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

```                                Intrinsic Energy of the Electromagnetic Field

(71) We have seen (33) that the intrinsic energy of any system
of currents is found by multiplying half the current in each
circuit into its electromagnetic momentum. This is equivalent
to finding the integral
E = [half][capital sigma](Fp' + Gq' + Hr')dV (37)
over all the space occupied by currents, where p q r are
the components of currents, and F G H the components
of electromagnetic momentum.

Substituting the values of p' q' r' from the equations of Currents (C)
this becomes 1/8 [pi] [capital sigma]{F(d[gamma]/dy - d[beta]/dz) + G(d[alpha]/dz - d[gamma]/dx) + H(d[beta]/dx - d[alpha]/dy)}dV
Integrating by parts and remembering that [alpha] [beta] [gamma] vanish at
an infinite distance the expression becomes
1/8[pi] [capital sigma]{[alpha](dH/dy - dG/dz) + [beta](dF/dz - dH/dx) + [gamma](dG/dx - dF/dy)}dV
where the integration is to be extended  over all space. Referring to
the equations of Magnetic Force (B) this becomes

E = 1/8[pi][capital sigma]{[alpha].[mu][alpha] + [beta]/[mu][beta] + [gamma].[mu][gamma]}dV (38)
where [alpha] [beta] [gamma] are the components of magnetic intensity or
the force on a unit magnetic pole, and [mu][alpha], [mu][beta], [mu][gamma] are
the components of the quantity of magnetic induction or the
number of lines of force in unit of area.
In isotropic media the value of [mu] is the same in all directions and
we may express the result more simply by saying that the intrinsic
energy of any part of the magnetic field arising from its magnetization
is [mu]/8[piI[sup]2<\sup>
per unit of volume, where I is the magnetic intensity

```
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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