Potential in Concentric Spherical Shells

[Nicolaus]

Homework Statement

Two grounded spherical conducting shells of radii a and b (a < b) are arranged concentrically. The space between the shells carries a charge density ρ(r) = kr^2. What are the equations for the potential in each region of space?

Homework Equations

Poisson's and LaPlace's in Spherical Coordinates

The Attempt at a Solution

I solved Poisson's Equation for the space between the shells, in spherical coordinates, and arrived at:
V(r) = (1/ε)kr^2/6 - (C1)/r + (C2)
where C1 and C2 are the constants of integration.
What would be the general solution for the potential in the other regions where ρ=0? Would I simply apply Laplace's equation in those regions, than apply the suitable boundary conditions?

 

[TSny]

I solved Poisson's Equation for the space between the shells, in spherical coordinates, and arrived at:
V(r) = (1/ε)kr^2/6 - (C1)/r + (C2)
where C1 and C2 are the constants of integration.

Your first term on the right does not have the correct dimensions for electric potential.

What would be the general solution for the potential in the other regions where ρ=0? Would I simply apply Laplace's equation in those regions, than apply the suitable boundary conditions?

Yes, that will work.

 

[Nicolaus]

I made a mistake, the equation in-between a and b should read: V(r) = kr^4/20 - c1/r +c2
The boundary conditions should be Vinside(a) = Vbetween(a) and Vbetween(b) = Vout(B) for continuity; am I missing any other conditions? I know there's the discontinuous derivative of potential = some surface charge, but I am not given such a surface charge.

 

[TSny]

I made a mistake, the equation in-between a and b should read: V(r) = kr^4/20 - c1/r +c2

I believe the sign of the first term is incorrect.

The boundary conditions should be Vinside(a) = Vbetween(a) and Vbetween(b) = Vout(B) for continuity; am I missing any other conditions? I know there's the discontinuous derivative of potential = some surface charge, but I am not given such a surface charge.

I don't understand your boundary conditions. Perhaps it's the notation you are using. The usual interpretation of "grounding a conductor" is to set the potential of the conductor to 0.

The potential is continuous everywhere. As you say, the derivative of V will be discontinuous at a surface containing surface charge.

You will be able to determine the surface charges after you find V.