Page:ThomsonMagnetic1889.djvu/5

$f=-{\dfrac {e}{4\pi }}{\dfrac {d}{dx}}{\frac {1}{r}},\quad g=-{\dfrac {e}{4\pi }}{\dfrac {d}{dy}}{\frac {1}{r}},\quad h=-{\dfrac {e}{4\pi }}{\dfrac {d}{dz}}{\frac {1}{r}}:$

hence by equations (1),

${\begin{array}{l}\alpha =ew_{0}\left(1+{\frac {1}{2}}{\dfrac {a^{3}}{r^{3}}}\right){\dfrac {y}{r^{3}}},\\\\\beta =-ew_{0}\left(1+{\frac {1}{2}}{\dfrac {a^{3}}{r^{3}}}\right){\dfrac {x}{r^{3}}},\\\\\gamma =0.\end{array}}$

Thus the lines of magnetic force are circles with their centres along and their planes at right angles to the axis of $z$.

At a distance from the centre large compared with the radius of the sphere the magnetic force is the same as that due to a current $ew_{0}$, but close to the sphere the relative motion of the sphere and æther causes it to be larger than this, and at the surface of the sphere it is the same as that due to a current ${\tfrac {3}{2}}ew_{0}$.

The energy due to this distribution of currents is ${\tfrac {3}{7}}{\dfrac {e^{2}w_{0}^{2}}{a}}\mu$.

Another case which can be easily solved is that of a right circular cylinder rotating with an angular velocity $\omega$, each unit length of the cylinder being charged with $\mathrm {E}$ units of electricity. If $a$ is the radius of the cylinder,

${\begin{array}{lll}u=-\omega {\dfrac {a^{2}y}{r^{2}}},&&v=\omega {\dfrac {a^{2}x}{r^{2}}},\\\\f={\dfrac {\mathrm {E} }{2\pi }}{\dfrac {x}{r^{2}}},&&g={\dfrac {\mathrm {E} }{2\pi }}{\dfrac {y}{r^{2}}};\end{array}}$

and by equations (1).

$\alpha =0,\quad \beta =0,\quad \gamma =-2\mathrm {E} \omega {\dfrac {a^{2}}{r^{2}}}$

Thus outside the rotating cylinder there is a magnetic force parallel to the axis of rotation.

If we assume that the æther outside the sphere is at rest, we can find the solution of the case of a charged metal sphere executing harmonic oscillations. Suppose the sphere to be moving parallel to the axis of $z$. the velocity at any time $t$ being represented by the real part of $\omega \epsilon ^{ipt}$. Then if we take rectangular axes passing through the centre of the sphere and moving with it, the following equations are true inside the sphere if $u,v,w$ are the components of the current, $a,b,c$ those of magnetic induction, $\psi$ the electrostatic potential, $\mathrm {F,}$