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

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                                Part VII. Calculation of the Coefficients of Electromagnetic Induction 
General Methods 

(109) The Electromagnetic relations between two conducting circuits 
A and B depend upon a function M of their form and relative 
position as has already been shown 
M may be calculated in several different ways, which must 
of course all lead to the same result 
1<sup>st<\sup> method. M is the electromagnetic momentum of the circuit 
B when A carries a unit current or 
M = [integral](F dx/ds' + G dy/ds' + H dz/ds') ds' 
where F, G, H are the components of electromagnetic momentum 
due to a unit current in A and ds' is an element of length 
of B and the integration is performed round the circuit of B 
To find F G H we observe that by (B) and (C) 
d<sup>2<\sup>F/dx<sup>2<\sup> + d<sup>2<\sup>F/dy<sup>2<\sup> + d<sup>2<\sup>F/dz<sup>2<\sup> = - 4[pi][mu]p' 
with corresponding equations for G & H, p' q' & r' being the 
components of the current in A 
Now if we consider only a single element ds of A we shall have 
p' = dx/ds ds q' = dy/ds ds r' = dz/ds ds 
and the solution of the equation gives 
F = [mu]/[rho] dx/ds ds G = [mu]/[rho] dy/ds ds H = [mu]/[rho] dz/ds ds 
where [rho] is the distance of any point from ds. Hence 
M = [integral][integral] [mu]/[rho] (dx/ds dx/ds' + dy/ds dy/ds' + dz/ds dz/ds') ds ds' 
= [integral][integral] [mu]/[rho] cos [theta] ds ds' 
where [theta] is the angle between the directions of the two elements ds, ds', 
and [rho] is the distance between them and the integration is performed 
round both circuits 
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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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