A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 14 of 27

THEORY OF ELECTRIC CIRCUITS.

578.] WE may now confine our attention to that part of the kinetic energy of the system which depends on squares and products of the strengths of the electric currents. We may call this the Electrokinetic Energy of the system. The part depending on the motion of the conductors belongs to ordinary dynamics, and we have shewn that the part depending on products of velocities and currents does not exist.

Let Al, AD &c. denote the different conducting circuits. Let their form and relative position be expressed in terms of the variables a?!, #2, &c., the number of which is equal to the number of degrees of freedom of the mechanical system. We shall call these the Geometrical Variables.

Let j/x denote the quantity of electricity which has crossed a given section of the conductor A1 since the beginning of the time t. The strength of the current will be denoted by y^, the fluxion of this quantity.

We shall call y^ the actual current, and y^ the integral current. There is one variable of this kind for each circuit in the system.

Let T denote the electrokinetic energy of the system. It is a homogeneous function of the second degree with respect to the strengths of the currents, and is of the form

T=±Llyl* + ±L2^+&c. + Ml2yly2 + &c.) (1)

where the coefficients L, M, &c. are functions of the geometrical variables #15 #2, &c. The electrical variables yl} y2 do not enter into the expression.

We may call Llt I/2, &c. the electric moments of inertia of the circuits Alt A2, &c., and M12 the electric product of inertia of the two circuits A^ and A2 , When we wish to avoid the language of

579-] ELECTROKINETIC MOMENTUM. 207

the dynamical theory, we shall call L^ the coefficient of self-induction of the circuit Alt and M12 the coefficient of mutual induction of the circuits A1 and A2. MlZ is also called the potential of the circuit A^ with respect to Az. These quantities depend only on the form and relative position of the circuits. We shall find that in the electromagnetic system of measurement they are quantities of the dimension of a line. See Art. 627.

By differentiating T with respect to y± we obtain the quantity _p1 , which, in the dynamical theory, may be called the momentum corresponding to y±. In the electric theory we shall call p± the electrokinetic momentum of the circuit A1 . Its value is

Pl = A ^1 + ^12^2 + &C"

The electrokinetic momentum of the circuit A1 is therefore made up of the product of its own current into its coefficient of self- induction, together with the sum of the products of the currents in the other circuits, each into the coefficient of mutual induction of A1 and that other circuit.

Electromotive Force.

579.] Let E be the impressed electromotive force in the circuit A, arising from some cause, such as a voltaic or thermoelectric battery, which would produce a current independently of magneto-electric induction.

Let R be the resistance of the circuit, then, by Ohm's law, an electromotive force Ey is required to overcome the resistance, leaving an electromotive force E — Ry available for changing the momentum of the circuit. Calling this force Y'9 we have, by the general equations, dp dT

JL = -j- -- ^— >

at ay

but since T does not involve y, the last term disappears. Hence, the equation of electromotive force is

or - =,+ •

The impressed electromotive force E is therefore the sum of two parts. The first, JRy, is required to maintain the current y against the resistance R. The second part is required to increase the elec tromagnetic momentum p. This is the electromotive force which must be supplied from sources independent of magneto-electric

208 LINEAR CIRCUITS. [580.

induction. The electromotive force arising from magneto -electric induction alone is evidently — -j-, or, the rate of decrease of the

(A' v

electrokinetic momentum of the circuit.

Electromagnetic Force.

580.] Let X' be the impressed mechanical force arising from external causes, and tending to increase the variable x. By the general equations ^ d dT dT

dt dx dx

Since the expression for the electrokinetic energy does not contain the velocity (#), the first term of the second member disappears,

and we find ^y

Ji. = -- 7 — •

dx

Here X' is the external force required to balance the forces arising from electrical causes. It is usual to consider this force as the reaction against the electromagnetic force, which we shall call X, and which is equal and opposite to X'.

•v AT

Hence X = -T- >

dx

or, the electromagnetic force tending to increase any variable is equal to the rate of increase of the electrokinetic energy per unit increase of that variable, the currents being maintained constant.

