Calculate the potential energy of a uniformly-charged sphere

Homework Statement

Using integration by volume calculate the potential energy of a uniformly-charged sphere with total charge Q. I assume the sphere is solid with uniform charge density.

Homework Equations

U=(1/8pi)*∫(E^2)dV

The Attempt at a Solution

My problem is that when I attempt to integrate from 0 to r, the 1/r^2 term of E blows up at 0 and I'm left with infinite potential energy. Is there another equation I can integrate? Which integral should I be taking to evaluate this problem, or what limits should I be using?

I'm trying to build up the total E-field layer-by-layer, by adding the E-fields of multiple overlapping spheres, but I still can't get the integral.

Edit: Nevermind I've got it. U=(3/5)*(Q^2/r)

Last edited:

Using integration by volume calculate the potential energy of a uniformly-charged sphere with total charge Q.

Assuming that the sphere is non conduction and let charge density is ρ

find charge in sphere when its radius becomes r

now find the potential at surface of the sphere (V)

now let by bringing some charge we increase its radius by dr
calculate the charge in this dr ... dq

now to bring this charge work done is dW = Vdq

Now integrate