Derive Electric Potential Energy

[SpringPhysics]

1. The problem statement, all variables and given/known
For a a uniformly charged sphere of radius R with total charge Q, show that the potential energy of a point charge q' varies with r

U(r) = Qq'/4$$\pi$$$$\epsilon$$R * (3/2 - r²/2R² if r < R

Homework Equations

$$\Delta$$U = - $$\int$$E(r)q'dr cos$$\theta$$

The Attempt at a Solution

I used that the electric field of q' when r < R is Qr/4$$\pi$$$$\epsilon$$R³. I tried to integrate from initial to final for r, but then I only get 1/2 as opposed to 3. I have no idea where the 3 comes from.

Can someone help please?

 

[ideasrule]

What did you take as the initial and final r's? You have to integrate from infinity to r because potential energies are always stated with the potential at infinity as 0. (Of course, the electric field outside the sphere is not E=kQr/R^3, which adds another complication.)

 

[SpringPhysics]

Ohh, I took the initial r to be R. Does that mean I have to add back the potential energy at R from infinity in order to get the expression for the potential energy at r when r < R?

 

[ideasrule]

Yes.

 

[SpringPhysics]

Thank you!