The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 20 of 35

form. In order to obtain a clear conception of (he inter- relation of the idle wheels or the electric particles and the revolving cells or lines of induction, we may construct a mechanical illustration of one element of the mechanism as it is supposed to exist in the dielectric.

Consider A and B (see Fig. 130) to he two wheels of india- rubber, and that C is another small wheel lying between A and B and transmitting motion from one to the other. Let C be tethered to a fixed point, D, by an elastic spring, and let C be at the same time capable of rotation round its centre. Suppose A is set in rotation, clock-hand wise, whilst B is held fast, and that the wheel C cannot slip on A, the result will be to drag down C to the position of C1, stretching the spring and displacing C. Let B be then set free ; the wheel C continues to roll on A, and transmits its rotation to B. Owing to the

FIG. 130.

assumed elasticity of the discs A and B, the wheel C can be drawn down between them, and yet within the limits of its displacement equally transmit the rotation of A to B without slip. The same action of a preliminary displacement of C and subsequent rotation of B will take place if the wheel B possesses inertia — that is, if we assume it to be a heavy wheel which cannot in virtue of its mass be set rolling with finite speed in an infinitely short time.

If, then, we suppose a long row of such wheels with inter- mediate displaceable idle wheels, the main wheels being heavy bodies, the result of causing the first wheel to rotate would be to propagate along the line a successive displacement of the idle wheels, and to set the main wheels successively in rota-

DYNAMICAL THEORY OF INDUCTION. 343

tion. Translating these mechanical concepts into their elec- trical equivalents, Maxwell considers that the heavy wheels are the analogues of the molecular vortices or lines of force, and that their density is determined by what we call the magnetic permeability of the medium ; the elastically dis- placeable idle wheels are the electricity in the dielectric ; and that when a line of force is brought into existence in a dielectric, or, in other words, when a line of cells is set rotating, this action propagates itself outwards, producing successive displacements of the electric particles, or generates a displacement wave, and is accompanied by the successive appearance of rotation in the cells, or by the propagation of a wave of electromagnetic force.

The velocity of propagation of this wave will depend on the elastic forces restraining displacement, and on the inertia of the revolving vortices. We have seen that the elasticity of

the dielectric is expressed by the quantity 7r> where K is the

K

specific inductive capacity. We shall see later on that the electromagnetic density of the medium is expressed by 4ir/i, where //, is the magnetic permeability.

The velocity of propagation of a disturbance through an elastic medium is numerically equal to the quotient of the square root of its effective elasticity e, by the square root of

its density d, or by v. v d

If, then, for the electromagnetic medium e = — and

1

d = 4:r p., we have v = , , or the velocity of lateral propaga-

tion of a wave of electric displacement or of magnetic force in a medium is numerically equal to the square root of the reciprocal of the product of its specific inductive capacity and its magnetic permeability. Such a mechanical hypothesis shows us how the spin of one line of vortices results in pro- ducing displacement of the idle wheels or electricity along lines which are circles described round the initial vortex as axis, and in propagating outwards the vortex spin or mag- netic force with a finite velocity from one line of molecular vortices to another.

344 DYNAMICAL THEORY OF I

By the aid of the ideas which were discussed in the last section we are enabled to arrive at a mechanical conception which helps us to connect together observed facts, and which, even if not a real representation of what is taking place, is at least a working model, which may assist us to correlate the actions taking place when an electric current is started in a wire.

