E L E C T K I C I T Y [ELECTROMAGNETIC INDUCTION. In the case where the field is due to a current r , we Lave by formula; (4) and (14) of last division N=i M ....... (24), where M now stands for "tended a11 over the two circuits. M, which depends merely on the configuration and relative position of the two circuits, is called the coefficient of mutual in duction. Electro . An application of the principle of the conservation of kinetic energy of great importance wa3 made by Sir William energy. Thomson to the case of two electric circuits of any form, Theory of ^ w hj c h fa e currents are kept constant. Let two such circuits, the currents in which are n , be displaced so that the coefficient of mutual induction M increases by dM. Let us suppose that the currents i and i are maintained by two constant batteries of electromotive forces E and E , and that the motion takes place so slowly that the currents may be regarded as constant throughout. If R and R be the resistances of the circuits, tidt the mechanical equivalent of the whole heat generated, and KM the whole expenditure of chemical energy in the batteries in time dt, H = Ri* + R i 2 , and K = Ei + E t", K - H = t(E - Rt) + i(E - RY). Now, applying (23), Case of two cir cuits. r = E-t ^-, and RV =-E -t dt ..,dM whence K - H = 2tt , or, as we may write it, (K -H)cft=;2t i"dM ... . (25). Now u dM is the work done by the electromagnetic forces during the displacement which we may suppose spent in lifting a weight. Hence, when two electric currents are allowed slowly to approach each other, being kept constant and doing work the while, over and above the work which is spent in generating heat in the conductors, an amount of energy is drawn from the batteries equivalent to twice the work done by the electromagnetic forces. There remains therefore an amount of work as yet unaccounted for. What becomes of it 1 The answer is, that the energy, or, as Sir W. Thomson calls it, the " me chanical value," of the current is increased. But how increased ? When a material system (and we may consider the two circuits, the batteries, the lifted weight, &c., as such) is left to itself, it moves so that its potential energy decreases. In this case, therefore, there must have been an increase of kinetic energy somewhere. This energy may be called the electrokinetic energy of the system; according to Maxwell s theory, this kinetic energy has its seat in the medium surrounding the wire. The energy thus stored up is accounted for in the increased development of heat, ifec., when the two currents are broken in succession. Returning now to our general law of induction, let us rite down in the most general form the equations which determine the course of the currents in two circuits (A, B), in which the form and relative positions of the circuits, as well as the current strengths, are variable. The number of lines of force which pass through a circuit depends partly on neighbouring circuits, partly on the circuit itself. Re taining the notation used above, we may, in the case of two circuits, write the first part Mi , and the second part Li; where L is a double integral of the same form as M, only both elements ds and da- now belong to the same circuit. We have, therefore, for the whole number of lines of force passing through the circuit A, Mi +Li. Similarly we have for B, M/ + Nf. We have therefore by our general law, d (26). divided into two parts : one of these, viz., 7i(Mz") is due
- o the circuit B, the other - (Li) is due to the circuit A
itself, and is called the electromotive force of self-induction. L is called the coefficient of self-induction for A. Similarly (Nf) is the electromotive force, and N the coefficient Force of self-induction for B. If we have only one circuit then M = 0, and the equation for the course of the current is here there is only self-induction. F. E. Neumann, to whom belongs the honour of first Theor stating with mathematical accuracy the laws of induction, Neu - adopted a foundation for his theory very different from the * c one chosen above. His method was based on the law of ], PT1Zt Leuz 1 , enunciated very soon after the great discovery of Faraday, which lays down that, in all cases of induction by the motion of magnets or currents, the induced current has a direction such that its electromagnetie action on the inducing system tends to oppose the motion producing it. Besides its historical importance, this law affords a very convenient guide in many practical applications of the theory of induction. The reader will find no difficulty in verifying it on the elementary cases given at the beginning of this division. It can be deduced at once from our general law. Consider any circuit in which a current i is flowing, and let the direction of the current be the positive direction round the circuit. Suppose the circuit to move so that the number of lines of force passing through it increases, this is the way the circuit would tend to move under the electromagnetic forces when traversed by a current i; but the electromotive force of induction is in the negative direction found the circuit by the general law, and would therefore produce a current opposite in direction to i. The electromagnetic action on this current would be opposite to that on i, that is, would tend to hinder the displacement. It is a curious fact that a law exactly like this had been announced shortly before Lenz by Ritchie, only with the direction of the action reversal in every case. The results of Neumann are identical with those given above. The double integral M, which is here called the co efficient of mutual induction of two circuits, Neumann calls the mutual potential of the two circuits, and what has been c.illed above the coefficient of self-induction of a circuit he calls the potential of the circuit on itself. Accounts of his theory will be found in Wiedemann s Galvani&nuf, and in most Continental works on electricity. Experimental Verification of the Laws of Mutual Induction. It will be observed that, in the law of induction for linear circuits, no statement is made respecting the material or thickness of the circuit in which the electromotive force of induction acts, or of the non-conducting medium across which induction takes place. Faraday showed that the material of the circuit lias no effect. 5 Eip He found, for instance, that when two wires of different metals were men joined and twisted up together, as in fig. 48, so as to be insulated Fan from each other, no in- duced current could be ?^~-^~?~**-*<^=<z3<?!>^^><^>5>+^> obtained by passing the -p[a. 48. arrangement between the poles of a powerful magnet. The same result was obtained when one of the branches of the circuit was an electrolyte. Lenz 3 connected two spirals of wire in circuit with each other, and placed first one then the other, on the soft iron keeper of a horse-shoe magnet; so Ion" as the number of turns on each spiral was the same, the induced These are the general equations for the induction of two j Circuits. The electromotive force of induction in A can be 1 Pogg. Ann., 1834. 2 Exp. Res., 193, &c., 1832 ; also 3143, &G., 1851.
3 Poyy. Ann., 1835.