distribution modified by the presence of the ring, and the intensity of magnetization will be simply expressed by 3 = /cOC, 3C representing the intensity of the field. A conductor coiled round an iron ring, and traversed by an electric current, approximately realizes the above condi- tion, as will be shown later. After the stoppage of the cur- rent, the annular core retains the greater part of its mag- netism in the permanent state, for, in the absence of free poles, there is no demagnetizing force. 55. Case of a Cylinder of Indefinite Extent. A third solution is furnished by a cylinder of indefinite extent, placed parallel to the lines of force of a uniform field. The intensity of the field in the interior of the cylinder is the re- sultant of the original field and of the action of the poles induced at the extremities of the cylinder. The longitudinal magnetization that can be given to an iron cylinder, whose axis is placed parallel to the direction of the field, increases as the length of the cylinder is in- creased, since the effect of its poles then produces less and less diminution of the intensity of the field inside the cylin- der. Short steel cylinders cannot, therefore, make good permanent magnets, for they become only slightly magnet- tized, and the reaction of the poles tends to rapidly change the molecular orientation after taking away the magnetizing 60 MAGNETISM. force. On the other hand, long cylinders become strongly magnetized and retain their magnetism. When an iron cylinder is of indefinite length, the demag- netizing action of its poles becomes negligible for points situated in the accessible region of the cylinder where the intensity of magnetization is uniform and expressed by 3 = xrOC. It has been shown experimentally that this formula is still applicable when the length of the cylinder is equal to 400 or 500 times its diameter. 56. Portative Power of a Magnet. Let us consider a cylinder of indefinite length, magnetized parallel to its axis. If we imagine a narrow crevasse cut out normal to the axis, the opposite walls will be covered with magnetic masses whose density is equal to the intensity of magnetization, (7 = 0. The force with which unit mass, situated near the face whose density is cr, is attracted by this latter, is expressed by 27Ti, is superior to zero, while the susceptibility of diamagnetic bodies is negative. The susceptibility and permeability of iron, cobalt and nickel at ordinary temperatures are so superior to those of INDUCED MAGNETIZATION. 7 1 other bodies, whether magnetic or diamagnetic, that there is no practical error in taking the permeability of all other bodies as equal to unity and their susceptibility as zero. The most diamagnetic body in existence, bismuth, has a permeability of 0.9991^ 61. Work spent in Magnetizing. As the magnetiza- tion of a body imparts to it a certain quantity of potential energy, it necessitates a certain expenditure of work. We shall show later on that this work is expressed, per unit volume of the magnetized body, by i C l C - I OCd(B=r / yuOC ^J ^J the integral being extended to the limits between which the induction of the magnet has been changed. If the values of the induction compared with the field-intensity be shown by the curve OA, Fig. 20, and if the magnetization reach the state denoted by the point A, the work expended will FIG. 20. be represented by the area, divided by 471-, of the surface comprised between the curve, the axis of ordinates and a parallel to the axis of abscissae drawn through A, 72 MAGNETISM, If jj. were a constant factor, as is the case for slightly magnetic substances, the integral would reduce to for a variation extending between o and 3C. Let us suppose that an indefinitely long bar, after having reached the magnetic state A, traverses a cycle ACA'C'A, Fig. 20, the intensity of the field passing from the value OB to o, and then returning to OB. The integral /&= AB I oed& 4 ^ = 7 kilogausses and (B = 16 kilogausses. Between these limits the mean values of the permeability are approximately given by the empirical formula, deduced from Hopkinson's curve, * = ~~ 3^5 + 48S * Figure 23 presents a curve determined by Ewing and show- ing the loss in a soft-iron bar subjected to increasing alter- nating magnetizing forces. The ordinates of the curve denote ergs per cm 3 and the abscissae give in C. G. S. units the extreme values, positive and negative, of the magnetic induction through the metal. w 10000 8000 6000 4000 2000 FIG. 23. According to Steinmetz, the loss of energy in ergs is represented by the expression* w 77 (B 1 - 6 , * Steinmetz, L' Industrie Jlectrique, March 6, 1892. INDUCED MAGNETIZATION. 77 where 77, the coefficient of hysteresis, may have the following values : Material. Composition and State. Coefficient of Hysteresis, i? Anneal 045 P .032 .032 4-73 3-35 3-45 ed ercent. c i < < < >f carbon, annealed " tempered manganese, forged tungsten, tempered carbon .00202 .OO262 .00598 .00954 .05963 .05778 .01826 Soft Bessemer steel . Manganese-steel. . . Gray cast-iron With the magnetizing forces that can be obtained in dynamo-electric machines, their iron cores seldom exceed an induction of 20,000 C. G. S. units, or gausses, but by estab- lishing particularly powerful fields Messrs. Ewing and Low have succeeded in communicating to very soft iron an in- duction of 45 kilogausses. Under high inductions, the in- tensity of magnetization has a constant value of about 1700 C. G. S. units, corresponding to saturation, and the per- meability falls to a constant value of between I and 2. We then get the relation &:=3e-|-47r3 = 3e-|- constant. In very intense fields cobalt is capable of attaining the same maximum intensity of magnetization as cast-iron, or about three fourths of the magnetization of soft iron. Nickel never exceeds one third of the maximum intensity of magnetization of soft iron. Lord Rayleigh has found that in very weak fields the per- meability can be expressed by a formula = a b 3C. For a specimen of soft iron he has found #=8i and =64. He has also established the fact that hysteresis is absent 7 MAGNETISM. when a bar is subjected to magnetizing forces varying be- tween very narrow limits, whether the metal already possesses any magnetization, or if it has been taken in the neutral state; in these conditions the permeability is constant. When these small variations occur near a magnetizing force of 29 C. G. S. units or gilberts, he has found that the per- meability of soft iron is only 80 per cent, of the permeability near the neutral state. 63. Effect of Temperature on Magnetism. Recales- cence. We have already made allusion to the influence of the temperature on the magnetism of iron and its deriva- tives, steel and cast-iron, whose magnetism disappears com- pletely at a bright red heat. Dr. Hopkinson, to whom we are indebted for precise experiments on this thermic effect,* has observed that in a feeble and constant field of 0.3 gauss the permeability of a soft-iron bar, heated gradually, increases progressively from 500 to 11,000; but at the temperature of 775 C., the permeability falls suddenly to a value very close to I. When the intensity of the field increases, the increase of permeability is much less sensible and the fall is less sudden. Finally, in an intense field, the permeability decreases con- tinuously with the rise of temperature. In every case the iron becomes completely demagnetized at a temperature in the neighborhood of 785 C.; Dr. Hopkinson calls this thermic point the critical temperature of the metal. For exceptionally soft iron the critical temperature may rise as high as 880 C., while in steel it falls to 690. For nickel the critical temperature is about 310 C. The critical temperature seems to correspond to a mo- lecular change in the substances, shown likewise by other * See Hopkinson, Magnetism, Journal of the Institution of Electrical Engineers, vol. xix. INDUCED MAGNETIZATION. 79 phenomena. Kohlrausch has observed that at this temper- ature the electrical -resistance of iron shows a sudden variation. According to Tait the tfyeftno-electric power of iron is also modified in a profound degree towards this point ; and lastly, Barrett has discovered a very characteristic effect, to which he has given the name of recalescence. If we allow a piece of iron or steel to cool down after having heated it to a bright red, there comes a certain stage where the process of cooling stops and where the piece becomes slightly heated again, after which the decrease of temperature goes on again regularly. This recalescence is shown in hard steel by a very visible luminous effect, the color of the metal passing from a dull red to very bright red at the moment when the critical temperature is reached. This experiment succeeds very well when a knitting-needle is used, first heat- ing it to a bright red by passing an electric current through it. It is very surprising that magnetic qualities should be clearly exhibited by only three metals iron, nickel, and cobalt. The other elementary bodies are so little capable of magnetization that they are ordinarily considered as non- magnetic. It may be that it is only a mere question of temperature, the three metals mentioned being the only ones which manifest decided magnetic properties at the ordinary temperatures. This was Faraday's opinion, who thought that all substances would become magnetic at a sufficiently low temperature. The following fact discovered by Dr. Hopkinson seems to support this opinion: An alloy of iron containing 25 percent, of nickel is non-magnetic like all alloys. But if this alloy be cooled to slightly below o C, it is capable of becoming magnetized in a very marked degree. It possesses, therefore, a low critical temperature. If the alloy be afterwards reheated, it remains magnetic and its susceptibility increases up to about 525 C., at this point 8O MAGNETISM. the susceptibility falls rapidly and becomes zero at 580 C. Upon recooling the metal, it does not reassume its suscep- tibility until below o C. 64. Ewing's Addition to Weber's Hypothesis.* In order to explain in Weber's hypothesis of the molecular constitution of magnets, 39, the coercive force and the loss due to hysteresis, it has been supposed that the ele- mentary magnets (or magnetic molecules) offer a resistance to orientation in the nature of friction and that it is the work spent in overcoming this friction which constitutes the loss by hysteresis. The existence of such a passive resist- ance enables us, up to a certain point, to account for the effect of vibrations and temperature on magnets, but it does not at all explain the changes in susceptibility especially shown in the regions A, B and C of the magnetism-curve, Fig. 24. FIG. 24. Ewing has found experimentally that the observed phe- nomena are to be explained without bringing in the supposi- tion of friction, by the simple effect of the mutual reactions of the elementary magnets. He has reached this conclusion by investigating the way in which a system of magnetic needles acts, when they are arranged regularly one next the * Ewing, Contributions to the molecitlar theory of induced magnetism, Roy, Soc. 1890 ; also Magnetic Induction^ etc., pp. 287-8 et seq. INDUCED MAGNETIZATION. 8 1 other so as to be able to oscillate in the same horizontal plane without touching each other. These needles are sub- jected to a magnetizing force obtained by rolling coils of wire, carrying an electric current, around the case enclosing them. When the needles are left to their own reactions, that is, when the field produced by the current neutralizes the earth's field, it is observed that they form more or less complex geometrical combinations with each other in stable equilibrium. If one of the elements of this combination is slightly altered from its position, it immediately returns to it ; but if the alteration of position is considerable, the com- bination is not formed again, and new combinations are formed between the neighboring magnets. If a progres- sively increasing directive force is applied to such a system, it is seen that the various combinations are at first slightly deformed without being destroyed. This, which might be termed an elastic deformation since it is reversible by with- drawing the magnetizing force, is comparable to the state of the molecules of a magnetized bar in the region A of the magnetism-curve, Fig. 24. If the current through the coils is continuously increased, a point is reached where one of the combinations of the magnets exceeds the limiting deformation which it can stand. There is then produced a sudden change in this combination, and, by the mutual action, the whole system enters on a state of unstable equilibrium, so that a very slight increase in the directive force is sufficient to align all the magnets in a direction approaching to that of the direc- tive force itself. This period corresponds to the region B of the magnetism-curve. The observation of the propaga- tion of the new groupings from one group to the next one is eminently suggestive as explaining the necessity of a definite interval of time for the molecules of a magnet to assume 82 MAGNETISM. their positions of equilibrium under the action of a magnet- izing force. When, after this, we still continue to apply increasing forces, we observe that the mutual reactions of the magnets are more and more overpowered and that they align them- selves in a direction which eventually coincides with that of the acting field. This is the state designated under the name of saturation and shown at C in the curve. Figures 25, 26, and 27 show three successive states of the system of magnets; Fig. 25 corresponding to the end of state A, Fig. 26 to the end of state B, and Fig. 27 to the end of state C. If we diminish the intensity of the field, the magnets still maintain their general orientation, but become slightly displaced by the effect of their own reactions. When the directing field becomes zero, the magnets remain more or less aligned in the direction of the field, which accounts for the residual magnetism. But if the directing field changes sign and increases in the opposite direction, we soon observe a sudden return to the state of unstable equilibrium, and then an orientation in the opposite direc- tion. According to the curve AC A', Fig. 18, annealed soft iron is in a state of unstable equilibrium when the magnetizing force becomes zero, since the elbow of the curve is on the side of the positive magnetizing forces. With tempered steel, on the contrary, this elbow is produced on the side of the negative abscissae, which accounts for the aptitude of this metal for retaining its residual magnetism. In fact, at the moment when the magnetizing force becomes zero, the state of the metal is shown by a point of the curve situated in the region corresponding to stable equilibrium. If, instead of placing the magnets regularly, we arrange them at varying distances, we observe that the duration of the state corresponding to unstable equilibrium (13) is in- INDUCED MAGNETIZATION. creased. This is observed in cold-hammered iron, the arrangement of whose molecules has been changed by mechanical means. The various combinations of elements which are formed in such, a" case are more independent of each other than in the homogeneous metal, and one or another of them can be modified without influencing the adjoining groupings. " 1111 I 1 ~ 1 1 1 1 1 1 I 1 1 1 1 1 1 - 1 1 1 1 1 1 - 1 1 1 1 11 an 1 ! 1 1 1 1 1 1 \ lilt 1 1 III lltl 1 1 1 1 ' nil FIG. 25. FIG. 26. FIG. 27. In steel and cast-iron, where the stage B is of great extent, the groupings of the molecules are influenced by the presence of foreign bodies. According to Ewing, the heating due to hysteresis cor- responds to the oscillations of the magnets on passing from one position of stable equilibrium to another. Vibrations diminish the stability of the combinations and consequently facilitate the orientation of the magnets placed under the action of the field, as well as their return to the neutral state when the magnetizing force has ceased to act. 84 MAGNETISM. A rise in temperature produces analogous effects when the magnetizing force is feeble ; we have already seen, how- ever, that heating reduces the permeability when the mag- netizing force is intense. Ewing explains this fact by considering that the molecular agitation caused by a rise in temperature corresponds to oscillations of the magnets about their axes. When the magnets are already oriented, these oscillations result in a diminution of the mean external action of the system. Or we can admit, with Dr. Hopkinson, that the magnetic moment of the elementary magnets decreases when the temperature increases. Lastly, the absence of hysteresis, observed by Lord Rayleigh in the case of very feeble variations in the magnet- izing force, can be explained if we observe that such varia- tions produce only feeble displacements of the elementary magnets about their positions of equilibrium ; such dis- placements are reversible without break of equilibrium and consequently without the extensive movements which give rise to the development of heat. The irreversible variations, which are made evident by the separation of the ascending and descending curves of magnetism, are the only ones which give rise to the evolu- tion of heat. 65. Equilibrium of a Body in a Magnetic Field. We have seen, 45, that a shell free to move in a magnetic field moves so that the flux entering by its negative face may be a maximum. This conclusion extends to any mag- netized body whatever which may be considered as formed by superposed shells. Thus in a uniform field the axis of an iron cylinder of elongated form aligns itself parallel to the lines of force of the field, in such a way that the flux of force enters by the induced south pole. INDUCED MAGNETIZATION. 85 It has been shown, 53, 55, that this position corresponds to a more intense magnetization of the metal than does any other position. In a uniform field an isotropic sphere is in equilibrium in all positions, ,while an anisotropic sphere, whose permeability varies in Different directions, orients * itself so that the direction of the field may be parallel to the axis of maximum permeability. The same considerations show that in the neighborhood of a magnet where the field is variable, magnetic bodies tend to move towards the poles so that the flux traversing them may be a maximum. These movements are clearly exhibited in a liquid placed in a watch-glass over the poles of a powerful electro-magnet. FIG. 28 A solution of sulphate of iron, S, Fig. 28, will present a concavity towards the centre. Diamagnetic bodies, on the other hand, appear to be re- pelled by the poles; thus a bar of bismuth assumes a posi- tion at right angles to an electro-magnet ; a solution of bisulphide of carbon, S', Fig. 28, is heaped up in the middle of the vessel. Becquerel and Faraday have found a simple explanation of diamagnetic repulsion, by means of comparing it with the action of gravity upon bodies plunged in a liquid denser than themselves. These bodies are apparently repelled by the earth ; in the same way it may be that the action of 86 MAGNETISM. magnets on diamagnetic bodies is simply due to the fact that they are less magnetic than the air or the medium which surrounds them. From this point of view there would be, properly speaking, no diamagnetic substances, but only degrees of permeability. This hypothesis, corn- batted by Tyndall, has been recently confirmed by Messrs. Parker and Duhem. HYSTERESIS. 87 HYSTERESIS AND MOCtCULAR MAGNETIC FRICTION.* 66. Hysteresis. Some materials, such as iron, nickel, etc., when exposed to the action of a magnetomotive force, that is, when in a magnetic field, have induced in them a mag- netic flux far in excess of that set up under the same conditions in air or other materials. The former are there fore called magnetic materials. In air and other non-magnetic materials the magnetic flux, OS, varies proportionally to the magnetomotive force, F, or to the field intensity, 5C. In magnetic materials, such as iron, the magnetic flux is proportional to the M. M. F. only for very low values of the latter. With increasing M. M. F.'s it begins to increase at a greater rate than the M. M. F., be- comes proportional again to it at still higher values, and for very high M. M. F.'s increases more and more slowly until ultimately its further increase with increase of M. M. F. approaches a limit where it is not greater than in non- magnetic materials ; or, in other words, the difference & 3C approaches a finite limit, called the absolute magnetic saturation of the material,-)- which in soft iron corresponds to about (& = 20,000, in nickel to ^ x f ' '' / 10000 9000 8000 7000 6000 5000 4000 3000 / / / / / / / 1 I \ \ \ 2000 1 1 .1000 n / x FIG. 29. RISING AND DECREASING MAGNETIC CHARACTERISTIC. shown in Fig. 29 in dotted line. Thus the magnetic char- acteristic is different for decreasing and for increasing mag- netism ; or, in other words, the magnetic flux, (B, in mag- netic materials, such as iron, depends not only upon the present value of M. M. F., but also upon the previous values, and thus lags behind the M. M. F. This lag of the magnetic flux (B behind the M. M. F. is HYSTERESIS. 8 9 practically independent of the time; that is, independent whether the change of M. M. F. takes place rapidly or very slowly. It is called hysteresis. The effect of hysteresis upofi the magnetic characteristic is most pronounced in .cyclic changes of flux as produced by cyclic changes of M. M. F. Thus if the M'. M. F., F, or the 7 -fUQO -KOOO +2000 -4000 -6000 -80/0 ^1200 KB, KJC, FIG. 30. HYSTERETIC LOOP OR MAGNETIC CYCLE. field intensity, 5C, is varied periodically between a maximum value +^1 and an opposite value 5C, , the magnetic flux will vary between corresponding maximum values + (B, and (B, , describing a loop-shaped curve called the magnetic cycle or hysteretic loop, as shown in Fig. 30. 90 HYSTERESIS MOLECULAR MAGNETIC FRICTION. It follows that when the M. M. F. has been reduced to zero, the magnetic flux has still a considerable value, which is called the remanent magnetism, R in Fig. 30. The magnetism will reach zero only after the M. M. F. has been reversed and increased to a considerable value , in opposite direction. Thus the iron acts as if an internal FIG. 31. MAGNETIC CYCLES OF SOFT SHEET-IRON OR SHEET-STEEL. M. M. F., (B, tends to maintain its magnetic flux. This M. M. F. (2. is called the coercive force of the iron. The shape of the hysteretic cycles varies with different magnetic materials, and even with the same magnetic material in different physical conditions, and is different for different values of maximum magnetic flux, &,. A number of such magnetic cycles are shown in Figs. 31, 32, and 33. Fig. H YS TERESIS. 9 1 31 gives the magnetic cycles of sheet iron or soft sheet steel for different values of magnetic flux, (B = 2000, 6000, lopoo, and 16000.* Fig. 32 gives cast-iron cycles for (B = 6800 and (B = iO3OO.f Fig. 33 gives cycles of tool-steel at different degrees of hard- ness. H is a sample hardened in water, O one hardened in oil, and 5 an annealed sample, all three of the same -9O-8O 7O-6O -5O 4O SO - 430 47O +8O * 9O FIG. 32. MAGNETIC CYCLES OF CAST-IRON. material.! As may be seen, the harder the material the lower and wider in general is the hysteretic loop ; that is, the lower the maximum and remanent flux, the higher the coer- * " On the Law of Hysteresis," Part III, A. I. E. E. Transactions, 1894, p. 717. \ Ibid., Part I, A. I. E. E. Transactions, 1892, p. 40. \ Ibid., Part II, A. I. E. E. Transactions, 1892, p. 653. 92 HYSTERESIS MOLECULAR MAGNETIC FRICTION. cive force. In the cycles of Figs. 31 to 33, the abscissae are not the field intensities 3C, but the ampere-turns per centi- metre length of the magnetic circuit, expressed by p 103C which is sometimes used as a practical unit of M. M. F. If an air-gap is introduced into the magnetic circuit that is, if the magnetic circuit is partly of iron and partly of air, F IG> 33. MAGNETIC CYCLES OF WELDED STEEL AT DIFFERENT DEGREES OF HARDNESS. as for instance in dynamo machinery the hysteretic cycle changes to the shape shown in Fig. 34, in which the straight dotted line represents the M. M. F. required for the magnet- ization of the air-gap, and the hysteretic loop has the same relative position that is, the same horizontal distance from this dotted line as it had from the vertical line in the circuit consisting entirely of iron. HYSTERETIC LOOP. 93 As may be seen, the effect of the introduction of an air- gap is to require for the same maximum flux (B x a much greater M. M. F., to reduce the remanent magnetism very greatly, but the coercive forge XB is not thereby affected. -50-45- 48000 XXK -14000 XX +1520+2543b44 450 FIG. 34. MAGNETIC CYCLE OF CIRCUIT CONTAINING AN AIR-GAP. 67. Hysteretic Loop. In the hysteretic loop of a mag- netic circuit, with M. M. F. as abscissae, and magnetic flux as ordinates, the area of the loop is M. M. F. X magnetic flux. Since M. M. F. has the dimension and magnetic flux the dimension where L = length, M = mass, jT time, the area has con sequently the dimension that is, the same as that of energy. 94 HYSTERESIS MOLECULAR MAGNETIC FRICTION. In the hysteretic loop with field intensity 3C as abscissae, of dimension and magnetic flux density (B as ordinates, of dimension the area has the dimension volume thus representing an energy per unit volume. Let/=: instantaneous value of M. M. F., $ = maximum " " " = instantaneous value of magnetism produced there- by, $ = maximum value of magnetism produced thereby. If the M. M. F./is produced by an alternating current, /, flowing through n turns, then e = - n^- = E. M. F. Qt induced by the magnetism m, and dw eidt = mdfi = energy expended by the change of magnetic conditions. Since ni /, dw = ~/d0, and r+s r-$ W= /d0+ MOLECULAR MAGNETIC FRICTION. 95 total energy expended during the cyclic change of mag- netism ; but / ycl0_|_ /. /d0 = area of the hysteretic loop. " Therefore, the area of the hysteretic loop, with the M. M. . in ampere-turns as abscissae, and with the magnetic flux in volt-lines (= io 8 lines, or one hundred mega- webers) as ordinates, is equal to the energy expended by hysteresis, in coulombs. With lines of force as ordinates, and tens of ampere-turns as abscissae, the area is the hyster- etic energy in ergs. The area of the hysteretic loop, with field intensity, JC, in tens of ampere-turns per unit length of magnetic circuit, as abscissae, and with magnetic flux density, (B, as ordinates, is equal to the loss of energy by hysteresis in ergs per unit volume. With field intensity JC = -- as abscissae, and with lines of force per cm. 2 , (B, as ordinates, the energy expended by hysteresis during a complete cycle of magnet- ization is = 47T X area of hysteretic loop. Hysteresis thus represents an expenditure of energy by the M. M. F. and is measured by the area of the hysteretic loop or magnetic cycle. 68. Molecular Magnetic Friction. If by an alternating M. M. F. an alternating magnetic flux is produced in iron or other magnetic material, a loss of energy takes place in the iron by a kind of frictional resistance of the molecules against the change of their magnetic condition. This phe- nomenon is called molecular magnetic friction. Therefore, to alternate a magnetic flux, energy has to be expended upon the iron. g HYSTERESIS MOLECULAR MAGNETIC FRICTION. If the alternation of the magnetic flux is produced by an alternating current, and the condition is such that no energy is expended upon the magnetic circuit by any other source, nor external work done by the magnetic circuit, the energy consumed by molecular magnetic friction has to be supplied by the alternating current. Consequently, the magnetic flux cannot follow the M. M. F., but must lag behind it so far that the hysteretic curve of magnetic flux and M. M. F. represents the energy expended by molecular magnetic friction. It follows that in an alternating magnetic circuit which neither produces external work nor receives energy from another source than the alternating M. M. F., the energy con- sumed by molecular magnetic friction is equal to the energy expended by magnetic hysteresis. If, however, external work is done by the magnetic circuit, or work expended upon it by an external force, the identity between the energy of molecular magnetic friction and the energy of magnetic, hysteresis no longer exists. Thus, if the magnetic circuit is vibrated mechanically during the cycle of magnetization, the hysteretic loop collapses more or less completely, and the rising and decreasing magnetic characteristics coincide ; the energy consumed by molecular magnetic friction being supplied in this case from the mechan- ical source vibrating the magnetic circuit. Conversely, if mechanical work is done by the magnetic circuit, as, for in- stance, if the magnetic circuit consists of iron filings or loose laminations which can vibrate and rearrange them- selves, the hysteretic loop is greatly extended and represents not only the energy consumed by molecular magnetic fric- tion, but also the mechanical work done.* * For proof and discussion of the distinction between hysteresis and mo- lecular magnetic friction see: "On the Law of Hysteresis," Part II, Chap. V, A. I. E. E. Transactions, 1892, p. 711, and "On the Law of Hysteresis," Part III, Chap. II, A. I. E. E. Transactions, 1894, p. 706. DETERMINATION OF VALUES. 97 It follows that in determining the energy loss bymolecular magnetic friction from the hysteretic loop of the material, care must be taken that neither external work is done nor absorbed by the magnetic circuit while the hysteretic loop is being determined. R -* 69. Determination of Hysteresis and Molecular Mag- netic Friction. The different methods of determining the value of hysteresis and molecular friction are as follows: (a) Ballistic Method. A magnetic circuit is built up of the iron to be tested, a magnetizing coil wound around it as uniformly as possible, and a second or exploring coil em- ployed, connected to a ballistic galvanometer. The current in the magnetizing coil is varied step by step, and the time integral of E. M. F. induced in the exploring coil by the variation of current, and consequently the change in magnetic flux, is observed by means of the ballistic galvanom- eter. In this way a complete cycle of magnetism is plotted and from its area the loss of energy determined. This method does not give the energy of molecular friction directly, but gives the energy expended by hysteresis. It can be used for the determination of the saturation curve also, and is suitable for the investigation of solid materials, as well as of laminations, etc. In determining the saturation curve by this method, it is desirable to dissipate the remanent magnetism previous to the test, by applying a strong alternat- ing current through the magnetizing coil, and gradually re- ducing this current to zero. In determining the hysteretic cycle, a greater number of cycles between the same maxi- mum values should be described before taking readings, so as to make the cycle symmetrical and independent of rema- nent magnetism due to the previous history of the iron. (b) Alternate Current Method. The energy expended by magnetic hysteresis can be determined directly by sending 98 HYSTERESIS MOLECULAR MAGNETIC FRICTION. an alternating current through a magnetizing coil surround ing the magnetic circuit, and taking readings of an ammeter, a voltmeter, and a wattmeter. The M. M. F. is determined from the ammeter reading, the magnetic flux by calcula- tion from the voltmeter reading, number of turns, fre- quency and shape of the magnetic circuit, and the energy loss by hysteresis from wattmeter readings. This method is applicable only to laminated material, and measures not only the energy expended by hysteresis but also the energy loss by eddy or Foucault currents. Conse- quently, either the material has to be subdivided so as to make the latter negligible, or hysteresis and eddy currents have to be separated from each other afterwards. Since the maximum flux and maximum M. M. F. in this method are obtained by calculation from the effective values read on the instruments, the wave of E. M. F. should be as near as possible a sine wave. Owing to the distortion of the current wave by hysteresis, the maximum value of current, even with a sine wave of E. M. F., is not through- out the whole range equal to ^/ 2 times the effective value. Calculating the M. M. F. under the assumption of a sine wave of current, therefore, gives a magnetic characteristic, which, while practically coinciding with the true magnetic characteristic within a range up to (B = 10,000 or 14,000, yet differs greatly therefrom beyond this range, showing appar- ently very high values of flux. In Fig. 35 is shown the true magnetic characteristic in full line, the magnetic char- acteristic as determined from an alternating current test in dotted line, and the loss of energy by hysteresis in ergs per cm 3 and cycle in the full line of single curvature.* Instead of connecting the voltmeter and the potential coil of the wattmeter across the magnetizing coil of the * "On the Law of Hysteresis," Part III, A I. E- E. Transactions, 1894, p. 722. DETERMINATION OF VALUES. 99 magnetic circuit, it is preferable to connect it across a second or exploring coil wound uniformly over the magnetic ,cr 4 II nf\ ... / 1 / 1 I I 19 / 1 1 1 1 1 1 Ifi / - 1 / / / 10 . / / 1 A // U / 1 1 1 / /, 1 1U / //' / / /, / / V' / // // // / ^ 4 / ^ ^ / rr-s 5 ^ ^ ^^ "^ ^^ 1 > ^ x* 1 / / ^ ^~ (B=l,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,00011,00012,00013,60014,00015,00016,00017,000 FIG. 35. MAGNETIC CHARACTERISTIC FROM BALLISTIC AND ALTERNATING CURRENT TESTS, AND MOLECULAR MAGNETIC FRICTION CURVE. circuit, as shown diagramatically in Fig. 36, and thereby eliminate the error due to the drop of voltage and loss of energy in the resistance of the magnetizing coil. 100 HYSTERESIS MOLECULAR MAGNETIC FRICTION. (c) Power and Torque Tests. The loss of energy by mo- lecular magnetic friction and eddy currents may be deter- mined by moving the material to be tested in a uniform magnetic field, and measuring the power required therefor. This method is commonly used in determining the hyster- etic loss in the armatures of dynamo machinery. In this case the armature is turned, first in an unexcited magnetic field, and then in a magnetic field of various degrees of FIG. 36. INSTRUMENT CONNECTIONS FOR ALTERING CURRENT HYSTERESIS TESTS. excitation. The difference in work consumed in these cases is the energy expended in molecular magnetic friction and eddy currents in the armature. A similar method suitable for testing of iron, is the following : A uniform magnetic field is set in rotation, and in the centre of the field the iron to be tested is suspended by a torsion spring. Owing to the hysteretic loss in the iron, the LOSS OF ENERGY. IOI rotating field tends to turn the iron in its direction of rota- tion, and the torsion given to the spring to counterbalance this torque is directly proportional to the hysteretic loss per cycle in the test-piece. The magnetic flux is measured by the E. M. F. induced in an coloring coil surrounding the test-piece.* This method has the advantage that the torque exerted upon the iron by molecular magnetic friction is independent of the speed of the rotating field. 70. Loss of Energy. The hysteretic cycle has been found identically the same from very slow alternations, up to the highest frequencies reached by dynamo-electric machin- ery beyond 200 cycles per second. It can, therefore, be said that hysteresis and the loss of energy by molecular magnetic friction per cycle are independent of the frequency, and consequently the loss of power is proportional to the frequency, that Is, to the number of cycles. No difference between the value of hysteretic energy and loss by moleculai magnetic friction has been found between rotating and reversing fields, and therefore it is permissible to use the values found in alternating fields for losses in rotating fields, and conversely. The hysteretic loop increases with increasing maximum magnetic flux density (B ; that is, the energy expended by hysteresis per cycle, and the energy absorbed by molecular magnetic friction, both increase with the magnetic flux. Plotting the loss of energy as a function of the magnetic flux, the curve thus obtained rises uniformly, and does not show any marked features at the critical points of the magnetic saturation curve, following the same law at satu- ration and below saturation. Approximately, the loss of energy by molecular magnetic * Holden, The Electrical World, June 15, 1895. 102 HYSTERESIS MOLECULAR MAGNETIC FRICTION. friction can be expressed by the empirical formula,* where W H = loss of energy in ergs per cycle and cm a , (B = maximum values of magnetic flux between which the mag- netic cycle is performed, and rj is the coefficient of molecular magnetic friction. The same empirical function of the 1.6 power holding for reversals of magnetism, holds also for cyclic changes of oc 1S.CCO / \ / / 10,000 8000 I / X / 2 / / / / / / / sooo X / ^ ^> ~Z? . " ,s >,= 2000 MOO 6000 8000 10,000 12,000 14,000 16,000 I8.0UO FIG. 37. CURVE OF HYSTERESIS OF SOFT IRON WIRE. flux between any two limits (B, and & 3 , whether these two limits be of the same direction and same sign or of opposite direction. Thus, in general form the empirical law of molecular magnetic friction can be expressed by the formula : *"On the Law of Hysteresis," A. I. E. E. Transactions^ 1892, p. 3 and p. 621; 1894, p. 702. LOSS OF ENERGY . 103 This equation can be considered as empirical only, and, in fact, does not hold for extremely low values of magnetic flux ; it is, however, correct with sufficient approximation through a wide range, as shown in f igs. 37 and 38,* which give the observed values of hysteretic^Joss in a sample of soft-iron wire, and in a sample of annealed steel wire, as taken from Ewing's tests. The curve of 1.6 power is shown in full 7 UV / 1 / Ma ed jnetic Cycles of Pianoforte Stee ;Ewing,p.109) . / 60,000 Ar nee I W re. / / 7 60,000 1 '/ / / 7 / 20,000 IQjOOO / 7 / / ' * ^ "' ^ "/? ^ / /_ ** X 1 1 1 I FIG. 38. CURVE OF HYSTERESIS OF ANNEALED PIANOFORTE STEEL WIRE. line in these figures, with the observed values marked by crosses. The dotted line represents the magnetic characteristic. In reality, the value (ft in the formula of hysteresis should probably not be the total magnetic fiux, but the " metallic magnetic flux," or & 5C, which reaches a finite value of * "On the Law of Hysteresis," Part II, A. I. E. E. Transactions, 1892, pp. 673, 674 IO4 HYSTERESIS MOLECULAR MAGNETIC FRICTION. absolute saturation, while (B increases indefinitely with in- creasing OC.* 71. Coefficient of Hysteresis. The coefficient of mo- lecular magnetic friction varies very greatly with different materials and even with different conditions of the same material. In iron, the loss by molecular magnetic friction seems to depend comparatively little upon the chemical constitution. In general, the purer the iron is, the lower the co- efficient ??, and the less the loss by molecular friction. Frequently, however, very pure samples of iron show com- paratively high values of 77, while impure samples show re- markably low hysteretic losses. A large percentage of carbon, silicon, and phosphorus seems to be objectionable, while manganese in small percentages appears comparatively harmless. Even here, however, the effect of impurities seems to be indirect, and due to changes in the physical constitution of the material. Of the greatest importance regarding hysteretic loss is the physical condition of the material, one and the same material in a hardened state occasionally showing a hysteretic loss many times larger than when in annealed state. Annealing is always found to reduce the loss by molecular friction more or less, while hardening increases it. With tool steel, for instance, the coefficient of hysteresis has been varied from 14.5 to 75 by annealing and hardening, respectively. (See Fig. 330 Break of continuity of the material usually greatly in- creases the loss by molecular magnetic friction. Thus, gray cast-iron, even when very soft, shows a comparatively high * " On the Law of Hysteresis," Part II, A. I. E. E. Transactions, 1892, Chap. Ill, pp. 678 seq. COEFFICIENT OF HYSTERESIS. 1 05 coefficient, 77, due to the interposition of graphite in the metallic structure. The temperature affects the loss of energy by molecular magnetic friction very little, within the range of atmospheric temperatures, the hysteretic^loss decreasing considerably only at high temperatures. If iron is heated and then cooled, the loss by molecular friction is much smaller when hot, and is not increased again to the same value as before when cooled ; a part of the decrease due to the heating be- comes permanent, probably due to a change of physical condition. This is especially noticeable with steel. After repeated heating and cooling, the variation becomes reversi- ble and the hysteretic loss decreases with increasing temper- ature and increases again to the same value with decreasing temperature, approximately as a linear function of the temperature.* Mechanical action affects the energy expended by hystere- sis very greatly, and causes the hysteretic loop to collapse more or less, but apparently does not affect the loss of energy by molecular magnetic friction ; a distinction thus exists between hysteretic loss and molecular magnetic friction loss. Very long-continued exposure of iron in alternating mag- netic fields seems in some cases to increase the loss by molecular magnetic friction through what has been called ageing of the iron, which has been observed especially in iron with very low coefficient 77. Other careful tests, how- ever, have not shown any trace of an increase of hysteretic loss during continuous use in alternating fields, and the ob- served increase thus appears to be due to secondary causes, probably to a continued heating in the alternating field beyond a critical point of the iron, and a change of the * W. Kunz, Electrotechnische Zeitschrift, Berlin, April 5, 1894. 