An electric current on this hypothesis is a flow or pro- gression of the electric particles which are free to move forward in a conductor, and which only can move steadily forward, owing to their incompressibility, when the circuit in which they flow is a complete circuit. Suppose a thin con- ductor bent into the form of a very large circle, and that an electromotive force urges a procession of electric particles round it. As these particles go forward they cause the electric cells next them to rotate, and the motion of this line of cells embracing the line of current will be just like that which would take place if a bracelet of spherical beads strung on an elastic thread were rolled along a round rod which it closely embraces. Each bead would turn over and over, rolling on the rod, and the motion of the whole bracelet would be like that of a tightly-fitting india-rubber umbrella ring pushed along a round ruler. The progression of the electric particles would start circular vortex rings revolving round the line of motion. This corresponds to the fact that a linear current creates a magnetic field composed of circular embracing lines of forces. The first or adjacent line of vortices would, by the intervention of the idle wheels, set in rotation another set of cells lying on a concentric line, and cause them to rotate in the same manner as the first ones. Also, it would cause a back- ward displacement of the intermediate idle wheels, if we con- sider that only the central line of electric particles are conduct- ing matter, and that the next and all succeeding rows are in a dielectric. The starting of the progressive movement of the line of electric particles in the conductor will result in an elastic displacement in the opposite direction of all surrounding electric particles in the dielectric along lines parallel to the line of current ; and also in setting up a system of molecular vortices composed of revolving cells, the axes of these vortices being co-axial circles described round

DYNAMICAL THEORY OF INDUCTION. 345

the line of flow, the rotations and displacements being propa- gated out laterally from the line of current. In consequence of the fact that the revolving cells are supposed to possess inertia or mass, and that all the mechanism is supposed to be rigidly connected together, a steady force applied to set the central line of electric particles in motion will not be able to produce in them the full velocity until time has elapsed suffi- cient to allow the inertia of the connected mechanism to be overcome. We are thus able mechanically to imitate the phenomena of self-induction of the circuit and the gradual rise of current strength in an inductive curcuit under the operation of a steady impressed electromotive force, and to deduce it as a consequence of the fundamental hypothesis.

Oar theory, then, points out that a current should rise gradually in strength, and also that the embracing lines of

B

magnetic force must be considered to come into existence successively as the rotation is taken up in ever-widening circles by the molecular vortices successively receiving motion of rotation. Also, on withdrawing the impressed electromotive force the inertia of the mechanism tends to make it run on for a little and the electric particles, which by their motion started the vortices, are now themselves urged forward for a little in the same direction, and this constitues the extra current at "break."

' Let us next endeavour to see what ought to happen on the supposition that there are two conducting circuits in the field, both forming closed circuits, and to one of which an impressed electromotive force can be applied. LetV1,V2,Vs,&c. (Fig. 131),

346 DYNAMICAL THEORY OF INDUCTION.

represent the sectional view of a series of vortex lines of electric cells, and let I1? I2, 13, &c., be the idle wheels or electric particles. Let the row of electric particles Ia be supposed to be lying inside a conducting circuit, A, represented by the dotted lines, and by our fundamental supposition, the particles Ia are quite free to move along the conductor, and to rotate on their axes. Let there be another conductor, B, placed parallel to A, and let I5 be the electric particles in it. The space C between is supposed to be occupied by a dielectric, and in it the electric particles can only be displaced elastically from a fixed position. We may regard these idle wheels I2 13 14 as tethered by springs- to one spot. Such being the mechanism, imagine that the row of particles Ix is urged forward in a downward direction. As the row of particles pass between the cells Vx V2 they will set them in rotation in opposite directions. Owing to the inertia of the vortices thi first effect of the rotation of V2 will be to cause I2 to roll over V3 and be displaced in an upward direction ; its disph cement is resisted by the elastic force of the spring. The rotation of 1^, however, sets V3 in rotation, and after a short interval V3 is rotating at the same speed and in the same direction as Y2. I2 then ceases to be displaced, because the action of V2 on I2, and the reaction of V3 on 1^ simply amount to a couple or twist on I.,. The same sort of action results in a gradual handing on of the rotation from vortex to vortex, and a propagation of displacement from one idle wheel to the other. When the motion reaches the conductor B, the first result is to cause a displacement of the electric particles upwards, the rotation of V5 not being instantly acquired by V6. This amounts to a current in the upward or opposite direction. As soon, however, as the vortex V6 has accepted the full speed of rotation, then the forces on the electric particles I5 amount only to twists, and not to forces of displacement ; hence the particles I5 cease to experience any forca impelling them forward, and come to rest in virtue of the fact that the conductor offers a resistance to their motion. They fritter down their energy of motion into heat, and come to rest. Hence the induction current in the conductor after a short flow ceases, and the vortex spin becomes equal in the vortices on either side of it. Suppose now that the impressed force in the circuit A is withdrawn, the electric-

