mining the magnitude of the electromotive force are decidedly simpler, we shall examine them first.

Going back to § 130, we know that, when work is done at the expense of the energy of an electric current, an inverse electromotive force is generated, the magnitude of which is expressed by the equation

W

where W is the work done in unit of time as the result of a current of strength C. If W is measured in watts, and C in amperes, e equation in volts.

is given by this

We have seen further (§ 252), that when a conductor carrying a current moves in a magnetic field so as to cut lines of force, work is done equal to the product of the strength of the current into the number of lines of force cut through; or, more generally (§ 261), that when the flux of magnetic force through the area of a circuit alters, work is done equal to the product of the strength of the current into the increment of the magnetic flux.

FIG. 269.

Apply these considerations to a case such as that represented in Fig. 269, where AB and A'B' are parallel metal rails connected by a cross-bar CC', which can slide along them at right angles to its own length, and P is a battery by which a current can be sent through the rails and cross-bar. Suppose the plane of the rails and bar to be perpendicular to the direction of a magnetic field of intensity H, let the direction of the current be from A to C and c' to A' in the figure, and the direction of the magnetic force to be as indicated by the arrow Z. Then the electro-magnetic forces will tend to enlarge the area of the figure, so as to make it enclose a greater number of lines of force, and the sliding bar will move towards BB'. Let be the force tending to displace the bar: while it moves through a small distance ds, the work done will be dW=$ds. But (§§ 252 and 261) we have also dW= CdQ, if C is the strength of current and do the increment of magnetic flux through the circuit caused by the displacement. Repeating the reasoning of § 130, we may write the following equation, which

states that the energy received by the circuit in an element of time, dt, in consequence of the electromotive force E of the battery, is equal to the energy expended in generating heat in and doing work:

ECdt=C2Rdt+CdQ.

If the sliding bar were not allowed to move, that is, if no work were done, the strength of the current would be C, and the above equation would become

EC dt=C2Rdt.

It is evident that C is not the same as C,, and, as the resistance is the same in both cases, a change in the strength of the current can only be due to a change in the effective electromotive force of the circuit in other words, to the generation of an additional electromotive force e. The effective electromotive force is the algebraic sum of the electromotive forces of the circuit, and by Ohm's law we may write

But if we divide the first of the above equations by C and by dt, we get

dQ
dt

It is to be noted that the induced electromotive force e is independent of the strength of the current, and therefore has the same value even if there is no other electromotive force, and therefore no current except that due to induction. It depends only on the rate of increase of the magnetic flux through the circuit. This, in the case supposed, is equal to the strength of the magnetic field H multiplied by the distance between the rails and by the is dt'

velocity, v = with which the bar moves.

The negative sign in the expression for e receives its interpretation from Lenz's law, of which, in fact, it is the expression. It indicates that the electromotive force, due to an increase of the magnetic flux in a given direction, would, if it acted alone in a circuit, produce a current causing a magnetic flux in the opposite

direction. Thus if, in the case already considered, the battery is replaced by a conducting wire R (Fig. 270), motion of the crossbar from A towards B will produce a current in the direction of the arrows, that is, opposite to the battery current in the previous case. The cur

rent is inverted if the motion is from B towards A. These effects can be rendered visible by including a galvanometer between A and A'.

293. General Law of InductionElectromotive Force.-The general law of induction is only a generalisation of the foregoing. Whether the circuit moves across the lines of force, or whether the lines of force move in reference to the circuit, whether or not the electro-magnetic forces do work, that is to say, whether there is or is not a displacement of their points of application, each element of the circuit which in the

B

B'

R

FIG. 270.

H

time at cuts a number of lines of force dq, is the seat of an electromotive force,

whose direction is from foot to head of an observer who, looking along the lines of force, sees them pass from right to left.

All these electromotive forces add up algebraically, like those of a number of galvanic cells connected in the same circuit.

If the conductor does not form a closed circuit, the two electricities are accumulated at the opposite ends until a difference of potential is produced equal to the algebraic sum of the elementary electromotive forces.

If the circuit is closed, the total electromotive force is likewise the algebraic sum of the electromotive forces corresponding to the different elements and since the algebraic sum of the lines of force cut by all the elements is equal to the variation dQ of the flux of force through the circuit, the electromotive force at the time t is

If R is the resistance of the circuit, this electromotive force produces at the moment in question a current of strength

The direction of the current is such that its axis (§ 250) is in the opposite direction to the flux if dQ is positive, and in the same direction if do is negative.

294. Coefficient of Self-Induction.—It has been shown (§§ 244 et seq.) that the existence of an electric current in a conductor implies the existence of magnetic force in the neighbouring space. Indeed, in order to ascertain the existence, direction, and strength of a current, we usually examine, not the conductor itself, but the magnetic properties of the space near it. Whatever the ultimate

nature of the phenomenon which we recognise and speak of as an electric current may be, it is an incomplete view of it to confine our attention to the process going on in the conducting circuit; it involves also, as an essential part, the existence of the correlative magnetic field.

Every complete circuit forms a closed curve, and the corresponding lines of magnetic force, which also form closed curves, are linked through it (see, for example, Figs. 218, 219). Hence, irrespective of any other magnet or circuit, there is, through the area of every circuit carrying a current, a flux of magnetic force depending on the current in the circuit. As already mentioned (§ 271), the magnetic field due to a current remains similar to itself whatever the strength of the current, but the force at any point of the field is proportional to the current. Hence the total flux through the circuit is proportional to the current, and may be represented by

Q = LC,

where L is a factor depending on the geometrical characters of the circuit, its shape, the area enclosed, and the number of times the current goes round it, and on the magnetic permeability of the surrounding medium.

The coefficient L is called the coefficient of self-induction of the circuit. It may be defined as the magnetic flux which, in consequence of a current of unit strength in the circuit, traverses the area of the circuit. It is a quantity of the same kind as the coefficient of mutual induction (§§ 184, 261) of two circuits..

295. Electromotive Force due to Self-Induction.-If we connect what has been said in the last paragraph with the result arrived at in § 292 as to induced electromotive force, we see that the self-induction of a circuit must give rise, in the case of any variation of the current, to an electromotive force tending to oppose the variation. In fact, the equations

Hence when

dc
dt

is positive, that is, when the current is increasing,

there is a negative electromotive force due to self-induction; and when is negative, or the current is decreasing, the electro

dC

dt

motive force of self-induction is positive. Whenever the strength

of the current is constant,

induction is nothing.

dC dt

=

o, the electromotive force of self

These results contain the proper interpretation of the phenomenon of the "extra current " (§ 291).

Intrinsic

296. Work Spent in Establishing a CurrentEnergy of a Circuit.-If there were no such thing as selfinduction, an electromotive force, E, applied to a circuit of resistance, R, would instantly give rise to a current of strength, C = E/R. The electromotive force of self-induction, however, causes the establishment of a current to be a more or less gradual process. It is true that in many cases it is so rapid as to need delicate observation to detect that it is not instantaneous, but in cases where the coefficient of self-induction has a high value, the growth of the current occupies a very appreciable time. This point is discussed more particularly in § 310.

During the variable period at the commencement of a current, more energy is imparted to the circuit by the battery than is given out by it in the form of heat or otherwise. The excess is consequently stored up, and goes to increase the energy of the circuit. Take again the equation