Page:SearleEllipsoid.djvu/13

that as far as terms in ${\displaystyle u^{2}/v^{2}}$ 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 ${\displaystyle b^{2}/a^{2}}$ may be neglected in comparison with unity we have

 ${\displaystyle \mathrm {W} ={\frac {q^{2}}{2\mathrm {K} a}}\left\{\left(1+{\frac {u^{2}}{v^{2}}}\right)\log {\frac {2a}{b{\sqrt {1-{\frac {u^{2}}{v^{2}}}}}}}-{\frac {u^{2}}{v^{2}}}\right\}}$. (26)

When ${\displaystyle u/v}$ is small, this becomes

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

(D) Energy of a Disk.

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

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

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

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

But

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

so that (23) becomes

 ${\displaystyle \mathrm {W} ={\frac {q^{2}}{4\mathrm {K} r{\sqrt {\alpha }}}}\left\{\left(1-{\frac {u^{2}a^{2}}{v^{2}r^{2}\alpha }}\right)\tan ^{-1}{\frac {r{\sqrt {\alpha }}}{a}}+{\frac {u^{2}a}{v^{2}r{\sqrt {\alpha }}}}\right\}}$. (27)

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

 ${\displaystyle \mathrm {W} ={\frac {q^{2}\pi }{4\mathrm {K} r{\sqrt {\alpha }}}}}$. (28)

In all these cases it will be found that when ${\displaystyle u=v}$ 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.