DYNAMICAL THEORY OF INDUCTION. 347

particles in the A circuit are driven forward for a short time by the energy stored up in the adjacent vortices ; these last, however, give up one by one their energy to the circuit A, where it is dissipated as heat. This surrender of velocity is propagated outwards until at the surface of the circuit B the state of things finally is, that when the vortex V5 has come nearly to rest, the motion of V6 still continues. The energy of V6 and of vortices beyond expends itself in moving forward the electric particles in circuit B in the same direction as that in which the current in A was travelling originally — in other words, part of the energy of the field is spent in making a transitory current in B as well as in A in the same direction. It follows, therefore, that there is a less induction current in A at breaking circuit when a closed circuit B is present than if B were not there — that is to say the presence of a closed secondary circuit B diminishes the self-induction of the primary circuit, as is known to be the case. We see, therefore, that the theory is so far in accordance with observed facts.

The theory must, however, be taken for no more than it is worth, viz. : an attempt to construct a mechanical system which shall act in the manner in which we find electro-magnetic fields and circuits do act. The true mechanism may be very different; the one described * has at least the utility that it shows a way in which the observed effects might be produced. The various dynamical elements in the supposed mechanism have their equivalents in the recognised electrical and electro-magnetic qualities. The angular velocity of the cells or vortices around their axis represents the intensity of the mag- netic force, or the strength of the magnetic field. The angular momentum of the vortices represents the magnetic induction, hence the mass of each cell, or the density of the medium, is the analogue of the magnetic permeability. This is greater in paramagnetic substances than in air or vacuum, and greatest of all in iron ; in fact, so exceptional is it in iron that Maxwell supposed the particles of the iron themselves to take part in the vortex action. Hence, the energy of a magnetic field is greater if that field contain iron, and accordingly the presence of iron in a core immensely increases the vortex energy for a given vortex velocity, that is, it increases the inductance of

348 DYNAMICAL THEORY OF INDUCTION.

the circuit. The energy associated with any revolving cell or vortex is proportional to the product of its velocity and momentum, or the product of the magnetic force, and the magnetic induction estimated in the same direction is a measure of the energy per unit of volume existing in that portion of the field. The " number of lines of force " passing through any circuit is on this theory to be identified with the whole momentum of the molecular vortices linked with that circuit. If any circuit is traversed by lines of force or linked with lines of molecular vortices, and the cause creating this field is removed, say, by withdrawing the magnet or repressing the electric current creating it, the vortices give up their energy gradually to this secondary circuit, and it appears there as energy of motion of the electric particles or as an electric cur- rent. When one system of bodies iu motion sets another set in motion by mutual action and reaction, and there is no loss of energy by anything like friction or imperfect elasticity, then the momentum gained by one must be equal to thai lost by the other, and the rate of gain of momentum of the one system is at any instant equal to the rate of loss of momentum by the other. Hence, if the vortices lose momentum their rate of loss of momentum — that is, the rate of withdrawal of lines of induction from the circuit, must he equal to the rate of gain of momentum of, or to the force acting on, the electric particles which are absorbing the momentum. Hence we see that the impressed electromotive force in the circuit must be equal to the rate of withdrawal of lines of induction, and the theory conducts us to Faraday's law of induction, as a necessary dynamical consequence of our fundamental assump- tion. Maxwell has extended the theory of molecular vortices to the explanation of electrostatic phenomena, with which we are not, however, here directly concerned. We have seen that the theory is capable of affording an explanation on mechanical principles, of self-induction, mutual induction, and the law of electro-magnetic induction. In order to complete the theory as far as regards the phenomena of magnetism, it is necessary to suppose that the particles of magneti sable metals, such as iron, are set in rotation by the molecular vortices which traverse them, and that an increase of speed of these vortices does not increase proportionallv the rotation of

