An introduction to the mathematical theory of attraction ...

268 Systems of Conductors.
9. Find the capacity of a conductor formed of two equal spheres in contact.
If a denote the radius of one of the spheres, and q the required capacity, by
(34), Art. 124, we have q = 2a log 2 = 1-386294a.
10. Find the capacity of a conductor formed of a large and a small sphere
in contact.
If a denote the radius, and Ea the charge of the small sphere, b and Eb the
radius and charge of the large, q the required capacity, L the potential, and E
the total charge, by (30) and (32), Art. 124, we find q = b, and therefore, if the
approximation be not carried beyond ( t), the capacity is the same as that of
the large sphere.
If the approximation be carried on so asto include terms containing ( b,by
(29), Art. 124, we have,a = Lb {vS2 - V3(2s2 - )},
where
a 1 I
v=b' 2 = X,' ^S3= i3 -Again, by (31), Art. 124, we have
Eb = Lb 1 - v2I2 + v3(2S2 + S3)};
whence
E = SE, + Eb = Lb(1 + 2v3S3).
Hence
=b 1 +2533)
q=b 43\
The approximate value of S3 is 1'202, and therefore
q= b (1 2404 - )
132. Coefficients of Potential.-The potential energy
W of an electrified system is always positive whatever be
the charges or the values of the potential, but if all the charges
except el be zero 2 W= p11e12, and therefore pl must be
positive. Similarly p2, 233, &c., are each positive.
Again, if el be positive and all the charges except el zero,
the number of unit tubes of force which terminate on one of
the uncharged conductors A2 must be equal to the number of
those which leave it; and as these tubes go from higher to
lower potential, the potential at A2 cannot be the highest or
lowest in the field. A similar result holds good for A,, A4,.. A,, and as the potential cannot be highest or lowest in
empty space, the highest potential is on A1, and the lowest