106 HYSTERESIS MOLECULAR MAGNETIC FRICTION. physical condition caused thereby. An effect similar to that observed in the crystallization of wrought-iron in bridges, etc., under vibrating stresses, may also account for the so-called ageing. The following values of the coefficient of molecular mag- netic friction have been observed in different materials.* COEFFICIENT OF HYSTERESIS, AND ABSOLUTE MAGNETIC SATURATION. Coefficient in Milliunits. Average Absolute Magnetic Satu- ration, (B - 3C = 4ir/ in Kilolines. Soft sheet-iron 'and ^heet-steel I 2J.-5 ^ 2 ^3 ^ 17 20 Cast-iron II .316.2 iq IO II Cast-steel of low permeability . 12 j j " of high " soft .... 332 Q 6 12 IQ ^ " of " hard .... 28 18.5 Welded steel annealed 14. ^ 17.4. " " oil hardened 27 16.7 " " very hard 7c 8 ^ Manganese steel, annealed, 4.7$ Mn. . . 41 8.74$ Mn.. 82 " " oil hardened, 4.7$ Mn 6 7 Chrome-steel, annealed I 2% Cr 16 " oil hardened, 1.2% Cr. . . . 44 Wolfram steel annealed 4 6$ W^o . . . . 14. " oil hardened, 3.44$ Wo. 4 8 Magnetite 20.4-23.5 4-7 Nickel wire soft . 12 215 6 I 3 " " hardened ...... 18 * 1 J v Cobalt cast II Q Amalgam of iron, 1 1% Fe 231 V While the loss of energy by hysteresis and by molecular magnetic friction is independent of the frequency within a very wide range, for extremely slow variations of M. M. F. a time lag exists, especially for very low M. M. F.'s. That * " On the Law of Hysteresis," Part II, A. I. E. E. Transactions, 1892, p. 680; Part III, 1894, p. 705. EDDY CURRENTS. IO/ is, after the application of the M. M. F. the magnetism rises quickly to a certain value and then keeps on rising slowly for seconds and even minutes until a final second value is reached. In alternating current machinery this phenomenon of magnetic sluggishness is o< no importance. jp{ 72. Eddy Currents. In magnetic materials exposed to an alternating magnetic field, besides the loss of energy by magnetic hysteresis or molecular magnetic friction, a further loss of energy takes place by eddy currents or Foucault currents. The eddy currents are not a magnetic phenomenon like hysteresis, but a purely electrical phe- nomenon. They are secondary currents induced in the iron by the alternating magnetic field. Thus, while the loss of energy by molecular magnetic friction is entirely independent of the shape of the magnetic circuit, and is the same whether the material is solid or subdivided, the loss of energy by eddy currents depends entirely upon the shape of the iron, and though often excessive in solid iron, may be made small or negligible by thorough subdivision. Eddy currents are true secondary or induced currents, and therefore their E. M. F. is proportional to the magnetic flux (B and the frequency N\ consequently the loss of energy by eddy currents is proportional to (B a and IV*. The loss of energy per cycle is proportional to the square of the magnetic flux, (B, and directly proportional to the fre- quency N, while the loss of energy of molecular magnetic friction is proportional to the 1. 6th power of the magnetic induction (B and independent of the frequency N. The loss of energy by eddy currents per cm* and cycle may, therefore, be represented by the formula W e = 108 HYSTERESIS MOLECULAR MAGNETIC FRICTION. and the total loss of energy in the iron by molecular mag- netic friction and eddy currents by the formula W= W H + W e = r}&'* + eN&\ where ?;= coefficient of hysteresis ; e = coefficient of eddy currents ; N = frequency ; OS = maximum magnetic flux density ; W = W H + W e = total loss of energy, in ergs per cm 8 and cycle.* This formula of the total loss of energy allows the separa- tion of the loss by molecular magnetic friction from the loss by eddy currents, by means of two experimental observa- tions. This can be done either by making two observations for the same flux (B and different frequencies TV, and N 9 , or by making two observations at the same frequency N for different magnetic fluxes (B, and (B a . In either case we get two equations with two unknown quantities rj and e. The latter method, with different magnetic fluxes and the same frequency, is very convenient, but depends upon the shape of the curve of molecular magnetic friction that is, upon the empiric law of the i.6th power while the former method of observation at different frequencies is independ- ent thereof. In general, it is preferable to make a considerable number of observations, plot them in a curve, and take two points from the curve as far apart from each other as is possible without introducing the increased error of observation at the limits of the range of test. * "On the Law of Hysteresis," Part I, A. I. E. E. Transactions, 1892, p. 22. For calculation of the coefficient of eddy currents for laminations and wire, see "On the Law of Hysteresis," Part III, A. I. E. E. Transactions, 1894, pp. 734-738. EDDY CURRENTS. 1 09 73. Effect of Molecular Magnetic Friction and Eddy Currents. Eddy currents can be eliminated by thorough subdivision of the magnetic material, and therefore no excuse exists for their presence in any first-class electrical machinery. Molecular magnetic friction represents a loss of energy which cannot be avoided by subdivision, but can be greatly reduced by the choice of the best available iron. This loss of energy is present wherever magnetic fields are alternating, and is, therefore, found in the armatures of continuous cur- rent dynamos and motors ; to a larger extent still in the armatures of alternators, due to the higher frequency at which they usually operate ; and in transformers and induc- tion motors. In continuous current machinery the loss by molecular magnetic friction is usually only a comparatively small part of the total loss. In alternators the molecular magnetic friction loss is one of the largest losses, and frequently even larger than all the other losses combined. It is therefore of the greatest importance in alternating practice to use the best possible iron. It is obvious that neglect of the molecular magnetic friction in the determina- tion of the efficiency of alternators causes the efficiencies to appear very much higher than they really are. Consequently the so-called electrical efficiency of alternators is of no value whatever, and in discussing the efficiency of alternators, care has to be taken to include the loss of molecular mag- netic friction ; this has not always been done, and as a result efficiencies have been, and still are being, quoted in excess of any reasonable value. When speaking of alternators, obviously alternate current generators as well as synchron- ous motors are understood. In transformers the loss from molecular magnetic friction is usually the largest of the several losses, especially in HO HYSTERESIS MOLECULAR MAGNETIC FRICTION. larger sizes of transformers, and is the more undesirable as it occurs equally at all loads and at no load, while the resist- ance loss in the electric conductors is significant only with a considerable load. Thus the all-day efficiency of a trans- former, or the ratio of the total amount of energy put into the primary of the transformer to the useful energy taken out at its secondary, depends almost exclusively upon the molecular magnetic friction loss. The same applies to induction motors, in so far as they are transformers. 74. Equivalent Sine Curves. A further effect of mo- lecular magnetic friction is its action on the shape of the current wave. If a sine wave of E. M. F. is impressed upon a magnetic circuit, a current will be produced in the circuit which is not a sine wave, but differs widely therefrom, being distorted by hysteresis. With a negligible internal resistance, the sine wave of im- pressed E. M. F. implies a sine wave of counter E. M. F. or induced E. M. F. that is, a sine wave of magnetism, or of magnetic flux, (B. Owing to the discrepancy between M. M. F. and magnetic flux as represented in the hysteretic cycle, the M. M. F., and therefore the current required to produce the sine wave of magnetic flux, is not a sine wave. It is determined by taking from the sine wave of mag- netic flux (B the instantaneous values of (B, and from the hysteretic curve the values of M. M. F., and therefore of current corresponding to the instantaneous values of magnetic flux (B, and plotting these currents as func- tions of the time in the same way as magnetic fluxes are plotted as sine waves. In this way different current waves are produced for different maximum magnetic fluxes. A number of such distorted current waves are shown in Figs. 39 to 42, corresponding to values of maximum flux EQUIVALENT SINE CURVES. Ill (B = 2000, 600O, IO,OOO, 16,000 and of M. M. F.'s, $ = 1.8, 2.8, 4.3, 20.0, 1 and also to the hysteretic cycles : -ghown in Fig. 31. As may be seen, all these curves are bulged out at the ascending, ^2000 = = 1.8 v/ = 6000 r2.8 ^ Bradley $ Poatet, Engr't, ff.f. FIGS. 39 AND 40. DISTORTION OF CURRENT WAVE BY HYSTERESIS. and hollowed in at the descending side. With increasing values of maximum magnetic flux, the maximum point of the current wave becomes pointed, as in Fig. 41, and ultimately a sharp high peak is formed, as in Fig. 42, due to magnetic 112 H YS TERESISMOLECULA R MA GNE TIC FRICTION. saturation.* Such distorted current waves can, like most distorted alternating waves, be replaced for all practical purposes by equivalent sine waves ; that is, sine waves equal in effective intensity and in energy to the distorted wave. These equivalent sine waves of exciting current are shown \\ -Cj 16000 20 \ 13 \ \ \J Bradley & Poatee, Engr's, X. Y. FIGS. 41 AND 42. DISTORTION OF CURRENT WAVE BY HYSTERESIS. in Figs. 39 to 42 ; the remainder, or the difference between the true distorted current wave and the equivalent sine wave, is shown also, and consists, as seen, essentially of a term of triple frequency. This means that hysteresis intro- On the Law of Hysteresis," Part III, A. I. E. E. 7*ransactions, 1894, p. 719- EQ UI VA L EN T SINE CUR VES. 1 1 3 duces higher harmonics in the current wave, amongst which the triple harmonic is especially pronounced. The equivalent sine wave of exciting current is not in phase with the wave of magnetfem, but leads it by an angle a, which is called the angle of feysteretic advance of phase. In consequence, the exciting current can be resolved into two components ; one, / cos a, in phase with the magnetism that is, in quadrature with the induced E. M. F., or wattless, which is called the magnet- izing current ; and into another component, /sin a in quadrature with the magnetism that is, in phase with the induced E. M. F., representing consumption of energy and called the magnetic energy current. The ratio of the magnetic energy current to the total exciting current is the so-called " power " factor of the electromagnetic circuit, whose value is sin a = that is, it depends upon the coefficient of hysteresis 77, the magnetic permeability // and the maximum magnetic flux (&, but is entirely independent of the shape of the mag- netic circuit and the volume of iron used ; in other words, it is a function of the iron alone and not of the electric circuit. Eddy currents have in general no effect on the shape of the wave, but follow the shape of the E. M. F. wave like any other secondary current. In their influence upon the pri- * " On the Law of Hysteresis," Part III, A. I. E. E. Transactions, 1894, p. 595- 114 HYSTERESIS MOLECULAR MAGNETIC FRICTION. mary current they also act like a secondary current that is, in the transformer like a partial load on the secondary cir- cuit, shifting the current wave ahead of the magnetism, and bringing it into lesser phase displacement from the E. M. F. M \ \ \ 7 \ M \ \ 7 i \ radley Poates, Engr's, Jf.f. FIGS. 43 AND 44. DISTORTION OF CURRENT WAVE BY HYSTERESIS. In Fig. 43 is shown the current wave in a transformer at partial secondary load, showing the reduction of the distortion under load. The dotted line M represents the magnetic fiux. HYSTERETIC LOSSES. 11$ In a magnetic circuit consisting partly of iron and partly of air, the curve of exciting current of an impressed sine wave of E.M.F. is the sum of the exciting currents of the iron and of the air portions of f the circuit. Since air has no hysteresis, but very low permeability, the exciting current of the air portion of the circuit is a sine wave of high ampli- tude, and therefore, when superimposed upon the wave of exciting current of the iron circuit, reduces the relative dis- tortion of the total exciting current, and makes it appear more sine-shaped ; this is shown in Fig. 44, which represents the exciting current of a magnetic circuit containing an air- gap of 1/400 of the length of the iron,* and corresponding to a maximum flux of (B = 6000, or to the curve shown in Fig. 40. The magnetic flux is shown by the dotted line M. 75. Hysteretic Losses. The loss of energy by hyster- esis depends upon the maximum value of the magnetic flux, while the energy transformed in a transformer depends upon the effective value of E. M. F. With a sine wave of E. M. F., the magnetic flux follows a sine wave also, and therefore the maximum value of the latter is proportional to the impressed E. M. F. If, however, the wave of impressed E. M. F. is dis- torted, or differs from the sine form, the wave of magnetic flux will be distorted also by the superposition of higher harmonics, and therefore, with the same effective value of impressed E. M. F., its maximum value in the case of a flat- topped wave of magnetic flux, can be lower than with a sine wave, or higher than with a sine wave with a peaked wave, of magnetic flux. In the first case the same amount of energy will be transformed with a lesser hysteretic loss, and in the latter case with a larger hysteretic loss ; or, in other * "On the Law of Hysteresis," Part III, A. I. E. E. Transactions, 1894, p. 720. Il6 HYSTERESIS MOLECULAR MAGNETIC FRICTION. words, the hysteretic loss in a transformer, at the same im- pressed effective E. M. F., depends upon the shape of the E. M. F. wave, and differs with different wave-shapes of E. M. F., by as much as 30$. Iron-clad alternators of " uni-tooth " construction in general give peaked waves of E. M. F., and, therefore, flat-topped waves of magnetic flux ; that is, give hysteretic losses in transformers lower than with sine waves by something like Distributed windings, that is, comparable to windings of a continuous current machine tapped at two diametrically opposite points of the armature, frequently give a pointed wave of magnetic flux, thus increasing hysteretic losses. Since eddy currents are secondary currents, the loss of energy by such currents in transformers is entirely inde- pendent of the wave-shape of impressed E. M. F., and there- fore proportional to the load on the secondary circuit at constant resistance ; or, in other words, if the secondary circuit of the transformer is closed by a given resistance, and the same effective E. M. F. impressed upon the primary cir- cuit, the energy of this E. M. F. will divide in the same pro- portion between the useful secondary circuit, and the secondary waste circuit of eddy currents, independently of the shape of the E. M. F. wave. PROPERTIES OF ELECTRIFIED BODIES. 1 1/ ELECTRICITY. PROPERTIES OF ELECTRIFIED BODIES. 76. Phenomenon of Electrification. When a glass or resin rod is rubbed with a piece of woollen cloth, it is ob- served that the rod attracts light bodies. The experiment may be made by suspending a ball of elder-pith by a silk thread. The ball is first attracted by the rod, then it is repelled after having touched the rod. But if a rod of rubbed resin is brought near the ball, after it has been repelled by the glass rod, it is observed that the ball is again attracted. The bodies between which similar actions are shown are said to be electrified, and the name electricity is given to the unknown agent which produces these phenomena. Observations hows that the electric properties observed in glass and resin are general. Any two bodies, A and B, attract each other after having been rubbed together. But two bodies of the same nature, A and A r , repel each other after having been rubbed by a third. To interpret these phenomena, opposite electrifications are attributed the bodies after rubbing; those which behave like glass, in regard to wool, are said to be positively electri- fied, or charged with positive electricity; those which act like resin are said to be negatively electrified, or charged with negative electricity. Electric actions are summed up in the following rule : Bodies charged with electricity of the same name repel each other and attract those charged with electricity of contrary name. 1 1 8 ELE CTRICIT Y. It must be observed, however, that these denominations do not imply the existence of two distinct kinds of elec- tricity, but that they are only forms of speech designed to denote different states of electrification. One must not extend their meaning further than is done in mathematics in the case of quantities of contrary sign. Following a hypothesis suggested by Franklin, electricity is generally compared to an imponderable fluid, of which every body contains a normal quantity. If the charge sur- passes this quantity, there is positive electrification. In the opposite case the electrification is negative. A body is in the neutral state when it has its normal quantity of electricity. According to some eminent physicists, Clausius among others, electricity is supposed to be the ether in which the molecules of all bodies are enveloped and which fills interplanetary space. The phenomena of electrification are, on this hypothesis, supposed to be due to particular conditions of the ether. 77. Conductors and Insulators. Different bodies act differently with regard to electricity. When one of the extremities of a metallic rod is electrified by friction, attractive properties are immediately manifested in all parts of the rod. In a rod of resin, on the contrary, this propagation of the phenomenon is extremely slow. Those bodies which propagate electric actions rapidly are called conductors, the rest are called insulators. Conductors, which include all metals and alloys, have to be supported by insulators in order to retain their attractive properties. From the point of view of the experiments which we are examining, the earth and all bodies impreg- nated or coated with moisture act like conductors. The choice of the insulating supports is therefore important for the success of these experiments. Air is frequently em- PROPERTIES OF ELECTRIFIED BODIES. 1 1 9 ployed as an insulator ; in order to prevent the moisture it contains from condensing on the solid insulators which support the electrified conductors, and thus producing a moist conductive coating, a support may be used whose base passes through air dried by concentrated sulphuric acid, Fig. 45. FIG. 45- Among solids the insulators most commonly used are glass, glazed porcelain, india-rubber, ebonite or india-rubber hardened by combination with sulphur, gutta-percha, paraf- fine, silk, cellulose, and shellac. Some of these substances, such as glass and cellulose, are hygroscopic. They should be covered with a coating of insulating varnish or im pregnated with paraffine. 78. Electrification by Influence. The experiment with the pith-ball, 76, has shown us that a body can be electri- fied by contact. Even the mere approach of an electrified body is sufficient to produce electrical manifestations in a neighboring conductor. Thus a sphere, A, charged with positive electricity, being brought over a conductor BC, Fig. 46, it is observed that the extremities of this latter act upon a freely suspended pith-ball. If this ball be charged beforehand with positive electricity, it is seen to be attracted by the extremity B, and repelled by C. From this we conclude that the former is negatively, the latter positively, electrified. When the influencing sphere, A, is withdrawn, the conductor, BC, no longer acts on a ball in the neutral state ; which shows that 120 ELECTRICITY. the contrary electricities accumulated in B and C have re-established the neutral state by recombining. FIG. 46. If the conductor BC is connected for a moment to the earth while it is under the influence of A, we observe after taking away A that every portion of BC is charged negatively. This method of charging the conductor is called electri- fication by influence. 79. Let us suppose that the framework of a quadrant electrometer, E, Fig. 47, be connected to the earth through FIG. 47- the suspension-wire, and that the two pair of quadrants are electrified, but of opposite sign. The framework will be electrified by influence ; it will be equally attracted in both PROPERTIES OF ELECTRIFIED BODIES. 121 directions and will remain motionless. But if it receives a positive charge, it will immediately be displaced towards the negative quadrants. If the charge be negative, the displacement will be towards '"the positive quadrants. We shall show, further on, that tfete swing of the framework is proportional to the charge that has been given to it ; we shall also see that in order to give the quadrants equal and contrary charges they need only be connected to the poles of a battery. This apparatus, which is suitable for measuring feeble electrifications, will enable us to proceed to some funda- mental experiments. 80, Experiments. Let us consider a metallic cylinder, A, Fig. 48, open at the top and supported on an insulating FIG. 48. base ; let us also connect the cylinder with the framework, c, in the electrometer by a thin wire, both pairs of quadrants 122 ELECTRICITY. being electrified in the manner indicated in the preceding paragraph. If the cylinder be in the neutral state, the framework will not change its position. I. Introduce into the cylinder a conductor, B, hung from a silk thread and charged with positive electricity. We know, by the phenomenon of influence, that the internal wall of the cylinder will be charged negatively, and that the electricity of the same sign as that of B will be repelled towards the ex- ternal wall and to the framework, c. This latter will exhibit a displacement which will increase as the conductor B de- scends, until this last has reached a certain depth in the cylinder. From this point onwards the displacement of the framework c remains invariable, whatever be the position of B; even if B be put in contact with the cylinder, the dis- placement of the electrometer remains the same. Upon withdrawing the body after contact it will be found, by means of a pith-ball electroscope or a second quadrant elec- trometer, that B has returned to the neutral state. Hence we conclude that the positive charge which it has parted with to the cylinder has been neutralized by the negative charge developed by influence, the cylinder preserving its induced positive charge. If the charge on B were negative, the electrometer would have shown a displacement in the opposite direction. II. Place inside the cylinder a metallic sphere charged with electricity and giving a displacement a. Touch this sphere with an equal one in the neutral state. Then the two spheres, if placed successively inside the cylinder, will give deviations equal to a/2. III. Suspend inside the cylinder two insulated bodies in the neutral state and rub them one against the other. The electrometer will show no deviation, but will remain in its first position. PROPERTIES OF ELECTRIFIED BODIES. 12$ IV. If we introduce separately into the cylinder various bodies, B, D, E, charged with electricity, we get deviations some of which are positive, others negative, according to the signs of the charges. , " On introducing simultaneously B, D, and E the devia- tion obtained is the algebraic sum of the preceding devia- tions and remains invariable, even if the bodies are put in contact or rubbed against each other. From these experiments we conclude that electric charges are quantities susceptible of measurement. We might take a given charge as unity and consider as double and triple those charges which produce a double or triple deviation in the electrometer connected with the cylinder. The two last experiments show that the total charge of a system of electrified bodies is invariable, and that friction produces on bodies equal and opposite electricities, capable of neutralizing each other. If, for example, we electrify a resin rod by aid of a piece of cloth, a quantity of electricity equal and opposite to the charge on the resin is produced on the cloth and on the neighboring conductors connected with it, such as the body of the experimenter, the table used for the experiment, and the walls of the room. 81. In the case of a conductor in equilibrium the elec- tricity is distributed on its surface. This important property can be demonstrated in various ways. I. For example, let A, Fig. 49, be a hollow metallic sphere. Touch the exterior surface of the sphere with a proof-plane formed of a disc of copper foil on an insulating handle ; the disc will take an electric charge, as can be dis- covered by means of an electroscope or electrometer. If, on the other hand, the point touched be on the interior sur- face of the sphere, the disc will show no electrification. 124 ELECTRICITY. The method of the proof-plane, though by no means ex- act, can be employed to compare the electric charges on the different parts of a body. If we touch successively various points on the surface of an electrified sphere, we find that FIG. 49. the charge carried away each time is constant, that is, that the sphere is uniformly charged. For a body of ovoid form we find that the charge increases inversely as the radius of curvature of the surface. II. The fact of the external distribution of the charge on a conductor has been established beyond doubt by Faraday, who had a chamber four metres square constructed, sup- ported on insulators and covered with metallic sheets. He entered this chamber while it was electrified and was unable to discover the least trace of electricity on its internal walls, though using the most delicate electrometers. 82. Law of Electric Actions. The preceding facts, on being summed up, show that the action between electrified bodies must be placed amongst the central forces, and that we can define by quantity of electricity, or electric mass, or charge of a body, a quantity proportional to the force which it exercises upon neighboring electrified bodies. The superficial density is the charge per unit surface. In virtue of this definition the elementary law of the force between two electric masses q and q f , at a distance /, will be of the form /= kqq'&l}. PROPERTIES OF ELECTRIFIED BODIES. 125 The function 0(/) is determined by the fact that the electric action is zero in the interior of an electrified con- ductor in equilibrium, 81. The reciprocal of the theorem demonstrated in 26 shows that, in order th#t a homogeneous spherical shell may have no action upon an internal point, the force must be in inverse ratio to the square of the distance. The law of electric attraction and repulsion is therefore in which we consider the force f as repulsive or attractive according as the masses q, q' are of the same or contrary sign. This law, experimentally demonstrated by Coulomb, en- ables us to apply to electric attraction the general properties of the Newtonian central forces. 83. Definitions. Electric Field. Electric Potential. By electric field we will designate a space or region where electric forces take origin. The intensity in a point of an electric field is the resultant of the forces exercised there upon a positive mass taken as unity. The direction of this resultant is called the direction of the field. Electric forces are defined by aid of a potential, 12, called electric potential, which has as its expression The intensity of the field in a direction s is expressed as a function of the potential by K da X ' = "57- 84. Potential of a Conductor in Equilibrium. Since the force is zero at the interior of a conductor carrying an elec- 126 ELECTRICITY. trie charge in equilibrium, 81, the electric potential has the same value at all internal points ; the surface of the con- ductor is equipotential, and the lines of force are normal to its surface. This constant value is called the potential of the con- ductor. The idea of potential may be distinguished from the idea of charge by the following experiment ; The method of the proof-plane demonstrates that the charge is variable at different points of a conductor of irregular shape. Nev- ertheless, if we successively connect these points by a wire with the electrometer, the deviation remains constant, for the force which urges the electricity to pass from the con- ductor to the electrometer depends only on the potential, which is invariable. In the case of a metallic sphere of radius R electrified to a density cr the potential at the centre is 26. K K. This quantity is the potential of the sphere. 85. Potential of the Earth. Electric forces do not depend at all on the absolute values of the potential, but only on its variation. Thus the force which impels the electricity of a conductor to flow to an adjoining conductor, such as the earth, depends on the difference between the potentials of the bodies under consideration ; positive elec- tricity tends to be displaced in the direction of decreasing potentials. The ground and the walls of an experimenting- room being conductors, there is no objection to considering their potential as zero and then taking as positive those potentials above that of the earth, the potentials being nega- tive in the opposite case. PROPERTIES OF ELECTRIFIED BODIES. I2/ 86. Coulomb's Theorem. The intensity of the field at a point infinitely near a conductor in equilibrium is equal to ^nk multiplied by the superficial density at that point. Let us consider an element ds of the surface charged with a quantity of electricity e^ual to ads, a being the den- sity. Let us suppose a tube of force, passing through the contour-line of ds, limited externally by an infinitely near equipotential surface U, and closed inside the conductor by any surface 5. Let us then apply Gauss's theorem, 20, to the volume thus enclosed. The flux of force is zero across the surface 5, since there is no electric force inside a con- ductor in equilibrium ; as there is no component across the lateral walls of the tube, the outflowing flux from the given volume is limited to the flux across the equipotential sur- face U r Let OC g be the intensity of the field normal to this surface ; the flux 3 e ds is equal to ^nk multiplied by the quantity of electricity ads. Consequently OC, == 47tk. The deviation is proportional to the potential of the needle. 140 ELECTRICITY. M. Gouy* has shown that the preceding formula does not hold when the potentials of the quadrants are consider- able relatively to that of the needle. In such case there exists a directive electric couple acting together with the torsion-couple, and whose action must be taken into account. It must be further remarked that, if the needle is not of the same metal as the quadrants, there arises a difference of potential by contact, 107, which slightly alters the results, and which is eliminated by making two successive experi- ments in which this extraneous difference of potential pro- duces equal and opposite effects. The mean of the devia- tions thus obtained is the result desired. 103. Specific Inductive Capacity of Dielectrics. We have so far paid no attention to the dielectric separating the condenser-plates. For the purpose of studying this ele- ment let us consider two identical plane condensers, the dielectric in the one being air, and in the other paraffine. The similar plates being connected to earth, electrify the free plate of the first condenser to the potential U, and con- nect it with the needle of an electrometer whose quadrants are supposed to be at equal and contrary potentials. Let a be the deviation of the needle. Then connect the free plates of the two condensers by a wire. As the initial charge q spreads over both condensers the potential diminishes and becomes /'; the deviation of the electrometer falls to a' . Calling c and c' the capacities of the two condensers, we get the condition whence U U' Journal de Physique, 1888. PROPERTIES OF ELECTRIFIED BODIES. 14! But jj-, = -^; consequently The ratio is called the Specific inductive capacity of the paraffine in comparison with air. This ratio is about 2.3 in the present case. We might compare the capacities of dielectrics to that of a vacuum by placing one of the condensers in a vacuum- chamber. We give here some of the values thus obtained, according to Boltzman: Sulphur ........................ 3.84 Glass ........................... 5.83 to 6.34 Paraffine ...................... 2.32 Ebonite ........................ 2.2 1 to 2.76 Essence of turpentine ........... 2.21 Air .......... . ................... 1.00059 Carbonic anhydride ............. 1.000946 The comparison of these values with the indices of re- fraction i and i' of the same substances for light shows that the former are sensibly proportional to the squares of the latter. As the index of refraction of a medium is inversely pro- portional to the velocity of light in it, it follows that the specific inductive capacities of bodies are inversely propor- tional to the square of the velocity of light in those bodies : 104. Nature of the Coefficient k in Coulomb's Law. Given an air-condenser charged with a quantity of electricity q at a potential V=k?l 142 ELE CTRICI T Y. the capacity is =* Let us substitute for air some other dielectric ; keeping the charge constant, we will get a new capacity, the potential necessarily varying and becoming ; ;:^r ^4 Hence we deduce L - - - - 7~~ u ~ k ~ ' ~tf Now the ratio of the capacities is the same as that of the specific inductive capacities. Consequently the coefficient in Coulomb's law is inversely proportional to the inductive capacity of the dielectric sep- arating the electrified bodies, and, if the deductions of the last paragraph are true, this coefficient is also proportional to the square of the velocity of light in the dielectric. It is because the coefficient k is not an arbitrary factor, but an actual physical quantity, that we have retained it in our formulae. The importance of this coefficient will be seen in the comparison of electrical units. 105. Role of the Dielectrics. Displacement. What we have just seen shows the importance of the role of the di- electric in electrostatic actions. We present here an ex- periment which leads to the consideration of the phenome- non of condensation (or accumulation) in a new light. A condenser formed of three detachable parts, a glass jar or plate and two metallic plates, is charged in the ordi- PROPERTIES OF ELECTRIFIED BODIES. 143 nary way. The three parts are then separated and the metal plates connected to earth ; then the condenser is put together again, and it can be discharged as if it had remained undisturbed. This experiment sho,ws that ^the charge is, at least in part, on the dielectric. Faraday has deduced an ingenious hypothesis from this fact. According to him, dielectrics present a polarization which recalls that which we have met with in the phenomena of magnetization. This polarization takes place along the lines and tubes of force which join the plates of condensers and electrified bodies in general. Take the case of an elementary tube between two con- ductors A and B joining corresponding elements ds 1 , dj, charged with densities cr l , 0* 2 , 88. This tube can be subdivided by equipotential sections into elements of volume presenting equal and contrary charges on their opposing ends, thus balancing each other. At the extremities of the tube exist quantities of free electricity, + q = o-.ds, , - q = a,ds, , which charge the conductors, these latter having no other function than to limit the dielectric. In this hypothesis the product ads = CT I ^S I is constant along the tube ; but at the surface of the conductors, 86, we have and as the product 3C/ta = 3t l ds 1 is constant in the tube, 21, we conclude that in any point of the field the quantity of electricity displaced per unit equipotential surface is 144 ELECTRICITY. equal to the intensity of the field at this point divided by the factor ^nk : Maxwell calls this expression the displacement of electricity in the dielectric. The analogy which exists between electric induction thus interpreted and magnetic induction will be recognized : electric displacement corresponding to intensity of magneti- zation, and the specific inductive capacity to the coefficient of magnetization (magnetic susceptibility}. This point of view gives us an explanation of the energy of electrified conductors ; we conceive, namely, that there results from the polarization of the dielectric molecules a state of tension along the lines of force, in virtue of which these lines tend to shorten themselves by causing the approach of the conductors which limit the dielectric. This tension, which is comparable with that of an elastic body, assimilates the energy of the dielectric to that of a spring. It can be readily verified that, if the electric field is uniform between the plates of a condenser, such as a guard- ring condenser, the total energy is expressed by the volume 3C a of the dielectric multiplied by --j., 3C, being the intensity of the field. In fact, keeping the previous notation, W=teU=i C7 t , the lines of force are parallel to the axis of the cylinder. At any point . AU " d/' d/ denoting an element parallel to the axis. As H is constant all along the cylinder, we have // />/, H d/=-/ t/o l/t/! whence Hl=U,-U,. Consequently the current I 5 8 ELECTRICIT Y. where the sign / must be extended to every section of the conductor ; consequently r_ ys The same method of reasoning can be applied to a con- ductor of any shape and section whatever, provided that the latter be constant throughout the conductor. The ratio is called the resistance of the conductor ; , ys y which measures the resistance of a conductor having unit length and unit section, is the specific resistance or resistivity. It is seen that the resistivity is the inverse of the con- ductivity, just as the resistance of a conductor is the inverse of its conductance. The above equation was discovered by Ohm. It is enunciated as follows : The strength of the current between two points of a con- ductor of any form, but of constant section, is directly propor- tional to the difference of potential and inversely proportional to the resistance between those points. By virtue of the definition given of strength of current the quantity of electricity which passes across a section of the conductor in a time t is This evident equation is sometimes called Faraday's law. 116. Graphic Representation of Ohm's Law. Ohm's law proves that in a conductor of constant section the poten- tial falls uniformly, for the relation ys LAWS OF THE ELECTRIC CURRENT. 1 59 shows that the variation of the potential is proportional to the variation of the length. Mark off on a horizontal line Ox a length OA, measuring, on a given scale, the resistance-^ of a conductor. FIG. 55- On the perpendiculars at the extremities of OA mark off OB and AC to represent the potentials at the two ends of the conductor; then the ordinates of the straight line BC represent the potentials of the intermediate points of the conductor. The strength of the current is measured by the tangent of the angle between BC and OA, for we have OB -AC U-U 117. Variable Period of the Current. Suppose a con- ductor traversed by a current and enveloped in a dielec- tric sheathing, which latter is surrounded by a conductive layer: this is the case with an insulated conductor under water. If the outside conductor is connected with the earth, its potential is zero, and an electric field is formed in the di- electric, the lines of force passing from the inside conductor to the outside or vice versa, according as the potential of the inner conductor is greater or less than zero. These lines of force join points carrying opposite charges ( 88), so that the adjoining surfaces of the dielectric as- sume contrary electrifications, as in the case of a condenser. 1 60 ELECTRICIT Y. The charge per unit length will, however, vary at different points of the conductor. If, for example, the potential de- creases uniformly from U to o in the conductor, the charge will also decrease regularly from one extremity to the other, and the total charge of the conductor will be equal to the capacity of the condenser multiplied by the mean difference of potential of the two plates. To explain the phenomenon of condensation we must admit that at the moment when a difference of potential occurs in the conductor a momentary flux of electricity traverses its surface to produce the surface-distribution. This variable period is followed by the permanent period, during which the electric flux is constant and passes in its entirety parallel to the walls of the conductor. 118. Application of Ohm's Law to the Variable Period of the Current in but Slightly Conductive Bodies. When the plates of a condenser are joined by an imperfect insulator, such as caoutchouc or gutta-percha, it is observed that they lose their electrification little by little. This loss can be attributed to a certain conduction through the given substance, and Ohm's law can be applied to this special case of a variable period. Let C be the capacity of the con- denser, R its resistance, ^the difference of potential between the plates. At any moment the current / is the ratio of the potential difference to the resistance, The quantity of electricity which passes during a time d/is - d? = 7d/ at that instant of time. LAWS OF THE ELECTRIC CURRENT. l6l But q = CU. (82.) Consequently d?