80 E L E C T R I C I T Y [ELECTROMAGNETIC INDUCTION. through a resistance equal to the effective resistance from E to K. Further details concerning the method and results of these experiments may be found iu Wiedemann, Bd. ii. 744, etc. A very convenient method for exhibiting and measuring t ue extra current is obtained by using a Wheatstone s bridge instead of a differential galvanometer. Let the bridge be balanced as usual, so that when the battery circuit is made, and the galvanometer circuit made after wards, there is no deflection. If one of the resistances be wound so as to have a large coefficient of self-induction, and the galvanometer circuit be completed before the battery is thrown on, then, owing to the self-induction, the galvanometer needle will be suddenly deflected. Let AC, CD, UB, BA be four conductors of resistance S, Q, P, R, arranged as a Wheatstone s bridge (see fig. 22), with a battery be tween A and D, and a galvanometer G between B and C. Let L be the coefficient of self-induction of the coil S. Then, A, C, &c., denoting the potentials at A, C, &c., x and y the currents in AC and AB, and z the current iu G, we have &c. Eliminating, as in Maxwell, vol. i. p. 399, or above, p. 43, we get (28), i "> ere dt IS a Sf T aratt d symbol, and D is the determinant of the system of resistances with S + j. written for S. We may there fore write -, at I) being the ordinary determinant, and H a function of PQR, &c., which ve need not determine. Equation (28) may therefore be written dt (29), provided the bridge be balanced, i.e. if PS- QR be zero. Suppose now the galvanometer circuit is closed, and then the battery circuit closed ; then, integrating equation (29), from the instant before the battery is thrown in up to a time r when all the currents have be come steady and no further current Mows through the galvanometer, we get PLE where z l denot r es / zdt , .e. (30), the whole amount of electricity that flows through the galvanometer owing to the induced current. If now we derange the balance in the bridge by increasing S by a small quantity x, and decreasing Q by as much, we get a steady current through the galvanometer given by Hence (P+R)seE ~D~ s, PL (31). Now, if be the first swing of the galvanometer needle, owing to the induction current, o the deflection under the steady current, and T the time of oscillation of the needle under the earth s force alone, (T is supposed to be so large that the duration of the induced current is very small compared with it); then it may be shown that -i_Tsin|/3 ! TT tan a Pirtano (33;. We thus get L in terms of quantities which can be easily measured. This methed of finding L is due to Maxwell. 1 Ortaiu corrections would in general be necessary in practice, but wo need riot discuss them here. The application of the equations (26) to determine the Calcul march of the current in certain simple cases leads to result- ^ns c of great interest. Suppose that an electromotive force E begins to act in a circuit of resistance R and coefficient of self-induction L. The equation for the current strength t at any time t after it has begun to act, is The integral of this is i "dt E I ^ - e ~ L ) (34). (35), the constant of integration being determined by the condition i n( c steady current) when t=<x>. Hence the current starts with the value zero, and in- E creases continuously till it reaches the steady value = E - (36). R The part - "e" 17 is due to self-induction, and is called _tv the extra current. The whole amount of electricity passing in this part of the current is EL " R The quantity 5 is of the same dimension as t, and is called the time constant of the coil. According as the time constant is greater or less, the longer or shorter time will the current take to rise to a given fraction of its steady value. If we desire therefore to prolong the induction and to increase it as well, we must make L large and R small, two conditions which in the extremes are inconsistent. Cal culations of the form of coil for maximum inductive effects might be made, but this is not the place to enter on them. Next, let the electromotive force E suddenly cease to act, the resistance of the circuit being unchanged. This may be realized experimentally within certain limits by throwing the battery out of the circuit, and at the same time substituting for it a wire of equal resistance. It is easy to show as above that the extra current at a time t after E ceases to act is E -*, 4. e L , Tii ue c stant c and the whole amount of electricity which passes is + 7:5 Helmholtz, 2 who was the first to treat this subject both experimentally and mathematically, operated as follows : (1) The battery was thrown into the circuit, and after a time t the circuit was broken. (2) The battery was thrown in, and after a time t replaced by a circuit of equal resistance. These changes were effected by means of a system of levers, which it is not necessary to describe here. An account of the apparatus will be found in the paper quoted. The amount of electricity which passes through the circuit in measured by a galvanometer whose time of oscillation is long com pared with t. In the first case the amount is E#_EL, "R in the second because here the two extra currents just counterbalance each other. The observed value of B in each case enables us to calculate /. E and R being found by separate observations, one observed value of A enables us to calculate L. Using these values of E, R and L, a series of values of f, and hence A, can be calculated from the observed values of B, and the result compared with the observed value of A. The agreement between theory and experiment was sufficiently close to justify the application of the principles from which the above formula? were deduced. Among these principles may be mentioned the validity of Ohm s law for transient currents. The reader will find in the original paper details concerning the above and other similar results arrived at by Helmholtz. Experi ments ( Helm holtz.
1 Pogg. Ann., 1851.