Insulating Spherical Shell Potential Problem question

[ramuramu]

Homework Statement

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.

Homework Equations

E=-(grad)V

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!

 

[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.

 

[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!

 

[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!