Page:ThomsonMagnetic1889.djvu/10

taking the real part

$=\omega e{\dfrac {d}{dy}}{\dfrac {\cos {\big (}pt-\lambda (r-a){\big )}}{r}}.$

Similarly the magnetic force parallel to the axis of $x$

$=-\omega e{\dfrac {d}{dx}}{\dfrac {\cos {\big (}pt-\lambda (r-a){\big )}}{r}},$

and the magnetic force parallel to $z$ vanishes. Thus the magnetic force is the same as that which would be produced by a current-element $\omega e\cos pt$ or $ev$, $v$ being the velocity of the sphere (see Proc. Math. Soc. xv. p. 214).

The magnetic force inside the sphere parallel to $x$ equals

${\begin{array}{c}-3\mathrm {D} \epsilon ^{ipt}{\dfrac {d}{dy}}\mathrm {S} _{0}(\lambda _{1}r),\\\\=3\mathrm {D} \epsilon ^{ipt}{\dfrac {d}{dr}}\mathrm {S} _{0}(\lambda _{1}r){\dfrac {y}{r}}.\end{array}}$

Substituting the value for $\mathrm {D}$ given by equation (7), this equals

${\dfrac {\omega e}{a}}p^{2}\mathrm {K} y\epsilon ^{ipt};$

or, taking the real part and writing $\lambda ^{2}$ for $\mathrm {K} p^{2}$,

${\dfrac {\omega e}{a^{3}}}\left(\lambda ^{2}a^{2}\right)y\cos pt.$

The component parallel to $y$ is

$-{\dfrac {\omega e}{a^{3}}}\left(\lambda ^{2}a^{2}\right)x\cos pt,$

and the $z$-component vanishes. Thus the maximum magnetic force inside the sphere is

${\dfrac {\omega e}{a^{2}}}\cos pt(\lambda a)^{2}.$

If $\lambda a$ is very small, this is very small compared with the force outside the sphere. If the velocity is uniform, $p$, and therefore $\lambda =0$, and the magnetic force inside the sphere vanishes. When there is no magnetic force inside the sphere its energy and the force acting upon it have the values assigned to them by Mr. Heaviside.

Let us next take the case where $\lambda a$ is small and $\lambda _{1}a$ large: in this case $\mathrm {C}$ and $\mathrm {D}$ have the same values as before, so that the magnetic force due to the moving sphere is the same.