Page:ThomsonMagnetic1889.djvu/11

We must now consider the case where $\lambda a$ and $\lambda _{1}a$ are both large; in this case we find from (6) and (7)

$\mathrm {C} ={\dfrac {\omega e}{\mathrm {K} p^{2}a^{3}}}{\dfrac {i\lambda a}{3\epsilon ^{-i\lambda a}}},$

if $\lambda _{1}a/\lambda a$ and $\lambda _{1}a/\lambda ^{2}a^{2}$ are both large.

Thus the magnetic force parallel to $x$ outside

$={\dfrac {1}{3}}{\dfrac {\omega e}{\lambda ^{2}a^{2}}}{\dfrac {d}{dy}}{\dfrac {\cos \left\{pt-\lambda (r-a)\right\}}{r}},$

and that parallel to $y$

$=-{\dfrac {1}{3}}{\dfrac {\omega e}{\lambda ^{2}a^{2}}}{\dfrac {d}{dx}}{\dfrac {\cos \left\{pt-\lambda (r-a)\right\}}{r}}.$

Thus, since $\lambda ^{2}a^{2}$ is large, the magnetic force, though in the same direction as that due to a current $\omega v$, is very much smaller in magnitude, and fades away to zero as $\lambda a$ increases without limit.

$\mathrm {D} =-{\dfrac {\omega e}{3a^{2}{\dfrac {d}{da}}\mathrm {S} _{0}(\lambda _{1}a)}},$

The maximum magnetic force inside the sphere

$3\mathrm {D} {\dfrac {d}{da}}\mathrm {S} _{0}(\lambda _{1}a)\epsilon ^{ipt}=-{\dfrac {\omega e}{a^{2}}}\cos pt.$

Thus in this case the magnetic force just inside the sphere is equal to $-ve$, while that outside the sphere is very much smaller. This is a striking contrast to the previous cases, where the magnetic force inside the sphere is very small compared with that outside. Thus, in this case, when the time of the oscillation is small compared with that of the electrical oscillations the distribution of magnetic force is turned inside out. The magnetic force diminishes very rapidly as we recede from the surface of the sphere. In this case the total current parallel to the axis of $z$ inside the sphere is finite, for this by equation (2) equals

${\begin{array}{l}-{\dfrac {ip}{\sigma }}2\mathrm {D} \displaystyle \int _{0}^{a}\mathrm {S} _{0}(\lambda _{1}r)4\pi r^{2}dr,\\\\={\dfrac {ip}{\sigma }}{\dfrac {2\mathrm {D} 4\pi }{\lambda _{1}^{2}}}\epsilon ^{ipt}a^{2}{\dfrac {d}{da}}\mathrm {S} _{0}(\lambda _{1}a)\\\\=-{\tfrac {2}{3}}e\omega \cos pt.\end{array}}$