An introduction to the mathematical theory of attraction ...

Concentric Spheres. 243
125. Concentric Spheres.-If two concentric spheres
(A) and (B) be at potential zero under the influence of a
charge e at a point O situated between them, the potential at
any point between the spheres is that due to the charge at 0,
together with charges at the successive images of O in the
spheres. If I,, I2, &c., denote the images in (A), 41,,, &c.,
their distances from C the centre, il, i2, &c., the charges at
these points, Ji, &c., ni, &c., and jy, &c., the corresponding
points and quantities for (B), denoting the radii of the
spheres by a and b, and putting O = f, and a = 1ub, the inner
sphere being (A), we have
(a2 b2 a2
[~=~,.'-, ~=Mf, 2.72-f.
Si = "5^=7? ^2 =1i -= r2 = 1,27.
Assuming then
2 b2
2n-i =2(-) 2',2 = 12nf, 2n-i = 1 -2f1 -1) r2n = c-2~f, (35)
a2 b2
\ since n+1 = -, and n+i = —
/ \n - ny
we see that, as the above, - J. I- -- r I I-C ) assumptions hold good for
\ \ / n = 1, they hold good in
general. Also, since by (1),
Art. 110, the charges at a
point and its image in a
sphere are proportional to the square roots of their distances
from the centre, we have
i,=+~ I e, jn=~,+ e;
4f f
whence
a b.2n-_ = - C- e, i2n = -1:e, j2n-1i = - ( j-l) e, jn = V- e. (36)
Since the images in (A) produce the same potential in
external space as the distribution on its surface, if Ea be the
total charge on (A), we have
a en
Ea = i-i= e ES _ a^ ^-; } = e =
Rt2