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StasKO
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Homework Statement
A spherical shell of radius R0 has a non-uniform surface charge density: η=η0*cos(2θ), where θ is the angle measured from the positive z axis and η0 is a constant. This shell is inserted into another spherical shell (this one has a volume) with its inner lip at radius R1 and outer lip at radius R2 and in the volume between there is a perfect conducting material (denoted σ→∞). this outer shell is electrically neutral.
The two spheres are concentric.
Except for the volume between r=R1 and r=R2, the whole space is characterized by the vacuum constants, ε0 and μ0
What is the electric scalar potential everywhere?
I attached a sketch of the problem.
Homework Equations
Laplace Equation in spherical coordinates with azimutal symmetry (∂/∂[itex]\varphi[/itex]=0):
[itex]\nabla[/itex]2ψ=0
The Attempt at a Solution
In the regions r<R0 and R0<r<R1 I write the general solution for the laplace equation:
∑(Anrn+Bnr-n-1)Pn(cosθ), n goes from 0 to ∞ and Pn() are lagandere polynomials.
In the region r<R0 i set Bn=0 so that the solution won't "explode" when r→0. I "stitch" the two potentials (from either side) at r=R0 through the continuity of the tangential electric field and the discontinuity of the normal electric field due to η.
Here i got stuck. I think that I need to find the boundary conditions at r=R1 and r=R2 but I can't find it.
Any help would be appreciated,
Thanks!