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

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                                Electromagnetic Relations of two Conducting Circuits. 
(28.) In the case of two conducting circuits A and B we shall assume 
that the electromagnetic momentum belonging to A is 
Lx + My 
and that belonging to B, Mx + Ny 
where L M N correspond to the same quantities in the dynamical 
illustration except that they are supposed to be capable of variation 
<s>according<\s> when the conductors A or B are moved. 
Then the equation of the current x in A will be 
[xi] = Rx + d/dt(Lx + My)  (4) 
and that of y in B [eta] = Sy + d/dt(Mx + Ny) (5) 
where [xi] and [eta] are the electromotive forces x and y the currents 
and R & S the resistances in A and B respectively. 

Induction of one Current by another 

(29) Case 1<sup>st<\sup> Let there be no electromotive force on B except that which arises 
from the action of A, and let the current of A increase from 0 to the 
value x then Sy + d/dt(Mx + Ny) = 0 
whence Y =[integral]ydt = - M/S x <s>(6)<\s> 
that is a quantity of electricity Y being the total induced current 
will flow through B when x rises from 0 to x. This is induction 
by variation of the current in the primary conductor. When M 
is positive, the induced current due to increase of the primary 
current is negative. 

Induction by motion of conductor

(30) Case 2<sup>nd<\sup> Let x remain constant and let M change from M to M' 
then Y = - M1 -M / S x <s>(7)<\s> 
so that if M is increased, which it will be by the primary and 
secondary circuits approaching each other, there will be a 
negative induced current, the total quantity of electricity passed 
through B being Y. 

This is induction by the relative motion of the primary and secondary conductors. 
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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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