Page:ThomsonMagnetic1889.djvu/4

and a mechanical force equal to

$c{\dfrac {dg}{dt}}-b{\dfrac {dh}{dt}}+g{\dfrac {dc}{dt}}-h{\dfrac {db}{dt}},$

if the electrified bodies are at rest.

The first of these corresponds to the well-known expression for the electromotive force on a conductor moving in a magnetic field; the second is the mechanical force on a current in a magnetic field plus the term $g{\dfrac {dc}{dt}}-h{\dfrac {db}{dt}}$.

We can deduce an important consequence of the assumption, if we consider the case of the æther moving with uniform velocity between two parallel planes charged, the one with positive, the other with negative electricity.

If $v$ is the velocity of the æther, $h$ the electric displacement at right angles to the planes, the magnetic force between the planes will be parallel to $x$, and equal to $-4\pi vh$; or if $\sigma$ is the surface-density of the electrification on the planes $-4\pi v\sigma$, the magnetic force vanishes except between the planes, so that on crossing the positively electrified surface there is an increase in the magnetic force parallel to $x$ equal to $4\pi \sigma v$. Thus the charged surface acts like a current sheet of intensity $-\sigma v$, but $-v$ is the velocity of the plane relatively to the æther; so that a charged surface moving with velocity $v$ relatively to the æther must act like a current sheet of intensity $\sigma v$.

We will now proceed to apply these results to some special cases. Let us suppose that we have a charged sphere moving along the axis of $z$ with the velocity $w_{0}$, and that it sets the æther around it in motion in the same way as an incompressible fluid is set in motion by a solid sphere of the same radius moving through it with the same velocity. If $a$ is the radius of the sphere,

${\begin{array}{l}u={\frac {1}{2}}w_{0}a^{3}{\dfrac {d^{2}}{dx\ dz}}{\dfrac {1}{r}},\\\\v={\frac {1}{2}}w_{0}a^{3}{\dfrac {d^{2}}{dy\ dz}}{\dfrac {1}{r}},\\\\w={\frac {1}{2}}w_{0}a^{3}{\dfrac {d^{2}}{dz^{2}}}{\dfrac {1}{r}},\end{array}}$