# Electrostatic Potential Energy of a Conducting Sphere

## Homework Statement

Determine the total electrostatic potential energy of a conducting sphere of radius r_0 that carries a total charge Q distributed uniformly on its surface. Give your answer in terms of Q, r_0, epsilon_0 and appropriate constants.

## The Attempt at a Solution

I know that U = QV and I know that V = kQ/r. I tried to answer it as U = (1/(4pi*epsilon_0))*Q^2/r_0 but that seems to be incorrect. Can anyone point me in the right direction?

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ehild
Homework Helper
The electrostatic potential energy of the sphere is equal to the work done while it is charged.
If there is q charge on the sphere, the potential is kq/r0 on it surface. The work needed to move a charge dq from infinity to the surface of the sphere is:

$$dW=kq*dq/r_0$$

To get the whole work, you have to integrate from q=0 to q=Q.

ehild

• snate
So if you integrate dW = kq *dq/r you end up getting W = k/r * the integral of q *dq from 0 to Q. Which just ends up being W = kQ^2/2r right?

ehild
Homework Helper
So if you integrate dW = kq *dq/r you end up getting W = k/r * the integral of q *dq from 0 to Q. Which just ends up being W = kQ^2/2r right?
Yes, but with r0, the radius of the sphere.

ehild

i have a question similar to this. I was wondering if the question was rephrased to say that the sphere has a charge density, p, instead of a charge q how you would answer it?

Would simply become a Q/(volume of sphere) instead of q in your integral equation with everything else remaining the same?

Ps. sorry if this is not the correct format to ask a question (im new on the forum). If you guys want me to make a new thread please let me know thanks!