Page:Motion of Electrification through a Dielectric.djvu/15

When P is equidistant from the edges, H is zero. There is therefore no H anywhere due to the motion of an infinitely large uniformly charged plane perpendicularly to itself. The displacement-current is the negative of the convection-current and at the same place, viz. the moving plane, so there is no true current.

Calculating E₁; the z-component of E, z being measured from left to right, we find

$cE_{1}=2q\left\{\tan ^{-1}{\frac {y_{1}}{z}}\left(1-{\frac {u^{2}}{v^{2}}}\right)^{\frac {1}{2}}-\tan ^{-1}{\frac {y_{2}}{z}}\left(1-{\frac {u^{2}}{v^{2}}}\right)^{\frac {1}{2}}\right\}$.

(45)

The component parallel to the plane is H/cu. Thus, when the plane is infinite, this component vanishes with H, and we are left with

$cE_{1}=cE=2\pi q$,

(46)

the same as if the plane were at rest.

A Charged Plane moving in its own Plane.

19. Lastly, let the charged plane be moving in its own plane. Refer to the first figure, in which let AB now be the trace of the plane when of finite breadth. We shall find that

$H=2qu\left[\tan ^{-1}{\frac {z}{h(1-u^{2}/v^{2})^{\frac {1}{2}}}}\right]_{z_{1}}^{z_{2}}$,

(47)

z₁ and z₂ being the extreme values of z, which is measured parallel to the breadth of the plane.

Therefore, when the plane extends infinitely both ways, we have

$H=2\pi qu$

(48)

above the plane, and its negative below it. This differs from the previous case of vanishing displacement-current. There is H, and the convection-current is not now cancelled by coexistent displacement-current.

The existence of displacement-current, or changing displacement, was the basis of the conclusion that moving electrification constitutes a part of the true current. Now in the problem (48) the displacement-current has gone, so that the existence of H appears to rest merely upon the assumption that moving electrification is true current. But if the plane be not infinite, though large, we shall have (48) nearly true near it, and away from the edges; whilst the displacement-current will be strong near the edges and almost nil where (48) is nearly true.

But in some cases of rotating electrification, there need be no displacement anywhere, except during the setting up of the final state. This brings us to the rather curious question whether there is any difference between the magnetic field of a convection-current produced by the rotation of electrification upon a good nonconductor and upon a good conductor respectively, other than that due to diffusion in the conductor. For in the case of a perfect conductor, it is easy to imagine that the electrification could be at rest, and the moved conductor merely slip past it. Perhaps Professor Rowland's forthcoming experiments on convection-currents may cast some light upon this matter.

December 27, 1888.