Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/361

current. The detect, as FitzGerald showed, may be immediately removed by assuming that a moving charge itself is to be counted as a current-element: the total current, thus composed of the displacement-currents and the convection-current, is circuital. Making this correction, FitzGerald found that the magnetic force due to a sphere of charge e moving with velocity v along the axis of z is curl(0, 0, ev/r)—a formula which shows that the displacement-currents have no resultant magnetic effect, since the term ev/r would be obtained from the convection-current alone.

The expressions obtained by Thomson and FitzGerald were correct only to the first order of the small quantity v/c. The effect of including terms of higher order was considered in 1889 by Oliver Heaviside,^[1] whose solution may be derived in the following manner:—

Suppose that a charged system is in motion with uniform velocity v parallel to the axis of z; the total current consists of the displacement-current Ė/4πc² where E denotes the electric force, and the convection-current ρv where ρ denotes the volume-density of electricity. So the equation which connects magnetic force with electric current may be written

$\mathbf {\dot {E}} /c^{2}={\text{curl }}\mathbf {H} -4\pi \rho \mathbf {v}$.

Eliminating E between this and the equation

${\text{curl }}\mathbf {E} =-\mathbf {\dot {H}}$,

and remembering that H is here circuital, we have

$\mathbf {\ddot {H}} /c^{2}-nabla^{2}\mathbf {H} =4\pi {\text{curl}}\rho \mathbf {v}$.

If, therefore, a vector-potential a be defined by the equation

$\mathbf {\ddot {a}} /c^{2}-\nabla ^{2}\mathbf {a} =4\pi \rho \mathbf {v}$,

the magnetic force will be the curl of a; and from the equation for a it is evident that the components a_x and a_y are zero, and that a_z, is to be determined from the equation

${\ddot {a}}_{z}/c^{2}-\nabla ^{2}a_{z}=4\pi \rho v$.

1. Phil. Mag. xxvii (1889), p. 324.