Page:SearleEllipsoid.djvu/13

that as far as terms in $u^{2}/v^{2}$ the electric part of the energy is unaltered by the motion.

(C) Energy of a very slender Ellipsoid. When the ellipsoid is so slender that $b^{2}/a^{2}$ may be neglected in comparison with unity we have

$\mathrm {W} ={\frac {q^{2}}{2\mathrm {K} a}}\left\{\left(1+{\frac {u^{2}}{v^{2}}}\right)\log {\frac {2a}{b{\sqrt {1-{\frac {u^{2}}{v^{2}}}}}}}-{\frac {u^{2}}{v^{2}}}\right\}$.

(26)

When $u/v$ is small, this becomes

$\mathrm {W} {=}{\frac {q^{2}}{2\mathrm {K} a}}\left\{\left(1+{\frac {u^{2}}{v^{2}}}\right)\log {\frac {2a}{b}}+{\frac {1}{2}}{\frac {u^{2}}{v^{2}}}\right\}$.

(D) Energy of a Disk.

When $a^{2}<\alpha b^{2}$ the ellipsoid is more oblate than Heaviside's, and $l^{2}$ becomes negative. In this case let us write

$r^{2}{=}b^{2}-{\frac {a^{2}}{\alpha }}$,

so that $r$ is the radius of the disk which is the "image" of the ellipsoid $a,b$. Then writing ${\sqrt {-1}}=i$ we have from (23)

$\mathrm {W} {=}{\frac {q^{2}}{4\mathrm {K} ir{\sqrt {\alpha }}}}\left(1-{\frac {u^{2}a^{2}}{v^{2}r^{2}\alpha }}\right)\log {\frac {1+i{\sqrt {a}}r/a}{1-i{\sqrt {\alpha }}r/a}}+{\frac {q^{2}u^{2}a}{2Kv^{2}r^{2}\alpha }}$.

But

${\frac {1}{i}}\log {\frac {1+xi}{1-xi}}{=}2\left(x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}\dots \right){=}2\tan ^{-1}x$,

so that (23) becomes

$\mathrm {W} ={\frac {q^{2}}{4\mathrm {K} r{\sqrt {\alpha }}}}\left\{\left(1-{\frac {u^{2}a^{2}}{v^{2}r^{2}\alpha }}\right)\tan ^{-1}{\frac {r{\sqrt {\alpha }}}{a}}+{\frac {u^{2}a}{v^{2}r{\sqrt {\alpha }}}}\right\}$.

(27)

When $a=0$ we find for the energy of a disk of radius $r$ moving along its axis

$\mathrm {W} ={\frac {q^{2}\pi }{4\mathrm {K} r{\sqrt {\alpha }}}}$.

(28)

In all these cases it will be found that when $u=v$ the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light.