= whence The time that the difference of potential takes to pass from 7, to / 3 is /",, Jo dt = - or 119. Application of Ohm's Law to the Case of a Heterogeneous Circuit. Suppose two conductors ab, be, of resistances R l and R^ , in contact at the point b, and whose free ends a and c are maintained, by any means whatever, at constant potentials U l and U^ . A current will proceed from the point of higher potential to that of lower potential. Suppose U,> U,. At the point of contact b an electromotive force E is produced, being the difference of the potentials /"/, U t ', of the two sides of the surface of separation. Suppose v; > u,\ we have E=U l ' - /,'. Let us apply Ohm's law to the two portions ab, be, observing that, as the electricity can be neither accumu- lated at nor abstracted from the point b during the perma- 1 62 ELECTRICIT Y. nent stage, the current must necessarily be the same in the two conductors. We shall therefore have /?, A. U,-U,-E R>+R* In the case where u, f > u T _V,-U, the electromotive force E being taken wjth the -f- or sign according as it gives an increase or decrease of potential in the direction of the current, that is to say, according as it tends to increase or diminish the current. By extension, if there are several conductors in contact, E l , E 9 , E 3 . . . being the electromotive forces, R 1 , R,, R t . . . the resistances, we have r &> the signs of the electromotive forces being obtained by the preceding rule. If we connect the end conductors directly, we will have We know that 2 is zero, unless there is a difference of temperature or chemical action at the points of junction (" 3). 120. Graphic Representation. Take the case of three conductors ; mark off successively on the axis of abscissae lengths proportional to their resistances R iy R^, R 3 . LAWS OF THE ELECTRIC CURRENT. 163 Let aa! = U l (Fig. 56). From the point a' draw the right line a'b' inclined at an angle a, such that = I. tan If the electromotive force ofc contact E l is positive, meas- ure it off along b'b" and from b" draw b"c parallel to a'b'. ~ 'd FIG. 56. The electromotive force E 9 , supposed to be negative, is marked off on c'c" and the new line c"d! will cut the poten- tial-line dd' at the point d' '. 121. Kirchhoffs Laws. This name is given to two laws, one of which is evident when we compare the electric current with a fluid current, the other deduced from Ohm's law ; the two together enabling us to solve the problem of even the most complicated electric circuits. First Law. When any number of conductors meet in a point, the algebraic sum of the currents at that point is zero. This rule simply expresses the fact electricity can be neither accumulated nor subtracted at the meeting-point of conductors. The currents are considered as of opposite sign, according as they flow into or away from the point. Second Law. In every closed circuit, the algebraic sum of the electromotive forces equals the algebraic, sum of the products of the currents by the resistances of the conductors. Let abed be a circuit in a network of conductors. Let us denote by i , , z a , i 3 , i< the currents whose direc- 164 ELECTRICITY. tion is shown by the arrows, and by r l , r^ , r a , r t the resist- ances, and by *, , * a , *, , ^ 4 the electromotive forces shown by the unequal parallel strokes. The -j- sign shows the FIG. 57. direction in which each electromotive force tends to produce an increase of potential. Let u l , & a , u a , & 4 be the potentials at the points a, b t c 9 d. We shall then have, by 113, whence or The signs which should be given to the currents and electromotive forces are easily determined, viz., follow the circuit round in the direction of movement of the hands of a watch ; give the sign -J- to those currents which move in^ this direction, and the sign to those moving in the contrary direction. As to the electromotive forces, give them -f- or signs, according as they cause an increase or diminution of potential in the given direction. LAWS OF THE ELECTRIC CURRENT. 165 The application of Kirchhoffs laws to a combination of n conductors gives n distinct equations between the currents, the resistances, and the electromotive forces, whence we can deduce n of these quantities if the others are known. This method will enable us^ for example, to determine the currents and their signs : we begin by supposing arbi- trary directions of current the true directions will be given by the calculation ; a positive value will show that the sup- posed direction was correct, a negative value will indicate that the current was in the opposite direction. 122. Application to Derived Circuits. The combina- tion of circuits shown in Fig. 58 is made up of homogeneous conductors having resistances r 1 ,r. lJ r aJ ending in the points a and b, which are connected by a conductor of the same nature, including a chemical source of electromotive force /. Let r be the resistance of this part of the circuit. FIG. 58. The electromotive force produces a total current 7, which divides itself among the three derived branches r l9 r 9 , r % into three partial currents i l , /, , i % , such that KirchhofTs second law gives the equations 1 66 ELECTRICIT Y. Eliminating successively i l , z, , z, from these four equa- tions, we get e The expression represents the combined resistance of the three derived con- ductors r 1 , r v and r 3 In general, the reciprocal of the combined resistance of a number of derived conductors is equal to the sum of the recipro- cals of the resistances of the component conductors. 123. Wheatstone's Bridge or Parallelogram. The arrangement in Fig. 59 has been designed by Wheatstone for the purpose of measuring electrical resistances. FIG. 59. Suppose there are six conductors having resistances a, b, f, d, g, p. The branch r contains an electromotive force e ; the branch g an apparatus to indicate the passage of a cur- rent. The total current / is divided into partial currents which we will denote by the capitals A, B, F, D, G. ENERGY OF THE ELECTRIC CURRENT. 1 67 The application of Kirchhoff's laws gives the six follow- ing equations: I-A-F =o; A-G^-B =o; E+G-&D =o; aA Eliminating A, B, F, D, we have I(ad-bf) In order that the current G may be zero, it is sufficient to have. ad=bf or |=- ENERGY OF THE ELECTRIC CURRENT. 124. General Expression. In consequence of the defi- nition of electric potential, when a quantity q of electricity passes from a potential U l to a lower potential 7 a , the work accomplished is In the case where a current 7 circulates between the given points the work per second, that is, the electric power de- veloped by the current, is therefore (U, - U,)f. If U l U^ represents an electromotive force E, thermic or chemical, the power will be expressed by EL 1 68 ELECTRICIT Y. 125. Application to the Case of a Homogeneous Conductor. Joule's Effect. If we take a homogeneous conductor of resistance R traversed by a constant current /, we have (v l -uj/=m. (115.) The work developed in a time / is W=I*Rt. Joule has experimentally proved that this work is en- tirely transformed into heat in the conductor. One of the most beautiful illustrations of Joule's effect is the incandescent electric lamp, in which the current heats a carbon filament placed in a glass bulb from which the air has been exhausted in order to prevent combustion. 126. Case of Heterogeneous Conductors. Peltier Effect. Suppose a number of conductors R l , R^ , ^ 3 ( 120), without chemical action on each other; denote by / the current which flows through them, by E l and E^ the elec- tromotive forces of contact. By Joule's law the heat de- veloped per second in each of the conductors is respectively At the points of junction there are, in addition, abrupt variations of potential E l , E 9 which correspond to amounts of electric energy .,/, EJ. The increase of energy of the current will be negative if the potentials fall in the direc- tion of the current ; it will be positive in the contrary case. Peltier has shown, in the first case, that the junction is heated ; in the second, that it is cooled. These calorific variations are equal and contrary to the variations in the energy of the electric flux. This phenomenon, known under the name of Peltier effect, enables us to measure exactly the electromotive force of contact. Contrary to the Joule effect, ENERGY OF THE ELECTRIC CURRENT. 169 we see that the Peltier effect depends on the direction of the current and that it changes sign with the current. To show the Peltier effect it is necessary to take steps to prevent the heat developed, in the conductors by the Joule effect from hiding the variations of temperature, generally feeble at the points of junction. This is effected by using feeble currents and coating the junctions with some readily fusible substance, such as wax. The wax is then observed to melt when a current flows in one direction, and to solidify when it flows in the opposite direction. From the law of successive contacts ( 112) it follows that in a closed circuit where no differences of temperature are maintained by external sources of heat the algebraic sum of the electromotive forces of contact is zero, and con- sequently the sum of the Peltier effects is also zero. 127. Chemical Effect of the Current. Faraday's and BecquereTs Laws. When an electric current passes through a. compound liquid, by means of conductors or elec- trodes dipping into the liquid and kept at different poten- tials, besides the heating due to the Joule and Peltier effects, decomposition of the liquid is observed to take place. The separated elements go to the electrodes, with which in certain cases they enter into combination. This decomposition is called electrolysis, and the decom- posed body the electrolyte. The electrode having the highest potential, by which the current enters, is the positive elec- trode or anode; the other is the negative electrode or cathode. The products of decomposition are the ions. Electrolysis takes place according to the following (Fara- day's) laws : I. The weights of the ions deposited and of the decomposed electrolyte are proportional to the quantities of electricity which have passed through the liquid. ELECTRICIT Y. II. When a number of electrolytes are traversed by the same current, the weights of the different ions set free are to each other as the chemical equivalents of these ions. The electrochemical equivalent of an ion or an electrolyte is the weight of this body deposited or decomposed per unit quantity of electricity. BecquerePs Law. In the case where two bodies form various combinations with each other the decomposition of these different combinations is dependent on the negative element. Thus in the electrolysis of the combinations /W t , PN % , /yV 8 , where P is a metal and N a metalloid, unit quantity of electricity sets free one electrochemical equiva- lent of N and weights of P equal to its electrochemical equivalent multiplied by I, -J, f. 128. Grothiiss' Hypothesis. The fact that the decom- position of an electrolyte is a necessity for the passage of the current has suggested the idea that the ions play the same part as the pith-ball in convective discharge. If we admit that the molecules of the electrolyte are formed of groups of elements having opposite charges of electricity (possibly, according to Maxwell, due to the electro- motive force of contact), then, at the moment of introducing electrodes, the positive elements or ions will turn towards the cathode and the negative elements towards the anode. This polarization will take place along the lines of force of the field produced in the liquid by the electrodes. If the intensity of the field is sufficient to overcome the chemical affinity of the compound, the ions near the electrodes are set free, while in the intermediate molecules of the liquid there is simply an exchange of elements. We have thus an explana- tion of the reason why the products of decomposition only make their appearance at the points where the current en- ters and leaves the liquid. ENERGY OF THE ELECTRIC CURRENT. I? I The electric charge carried per second by the positive ions to the cathode represents the amount of the current, which accounts for Faraday's first law. To account for the second law we need only suppose that the electronegative elements of different electrolytes have all the same electric charge. According to Clausius' kinetic theory, the molecules are in motion and their impact causes their disassociation into the component atoms. But these atoms recombine with those liberated from the adjoining molecules, so that there are continual changes in all directions. The electric current causes an orientation of these movements and brings about a final decomposition at the electrodes. 129. Application of the Conservation of Energy to Electrolysis. Voltaic Cell. The phenomenon of electroly- sis can be considered from the point of view of the con- servation of energy. In an endothermic electrolytic reaction which absorbs energy, as is the case when acidulated water is decomposed between platinum electrodes, the effective energy of the current is diminished ; a fall of potential takes place in the direction of the current, equal to what is called the elec- tromotive force of polarization of the electrolyte. This electromotive force is negative ( 119), and tends to produce a counter-current. The existence of this electromotive force can be shown by connecting the platinum conductors, imme- diately after electrolysis, to an apparatus for showing the passage of a current and then closing the circuit. A current will be observed directed from the cathode to the anode in the electrolyte, and at the same time the liberated elements, oxygen and hydrogen, will recombine. Lord Kelvin has shown that the electromotive force of polarization can be calculated when the energy of combina- ELECTRICITY. tion of the electrolyte is known. In fact, if no secondary action takes place, the electrical energy absorbed (represented by the product ie of the electromotive force of polarization into the current) is equal to the heat of combination of the weight of electrolyte decomposed per second, expressed in absolute units. Let z be the electrochemical equivalent of the electrolyte, h the heat of combination of unit weight of the same. We then have ie = zhi ; whence This expression gives the minimum difference of potential between the electrodes necessary to produce decomposition. From these considerations we deduce a means of separating the elements of several electrolytes mixed together. Sup- pose, for example, a solution containing sulphate of zinc and sulphate of copper ; as the heat of combination of the sec- ond salt is less than that of the first, we can, by suitably graduating the potential difference of the electrodes, deposit first copper and then zinc upon the cathode. There are cases where the ions react on the electrodes, giving rise to new compounds. The energy set free in these reactions must be taken into account in calculating the elec- tromotive force necessary for decomposition. Take the example of the electrolysis of a solution of cop- per sulphate between copper electrodes. The liberated cop- per will go to the cathode and the acid to the anode, which it will dissolve equivalent for equivalent. The reaction at the electrodes thus neutralizes the chemical effect of the current, so that the electromotive force of decomposition is zero. The whole energy of the current is used in the Joule effect, that is, in heating the bath. ENERGY OF THE ELECTRIC CURRENT. 1/3 Suppose that water acidulated with sulphuric acid is de- composed between a zrnc anode and a copper cathode. The hydrogen will be deposited on the copper, while the oxygen will form oxide of zinc with, the anode, which will then be dissolved in the state of sulphate of zinc. But as the heat of combination of zinc sulphate is higher than that of sul- phuric acid, a quantity of energy is consequently liberated which shows itself by an increase of potential in the direction of the current equal to the effective difference of the elec- tromotive forces. This electromotive force is E = ph p'h' , ph denoting the heat of formation of one electrochemical equivalent of zinc sulphate, p'h' that of one equivalent of sulphuric acid. Such a combination, called voltaic element or couple, is a source of electricity ; and if we connect the two electrodes by a copper wire it will be found that it is traversed by a cur- rent flowing from the zinc to the copper in the electrolyte, and from the copper to the zinc in the external circuit formed by the wire. The direction of the current in the external circuit shows that the copper is at a higher potential than the zinc, whence the name of positive pole or plate given to the plate of cop- per. The zinc plate is, conversely, called the negative pole or plate. The current is where E represents the electromotive force and R the re- sistance of the circuit, including the resistance of the liquid as well as that of the wire and the electrodes. 174 ELECTRICITY. THERMO-ELECTRIC COUPLES. 130. Seebeck and Peltier Effects. Seebeck has proved (113) the production of an E. M. F. in a chain of metals whose junctions are maintained at unequal temperatures. Thus on forming a circuit of an iron and a copper wire the latter including a galvanometer coil, and on raising the tem- perature of one of the points of junction, a current is set up in the system going from the copper to the iron across the hot junction. Such a circuit is called a thermo-electric couple. The Seebeck effect is reversible, as has been shown by Peltier ; when a current due to an outside E. M. F., traverses the junction of the two metals, from the copper to the iron, the junction is cooled ; if the direction of the current is re- versed, it grows hot. This phenomenon is distinct from the Joule effect ; but, as the two effects occur simultaneously, certain precautions must be taken in order to distinguish one from the other. The development of heat due to the Peltier effect is pro- portional to the first power of the current, while the Joule effect depends on its square ; it is therefore advantageous to employ feeble currents in order to distinguish the two actions. According to Maxwell, the Peltier effect is a measure of the E. M. F. of contact, 107. In fact the heat developed at the junction by a current i in one second is expressed by the product ei, in which e represents the difference of potential which is set up on the contact of the given bodies. If this heat n is given in gramme-degrees, e in volts, and i in am- peres, we have 4.2^ = ie ; whence 4.2 . = 2r- VOltS. THERMO-ELECTRIC COUPLES. 1/5 The values found by this means depend on the absolute temperature of the junction e = f (T) ; they are very slight compared with the potential differences observed on con- necting to the terminals of an*electrometer two points of the conductors taken on each sig(e of the junction and close to it. Thus the E. M. F. of contact of zinc and copper measured by the first method is, at 25 C, 0.00045 v ^ while the electrometer indicates about 0.8 volt ; but Maxwell has observed that, in the latter case, we have not only to do with the contact Zn \ Cu, but that these metals form, with the air which separates the fixed parts from the movable parts of the electrometer, a closed chain : Zn | Cu + Cu | air -f- air | Zn. Now the E. M. F. of contact of the air with the metallic parts of the instrument, by virtue of which they assume electric charges, may be much greater than that which is developed in an entirely metallic contact, which would ex- plain the observed anomaly. In the thermo-electric series a metal is said to be positive with regard to another when the E. M. F. of contact is directed from the first to the second across the heated junc- tion. The E. M. F. which is set up in a metallic arc formed of two dissimilar metals depends, as might be expected, on the heat communicated to one of the junctions, and the simplest way is to express this E. M. F. as a function of the difference be- tween the temperatures 0, 0' of the two junctions of the metals forming the circuit. But we have seen that this E. M. F. is also dependent on the absolute temperature of the two junctions, or, in other words, on the mean of their tem- peratures : 1 76 ELECTRICIT Y. Thus, when, in the case of the above-mentioned couple, we make one of the junctions continually hotter while keeping the other at a constant temperature, the thermo- electric current increases to a maximum, then decreases, becomes zero, and ends by changing direction. 131. Kelvin Effect. In seeking for the cause of this peculiarity Lord Kelvin discovered that the E. M. F. which gives rise to the current is not situated at the junctions alone, as was long believed, but that the homogeneous wires, unequally heated, which form the circuit are also the seat of electromotive forces. Thus in a metal bar an unequal distribution of tempera- ture causes differences of potential between the various points. If the temperature rises from one end of the bar to the other, a continual rise of potential is observed for some metals, while others give a fall of potential in the direction corresponding to the rise of temperature. According to M. Leroux, lead is the only metal in which similar electric phenomena are not manifested when various parts of a piece of this substance are put in different thermal conditions. These E. M. F.'s combine with those which are set up at the points of junction and give a resultant E. M. F., whose ratio to the resistance of the circuit is the strength of the thermo-electric current. If the sum of the potential differ- ences set up in the metals is opposite to the sum of the potential differences at the junctions, it may be assumed that there exist temperatures for which these sums are equal, and the thermo-electric current ceases. The mean of the tem- peratures of the junctions for which this phenomenon occurs is called the neutral temperature or temperature of reversal. The properties discovered by Lord Kelvin, and which are THERMO-ELECTRIC COUPLES. 1 77 called the Kelvin effect, are, from a certain point of view, reversible. Thus, when a current is passed through a wire whose ends are maintained at different tqmPperatures, and which conse- quently has a distribution of potential of its own, the current cools the wire if it is directed towards increasing potentials, and heats it in the contrary case. In order to exhibit this effect independently of the Joule effect proceed as follows : A metal bar is heated towards the middle, while its extremities are kept at o in melting ice. When a current passes in the bar, it is found that points situated symmetrically with respect to the middle are not at the same temperature ; in fact there is a greater heating in one of the halves where the Joule and Kelvin effects are added together ; in the other half the Kelvin effect absorbs part of the heat due to the Joule effect. The action is the same as if there were a transference of heat in the bar ; the transfer is made in the direction of the current for certain metals, and in the opposite direction for others. Lead is the only metal which preserves a perfect symmetry as re- gards the distribution of the temperature. It is interesting to observe that, in a thermo-electric chain whose junctions are maintained at unequal temperatures, the algebraic sum of the potential differences is zero, as in- deed is the case in every closed electric circuit. It follows that the heating observed at the points where the current experiences a drop in potential exactly compensates the corresponding cooling at the rises of potential; in other words, the total heat produced in the circuit by the Pel- tier and Kelvin effects is zero. This is not the case with the heat corresponding to the Joule effect, whose value, necessarily positive, represents the quantity of heat supplied at the hot jnnction, excepting for losses by radiation and conductivity. 1 78 ELECTRICIT Y. 132. Laws of Thermo-electric Action. The two fol- lowing laws were discovered experimentally by Becquerel : Law of Successive Temperatures. In a thermo-electric couple formed of two dissimilar bodies the E. M. F. corre- sponding to two temperatures #, and 0, of the junctions is equal to the algebraic sum of the E. M. F.'s corresponding to the temperatures 0, and # on the one hand, and and #, on the other. Law of Intermediary Metals. If two metals in a circuit are separated by one or more intermediary metals, all kept at the same temperature, the E. M. F. is the same as if the two metals were directly united and their junction raised to the same temperature ; consequently the solder placed between two metals is without effect on the E. M. F. of the couple. These laws are supplemented by those of Kelvin and Tait, which may be announced as follows : Kelvin s Law. If the extremities of a homogeneous bar are kept at temperatures 6 and 8' ', an E. M. F. exists in the bar proportional to 0' 8, the coefficient of proportionality, itself variable with the temperature, is called by Kelvin the specific heat of electricity. These laws being granted, let us consider a couple formed of the metals A and B, whose junctions are maintained at temperatures B and Q'\ let U and V be the sudden variation of potential at junctions, cr and a*' the specific heats of elec- tricity of A and B. The total E. M. F. will be e=U-U' Taifs Law. The specific heat of electricity of a body is proportional to its absolute temperature, cr = K6. 133. Thermo-electric Powers. The E. M. F.'s which are set up in thermo-electric couples by the effect of progres- THERMO-ELECTRIC COUPLES. 1 79 sive variations of temperature are determined by observing the currents which result from them in circuits of known re- sistance. The increase of resistance caused by the rise of temperature of one of the junctions is rendered negligible by introducing into the circuit ait large supplementary resist- ance, which may be of the same substance as one of the metals in the couple, or of a different substance on condition that it is kept at the same temperature in all its points (law of intermediary metals). Suppose that one of the junctions be kept at an invariable temperature by immersion in melting ice, for example, and the other junction raised to increasing temperatures by im- mersion in a heated bath containing a thermometer. The difference of temperature and the total E. M. F.'s are to be observed simultaneously. To represent the phenomenon graphically the values of the former may be made abscissae and those of the latter ordinates ; the curves thus drawn are very sensibly parabolas with vertical axes, their apex corre- sponding to the temperature of reversal (Gaugain). Their equation is of the form From this we deduce d This derivative, which is the angular coefficient of the tan- gent to the parabola, is the E. M. F. corresponding to a difference of temperature of i between the junctions at the mean temperature 0. This value has by Lord Kelvin been given the name of thermo-electric power of the couple at the given temperature. 1 8O ELECTRICIT Y. At the neutral temperature the tangent is parallel to the axis of the abscissae, whence The total E. M. F. corresponding to temperatures and 6' at the junctions is given by the knowledge of the coefficients a and b : This formula shows that, when the temperatures B and 0' are equidistant from / , the E. M. F. is zero. In order to determine the pairs of parameters a and b or b and t n we need only perform experiments at temperatures / and t' , ^ and // ; we thus get two equations in which the quantities sought are the only unknown ones: In order to represent graphically the variations of the thermo-electric power of a coupled \ B with the temperature, we need only draw the right line MM ', whose equation is For another couple A \ C we shall have a second right line NN' usually cutting the first. THERMO-ELECTRIC COUPLES. 181 Now by the law of intermediary metals the thermo-elec- tric power of the couple C \ B will be given by the difference of the ordinates of MM' and NN r . FIG. 60. Knowing the parameters of the two couples A \ B and A | C, (gj =***% \atrj A c we deduce those of the couple C \ B from the formula We need therefore only draw diagrams of the thermo- electric powers of all the metals taken separately with one of their number in order to learn the values of the thermo- electric powers of all the metals taken in pairs in any com- bination. Lead is generally adopted as the metal of comparison, be- cause its specific heat of electricity is zero. In these diagrams the intersection of two right lines has as its abscissa the value of the temperature of reversal. It will be observed that the E. M. F. of a couple A | B between two temperatures 6 and & is expressed by 1 82 ELECTRICITY. being represented by the area included between the right line MM' y the axis of abscissas and the extreme ordinates corresponding to 6 and 0'. Likewise the E. M. F. of C \ B, between the same temperatures, is represented by the area MM'N'N. This area can also represent the work done by a quantity of electricity equal to one coulomb traversing the circuit C | B. We give here the values of the parameters a and b y which enable the thermo-electric powers of various bodies, taken with lead, to be calculated in microvolts : a b Copper 1.34 0.0094 Alloy (90 Pt + 10 Ir).. 5.90 + i -Oi 33 Iron 17.15 +0.0482 German-silver -["11.94 +0.0506 From these figures it is seen that an iron-german-silver couple has a thermo-electric power of ( 29.09 0.00246) microvolts. The current goes from the german-silver to the iron across the hot junction. The E. M. F. for the temperatures o and 200 at the junctions is 5.866 millivolts. These results show that thermo-electric couples only pro- duce extremely feeble E. M. F's, and that it is consequently necessary to join up a large number of them in series in order to obtain differences of potential comparable to those obtained in hydro-electric batteries. It is true that, as the couples are formed of very good conductors, they are ca- pable of producing tolerably strong currents in an external circuit of small resistance. Various bodies give E. M. F.'s very much higher than those of the common metals, but they cannot stand as high temperatures. According to Becquerel, at a temperature of THERMO-ELECTRIC COUPLES. 183 50 C. the thermo-electric power of the bismuth-lead couple is -|- 40 microvolts and that of the fused copper-sulphate- lead couple is 352 microvolts. Antimony-zinc alloy (in equal proportions) gives witn lead 98 microvolts. The conductivity of these different Bodies is very inferior to that of the metals, and it is necessary to use them in the form of tolerably thick bars. 134. Thermo-electric Pile. To form a thermo-electric pile a chain is made in which the alternate links are formed of one of the two metals chosen to form a couple, and all the odd (or all the even) junctions are heated. In order not to need to use as many heaters as there are couples the chain is folded zigzag, the consecutive links be- ing isolated with asbestos; in this way a solid block is ob- tained, with the even junctions on one side and the odd ones on the opposite side ; then only one source of heat is needed to heat all the junctions on one side. The opposite junctions can be cooled by a current of air ; they are also often fur- nished with expansions intended to aid the radiation of the heat transmitted across the couples by conductivity ; for this purpose are used thin sheets of copper or iron, blackened in order to increase their emissive power. It frequently happens that the substances used to make the couples are not very capable of supporting the direct action of the flames which heat the junctions. In this case the latter are covered with a solid envelope upon which the flame plays, and which transmits the heat to the couples by conductivity. This arrangement has also the advantage of rendering the variations of temperature in the couples less sudden when the fire is lighted or extinguished, and con- sequently of diminishing the disintegration which takes place, in consequence of these sudden changes, in the bars of alloy employed. ELECTROMAGNETISM. MAGNETIC PHENOMENA DUE TO CURRENTS. J 35- Oersted's Discovery. Oersted established in 1820 the action of an electric current upon a magnet- needle. This discovery was the point of departure for the theory of electromagnetism, which was established almost entirely by Ampere. According to the practical rule pointed out by this physicist, the north pole of the needle FIG. 61. tends to move towards the left hand of an observer who looks at the needle when he is placed in the direction of the current, so that it enters at his feet (Fig. 61). This fundamental action shows that the current produces a mag- netic field, which fact can be also shown by the use of iron- filings ( 47). Upon dusting iron-filings on a sheet of paper traversed by a current perpendicular to the plane of the paper it is seen that the particles form circles whose centre is in the axis of the conductor. A magnetic pole free to move 184 MAGNETIC PHENOMENA DUE TO CURRENTS. 185 about the conductor would consequently tend to turn around it. The direction of this movement can be deter- mined by Ampere's rule, or by that of Maxwell, which is often more convenient to apply. The direction of rotation FIG. 62. of the north pole and the direction of the current are indi- cated by the relative movements of rotation and translation of a corkscrew. The circular form of the magnetic lines of. force due to a rectilinear current explains why a magnet-needle tends to place itself transversely to the current, so that its magnetic axis may be tangent to the line of force which passes through its support. 136. Magnetic Field due to an Indefinite Rectilinear Current. The intensity at different points of the field may be studied by the method of oscillations ( 47). By applying this method, Biot and Savart have found that the intensity of the field due to a rectilinear current, sufficiently long and distant from the rest of the circuit to be consid- ered as indefinite, is proportional to the current and in- versely proportional to the distance from the conductor. The direction of the field is normal to the plane passing through the conductor and the point under consideration. The force exercised on a positive pole m can therefore be expressed by kim 1 86 ELECTROMA GNE TISM. The intensity of the field at a distance / is consequently As the reaction is equal and contrary in direction to the action, a pole m exercises upon the current a force equal to kim This force is directed towards the right-hand of Am- pere's manikin when he faces the pole, or towards his left if he is looking in the direction of the lines of force emerg- ing from the pole. 137. Laplace's Law. Biot has investigated the action of a current traversing two indefinite rectilinear conductors, AB, AC, placed at an angle, upon a pole m situated on the FIG. 63. line bisecting the angle (Fig. 63). He found that the force can be represented by k being a coefficient of proportion, / the distance from the pole P to the apex A, a the half-angle between the conduc- MAGNETIC PHENOMENA DUE TO CURRENTS. 1 8? tors. This expression reduces to that of the preceding par- agraph when ex = 90. . The direction of the force is normal to the plane of the two conductors. Laplace has Deduced from this expression the action of an element of current on a pole.* ll By reason of symmetry, the effect of one of the branches ABis (i) a being the angle made by AB with PA. Prolong the branch BA by a quantity AA' = ds, and find the action of this element of current on the pole situ- ated at P. FIG. 64. It will be observed that F is a function of two variables, r and a, which determine the relative positions of m and ds. We can then write the identity ,., dF , (dFda . dFdr\ , dF= ds = + - - ) ds. . . . (2) ds \dads dr dsJ In order to obtain the expression for df, we need only substitute in (2) the values of the four derivatives, deducing *The following demonstration is due to M. de Weydlich, formerly assist- ant at the Liege Electrotechnical Institute. 1 88 ELECTROMAGNETISM. them from the results of experiments and from geometrical considerations. Drawing from the point P as centre the arc AA", and observing that the angle APA" = da, since it is the increase of the angle between the directions of AB and PA, we will have in the infinitely small triangle AA' A" A A" = ds sin a = Ida, whence da _ sin a d7 = ~~T and whence dl - = cos a. ds On the other hand, equation (i) gives directly dF k'im i ac** / ~^ cos 3 -. dF k'im t a r-f = -- ^ tan dl ? 2- Substituting these expressions in (2), we find i dF = TJ- [j"- - sin a -\- tan -. cos on d^ cos'f . , . u , x = -- sin ^d^ = -- dj sm (/, ds). MAGNETIC PHENOMENA DUE TO CURRENTS. '" 189 The elementary force is normal to the plane of the cur- rent and of the pole. . If we consider the reaction of the pole on the element ds, we find it directed towards the right hand of Ampere's manikin when placed in the direc- tion of the current and facing r< the pole ( 136), or towards his left if turned so as fo face in the direction of the lines of force produced by the pole. 138. Action of a Magnetic Field on an Element of Current. It will be noticed that in Laplace's law the fac- 7/2 tor -j- represents the intensity of the field JC due to the pole ^2 m, at the point where the current-element is situated. We can therefore write dF = ki JC ds sin (JC, ds). It is easy to generalize the law for the case of a r umber of poles. The total force dF is the product of kids by the resultant of the terms such as m . ,* ,\ sin (/, ds), which represents the product of the intensity of the field JC by the sine of the angle between the direction of the field and the direction of the element, since the projection of the resultant is equal to the sum of the projections of the components. Consequently the resultant is, in absolute value, dF= ki dsW, sin (JC, ds). From what we have learned in the preceding paragraph, this force, applied to an element of current, is normal to the plane of the current and the field, and directed towards the 190 ELECTROMA ONE TISM. left of Ampere's manikin when he faces in the direction of the lines of force of the field. Fleming has pointed out another way of determining the direction of the electromagnetic force. If the thumb and first two fingers of the left hand be pointed in three directions perpendicular to each other, pointing the fore and middle fingers respectively in the direction of the magnetic lines of force and the current, then the thumb points in the direction in which this last tends to be displaced. 139. Work due to the Displacement of an Element of Current under the Action of a Pole. Suppose a cur- rent-element ds = ab (Fig. 65) acted on by a pole situated in a point P. FIG. 65. The electromagnetic force tends to displace the element ds perpendicularly to the plane /, d5 following a direction af. If the element moves in a direction ag, the work per- formed during a displacement ds' ag is equal to the prod- uct of the force by the projection of the displacement along the direction of the force. MAGNETIC PHENOMENA DUE TO CURRENTS. We then have dW= dssm(Pal>)ds' cos(gaf). . . (i) t Now construct a parallelogram on ag and ab ; pass through af a plane fam normal to /and determine the in- tersections e, n of this plane with the right lines gP, dP. As ag and ab are infinitely small in comparison with r, the plane gdP is normal to fam and contains the right line gf which is projected in ag on af. Equation (i) can consequently be written ~X^Xtf. But the product am X af measures the surface of the parallelogram aenm, of which af is the altitude, this paral- lelogram being capable of being considered as the projec- tion of agdb on a sphere of radius / and centre P. Dividing this projection by F, we obtain the projection on a sphere of unit radius, or the solid angle subtended by the parallelogram agdb, that is to say, the area described by the element ds. Calling this solid angle do?, d W kimd GO. 140. Work Due to the Displacement of a Circuit under the Action of a Pole. To find the work performed by a current of finite length displaced under the action of a pole, we need only consider the sum of such terms as kimdoo. We find the product of kim by the solid angle subtended at the pole by the surface described by the given current. In the case of a closed circuit, abgd. Fig. 66, which as- sumes a position a'b'g'd' under the action of a pole situated 1 92 ELECTROMA GNE TISM. in the direction of the reader, we can consider separately the segments abg, gda traversed by opposite currents. The work accomplished by abg is proportional to the solid angle subtended by the area aa'b'g'gb ; that by adg is propor- tional to the apparent surface of aa'd'g'gd. The resultant work will be proportional to the difference between these apparent surfaces, that is, to the' difference between the solid angles subtended by the two contours a' b' g' d' ,' abgd of the circuit. It follows from the preceding, that to bring a pole m from an infinite distance to a point at which the outline of the circuit subtends an angle GO, the work performed is GO being the solid angle subtended by that face of the cur- rent which attracts a positive pole. This expression therefore represents the relative energy of the current and the pole. If the latter were a unit pole, the work would be kica. 141. Magnetic Potential Due to a Circuit. Unit of Current Ampere's Hypothesis on the Nature of Mag- netism. It will be observed that the expression ktGo an- swers to the definition of potential in terms of work ( 12). MAGNETIC PHENOMENA DUE TO CURRENTS. 