Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/307

sphere is due partly to the direct action of the fixed sphere, but partly also to the electrification, if any, of the sides of the case.

If the case is made of glass it is impossible to determine the electrification of its surface otherwise than by very difficult measurements at every point. If, however, either the case is made of metal, or if a metallic case which almost completely encloses the apparatus is placed as a screen between the spheres and the glass case, the electrification of the inside of the metal screen will depend entirely on that of the spheres, and the electrification of the glass case will have no influence on the spheres. In this way we may avoid any indefiniteness due to the action of the case.

To illustrate this by an example in which we can calculate all the effects, let us suppose that the case is a sphere of radius $b$, that the centre of motion of the torsion-arm coincides with the centre of the sphere and that its radius is $a$; that the charges on the two spheres are $E_{1}$ and $E_{2}$ and that the angle between their positions is $\theta$; that the fixed sphere is at a distance $a_{1}$ from the centre, and that $r$ is the distance between the two small spheres.

Neglecting for the present the effect of induction on the distribution of electricity on the small spheres, the force between them will be a repulsion

$={\frac {EE_{1}}{r^{2}}}$

and the moment of this force round a vertical axis through the centre will be

${\frac {EE_{1}aa_{1}\sin \theta }{r^{3}}}$

The image of $E_{1}$ due to the spherical surface of the case is a point in the same radius at a distance ${\frac {b^{2}}{a_{1}}}$ with a charge $-E_{1}{\frac {b}{a_{1}}}$, and the moment of the attraction between $E$ and this image about the axis of suspension is

${\begin{array}{c}EE_{1}{\frac {b}{a_{1}}}{\frac {a{\frac {b^{2}}{a_{1}}}\sin \theta }{\left\{a^{2}-2{\frac {ab^{2}}{a_{1}}}\cos \theta +{\frac {b^{4}}{a_{1}^{2}}}\right\}^{\frac {3}{2}}}}\\\\=EE_{1}{\frac {aa_{1}\sin \theta }{b^{3}\left\{1-2{\frac {aa_{1}}{b^{2}}}\cos \theta +{\frac {a^{2}a_{1}^{2}}{b^{4}}}\right\}^{\frac {3}{2}}}}\end{array}}$

If $b$, the radius of the spherical case, is large compared with $a$