Insulating Spherical Shell Potential Problem question

1. Apr 19, 2010

ramuramu

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

I have a question about an upper-division E&M HW
problem I'm stuck on. Any help would be GREATLY APPRECIATED!!!

Problem Statement:

An insulating spherical shell of radius R was set up to have a potential
on its surface of V=A*cos^2(theta), where A is a constant. The potential
at a distance very far away from the shell is defined to be zero.

(a) Find the potential V(r,theta) both inside and outside the shell.

2. Relevant equations

3. The attempt at a solution

Since E=-(grad)V , I took the gradient in spherical coordinates of the V expression above
and got

E={[(2*A)/R]cos(theta)sin(theta)}theta-hat

for the E field on the surface of the shell. I then let R become the
variable r and tried to do the line integral from "infinity" to R
to get the potential outside the shell. But this involves a dot product with dr
which is orthogonal to theta-hat so I got ZERO!! which certainly
doesn't make sense...I would REALLY appreciate any advice someone might have on
this problem!!!

2. Apr 19, 2010

fantispug

Think carefully about what you're doing here.

Here's how I have read the question: You have a charged insulating spherical shell and no other sources of charge. You know the potential on the shell; deduce the potential at all other points.

Now the equation E=-grad(V) tells you how to get the electric field from the potential. I couldn't follow exactly what you were doing but it seemed like you calculated the electric field along the surface of a shell (you don't know what it is perpendicular to the shell yet) trying to add together infinitely many spherical shells to find the electric field of a solid sphere of infinite radius?

I think you need to consider general techniques for obtaining the potential due to a charged spherical shell, knowing the potential on the surface of the shell.

3. Apr 19, 2010

ramuramu

THANK YOU very much for your response.

Yes, I agree. I think I need to solve Laplace's Equation in spherical subject to the
boundary conditions:

(i) Vin = Vout
(ii) (dVin/dr - dVout/dr)|r=R = -(surface charge density)/epsilon-knot
(iii) As r-> infinity , V-> 0

Does this sound right to you?

Thank you so much again in advance!

4. Apr 19, 2010

ramuramu

I solved the problem using Laplace's Equation - the only problem is the Dielectic aspect of the shell. I suppose my boundary condition (ii) above:

(ii) (dVin/dr - dVout/dr)|r=R = -(surface charge density)/epsilon-knot

is wrong? Should I just change the epsilon-knot to the epsilon of the dielectric surface?
Although that doesn't seem to make too much sense to me. If there's no dielectric
VOLUME then does it not "matter" and I should just use epsilon-knot for the vacuum around???

Anyway, thank you so much again for your help! I REALLY appreciate it!!!!!