1 93 We are moreover justified to define the magnetic forces due to the current by -a potential, since the work accom- plished in the field of the current is a function of the co- ordinates of the circuit traversed by the current, and of the point where the magnetic pole; t is supposed to be placed. The expression kioD can therefore be called the magnetic potential due to the current at the given point where the unit pole is placed : U= -kioo. (i) On comparing the potential kioo due to the current with the potential F.oo due to a magnetic shell ( 43), we recognize the identity of the two expressions. A current gives the same potential, and consequently produces the same magnetic forces, as a shell of the same outline, whose strength F s would be equal to ki. To determine the direc- tion of the forces and the sign of the potential, it is neces- sary to distinguish the two faces of the circuit. According to Ampere's rule ( 135), the face of the circuit which pro- duces the same action as the negative side of a shell (that is, attracts a north pole) is that around which the current seems to go in the direction of the hands of a watch : this is the negative or 5 face of the current, the other is the posi- tive or N face. The numerical coefficient k of equation (i) depends on the unit chosen to measure the current. It could be put equal to unity, and the unit current defined as that current which produces unit magnetic potential at a point where the cir- cuit subtends unit solid angle. The unit thus chosen has the same dimensions as the unit of strength of a shell This is the unit which we shall adopt in future to express the 194 ELECTROMAGNETISM. current ; we will later on come across more tangible defini- tions of it. Ampere, who discovered the identity of effect of a shell and a current, interpreted this identity in the following manner : Experiment shows that a small closed current acts like a small magnet normal to the plane of the current on condition that the moment of the magnet be equal to the current multiplied by the surface of the circuit. Suppose any finite circuit divided by lines drawn across it interiorly in the form of a network with an infinite number of meshes, and suppose that the edges of these meshes are traversed by equal currents in the same direction. The result will be currents in the interior lines of the network which annul each other in pairs ; the outer contour of the circuit will be the only seat of a current. Replacing each mesh by an equivalent elementary magnet, the combination of these ele- ments forms a magnetic shell whose effect is identical with that of the current having the same contour. From the preceding comparison Ampere deduced an hy- pothesis which refers magnetic phenomena to electric phenomena. It need only be admitted that each atom of a magnet is the seat of a circular current ; the orientation of these currents will produce effects identical with those of magnets. We must then adopt as a postulate that such an elementary current can exist without expenditure of work, that is, that electric resistance is only manifested in travers- ing interatomic spaces.* Fleming and Dewar have recently advanced an experi- mental proof in favor ofth is latter hypothesis. They showed that if the variations of the resistance of pure metals are represented graphically in terms of their temperatures, curves are obtained which appear to converge toward a point of no resistance, or absolute zero. * See Ampere, Alemoires public's pur la Societd de Physique. MAGNETIC PHENOMENA DUE TO CURRENTS. 1 95 There is, however, a distinction to be established between the potential due to a current and that given by a shell Suppose a positive magnetic mass equal to unity be placed against the positive face of a fetiell. It will be repelled, will follow a curved trajectory, called line of force, and will bring up against the negative face, where it will remain in stable equilibrium. The work accomplished during this revolution is 471 F s ( 44). In the case of a current, the N pole will also follow a line of force, but as this latter is a continuous curve, the pole will continue to move in this orbit as long as the current lasts. Each revolution will increase the work done by the circuit by 4?n, and consequently, by definition, the po- tential will be expressed by U = t(a> n denoting the number of revolutions described by the unit pole. If the work has been performed by the magnetic force due to the current, we must take the sign -f, for the potential will then have decreased ; in the contrary case, that is when the pole has been forced to move backwards, the sign must be chosen. It follows from the preceding that the magnetic potential due to a current contains one constant more than the po- tential of a shell. Nevertheless, as regards the determination of the forces, this constant is eliminated, since the intensity of the current's magnetic field is, in a direction /, We can therefore say that as regards exterior magnetic actions a current i is comparable to a shell F 8 having the same contour. 1 96 ELECT ROM A ONE TISM. The total magnetic flux produced by the current is the sum of the terms of the same sign, such as JCd^, which can be formed in an equipotential surface. The electromagnetic action of a current can be summed up by saying that the current magnetizes the medium which surrounds it, and develops a flux of magnetic force propor- tional to the intensity of the electric flux and the permeabil- ity of the medium. A current surrounded with iron will pro- duce a very much greater flux than if it were surrounded by air or some feebly magnetic substance. 142. Energy of a Current in a Magnetic Field. Max- well's Rule. Following up the comparison between shells and electric circuits, and extending the expression imco which has been found for the relative energy of a current and a pole, it is easy to see that the relative energy of a cur rent and a field is W= -i$, denoting the flux of force across the negative face of the circuit. If the current is displaced in the field, the work accom- plished is measured by the variation of potential energy. When this latter becomes a minimum the circuit reaches a position of stable equilibrium, which corresponds to a max- imum flux of force penetrating across the negative face of the current. Hence Maxwell's rule: A current free to move in a magnetic field tends to place it- self so as to receive the greatest possible flux of force across its negative face. Thus, a circular current movable about one of its diame- ters which is perpendicular to the direction of the earth's field turns so as to point its positive face towards the north : the lines of. force then penetrate perpendicularly by its neg- MAGNETIC PHENOMENA DUE TO CURRENTS. ative face. If, at starting, the flux entered by the positive face of the circuit, this movement would at first have the result of reducing the number of lines traversing it ; then towards the end of the movement the flux would enter by the negative face. 143. Relative Energy of Two Currents. To complete the identity of currents and shells, the relative energy of two circuits traversed by currents t, i' must be expressed by W=-ii'L m , 46. The factor L m has the dimensions of a length and is called the coefficient of mutual induction of the two circuits. By definition ( 46), L m i is the flux sent by the current i across i', and L m i r the flux sent by i' across i. This last deduction from the properties of shells was not a priori evident, for it does not necessarily follow, from the fact that two currents act upon a pole, that they act upon each other; e.g., two pieces of soft iron act on a magnet, but taken separately they have no influence upon each other. It was Ampere that discovered the existence of the forces, called electrodynamic, exercised between currents. 144. Intrinsic Energy of a Current. A circuit carrying a current is traversed by the lines of force which it gener- ates, and which form closed curves around the conductor. The figure assumed by these lines in a plane can be shown by the use of iron-filings strewn on a sheet of paper through which the current passes. This figure is analogous to that fora lamellar magnet with the same contour, and magnetized on its opposite faces. Suppose the circuit be placed in a medium of constant permeability, for example air, and denote by L t the flux of 1 98 ELECT ROM A GNE TISM. force passing in the circuit when the current is equal to unity. For a current c the flux will be L s i = 0. Now a current traversed by a flux possesses a reserve of potential energy, the variation of which measures the work performed. In the case we are considering, the flux is de- pendent on the current ; in order to find the expression for the energy it is necessary to use the method of reasoning employed in regard to the phenomena of electrification or magnetization ( 25). When the current varies by dz, the flux varies by d#, and the potential energy by This energy is essentially positive, for the setting up of the current demands an expenditure of energy. If, then, the current passes from o to c, the energy, which at first is zero, becomes or This expression represents the intrinsic energy of the cur- rent. The coefficient L s , whosed imensions like those of the co- efficient of mutual induction reduce to a length, is called the self -inductance ) or coefficient of self-induction, of the circuit. 145. Faraday's Rule. Before passing to the applications of these various formulae, let us try to find another expres- MAGNETIC PHENOMENA DUE TO CURRENTS. sion besides Maxwell's ( 140) for the work done in displace- ments of a circuit in a field. Maxwell considers the circuit as a whole, and includes in one simple formula the work done in a deformation or a displacement of the conductors. It is often useful to analyze Separately the action of the various parts of a circuit, and to determine the part which belongs to each one of them in the work accomplisned. For this purpose let us consider again the expression for the work of an element of current ds ( 139), which is dis- rH, placed by ds' in a field of intensity 3C = -^-, due to a pole m, dW = iWds sin (/, ds)ds' cos (d/, ds'). Now the product OCdj sin (/, ds)ds f cos (df,ds')> which represents the product of the field-intensity by the projection of the area described by the given current-ele- ment upon a plane normal to the direction of the field, is simply the flux of force swept over by the conductor, for JC is the flux per unit equipotential surface. It is easy to extend this to a conductor of finite length and to obtain the following rule, first pointed out by Fara- day : The ^vork accomplished by a conductor which is displaced in a field is equal to the product of the current by the flux of force (or number of lines of force] cut by the conductor. It should be further remembered that the current tends to move towards the left-hand of Ampere's manikin when he faces in the direction of the field. If the conductor is moved in such a way as to cut no lines of force, the work accomplished is zero. This is the case when the conductor is displaced parallel to the direction of the field. 200 ELECTROMAGNETISM. APPLICATIONS RELATING TO THE MAGNETIC POTENTIAL OF THE CURRENT. 146. Case of an Indefinite Rectilinear Current Let us see whether the application of the idea of potential leads to the expression for the electromagnetic force found by Biot and Savart ( 136), in the case of an indefinite rectilinear current acting on a neighboring pole. FIG. 67. Such a current projected on O can be considered as the limit of a plane circuit projected along OO' t and extended in- definitely towards the right. The conductors which com- plete the circuit being infinitely removed from the pole, supposed to be at P, have no action upon it. Let us now interpret the expression for the potential U i(a> The solid angle GO, subtended by the circuit at the point P, is obtained by cutting the sphere of unit radius, drawn about P, by a diametral plane OP, which includes all the right lines joining the point/* to the conductor projected on O, and by a second plane PP' 9 which likewise includes MAGNETIC POTENTIAL OF THE CURRENT. 2O1 all the lines drawn from P 'to the infinitely distant limit of the given imaginary circuit. The portion of the spherical surface thus cut off is a segment which is measured by twice the dihedral angle a between, the planes OP, PP. We have therefore ^ U= i( 2a the sign of 2a being positive or negative according as the current flows upwards or downwards. The potential has consequently a constant value in the plane OP, which is equipotential. The intensity of the magnetic field produced by the cur- rent is - ds In any point of the plane OP the forces of the field are directed perpendicularly to this plane ; let us now try to find the value of the intensity in this direction. Let we have ds ld(n a) = /da, whence This expression conforms with Biot and Savart's law. If the current O flows upwards, P tends to approach the shell and the sign of the force is negative ; the sign is positive in the contrary case. The dotted circumferences around the point O show lines of force which correspond to field-intensities decreasing in geometrical progression. The planes passing through the 202 ELECT ROM A GNE TISM. axis of the conductor are normal to the lines of force, and consequently equipotential. 147. Case of a Circular Current. Tangent-galvanom- eter. A circular current of radius R subtends, at a point P on its axis OP, a solid angle GO measured by the spherical segment + 27i(\ cos a). If, for an observer situated at P, the movement of the cur- rent is in the same direction as the hands of a watch, the potential at the point P is U = i(a> 47rn) = 2ni(i cos a 2n). ffk FIG. 68. The intensity of the field due to the current must be di- rected along the axis OP by reason of symmetry; it is therefore expressed by dr (r* + *)! That is, on the above hypothesis, P is attracted towards O. If the point />were at the centre of the circle, the intensity would become Ic I being the length of the current. MAGNETIC POTENl^IAL OF THE CURRENT. 203 If there were n circular currents, so near together that their mutual distances were negligible compared to the ra- dius R, the intensity at the centre would be Fig. 69 shows the distribution of equipotential lines and lines of force (marked by arrows) in a field due to a circular FIG. 69. FIG. 70. current. The intensity varies in inverse ratio to the dis- tance between the equipotential lines, and directly as the density of the lines of force. Suppose a magnet-needle of very small dimensions be 204 ELECTROMAGNETISM. hung by a silk fibre of negligible torsion in the centre of a vertical circular frame, round which are wound n spiral coils of wire, very close together. Suppose, moreover, that the frame be oriented in the plane of the magnetic meridian. When a current i is sent through the spirals, the needle is acted on by the electromagnetic force, on the one hand, which tends to put it crossways to the current; on the other hand, by the terrestrial magnetism, the action of which op- poses this movement. Under the influence of these contrary actions, the needle takes up a position of equilibrium corresponding to an angle a with the meridian. Denoting by 2fft the magnetic moment of the needle, by 5C, the horizontal component of the earth's field, the couple due to the earth is 3710C sin a ( 37). The couple due to the current is SfliaC" cos a = 371 - cos a. JK. As these two couples are in equilibrium, 2nni whence cos a X sin a, K RW, = tan a. 2nn Knowing 3C, R, and n, and measuring a by one of the methods shown in 50, we can deduce from the preceding -expression the strength of the current around the frame. This apparatus is called the tangent-galvanometer. The spirals of wire are called the galvanometer-coil or multi- MAGNETIC POTENTIAL OF THE CURRENT. 2O$ r> plier; the quantity is the reduction factor of the galvano- meter. 148. Thomson Galvanometers. The application of the above simple formula "necessitates such a displacement be- tween the coil and the poles of the needle that the tangent- galvanometer is by no means sensitive. In order to measure weak currents, the multiplier must be wound very close to the needle. In order to determine the best form to give it with a view to economize the wire and diminish the resistance, let us take up again the expres- sion for the action of a circular current on unit pole situated in 0, Fig. 71. OC 2ni (r> + If we put 3C = const., and consider r and R as variable, we get the equation of a curve having two symmetrical parts with regard to o, Fig. 71. FIG. 71. The area limited by this curve is the meridian section of a volume of revolution about which the wire can be wound. All the spirals composing such a volume will have an action on the unit pole at least equal to JC. All the spirals exterior 2O6 ELECTROMA GNE TISM. to this volume will have a less action. The cross-hatched section (Fig. 59) which allows for a cylindrical cavity in which to place the needle represents, therefore, the rational form for the coil of a very sensitive galvanometer. Lord Kelvin has approached as nearly as possible to this form in the construction of his galvanometers. In order to increase the sensitiveness, it is possible, in addi- tion, to diminish the action of the earth on the needle. Two means are employed for this purpose. The rod which sup- ports the needle and passes through the coil carries a second needle, Y, oriented in the opposite direction to the first (Fig. 72). By choosing two needles of slightly different magnetic moments, the directive couple due to the earth can be diminished as much as desired. Such a system of of needles is called astatic. The earth's action may likewise be modified by a compensating magnet NS, placed above the needle. By changing the position and orientation of this magnet it is possible to diminish or increase its compen- sating, or directive, effect at will. These various methods are often employed together in Lord Kelvin's galvanometers. In consequence of the complex form of the coil and the nearness of the current to the poles of the magnet, the cur- MAGNETIC POTENTIAL OF THE CURRENT. 2O/ rent c is not related to the deviation a by a simple formula, as in the case of the tangent-galvanometer. If i is developed as a function of a, according to Mac- laurin's series, it will take the fprm The function must become zero when a is zero, whence it follows that S(o) = o. Moreover, if the deviation is very small, we can neglect the third and subsequent terms, and take the current as pro- portional to the deviation i = ka. The coefficient k, which is the reduction factor, is deter- mined by sending a current of known strength through the coils. The last formula is admissible for deviations of less than 3 ; the readings are made, necessarily, by the method of reflection ( 50). 149. Shunt When the deviation exceeds this limit, it is reduced by an artificial resistance called a shunt placed around the galvanometer. The partial current through galvanometer is then, de- noting by g the resistance of the apparatus, and by s that of the shunt ( 121), whence = g-= mg. 208 ELECTROMA GNE TISM. The factor m is called the multiplying power of the shunt. It is the factor by which the value of the current as found by the galvanometer must be multiplied, in order to obtain the total current. 150. Measurement of an Instantaneous Discharge. Suppose a quantity of electricity q traverses the coil of a tangent-galvanometer with such rapidity that the needle is not displaced by an appreciable quantity during the dis- charge. Suppose, too, that the movement of the needle is not damped, so that a double oscillation is obtained having a duration "'" (37) -d) Let us express the fact that the kinetic energy of the needle is equal to the terrestrial couple, and that its momen- tum represents the impulse that has been given to it. In expressing these facts it must be remembered that equations relative to movements of translation are appli- cable to movements of rotation, if the masses be replaced by the moments of inertia, the forces by the couples, and the linear by the angular velocities. Let GO be the initial angular velocity, and a the maxi- mum deviation of the needle. The terrestrial couple is, for an angle , sn a. The equation of the kinetic energies gives = J 'sfiwe sin ad)oj, .... (3) T representing the duration of the discharge. Now fi t/ idt is the quantity of electricity q which was discharged. Eliminating ^mr* and 09 between equations (i), (2), and (3), we get T LW . a, q = -- - sin - n 2nn 2 In the case of a very small deviation we get simply T , q -- = Aa 7t 2nn 2 The quantity of electricity is then proportional to the arc of the needle's swing. Under the "condition that the devia- tions are slight, this formula may be extended to any form of mirror-galvanometer whatever. In order to satisfy the condition at the beginning of this paragraph, galvanometers are chosen for this purpose with a heavy needle having considerable moment of inertia. These needles swing slowly, and enable us to read the limit of the swing exactly. 2 1 ELECTROMA GNE 7 'ISM. 151. Solenoid. Cylindrical Bobbin. Under the name of solenoid Ampere defined a series of equal circular cur- rents, placed very close together and normal to a rectilinear or curvilinear axis passing through the centres of gravity of the surfaces which have these currents as their edges. Denote by s the surface of the circuits, by e their distance apart, and by i the current. Each of them can be replaced by a shell of the same contour, and having a strength i. We may choose the thickness of the shells arbitrarily: we will take it equal to e. Denoting by a the magnetic density of the faces of the shells, and observing that ea = i, we have n l representing the number of currents per unit length. The adjoining faces of neighboring shells counterbalance each other, and these remain, at the extremities of the series, poles whose mass is m = A bobbin formed on a layer of insulated wire wrapped round a cylindrical core may be considered as a solenoid when the wire is traversed by a current. As a result, how- ever, of the obliquity of the spirals of the bobbin there is an exterior action, which may be obviated by bringing back the ends of the wire along the axis of the bobbin (Fig. 73). If there are an even number of layers of wire on the core with their spirals inclined in opposite directions, the effects due to their obliquity are rendered null. Supposing the thickness of the layers to be negligible MAGNETIC POTENTIAL Of THE CURRENT. 211 compared to their diameter, the resultant poles of the bob- bin are given by m = n l being the number of turns pr unit length, and s their mean surface. The magnetic moment of the solenoid is ml = ins, n representing the total number of turns of wire Such a solenoid has all the magnetic properties of a uni- form cylindrical magnet ( 42). The earth's action on the solenoid is shown by hanging it from the ends of the wire (plunged into cups of mercury), which serve as pivots, and at the same time give access to the current from a battery, Fig. 73. By this means it is shown that the positive face of the solenoid turns towards the north. To determine the intensity of the field due to the sole- noid, it need only be remembered that it can be replaced by a uniform cylindrical magnet. Its action on unit pole situated on its axis, at some ex- ternal point, is 5C = o-(2oo a/), ( 31) 212 ELECTROMAGNETISM. GO and GO' being the solid angles subtended by the bases of the cylinder at the given point. If the unit pole is situated in the plane of one of the bases, the above equation becomes OC = a(27t GO'). Finally, in order to obtain the expression for the field at a point inside the solenoid, suppose the latter to be cut into two portions by a plane through the given point. The total action is the sum of the actions of the two portions. Now the effect of the first is s = Calling / the length of the circular axis of the ring, n the total number of turns in the solenoid, whence (i) This form is analogous to Ohm's law ; the magnetic flux is proportional to the expression ^uni, which by similitude is called magnetomotive force* and inversely proportional to , which is given the name of magnetic resistance of the circuit, on account of its likeness to electric resistance (i us). Heaviside, in consideration of the fact that there is in magnetism nothing analogous to the Joule effect, which is work accomplished by the electric current to overcome the electric resistance, has substituted for the expression "mag- netic resistance " that of reluctance. We can get equation (i) by a direct proceeding. We have seen ( 141) that the work performed by a unit pole * Bosanquet, Philosophical Magazine, 1883, Vol. XV, -p. 205. 232 ELECTROMA ONE TISM. in moving across a circuit is equal to ^ni times the number of revolutions made, ^ni representing the difference of mag- netic potential between the two faces of the circuit. Now if the unit pole moves along the internal axis of the ring, it traverses the current n times in one revolution. The work accomplished, ^.nni is also expressed by the product of the mean intensity 3C into the path / traversed by the unit pole, whence but whence # = The magnetomotive force ^nni is therefore measured by the sum of the differences of magnetic potential produced in the solenoid. If the core were composed of various segments having lengths /, /', I", sections s, s', s", and permeabilities //, //, /*", we would have There is, however, an essential distinction to be made be- tween the electric and magnetic circuits. The resistance of the former is independent of the current, while that of the latter is a function of the permeability, which depends not only on the actual flux, but also on the preceding fluxes ( 57)- O n tne ther hand, the flux and the quantity of magnetism are not connected by a law similar to that which connects the quantity of electricity to the current and the ELECTROMAGNETS. 233 time. Finally, in consequence of the residual magnetism, a magnetic flux can exist-in a circuit without any magnetomo- tive force, or even a flux in a direction opposed to the mag- netomotive force. We must therefore be careful not to con. elude an analogy of fact from a^i analogy of form, and only look on the extension^of Ohm's law to the magnetic circuit as an artifice to facilitate the investigation of the question as well as the calculations connected with it. Following this order of ideas, we can push the compari- son still further and treat cases of complex magnetic cir- cuits by Kirchhoffs laws. FIG. 85. Let us take the case of a magnetic circuit, placed in a medium supposed to be impermeable to the lines of force, and divided into two portions. The flux $ generated by a solenoid of n turns traversed by a current i, divides itself into two derived fluxes, $' and 3>". We shall have $=&+ <". Let / be the length of the common branch, s its section, and /i its permeability. Let I' , /, // ; I" , s" , //' be the corresponding elements for the derived branches. 2 34 ELECTROMA GNE TISM. Kirchhoff's second law will give the equations The case that we have just considered is imaginary, for there is no medium in existence which is impermeable to lines of force. In the phenomena of the electric current we practically take air at the ordinary pressure as a perfect in- sulator for the electric flux ; but the same does not hold for magnetic phenomena. If the permeability of air and other gases is negligible compared with that of iron when traversed by a mean magnetic induction, it becomes de- cidedly comparable with it for intense magnetic fluxes. It follows that in the case of Fig. 85, for example, a part of the flux generated by the solenoid will go off into the sur- rounding medium. The ratio of the flux into the air to the flux traversing the iron rings will increase with the current in the solenoid in consequence of the gradual weakening of the permeability of iron and the constancy of that of air. The case is not without analogy with that of an electric circuit placed in the midst of a liquid having a certain rela- tive conductivity. A part of the current furnished by the electric generator will pass through the liquid without fol- lowing the line of the metallic circuit. We shall see, when studying dynamos, how the derived flow of the magnetic flux in the surrounding medium can be experimentally es- timated. 165. Forms and Construction of Electromagnets. When it is desired to utilize the portative power of an elec- tromagnet, the horseshoe form is used, consisting of two straight cores wound with wire and connected at the base ELECTROMA GNE TS. 235 by an iron yoke. The free poles attract the armature (or keeper], which, with the air-gap, completes the magnetic circuit. The attractive force is proportional to the cross-section and the square of the magnetic induction. We have seen ( 56) that the portative' power of an electromagnet per unit /T>2 section is 2n?? + OC3, which expression becomes - - if we O7t 'TO 2 add the very small term . It is therefore advantageous Q7T to make electromagnets of short and massive pieces. When we wish to calculate an electromagnet to obtain a given portative power we take a magnetic induction (B, which can be obtained without too great expenditure of electric energy. The magnetism-curves ( 57) show that beyond (E = 16000 the increase in induction is very feeble relatively to the ex- penditure : consequently we choose a section s about 16000 Next we seek the dimensions of the bobbins capable of pro- ducing the flux <$>s ; we take a given length, /, for the cores, and, after having joined them by a yoke and a keeper of section s, we calculate the magnetomotive force qnni that the coils must develop, by the formula = ($>s X An example of such a calculation will be seen further on. Instead of using the horseshoe form in order to obtain a ELECTROMA GNE TISM. closed circuit, the helix of a straight electromagnet may be enclosed by an iron sheath joined to the core at one end by the yoke, and at the other by the keeper, Fig. 86. This FIG. 86. form is very compact, but it does not lend itself readily to the employment of strong currents, for the radiation of the heat produced in the helix by the Joule effect does not take place so well as in the other form. When the keeper is moved away from the poles, the at- tractive force diminishes very rapidly. In fact the attrac- tion is inversely proportional to the square of the distance between the poles (of the keeper and magnet) which are opposite each other, and the poles themselves decrease with the magnetic flux, which grows very suddenly less on ac- count of the feeble permeability of the air. It follows that at quite a short distance the attraction exercised by the mag- net-poles on the keeper becomes negligible ; we have, conse- quently, to limit the amount of play of this movable part. In order to diminish the weakening of the flux, caused by the large reluctance of the air-gap, we can increase the magnetomotive force by increasing the number of turns in the helix. This leads to the lengthening of the cores ; this lengthening produces in itself an increase in the magnetic resistance, but in view of the high value of the permeability of iron compared with that of air, the total resistance of the circuit is not notably increased. This means is made use of in telegraphic electromagnets. In many applications it is necessary to increase the play ELECTROMA GNE TS. % of the keeper ; this is managed very simply by giving it, in- stead of a movement perpendicular to the line of the poles, an oblique movement, which augments its play without altering the work done in displacement. One solution con- sists in furnishing the keeper w^h conical projections which penetrate into conical cavities cut in the cores. If it is desired to obtain considerable displacements, with a small variation in the attractive force, we must make use of the suction-effect of a long solenoid upon a cylindrical core placed at the open end of the solenoid. The core tends to place itself in such a position that the flux traversing it is a maximum ( 65, 151). Before analyzing the effect of a long bobbin or solenoid, placed vertically, upon a core acting as a plunger, it should be recalled that the field is sensibly uniform in the middle region of the solenoid, and that its intensity decreases rap- idly towards the ends ( 151). A pole of invariable strength is therefore urged through the solenoid by a force which in- creases from the point of entry, becomes constant in the middle, and decreases towards the lower end. The force is in each point equal to the intensity of the field multiplied by the strength of the given pole. When a soft iron core is presented at the opening of a solenoid, the phenomenon is more complex. The attractive effort depends on the magnetization induced in the core, which itself varies with the intensity of the field. To sim- plify the matter, let us consider a cylindrical core whose length is very great compared to that of the solenoid, so that we have only to consider the action exercised upon the pole induced in its lower end. The strength of this pole in- creases as the core descends, for the total flux produced by the solenoid increases in consequence of the gradual dimi- nution of the resistance of the magnetic circuit due to the insertion of the iron core. When, however, its lower end 238 ELECTROMAGNETISM. reaches the bottom of the solenoid the force tends to de- crease, for the induced pole reaches a region of the field where the intensity is diminishing. If the length of the core equals that of the solenoid, we must take into consideration the antagonistic action exer- cised on the pole induced in the upper end of the core. Experiment shows that in this case the resultant effort is maximum when this latter reaches the middle of the sole- noid ; it then decreases and becomes zero when the core is in a symmetrical position with regard to the solenoid, this being the position in which the flux produced by the cur- rent and traversing the iron is a maximum. In order to regulate the attractive effort exercised on the core, when this latter is subject to considerable displace- ments, it is given the form of a very long cone with the apex pointing downwards. In this case the total flux increases as the core enters into the solenoid, even when the apex of the cone has passed the lower opening. Solenoids develop a very much lower attractive effort on their cores than that exercised by electromagnets of horse- shoe shape on their keepers, for in the latter case the mag- netic circuit has much less reluctance. To increase the suction-effect we can put an iron sheath- ing on the solenoid, which presents a very permeable path for the lines of force. A solenoid armored in this way gives a very intense, uniform field in its interior, but its external effects are negligible : the core is not attracted, in this case, until it is introduced into one of the openings in the sheath- ing. In order to reduce as much as possible the perimeter of the wire in an electromagnet, the core must have a circular section ; sometimes, however, a square or rectangular sec- tion is chosen to gain compactness. Such a shape increases not only the length and consequently the resistance of the ELECTROMA GNE TS. 239 magnetizing wire, but also the losses of flux which take place between the lateral faces of the adjoining cores, par- ticularly in horseshoe electromagnets, by bringing the cores closer together and increasing theJr surface. As far as possible sharp edges on the polar surfaces must be avoided, since the liftes of force have a tendency to crowd together at the edges and escape by them into the surrounding air. This effect is not without analogy with the effect of points in the case of electrified bodies. The wire used in electromagnets is generally of the purest possible copper so as to reduce the heating. If the cross- section of the conductor has to be large, it is made more manageable by using cords of twisted copper strands or bundles of copper strips. These latter have the advantage of diminishing the interstices between the consecutive turns. For mean-tension currents the wire is covered with two or three layers of cotton impregnated with varnish or shellac. If the coils are very small, the wire is generally insulated with silk. For high-tension currents the successive layers of wire must be separated by vulcanized fibre, cotton tape impregnated with shellac, Wellesden paper, mica, or ebon- ite. In special cases the windings are even separated into parts by ebonite partitions perpendicular to the axis of the core, the spaces between the partitions being filled with flat bobbins connected in series. In this manner conductors at very different potentials are kept apart and the danger of disruptive sparks avoided. If electromagnets are exposed to excessive heating, the wire is insulated with asbestos or mica. The conductor joining the inner layer of a coil with the outer < layer should be made extra strong, for, if it breaks, the coil has to be unwound in order to repair it. The shape given to the keeper of an electromagnet has a marked influence on the magnet's portative power, which is 24O ELECTROMA GNE TfSM. (B 1 expressed by s ( 165). If the polar surface is dimin- O7f ished, the induction (B is increased, for the lines of force, having a tendency to make their way across the iron rather than across the surrounding air, will go more and more to- wards this surface in proportion as it grows smaller near the keeper. It follows that the product ($>*s will be a maximum for a value of s, generally much smaller than the section of the cores. It is for this reason that keepers are frequently given a convex form in the part adjoining the poles. 166. Magnetization of a Conductor. When a perma- nent electric current traverses an iron or steel wire, the mole- cules situated on the surface of the wire align themselves circularly along the lines of force created by the interior electric flux, forming closed magnetic filaments without ex- ternal action. To exhibit this transverse magnetization, we need only cut a longitudinal groove in the wire ; its edges will exhibit opposite polarities. 167. Modifications in the Properties of Bodies in a Magnetic Field. A magnetic field causes perturbations in the propagation of light in bodies placed in the field. This discovery, due to Faraday, is proven by the aid of polarized light, whose plane of polarization is altered when it trav- erses, in the direction of the lines of force, a solid, liquid, or gaseous body placed in a magnetic field. Faraday used, in his demonstration, an electromagnet whose field-cores are hollowed out along their axis and joined by right-angled iron supports and an iron base. The substance to be tried is placed between the poles of the magnet. The field-cores are traversed by a ray of light polarized by passing through a Nicol's prism ; on leaving the field-cores the ray is extinguished by an analyzer. When a current is sent through the field-coils the ray reappears : ELECTROMAGNETS. 24! it can be re-extinguished by giving the analyzer a rotation which measures the angle by which the plane of polariza- tion has turned. Verdet has shown that this angle is proportional to the difference of the magnetic potentials at the extreme points of the path of the ray trlrough the substance affected. The direction of rotation, however, is independent of the direc- tion of the ray of light, but is different in magnetic and dia- magnetic bodies. This experiment can also be made by the aid of a sole- noid traversed by a current and having placed along its axis a tube, closed by glass ends, in which the liquid to be ex- perimented on has been poured. The polarizer is placed at one of the ends of the tube and the analyzer at the other. If the length of the tube greatly exceeds that of the sole- noid, the magnetic potential is practically zero towards its extremities ( 151); the difference of magnetic potential is equal to the work accomplished by unit pole in moving from one of these extremities to the other, or ^nni', n denoting the total number of turns, and i the current. a being a constant, called Verdet's constant, for a given body at a definite temperature, the rotation of the plane of polarization is = This relation gives the means of measuring the current as a function of the deviation of the polarized ray. 168. Hall Effect. Hall has discribed a phenomenon which, like the preceding, probably owes its origin to a physical modification in the bodies placed in a field. If we connect with the poles of a battery the points a and b (Fig. 87) of a thin conductive plate of circular form, we can trace equipotential lines and lines of electric flux dis- tributed as shown in the figure. In order to obtain one of 242 ELECTROMA GNE TISM. the equipotential lines, we need only affix at one point of the plate the extremity of a wire connected with a galva- nometer, and move about over the plate, another wire con- nected to the second terminal of the same instrument. Each time that we meet with a point at the same potential as the first, the galvanometer will show no deviation. The equipotential lines, marked by dots, are symmetrical to the diameter cd, whose extremities are at the same po- tential. Suppose now that the disc be placed between the poles of an electromagnet so that the lines of force traverse it perpendicularly. The points c and d immediately cease to be at the same potential, and a new distribution of the lines of flux is pro- duced as shown in Fig. 88. FIG. 88. At first sight it seems as if the displacement of the lines of flux ought to be attributed to the direct action, governed ELECTROMAGNETS. 