An introduction to the mathematical theory of attraction ...

274 Systems of Conductors.
EXAMPLES.
1. Find the electric energy due to two charged spheres at an infinite distance
apart.
In this case if e2 = 0, we have V2 = 0, and therefore pl2 = 0, also
1 1
Pl 1=, P22= =b
where a and b are the radii of the spheres. Ience
W=~ {,~'-1 e~22)
2. Find the work iM required to bring together, from an infinite distance, two,equal charged spheres.
If a denote the radius of one of the spheres, the potential electric energy W
when the spheres are at an infinite distance apart is given by the equation
e12 + e22 (el + e2)2
2 /1 = e ----, and when they are in contact by the equation 2 W = 2
a 2a log 2
See Ex. 9, Art. 131. Since M.= W' - W, we have, therefore,
1
M 4 log {2ele2 - (2 log 2 - l)(e2 + e22)};.and substituting for log 2 its approximate value 0-693, we get
e1 1 e2 01 22) +
a (1-386 ei,2'
M is approximately zero if el and e2 have like signs and = 5. If e2: el > 5,
el
the value of M is negative, and the spheres tend to approach each other without
the expenditure of any. external work.
If e2 el < 5, the value of JM is positive till e2: el = 1: 5, when M is again
zero; and if e2: el < 1: 5, the value of M is negative. The last two results are
of course an immediate consequence of the former.
When el and e2 have unlike signs, Mis always negative.
3. Find the energy due to a charged conductor formed of a large and a small
sphere in contact.
If a denote the radius of the small sphere, b that of the large, E the total
charge, and Wthe energy, we have, Ex. 10, Art. 131,
BE2 a, 3 -1 _Ez a
W=2b (1 2S3 (b) = (- 1-202 ())
4. Find the work required to bring together, from an infinite distance, a
small and a large sphere, each charged with a given quantity of electricity.