Electrostatic Self-Energy of a Uniform Density Sphere of Charge

[harshey]

Find an expression for the electrostatic self-energy of an arbitrary spherically symmetric charge density distribution p(r). You may not assume that p(r) represents any point charge, or that it is constant, or that it is piecewise constant, or that it does or does not cut off at any finite radius r. your expression must cover all possibilities. your expression may include an integral or integrals which cannot be evaluated without knowing the specific form of p(r).

I had no idea how to start this problem because i couldn't figure out what my professor meant by electrostatic self-energy of an arbitrary spherically symmetric charge density distribution.

 

[Fungiss]

Well, self-energy is potential energy of unbalanced charge system (in this case the sphere had just charged and not in equilibrium state).

Your answer:
Ws = 1/(4pi*epsilon0)*3/5*p(r)*V/R.

You could start from general definition formula of Ws = 1/2*q*V, then calculate for 3 case: r < R, r > R, and r = R. The cases r = R and r > R help define constant C in case r < R.