243 by Ampere's rule, of the magnetic force on the electric cur- rent. But experiment shows that the deviations are not in the same direction in different bodies. For example, the defor- mation of the lines of flux follow^ Ampere's rule in iron and zinc ; it is the opposite In bismuth and nickel. At the same time with the above-mentioned displace- ment, there is observed an apparent increase in the electric resistance of the conductor, attributable to the fact that the mean lines of flux are lengthened by the twisting. This in- crease of resistance, which is more manifest in conductors of lengthened form placed perpendicularly to the lines of force, can be used, as M. Leduc has shown, to measure the inten- sity of a magnetic field. UNITS AND DIMENSIONS* GENERAL THEORY. 169. Units. In every quantitative statement of the value of a physical quantity two factors are involved the unit of the same concrete kind as the quantity, and a numerical fac- tor. The numerical factor, called the " numeric " for short, is the ratio of the concrete quantity to its unit ; or, it is the number expressing how many times the unit is contained in the concrete quantity. For instance, if / is a definite length and L the unit length, then l/L is the numerical value, or numeric, of this length /. It is clear that when the unit chosen as a measure of length is changed, the numeric of any concrete length will change inversely ; any length contains twelve times as many inches as feet, the unit being diminished from feet to inches in the ratio of I : 12, the numeric increases in the ra- tio of 12: i. A unit, such as that of length, which involves in its defi- nition no reference to other units, is called a fundamental unit. From the fundamental unit others flow, dependent upon the fundamental units ; these are called derived units. Such derived units are the unit of area, the area of a square whose side is the unit length ; the unit of volume, the volume of a cube whose edge is the unit length ; * By Gary T. Hutchinson, Ph.D. 244 GENERAL 7'HEORY. 245 the unit of velocity, the velocity with which unit length will be traversed in unit- time: these are all simple derived units. Others are more involved, requiring several steps in their reference to the fundamental units. For instance, the unit of acceleration is defmed^to be the acceleration with which the velocity is increased by unity in unit time ; this involves a reference to time and velocity which in turn im- plies length and time. The choice of fundamental units is to a great extent ar- bitrary. Rational systems of units can be built up, start- ing from many different sets of fundamental units ; the units of length, mass, and time are, however, almost uni- versally adopted as fundamental. Practically all physical quantities can be expressed in terms of length, mass, and time. 170. Dimensions. A change in the fundamental units will obviously change a derived unit. The amount by which a derived unit is affected by given changes in the fun.da- mental units is determined by what are called the "dimen- sions," or the dimensional equation of the derived unit. These dimensions express the manner in which the funda- mental units enter into the derived unit. A simple illustration of the method of deriving dimen- sions is that of velocity. Let v, /, and / denote respectively a concrete velocity, length, and time, connected by the rela- tion that the numerical value of the velocity, or its numeric, is equal to the numerical value of the length divided by that of the time ; that is, that the given length / would be de- scribed in the given time t by a body moving with the given velocity v. Let V, L, T be the units of velocity, length, and time, respectively; then the numerics of ^, /, and / are v/V, l/L, t/T, and by assumption, v /r=i/L+ 246 UNITS AND DIMENSIONS. or This equation shows that the numeric of the velocity v varies inversely as the unit of length Z, and directly as the unit of time T. It also shows that the unit velocity V varies directly as the unit length Z, and inversely as the unit time T, since the concrete quantities v, /, and / are not affected by changes in the units F, Z, 7". The equation is known as the dimensional equation of F, and LT~ l are the dimensions of V. It is clear that these dimensions are dependent entirely upon the fundamental units adopted and upon the defini- tion given to velocity ; also, that they have no connection with the real physical nature of the quantities to which they refer. As an illustration, suppose force, mass, and time were chosen fundamental units ; then since force = mass X acceleration, or f=mXa=mX v/t t v=/X t/m, and an entirely different set of dimensions. In the same manner the following dimensions are de- duced from the ordinary definitions of the quantities in- volved. DIMENSIONS OF SOME PHYSICAL QUANTITIES. 247 DIMENSIONS OF SOME PHYSICAL QUANTITIES. Physical Quantity. Symbol. Length /' Mass r. . . m Time / Area s Volume V Velocity v Angular velocity GO Density Force f Work W Power P Pressure / Moment of inertia. , K Defining Equation. Dimensions L M T GO = V/l = m/T f m x a w=/xt P= W/t LMT-* DMT-* UMT-* DM The other mechanical and dynamical quantities can read- ily be deduced from these in the same manner. 171. C. G. S. System of Units. The system of units adopted by the entire scientific world is that based upon the centimetre as the unit length, the gramme as the unit mass, and the second as the unit time. These units were chosen after long consideration because: they admit of accurate comparison with quantities of their own kind, the units can be determined easily and accurately and standardized at all times and places, and they lead to simple definitions of the most important physical quantities, with their relations to one another. In all that follows L will denote unit length or one centi- metre, J/the unit mass or one gramme, and T the unit time or one second. The C. G. S. system of units applies to mechanical as 248 UNITS AND DIMENSIONS. well as electrical quantities, although it has been given prominence mainly through its electrical applications. ELECTRICAL AND MAGNETIC UNITS. 172. General Considerations. A consistent system of electrical and magnetic units can be devised with any one of the equations defining an electric or magnetic relation as a starting-point. Each system so formed would have differ- ent dimensions for the same concrete quantity, and these dimensions, as explained above, would serve to show the changes in the various units when the fundamental units were changed, and consequently the ratio of the various units in the different systems. Two systems have been built up in this way, and are in use to-day : the first, the Electrostatic System (E. S.), is based on the definition of unit quantity of electricity ; the second, the Electromagnetic System (E. M.), is based on the definition of unit magnetic pole. In order that mag- netic quantities may be expressed in terms of the electro- static units, and electric quantities in terms of electromag- netic units, some connecting link between the two sets of phenomena must be used : this link is the relation devel- oped in 136, that the magnetic force due to an unlimited rectilinear current i at a point at a distance /, is proportional to the current i, and inversely proportional to the distance /. That is, Magnetic force = a X i/l. If 3C is the unit magnetic force and /the unit current, OC = /-' (A) There is only one known relation connecting electric cur- rent and magnetic force ; equation (A) is one expression ELECTRICAL AND MAGNETIC UNITS. 249 of this relation. Another common form of expressing the same relation is, the work done by unit pole in passing around any closed curve that threads the electric circuit equals 47*7. These two relations, and others of the same kind, are interchangeable. Coulomb's experiments connect mechanical force with electric quantity ; as ordinarily stated, f= q -/r. This statement assumes the action to take place in air (or vacua), and does not bring the medium into evidence; for greater generality the expression should be f=f/(l'K), . . ..-. , . (B) where K is the specific inductive capacity or permittivity of the medium. Similarly, for magnetic quantities, /=*/(/';), . . . . . . . (C) where JA is the permeability of the medium. 173. Systems in Terms of K and /*. Starting with equa- tions (A) and (B), a system of units can be developed in which all the electric and magnetic quantities are expressed in terms of LMT and K. Similarly, starting with (A) and (C), a system can be developed in which all the quantities are expressed in terms of LMT and ju. These two systems are the only ones in use : the first, the system in terms of K, leads directly to the Electrostatic System ; the second, the system in terms of yu, leads to the Electromagnetic System. The two are herein developed side by side ; the quantities are chosen in pairs so as to show, as far as possible, the similarity of each in the AT-sys- 250 UNITS AND DIMENSIONS. tern to its fellow in the yw-system. Units in terms of K will be distinguished by a line over the symbol. Electric Quantity (q) : In terms of K. Unit quantity is defined by equation (B); that is, the mechanical force /acting between two equal quantities q at distance / is = FDK ~Q = Electric Current (Y) : Electric Current (Y) : In terms of K. Current is defined as the quantity of electricity carried across the section of a conductor in unit time ; or, i = q/t, ~T Strength of Magnetic Pole (m) : In terms of ^. Pole strength is defined by equation (C), similar to equation (B) for quantity. Therefore, m = Magnetic Force (3C) : In terms of //. Magnetic force is defined as the force acting on unit pole ; therefore, X = F/m ELECTRICAL AND MAGNETIC UNITS. 251 Electric Current (i) : In terms of /*. Equation (A), the linking equation, gives the magnetic force due to unit current i at distance /as OC = a X *//, . Electric Quantity (q) : In terms of //. q = i x /, as above Magnetic Force (X) : In terms of AT. Strength of Magnetic Pole (m) : In terms of K. f JC;, as above ; All the electric magnitudes depend on q, from which they are easily derived, following the usual definitions; hence, having ^expressed both in terms of A^ and of /*, the expression for all the electric magnitudes can be deduced in 252 UNITS AND DIMENSIONS. terms, both of K and of /*. Similarly, having pole strength m, expressed both in terms of K and of /*, the expression for all the other magnetic magnitudes can be deduced both in terms of K and /*. The accompanying table gives the symbol, the defining equation, and the dimensions, both in terms of K and of yw, of all the common electrical and magnetic quantities ; it gives also the ratio of the dimensions in terms of K to the dimensions in terms of /*. In the table, the symbols and names recommended by the Committee on Notation of the Chamber of Delegates of the International Electrical Congress of 1893 are used, this system having been adopted throughout the book. 174. Proposed Nomenclature. The table contains the names and symbols of the quantities in common use; names have been proposed for the ratios of Flux/Force and Flux-Density/Force-Intensity for the dielectric flux to correspond to those already in use for the conductive and magnetic fluxes. For the conductive flux : Flux/Force = i/e = Conductance {g) = I/Resistance (r). Flux-Density/Force-Intensity = i/T -z- e/l = Conductiv- ity (y) = I/Resistivity (p). For the magnetic flux : Flux/Force = $/JF = Permeance = i/Reluctance ((R). Flux-Density/Force-Intensity = (B/OC Permeability (/*) = i/Reluctivity (v). For the dielectric flux two systems of names have been proposed, the first by Oliver Heaviside and the second by F. E. Nipher. Heaviside's * notation is as follows : Flux/Force = N/e Permittance = i/Elastance. * Electromagnetic 7^heory, Vol. I. ELECTRICAL AND MAGNETIC UNITS. Ratio of - 's \ " 3! "i ^ 1^1" 7" 7 7 tential. ^raday Tubes, agnetic Force. ta* o .-o 1 1 1 1 - 1 1 QJ *-> oj &g| S"J EN ONM W-lO-< OM, s=i ^ MMMN WWNM ciwOO OOON OOOO OO =1 " VH wr; >% in terms o & C1MPIC4 C4OMC4 i * III II -c^ a 1 . T rt . _ V -CX : : g 1 : S ttiiiliiiliHi ,*HK "t;" 4 -' o*-' 1 -'-^ o-ooj 3 3 ^5 oo c^ :ll i 111 nil iisg Jin si 1 Electric Field Strength a Magnetic Field Strengt 8 Electric Potential ; Vol WCL.H2 W2Q^ tLJ2o)d< CudiCtiU D^C^CJU >Su 254 UNITS AND DIMENSIONS. Flux-Density/Force-Intensity = D/F = Permittivity = i/Elastivity. Permittance is capacity ; Permittivity is specific inductive capacity. Nipher's * notation is as follows : Flux/Force = N/e Perviance = i/Diviance. Flux-Density/Force-Intensity = Perviability = i/Divia- bility. Perviance is capacity ; perviability is specific inductive capacity. Neither of these last sets of names is in use. 175. Dimensions of K and /*.- The table shows that the dimensions of the unit of every quantity in the AT-system are related to its dimensions in the /^-system by a simple function of LT~\K^, in which Yl/does not enter, namely, the quantity, its reciprocal, its square, and the reciprocal of its square. The dimensions thus deduced are entirely inde- terminate so long as K and yw have no dimensions in LMT. To give K and yw, coefficients depending on the nature of the medium, dimensions in LMT, implies a mechanical theory of electricity and magnetism. If the phenomena of electricity and magnetism are differ- ent manifestations of mechanical action of one kind or an- other, then the quantities AT and /i must have dimensions in LMT, and the expression of the dimensions of any quan- tity, as q, in these two systems must be identical; that is, their ratio, a function of LT~*(K)ty must be zero dimen- sions, or L Q MT\ or, (ATyw)"3 = LT~ l = v, a. velocity. Maxwell, in his electromagnetic theory, which does as- sume all the phenomena of electricity and magnetism to be * Electricity and Magnetism. ELECTRICAL AND MAGNETIC UNITS. 255 manifestations of ordinary mechanical forces, shows that the velocity of propagation of electric disturbances in a medium of specific inductive capacity, K, and permeability, M, is (K^)-*. The ratio of the dimensions^being in all cases a function of F(Ajw)*, any values'of K and /* that make this expres- sion unity will result in identical dimensions in the two sys- tems for every quantity. Any values of K and yw that are based on the assumption of a mechanical theory of elec- tricity will fulfil this condition. Many different systems of dimensions have been pro- posed of late, all based on certain assumptions for the values of K and /* ; the number that can be devised is with- out limit, but as there is no reality behind the assumptions it is useless to record them. The two systems of units ordinarily met with are the Electrostatic (E. S.) and the Electromagnetic (E. M.) : the first is based on the assumption that the specific inductive capacity of vacuum is unity, and is therefore deduced from the AT-sy stem by making K unity in all the formulae; the second is based similarly on the assumption that the per- meability of vacuum is unity, and is therefore deduced from the yu-system by making ^ unity in all the formulae. These two systems are the only ones that are in use, and of these the Electrostatic system has no practical impor- tance ; all electrical measurements as ordinarily made are in terms of the Electromagnetic system, or in the " Practical System," of which the units are merely multiples of the units on the Electromagnetic system. The values of unity for both K and /f were chosen merely for simplicity; they assume vacuum as the standard medium, and ignore the physical character of the two quan- tities ; that is, they assume them to be merely numerical coefficients. As simultaneous values, they do not satisfy 256 UNITS AND DIMENSIONS. the condition (K^)-* = velocity, and hence do not lead to identical dimensions. 176. Value of Ratio "z/." The ratio of the dimensions of any unit in the Electrostatic system to its dimensions in the Electromagnetic system that is, the function of V(K}*)* now reduces to a function of "#"; this "v" is a definite concrete velocity, and its numerical value can readily be found by determining the value of any electric or magnetic quantity, first in one system and then in the other, the ratio of the numerical values representing a func- tion of "v." This velocity "v" has been determined ex- perimentally by comparing the two measures of many of the electric units. The best experiments give for the value of "z/" very nearly 3 X IO 10 centimetres per second, the same as the value of the velocity of propagation of light in vacua. As "z/," on Maxwell's theory, is the velocity of propagation of an electric disturbance, these disturbances are then propagated with the velocity of light. But, all theory aside, experiment shows this ratio "z;" to be practi- cally the velocity of light, whatever it may represent physi- cally. There are a number of ways in which the physical signifi- cance of-"?;" can be illustrated; one of the simplest is shown in the following ideal experiment : Assume a conductive sphere, which is at the same time compressible ; the capacity of a sphere is equal in E. S. measure to its radius. That is, q = I X e, where / is the radius and e the potential to which it is charged. Suppose this sphere to be connected to the ground through a resistance of which the value in E. S. units is r, ELECTRICAL AND MAGNETIC UNITS. the sphere will then discharge and its potential will fall ; but imagine the sphere- to be compressed at the same time at such a rate that the increase of potential due to compres- sion equals the diminution due.-to the discharge. Then as e is constant, a constant current will flow, rep- resented by i = e/r. But = / x *, da *. edl , and dt = ' = Tt = e/r - That is, r i ^ dl/dt. Or, the velocity dl/dt with which the radius of the sphere must be shortened is numerically equal to the conductance of the discharge circuit in E. S. units. The chief use of these dimensions is to show the manner in which the fundamental units enter into the composition of the unit of any quantity, and thus show the changes in the derived unit flowing from given changes in the fun- damental units. For instance, if LMT are changed to L ' M 1 ' T' ', the new E. S. unit of quantity will be equal to times the old E. S. unit"; similarly the new E. M. unit of quantity will be equal to fL'\*(M'\* fflSI times the old E. M. unit. 177. Practical System. In the E. M. system the units of the quantities in common use are either impracticably 258 UNITS AND DIMENSIONS. large or small for convenient use. This fact had led to the adoption and use of a so-called " Practical " system, based on a unit of length equal to io 9 centimetres, or an earth's quadrant ; a unit of mass equal to io- 11 grammes, and a unit of time equal to the second. That is, in the Practical system, L' = io 9 centimetres, M' = ro~ n grammes, T' = I second. From the table of dimensions of the various quantities the accompanying table is derived, showing the relation between the values of the various units in this Practical sys- tem and in the C. G. S., Electromagnetic system, for electric and magnetic quantities. RELATION BETWEEN PRACTICAL AND C. G. S. ELECTRO- MAGNETIC SYSTEMS. n Name of Unit Practical Unit ua in Practical System. equals Length Quadrant io 9 centimetres Mass io~ n grammes Time. . Second 10 seconds Area io 18 cm 2 Velocity io 9 cm/s Force .... io~ 2 dyne Work Joule io 7 erg Power Watt io 7 erg/s Resistance Ohm io 9 cm/s Current Ampere io- 1 C. G. S. Quantity Coulomb io- 1 C. G. S. Electromotive force Volt io 8 C. G. S. Magnetic flux .... io 8 Weber Magnetic flux-density io- 10 Gauss Reluctance io~ 9 Oersted ELECTRICAL AND MAGNETIC UNITS. 2$$ Name of Unit Practical Unit y uantlt y- in Practical System. equals Permeance IO 9 C. G. S. Magnetic force io~ 10 C. G. S. Magnetomotive force .... A . io -1 Gilbert Capacity ... . . . Farad io~ 9 C. G. S. Inductance Henry io 9 C. G. S. Specific inductive capacity .... io~ 18 C. G. S. 178. Nomenclature of Practical Units. The more common units in the Practical system have long had the names of distinguished men of science, such as Ohm, Volta, Ampere, Coulomb, Joule, Watt, Faraday. These names have answered all ordinary purposes until the last few years ; but now a demand has arisen, owing to the more common use of the quantities, for names for the units of flux, induction, magnetomotive force, reluctance, induc- tance, and some others. The matter has been discussed widely at electrical congresses and before electrical socie- ties. The units of the Practical system were selected to be of convenient size for the common electric quantities ; they serve fairly well for this purpose, with the exception of the Farad, which is about one million times too large, and which is replaced in practice by the microfarad. But the Practical system does not give magnetic units of convenient magnitude ; for instance, the unit of flux would be io 8 C. G. S. units, which is about one hundred times too large ; the unit of flux-density or induction would be io~ 10 C. G. S. units, whereas the C. G. S. unit itself is more nearly right ; the unit of reluctance is io~ 9 C. G. S. units, whereas the C. G. S. unit is again approximately right. The latest official body to pass upon the question, the International Electrical Congress of 1893, refused to adopt new magnitudes or names for the magnetic units, but recommended the adop- 260 UNITS AND DIMENSIONS. tion of the C. G. S. system. This same congress, however, adopted the Henry as the unit of induction in the Practi- cal system, and defined it as follows : " As a unit of induc- tion the henry is recommended, which is the induction in a circuit when the electromotive force induced in this circuit is one international volt, while the inducing current varies at the rate of one ampere per second." It may be con- sidered official. Since this date, however, the American Institute of Elec- trical Engineers has adopted, " provisionally," the names Weber, Gauss, Gilbert, and Oersted for the C. G. S. units of flux, flux-density, magnetomotive force, and reluctance, re- spectively, as shown by the table. It adopts Gauss for the unit of magnetic force as well as for magnetic flux-density; these two quantities are different physical magnitudes, for which the same name should not be used. These names have not been generally accepted, and should be considered merely provisional. The need for such names is in many cases not clear ; the giving of names of individuals to C. G. S. units is not in conformity with previous usage in the matter. The British Association has recently proposed a different system of names and magnitudes. For unit flux it recom- mends io 8 C. G. S. units, and calls it a Weber, thus agreeing in name only with the American Institute of Electrical Engineers ; for unit magnetomotive force, to be called the Gauss, two magnitudes are suggested, one, ^.nni, and the other, ni, or the ampere-turn. There are other recom- mendations of minor importance. 179. "Rational" System. Another system of units dif- fering fundamentally from these has been developed by Oliver Heaviside, and christened by him the " Rational " System. ELECTRICAL AND MAGNETIC UNITS. 26 1 The underlying assumption in the E. S. and E. M. systems is that the action takes place directly between unit quanti- ties of electricity or unit poles of magnetism ; they ignore the existence of the medium "entirely. Heaviside's system is based on the assumption thaj^all the phenomena are mani- festations of stresses and strains in a medium ; the medium is therefore brought into evidence. This plan eliminates the factor 471 from the expression of magnetomotive force, making it correspond to ampere-turns ; this is merely an incidental advantage of the " Rational " system, but it has been seized upon as the main reason for its adoption. Coulomb's law for the force existing between two quanti- ties m and m' at distance /, or, in the E. M. system, f= mm'/l\ j4= I, Heaviside puts OC = w/(4*O and /= X,m f = mm' f^itT. That is, instead of making the flux from a pole of strength m equal to ^itm, he makes it equal to m. The question is simply, What is the rational or natural method of measuring the strength of a source at a given distance from its centre ? Coulomb's formula follows the ordinary astronomical or gravitational method ; but this is clearly illogical : the strength of a source m at a distance / is the quantity of m per unit surface at that point. The total flux from the source is necessarily, by the principle of con- tinuity, the quantity at the source. Its strength at distance /, when the law of variation is that of the inverse square, is w/47T/ 2 , or the strength of the field ; the force which the 262 UNITS AND DIMENSIONS. field exerts on another source m' is equal to the product of m' by the strength of the field at the point, that is, mm This simply means in ordinary language that unit pole sends out one line of force, not ^.n lines. A similar irrationality would be introduced into the geo- metrical measures if unit area were defined to be the area of a circle whose diameter is unity, instead of the area of a square whose side is unity. If this were done, the expres- sion for areas and volumes of rectangular surfaces and solids would involve ^.n as a factor, while ^n would be eliminated from the expression for area of circle and volume of sphere, where they properly belong. The changes that this makes in the various units are readily obtained by giving to // the value 4^ in the table showing the dimensions in the w-system. Denoting units in the Rational system by the subscript " r," then these relations are found : ( 4 7T) = r r /r = L r /L = We see that the term -j- plays the part of an electro- motive force which tends to diminish the current ; it is the general expression for the electromotive force of induction. If we discard the cell and then displace the circuit in the same way as before, by an expenditure of mechanical en- d<& ergy, an electromotive force of induction, _= , is set up, which produces a current z = Equation (2) shows that an immovable circuit traversed by a variable magnetic flux is equally the seat of an electro- motive force of induction which is in every case equal and of contrary sign to the rate of variation of the flux with regard to the time. 185. Maxwell's Rule. The direction of the electromo- tive force is readily deduced from Maxwell's rule ( 135). If we suppose a corkscrew to move forward, rotating in the direction of the field, the direction of rotation indicates the positive direction of the electromotive force. Now, when the flux or the number of lines of force decreases, the elec- tromotive force -j- is positive. If the flux increases, the electromotive force is negative, i.e. it is in the opposite direction to the movement of rotation of the corkscrew. LAWS OF ELECTROMAGNETIC INDUCTION. 2? I To sum up : the electromotive force is in the direction of the rotation of a corkscrew advancing in the direction of the lines of force, when the flux decreases ; when the flux increases, it is in the opposite direction. Apply this rule to a solenoid into which a magnet enters by its TV end. The flux of force across the solenoid is increased, consequently the electromotive force is negative. When the magnet is drawn out, the flux decreases and the electro- motive force becomes positive. 186. Faraday's Rule. In the case of induction by dis- placement, it is often useful to determine the electromotive force due to the movement of the various parts of the circuit acted upon. Now it will be remembered ( 145) that the total variation of the flux across a circuit is the algebraic sum of the elementary fluxes cut by its various parts. We can, therefore, interpret the general law of induction by saying that the electromotive force set up in a conductor is measured at each instant by the flux of force cut in unit time. In other words, the electromotive force equals the length of the conductor multiplied by the field-intensity, and by the projection normal to the lines of force of the displacement made in unit time. The direction of the electromotive force is obtained by Lenz' law ( 183). Thus the axle of a car, which, in the northern hemisphere, moves from east to west cutting the terrestrial lines of force, is the seat of an electromotive force of induction di- rected from north to south. To find this direction, we need only suppose a figure lying along the axle and looking down- wards (since in the northern hemisphere the lines of force go downwards), and that the figure is displaced to the left ; the induced current then tends to pass from the head to the feet. 2/2 ELECTROMAGNETIC INDUCTION. Faraday's rule has the advantage over Maxwell's of apply. ing even in the case of an open circuit, and of showing that an electromotive force of induction exists in conductors cutting the lines of force of a magnetic field. Fleming's mnemonic rule can also be used ( 138). In the present case the right hand should be used : The index finger and thumb being respectively in the direction of the lines of force and the displacement, the middle finger will indicate the direction of the induced E. M. F. It follows from the above rules that if a circuit is dis- placed in a field in such a way that the flux embraced by it remains constant, or when a conductor moves parallel to the lines of force in a field, no induced currents are gener- ated. 187. Seat of the Electromotive Force of Induction. The electromotive force takes its rise in all the parts of the circuit which cut lines of force, and in these only. Thus, in the above example of an axle moving upon rails, the E. M. F. is developed in the axle. Suppose that a magnetized bar be displaced along the axis of a metallic ring. All the elements of the ring cut in each instant the same number of lines of force, and in each element, with a resistance dr, is set up an E. M. F. de and a current . de By reason of symmetry there can be no difference of po- tential in the ring; but if the ring be cut, there immediately occurs at the points of separation a difference of potential representing the sum of the electromotive forces of the dif- ferent elements. This case shows that an E. M. F. can ex- ist without difference of potential. Hydrodynamics offers analogous examples : a circular LAWS OF ELECTROMAGNETIC INDUCTION. 2/3 trough filled with water, in which a solid ring is revolved, would show a current due to the friction of the liquid against the ring ; the motive force, being uniformly distributed, would not produce any difference of level between the vari- ous parts of the water in^the trough. 188. Flux of Force Producing Induction. It is impor- tant to determine the expression for the flux traversing an induced circuit. In a general way the flux can be separated into two parts, the first due to the current itself which traverses the cir- cuit, the second to the external field produced by currents or magnets. In 163 we defined, under the name of coefficient of self-induction of a circuit, the ratio of the flux traversing it to the current. This coefficient depends on the form of the circuit and on the medium in which it is placed. In fact, the flux of magnetic force generated is proportional to the permeability of the surrounding medium. If the circuit is completely surrounded with iron, or if it simply contains a core of iron, the flux of force has a very much higher value than if the circuit is simply surrounded by air ; the perme- ability of iron for moderate inductions being very much greater than that of air. If, then, we denote by L s i the flux of force produced by the current across its own circuit, the coefficient of self-induction L s has a constant value only on condition that the circuit is invariable in form and placed in a feebly magnetic medium. Where the circuit is near very magnetic bodies, its coeffi- cient L s becomes variable with the current. We cannot, therefore, specify the coefficient of self-induc- tion of an electromagnet without specifying the current which traverses the field-coils, together with the former magnetic condition of the core. 2/4 ELECTROMAGNETIC INDUCTION. As an application of the above, let us take an annular solenoid, in which the thickness of the section is negligible compared to the diameter, 153. Denoting by n the number of turns of wire per centi- metre, measured along the circular axis of the solenoid, and by s its cross-section, the internal magnetic flux, for a cur- rent 2, is expressed by This flux traverses successively the n } turns of the sole- noid ; consequently the total flux across the solenoid is # = nX 3C.y, and the coefficient of self-induction has the value Such a solenoid having 20 turns per unit of length and a section of 100 cm 2 would have a coefficient of self-induction per centimetre equal to i6o,ooO7rC. G. S. units or 0.503 X 10 * quadrants. If there is an annular iron core of permeability // in the solenoid, the coefficient becomes The permeability sometimes exceeds 3000, which explains why an electromagnet produces extra-currents very much greater than those of a solenoid without a core. The preceding expression also represents the coefficient of self-induction of a straight electromagnet of great length, if we neglect the influence of the extremities. When a coil is wound with wire doubled on itself, a gen- erator of electric energy sets up in it helicoid currents of opposite directions whose resultant magnetic effect upon an interior core, as well as on the surrounding medium, is zero. It follows that such a solenoid has a negligible coefficient f.AWS OF ELECTROMAGNETIC INDUCTION. 2?$ of self-induction. The same result is reached with a single wire if the successive layers are wound in the opposite di- rections, supposing that there are an even number of layers having each the same number of turns. So, too, two straight conductors near together, or two wires twisted together, do not give rise to effects of lateral induction when they are traversed by currents equal and opposite in direction. When the current passing in an electromagnet does not exceed the value which corresponds to the " elbow " in the core's magnetism curve, 57, we may take for granted, in approximate calculations, that the permeability is constant and consequently the coefficient of self-induction also. According to Lord Rayleigh this hypothesis always holds good, whatever be the magnetization of the core, when we are treating small variations of the magnetizing force, i.e. of the current. To sum up, the total flux across an isolated circuit trav- ersed by a current i is = Lj. If the current varies, an E. M. F. of self-induction is set up, equal to d$ d , T .. -37 = -d/*^)' or simply * if L s is constant. When the current increases, dt is positive and the E. M. F. negative ; whence an inverse extra-current. When the cur- rent is decreasing, the E. M. F. is positive and gives rise to a direct extra-current. 276 ELECTROMAGNETIC INDUCTION. If the circuit, instead of being isolated, is near magnets or currents, to the flux due to the current itself is added a supplementary flux, comprising lines of force in the mag- netic field produced by these external causes. In the case of currents, we have defined ( 142), under the name of coefficient of mutual induction of two circuits, the ratio of the flux traversing one of these circuits to the cur- rent in the other circuit. We must introduce the same condition here as in the case of self-induction ; when the permeability of the medium is variable, L m varies with the currents under consideration. Suppose a circuit whose own flux is L s i and which is near another circuit whose coefficient of mutual induction and current are respectively L m and i' . The total flux will be The E. M. F. of induction will in this case be expressed by d$ d , If the two circuits are of invariable form, and if we sup- pose that the permeability of the medium surrounding them is constant, L s and L m are constant factors, and we have d/ T oV The secondary circuit may, moreover, be situated in a field produced by the earth and magnets. If we denote by 3s-}-2(m(*)) the total flux set up by these causes across the circuit, we get the general expression <2> = L s i + LJ + Ws + whence LAWS OF ELECTROMAGNETIC INDUCTION. 2/7 NOTE. Let us take up again for a moment the case of an annular bobbin, with an iron core closed upon itself and traversed by currents whose direction varies periodically, passing from -|- i to i and inversely. At every change of direction the magnetization oLthe core is reversed and the coefficient of self-induction for the bobbin takes the value L s = ^$nn^s. But if the currents traversing the coils vary from o to a value i without assuming negative values, i.e. if the current is simply intermittent, the magnetic filaments formed in the core remain oriented by virtue of their coercive force, and the coefficient of self-induction of the bobbin has, for the currents following the first, sensibly the same value as if the core did not exist. Matters go on as if the core were a permanent magnet and the coefficient of self-induction sim- ply equal to When the core is not continuous, no closed filaments are formed, so that the effect of the coercive force is consider- ably weakened. 189. Quantity of Induced Electricity. Suppose the flux across a circuit varies from o to a quantity . The in- duced current at any instant is and the total quantity of induced electricity /*. /* d$ $ q = / fat = I = -. 3 t/o /o r r 278 ELECTROMAGNETIC INDUCTION. If the flux then returns to o, the quantity of electricity is /-"=* it is equal to the preceding, and is displaced in the opposite direction. APPLICATIONS OF THE LAWS OF INDUCTION. 190. Movable Conductor in a Uniform Field. Suppose that a conductor of length /, such as the axle of a railway car, is displaced horizontally with a velocity v. It cuts the terrestrial lines of force, and the E. M. F. of induction is, denoting by 3C the vertical component of the magnetic field, The direction of this E. M. F. is given by Faraday's rule ( 186). Put /= 150 cm, v = 1666 cm per second, oe = 0.44. We get e = 150 X 1666 X 0.440. G. S. units, or o.oon volt. 191. Faraday's Disc. If Barlow's wheel is caused to rotate ( 157), there arises an E. M. F. directed from the periphery to the centre or inversely, according to direction of rotation. Keeping the notation used in the above-mentioned para- graph, and denoting by n the number of rotations per sec- ond, and by <*> the angular velocity of the wheel, GO = 2nn ; APPLICATIONS OF THE LAWS OF INDUCTION. 2/Q the E. M. F. of induction is Faraday, to whom this experiment is due, thus discovered the simplest known induction-machine. We also are in- debted to him for the following arrangement, which is the principle of the so-called unipolar machines. Let us consider the apparatus shown in Fig. 75, and sup- pose that we leave out the cell, joining the trough directly to the middle of the magnet. If the conductor is now made to revolve, it becomes the seat of an E. M. F. of induction, whose value is, denoting by the GJ angular velocity of the conductor, n the number of turns per second, and m the magnetic mass of the pole which serves as pivot, e = ^nm X n = 2moo. Edlund has pointed out that, by leaving the conductor at rest and rotating the magnet, there is also an induced current set up. This can only be attributed to the de- velopment of an E. M. F. of induction in the magnet itself in consequence of its rotation in its own field. 192. Measurement of the Intensity of the Magnetic Field by the Quantity of Electricity Induced. Suppose a uniform field of sufficient extent to allow a flat bobbin to turn upon itself in the field, its axis of rotation being along a diameter normal to the direction of the field. The ends of the wire in the coil are connected by sliding contacts to a ballistic galvanometer ( 150). Let R be the total resistance of the circuit, 3C the field- intensity. If the bobbin, containing n turns of surface a, is made to perform half a revolution, starting from a position with its plane normal to the direction of the field, the flux 28O . ELECTROMAGNETIC INDUCTION. traversing it passes from 3an to o, then from o to The total variation is 23an, and the quantity of electricity induced ( 171) Again, denoting by a the swing of the galvanometer, by K a constant, whence _KaR 2an ' Weber s inclinometer, which serves to determine the inten- sity of the earth's field, is based upon this method. If, by small equal advances, we slide a test-coil along a magnet which it exactly encloses, we will get in a ballistic galvanometer connected with the test-coil, swings propor- tional to the field-intensity in the various regions of the magnet. This is a convenient way to obtain the distribu- tion-curve of magnetism in a magnet ( 47). 193. Expression for the Work Absorbed in Magneti- zation. Loss due to Hysteresis. Take the case of an annular electromagnet ( 142) of sufficient diameter to allow the supposition that the internal field is uniform and ex- pressed by 3C == n 1 being the number of turns per unit length. The work done to magnetize the electromagnet is neces- sarily equal to the energy given back by it under the form of the extra-current. Now the energy of this latter is given by W=f t eiAt =/"- id/ - /- APPLICATIONS OF THE LAWS OF INDUCTION. 28 1 But, denoting by s the cross-section of the magnet, / its mean length, and (B the induction traversing it, we have 1 3C Substituting for > this valu^e, and for i, - , we get The energy expended per unit of volume will be i f JCd(B. V Now (B = 3C whence / 3Cd& = t/ The integral represents the energy restored by the solenoid, that is to say its intrinsic energy. The integral is therefore the energy restored by demagnetizing the core. If the magnetizing force varies between values 3C, lt 5C 2 , this last expression of energy will be It would be represented by the area comprised between the curve 3 =/(3e), the axis of and two parallels to the 282 ELECTROMAGNETIC INDUCTION. axis of abscissae drawn through the points of the curve cor- responding to 5Cj and 3C 2 . Where the current successively assumes values -f- z, i, the energy restored by the solenoid equals the energy ab- sorbed, for we have But the integral is not zero, since it represents the area comprised between the curves AC A' and AC' A of Fig. 90. FIG. 90. This integral expresses the loss per cycle and per cm 3 of the core. APPLICATIONS OF THE LAWS OF INDUCTION. 