ELECTKOM. GNET1C INDUCTION.] ELECT R I C I T Y 81 o fixed The case of two circuits of invariable form and position
- uits. i s of great interest, from the light it throws on the action
of the induction coil. We shall suppose that we have no soft iron core, and that the break in the primary is instan taneous. The latter condition is very far from being realized in practice, even with the best arrangements, so that our case is an ideal one. Let i and j be the current strengths in primary and secondary, R and S the resistances, L, M, N the coefficients of induction, E the electromotive force in the primary. The equations are M^+N^+S/-0 (38). It is easy, in the first place, to show that the whole amounts of electricity which traverse the secondary at make and break of the primary are equal but of opposite signs. In fact, if we integrate (38) from the instant before make to a time when the induction currents both in primary and secondary have subsided, we get f ,. M T ME BS (39). where 1 denotes the steady current in the primary. Similarly inte grating over the break, we get /.,, , M ME In the second place, if we assume the break instantaneous, we can find the initial value of the direct current in S. Thus integrate 1 (38) from the instant before break to the time r after it, T being infinitely short compared with the duration of the induction currents, then Now the last term may lie neglected, because r is infinitely small and/ is not infinite, hence we have, for the initial value of/, . _M, ME
- "If -NR
(40). It is very e;.sy now to determine the farther course of the current in S. The equation for/ reduces to and we get, using (40), . ME -A, A _ / >
- NR
(41). The direct induced current (current at break), therefore, starts in our ideal case with an intensity which is to the intensity of the steady current in the primary as the coefficient of mutual induction of the coils is to the coefficient of self-induction of the secondary, and then dies away in a continuous manner like any other current left to itself in a circuit of given resistance and self-induction. Since we have already given enough of these calculations to serve as a specimen, we content ourselves with stating the result for the current at make. Owing to the self-induction of R, itc., the current in R rises continuously from zero to the value I : the induced current in S therefore begins also from zero, rises to a maximum, and then dies away. The mathematical expression for it contains, as might beexpected, two exponential terms. It is instructive, in connection with what has already been said concerning the electrokiuetic energy of two moving circuits, to examine what becomes of the energy in the case of two fixed circuits of invariable form. Equations (37) and (38) may be used if, for generality, F be written iustead of in (38), so that there is electromotive force (say of constant batteries) in both circuits. Multiplying (37) by i and (3S) by/, adding, and integrating from the time before E and F begin to act to a time T when the currents have all become steady, we get 1 The reader might suppose that this process of integration might be equally applied to (37). This is not so, however, owing to the vari ability of R at the break. J (J- (42). In words, the excess of the chemical energy exhausted in the batteries over the amount of energy which appears as heat in the circuits is J(Lt s + 2Mv +N7 s ), which we denote by K. Similar remarks to those made at p. 7G apply here. K is the amount of electrokiuetic energy stored up in the medium surrounding the circuits during the time that E and F are raising the currents against self and mutual induction. If we integrate .similarly over the break of both ciirrents, we find the defect of the chemical energy exhausted under the heat evolved in the circuit to be again K. Much of the energy thus discharged frum the system at break usually appears in the spark. Electrical Oscillations. Helmholtz 2 seem? to have been Electri- the first to conceive that the discharge of a condenser might ca osci1 consist of a backward and forward motion of the electricity latlolls - between the coatings, or of a series of electric currents alternately in opposite directions. Sir William Thomson 3 TL , m took up the subject independently, and investigated mathe- t ueory niatically the conditions of the phenomenon. Let q be the charge of the condenser at time t, C its capacity, E the difference of potentials between the armatures, i the current in the wire connecting the armatures, R its resistance, 4 L the co efficient of self-induction. Then we have and t- at d-q R dq - ~ The solution of this equation is (43). (44), where _ R / R 2 _ l_ ~2L W V ^L 2 LC A and B are constants to be determined by the conditions q = Q and ~ = when t = 0. Two distinct cases arise. (1.) Let R be greater than / ; then the exponentials in (44) are real, the discharge is continuous, all in one direction, and in volves no essentially new features. n / - ; V then the appropriate form of the (2.) Let R be less than / " solution is q = e~*" (A cos Ji< + B sin nt), where m has the same meaning as before, but n stands now for / j-_- - jp, - If we determine A and Bby the initial conditions, )Q (^). V we get g = e "( V Tlie current is iven by (46). It follows from these equations that, when R <C / ; > * ne charge of one armature of the condenser passes through a series of oscillations. The different maxima are Q, occurring at times 0, - Die ErJiaUung dcr A raft, 1847. 3 Phil. Mag., 1855. This paper is a very remarkable one in many respects. The methods used in the beginning to arrive at the equation (43) are well worth the reader s study. 4 R here must be understood to represent the mean resistance of j the circuit during the discharge.
VIII. ji
