ELECTROMAGNETIC INDUCTION.] In these expressions Q is a functiun to be determined only by further assumption. Q = constant gives Ampere s formula ; Q = - gives the formula of Grassmann, and so on. We may in fact construct an infinite variety of different elementary formula;. The reader interested in this subject may consult Wiedemann, Bd. ii. 26, 27, 45-54, <kc., and Tait, Proc. R.S.fi., 1873. in of In our account of the magnetic action of electric cur- irou. rents no mention has been made of the effect of the proxi mity of soft iron. Under the magnetic action of the electric circuit soft iron is magnetized inductively. The distribution of the lines of force is in general greatly affected thereby. The general feature of the phenomenon is a concentration of the lines upon the iron. By the proper use of this effect electromagnetic forces of great power may be developed. It is not easy to give a mathe matically accurate account of the action, owing to our ignorance of the exact law of magnetic induction in power fully paramagnetic bodies. The discussion of this subject, however, belongs to MAGNETISM (which see). The Induction of Electric Currents. A brief account has already been given (see Historical 1 Sketch, p. 11) of Faraday s discovery 1 of the induction of electric currents. The results he arrived at may be summed up as follows. Let there be two linear circuits, ABKE (the primary) and CDG (the secondary), two portions of which, AB and CD, are parallel, and near each other. I. When a current is started in AB, a transient current flows through CD in the opposite direction to the current in AB; when the current in AB is steady, no current in CD can be detected ; when the current in AB is stopped, a transient current flows through CD in the same direction as the current in AB. These currents in CD are said to be induced, and may be called inverse and direct currents respectively, the reference being to the direction of the primary. Both inverse and direct currents last for a very short time, and the quantity of electricity which passes in each of them is the same. II. If the circuit AB, in which a steady current is flowing, be caused to approach CD, an inverse current is thereby induced in CD; when the circuit AB, under similar circumstances, recedes from CD, a direct current is induced in CD. We have already mentioned that when AB is at rest, and the current in it does not vary, there is no current in CD. AB has been supposed to approach and recede from CD, but the same statement applies when CD approaches and recedes from AB. III. When a magnet is magnetized or demagnetized in the neighbourhood of a circuit, or approaches or recedes from the cir cuit, the effect is the same as if an equivalent 2 current approached or receded from the circuit. For example, imagine a small circular circuit placed horizontally, and a vertical bar magnet lowered in the axis of the circuit with its north pole pointing down upon the ircuit, the magnet may be replaced by a series of coaxial circular currents (see above, p. 71), and the motion will induce a current passing round the circuit against the hands of a watch. Faraday showed how the direction of the induced current can be predicted when the variation of the magnetic field or the motion of the conductor in it is known, and he gave, in his own manner, indications how the magnitude of the current could be inferred. Maxwell has thrown the law of Faraday into the follow ing form : " The total electromotive force acting round a circuit at any instant is measured by the rate of decrease of the number of lines of magnetic force which pass through it." Or, integrating with respect to the time : " The time integral of the total electromotive force acting round any with Ampere s theory, may be found in Maxwell, vol. ii. 502, &c., ami in almost any Continental work on experimental physics. 1 Kxp. Res., ser. i., ii., (ix.), xxviii., xxix., 1831-32, 1851. The general statement in the text is given for the reader s convenience, ! and is not meant to be historical. Equivalent in the sense of producing the same magnetic field. 75 circuit, together with the number of lines of magnetic force which pass through the circuit, is a. constant quantity." For "number of lines of force" may of course be sub stituted the equivalent expressions, " induction through the circuit," or " surface integral of magnetic induction," taken over any surface bounded by the circuit. Some care must be taken in determining the positive direction round the circuit. The following is a correct process : Assume one direction (D, fig. jt 46) through the circuit as positive, then the positive direction round (II) is deter mined by the right-handed screw rela tion ; if the number of lines of force reckoned positive in direction D is de- Fig. 46. creasing, then the electromotive force is in direction R ; if that number is increasing, the electromotive force is in the opposite direction. This will be clearer if we consider the following simple example. Let ABCD (fig. 47) be a horizontal rectangular circuit (AB next the reader). In a northern lati tude, the vertical component Z of the earth s magnetic force is downwards ; if, therefore, the positive direction through the circuit be taken down wards, the positive direction round is ADCB, and the number of lines of force through it is Z.AB.BC. If BC slide on DC and AB parallel to itself through a small distance BB in time T, Z.AB.BC increases by Z.CB.BB ; hence the elec tromotive force is Z.BC.BB -^-r, and acts in the direction ABCD. If v be the velocity of BC, we may write for the electromotive force Z. BC.. That is, the electromotive force at any instant is propor tional to the velocity. The law of Faraday loads to a complete determination of the induced current in all cases. We may regard it as resting on the experiments of Faraday, and of those who followed out his results. Another view of the matter of great importance was enunciated independently and about the same time by llelmholtz 3 and Sir William Thomson. 4 Let a circuit carrying a current 5 i move in an invariable mag netic field, so that the number of lines of magnetic force passing through it is increased by rfN, then the vork 6 done by the electro magnetic forces on the circnit is by Ampere s theory idN ; also, if R be the resistance of the circuit, Ei 2 dt is the heat generated in time dt. Now if E be the electromotive force of the battery which maintains the current i, the whole energy supplied is Eidt; hence we must have Theory of Helni- holtzand Thorn- son _ Hence there is an electromotive force - -JT- in the moving circuit. Now -37 is the rate of increase of the number of lines of force pass ing through the circuit. We have therefore deduced the law stated above from Ampere s theory .and the principle of the conservation of energy ; at least we have done so for the case of induction by permanent magnets, and the same reasoning will also apply to the case where the alteration of the magnetic field, owing to the induced current in the primary circuit, is so small that it may be neglected. W T e have now the means of stating in a convenient form Electro- the electromagnetic unit of electromotive force. It is the magnetic electromotive force of induction in a circuit the number of V 1 ot lines of magnetic force through which is increasing at luot i ve the rate of one per second. force. 3 Ueber die Erhaltunfj dcr Kraft, 1847. 4 Rep. Brit. Ass., 1848, and Phil. Mag., 1851. 5 All the quantities are supposed to be measured in electromagnetic absolute units.
6 We may suppose this work spent in raising a weight, &C.