283 The work expended, equal and of contrary sign to the work restored by the core, is represented by 194. Self-induction in a Circuit composed of Linear Conductors. Case of a Constant Electromotive Force. Time-constant. When a circuit containing an element having constant E. M. F. is closed, the current does not instantaneously attain its normal value, especially if the cir- cuit contains an electromagnet. In the same way, when the circuit is broken the current does not stop suddenly, but is prolonged by the extra-current which appears in the spark at the point of interruption. Faraday showed the analogy existing between these phe- nomena and those caused by the inertia of fluids. A liquid current cannot be set up or cease suddenly in a pipe, and at the moment of stoppage we observe a sudden blow due to the momentum of the moving fluid ; but there are pro- found divergences between these two cases. While the blow given by a liquid is diminished by bends in the pipe, the extra-current is much more marked in a coil than in a straight wire of equal length. It will be seen fur- ther on that this difference is explained by placing the seat of the current's energy in the medium surrounding the con- ductors. There exist, however, even in the expressions for the phe- nomena, analogies which it is useful to bring out. Thus, when a fluid is urged to move in a pipe the force expended is utilized on the one hand to overcome the fric- tion against the walls, on the other to increase the momen- tum of the movable mass. If the velocity v is small, the friction can be expressed by Av, when A is a constant. 284 ELECTROMAGNETIC INDUCTION. The increase of momentum of the mass m is in unit time l m-r~. The total force is therefore d/ (i) Now take the case of a circuit with a resistance r, whose coefficient of self-induction L s is constant ( 198), and which includes a cell with an E. M. F. denoted by E. At the moment of closing the circuit there is set up an E. M. F. of induction A~r so that the current is given by the equation I = -. From this we deduce R ri-\- T~ (?\ ^a? an expression analogous to (i). ri represents, likewise, the portion of E. M. F. used to overcome the friction of the conductor and L s -r~ = e the portion used to increase the in- trinsic energy of the circuit, for, multiplying e by t, we have r di . d It is taken for granted in equation (2) that the distribu- tion of the current is uniform over the section of the con- ductor, and that the resistance of the latter to a variable current is the same as to a steady current. Further on we shall see that this is not the case for linear conductors, i.e. those of very small cross-section. At the end of a time t the current will attain a value * APPLICATIONS OF 7 'HE LAWS OF INDUCTION. 285 given by the integration of the differential equation (2). We have whence /* d* r'dt / E-fa-tJ. L; and lastly / -1\ I = I I = e La], r \ I in which.* represents the base of the Naperian logarithms. p Theoretically, the current does not reach its full value _ rt until an infinite time has elapsed, but as the value of e L S decreases rapidly, this term becomes negligible, compared with unity, in a short time. The ratio , which is homogeneous with a time, is called the time-constant of the circuit ; we will denote it by r. Suppose, for example, an annular solenoid of 100 cm 2 sec- tion, having 20 turns per cm, measured on the axis, and whose length when developed is 100 cm. If the resistance be i ohm or 10 C. G. S. units, the exponential factor as- sumes the value 47T X 20X2000 X 100 l67T It is seen that for a comparatively small value of / this term may be neglected. The variation of the current in terms of the time is rep- resented by a curve which rises rapidly from the origin, then tends towards an asymptote parallel to the axis of the 286 ELECTROMAGNETIC INDUCTION. times. The first part of the curve, corresponding to small values of the time, can be replaced by a right line, Et This equation shows that during the first instants the current depends, not on the resistance of the circuit, but on its self-induction. The quantity of electricity which passes in the circuit during the variable period is C* . , r*Et _*\ , E E r* * * q = I idf= / ~~ I - e 7V d* 55 I / e'^dt i/o e/o r \ j r TV O E,. E -*- Beyond a certain value of / the second term becomes negligible, and we have simply / is the quantity of electricity which would have passed during the time t if the current had instantaneously assumed its permanent value. The term r represents the quantity due to the inverse extra-current or make-induced current. We come now to the break-induced current, and, to sim- plify the calculation, suppose that the resistance of the cir- cuit be maintained constant by substituting for the cell a wire having the same resistance. The break extra-current is given by the equation dt APPLICATIONS OF THE LAWS OF INDUCTION. 28/ Integrating between -- and t, we find E _ i = e T. r , It is easy to see that this current decreases rapidly. The quantity of electricity due to the extra-current on breaking is the same as that of the extra-current on making the circuit. The total quantity of electricity produced by the cell is, moreover, the same as if there were no induction-effects, for We can find directly the expression for the quantity of induced electricity by applying the equation q = T * (8 189) and observing that $ is the product of the current, when fully established, into the coefficient of self-induction. We see that the induction phenomena which occur in a circuit during the variable period of the current have the effect of causing an apparent increase in the resistance of the conductors. The total quantity of electricity put in motion is, however, the same whatever be the coefficient of self-induction, for the direct extra-current restores the quantity of electricity abstracted at the beginning of the current. 195. Work accomplished during the Variable Period. By Joule's law, the work accomplished during the vari- able period on closing the circuit is expressed by w,= 2 r 288 ELECTROMAGNETIC INDUCTION. If t is large enough, we have simply E" r Now the cell produces during the variable period a total quantity of electricity _ & This quantity exceeds the Joule effect by LJ' denoting by /the current when fully established. This dif- ference represents the intrinsic energy of the circuit trav- ersed by the electric flux ( 144). We see, then, that when a current is set up in a circuit, the energy furnished by the source of electricity is composed of two parts, the one transformed directly into heat by the Joule effect, the other stored up in the potential state. This intrinsic energy is in its turn transformed into heat during the extra-current on breaking the circuit, for there is then a Joule effect represented by /oo 772 /oo 2t ?rdt = ~ ,-7d/ = r / T n Eidt- / ?rdt = I Ljdi = . A t/o 2 196. Application to the Case of Derived Currents. Suppose two conductors joined in parallel, the self-induction of the first being L s , that of the second having a negligible value ; suppose further that the two branches have no mu- tual inductive influence on each other. When we send a current into the two conductors, whose resistances are r l and r a , the division of electricity between them is influenced by the reaction of the self-induction in one of them. Let i l be the current in this one at any instant of the variable period, and z' 2 the current in the other at the same instant. The first one is the seat of an E. M. F. equal to ATT, so that on applying Kirchhoff's second law to the closed cir- cuit including r l and r 2 , we get whence & - r*fi& = - L,fdt lt the integration being comprised between corresponding lim- its of time and current. If the circuit of the cell is closed and then opened, the 2QO ELECTROMAGNETIC INDUCTION. initial and final currents are zero in the branch r^\ conse- quently the second member and q^ and q t denoting the quantities of electricity that have traversed the two branches. These quantities are exactly the same as if there had been no induction effect in the two branches. The same effect would occur if a condenser were dis- charged through the branches. The division would occur as if there were no self-induction currents in one of them. It is taken for granted, in these deductions, that the con. ductors have a sufficiently small cross-section for the cur- rent to be uniformly spread over every part of it during the variable period. 197. Discharge of Condenser into a Galvanometer with Shunt. The above calculations cannot be applied without reserve in the case of discharging a condenser into a galvanometer furnished with a shunt. If the galvanometer remains motionless during the whole period of discharge, the subdivision of the quantities of electricity takes place according to the above simple law. But it frequently hap- pens that the needle commences to move before the end of the discharge ; in which case it produces in the galvan- ometer-coil an induced current opposite in direction, by Lenz' law, to that which would itself have produced the movement. It follows that the total quantity of electricity which traverses the coils is diminished. To calculate this diminution we may suppose, with M. L. Clark, that the flux of magnetic force produced by the APPLICATIONS OF THE LAWS OF INDUCTION. 291 needle across the coils is proportional to the sine of the angle described. Denote by i^, g, L s the current in the galvanometer, its resistance and coefficient of sejf-induction ; by z a , s, the cur- rent in the shunt and its resistance, and suppose it to be formed of a straight wire or bobbin with double winding. KirchhofFs second law shows that T di l . ^d sin a v - t =L sW +K-^ r , whence sJ 9 'i*dt - gi& = L s f o di, + K^"d sin a. The current in the galvanometer is zero at the beginning and end of the discharge, whose duration is /, and the supe- rior limit of OL is the deviation tf of the needle. Let we will have But by the theory of the ballistic galvanometer ( 150), if the arc of swing is small enough PL sin d = 2 sin = a being a constant. Consequently Denoting by Q the total discharge equal to q l -\- S (Q + ^d/) + v^u j^ smatdt. . . (4) \ L, s I L> s Now make dv -\--j-vdt = o, L s and for simplicity put -j- =b, when we get log e K being a constant of integration ; whence v = Ke~f tdt ....... (5) Now equation (4) reduced to E vdu = ^ s APPLICATIONS OF THE LAWS OF INDUCTION. 2$$ gives d/ . ... du = -^e j -= sin atdt ; A L, s whence u = sin atdt; * A and putting KK' = D, f*- si" atdt + D \ = /- sn Integrating by parts, we get / whence sn ^// = a , t , sn ^/ cos ,' = f " (^ sin tf / - cos a*) + ^-. . (6) ^sV" 1 "I ^ / We can substitute for the difference a sin at -- cos ^f by sin (^/ 0), the value of being determined by the condition that the equality sin at -- . cos at = sin (at 0) * be verified for all the values of /. Now for t=o we get sin

^ T ' E. COS . /27T* A - -- sm - <>} (10) Equation (10) shows that the maximum current is E cos _ APPLICATIONS OF THE LAWS OF INDUCTION. It is observed that the E. M. F. of self-induction reduces the maximum current, which would be if L s were zero. E cos represents the maximum effective E. M. F. re- sulting from the combination Q the electromotive force E and the reaction of the self-induction. The subtractive term 0, which is not found in equation (i) of E. M. F., shows that there is a retardation of phase between the maximum values of the current and the E. M. F. due to the field. T This retardation is in duration. Remarks. I. These results are applicable to a coil com- posed of a number of turns, and, for approximate calcula- tions, to a coil with an iron core, provided the magnetiza- tion be sufficiently feeble to allow us to take the permeabil- ity as constant, without much error. II. We denote by the name of apparent resistance of a circuit the radical by which the E. M. F. must be divided in order to find the current : Heaviside has called this the impedance of the cir- cuit. It will be noticed that the radical is homogeneous with a resistance and can be expressed in ohms. The term =-* is called the reactance. III. Denoting by t the time-constant of the circuit equal to , the result can also be put in the form 298 ELECTROMA GNE TIC IND UCTION. These formulae show that the apparent resistance (impe- dance) and the retardation of phase depend essentially on the time-constant. A large self-inductance may only produce a minimum apparent increase of resistance, if the resistance is already considerable in itself. 199. Graphic Representations. The axis Oy represent- ing the direction of a uniform field which develops induc- tion-currents in a coil turning in the direction of the arrow, let us represent by the right line OM, which makes an angle a = -=- with Ox, the maximum electromotive force E . The E. M. F. due to the field is represented at the instant / by the projection oa = E Q sin a. The maximum effective electromotive force, E cos 0, is represented by ON, the projection of OM on a straight line making an angle with the latter. Now as the effective E. M. F. is the resultant of E n and cl-- v P FIG. 92. the E.M. F. of self-induction, the maximum of this last will be shown by a right line OP = E sin 0, which completes the parallelogram OM. NP. The projections of OM, ON, APPLICATIONS OF THE LAWS OF INDUCTION. 299 and OP on Oy represent the values of the different E. M. Fs. at the moment of rotation when the coil makes an angle a with Ox. The effective E. M. F. at tfyis* moment is Ob = ON sin (a^ (p) = 7s cos sin (a 0); it is less than the E. M. F., Oa, due to the field, by a quan- tity ab = Oc which measures the reaction of self-induct- ance. But when the coil OM has passed the axis Oy by an angle 0, OP comes above the axis of x and the projection of the resultant, ON, is greater than the projection of OM, for the action of the self-induction is now added to the E. M. F. due to the field. On rotating the parallelogram OMNP about the point O, the projections of OM, ON, and OP on Oy show, at each instant, the relative values of the various E.M.Fs. in action. The current in the coil is given at any given instant by the ratio of the length of Ob to the resistance r of the coil. We can also show the variations of the E. M. Fs. in terms of the time, by drawing the curves represented by the equations 271 't E = E Q sin -=- = E sin at, d* e= L,-^ = E sin cos (at 0), E' = E Q cos sin (at 0). We will then get three sinusoidal curves like those in Fig. 93; the sine-curve E represents the E. M. F. due to the field; e is the E.M.F. of telf-induction and E' the effective or resultant E. M. F. The ordinates of E' are the differences between those of E and e. 300 ELECTROMAGNETIC INDUCTION. The current at any moment is found by dividing the or- dinates of the curve E' by the resistance of the circuit ; the phase of the current, moreover, coincides with that of E'. FIG. 93. The retardation of the current-phase behind that of the E. M. F. due to the field is where The maximum possible retardation corresponds to a.L s = oo . We have then = and 0= 2 4' It will be observed that the curve e is a quarter phase, or T , behind the curve E ' . Consequently if the latter is itself 4 T behind E, the positive waves in E will be exactly over 4 the negative waves in e. The effective E. M. F. ought then to be zero (and also the current) for E cos 0=o. APPLICATIONS OF THE LAWS OF INDUCTION. 3OI 200. Mean Current and Effective Current. Measure- ment by Dynamometer. The quantity of electricity which passes through the circuit in a half-period is independent of the lag ; it is expressed by =. f'idt = " P si J^ y r ' + a ' L ,'J t The mean current, during a half-period, is - 9 - 2 E > 2 / being the maximum current. The mean current-strength, which may be represented by i r kr T^F MI, is therefore equal, in the case under consideration, to the product of the maximum current by the factor . It is to be observed that the needle of a galvanometer gives no deviation when the coil of the instrument is trav- ersed by an alternating current of short period, for it then receives equal and contrary impulses ; but it is possible to use the electrodynamometer, 152, whose readings are pro- portional to the square of the current. With this instrument we get a deviation proportional to the mean of the squares of the current. This mean is expressed by 302 ELECTROMAGNETIC INDUCTION. and its square root is called the effective current, the lumin- ous effects of the current in electric lamps, for example, depending upon it. The effective . M. F. is likewise defined by the square root of the mean square of the E. M. F. The impedance of the circuit is the factor by which the effective current must be multiplied to obtain the effective E. M. F. Sometimes the name of inductive r:sistance is given to a resistance having a coefficient of self-induction. In the present case, the mean square of the current is It will be observed that the square root of the mean square is different from the mean current, = 0.9; n*% consequently i m = 0.9 ^(TJT; that is to say, the readings by the electrodynamometer must be reduced by a tenth to get the mean current. If we denote by E Q and /the E. M. F. and the maximum current, it is easy to express in terms of these quantities both the mean E. M. F. and current and the effective E. M. F. and current. APPLICATIONS OF THE LAWS OF INDUCTION. 303 We have then : Mean E. M. F. e = . Mean current i m = /. Effective E. M. F. V(?) m = E eff = ^. 4/2 Effective current S= I= - = ., Impedance V r* -\ ^j^- It must not be forgotten that these various relations do not hold except in the case where the periodic E. M. F. is a simple sine-function of the time and where the self-in- ductance L s is constant ; which makes it necessary that the permeability of the surrounding medium be invariable. If the periodic function were more complex, it might, by Fourier's theorem, be represented by a sum of sinusoids, but the above coefficients of reduction would be changed. The mean heat developed by the current in one second is i r i r Y frdf =-r X *'' I /; *J I/ i that is, the product of the real (or " ohmic ") resistance of the circuit by the square of the effective current. We have . = -Ej cos 0, 304 ELECTROMAGNETIC INDUCTION. whence P m = \/(e*) m V(i\ cos = E eff l eff cos 0. We see that the heat developed in a circuit where a peri- odic E. M. F. is acting varies with the lag of the current- phase behind the E. M. F., and consequently with the self- induction of the circuit. The calorific power is zero for = , i.e., for a lag of \ of a period. In the expression p -1 *' 2 r i a * L * r if L s is constant, the denominator is minimum for consequently the value of the resistance of the circuit which renders the mean calorific power a maximum is r = aL s . As then we have and the lag corresponding to the maximum power is equal to period. This power is expressed by APPLICATIONS OF THE LAWS OF INDUCTION. 305 201. Mutual Induction of Two Circuits. Suppose that two circuits of invariable form, having resistances r and r' and a coefficient of mutual induction equal to L m ( 143), approach each other so slowly, that the currents z, i' y which traverse them, can be considerecLas constant. For an elementary displacement, denoting by E, E' the electromotive forces of the sources producing the currents, we have dt ., = - '= From these equations we get (Ei + E'i'}dt - (i*r -f *'V)d/ = 2u'dL m . (Ei + E f i')dt expresses the energy furnished by the gen- erators during a time dt ; (fr -{- i'^r')dt is the portion of this energy which is transformed into heat in the conductors. oy ii'dL m is the work of the electrodynamic forces. As the excess of the energy expended over that transformed into heat is double this work, we conclude that a portion equal to ii'dL m is stored up in the system in the state of potential, or intrinsic, energy. It will be remembered, in fact ( 143), that the mutual energy of two circuits is expressed by ii'L m ; its variation is therefore, of course, ii'dL m . 202. Mutual Induction of Two Fixed Circuits. Two circuits, invariable in form and position, have resistances r and r' , coefficients of self-inductance L s and Z-/, and a coeffi- cient of mutual inductance L m ( 143). If cells having electromotive forces E and E' are in the circuits, the currents at any given instant of the variable period will be ELECTROMAGNETIC INDUCTION. i= , . . . . (I) (2) As the circuits are fixed, L m , L s and LJ are constants, so that on multiplying (i) and (2) respectively by idt and i'dt and then adding, we get (Ei+E'i')dt - (i*r + i'*r')dt = L,idt + L.'t'dt' + L m (idi' + i'di). We see that, in this case, the excess of energy furnished by the cells over the energy transformed into heat is L t idi + L s 'i f di' + LJJdi' + i'di). This expression is the exact differential of which represents the potential energy of the circuits when the currents have values z', i' '. The first two terms represent the intrinsic energy of each circuit, and the third is their mutual energy. 203. Quantity of Induced Electricity. Let us take the case where E' = o; the current of the second circuit is then alone the cause of the mutual induction, and equation (2) gives by integration for the induced current is zero at the beginning and end of the variable period of the inducing current. APPLICATIONS OF THE LAWS OF INDUCTION, 3O/ Consequently A-M/ *'- L i- L - E J, *&- - --?- 1 - --fr- \ When the inducing circuit is broken, we have likewise The quantities of electricity induced are equal and of opposite sign in the two cases. 204. Expression for Mutual Inductance. We have shown ( 163) that the sum of the magnetic fluxes which traverse the coils of a cored annular bobbin, divided by the current in the coils, is expressed by L, = ^Ttn^^s = ^7tn*}Jils t when the diameter of the ring is very great compared to its thickness. If we suppose that the first bobbin is entirely covered by a second, having n' turns per unit of length along the axis, and n' total turns, the coefficient of induc- tion of this second bobbin is L t ' = 47rn l f n r ^is = qnn^pls. Now the coefficient of mutual induction or the mutual inductance is the ratio of the flux produced by one of the bobbins across the coils of the other, to the current which traverses the first. L m ^nn^s X ri = ^nn{ f*s X n We see therefore that we then have the relation 3O8 ELECTROMAGNETIC INDUCTION. This simple expression is applicable whenever the lines of force generated by one of the bobbins traverse all the turns of the other. It is the maximum value, therefore, of the mutual induction of the two circuits. This condition is realized in the middle region of two concentric solenoids of great length and with a rectilinear axis.* 205. Induction in Metallic Masses. In what has pre ceded, we have had particularly in view the phenomena of induction developed in linear circuits ; but it is evident thai- induction takes place in metallic masses of any shape when they cut lines of force in a magnetic field. Faraday's disc ( 191) shows the development of induced currents in the case of a solid disc turning between the poles of an electromagnet. The determination of the lines of electric flux or current in such a case presents great complexity. To resolve the problem, we put against the disc copper points connected with a galvanometer, proceeding as shown in 168. In this way we get series of points at the same potential, which enables us to draw equipotential lines. The lines of flux are perpendicular to these. If Faraday's disc is made to revolve without connecting the sliding contacts by a conductor, the lines of current be- come closed on themselves in the disc, producing curves which envelop each other without intersecting, and which are divided into two separate groups by a vertical plane passing through the axis of rotation. By Lenz' law the direction of these currents is such that they tend to oppose the movement of the disc ; consequently the currents which approach the poles are in the opposite di- rection to the currents of a solenoid equivalent to the induc- * See Mascart and Joubert, Lemons sur FElectridtt et le Magnetisme for the working out of calculations on inductances of solenoids. APPLICATIONS OF THE LAWS OF INDUCTION. 309 ing magnet. The currents which move away from the poles are in the same direction' as would be the solenoidal ones. 206. Foucault Currents. jThis consequence of Lenz* law is shown in various ways ; t^e following experiment is due to Foucault. A rapid movement of rotation is given to a disc embraced by the pole-pieces of a powerful electro- magnet. At the moment when a current is passed through the latter, the mechanical resistance caused by the induced currents causes the stoppage of the disc ; if the motion is continued by a sufficient expenditure of motive power, the disc grows hot in consequence of the Joule effect. The induced currents are very greatly reduced, and consequently the resistance to rotation and the heating, by dividing the disc by slits normal to the direction of the E. M. Fs. of induction and thus cutting the lines of flux. In the present case these divisions would be circles concentric with the disc, and this latter would have to be made of rings of increasing diameter, separated by an insulating material. The currents induced in metallic masses are usually com- prised under the name of Foucault or eddy currents. The mechanical resistance caused by the induced currents in these masses is utilized to deaden the movement of galvanometer-needles. If, for example, we surround .a magnet, movable around an axis of suspension, with a mass of copper in which is hollowed a cavity sufficient to permit the swings of the magnet, the latter will develop in the copper induced cur- rents which oppose its movement and cause its stoppage. When a conductive rod which forms part of a circuit is displaced across the lines of force of afield presenting varia- tions of intensity at different points, besides the E. M. F. of induction observed in the circuit, Foucault currents are generated in the substance of the conductor. In fact the 310 ELECTROMAGNETIC INDUCTION. elementary filaments which constitute the conductor cut at the same instant different numbers of lines of force. These subsidiary currents heat the rod without doing useful work in the circuit. 207. Cores of Electromagnets Traversed by Variable Currents. Calculation of the Power Lost in Foucault Currents. Foucault currents tend to be set up in the core of an electromagnet whose coils are traversed by a periodic current. To diminish these currents, which heat the iron uselessly, the core is made of thin plates insulated from each other by varnish, or varnished or paraffined paper, and put together in such a way that their surfaces of separation are parallel to the axis of the coils and consequently cut the directions of the E. M. Fs. of induction. The bolts used to fasten the plates together should be insulated by tubes of vulcanized fibre and washers of the same put under the heads of the bolts and the nuts. It is a good plan to wrap the whole core round with varnished cloth, so that the edges of the plates may not pierce the insulation of the wires rolled on the iron. A core is sometimes made of a bundle of varnished iron wires, but it is to be remarked that the space lost by the interstices between the wires is much greater than in the case of a laminated core. Moreover this arrangement is only applicable to small, straight electromagnets. As the wires do not touch except along single lines, the insulation need not be so careful as that of the plates. The division of the core parallel to its axis enables the heating by Fou- cault currents to be avoided, but not that which results from hysteresis ( 6 1 ). Under the action of the periodic current of the coils, the core is, practically, subjected to successive magnetizations in opposite directions which cause a loss of energy in proportion to the variations of the mag- netizing force and the coercive force of the core. APPLICATIONS OF THE LAWS OF INDUCTION. 311 It is possible to calculate the loss occasioned by Foucault currents in an iron plate or wire taken in the core of an electromagnet traversed by periodical currents. Take the case of a cylindrical wire of length / and radius R, traversed longitudinally by ^variable flux which tends to develop eddy currents parallel to the edge of a right section of the wire. Let us take inside the wire a concentric tube infinitely thin, of radius r and thickness dr. Every variation of the flux along the wire sets up in the tube an E. M. F. of induction expressed by The resistance opposed to the current by the tube is 2nrp p being the specific resistance of the metal. The power set free in the form of heat in the tube is the ratio of the square of the E. M. F. to the resistance, or v~ d t , / /day, dp = - - = nr* l-r-J dr. 27i rp 2p\dt I ~~ldr~ The total loss in the wire is, consequently, r* , / /day, m?t/d traversed by the undulatory currents. The disc, being placed at aa' at an angle of 45 with the coil, is the seat of induced currents, and, in consequence of the repulsion exercised by the primary current, tends to place itself in the position bb' , for which the periodic flux traversing it is a minimum. The mirror M enables us to read by reflection the angle at which the disc comes to rest, under the influence of the electrodynamic forces on one hand and the torsion of the suspension-wire on the other. APPLICATIONS OF THE LAWS OF INDUCTION. 323 The disc must be placed obliquely to the coil, for if it were parallel to the turns of the coil it would oscillate con- FIG. 97. tinuously, the current giving it a new impulse at each cross- ing of the plane normal to bb f . IMPEDANCE* 213. Inductance. A sinusoidal E. M. F. may be ex- pressed by e = E sin cot, where e is the instantaneous E. M. F. in volts ; E is the maximum cyclic E. M. F. in volts ; GO is the angular velocity of the E. M. F. in radians per second, and is 2nn, where n is the numb.er of cycles of the E. M. F. per second ; t is the time expressed in seconds, starting from some suitable instant at which E is zero and changing from negative to positive. When such an E. M. F. is impressed upon a circuit hav- ing a resistance of R ohms and an inductance of L henrys, a certain current strength will pass through the circuit. We assume that the inductance is non-ferric, that is to say, that the circuit is not linked with, or situated in the neigh- borhood of, iron, so that if there are coils of wire in the circuit, these coils are without cores. Then the current in the circuit will also be sinusoidal. This may be shown as follows: The counter E. M. F. (C. E. M. F.) in the circuit at any instant due to the action of the inductance will be L-r at volts, where i is the instantaneous current strength in am- peres. The apparent C. E. M. F. due to fall of potential * By A. E. Kennelly. 324 IMPEDANCE. 325 in the ohmic resistances will be iR volts; and if no elec- trostatic capacity exists in the circuit, these will be the only sources of C. E. M. F. present. But the impressed E. M. F., e, must be equal and opposite at any and every instant to the tota.1 C. E. M. e = E sin cot = iR + L-j- volts. The solution of this differential equation within a con- stant which is negligible as soon as the current has become steady, is E sinf oot tan * = If the circuit contains resistance only, or L =. o, this be- comes sin cot e t ' t =^ -j) ^> amperes . (2) That is, the current i would be in phase with the E. M. F. and would have at each instant the value -5. The intro- J\ duction of the inductance into the circuit has altered both the magnitude of the current and its phase. The magni- tude becomes which is always less than ^, while the phase has become retarded by the angle tan" 1 -^-, which is always less than 326 IMPEDANCE, This effect of inductance may be considered from two distinct points of view, namely : 1st, and fundamentally, we may regard the circuit as having simply a resistance R, but as being the seat of two separate although interdependent E .M. Fs., one being the impressed sinusoidal E. M. F., e volts, and the other being the self-induced E. M. F., L~ volts. at Since i = / sin oot amperes, (3) IGD cos Got = loo sin f oat -| ) , . . . (4) /. L-r = ILdD sin f oat -\ j volts, (5) This shows that the self-induced E. M. F. has the maxi- mum value ILoo volts, that the effective value (square root of mean square) is -~=Lo) volts, and that the phase of this E. M . F. is 90 is rr 1 ahead of the current /. If, therefore, we represent the counter E. M. F. due to the fall of potential (iR volts) at any instant, say for con- venience at its maximum value, by the straight line 01, Lie) FIG. 97. Fig. 97, then the product of this value into will give the induced E. M. F. when turned through a right angle, or di- rected as shown by the line IE. IMPED A NCR. 327 The geometrical sum of these two E. M. Fs. will be OE in both direction and magnitude, and this will be the im- pressed E. M. F., e, at the instant considered. According to this conceptiqn;* the E. M. F. iR which acts upon the resistance R so as tq^orce through it the current t, is the geometrical difference OI of the impressed E. M. F. OE and the self-induced E. M. F. IE. In order to deter- mine the current strength z, we have to compute the influ- ence of i upon the C. E. M. F. of the circuit. 214. Inductance and Capacitance. Extending this method to the case of a circuit in which a condenser of capacity C farads is connected with the sinusoidal E. M. F. e through an inductant resistance of R ohms and L henrys, we find the C. E. M. F. of self-induction to be L -=- volts, at and the C. E. M. F. of capacity to be -^J^ idt volts, while the apparent C. E. M. F. of resistance is iR volts. The total C. E. M. F. in the circuit is consequently - iR -f Z + VoltS, so that, equating the impressed and reversed C. E. M. F., we have e = E sin tot iR + L~ + ^J o idt volts, . (6) The solution of this equation is -} - i)' 328 IMPEDANCE. The effective current strength will, therefore, be E i /= -=:-. : ITT amperes. . . (8) Plotting this case graphically in Fig. 98, OI is the r,. M. F. iR, active in producing the current i amperes, through the resistance R ohms. IA is the E. M. F. active in overcoming the C. E. M. F. of self-induction, and AE is the E. M. F. exerted in overcoming the C. E. M. F. of ca- pacity, so that EA is the C. E. M. F. of capacity, while OE is the impressed E. M. F. It is evident from the preceding that even with so simple a circuit as that comprising a sinusoidal E. M. F., resistance, inductance and capacity, the differential equation which has to be solved in order to evaluate the current strength is already formidable, and when any further degree of com- plexity is introduced, such as branch circuits, the difficulty of effecting the solution of the problem becomes very great. 215. Inductance and Reactance. 2d. The alternative method of dealing with sinusoidal alternating-current cir- cuits is to assume that no E. M. Fs. act in the circuits other than the impressed E. M. Fs, In other words, the C. E. M. Fs. of inductance and capacity are ignored, but their IMPEDANCE. 329 effects are considered as resistances added to the circuit and opposed to the impressed E. M. Fs. Thus, an inductance of L henrys is considered as a react- ance of Loo ohms, and any capacity of C farads is considered i as a reactance of ~~*- ohms."" CGO A reactance is reckoned in a direction making an angle of 90 with that of resistance, considered as a definite straight line or axis. Thus, if a resistance of R ohms, Fig. 99, be connected in A Loa if" FIG 99. circuit with an inductance of L henrys, the reactance of the latter will be Loo ohms, and the combined resistance and re- actance will be OR + RA, where OR is the resistance and RA the reactance. Consequently, OA will be the impe- dance, or geometrical sum of OR and RA. If j denotes V i, or the quadrantal versor operator, RA =.jLa>, and the impedance OA Z R -}-jLoo ohms (9) The effective current strength in the circuit will be in amperes impressed effective E. M. F. in volts impedance in ohms where E is the effective E, M. F. 33 IMPEDANCE. From the point of view of impedance, therefore, we con- sider a sinusoidal current circuit as acted upon by impressed E. M. Fs. only, and consider the resistance as modified by the presence of inductance or capacity. Taking again a circuit of inductance, resistance and ca- pacity, The resistance is R ohms ; The reactance of inductance isjLca ohms ; The reactance of capacity is -^ ohms ; The impedance is R -\-J\LGO ~ ) ohms ; L,Gd ' so that the current strength is E i 7 ~\ amperes (n) This expression is equivalent to that of (8). When a sinusoidal current circuit contains any number of inductances, resistances and condensers in series, the impe- dance is the geometric sum of all the resistances and react- ances. That is, 1= - . amperes. . . (12) ^A. r- 7 \ ^J-^Gd -^~7^ ] J \ CooJ The computation may be carried out by algebraic rules or by geometry. It is to be observed that the above equation is a versor equation, and that*/ is no longer a simple number of amperes, but a number of amperes turned through a certain angle. The execution of the operations demanded by equation (12), namely, the division of the quantity E by the complex quantity representing the impedance, results in a quotient /, IMPEDANCE. 331 which is also a complex quantity and should be represented by a line having both direction and magnitude, the direction being confined to a single plane and to an angle less than 90 from E. . $ 216. Joint Impedances. When sinusoidal current cir- cuits are connected in parallel, their joint impedance pre- sents, symbolically, no more difficulty than the treatment of joint resistances in continuous-current circuits. Thus if Z^ , Z^ , Z 3 . . . Z n be the impedances of n circuits connected in parallel, each impedance being a plane vector, or versor quantity, expressible in the form Z n = R n +j\L n (v - ^-^1 ohms, ... (13) then the reciprocal of Z n may be called the admittance of the circuit n. The admittance of a circuit is a plane vector or directed quantity expressible in mhos. Thus if Y H be the admittances of circuit n, Y n = -^-=- -7- -^ mhos. . . (14) *m - The joint admittance of the group of multiple circuits will then be the geometrical sum of all the admittances, or F= The joint impedance will be the reciprocal of the joint admittance, or Z will obviously be a plane vector quantity. 332 IMPEDANCE. When any inductant resistance in a sinusoidal circuit is linked magnetically with a secondary circuit, through a mu- tual inductance of L^ henrys, the presence of the secondary circuit will modify the impedance of the inductant resistance in a definite manner. Let R s and X s be the resistance and total reactance of the secondary circuit ; RP and X p be the corresponding resistance and react- ance of the inductant resistance forming the primary; then if n = ^- , where Z s = R s +JX, , the effective impedance of the inductant resistance acting as the primary is changed from R p -\-jX p in the absence of the secondary circuit to Z=Rp + n*R s + j(X p - n*X s ) ohms in the presence of the secondary circuit. ? ' ' THE PROPAGATION OF CURRENTS. GENERAL CONSIDERATIONS. 217. Phenomena which Accompany the Propagation of the Current in a Conductor. Ohm has furnished an expression for a continuous current in a conductor, relying on an assimilation between the electric and the calorific fluxes. If we call r the resistance of a conductor per unit length, -TJ- the variation of potential per unit length in the di- rection / of the conductor, we have dU Lord Kelvin has carried this law further so as to make it applicable to the variable period of a current which is being set up in a conductor of given capacity. Imagine an in- sulated cable submerged in water. This cable forms a cylindrical condenser, with the water, supposed to be at zero potential, as its outer plate. The moment a difference of potential is established between the two ends of the con- ductor, a current is set up; but at the same time every portion of the conductor is charged with a quantity of electricity in proportion to its capacity and the difference between the potentials of the plates. There then occurs, across the dielectric of the cable, a charge, or displacement- current (Maxwell), which may be considered as a derived current from that in the conductor. This displacement- 333 334 THE PROPAGATION OF CURRENTS. current ceases when the tension of the dielectric balances the potential-difference of the condenser-plates. Then there exists in the dielectric an electric field whose lines of force connect the condenser-plates and end in equal and opposite quantities of electricity ( 88). When the period of charge is past, there only remains the permanent current expressed by equation (i). The same phenomenon occurs, but in a lesser degree, when a current is sent through a circuit in the air, for the charge in the conductor excites a contrary charge upon the neighboring conductors separated by air or other dielectric. Denote by c the capacity of the conductor per unit of length ; the corresponding charge for a difference of poten- tial U is q=cU. ........ (2) The energy of the electric field per unit length is %cU*. Let 'us express the fact that the quantity of electricity which enters the given segment of the cable is equal to that which flows out, plus the current across the dielectric. The variation of the current in the conductor per cm is repre- sented by -r ; the displacement-current is , whence the condition _ di L = dq^ dU dl dt ~ C dt (3) Combining (i) and (3), we get the equation This elementary law has enabled Lord Kelvin to investi- gate the variable period in submarine cables, where conden- ser-phenomena play an all-important part. GENERAL CONSIDERATIONS. 