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

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                                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

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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