A sphere of radius a has a charge density that varies with d

A sphere of radius a has a charge density that varies with distance r from the center according to p_v = Ar^n where A = const. and n greaterthanorequalto 0. Find V at all points inside and outside the sphere and express your results in terms of the total charge Q of the sphere. b) Find the energy of the charge distribution 2 ways by using (1, 2). To what should your result reduce when n = 0? Does it? c) What fraction of the total energy is outside the sphere?

Solution

use the differential form of Gauss\' Law.
DivergenceE = D/eo (D=charge density)

Write the divergence in spherical coordinates and assume that E only has a radial component to get;
(1/r^2)d(r^2E)/dr = D/eo

Expand the derivative
(1/r^2)[2rE + r^2dE/dr] = D/eo

If you require E to be constant this reduces to;
2E/r = D/eo
D = 2eoE/r

I really don\'t know of any other way to do the problem.

I did think of a way to use the integral form of Gauss\' Law.

Choose a Gaussian sphere of arbitrary radius \"r\" inside the charged sphere. You have to assume spherical symmetry so E is constant on the choosen surface. You then get;
eoE4pir^2 = INTEGRAL[D r^2dr sin()d() d(phi)]

Now assume that the density depends on \"r\" to some power \"n\" as;
D = Ar^n (A & n are to be determined)

The angles integrate to 4pi and you have;
eoE4pir^2 = 4piINTEGRAL[Ar^(n+2)]
eoEr^2 = (A/n+3)r^(n+3)
eoE = (A/n+3)r^(n+1)

Now in order for E to be constant thruout the sphere it can\'t depend on \"r\" so you must have n=-1
and
eoE = A/2

So
A = 2eoE