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

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                                In this method we confine our attention during integration to 
the two linear circuits alone 
[(1]10) 2<sup>nd<\sup> method. M is the number of lines of magnetic force which 
pass through the circuit B when A carries a unit current, or 
M = [capital sigma]([mu][alpha]l + [mu][beta]m + [mu][gamma]n) dS' 
where [mu][alpha], [mu][beta], [mu][gamma] are the components of magnetic induction 
due to unit current in A, S is a surface bounded by the current 
B and l m n are the direction-cosines of the normal to the 
surface, the integration being extended over the surface 
We may express this in the form 
M = [mu][capital sigma] 1/[rho]<sup>2<\sup> sin[theta] sin [theta]' sin [phi] dS'ds 
where dS' is an element of the surface bounded by B, ds is an element 
of the circuit A [rho] is the distance between them theta] and [theta]; are 
the angles between [rho] and ds and between [rho] and the normal to dS' 
respectively and [phi] is the angle between the planes in which [theta] & [theta]' 
are measured. The integration is performed round the circuit A 
and over the surface bounded by B 
This method is most convenient in the case of <s>plane<\s> circuits 
lying in one plane, in which case sin[theta]' = 1 and sin[phi] = 1 

(111) 3<sup>rd<\sup> method. M is that part of the intrinsic magnetic energy 
of the whole field which depends on the  <s>combina<\s> product of 
the currents in the two circuits each current being unity 
Let [alpha] [beta] [gamma] be the components of magnetic intensity at any point due 
to the first circuit [alpha]' [beta]' [gamma]' the same for the second circuit 
then the intrinsic energy of the element of volume dV of the field is 
[[mu]/8{pi] (([alpha] + [alpha]')<sup>2<\sup> + ([beta] + [beta]')<sup>2<\sup> + ([gamma] + [gamma]')<sup>2<\sup>) dV 
The part which depends on the product of the circuits is 
[mu]/4[pi] ([alpha][alpha]' + [beta][beta]' + [gamma][gamma]') dV 
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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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