Page:Electromagnetic effects of a moving charge.djvu/10

In addition, the divergence of D must be q at the origin, and the divergence of B must be zero. The latter gives, applied to (9),

$\scriptstyle {\mathbf {H} =c\mathbf {VuE} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(10)}}}$

which gives H fully in terms of E. Eliminate H from (8) by means of (10), and we get

$\scriptstyle {\mathrm {curl} (\mu c\mathbf {VuVEu} -\mathbf {E} )=0,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(11)}}}$

or

$\scriptstyle {\mathrm {curl} \left[{\frac {u^{2}}{v^{2}}}\left(\mathbf {E} -E_{3}\mathbf {k} \right)-\mathbf {E} \right]=0,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(12)}}}$

where E₃ is the z-component of E and k a unit vector along z; or, integrating, and writing the three components,

$\scriptstyle {E_{1}=-{\frac {dP}{dx}},\quad E_{2}=-{\frac {dP}{dy}},\quad E_{3}=-\left(1-{\frac {u^{2}}{v^{2}}}\right){\frac {dP}{dz}},\quad \cdot \cdot {\text{(13)}}}$

where P is a scalar potential. Here is the new point. There is a potential, of a peculiar kind. The displacement due to the moving charge is distributed in precisely the same way as if it were at rest in an eolotropic medium, whose permittivity is c in all directions transverse to the line of motion, but is smaller, viz., c(1-u²/v²), along that line and parallel to it. The potential P is given by

$\scriptstyle {P={\frac {q}{c\left[\left(x^{2}+y^{2}\right)\left(1-u^{2}/v^{2}\right)+z^{2}\right]^{\frac {1}{2}}}}.\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(14)}}}$

It is a particular case of eolotropy. In general, c₁, c₂, c₃, the principal permittivities, are all unequal. Then, with q at the origin, the potential is

$\scriptstyle {P={\frac {q}{\left(c_{1}c_{2}c_{3}\right)^{\frac {1}{2}}\left({\frac {x^{2}}{c_{1}}}+{\frac {y^{2}}{c_{2}}}+{\frac {z^{2}}{c_{3}}}\right)^{\frac {1}{2}}}}.\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(15)}}}$

Observe that although the electric force in the substituted problem of a charge at rest in an eolotropic medium is the slope of a potential; yet it is not so when the medium is isotropic, and moves past the fixed charge, or vice versâ, although the distributions of displacement are the same.

When u=v, we abolish the permittivity along the z-axis in the substituted case, so that the displacement must be wholly transverse. We then have the plane electromagnetic wave. When u is greater than v it makes the permittivity negative along z; this is an impossible electrical problem, and furnishes another reason for supposing that there can be no steady state in the corresponding electromagnetic problem.

It now remains to find what would happen if electrification were conveyed through a medium faster than the natural speed of propagation of disturbances. There is the cone; but what takes place within it?

⁠Aug. 25, 1889.