J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’

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                                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

James Clerk Maxwell
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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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