335 It must be noted that the current traversing the conduc- tor is manifested by a loss of energy transformed into heat by the Joule effect. The displacement-current ^n-the dielectric represents, on the contrary, a storage of energy in the potential state, shown by the tension of the dielectric. This energy in its turn produces heating effects when the cable is discharged. In order to investigate the manifestations, observed dur- ing the variable period, in a general way, it is necessary to call in the magnetic phenomena produced by the current in the surrounding medium. We have just seen that the current develops an electric field whose intensity depends on the specific inductive ca- pacity of the dielectric and whose lines of force are normal to the conductor. But besides this, the current creates a magnetic field characterized by lines of force forming closed curves around the conductor ; the intensity of this field is proportional to the permeability of the surrounding medium magnetized by the current. The intensity of the magnetic field diminishes rapidly with the distance from the conductor. We may therefore say that the self-induction of a circuit is sensibly propor- tional to the length of the conductors composing it, on con- dition that they are far enough apart so that the lines of force developed by them do not encroach on each other. Such an encroachment occurs in the case of two wires stranded together and connected in series. They tend, under the influence of an electric current, to produce lines of force opposite in direction in the surrounding medium, so that their coefficient of self-induction is practically zero. The same holds for a bobbin wound with a wire bent double ( 1 88). Putting these cases aside, denote by L s the coefficient of self-induction of a circuit per unit length of the conductors. 336 THE PROPAGATION OF CURRENTS. The magnetic energy of the current, represented by the magnetization of the medium, is \Lf per centimetre. The medium opposes a certain inertia to magnetization, which has the effect of developing an E. M. F. contrary to that which gives rise to the current. This E. M. F. is L s per centimetre, so that Ohm's formula when corri- da pleted becomes Equations (3) and (5) enable us to treat the problem of the variable period in its whole extent, taking into account the production of the electric and the magnetic fields cre- ated by the current. We have supposed the current to be surrounded by a perfect insulator. If there were any losses of electricity across the dielectric, we would have to add to the second member of the preceding equation a term accounting for the derived currents due to the electric conductivity of the medium. Equations (3) and (5) are analogous to those met with in the theory of the propagation of sound-waves, when we admit that the passive resistance of the medium is propor- tional to the first power of the velocity, and, moreover, that electric resistance corresponds to friction, self-induction to the inertia of the medium, and capacity to the reciprocal of a pressure. From this analogy it follows that, if a circuit be subjected to a periodic E.M.F., the electric waves generated are propa- gated in accordance with laws identical with those of the propagation of sound. In particular, if a periodic E. M. F. is applied to one of the ends of a line insulated at the other end, the electric waves thus created are reflected at the insu- lated end and return to their starting-point, where they are GENERAL CONSIDERATIONS. 337 reflected again, just like sound-waves sent into a tube which is closed at one end. The analogy noticed above is of particular interest in telephony, for it shows that "the electric waves transmit speech according to laws identical with those which govern its propagation in a ponderable medium.* 218. Special Characteristics shown by Alternating Currents. As has been shown in 199, alternating cur- rents do not add together as do continuous currents, but are compounded like vectors according to the parallelogram of forces : it is for this reason that two equal periodic E. M. Fs. acting in a circuit do not usually give a resultant current double that which could be produced by each one of them. The mean resultant is not equal to the sum of the mean component currents unless the phases of the latter are to- gether; it is zero if the phases differ by 180, just as the resultant of two forces is zero when they are equal and in opposite directions. This view explains certain peculiar effects produced by periodic electromotive forces. Consider, for example, an alternating current split into two branches having different resistances and self-induc- tions. f The branch currents will present differences of phase with regard to each other and to the total current, 198; at each instant the total current will be equal to the sum of the branch-currents, but the mean total current will by no means be equal to the sum of the mean branch- currents. If the difference of phase of the latter is great enough, it may even happen that the mean current of each one of the branches will be greater than the whole current. * See Demany, Thdorie de la propagation de Velectricitt. Bulletin de L'association des ingenieurs sortis de 1'Institut Montefiore, 1890. f Lord Rayleigh, On Forced Harmonic Oscillations. Phil. Mag., May, 1886. THE PROPAGATION OF CURRENTS. It is sufficient, for this purpose, to have the angle of lag more than 120, for the parallelogram of forces shows that when two equal vectors make an angle of 120, their result- ant is equal to one of the component vectors. The sine- curve representing the total current will have, at each instant, its ordinates equal to the algebraic sum of the ordi- nates of the two component sine-curves. These latter may therefore have very much greater coordinates of the result- ant curves. Take another case, pointed out by M. Smith. Between two points a and d subjected to an alternating potential- difference of constant mean value, put two resistances ab and bd in series, one having a considerable coefficient of self-induction, the other non-inductive. On each resistance put a glow-lamp on a shunt. The two lamps are alike, and the non-inductive resistance is adjusted until they both burn with the same brilliancy; this result shows that equal currents are traversing the filaments, and that the potential- difference of the points #, b is the same as that of b, d. Next, the wire joining the lamps is separated from that connecting the resistances, and the brilliancy of the lamps is seen to de- crease, although the potential-difference of the points a, d retains the same mean value ; which proves that this latter is less than the sum of the mean differences found between the points a, b and b, d. This fact is also explained by a differ- ence of phase in the potential-differences to which the two resistances in series are subjected : as the resultant has re- mained constant, the two components, whose phases are not together, must have assumed values greater than half the acting potential-difference. Instead of using glow-lamps to indicate potential-differ- ences, we might have used the quadrant electrometer as arranged in 102 (II). It is unnecessary to state that there is no contradiction of GENERAL CONSIDERA7UONS. 339 the principle of the conservation of energy in these experi- ments. If we find an~ increase of power consumed in one branch of a circuit, we observe a corresponding expenditure at the source of electricity, by the increase in the mean power of this source, P m = E eff .I eff . cos0. 219. Comparative Effects of the Self-induction and Capacity of a Circuit. It is interesting to observe that in considering the flux of electricity which flows during the variable period, the capacity and self-induction play opposite parts. In fact, the displacement-current due to the capacity is added to the current which traverses the conductor, so that the phenomenon of condensation is equivalent to an apparent diminution of the resistance of the circuit during the variable period. Suppose a conductor of resistance r, whose ends are at a potential, one of U, the other zero. Insert, between the extremities of the conductor, a condenser of capacity c\ its charge at the end of the variable period is q = cU = dr. This charge is added to the quantity of electricity traversing the circuit. On the contrary, the electromagnetic induction produces an increase of the impedance and a diminution of the flux of electricity, during the variable period of closure, equal to ( 194) From these opposite effects follows a certain compensation which can be made use of in the transmission of signals in cables. The difference of the fluxes of the extra-current and dis- placement-current is L s i il r \ q q cir -{ L 8 cr 1. r r\ I 340 THE PRO PAG A TION OF CURRENTS. We see that, as regards the flux of electricity transmitted during the variable period, the effect of the condensation corresponds to a diminution of the self-induction equal to the product of the capacity by the square of the resistance of the conductor. This observation is due to Sumpner. 220. Effect of a Capacity in a Circuit Traversed by Alternating Currents.* A condenser may be inserted in series in a circuit traversed by alternating currents with- out interrupting the passage of these currents, as would be the case if we were dealing with a continuous E. M. F. In fact, at each reversal of the current the condenser is dis- charged, and recharged in the opposite direction. It is nec- essary, however, if a powerful mean current is desired, that the capacity of the condenser should be great enough to absorb the electric flux transported by the waves of the current. Suppose that in a circuit of resistance r and without self-induction, we insert a condenser of capacity c. Denoting by E E Q sin at the periodic E. M. F., and by u the potential-difference of the plates at an instant /, we have E = E sin at u -f- ri. . . . . . (i) But cdu = tdt. . . .-.- . V . . (2) Differentiating (i) and substituting for du its value ob- tained from (2), we arrive at a cos at d/ = - d/ + rdt. .... (3) This equation has a form similar to equation (3) of 1 80. Resolving it by the same method and suppressing the ex- ponential term, which becomes zero at the end of a very * Boucherot, Electricien, Nov. 15, 1890. GENERAL CONSIDERATIONS. 34! short time, we find for the expression of the regular current i .sin (at + 0) .... (4) i/ r '4__L_" -2gi on condition that = tan" 1 . ...... (5) acr This result shows that the capacity has the effect of advancing the phase of the current ahead of the E. M. F. We have moreover V which proves that the condenser reduces the current the more, the less its capacity is. The current becomes zero for c = o. An infinite capacity would produce the same effect as taking the condenser away. We could have found the preceding equations directly by substituting, in the equations of . 1 80, - for the factor aL s . This fact is explained if it is observed that the self- induction introduces an electromotive force e = AT- , while a condenser brings in a difference of potential (taken from (2) ) e' = - u = - - - A'd/. Substituting for i its value / sin at in these expressions, we find e = aLJ cos at, e' = -\- / cos at ; ac values which are equal if we put aL s = . 34 2 THE PROPAGATION OF CURRENTS. 221. Combined Effects of a Capacity and a Self- induction in a Circuit Traversed by Alternating Cur- rents. Ferranti Effect. If we introduce in a circuit traversed by alternating currents a self-induction and a capacity, as the first tends to make the phase of the current lag, and the second tends to advance it, there will be a more or less complete neutralization of the two effects. Let us analyze this combination : Keeping the preceding notation, we will have the two equations E = E Q sin at = A^ + ri+u, . . . (i) cdu = tdt, ....... (2) and, combining them, L s c-^+rc~ + i- E,accvsat = o. . . (3) The general solution of this differential equation is of the form i = Ae mt + B sin at + D cos at. . . . (4) Differentiating this equation twice and introducing in di dV (3) the values of / and -7-3 thus found, we can determine the arbitrary coefficients m, B, and D. If we notice more- over that the exponential term is very quickly zero with /, the value of the established current / reduces to * = - --- - ' - -- sin (at - 0) o . (5) on condition that ,A-- :.. 0=tan-' - ~. .... (6) GENERAL CONSIDERATIONS. 343 The effective current is % ' s ~~^~r) ac m This expression could have been found directly by sup- posing a circuit with two self-inductions L s and Z/ and substituting for the second one a capacity such that Equation (6) shows that the current-phase will be behind or ahead of the E. M. F. phase according as aL, is larger or smaller than , or a*L s c^ I. In every case the current will be less than if there were neither self-induction nor capacity, unless d*Lc I. If we determine from equation (5) the expression for the E. M. Fs. e = L s -r- and e f / idt at cj due to the self-induction and the capacity, we find = -A It will be noted that these values may be very much higher than E ; in particular, if aL s = and r tends ac towards o, it is easily seen that the potential differences at the terminals of the induction-coils and the condenser tend 344 THE PROPAGATION OF CURRENTS. towards infinity. These peculiarities are explained by the composition of the E. M. Fs., as has been seen in 218. It is to a similar cause that the effect must be attributed which was observed in Ferranti's cables in London : when these cables, which are concentric and have a considerable capacity, are traversed by alternating currents, an elevation of tension is observed at the end of the line. From all the above deductions it follows that the capacity of a circuit corrects the effects of the self-induction by de- creasing the impedance created by the latter, as well as the phase-lag. The capacity to give a circuit in order to com- pletely neutralize the effects of its self-induction is shown by the equation c?L s c I. If a = 100, for example, and L s = one quadrant, we will have c = 100 microfarads. This capacity, which can only be obtained by the use of expensive condensers, could be reduced by artificially increasing the self- induction by electromagnets with closed magnetic circuits. The problem of the subdivision of a periodic current in a branch having self-induction, and another with a condenser, also gives rise to interesting observations. The current in the inductive resistance will have a phase-lag behind the resultant current, while the flux in the condenser will be ad- vanced. Consequently the sum of the two derived fluxes will be greater than the resultant flux, and each of the branches may even be traversed by a mean current superior to the total current. 222. Oscillating Discharge. Let us take up again the discharge of a condenser, by investigating which we ap- proached the study of the electric current ( 103), being guided by our knowledge of the laws of induction. Suppose a condenser of capacity c, whose plates are at a difference of potential U. Call r and L s the resistance and self-induction of the circuit of discharge. It is well to note GENERAL CONSIDERATIONS. 345 that r expresses the metallic resistance of the discharge- circuit. The discharge-current is equal to the rate of variation of the charge, or f but q =. cU, whence and To resolve this equation, put q = e mf , and we get The general integral is of the form q = Ae m *' + Be**, ...... (l) A and B being constants of integration; m l and m t , the roots of the equation obtained by equating to zero the tri- nomial in parenthesis, or 2L, Substituting these values in (i) and putting = r, we get V ' - . __. If the roots of the trinomial are imaginary, the radical exponent takes the form _ err 4r 8 346 THE PROPAGATION OF CURRENTS. and equation (2) reduces, by Ruler's formula, to The constants of integration are determined by the simul- taneous conditions / = o, i = o, q = Q introduced in equations (2) or (3) and in the expression Then substituting in this last the values found, we get, in the case of real roots, i = e 2T V - e j. (4) 4 T * err In the case of imaginary roots, the value of the current assumes the form i = JL=r^ sin /-L --!.,. . (5) T 4r 3 - Equation (4) shows that, for or the discharge occurs in the form of a continuous current, constant in direction, beginning with a value of zero, rising rapidly to a maximum, and then rapidly decreasing. When the discharge-current oscillates periodically between positive and negative values which decrease rapidly. GENERAL CONSIDERATIONS. 347 Equation (5) shows that the oscillating current passes through the same phases for ..t =j d, 27r, err Hence we deduce that the period of the oscillating cur- rent is 27T 27T T _ If r is negligible compared with 2\ , we get If, for example, k= i microfarad, L f = i.io" 4 quadrant, T = 2n Vi.io~ X i.io~* = 0.000063 second. Figures 100, 101, and 102 give a graphical representation of the above phenomena. The curve (I) represents the variations of the charge in terms of the time in the case of a continuous discharge, and FIG. 100. curve (II) shows the variations of the current which, zero at the beginning of the discharge, rapidly assumes a maximum value and then decreases asymptotically towards zero. 348 THE PROPAGATION OF CURRENTS. Fig. 101 shows the variations of the charge in the case of an oscillating discharge, and Fig. 102 shows the current- FIG. IOT. variations. These variations are shown by decreasing undu- lations whose period is equal to T. We are indebted to Lord Kelvin for the first analytical TIME X FIG. 102. investigation of the oscillatory discharge, which of late years has been the subject of experiments performed by, among others, Hertz and Lodge. To understand these phenomena, we must recall the fact that the dielectric of a charged condenser is subjected to a tension which may be compared to that of a spring. If the cause which produces the tension disappears suddenly, the GENERAL CONSIDERATIONS. 349 dielectric returns to its original position after having per- formed oscillations comparable to those of a spring when suddenly released. To keep the spring from oscillating we must oppose a re- sistance to its movement, by plunging it, for example, in a viscous fluid. Thus, too, by presenting an electrical resist- ance to the discharge of a condenser, it is rendered contin- uous. The period of the oscillations of a spring depends upon its mass or its inertia. So the period of the electric dis- charge varies with the self-induction, which represents the magnetic inertia of the medium surrounding the circuit. By increasing the self-induction we increase the length of the period more and more ; to reach this result, it is suffi- cient to pass the discharge through a bobbin, the number of whose turns increases progressively. Curiously enough, it is useless to put an iron core in the bobbin, for, in conse- quence of the rapidity of the reversals of the discharge, the result is zero as far as affects the magnetization. By proceeding in this way, i.e., by interposing an increas- ing number of turns of wire between the condenser-plates, we get an oscillating discharge of increasing period. The discharge-spark, which occurs at every break in the conduc- tors, appears single on account of the rapidity of the phe- nomenon, but, if it is reflected from a rotating mirror, we see that it seems composed of a succession of luminous points. On increasing the period sufficiently, the chain of luminous points becomes visible to an observer who looks at the spark through a glass which he moves rapidly. Finally, when the self-induction of the discharge-circuit and the capacity of the condenser are sufficiently great, the impulses communicated to the air by the. electric undulations reach the limit of vibrations perceptible by the ear, and the spirals give a note whose height can be diminished at will. 350 THE PROPAGATION OF CURRENTS. Dr. Lodge, by suitably varying the values of r, k, Z 5 , has succeeded in obtaining a scale of electrical vibrations whose periods extended from one hundred-millionth to one five- hundredth of a second. 223. Transmission of Electric Waves in the Surround- ing Medium. The comparison of the oscillations of an electric discharge with those of a vibrating elastic body can be carried still further. A tuning-fork generates sound- waves in the surrounding air, which spread out in space and can be shown to be present by means of a resonator tuning- fork in pitch with the first. If sound-waves hit perpendicu- larly against a solid wall, they are reflected ; the incident waves interfere with those sent back from the wall, and nodes and loops of vibration are formed which follow each other alternately. The resonator remains mute at the nodes and marks the position of the loops loudly. The length of the wave is equal to double the distance between two con- secutive nodes. The velocity of transmission of the waves is the quotient of the length of a wave by its duration. Analogous phenomena appear in the propagation of calorific and luminous radiations, formed by vibrations transverse to the direction of the rays. Dr. Hertz has experimentally demonstrated that dis- charge-oscillations between electrified bodies produce, in the surrounding medium, electric waves whose properties are identical with those of the radiations emitted by bodies at high temperatures, that is to say that they give rise to phenomena of reflection, interference, refraction, polariza- tion, and diffraction. In order to exhibit these properties, it was necessary, first of all, to have an apparatus producing continual electric oscillations. Dr. Hertz made use of a Ruhmkorff coil for this purpose, Fig. 103, the secondary terminals of which are connected GENERAL CONSIDERATIONS 351 to two conductors which constitute an electric vibrator or exciter and which have been given a number of shapes. In Fig. 103, the vibrator V is formed by two conductive rods, situated in the line of each other's prolongation and terminated at their ^adjoining ends by metallic knobs, the opposite ends carrying metal spheres. These latter can be replaced by discs or by metal plates hung from the rods. The induction-coil is intended to keep up the elec- tric charges on the two parts of the vibrator. In virtue u i FIG. 103. of these alternately opposed charges, the medium sur- rounding the vibrator is the seat of an electric field whose lines of force, periodically reversed in direction, connect the two parts of the apparatus. The mean direction of the field, that is to say of the electric forces, is the axis of the conductive rods. In consequence of the high potential of the charges, they recombine under the form of sparks leap- ing between the two metallic knobs. The oscillations of this prolonged discharge are dependent on the capacity of the vibrator and the resistance and self-induction of the conductors composing it: note that here the resistance in question is that of the conductors connected to the spheres. When the charge of these latter is sufficient, a spark occurs between the adjoining knobs, and it is only then that electric oscillations take their rise between each sphere and the corresponding knob. The duration of these oscillations is very small on account of their rapid decrease, 352 THE PROPAGATION OF CURRENTS, Fig. IOI ; the induction-coil must, therefore, renew the charges of the spheres very rapidly, and the sparks succeed each other at very short intervals. The oscillations, which produce rapid alternating currents in the rods of the vibrator, are propagated in the surround- ing medium. To demonstrate this, Dr. Hertz made use of an apparatus capable of vibrating in unison with the electric vibrator and which is called an electric resonator. The resonator can be identical with the vibrator, as in Fig. 103, or else it may have an entirely different form, provided that the three characteristic magnitudes, capacity, resistance, and self-induction of the conductors composing it, satisfy the same conditional equation as the correspond- ing quantities of the vibrator, a fact which can also be experimentally verified. This being so, if we place the resonator in the vicinage of the vibrator so that their axes are parallel, a flow of sparks is observed between the knobs of the first-named apparatus. These sparks, caused by the variations of the electric force parallel to the axis of the conductors, diminishes steadily as the resonator is moved further off. The explosive distance naturally decreases with the maxima of the electric force. The phenomenon of electric resonance is produced even if a solid insulating wall is interposed between the vibrator and the resonator, e.g., a vertical partition of wood or masonry; but it ceases when the partition is a conductor. In this latter case, if the resonator be moved between the vibrator and the partition, it is found that in certain points the spark ceases and that in others it is re-enforced. These extinctions, which are repeated at intervals of equal length, prove that the transmission of electric forces takes place in the form of waves capable of being reflected by a conductive wall, the waves sent back by the wall being GENERAL CONSIDERATIONS. 353 capable of interfering with the incident waves to form the observed nodes and loops of vibration. The distance between two nodes corresponds to half a wave-length. To obtain a number of> rfodes in the limits of the room used for experimenting in, i^is necessary to produce short waves. By using as vibrator a brass tube 26 cm. long and 3 cm. in diameter, divided into two parts with the adjoining ends terminating in segments of spheres, Hertz succeeded in exciting waves only 30 cm. in length. The resonator was a straight wire I metre long, also divided in two parts and furnished with metallic knobs between which the sparking takes place. Hertz has likewise performed a very interesting series of experiments with the aid of a metallic parabolic reflector designed to concentrate the electric undulations in a given direction and to thus produce more marked effects. The reflector is composed of a sheet of zinc 2 metres in height, curved and held on a wooden frame so as to have the form of a cylindrical surface having a parabolic section. The focal line is then parallel to the generatrices of the cylinder. If the vibrator is placed so that its axis coincides with the focal line, the waves are propagated in the direction of the reflector's plane of symmetry and are perceptible by the resonator at a much greater distance than before. The distance at which they can be perceived is still further increased by placing the resonator along the focal line of a reflector similar to the first, so situated that the planes of symmetry of the two coincide. An insulating screen, such as a wall, placed between the two parabolic mirrors, does not intercept the undulations ; but a conducting screen stops them and casts a shadow behind it. It will be observed that we are naturally led 354 THE PROPAGATION OF CURRENTS. to employ the language of optics to denote electric un- dulations. These phenomena are identical with those produced by luminous rays ; the only difference is in the order of magnitude of the wave-lengths; luminous waves are nearly a million times shorter than the electric waves produced by means of the vibrators used by the German scientist. The fact that Hertzian vibrations are not reflected by a wall may be compared to the fact that light is not reflected by a very thin transparent body : we know, indeed, that soap- bubbles no longer reflect the surrounding objects at the moment of their bursting ; now the ratio of the length of light-waves to the thickness of the bubble is then of the same order as the ratio of the lengths of the Hertzian waves to the thickness of a wall. Electric undulations are propagated in a straight line, as is shown by the fact of their stoppage by a metal screen. The mode of producing vibrations shows that they are parallel to the axis of the vibrator, i.e., that they are trans- versal and, to use the corresponding optical expression, polarized rectilinearly. If we place in the path of the rays transmitted by the first mirror a screen made of parallel wires, the effect of this screen is very different according as the wires are parallel or perpendicular to the axis of the vibrator. In the first case the electric waves pass without difficulty ; in the second, the electric force is absorbed by the wires which are normal to it and the rays are extinguished. The effect is similar to that of a tourmaline plate in optics. If, after having removed the screen, we turn the resonator and its mirror by an angle of 90 about the direction of the rays, no sparks are observed. But if the above-men- tioned screen be interposed normally to the transmitted rays and inclining the wires 45 to the directions of the GENERAL CONSIDERATIONS. 355 focal lines of the mirrors, the screen decomposes the incident waves and lets the vibrations inclined 45 to the axis of the resonator go past. These can then act on the" resonator. This phenomenon recalls the lighting up of tlsue field of two crossed Nicol's prisms by the interposition of a crystal plate. Lastly, the two mirrors enable us to exhibit cleaijy the phenomenon of the reflection and refraction of electric waves. For example, if we send an electric ray against a plane conductive partition, we can perceive the reflected waves by the aid of a resonator, provided that the planes of symmetry of the two parabolic mirrors be placed so that they intersect each other at the partition and that the normal plane through their intersection makes two equal plane angles. To produce refraction Hertz made use of a large asphalt prism having a refringent angle of 30. The incident rays directed upon the prism by the vibrator make with the refracted rays, which are received in the resonator, an angle indicating an index of refraction 1.7, only a slightly higher value than that given by optical experiments. Hertz- ian waves are therefore refracted by an insulating prism as light waves are by a glass prism. These experiments by Hertz have been repeated by various physicists. MM. Sarrazin and de la Rive, of Ge- neva,* have shown that the form of the resonators can be modified without ceasing to perceive the sparks, and that the observed wave-length depends much more on the dimen- sions of this apparatus than on those of the exciter, which seems to show that the latter gives rise to complex waves which can be picked out by suitable resonators. This ex- plains why, when an oscillating discharge is kept up in a * Archives de Geneve, June, 1890. THE PROPAGATION OF CURRENTS. room, sparks are seen to fly between metallic objects which are near together. Lecher of Vienna has investigated the propagation of Hertzian waves along two parallel conductors, placing over their ends a Geissler tube to serve as a resonator. When a metallic bridge is slid along the conductors, the brilliancy of the tube grows less for certain positions of the bridge, and stronger for others. This phenomenon is comparable to the propagation of sound in a closed tube ; at the nodes the pressure of the air is different from the atmospheric pressure, at the loops it is equal to this pressure. Consequently the sound is not altered if the tube be pierced at a loop, but it changes if the orifice is opposite a node. Thus at the nodes of Hertzian waves the difference of potential is maximum, and a metal bridge placed at these points over the two wires hinders the discharge from traversing the Geissler tube. If the bridge connects the wires at loops, its influence is null and the tube lights up. Lecher has found that the ve- locity of propagation of Hertzian waves in conductors is close to the velocity of light ; Hertz had already found the same number for the velocity in air. Hitherto we have only considered the electric field whose mean direction coincides with that of the axis of the vibra- tor. But the periodic currents of the oscillating discharge create, in addition, a magnetic field whose lines of force sur- round the conductors traversed by the undulatory electric flux( 135). For a complete investigation of the phenomenon we must therefore examine at the same time both the effects of the electric forces propagated in form of waves, and of the per- pendicular magnetic forces which accompany these disturb- ances in the surrounding medium, and whose effect are added to the first-mentioned.* * For the details of Hertz's experiments see : Roosen, Oscillations ttectri- GENERAL CONSIDERATIONS. 357 224. Present Views on the Propagation of Electric Energy. Hertz's experiments are a brilliant confirmation of the views, set forth by Faraday and defined by Maxwell, concerning the part played .by the medium across which electric energy is transmitted^ It will be remembered ( 109) that one of the hypotheses presented to account for electric discharge in conductors consists in supposing the latter to be the seat of a displace- ment of electricity comparable to the movements of fluid in pipes. Hence the expressions electric current, flux of elec- tricity, and the various images borrowed from the dynamic theory of fluids in order to render the phenomena more easily understood at the outset. But the investigation of the properties of the electric cur- rents shows that there is a profound difference between it and fluxes of ponderable matter, in spite of the analogies met with in the laws governing these fluxes ( 217). When a fluid current circulates in a pipe, no external effect shows its presence ; the phenomenon is entirely con- centrated inside the pipe itself. The electric current, on the contrary, which makes itself evident in a conductor by giving out heat, exercises peculiar and very striking effects in the surrounding medium. It magnetizes the medium, as is shown by iron-filings figures ; it modifies the optic properties of bodies ( 167), and, lastly, it produces induction-phenomena in conductors displaced in its field. Bends in a pipe cause a loss of kinetic energy and dimin- ish the blow given at the moment of stopping the flow ; the twisting of a conductor into spirals or a solenoid, on the ques (Bull. Assoc. ing. sort, de 1'Inst. Mont., 1890); PoincarS, Electricitt et Optique, Paris, Carr6, 1891 ; Lodge, The Work of Hertz and some of his Successors, Lond., 1894. 35$ THE PROPAGATION OF CURRENTS. other hand, increases the energy of the extra-current on breaking the circuit. A fluid current may be alternating ; this is the case in a pipe where the sound-waves are propagated in the form of longitudinal vibrations. We are tempted to compare such a movement to an alternating electric current ; but here, too, differences are evident, not only in the surrounding space, but even inside the conductor in which such currents pass. The sound-wave presents maximum displacements along the axis of the pipe, while alternating currents have their greatest strength towards the surface of the conductor ( 208). An electric current must be considered as the centre of a disturbance which affects all or part of the conductor by the Joule effect, and which extends out into the surrounding medium. As this propagation takes place in a vacuum, it follows that it is the ether that serves as vehicle for electric waves. A current, at the moment it starts, excites an electromag- netic wave which is transmitted in the space surrounding the conductor with a velocity equal to that of light. When the current has reached its full strength, i.e., when it has ac- quired a constant value in all the points of a section of the conductor, the surrounding medium is in a state of tension which is manifested by a tendency to contract in the direc- tion of the magnetic lines of force and to dilate in a direction normal to them. The ether about the conductor is then in a state of equi- librium characterized by cylindrical layers under a stress concentric to the conductor. When the current ceases, the ether, being suddenly un-stressed, falls back on the conduc- tor, giving up to it its potential energy, which is shown in the extra-current. An alternating current excites continuous waves, which as in the preceding case are propagated in space like luminous GENERAL CONSIDERATIONS. 3 $9 waves ; the only difference lies in the duration of the period of the ether-vibrations. It will be seen further on that alternating-current dynamos give from 50 to 200 vibratkms per second. Given the enor- mous velocity of propagation in the ether, these currents produce waves several hundreds of kilometres long. The luminous vibrations of a lamp's flame amount to about 50 trillions per second, so that their wave-length is only some hundred-thousandths of a centimetre. When these luminous radiations strike a body which in- tercepts them, it is found that their absorption causes a de- velopment of heat. So, too, conducting bodies which stop electric radiations are the seat of induced currents, made evident by calorific phenomena. The induced flux of elec- tricity is in the same direction as the electric force and nor- mal to the magnetic force of the wave. This latter penetrates more or less into conductors, according as its length is greater or smaller. Short electro- magnetic waves only affect the external layers of the con- ductors on which they fall. Hence the necessity of modify- ing the conductor's form, and adopting for short-period cur- rents tubes, strips or ropes of metal in preference to solid conductors. Electromagnetic waves, like radiations of heat and light, necessarily carry off with them a part of the energy from the source they leave to the conductors on which they impinge. To avoid this threatened loss in alternating-current circuits, the circuit is made of an outgoing and a return wire placed very close together. The waves emitted by the first go directly to the second, to which they restore, in the form of induced currents, the energy radiated off. This result is most completely obtained when two concentric conductors are used, a wire and a tube for example, for then the circuit has no action upon a neighboring magnet, and the magnetic 360 THE PROPAGATION OF CURRENTS. field is rigorously limited to the space occupied by the con- ductors and the dielectric. The discovery of the propagation of electromagnetic actions in the form of waves, similar to those of light, is of capital importance. It establishes an intimate bond between electricity, light, and heat, and will doubtless lead to im- portant progress in the knowledge of the laws which govern these physical agents. But so far only a corner has been lifted of the veil which hides the mechanism of the transmission of electric energy. We have learned that it is propagated without loss in dielectrics, while conductors are the seat of heat-effects which entirely or partly absorb the effective energy. But the phenomenon of the electric current still remains unex- plained, even in its most simple form, viz., after the variable period is over. From all modern experiments it is evident that an electric current is the manifestation of a transfer of energy which is taking place in the medium surrounding the conductors. These latter serve only to direct the propagation, and they perform this part at the expense of the absorption, under the form of heat, of part of the transmitted energy. A conductor should therefore be considered as the directrix along which the transfer takes place, just as the wick of a lamp is the centre of the flame without being the seat of the illuminating effect. The seat of the propagation of electric energy is in the electromagnetic whirls which en- circle the conductors. As to the real mechanism of the transmission, it is as mysterious as the mechanism of gravitation.* * Consult : Lodge, Modern Views of Electricity ; Stoletow, Ether and Electricity (The Electrical World, Jan. 28 et seq., 1893). ELECTRICAL MEASUREMENTS. MEASUREMENT OF ELECTRIC POWER. 225. Case of a Continuous Current. The electric power developed in a conductor, subjected to a potential- difference e and traversed by a current z, is expressed by the product ie. This product can be determined indirectly by measuring the factors e and i separately, by means of one of the methods already given, or directly, by means of instruments called wattmeters. 226. Siemens Wattmeter. This apparatus, which is applicable to the measurement of power developed by a continuous current, is made like the electrodynamometer, 152, except that the circuits of the two coils are separate. The fixed coil, formed of a great number of turns of fine wire, is placed in shunt to the conductor in which the power to be measured is developed ; the moving coil, comprising a few turns of thick wire, is in series with this conductor and traversed by the same current as it. The mutual action of the two coils is proportional to the product of the currents which traverse them ; but in con- sequence of the high resistance of the stationary coil it may be assumed, as with voltmeters, that the current traversing it is proportional to the original potential-difference e of the ends of the conductor. The electrodynamic couple, measured by the torsion-angle 8 through which the suspen- 361 362 ELECTRICAL MEASUREMENTS. sion-spring must be turned in order to bring the movable coil back to its initial position, represents very nearly, therefore, the product ei\ consequently e = kei. The factor k is determined by connecting the apparatus to a conductor of known resistance r, traversed by a cur- rent of known strength, a being then the angle of torsion, 227. Case of a Periodic Current. When the electric energy is transmitted in the form of periodic currents, the choice of a method of measuring the power demands special attention. There are then two cases to be considered : I. The conductor in which is developed the power to be measured has a negligible self-induction ; this is the case with straight or zigzagged wires and coils with special windings. II, The self-induction is not negligible. 