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
.
Eliminating E between this and the equation
,
and remembering that H is here circuital, we have
.
If, therefore, a vector-potential a be defined by the equation
,
the magnetic force will be the curl of a; and from the equation for a it is evident that the components ax and ay are zero, and that az, is to be determined from the equation
.
- ↑ Phil. Mag. xxvii (1889), p. 324.