Page:Motion of Electrification through a Dielectric.djvu/7

 distribution of electric force, which has to be found by the additional datum

where E is the real force.

9. "Symbolically" expressed, the solution of (16) is

Here the numerator of the fraction to the right is the vector-potential of the convection-current. Calling it A0, we have

Inserting in (18) and expanding, we have

Given then ρu as a function of position and time, A0 is known by (19), and (20) finds A, whilst (15) finds H.

Complete Solution in the Case of Steady Rectilinear Motion.Physical Inanity of Ψ.

10. When the motion of the electrification is all in one direction, say parallel to the z-axis, u, A0, and A are all parallel to this axis, so that we need only consider their tensors. When there is simply one charge q at a point, we have

$\scriptstyle{A_{0}=qu/r,}$

and (20) becomes

at distance r from q. When the motion is steady, and the whole electromagnetic field is ultimately steady with respect to the moving charge, we shall have, taking it as origin,

$\scriptstyle{p=-u(d/dz)=-uD\,}$

for brevity; so that

Now the property

brings (22) to

and the property

where ν=sin θ, θ being the angle between r and the axis, brings (24) to