If the currents are maintained constant by a battery during a displacement in which a quantity, W, of work is done by electro motive force, the electrokinetic energy of the system will be at the same time increased by W. Hence the battery will be drawn upon for a double quantity of energy, or 2 W, in addition to that which is spent in generating heat in the circuit. This was first pointed out by Sir W. Thomson*. Compare this result with the electrostatic property in Art. 93.

Case of Two Circuits.

581.] Let AI be called the Primary Circuit, and A2 the Secondary Circuit. The electrokinetic energy of the system may be written

where L and N are the coefficients of self-induction of the primary

• Nichol's Cyclopaedia of Physical Science, ed. 1860, Article, ' Magnetism, Dy namical Relations of.'

582.] TWO CIRCUITS. 209

and secondary circuits respectively, and M is the coefficient of their mutual induction.

Let us suppose that no electromotive force acts on the secondary circuit except that due to the induction of the primary current.

We have then ci

E2 = B2fa+ (My, + Ny2] = 0.

Integrating this equation with respect to t, we have

Ry2 + Hjfi + Ny2 = C, a constant, where y.^ is the integral current in the secondary circuit.

The method of measuring an integral current of short duration will be described in Art. 748, and it is easy in most cases to ensure that the duration of the secondary current shall be very short.

Let the values of the variable quantities in the equation at the end of the time t be accented, then, if y^ is the integral current, or the whole quantity of electricity which flows through a section of the secondary circuit during the time t,

If the secondary current arises entirely from induction, its initial value jr. 2 must be zero if the primary current is constant, and the conductors at rest before the beginning of the time t.

If the time t is sufficient to allow the secondary current to die away, y£y its final value, is also zero, so that the equation becomes

The integral current of the secondary circuit depends in this case on the initial and final values

Induced Currents.

582.] Let us begin by supposing the primary circuit broken, or y^ = 0, and let a current y{ be established in it when contact is made.

The equation which determines the secondary integral current is

When the circuits are placed side by side, and in the same direc tion, M is a positive quantity. Hence, when contact is made in the primary circuit, a negative current is induced in the secondary circuit.

When the contact is broken in the primary circuit, the primary current ceases, and the induced current is y^ where

The secondary current is in this case positive.

VOL. II. P

210 LINEAR CIRCUITS.

If the primary current is maintained constant, and the form or relative position of the circuits altered so that M becomes M', the integral secondary current is y2, where

In the case of two circuits placed side by side and in the same direction M diminishes as the distance between the circuits in creases. Hence, the induced current is positive when this distance is increased and negative when it is diminished.

These are the elementary cases of induced currents described in Art. 530.

Mechanical Action between the Two Circuits.

583.] Let x be any one of the geometrical variables on which the form and relative position of the circuits depend, the electro magnetic force tending to increase x is

dL . dM . dN

If the motion of the system corresponding to the variation of x is such that each circuit moves as a rigid body, L and N will be independent of %, and the equation will be reduced to the form

dx

Hence, if the primary and secondary currents are of the same sign, the force X, which acts between the circuits, will tend to move them so as to increase M.

If the circuits are placed side by side, and the currents flow in the same direction, M will be increased by their being brought nearer together. Hence the force X is in this case an attraction.

584.] The whole of the phenomena of the mutual action of two circuits, whother the induction of currents or the mechanical force between them, depend on the quantity Jf, which we have called the coefficient of mutual induction. The method of calculating this quantity from the geometrical relations of the circuits is given in Art. 524, but in the investigations of the next chapter we shall not assume a knowledge of the mathematical form of this quantity. We shall consider it as deduced from experiments on induction, as, for instance, by observing the integral current when the secondary circuit is suddenly moved from a given position to an infinite distance, or to any position in which we know that M= 0.

CHAPTER VIII.

EXPLORATION OF THE FIELD BY MEANS OF THE SECONDARY

CIRCUIT.

585.] We have proved in Arts. 582, 583, 584 that the electro magnetic action between the primary and the secondary circuit depends on the quantity denoted by M, which is a function of the form and relative position of the two circuits.

