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342

The Followers of Maxwell.

Now, let $(x, y, ζ)$ denote coordinates relative to axes which are parallel to the axes $(x, y, z)$ , and which move with the charged bodies; then $(a z$ , is a function of $(x, y, ζ)$ only; so we have

${\frac {\partial }{\partial z}}={\frac {\partial }{\partial \zeta }},\qquad {\text{and}}\qquad {\frac {\partial }{\partial t}}=-v{\frac {\partial }{\partial \zeta }}$ ;

and the preceding equation is readily seen to be equivalent to

${\frac {\partial ^{2}a_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}a_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}a_{z}}{\partial \zeta _{1}^{2}}}=-4\pi \rho v$ ,

where $ζ 1$ denotes $(1 - v 2 / c 2) - 1 / 2 ζ$ . But this is simply Poisson's equation, with $ζ 1$ substituted for $z$ ; so the solution may be transcribed from the known solution of Poisson's equation: it is

$a_{z}=\iiint {\frac {\rho ^{\prime }v^{\prime }\ dx^{\prime }\ dy^{\prime }\ d\zeta _{1}^{\prime }}{\{(\zeta _{1}-\zeta _{1}^{\prime })^{2}+(x-x^{\prime })^{2}+(y-y^{\prime })^{2}\}^{\frac {1}{2}}}}$ ,

the integrations being taken over all the space in which there are moving charges; or

$a_{z}=\iiint {\frac {\rho ^{\prime }v^{\prime }\ dx^{\prime }\ dy^{\prime }\ d\zeta ^{\prime }}{\{(\zeta -\zeta ^{\prime })^{2}+(1-v^{2}/c^{2})(x-x^{\prime })^{2}+(1-v^{2}/c^{2})(y-y^{\prime })^{2}\}^{\frac {1}{2}}}}$ .

If the moving system consists of a single charge $e$ at the point $ζ = 0$ , this gives

$a_{z}={\frac {ev}{r(1-v^{2}\sin ^{2}\theta /c^{2})^{\frac {1}{2}}}}$ .

where $sin 2 θ = (x 2 + y 2)/ r 2$ .

It is readily seen that the lines of magnetic force due to the moving point-charge are circles whose centres are on the line of motion, the magnitude of the magnetic force being

${\frac {ev(1-v^{2}/c^{2})\sin \theta }{r^{2}(1-v^{2}\sin ^{2}\theta /c^{2})^{\frac {3}{2}}}}$ .

The electric force is radial, its magnitude being

${\frac {e^{2}c(1-v^{2}/c^{2})}{r^{2}(1-v^{2}\sin ^{2}\theta /c^{2})^{\frac {3}{2}}}}$ .

The fact that the electric vector due to a moving point-charge is everywhere radial led Heaviside to conclude that the same solution is applicable when the charge is distributed over