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

```                                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
```
images

## 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