Case of Two Circuits (37) Let R be the primary circuit and S the secondary circuit then we have a case similar to that of the induction coil The equations of currents are those marked A & B and we may here assume L M N as constant because there is no motion of the conductors The equations then become Rx + L dx/dt + M dy/dy = [xi] Sy + M dx/dt + N dy/dt = 0 (13) To find the total quantity of electricity which passes we have only to integrate these equations with respect to t, then if x<sub>0<\sub> y<sub>0<sub> be the strengths of the currents at time 0 and x<sub>1<\sub> y<sub>1<\sub> at time t and if X Y be the quantities of electricity passed through each circuit during time t X = 1/R {[xi]t + L (x<sub>0<\sub>  x<sub>1<\sub>) + M(y<sub>0<\sub>  y<sub>1<\sub>)} Y = 1/S {M(x<sub>0<\sub>  x<sub>1<\sub>) + N(y<sub>0<\sub>  y<sub>1<\sub>)} (14) When the circuit R is completed, then the total currents up to time t, when t is great, are found by making x<sub>0<\sub> = 0 x<sub>1<\sub> = [xi]/R y<sub>0<\sub> = 0 y<sub>1<\sub> = 0 then X = x<sub>1<\sub>(t  L/R) Y = M/S x<sub>1<\sub> (15) The value of the total counter current in R is therefore independent of the secondary circuit, and the induction current in S depends only on M the coefficient of induction between the coils, S the resistance of the secondary coil and x, the final strength of the current in R When the electromotive force [xi] ceases to act there is an extra current in the primary circuit and a positive induced current in the secondary circuit whose values are equal and opposite to those produced on making contact. (38) All questions relating to the quantity of transient currents as measured by the impulse given to the magnet of the galvanometer may be solved in this way without the necessity of a complete solution of the equations. The heating effect of the current, and the impulse it gives to the suspended coil of Weber's dynamometer depend on the square of the current at every instant during the short time it lasts Hence we must obtain the solution of the equations and from the solution we may find the effects both on the galvanometer and dynamometer and we may then make use of the method of Weber
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Manuscript details
 Author
 James Clerk Maxwell
 Reference
 PT/72/7
 Series
 PT
 Date
 1864
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Cite as
J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7
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