# Finding the potential outside a charge sphere

1. Oct 26, 2008

### Benzoate

1. The problem statement, all variables and given/known data

Find the potential outside a charged metal sphere(charge Q , radius R) placed in an otherwise uniform electric Field E0. Explain clearly where you are setting the zero of potential .

2. Relevant equations

3. The attempt at a solution

At the boundaries of the sphere V=0. Since a sphere is a 3 dimensional object,

1/r^2*d/dr(r^2*dV/dr)+(1/r^2*sin(theta))*d/d(theta)*(sin(theta)*dV/d(theta))+1/r^2*1/(sin(theta)^2)*d^2V/dphi^2=0
V(r,theta)=R(r)*$$\Theta$$($$\theta$$R

1/R*d/dr*(r^2*dR/dr)+(1/($$\Theta$$*sin$$\theta$$)*d/d$$\theta$$*(sin($$\theta$$*d$$\Theta$$/d$$\theta$$)=0

1/R*d/dr*(r^2*dR/dr)=l(l+1)
(1/($$\Theta$$*sin$$\theta$$)*d/d$$\theta$$*(sin($$\theta$$*d$$\Theta$$/d$$\theta$$=-l(l+1)

V(r,$$\theta$$)=(Arlrl+Bl/(rl+1))*Pl*cos($$\theta$$)

wouldn't I be setting the potential to zero because the sphere might be an equipotential?

Wait a minute , in the exterior region, wouldn't V(r,theta)=kR^3*cos(theta)/3epilison0*r^2

Last edited: Oct 26, 2008
2. Oct 26, 2008

### gabbagabbahey

The potential on the surface of the sphere certainly will be an epuipotential; and you could set it to zero if you choose to; however in this case the problem is much easier if you don't.

Have you seen the solution for an uncharged metal sphere in a uniform electric field? What is the solution for that problem and where did they set the potential to zero?

If you choose a clever spot to set the potential to zero in this problem, you can use the superposition principle with the solution for the uncharged sphere and a uniformly charged spherical shell of charge Q.

3. Oct 26, 2008

### Benzoate

Couldn't I set the potential to be zero at the boundary condition? Maybe at the edge of the sphere. Since the shape is the surface of a sphere, it wouldn't matter where I would set my boundary condition at the edge of the spheres?

Outside a sphere V(r,theta)= (R^3/r^2)*cos(theta)

Law of superposition principle :

V=V(outside)+V(inside) ?

I don't know the solution for a uniformly uncharged sphere. Why is the solution of a uncharged sphere relevant since I am asked to find the solution for a uniformily charged sphere?

4. Oct 26, 2008

### gabbagabbahey

Start with the solution for the uncharged sphere in an uniform electric field... What is that solution? What are the boundary conditions for that problem?

5. Oct 19, 2010

### erogard

help anyone?

I was wondering if we should simply add kQ/r to the potential given for the uncharged sphere or if there was a way to obtain this result more formally.

6. Oct 20, 2010

### gabbagabbahey

Does your total potential satsify Laplace's equation then? Does it satsify the correct Boundary conditions? What does the uniqueness theorem tell you about that solution?