Although this quantity M is in fact the same as the potential of the two circuits, the mathematical form and properties of which we deduced in Arts. 423, 492, 521, 539 from magnetic and electro magnetic phenomena, we shall here make no reference to these results, but begin again from a new foundation, without any assumptions except those of the dynamical theory as stated in Chapter VII.

The electrokinetic momentum of the secondary circuit consists of two parts (Art. 578), one, Milt depending on the primary current ilt while the other, Niz, depends on the secondary current i2. We are now to investigate the first of these parts, which we shall denote by j?, where n _

We shall also suppose the primary circuit fixed, and the primary current constant. The quantity jt?, the electrokinetic momentum of the secondary circuit, will in this case depend only on the form and position of the secondary circuit, so that if any closed curve be taken for the secondary circuit, and if the direction along this curve, which is to be reckoned positive, be chosen, the value of p for this closed curve is determinate. If the opposite direction along the curve had been chosen as the positive direction, the sign of the quantity jo would have been reversed.

586.] Since the quantity p depends on the form and position of the circuit, we may suppose that each portion of the circuit

212 ELECTROMAGNETIC FIELD.

contributes something1 to the value of p, and that the part con tributed by each portion of the circuit depends on the form and position of that portion only, and not on the position of other parts of the circuit.

This assumption is legitimate, because we are not now considering a current, the parts of which may, and indeed do, act on one an other, but a mere circuit, that is, a closed curve along which a current may flow, and this is a purely geometrical figure, the parts of which cannot be conceived to have any physical action on each other.

We may therefore assume that the part contributed by the element ds of the circuit is Jds, where J is a quantity depending on the position and direction of the element ds. Hence, the value of p may be expressed as a line-integral

(2)

where the integration is to be extended once round the circuit. 587.] We have next to determine the form of the quantity «7~.

In the first place, if ds is reversed in direction, / is reversed in sign. Hence, if two circuits ABCE and AECD have the arc AEG common, but reckoned in opposite directions in the two circuits, the sum of the values of p for the two circuits

Fl*g- 35- and AECD will be equal to the value of p for

the circuit AJBCD, which is made up of the two circuits.

For the parts of the line-integral depending on the arc AEG are equal but of opposite sign in the two partial circuits, so that they destroy each other when the sum is taken, leaving only those parts of the line- integral which depend on the external boundary of ABCD.

In the same way we may shew that if a surface bounded by a closed curve be divided into any number of parts, and if the boundary of each of these parts be considered as a circuit, the positive direction round every circuit being the same as that round the external closed curve, then the value of p for the closed curve is equal to the sum of the values of p for all the circuits. See Art. 483.

588.] Let us now consider a portion of a surface, the dimensions of which are so small with respect to the principal radii of curvature of the surface that the variation of the direction of the normal within this portion may be neglected. We shall also suppose that if any very small circuit be carried parallel to itself from one part of this surface to another, the value of p for the small circuit is

589.] ADDITION OF CIRCUITS. 213

not sensibly altered. This will evidently be the case if the dimen sions of the portion of surface are small enough compared with its distance from the primary circuit.

If any closed curve be drawn on this portion of the surface, the value of p will be proportional to its area.

For the areas of any two circuits may be divided into small elements all of the same dimensions, and having the same value of p. The areas of the two circuits are as the numbers of these elements which they contain, and the values of p for the two circuits are also in the same proportion.

Hence, the value of p for the circuit which bounds any element dS of a surface is of the form IdS,

where / is a quantity depending on the position of dS and on the direction of its normal. We have therefore a new expression for p,

(3)

where the double integral is extended over any surface bounded by the circuit.

589.] Let ABCD be a circuit, of which AC is an elementary portion, so small that it may be considered straight. Let APB and CQB be small equal areas in the same plane, then the value of p will be the same for the small circuits APB and CQB, or

p (APB) = p (CQB).