DYNAMICAL THEORY OF INDUCTION. 349

the iron molecules. These last behave like wheels slung loooely on a shaft, between which shaft and the wheel there is friction decreasing as the speed of rotation of the shaft increases. If, then, the wheel experiences a constant fric- tional resistance from external causes, indefinite increase of speed of the shaft would accelerate the wheel's rotational velocity up to a certain point, and the wheel would then cease to rotate. This supposition would enable us to make our theory agree with the fact that increase of magnetic force does not increase indefinitely the magnetic induction through iron, but brings is up to a point at which, approximately speaking, the induction remains stationary. To sum up, we- may say that the hypothesis of molecular vortices is an endeavour to imagine a mechanism capable of accounting for electro-magnetic induction on dynamical principles, and on the assumption that the energy of a magnetic field is energy stored up in a medium in virtue of a particular kind of rotation of its parts.

This medium consists of portions capable of elastic displace- ment when we consider parts of it lying in dielectrics or capable of progressive movement when in conductors, and these portions constitute what we call electricity. Other por- tions are capable of rotation round closed axes of rotation, and these constitute what we call " lines of force." The medium possesses, therefore, an elastic resilience, and the reciprocal of this quality, or its freedom of yielding to electromotive force, is recognised as the specific inductive capacity. The medium possesses also density, and we call this its magnetic permeability, or magnetic inductance. The mass of unit of length of the vortices is equal for all vortices, whether in vacuum, air, or non-magnetic bodies, but in iron the vortices are loaded by the adhesion to them of the molecules of the metal, and the density is increased, and hence the permea- bility ; but for very great angular velocities — that is, for great, magnetic forces — the adhesion of the molecules and vortices must be supposed to cease, and the permeability approximates to unity. The magnetic force at any point in a field is the angular velocity of the vortex motion at that point, and the magnetic induction is the angular momentum. Magnetic attraction and repulsion is due to the tension set up along a

350 DYNAMICAL THEORY OF INDUCTION.

vortex line by the polar contraction and equatorial expansion of the vortex cells. At places where there is magnetic polarity or free magnetism there is a discontinuity in the angular velocity of the vortices within and without the iron. Self-induction is the result of the inertia of the molecular vortices, whereby motion set up in them cannot be generated or checked instantaneously. Mutual induction, or the pro- duction of induction currents, is due to the fact that differences in the angular velocity of adjacent vortex filaments or cells cause a displacement of the electric particles or idle wheels. Finally, electromotive force is the force causing displacement of the electric particles, and electric currents consist in con- tinuous or periodic movements of these electric particles. Electric currents always produce magnetic fields because there is nothing of the nature of slip between the particles and cells, and, therefore, any progressive movement of the first sets up rotation in the second, and conversely differential rotations or spins of the cells or vortices sets up displacement of the electric particles, causing either electric strain in a dielectric or electric current in a conductor.

§ 4. Comparison of Theory and Experiment. — The test of any physical theory is its power to predict new phenomena .as well as to interpret ascertained experimental results. The theory of molecular vortices leads to the conclusion that electro-magnetic induction must be propagated through the medium with a finite velocity, and that in dielectrics of unit permeability the velocity of propagation is inversely as the square root of the specific inductive capacity. In the -dynamical theory of light it is shown that the ratio of the velocity of light in vacuo to its velocity in any given transparent medium is a constant quantity for each definite wave length, and is called the index of refraction of that body for that wave length, and is denoted in physical optics by the symbol p. Hence, the velocity of light of definite wave-length is inversely as the refractive index for that wave-length. The refractive index for very long wave-lengths can be calculated from observed values of /* for definite rays, and hence numbers obtained representing the relative velocity of these undulations in various transparent bodies. The values of the dielectric

DYNAMICAL THEORY OF INDUCTION.

