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
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