Page:SearleEllipsoid.djvu/13

From Wikisource
Jump to: navigation, search
This page has been validated.


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.