351

constants, or reciprocal of the electric elasticities, of various transparent and semi-transparent bodies have also been deter- mined, and it has been found that for a large group of bodies there is a tolerably close agreement between the values of the square root of the dielectric constant and the index of refraction /tx for very long waves, as shown by the selection from the results of some experimental determinations given in Table A.

Table A.

K

(Dielectric Constant).

-v/K

fJ.-X>

(Refrac- tive Index).

Authority.

Reference.

Sulphur 3'84

1-96

2-041

Colophonium.. .. 2'55 Paraffin 2'32

1-59 1-52

1-54 I 1-54

Boltzmann

/ Pof/^.Ann.,CLl., \ Ib74, p. 482.

Pure rubber 2-12

1-45

1-50

Schiller

/ Pony. Ann., \ CLII.,p.535.

Oil of turpentine 2'21 Petroleum 2'037

1-49 1-43

1-461 T46 I

Silow

( Por/g. Ann., I CLVI., 1875,

Benzine 2198

1-48

1-43 j

1 p. 395.

Petroleum spirit T92 Petroleum oil ...2'07 Ozokerite 2'13

1-38 1-44 i-46

1-38 \ 1-44 1-44 f

J. Hopkinson

(Trans. Roy. Soc, 1877, 1878 and

Turpentine 2'23

1-49

1-4&J

For some other dielectrics, such as glass and the vegetable and animal oils, the agreement is not by any means so close but for gases, as determined by Boltzmann (Pogy. Ann., CLL, 1875, p. 403), there is a fair coincidence. (See Table B.)

Table B.

Gas.

Air

! r00059

1-000295 1-000294

1-000946

1-000473 1-000 149

1 1-000264

1-000132 1-000138

Carbonic oxide

| 1-OOC69C

1-000345 1-000340

i 1-000994

1-000497 1-000503

1 1-001312

1-000656 1-000678

Marsh gas

! 1-000944

1-000472 1-000443

The gases are taken at 0°C. and 760 millimetres pressure. Accordingly, we can say that, for a large group of dielectrics, of which the magnetic permeability is unity, and hence the velocity of propagation of an electro-magnetic impulse proper-

352 DYNAMICAL THEORY OF INDUCTION.

tional to the square root of the electric elasticity or to the reciprocal of the square root of the dielectric constant, we do find a fair agreement between these numbers and the numbers representing the refractive indices or the relative velocities of propagation of very long waves or disturbances in the ethereal medium postulated to account for the phenomena of light. The imperfect agreement between the values of the refractive index for long wave-lengths and the square root of the dielectric constant for some other bodies shows that the theory is only approximately in agreement with fact, and that the results obtained by the methods adopted for deter- mining the dielectric constant are perhaps impure, and do- not give the true value of the electric elasticity. When we consider that the displacements which constitute the light wave motion of the luminiferous ether are changed some billions of times per second, it is seen to be highly prob- able that measurements of the specific inductive capa- city in which the electric stresses are only reversed tens or hundreds of times in a second may be rendered impure or mixed owing to the presence of effects due to an imperfect electric elasticity introduced by the superposition of electric conduction or of electrolytic transport upon the true or elastic displacement effect. In fact those bodies, such as glass and the vegetable oils, which exhibit the greatest discrepancy, are those in which the chemical composition indicates a possibility of electrolysis. There may be an electro displacement in such electrolisable bodies over and above the true electrostatic displacement which is engendered by a molecular change in the body, which change results in actual decomposition when the electric force reaches a certain limit. Put broadly, it may amount to this, that the true electric displacement is a displacement of electricity within the molecule, but that in electrolisable bodies electric stress sets up a strain of the molecule itself which, within certain limits, is an elastic- strain, and disappears with the removal of the stress, but that beyond these limits molecular disruption takes place. In these cases the displacement measured in taking the specific inductive capacity is the true or dielectric displacement plus a displacement due to strain of the molecule, and the result would be to make K appear too great, and, in fact, for glass.