Hence p (APBQCD) = p (ABQCD) + p (APB), = p (ABQCD) + 1

= p (ABCD), Fig. 36.

or the value of p is not altered by the substitution of the crooked line APQCfor the straight line AC, provided the area of the circuit is not sensibly altered. This, in fact, is the principle established by Ampere's second experiment (Art. 506), in which a crooked portion of a circuit is shewn to be equivalent to a straight portion provided no part of the crooked portion is at a sensible distance from the straight portion.

If therefore we substitute for the element ds three small elements, dx, dy, and dz, drawn in succession, so as to form a continuous path from the beginning to the end of the element ds, and if Fdx, G dy, and II dz denote the elements of the line-integral cor responding to dx, dy, and dz respectively, then

Jds = Fdse+ Gdy + Hdz. (4)

214 ELECTROMAGNETIC FIELD. [59°-

590.] We are now able to determine the mode in which the quantity / dep3nds on the direction of the element ds. For,

by (4), f=P%. + 0*+H%. (5)

ds ds ds

This is the expression for the resolved part, in the direction of ds, of a vector, the components of which, resolved in the directions of the axes of x, y^ and z, are F, G, and H respectively.

If this vector be denoted by 51, and the vector from the origin to a point of the circuit by p, the element of the circuit will be dp, and the quaternion expression for / will be

We may now write equation (2) in the form

'

(7)

The vector 51 and its constituents F, G, H depend on the position of ds in the field, and not on the direction in which it is drawn. They are therefore functions of x, y, z, the coordinates of ds, and not of I, m} n, its direction-cosines.

The vector 51 represents in direction and magnitude the time- integral of the electromotive force which a particle placed at the point (x, y, z) would experience if the primary current were sud denly stopped. We shall therefore call it the Electrokinetic Mo mentum at the point (x, ?/, z}. It is identical with the quantity which we investigated in Art. 405 under the name of the vector- potential of magnetic induction.

The electrokinetic momentum of any finite line or circuit is the line-integral, extended along the line or circuit, of the resolved part of the electrokinetic momentum at each point of the same.

591.] Let us next determine the value of p for the elementary rectangle ABCD, of which the sides are dy and dz, the positive direction being from the direction of the axis of y to that of z.

Let the coordinates of 0, the centre of gravity of the element, be a?0, yQ, ZQ, and let -p. 37 GQ> HQ be the values of G and of H at this

point. The coordinates of A, the middle point of the first side of the

MAGNETIC INDUCTION. 215

rectangle, are yQ and ZQ — - dz. The corresponding value of G is

(8)

and the part of the value of p which arises from the side A is approximately i dG

1 rlTT Similarly, for B, H0dz+--^- Ay dz.

For (7, -G,dy-\d^dydz.

For D, — H0 dz + - — Ay dz.

2 cly

Adding these four quantities, we find the value of p for the rectangle m da

If we now assume three new quantities, #, b, c, such that

dH dG i

(A)

a — -= -- -=-9

d dz

dF dH

-j --- j- dz dx

dG dF

7 ~~ 7 '

dx dy J

and consider these as the constituents of a new vector 33, then, by Theorem IV, Art. 24, we may express the line-integral of 51 round any circuit in the form of the surface-integral of 33 over a surface bounded by the circuit, thus

p = F~-^G +H~ds=(la + mb + nc}dS, (11) J ^ ds ds ds' JJ

or p = JT 2t cose ds = f j T<& cos TJ d8, (12)

where e is the angle between 5( and ds, and rj that between 33 and the normal to dS, whose direction-cosines are I, m, n, and T 51, T 33 denote the numerical values of 51 and 33.

Comparing this result with equation (3), it is evident that the quantity / in that equation is equal to 33 cos r;, or the resolved part of 33 normal to dS.

592.] We have already seen (Arts. 490, 541) that, according to Faraday's theory, the phenomena of electromagnetic force and

216 ELECTROMAGNETIC FIELD. [593-

induction in a circuit depend on the variation of the number of lines of magnetic induction which pass through the circuit. Now the number of these lines is expressed mathematically by the surface-integral of the magnetic induction through any surface bounded by the circuit. Hence, we must regard the vector 23 and its components a, b, c as representing what we are already acquainted with as the magnetic induction and its components.

