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—a formula which shows that the displacement-currents have no resultant magnetic effect, since the term 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. The effect of including terms of higher order was considered in 1889 by Oliver Heaviside, 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 where  denotes the electric force, and the convection-current  where  denotes the volume-density of electricity. So the equation which connects magnetic force with electric current may be written

Eliminating between this and the equation

and remembering that is here circuital, we have

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

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