An introduction to the mathematical theory of attraction ...

Examples. 61
5. Show that the electricity on a charged insulated ellipsoidal conductor is in
equilibrium when it is so distributed that its density at each point of the external surface of the ellipsoid is proportional to the normal thickness of the
coincident homoeoid.
If the electricity be distributed in this manner, by Art. 18 the force at a
point in the substance of the conductor is zero; and therefore, by Art. 29,
at the external surface, the resultant force is in the direction of the normal.
The conditions for electric equilibrium are therefore, Art. 30, fulfilled.
6. In Ex. 5 prove that o the electric density at any point Q of the surface
of the conductor, R the resultant force at a point outside the surface and
infinitely near Q, and -F the force per unit of area acting in the surface of
the conductor at Q, are given by the equations
Ep Ep 1 E2 p2
R — = -F =
47rabc' abc' 87r a2 b2 c2'
where E is the total charge on the conductor, a, b, c its semi-axes, and p the
perpendicular from its centre on the tangent plane at Q.
By Ex. 4, Art. 24, we have r = cp, where c is constant, but f odS = E, and
pdS = 4irabc; hence we obtain o, and the expressions for 1R and F follow from
Arts. 29 and 35.
From the expressions for o, R, and F, it follows that these quantities become
very large at points near the extremities of an elongated ellipsoidal conductor
for which b and c are very small compared with a.
7. Determine the distribution of electricity on an ellipsoidal conductor when
one axis becomes evanescent.
If p be the central perpendicular on the tangent plane at the point x, y, z on
the surface, we have, in general,
1 X2 y2 z2
p2 - ~ + b4 + C4;
c2 x2y2\ z2
whence = 2 - -+ _ +-;
when z and c are each zero, this becomes
c2 z2 X2 y2
2 - 2 - a2 b2'
p =p ab
c /(a2 b2 - bx2 2- a2 y2)'
and ~
4and (a2 b2 - b22 x2 2 y2)'
It is plain that the ellipsoid is transformed into an elliptic plate whose thickness at the edge is an infinitely small quantity of the second order. The density
at the edge is infinite, but on that part of the surface of the plate where the
density is infinite the total mass is infinitely small.