In the present investigation we propose to deduce the properties of this vector from the dynamical principles stated in the last chapter, with as few appeals to experiment as possible.

In identifying this vector, which has appeared as the result of a mathematical investigation, with the magnetic induction, the properties of which we learned from experiments on magnets, we do not depart from this method, for we introduce no new fact into the theory, we only give a name to a mathematical quantity, and the propriety of so doing is to be judged by the agreement of the relations of the mathematical quantity with those of the physical quantity indicated by the name.

The vector 33, since it occurs in a surface-integral, belongs evidently to the category of fluxes described in Art. 13. The vector 51, on the other hand, belongs to the category of forces, since it appears in a line-integral.

593.] We must here recall to mind the conventions about positive and negative quantities and directions, some of which were stated in Art. 23. We adopt the right-handed system of axes, so that if a right-handed screw is placed in the direction of the axis of x, and a nut on this screw is turned in the positive direction of rotation, that is, from the direction of y to that of z, it will move along the screw in the positive direction of x.

We also consider vitreous electricity and austral magnetism as positive. The positive direction of an electric current, or of a line of electric induction, is the direction in which positive electricity moves or tends to move, and the positive direction of a line of magnetic induction is the direction in which a compass needle points with the end which turns to the north. See Fig. 24, Art. 498, and Fig. 25, Art. 501.

The student is recommended to select whatever method appears to him most effectual in order to fix these conventions securely in his memory, for it is far more difficult to remember a rule which determines in which of two previously indifferent ways a statement is to be made, than a rule which selects one way out of many.

594-] THEORY OF A SLIDING PIECE. 217

594.] We have next to deduce from dynamical principles the expressions for the electromagnetic force acting on a conductor carrying an electric current through the magnetic field, and for the electromotive force acting on the electricity within a body moving in the magnetic field. The mathematical method which we shall adopt may be compared with the experimental method used by Faraday * in exploring the field by means of a wire, and with what we have already done at Art. 490, by a method founded on experiments. What we have now to do is to determine the effect on the value of ji, the electroldnetic momentum of the secondary circuit, due to given alterations of the form of that circuit.

Let AA', BB' be two parallel straight conductors connected by the conducting arc (7, which may be of any form, and by a straight

Fig. 38.

conductor AB, which is capable of sliding parallel to itself along the conducting rails AA and BB'.

Let the circuit thus formed be considered as the secondary cir cuit, and let the direction ABC be assumed as the positive direction round it.

Let the sliding piece move parallel to itself from the position AB to the position AB'. We have to determine the variation of _p, the electrokinetic momentum of the circuit, due to this displacement of the sliding piece.

The secondary circuit is changed from ABC to A'IfC, hence, by Art. 587, p (AB'C)-p (ABC) = p (AA'B'B). (13)

We have therefore to determine the value of p for the parallel ogram AA'BB. If this parallelogram is so small that we may neglect the variations of the direction and magnitude of the mag netic induction at different points of its plane, the value of p is, by Art. 591, 33 cos r\ . AA'ffBj where 33 is the magnetic induction,

• Exp. Res., 3082, 3087, 3113.

218 ELECTROMAGNETIC FIELD. [595-

and 77 the angle which it makes with the positive direction of the normal to the parallelogram AA'B'B.

We may represent the result geometrically by the volume of the parallelepiped, whose base is the parallelogram AA'B'B, and one of whose edges is the line AM, which represents in direction and magnitude the magnetic induction 33. If the parallelogram is in the plane of the paper, and if AM is drawn upwards from the paper, the volume of the parallelepiped is to be taken positively, or more generally, if the directions of the circuit AB, of the magnetic in duction AM, and of the displacement AA', form a right-handed system when taken in this cyclical order.

The volume of this parallelepiped represents the increment of the value of p for the secondary circuit due to the displacement of the sliding piece from AB to A'B'.

Electromotive Force acting on the Sliding Piece.

595.] The electromotive force produced in the secondary circuit by the motion of the sliding piece is, by Art. 579,

If we suppose AA' to be the displacement in unit of time, then AA' will represent the velocity, and the parallelepiped will represent

~, and therefore, by equation (14), the electromotive force in the

Ctu

negative direction B A.

