Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/268

or, integrating by parts,

$-{\tfrac {1}{2}}\iiint (\mathbf {I} _{0}.\mathrm {grad} \ \phi )\ dx\ dy\ dz$, or $-{\tfrac {1}{2}}\iiint (\mathbf {H.} \mathbf {I} _{0})\ dx\ dy\ dz$.

Since B = μH + 4πI₀, this expression may be written in the form

$-{\frac {1}{8\pi }}\iiint (\mathbf {H.B} )\ dx\ dy\ dz+{\frac {1}{8\pi }}\iiint \mu \mathbf {H} ^{2}\ dx\ dy\ dz$;

but the former of these integrals is equivalent to

$\iiint (\mathbf {B.} \mathrm {grad} \ \phi )\ dx\ dy\ dz$, or $-\iiint \phi \mathrm {div} \ \mathbf {B} \ dx\ dy\ dz$,

which vanishes, since B is a circuital vector. The energy of the field, therefore, reduces to

${\frac {1}{8\pi }}\iiint \mu \mathbf {H} ^{2}\ dx\ dy\ dz$,

integrated over all space; which is equivalent to Thomson's form.^[1]

In the same memoir Thomson returned to the question of the energy which is possessed by a circuit in virtue of an electric current circulating in it. As he remarked, the energy may be determined by calculating the amount of work which must be done in and on the circuit in order to double the circuit on itself while the current is sustained in it with constant strength; for Faraday's experiments show that a circuit doubled on itself has no stored energy. Thomson found that the amount of work required may be expressed in the form 1/2Li², where i denotes the current strength, and L, which is called the coefficient of self-induction, depends only on the form of the circuit.

It may be noticed that in the doubling process the inherent

1. The form actually given by Thomson was

   ${\frac {1}{8\pi }}\iiint \left(H^{2}+{\frac {4\pi }{R}}\mathbf {I} ^{2}\right)\ dx\ dy\ dz$,

   which reduces to the above when we neglect that part of I² which is due to the permanent magnetism, over which we have no control.