DYNAMICAL THEORY OF INDUCTION. 353

and certain oils the values in Table C have been obtained, which in all cases are such that /K" exceeds the value of IJL^ , or the refractive index, for very long waves of light.*

Table C.

Substance.

K

x/K

Vx (approx.)

Glass, extra dense flint
light flint .

9-896 6-72

31

2-59

1

6-96

2'63

}• 1-5 to 1-6

8-45

2'90

J

Castor oil ...

4-78

2-18

T46

3-02

1-73

1'46

Olive oil

3-16

1-77

1'46

3-07

1'75

1'45

J. Klemencic (abstract in the Journal of the Society of Tele- graph Engineers, 1886, p. 108) has experimented also on the specific inductive capacity of gases and vapours, and given a table (see Table D) in which he compares /K with /* (refrac- tive index) of these same bodies. It is seen that the agree- ment of vK and /* is very close for the simple gases, but that a marked difference exists in the case of more complicated molecules.

Table D.

Gas.

v/K

Boltzmann.

N/K Klemencic.

H Refractive index.

Air

1-000295 1-000132

1-000293 1-000132

1-000295 1-000139

1-000473

1-000492

1-000454

Carbonic oxide

1-000345

1-000347

1-000335

Nitrous oxide

1-000497

1-000579

TOO 516

1-000656

1-000729

1-000720

Marsh gas
Carbonic bisulphide ..

1-000472

1-000476 1 001450

1-000442 1-001478

1-OC477

1-000703

Ether

1-00372

1-00154

Ethyl chloride

1-00776

1-001174

1-00773

1-00122

The specific inductive capacity of a vacuum is taken as unity, and Boltz- mann's values are given for comparison.

• See Dr. J. Hopkinson, Phil. Trans. Royal Society, Vol. CLXXIL, 1881, p. 372.

A A

S54 DYNAMICAL TBEOKY OF INDUCTION.

§5. Velocity of Propagation of an Electromagnetic Dis- turbance.— There is another line of experimental enquiry which leads to an important relation between electric and optic phenomena. This is the comparison of electrostatic and electromagnetic measurements. If two very small spheres are electrostatically charged and placed with their centres at a unit of distance apart, the stress between them may be mechanically measured. If the conductors are equally charged with opposite kinds of electricity, and the stress when at a unit of distance in air is one unit, the electric quantities are said to be unit electrostatic quantities. If such unit quantities are discharged through a conductor at the rate of one discharge per second, the resulting flow or current is called an electrostatic unit of current.

In the above definition we suppose the dielectric to be a vacuum or some substance such as air, of which the dielectric constant does not differ sensibly from unity. If q and ql be two quantities measured electrostatically, and then be placed on small conductors separated by a distance r in a dielectric of constant K, the dynamical force between them will be nu-

numerically equal to <~.i ', and if g = ql, then the force is ^-^.

Hence, if r is always taken equal to unity, the real quantity of electricity producing by its action on another equal quantity a unit of force will vary as the square root of K when the experi- ment is performed in various dielectrics. In other words, the absolute magnitude of the electrostatic unit of quantity, and therefore also of the current, will vary as the square root of the specific inductive capacity of the medium in which the charges exist. There is another mode in which a unit of current may be defined, and this depends on the definition of a unit magnetic pole. If two magnetic poles of equal strength, 7??, are placed at a distance r apart in a magnetic medium of permeability p, the

stress or force between them will be numerically equal to ^—,

^7-2

in which expression it is seen that m and p. appear as quantities analogous to q and K in the electrostatic analogue. Hence, when r is unity, we see that to produce a unit stress between the poles m the pole strength must vary as the square root of /*, or the absolute magnitude of the unit magnetic pole varies directly

DYNAMICAL THEORY OF INDUCTION. 355

as the square root of the magnetic inductive capacity of the medium in which the experiment is performed, the absolute unit magnetic pole being denned as a pole which at a unit of distance acts on another like pole with a unit of force in a magnetic medium, assumed to be vacuum, or some standard substance of unit permeability.