228. Non-inductive Conductors. When a conductor of the first of the above classes is subjected to a periodic potential-difference, it becomes the seat of a current whose phases coincide with those of the potential-difference. Con- sequently, where the connection between the current and the time is a simple sine-curve, the mean power is equal to the product of the effective difference of potential by the effective current ; for we then have, 198, sin - T The effective current can be determined separately by means of an electrodynamometer, or by a voltmeter having MEASUREMENT OF ELECTRIC POWER. 363 a negligible coefficient of self-induction the Cardew volt- meter, for example. If we have to deal with a complex periodic function, or one the form of which is unknown, the preceding method should not be used. On the hypothesis that the power under consideration is completely transformed into heat by the Joule effect, we can then deduce the quantity of this heat from the current or potential-difference given respectively by the readings of an electrodynamometer or a Cardew voltmeter. Denoting by r the resistance, supposed to be known, of the conductor, we have 1 fat = rl f?dt = - T t/o T *A) r The Siemens wattmeter is not applicable in the case of periodic E. M. Fs. on account of the considerable self- induction of the fixed coil. The current produced in this coil would depend on its impedance, that is, on the fre- quency of the periods. Zipernowski has sought to escape this difficulty by making each of the coils of a small number of turns, so as to render their self-induction feeble. At the end of the coil which is to be placed in shunt are placed supplementary coils, wound double, whose only function is to supply the total resistance desired in the shunt. Call this resistance p, r that of the conductor in which we are trying to find the power developed, and R the resistance of the wattmeter's second circuit, The connections of the ap- paratus are generally arranged so that the resistance p is in parallel to the two resistances r and R placed in series. If the self-induction L s of the branch p were zero, or, more exactly, if the time-constant - were negligible, the current traversing this branch would be consonant in phase with 364 ELECTRICAL MEASUREMENTS. the potential-difference e, and the reading of the apparatus would be directly proportional to the mean power T r T = -1 / Mt, 1 /0 the constant k being determined in the same manner as in the Siemens wattmeter. But the time constant of the shunt is not negligible. The result of this is a diminishing of the current and a lag of the current-phase behind that of the potential-difference, which effects cause a reduction in the indications of the wattmeter below the value corresponding to the power actu- ally developed in the conductor r. As Mather has shown, the indications of the wattmeter may be rendered correct by winding the coil, which is put in series with the movable coil, in such a way that its ca- pacity neutralizes the effect of the movable coil's self- induction. 229. Conductors having Self-induction. The preced- ing methods are still at fault when treating a conductor whose self-induction is not negligible. In this case there is a phase-lag between the current and the periodic E. M. F. which sets it up in the conductor, such that we have We have seen, for a simple periodic function that cos being the angle of lag. If Zipernowski's modified wattmeter is applied to a con- ductor having self-induction, there is produced a retardation MEASUREMENT OF ELECTRIC POWER. 365 of phase both in the main circuit of the apparatus, in series with the conductor, and in the shunt. The result of this is a tendency towards consonance of phase in the two cir- cuits, which may have th/e "effect of raising the readings of the instrument beyond the^value corresponding to the real power, contrary to what happens in the case of a conductor without self-induction. This error is avoided if Mather's suggestion is utilized. In the case under consideration we can arrive at rigorous results by the use of the calorimeter. Fig. 104 gives a general idea of the calorimeter used by M. Roiti. The conductor TT, in which the power to be FIG. 104. measured is developed and, by hypothesis, entirely trans- formed into heat, is enclosed in a brass vessel placed inside another vessel of the same metal: connecting wires enter through tubes. The space between the two vessels is trav- ersed by a current of water which circulates under constant pressure and empties into a gauged receptacle R. Ther- mometers indicate, within a tenth of a degree, the tempera- ture of the water on entering and leaving the double covering around the conductor. When the temperatures have become constant, which is sometimes a matter of many 366 ELECTRICAL MEASUREMENTS. hours, the number of grammes of water flowing through per second and the difference in reading of the two thermome- ters are noted. The product of these quantities multiplied by the mechanical equivalent of the gramme-degree is the power sought. Messrs. Ayrton and Sumpner have worked out a method which allows of the exact determination of the energy consumed in an inductive resistance R, by means of a volt- meter and ammeter for alternating currents. A non-induc- tor resistance r is put in series with R, causing a fall of potential comparable with that caused by R. Let u l be the instantaneous potential-difference at the ends of R, & 2 that at the ends of r. The total fall of potential caused by R -\- r will be at the given moment u=u, + u t ........ (i) If a is the current at the time, the power at the given in- stant is p = au l = u^. But from (i) we deduce *.*. = K* - u ' ~ u ?\ whence / = ('-.'- q = .4rin \OL. But whence kR sin If the ballistic constant k of the galvanometer is unknown, it is determined by discharging into the apparatus a stand- ard condenser charged by means of a cell of known E. M. F., or by introducing into the galvanometer-circuit an inclinom- eter ( 192) which is made to turn in the earth's field so as to generate a calculable flux of electricity. * MEASUREMENT OF MAGNETIC PERMEABILITY. 233. Methods Based on Induction. I. The apparatus shown in Fig. 106 has been used by Dr. Hopkinson. The FIG. 106. iron bar, whose permeability is to be measured, is formed of two sections, C and C' ; it passes through a mass of wrought iron A, two magnetizing coils BB, and a small coil D con- nected with a ballistic galvanometer. This coil D is pulled laterally by an elastic thread, so that if the two sections C and C' be slightly separated the coi] is 37 ELECTRICAL MEASUREMENTS. drawn out of the apparatus and traversed by an electric flux in proportion to the magnetic induction of the bar. The iron mass and the bar form a magnetic circuit trav- ersed by a flux produced by the magnetomotive force ^nnc, n and c being respectively the total number of turns and the current of the coils BB. Let # be the flux, / that length of the bar inside the open space in A, s its section, /* its permeability; also, let I' be the mean path of the lines of magnetic force in the mass A, s' its section, X its permeability. We have ( 164) In commercial tests, where a very close approximation is not' called for, the second term of the binomial may be neglected in comparison with the first, and we have then simply . a differ only by a negligible quantity, and their ratio is practically equal to unity ; the formula then reduces to k sin l>a The constants k and K are determined by preliminary experiments. 239. Maxwell's Method, modified by Pirani, for Meas- uring a Self-induction in Terms of a Capacity. The relation existing between the effects of a self-induction and a capacity in a circuit traversed by a variable current ( 219 et seq.) has suggested a number of methods for meas- uring one of these quantities in terms of the other. The following one is especially convenient. A Wheatstone bridge (Fig. no) has three non-inductive arms, a, b, s. In the fourth arm is placed the coil r y whose coefficient of self-induction is sought, in series with a non- inductive resistance r' . A condenser of capacity C is put in shunt on this latter. First of all, the four arms are balanced for a fully estab- lished current; then they are balanced during the variable period by varying, for example, the point of connection of the condenser to the non-inductive resistance. Let R be the resistance of this fraction when the galvanometer remains at zero with the battery-circuit either open or closed. It was shown in 219 that, in regard to the flux of electricity 378 ELECTRICAL MEASUREMENTS. transmitted during the variable period, the condenser-effect corresponds to a decrease of self-induction equal to the pro- duct of the capacity by the square of the resistance of the conductor in parallel with the condenser. We shall then have in the present case A = kR\ where k is the capacity of the condenser. 240. Ayrton and Perry's Method for Comparing a Coil's Self-induction with that of a Standard Coil. Ayrton and Perry's standard of self-induction enables the self-induction of a coil to be determined in the quickest possible way. i > FIG. in. For this purpose a bridge is made (Fig. 1 1 1), by means of two non-inductive arms b, d, the coil to be standardized c, and the standard a. The galvanometer is brought to zero for a MEASUREMENT OF COEFFICIENTS OF INDUCTION. 379 fully established current, then the battery-circuit is opened and closed while altering the position of the movable coil in the standard until the galvanometer remains at zero during the variable period. We ,then have the proportion ~ "a This method does not necessitate the long series of small adjustments needed by the other methods in order to ob- tain a balance in the two states (fully established and varia- ble) of the current. If the self-induction of the resistance c is beyond the bounds of the standard, it is only necessary to modify the ratio of the arms b and d, as is done in compar- ing, by Wheatstone's bridge, very different resistances. Messrs. Ayrton and Perry have, moreover, designed a commutator which markedly increases the sensitiveness of both this and the preceding methods, by accumulating the effects on the galvanometer of the impulses due to repeated making and breaking of the battery-circuit, For this pur- pose the connections of the bridge with the galvanometer and the battery are made by means of metallic brushes bear- ing on two movable discs B, R, composed of annular con- ducting segments separated by insulating spaces. The movements of these discs are interdependent and adjusted so that the battery-circuit may be alternately open and closed, while the galvanometer is successively connected to the bridge and short-circuited. The periods of connection of the galvanometer to the bridge correspond to the variable periods of opening and closing, and the connections are alternated in such a manner that the momentary currents traverse the galvanometer-coil in the same direction. With this cumulative method a slight error in equilibrium is shown by a perceptible deflection of the needle. 380 ELECTRICAL MEASUREMENTS Note. When investigating an electromagnet by the preceding methods, the results found for the self-induction vary with the current which traverses the field-coils, since the coefficient of self-induction is a function of the permea- bility of the core ( 188). It is therefore necessary to indi- cate the maximum value of the current traversing the field- coils for each value of the coefficient of self-induction. 241. Mutual Induction Carey-Foster Method. Sup- pose two concentric coils with or without an iron core ; de- note by L m their mutual induction ; insert one of them in a circuit through which a constant current i is sent ; a flux of magnetic force L m i then traverses the second coil. If this latter is in communication with a ballistic galvanometer so as to form a circuit of total resistance R, the galvanometer is traversed by a flux of electricity, measured by a deflection or, when the current in the first coil is stopped. We have (i) A condenser of capacity c is next charged by connecting its plates to the extremities of a resistance traversed by the same current i. The discharge of the condenser into the same galvanometer gives a deflection a', such that (2) Equations (i) and (2) give sn Carey-Foster has given an arrangement enabling the two preceding combinations to be performed simultaneously so as to balance the effects of the two electric fluxes on the MEASUREMENT OF COEFFICIENTS OF INDUCTION. 381 needle. The galvanometer G, Fig. 112, is connected to the condenser and to the resistance r, on the one hand, and to the circuit of resistance R on the other. By varying r until S^ 5 l..K.nAftAA\nA/N(WU- i $ P],|,|L_ L*__M^J FIG. 112. the needle remains at zero, both on making and breaking the battery-circuit, we have the relation L m = crR. INDEX. NUMBERS REFER TO ARTICLES, AND NOT TO PAGES. Absolute electrometer, 96 Action at a distance, 9 " of a magnetic field on an element of current, 138 " n ic un iform field on a magnet, 37 " " earth on a magnet, 33 " " homogeneous sphere on external point, 28 " " homogeneous sphere on internal point, 291 " " spherical shell on external point, 27 * " spherical shell on internal point, 26 Actions, Law of electric, 82 Alternating current, Method for determin- ing hysteresis, 69 Alternating currents. Combined effects of a capacity and a self-induction in a cir- cuit traversed by, 221 Alternating currents, Effect of capacity in a circuit traversed by, 220 Alternating currents, Special characteris- tics shown by (Mode of combining), 218 Ampere's hypothesis on the nature of mag- netism, 141 Angles, Measurements of, in radians, 50 Annular coil, 153 " magnet, 54 Application of theorems relative to central forces, 26 to 31 Arago's disk, 209 Artificial magnets, 47 Attraction, Law of magnetic, 34 Ayrton and Perry's method for comparing self-induction of coils, 240 Ballistic method for determining hystere- sis, 69 Barlow's wheel, 157 Becquerel's law, 127 Bobbin, cylindrical, 151 " ring-shaped, 153 Bodies in a magnetic field, Modifications of properties of, 167 " Magnetic and diamagnetic, 51 Body in a magnetic field, Equilibrium of, 65 Bridge, Wheatstone's, 123 British Association unit, 180 Brush-discharge, Rotation of, by a magnet, 156 " Electric, no C Capacitance and inductance, 214 Capacity and self-induction, Combined ef- fects of, in a circuit traversed by alter- nating currents, 221 Capacity and self-induction of a circuit, Comparative effects of the, 219 Capacity (Electrostatic) of Conductors, 92 " Effect of, in a circuit traversed by alternating currents, 220 " of condenser not connected to earth, 98 Carey-Foster method for mutual induc- tion, 241 Cell, Voltaic, 129 Central forces, general theorems, 9 to 25 " " definitions, 9 " " elementary law governing, 10 C. G. S. and practical systems, Relation be- tween, 177 " system of units, 171 Charged conductor in equilibrium, 81 Chemical effect of the current, 127 " electromotive forces, 113 Circuit, Coefficient of self-induction of, 163 " composed of linear conductors, Self-induction in a, 194 Circuit composed of linear conductors. Self-induction where there is a periodic or undulatory electromotive force, 198 383 3^4 INDEX. Circuit, electric, Magnetic potential due to an, 141 Circuit electric, Work due to the displace- ment of, under the action of a pole, 140 Circuit, Magnetic, 164 " traversed by a current, Reactions produced in a, 160 Circular electric current, Magnetic poten- tial due to a, 147 Closing a circuit, Work accomplished on, '95 Coefficient k in Coulomb's law, Nature of, 104 " of hysteresis, 71 " " " Table of values, 71 " " self-induction of a circuit, 163 Coil, Annular, 153 " Cylindrical, 151 Combined effects of a capacity and a self-in- duction in a circuit traversed by alter- nating currents, 221 Comparative effects of the self-induction and capacity of a circuit, 219 Condenser, Capacity of, when not con- nected to earth, 98 " Cylindrical, 97 " Discharge of, into galvanome- ter with shunt, 197 " guard-ring, 95 plate, 94 " Residual discharge of a, 106 " Spherical, 93 Conductive discharge, 109 Conductor, cylindrical, Self-induction in the mass of a, 208 Magnetization of, by a current, 1 66 Conductors and insulators, 77 " Electrostatic capacity of, 92 Congress, electrical, of 1893, Recommenda- tions of, 181 Conservation of energy; application to elec- trolysis, 129 " " " Principle of, 6 " matter, 6 Constant difference of potential in a con- ductor, Means of keeping up, 113 Constant electromotive force, Case of, 194 Construction and forms of electromagnets, 165 Contact electromotive force, 107 Continuous current, Measurement of pow- er of, 225 Convective discharge, to8 Cores of electromagnets traversed by vari- able currents. 207 Corresponding elements of tube of electric force, 88 Coulomb's law, Nature of coefficient k in, 104 " Proof of, 49 " theorem, 86 Current, circular electric, Magnetic poten- tial due to a, 147 ' continuous, Measurement of power of, 225 " electric, 109 " " Action of a magnetic field on an element of, 138 " electric, Magnetic field due to an infinite rectilinear, 136 44 electric, in a magnetic field, En- ergy of, 142 " electric, Intrinsic energy of, 144 " " Rotation produced byre- versing, 158 Current, element of. Work due to the dis- placement of, under the action of a pole, 139 Currents, infinite rectilinear, Potential due to, 146 Current, mean and effective. Measurement by dynamometer, 200 Current-meter, Shallenberger's, 211 Current periodic, power of, 227 " " power of inductive con- ductors, 229 " power of non-inductive conductors, 228 Current, phenomena w.^.ich accompany the propagation of in a conductor, 217 Current, Rotation of, by a magnet, 155 " Variable period of, 117 " Variable period of application of Ohm's law in but slightly conductive bodies, 118 Currents, alternating, Combined effects of a capacity, and a self-induction on a circuit traversed by, 221 Currents, alternating, Effect of capacity in a circuit traversed by, 220 Currents, alternating, Special characterist- ics shown by (Mode of combining), 218 Currents, Eddy, 72 " " effects of, 73 " electric, Induced, 182 " " Mutual action of, 159 " " Relative energy of two, H3 " Foucault, 206 " " Calculation of power lost in, 207 INDEX. 385 Currents induced, Rotation due to, 209 Cylindrical bobbin, 151 " condenser, 97 " conductor, Self-induction in the mass of, 208 " (infinite) magnet, 55 Definitions of magnetic quantities, 36 Density, Surface and volume (defined), 9 Derived circuits, Application of Kirchhoff s Laws to, 122 Determination of magnetic moment of a magnet, 48 Diamagnetic and magnetic bodies, 51 Dielectrics, Specific inductive capacity of, 103 Difference of potential and electromotive force, Distinction between, 107 Difference of potential in a conductor, Means of keeping up, 113 Dimensions of derived unit, 4 " " A' and ju., 175 " " units, 170 " " " applications of, 8 " Theory of, 4 Disc, Arago's, 209 " Faraday's, 191 " Infinitely thin potential due to an, 31 " -magnet, 53 Discharge, Conductive, 109 " Convective, 108 " Disruptive, no " Instantaneous electric meas- urement of an, 150 " of a condenser into a galvanom- eter with shant, 197 " Oscillating, 222 Discharging power of electrified points, 89 Discovery, Oersted's, 135 Displacement, Electric, in dielectrics, 105 Disruptive discharge, no Distinction between electromotive force and difference of potential, 107 E Earth, Electric potential of the, 85 Eddy currents, 72 " " Effects of, 73 Effect, Feranti, 221 " Hall, 168 " Joule's, 127, 196 " Kelvin, 131 ' ' of capacity in a circuit traversed by alternating currents, 220 Effect, of the current, Chemical, 127 " Peltier, 126, 130 ' Seebeck, 130 Effects, Combined, of a capacity and a self-induction in a circuit traversed by alternating currents, 221 Effects, Comparative, of the self-induction and capacity of a circuit, 219 Electric actions, Law of, 82 " circuit, Magnetic potential due to an, 141 Electric circuit, Work due to the displace- ment of,under the action of a pole, 140 " conductor, Magnetization of, by a current, 166 " current, 109 " u Action of a Magnetic field on an element of, 138 " " Chemical effect of, 127 " " circular, Magnetic poten- tial due to an, 147 Electric current, element of, Work due to the displacement of, under the action of a pole, 139 Electric current, Energy of, general ex- pression, 124 " " Energy of heterogene- ous conductor, 126 " " Energy of homogene- ous conductor, 125 " " in a magnetic field, Energy of, 142 Electric current, infinite rectilinear, Mag- netic potential due to an, 146 Electric current, Intrinsic Energy of, 144 " " Laws of (preliminary note), in " " Magnetic field due to an infinite rectilinear, 136 Electric current, mean and effective, Measurement of, by electro-dynamom- eter, 200 Electric current, phenomena which ac- company the propagation of, in a con- ductor, 217 Electric current, Rotation of, by a mag net, 155 " " Rotation produced by reversing, 158 " currents, Induced, 182 " " Mutual action of, 159 u " Relative energy of two, i43 Electric discharge, instantaneous, Meas- urement of, 150 " displacement in dielectrics, 105 386 INDEX. Electric energy, Present views of the propagation of, 224 " field (defined), 83 44 potential (defined), 83 " of the earth, 85 41 screen, go " spark and brush, 11 " waves, transmission of, in the sur- rounding medium, 223 Electrical congress of 1893, Recommenda- tions of, 181 " standards of measure, 180 Electricity, induced, Quantity of, 189, 203 Electrification by influence, 78 phenomena of, 76 Electrified conductors, Energy of, 100 Electrodynamometer, 152 Electrodynamometer, Measurement of mean and effective current by an, 200 Electrolysis, Application of the conserva- tion of energy to, 129 Electromagnet (defined), 162 44 cores traversed by variable currents, 207 Electromagnetic displacements, Explana- tion based on the properties of lines of force, 161 Electromagnetic induction, Flux of force producing, 188 Electromagnetic induction, General law of, 184 Electromagnetic induction, Graphic repre- sentation of cases in, 199 Electromagnetic induction, Seat of electro- motive force in, 187 Electromagneti rotation, 154 Electromagnets, Energy expended in, 163 44 Forms and construction of, 165 Electrometer, Absolute, 96 44 Quadrant, 47 Electromotive force and difference of po- tential, Distinction between, 107 Electromotive force constant, ease of, 194 of contact, 107 Electromotive force of induction, Seat of, 187 forces, Thermal and chem- ical, 113 Electrostatic capacity of conductors, 92 pressure, 87 Elementary magnets, 40 Element of current, Action of a magnetic field on an, 138 Element of current, Work due to the Dis- placement of, under the action o! a pole, 139 Energy, Conservation of (potentialkinetic), 6 44 conservation of ; application to electrolysis, 129 44 electric, Present views of the prop- agation of, 224 44 expended in electromagnets, 163 44 of a current in a magnetic field, 142 44 of a magnetic shell in a field, 45 44 of an electric current, Intrinsic, 144 " of electrified conductors, 100 44 of the electric current (general ex- pression), 124 44 of the electric current homogene- ous conductor, 125 44 of the electric current heterogene- ous conductor, 126 44 of two currents, Relative, 143 44 of two magnetic shells, Relative, 46 Equilibrium of a body in a magnetic field, 65 Equipotential surfaces, 13 Equivalent sine curves, 74 Ether, Modes of motion of, 6 Ocean of, 9 Ewing's addition to Weber's hypothesis, 64 Experiments with static electricity, 80 Faraday's disc, 191 " law, 127 44 rule for energy of current in magnetic field, 145, 186 Ferranti effect, 221 Ferraris' arrangement for obtaining rota-" tion, 210 Ferromagnetic bodies, 51 Field, Electric, (defined), 83 44 magnetic, Action of, on an element of current, 138 " Magnetic, due to an infinite rectilin- ear current, 136 44 magnetic, Energy of electric current in, 142 44 magnetic, Measurement of, by quan- tity of electricity induced, 192 44 magnetic, Measurement of intensity of electromagnetic method, 231 44 magnetic, Method based on induc- tion, 232 44 Method by oscillation, 230 44 Modifications of the prop- erties of bodies in a, 167 INDEX. 387 Field of force, (defined), n " " u of single mass, 14 " " " of two acting masses, 16 " " " Uniform, 15 " " " " action of, on mag- net, 37 " uniform magnetic, Movable conduc- tor in, 190 Filament, Magnetic or sdfenoidal, 41 Flux of force, 18 " " " producing induction, 188 Force, Field of, n " Flux of, 18 " Line of, (defined), 13 " Magnetomotive, 164 " Tubes of, 17 Forms and construction of electromagnets, 165 Formula, Frolich's, 58 " Muller, von Waltenhofen, and Kapp, 59 Foucault currents, 206 " " Calculation of power lost in, 207 Frolich's formula, 58 Galvanometer, Tangent, 147 shunt, 149 " Thomson, 148 with shunt, Discharge of condenser into, 197 General theory of units, 169 Graphic representation of Ohm's law, n6, 1 20 " representations of cases in electro- magnetic induction, 199 Grothiiss' hypothesis, 128 Guard-ring condenser, 95 H Hall effect, 168 Heterogeneous circuit, Application of Ohm's law to, 119 Hypothesis, Ampere's, on the nature of mag- netism, 141 " Grothiiss', 128 " Weber's, 39 " Ewing's addition to, 64 Hysteresis, 57, 66-75 " Coefficient of, 71. " Determination of, and molecular magnetic friction, 69 " Loss due to, in magnetization, Hysteretic loop, 67 " loss of energy, 70, 75 I Impedance, 213 216 Impedances, Joint, 216 Induced currents, Rotation due to, 209 '* electric currents, 182 " electricity, Quantity of, 189, 203 Inducing and induced currents, Mutual re- pulsion, 212 Inductance and capacitance, 214 " and reactance, 215 " (defined), 213 " mutual, Expression for, 204 Induction, electromagnetic, Flux of force producing, 188 " electromagnetic, General law of, 184 " electromagnetic, Graphic repre- sentation of cases in, 199 " in metallic masses, 205 " Mutual, of two circuits, 201 " " " fixed circuits, 202 " Seat of electromotive force of, 187 Infinitely thin spherical shell; no action on masses within it, 26 " Action on external masses of, 27 Infinite rectilinear current, Magnetic field due to an, 136 " rectilinear current. Potential due to, 146 Influence, Electrification by, 78 Instantaneous electric discharge. Measure- ment of an, 150 Insulators and conductors, 77 Intrinsic energy of an electric current, 144 J Jar, Leyden, 99 Joint impedances, 216 Joule's effect, 127, 196 K /iTand /m, Dimensions of, 175 Kelvin effect, 131 Kirchhoff s laws, 121 " " Application of, to derived circuits, 122, 196 L Laplace's law, 137 Law, Becquerel's, 127 " Coulomb's, Nature of coefficient K in, 104 " " Proof of, 49 388 INDEX. Law, Elementary, governing theNewtonian forces, 10 11 Faraday's, 127 " of electric actions, 82 " of electromagnetic induction, Gen- eral, 184 " of magnetic attractions, 34 " of successive contacts, 112 " Laplace's, 137 " Lenz's, 183 " Ohm's, 114 " " Application of, to the case of a heterogeneous circuit, 119 " " Case of a conductor of con- stant section, 115 " " Graphic representation of, 1 16, 120 Laws, Kirchhoff' s, 121 Application of, to de- rived circuits, 122, 196 " of the electric currents; preliminary note, in " of thermo-electric action, 132 Lenz's law, 183 Leyden jar, 99 Lightning-rods, 91 Lines of force, Explanation of electromag- netic displacements based on proper- ties of, 161 Loop, Hysteretic, 67 Loss of energy, Hysteretic, 70, 75, 193 M Magnet, (defined), 32 " Action of earth on a, 33 " Annular, 54 " Cylindrical, (infinite), 55 " Disc, 53 " Portative power of, 56 *' Rotation of a brush discharge by a, 156 " Rotation of an electric current by a. 155 " Sphere, 53 Magnetic and diamagnetic bodies, 51 " circuit, 164 *' field, Action of, on an element of current on a, 138 " " due to an infinite rectilinear current, 136 " " Energy of electric current in a, 142 " " Energy of shell in a, 45 " " Equilibrium of a body in, 65 Magnetic field, Measurement of, by quan- tity of electricity induced, 192 Magnetic field, Electromagnetic Method of measurement of intensity, 231 Magnetic field, Measurement of intensity of method based on induction, 232 Magnetic field, Measurement of intensity of method by oscillation, 230 Magnetic field, Modifications of the proper- ties of bodies in a, 167 Magnetic field of force, Terrestrial, 38 ' uniform, Movable conductor in, 190 Magnetic induction, magnetization, and permeability, Another way of looking at, 60 Magnetic moment of a magnet, Determi- nation of, 48 " or solenoidal filament, 41 Magnetic permeability; measurement of magnetometric method, 236 Magnetic permeability; measurement of methods based on induction, 233-235 Magnetic permeability; measurement of method by portative power, 237 Magnetic pole, Work due to the displace- ment of an electric circuit under the action of a, 140 Magnetic pole, Work due to the displace- ment of an element of current under the action of, 139 Magnetic potential, 36 " " due to a circular elec- tric current, 147 Magnetic potential due to an infinite recti- linear electric current, 146 Magnetic potential due to an electric cir- cuit, 141 " quantities, (definitions of), 36 " resistance, or reluctance, 164 " shell, (defined), 43 " in a field, Energy of a, 45 " " mentioned, 42 " Potential due to, 43 Magnetic shell, Work done on unit posi- tive magnetic mass in passing from one side to the other of a, 44 Magnetic shells, Relative energy of two, 46 Magnetization, magnetic induction, and permeability, Another way of looking at, 60 Magnetization of a conductor by a current, 1 66 " Variations of, with the mag- netizing force, 57 Work absorbed in, 193 Magnetizing force, Variation of magnetiza- tion with, 57. INDEX. 389 Magnetizing force, Work done in, 61 Magnetism, Effect of temperature on, 63 " Nature of, Ampere's hypothe- sis, 141 Magnetometer, 48 Magnetomotive force, 164 Magnets, Artificial, 47 Elementary, 40 d, Uniform, 42 * Maxwell and Pirani's method for measuring self-induction in terms of a capacity, 239 Maxwell and Rayleigh's method for meas- uring a coefficient of self-induction in terms of a resistance, 238 Maxwell's rule for energy of current in a magnetic field, 142 Maxwell's rule for relation between the directions of current and rotation of magnetic north pole, 135, 185 Mean current and effective current, Meas- urement of, by dynamometer, 200 Measurement of a coefficient of self-induc- tion, Ayrton and Perry's method for comparing with a standard coil, 240 Measurement of a coefficient of self-induc- tion in terms of a resistance, Maxwell and Rayleigh's method, 238 Measurement of a coefficient of self-induc- tion, Maxwell's method, in terms of a capacity, 239 Measurement of angles in radians, 50 of an instantaneous electric discharge, 150 of power, continuous cur- rent, 225 Measurement of magnetic field intensity; electromagnetic method, 231 Measurement of magnetic field intensity; method based on induction, 232 Measurement of magnetic field intensity; method by oscillation, 230 Measurement of magnetic permeability; magnetometric method, 236 Measurement of magnetic permeability; methods based on induction, 233-225 Measurement of magnetic permeability; method by portative power, 237 Measurement of mutual induction, Carey- Foster method, 241 Mechanical action, Effect of, on hysteresis, 71, 105 Metallic masses, Induction in, 205 Method, Alternating current for determin- ing hysteresis, 69 " Ballistic current for determining hysteresis, 69. Method, Ayrton and Perry's, for compar- ing a coil's self-induction with that of a standard coil, 240 Method, Carey-Foster, for mutual induc- tion, 241 Electromagnetic, for measuring magnetic field, 231 Induction, for measuring mag- netic field, 232 Oscillation, for measuring mag- netic field, 230 Methods for measuring magnetic permea- bility; induction, 233-235 Methods for measuring magnetic permea- bility; magnetometric, 236 Methods for measuring magnetic permea- bility; portative power, 237 Methods for measuring self-induction, Maxwell and Rayleigh's, in terms of a resistance, 238 Methods for measuring self-induction, Maxwell and Pirani's, in terms of a ca- pacity, 239 Molecular magnetic friction Effect of, 73 (defined), 68 determination of, 69 Moment, magnetic, Determination of, 48 Motor, Ferraris', 210 Movable conductor in a uniform field, 190 Mutual action of electric currents, 159 " inductance, Expression for, 204 " induction, Carey-Foster method, 241 of two circuits, 201 " " fixed circuits, 202 N Nature of magnetism; Ampere's hypothe- sis, 141 Nomenclature of practical units, 178 Proposed, for units, 174 Number of lines of force, 24 Numerical relation of B and /*, 62 " value and magnitude of units, reciprocal relation of, 2 Objective method, Sir Wm. Thomson's, 50 Oersted's discovery, 135 Ohm, International, 180 " Legal, 1 80 Ohm's law, 114 " " Application to the case of a heterogeneous circuit, 119 " " case of conductor of constant section, 115 390 INDEX. Ohm's law, Graphic representation of, 116, 120 Oscillating discharge, 222 Peltier effect, 126, 130 Periodic current, Power of, 227 " " " inductive con- ductors, 229 Periodic current, Power of, non-inductive conductors, 228 Period of the current, Variable, 117 Period of the current, Variable; applica- tion of Ohm's law in but slightly con- ductive bodies, 118 Permeability, magnetic, Measurement of; magnetometric method, 236 Permeability, magnetic, Measurement of; methods based on induction, 233-235 Permeability, magnetic, Measurement of; method by portative power, 237 Permeability, magnetization and magnetic induction, Another way of looking at,6o Pile, Thermo-electric, 134 Pirani's modification of Maxwell's method of measuring a self-induction in terms of a capacity, 239 Phenomena of electrification, 76 Phenomena which accompany the propaga- tion of current in a conductor, 217 Plate condenser, 94 Points, electrified, Discharging power of, 89 Pole, magnetic, Work due to the displace- ment of an electric circuit under the action of, 140 Pole, magnetic, Work due to the displace- ment of an element of current under the action of, 139 Pole, Unit, magnetic, (defined), 35 Portative power of a magnet, 56 Potential (defined), 12 44 constant at all points within a spherical shell, 26 * electric, (defined), 83 44 of conductor in equilibrium, 84 44 Electric, of the earth, 85 ** energy of masses subjected to Newtonian forces, 25 ** in a conductor. Means for keeping up a constant difference of, 113 *' due to infinitely thin disc uni- formly charged, 31 " Magnetic, due to a circular elec- tric current, 147 *' Magnetic, due to an electric cir- cuit, 141 Potential Magnetic, due to an infinite recti- linear current, 146 Power and torque tests for hysteresis, 69 " of periodic current, 227 " inductive con- ductors, 229 44 " periodic current, non-inductive conductors, 228 14 * l continuous current, Measure- ment, 225 44 " points on electrified conductor, 89 44 Portative, of a magnet, 56 Powers, Thermo-electric, 133 Practical and C. G. S. systems, Relation be- tween, 177 *' system of units, 177 " units, Nomenclature of, 178 Present views on the propagation of elec- tric energy, 224 Pressure, Electrostatic, 87 Proof of Coulomb's law, 49 Propagation of current in a conductor, Phenomena which accompany, 217 Propagation of electric energy, Present views of, 224 Properties of bodies in a magnetic field, Modifications of, 167 Q Quadrant electrometer, 47 charged at high potential, 102 " " Theory of, 101 Quantity of induced electricity, 189, 203 Quantity of electricity induced, Measure- ment of intensity of magnetic field by, 192 R Radians, 50 Ratio " z/," Value of, 176 " Rational " system of units, 179 Rayleigh and Maxwell's method for meas- uring self-induction in terms of a re- sistance, 238 Reactance and inductance, 215 Reactions produced in a circuit traversed by a current, 160 Recalescence, 63 Rectilinear current, infinite, Magnetic field due to, 136 44 infinite, Potential due to, 146 Relation between practical and C. G. S. sys- tems, 177 INDEX. 39* Relation between the directions of current and rotation of magnetic north pole, i35. l8 5 Relative energy of two currents, 143 " " " " magnetic shells, 46 Reluctance or magnetic resistance^ 164 Repulsion exercised by an inducing current on an induced current, 212 Residual discharge of a condenser, 106 Resistance, Magnetic, or reluctance, 164 Reversing a current, Rotation produced by, 158 Ring-shaped bobbin, 153 Rotation, Electromagnetic, 154 '* of a brush discharge by a magnet, 156 " of a current by a magnet, 155 " produced by reversing a current, iS3 " under the action of induced cur- rents, 209 Rule, Faraday's, for energy of current in magnetic field, 145, 186 " Maxwell's, for energy of a current, in a magnetic field, 142 Rule, Maxwell's, for relation between the directions of current and rotation of magnetic north pole, 135, 185 Screen, Electric, 90 Seat of electromotive force of induction, 187 Seebeck effect, 130 Self-induction and capacity, Combined effects of, on a circuit traversed by al- ternating currents, 221 Self-induction and capacity of a circuit, Comparative effects of the, 219 Self-induction, Ayrton and Perry's method for comparing with a standard coil, 240 Self-induction in a circuit composed of linear conductors, 194 in terms of a capacity, Max- well's method for measuring, 230 Self-induction in terms of a resistance, Max- well and Rayleigh's method for measur- ing a coefficient of, 238 Self-induction in the mass of a cylindrical conductor, 208 Self-induction of a circuit composed of linear conductors, where there is a peri- odic or undulatory electromotive force, 198 Self-induction of a circuit, 163 Shallenburger's meter, 211 Shell magnetic, Energy of, in field, 45 Shell magnetic, Work done on unit positive magnetic mass in passing from one side to the other of, 44 Shells magnetic, Relative energy of two, 46 " " (defined), 43 " " mentioned, 42 " " potential due to, 43 Shunt, Galvanometer, 149 Siemens unit, 180 " wattmeter, 226 Sine curves, Equivalent, 74 Slightly conductive bodies, Application of Ohm's law to the variable period of current in, 118 Solenoid, Cylindrical, 151 Solenoidal or magnetic filament, 41 Spark, Electric, no Special characteristics shown byalternating currents (mode of combining), 218 Specific inductive capacity of dielectrics, 103 Sphere, homogeneous, Action of, upon ex- ternal point, 28 Sphere, homogeneous, Action of, upon in- ternal point, 29 Sphere, homogeneous, Surface pressure on, 3 Spherical condenser, 93 magnet, 53 Static electricity, Experiments with, 80 Standards of measure, Electrical, 180 Successive contacts, Law of, 112 Surface density of magnetism, 40 " distribution of electric charge in equilibrium, 81 " pressure on homogeneous sphere, 3 System of units, C. G. S., 171 Practical, 177 "Rational," 179 Systems of units in terms of K and /x, 173 Table of coefficients of hysteresis, 71 " " values of B and /x, 62 " " " " coefficients of hysteresis, 62 Tangent galvanometer, 147 Temperature, Effect of, on magnetism, 63 Terrestrial magnetic field of force, 38 Theorem, 19 " Coulomb's, 86 " Gauss', 20 " corollary I., 21 11,22 III, a* 392 INDEX. Theory of units, 169 " " quadrant electrometer, 101 Thermal electromotive forces, 113 Thermo-electric action, Laws of, 132 pile, 134 " powers, 133 Thomson galvanometers, 148 " Sir Wm., Objective method, 50 Time-const-nt, 194 Transmission of electric waves in the sur- rounding medium, 223 Tube of electric force, Corresponding ele- ments of, 88 " " force, 17 " " " unit, 24 U Uniform magnetic field, Movable conductor in, 190 " magnets, 42 Unit, British Association, 180 " Siemens, 180 " magnetic pole, (defined), 35 Units, C. G. S. system of, 171 " derived, 2 " example of, 3 " dimensions of, 4 " mechanical, 5 " multiples and submultiples of, 7 " C. G. S. system defined, 2 " Dimensions of, 170 " General considerations regarding, 172 Units, General theory, 169 practical, Nomenclature of, 178 proposed, Nomenclature of, 174 Practical system of, 177 " Rational " system of, 179 " Systems of, in terms of JiT and p, 173 Value of the ratio " z/," 176 Variable period of the current, 117 Voltaic cell, 129 W Wattmeter, Siemens, 226 Waves, electric, Transmission of, in the surrounding medium, 223 Weber's hypothesis, 39 Ewing's addition to, 64 Wheatstone's bridge, 123 Wheel, Barlow^, 157 Work absorbed in magnetization, 193 " accomplished on closing a circuit, 195 Work done on unit positive magnetic mass in passing from one side Jof magnetic shell to the other side, 44 Work due to the displacement of a circuit under the action of a pole, 140 Work due to the displacement of an element of current under the action of a pole, 139 Work spent in magnetizing, 61 McGraw Publishing Co., Publishers, Importers and Booksellers, 114 Liberty Street, ' :: :: New York City. 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American Plumbing Practice Descriptive articles upon current practice with questions and answers regarding the problems involved. 268 pages, 536 illustrations 2 50 American Steam and Hot-Water Heating Practice Descriptive articles upon current practice with questions and answers regarding the problems involved. 268 pages. 536 illustrations 3 oo American Street Railway Investments Financial data of over 1,300 American city, suburban and inter- urban electric railways, statistics of operation, details of plant and of equipment, and names of officers. Published annually. Vol. X, 1904 edition $5 oo We can still supply Vols. I (1894) to X (1903) at $5.00 each. Badt, F. B. Incandescent Wiring Handbook With 42 illustrations and 5 tables. Fifth edition i oo New Dynamo Tenders' Handbook With 140 illustrations, 226 pages i oo Bell Hangers' Handbook With 98 illustrations. Third edition. 105 pages i oo Baldwin, William J.-Hot-Water Heating and Fitting 385 pages, 200 illustrations 2 50 Baum, F. G. 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