Hence, the electromotive force acting on the sliding piece AB, in consequence of its motion through the magnetic field, is repre sented by the volume of the parallelepiped, whose edges represent in direction and magnitude — the velocity, the magnetic induction, and the sliding piece itself, and is positive when these three direc tions are in right-handed cyclical order.

Electromagnetic Force acting on the Sliding Piece.

596.] Let i2 denote the current in the secondary circuit in the positive direction ABC, then the work done by the electromagnetic force on AB while it slides from the position AB to the position A'B' is (M'—M)ili2, where M and M' are the values of M12 in the initial and final positions of AB. But (M'—M)^ is equal to//— p, and this is represented by the volume of the parallelepiped on AB, AM, and AA'. Hence, it' we draw a line parallel to AB

598.] LINES OF MAGNETIC INDUCTION. 219

to represent the quantity AB.i2, the parallelepiped contained by this line, by AM, the magnetic induction, and by A A, the displace ment, will represent the work done during- this displacement.

For a given distance of displacement this will be greatest when the displacement is perpendicular to the parallelogram whose sides are AB and AM. The electromagnetic force is therefore represented by the area of the parallelogram on AB and AM multiplied by ?/2, and is in the direction of the normal to this parallelogram, drawn so that AB, AM, and the normal are in right-handed cyclical order.

Four Definitions of a Line of Magnetic Induction.

597.] If the direction AA ', in which the motion of the sliding piece takes place, coincides with AM, the direction of the magnetic induction, the motion of the sliding piece will not call electromotive force into action, whatever be the direction of AB, and if AB carries an electric current there will be no tendency to slide along AA.

Again,, if AB} the sliding piece, coincides in direction with AM, the direction of magnetic induction, there will be no electromotive force called into action by any motion of AB, and a current through AB will not cause AB to be acted on by mechanical force.

We may therefore define a line of magnetic induction in four different ways. It is a line such that —

(1) If a conductor be moved along it parallel to itself it will experience no electromotive force.

(2) If a conductor carrying a current be free to move along a line of magnetic induction it will experience no tendency to do so.

(3) If a linear conductor coincide in direction with a line of magnetic induction, and be moved parallel to itself in any direction, it will experience no electromotive force in the direction of its length.

(4) If a linear conductor carrying an electric current coincide in direction with a line of magnetic induction it will not experience any mechanical force.

General Equations of Electromotive Force.

598.] We have seen that E, the electromotive force due to in duction acting on the secondary circuit, is equal to j- , where

220 ELECTROMAGNETIC FIELD.

To determine the value of E, let us differentiate the quantity under the integral sign with respect to ^, remembering that if the secondary circuit is in motion, as, y, and z are functions of the time. We obtain

f(dF tfa dG_dy dH dz. J^dt ds + dt r& + <fc «fr'

C,dF dx dG dy dH dz^ dx

J ^ dx ds dx ds dx ds ' dt

dF dx dG dy dHdz^ dy

dy ds dy ds dy ds' dt

dF dx_ dG_dy dH dz. dz

ds dz ds dz ds' dt

-/<

ds dt ds dt

,2,

Now consider the second term of the integral, and substitute

from equations (A), Art. 591, the values of — and -7- . This term

dx dx

then becomes,

[( ^ 7)^z dF dx dF dy dF dz^dx J\Cdi" ds "f 'das ds + ^7 Ts + Hz ds' di •

which we may write

f f (ty 7 dz dF^ dx _

— / (C '/ — ^ 7- + T-J -T7 ^-

J ^ ds ds ds ' dt

Treating the third and fourth terms in the same way, and col-

i ,. .-, . dx dy - dz

lectmg the terms m - , ^ , and — , remembering that

dx

^ = F~7- , (3)

dt ~ dsdt>" L dt

and therefore that the integral, when taken round the closed curve, vanishes,

f ( dz dx dG. dy

/ (a^7 ~c^7 7-^ 7

J ^ dt dt dt ) ds

dx d dH dz

598.] ELECTROMOTIVE FORCE. 221

We may write this expression in the form

Equations of Electromotive (-B)