Since an electric current produces a magnetic force, it may be defined as to magnitude by agreeing that the unit of current is to be one which, when flowing in a circular circuit of unit radius, acts for every unit of length of that circuit with a unit of force on a unit magnetic pole placed at the centre of that circle. The magnitude of the force on the magnetic pole is proportional to the product of the strength of the pole and the strength of the current. Hence, if the magnitude of the unit pole is varied the magnitude of the unit of current will vary inversely as the magnitude of the strength of magnetic pole which is taken as the unit pole. When the medium is varied, the magnitude of the unit magnetic pole, or of the pole which fulfils the condi- tion of acting on another equal pole at a unit of distance with a unit of force varies directly as the square root of the permea- bility of the medium. It follows, then, that the magnitude of the electro-magnetic unit of current varies inversely as the square root of the magnetic permeability of the medium in which the experiment is made.

We have, then, that the electrostatic unit of current is a quantity which varies directly as the square root of the electro- static inductive capacity of the medium, or as /K, and the electromagnetic unit of current is another unit of current which varies inversely as the square root of the magnetic induc- tive capacity of the medium, or as >//*. The electrostatic unit of current represents a much smaller quantity of electricity per second than the electro-magnetic — in other words the value of the ratio of the magnitude of the unit electro-magnetic current based on the definition of a unit magnetic pole, to the magnitude of the unit electrostatic current, based on the definition of a unit of electrostatic quantity, is an integer number, and a large one. This ratio of the two units of current varies when the fundamental inductive capacities of the medium is changed, but so that the ratio of the electro- magnetic to electrostatic unit varies inversely as the square

AA2

356 DYNAMICAL THEORY OF INDUCTION.

root of the product of K and p. If C,n is the magnitude of the electro-magnetic unit of current, and Cg is that of the electro-

static unit for the standard dielectric, in which K = 1 and /* = !,

r\ then, when the dielectric is. changed, -^ is changed in the ratio

Cg

of 1; «/K p. Let Ewac denote the value of the ratio for vacuum or for a standard dielectric, of which K = 1 and p. = 1, and EOT denote its value for any other medium of which the dielectric constant is K and the magnetic constant /x, then

We have next to consider what is the physical meaning of this ratio of the electro-magnetic and electrostatic units.

The degree in which one quantity is greater or less than another, or to put it more precisely, that amount of stretching or squeezing which must be applied to the latter in order to produce the former, is called the ratio of the two quantities.* The ratio of two physical quantities is therefore the expres- sion of the operation which must be performed on the one to make it the physical equivalent to the other. What operation must be performed on an electrostatically measured unit of electricity to make it the equivalent in every way of an electro- magnetically measured unit of electricity? The reply is, it must be set in motion with a definite velocity. The electric current produces a magnetic field. The electro-magnetic mea- sure of current is obtained by defining the field by stating its dynamical effect on a defined magnetic pole, and the unit of electric quantity measured electro -magnetically is the quantity conveyed by the unit current so measured in a unit of time. If we imagine a circular or other conductor conveying a unit (electro-magnetic) current to have stretched alongside of it another closely adjacent conductor of like form, each unit of length of which is charged electrostatically with a unit (electro- static) of electric quantity, we might submit the following question : — The current flowing in the first named conductor transmits a unit (electro-magnetic) quantity of electricity across each section of it per unit of time : with what velocity must electricity in the second conductor be set flowing in order that

• W. K. Clifford, " The Common Sense of the Exact Sciences," p. 99.

DYNAMICAL THEORY OF INDUCTION. 357