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