Force.

dy -.dz dF d^ where P = c -~ — o-j = =-

dz dx dG d^J

dt dt dt dy

_ dx dy dH d^

The terms involving the new quantity ^ are introduced for the sake of giving generality to the expressions for P, Q, R. They disappear from the integral when extended round the closed circuit. The quantity ^ is therefore indeterminate as far as regards the problem now before us, in which the total electromotive force round the circuit is to be determined. We shall find, however, that when we know all the circumstances of the problem, we can assign a definite value to ^, and that it represents, according to a certain definition, the electric potential at the point x, y, z.

The quantity under the integral sign in equation (5) represents the electromotive force acting on the element ds of the circuit.

If we denote by T @, the numerical value of the resultant of P, Q, and R, and by e, the angle between the direction of this re sultant and that of the element ds, we may write equation (5),

JT<$ cost els. (6)

fi =JT<$ cost els.

The vector @ is the electromotive force at the moving element ds. Its direction and magnitude depend on the position and motion of ds, and on the variation of the magnetic field, but not on the direction of ds. Hence we may now disregard the circum stance that ds forms part of a circuit, and consider it simply as a portion of a moving body, acted on by the electromotive force Q. The electromotive force at a point has already been defined in Art. 68. It is also called the resultant electrical force, being the force which would be experienced by a unit of positive electricity placed at that point. We have now obtained the most general value of this quantity in the case of a body moving in a magnetic field due to a variable electric system.

If the body is a conductor, the electromotive force will produce a current ; if it is a dielectric, the electromotive force will produce only electric displacement.

222 ELECTROMAGNETIC FIELD. [599-

The electromotive force at a point, or on a particle, must be carefully distinguished from the electromotive force along an arc of a curve, the latter quantity being the line-integral of the former. See Art, 69.

599.] The electromotive force, the components of which are defined by equations (B), depends on three circumstances. The first of these is the motion of the particle through the magnetic field. The part of the force depending on this motion is expressed by the first two terms on the right of each equation. It depends on the velocity of the particle transverse to the lines of magnetic induction. If © is a vector representing the velocity, and 33 another repre senting the magnetic induction, then if (^ is the part of the elec tromotive force depending on the motion,

^ = V. ® 33, (7)

or, the electromotive force is the vector part of the product of the magnetic induction multiplied by the velocity, that is to say, the magnitude of the electromotive force is represented by the area of the parallelogram, whose sides represent the velocity and the magnetic induction, and its direction is the normal to this parallel ogram, drawn so that the velocity, the magnetic induction, and the electromotive force are in right-handed cyclical order.

The third term in each of the equations (B) depends on the time- variation of the magnetic field. This may be due either to the time-variation of the electric current in the primary circuit, or to motion of the primary circuit. Let (£2 be the part of the electro motive force which depends on these terms. Its components are dF dG dH

-w ~w and -w

and these are the components of the vector, — — or 21. Hence,

dt

6, = -& (8)

The last term of each equation (B) is due to the variation of the function ^ in different parts of the field. We may write the third part of the electromotive force, which is due to this cause,

@3 = - V*. (9)

The electromotive force, as defined by equations (B), may therefore be written in the quaternion form,

@= r.® 33-21- V*. (10)

600.] MOVING AXES. 223

On the Modification of the Equations of Electromotive Force when the Axes to which they are referred are moving in Space.

600.] Let #', y', / be the coordinates of a point referred to a system of rectangular axes moving- in space, and let #, ?/, z be the coordinates of the same point referred to fixed axes.

Let the components of the velocity of the origin of the moving system be u, v, w, and those of its angular velocity w^ o>2, co3 referred to the fixed system of axes, and let us choose the fixed axes so as to coincide at the given instant with the moving ones, then the only quantities which will be different for the two systems of axes will be those differentiated with respect to the time. If

bx

•— denotes a component velocity of a point moving in rigid con-

o t

nexion with the moving axes, and - - and -j- that of any moving

ci/t civ