Electric Potential Due to Spherical Shells and Non-Uniform Surface Charge Density

Thread Starter


Joined Apr 28, 2012

I need some help with this problem.
First of all i want to say sorry that i dont have an image of the problem but ill try to explain it as best as i can.

So we have a spherical shell of radius R0 and this shell has a non-uniform surface charge density: η=η0*cos(2θ), where θ is the the angle measured from the positive z axis and η0 is a constant.

now before i proceed i want to point that a previous question stopped here and asked what is the electric scalar potential everywhere. that problem i solved using laplace equation with azimutal symmetry (∂/∂\(\varphi\)=0).

now the current problem adds another spherical shell 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\(\rightarrow\)∞).

the spheres are concentric.

what is the electric scalar potential everywhere?

my attempt at the solution:

1) write laplace equation for the area r<R0. simplify it by eliminating the term that "explodes" when r goes to zero (by setting its coefficient to zero).

2) write laplace equation for the area R0<r<R1. no terms are eliminated (in the previous problem you could have eliminated the term that "explodes" when r goes to infinity).

at this point i got stuck. i know that i can continue as in the previous problem by "stitching" the potentials at r=R0 by using the continuity of the tangential electric field and the discontinuity of the normal electric field due to η. but this doesnt suffice to find all the unknown coefficients.

i think i need to find some boundary conditions at r=R1 and r=R2.

in the solution that was given to us by the teacher he uses gauss's law for the electric field in integral form in the outer most region (r->R2) and then derives the potential through E=-\(\nabla\)ψ. then he says that because the area between R2 and R1 is perfecly conducting then the potential there is constant and equals to the potential at r=R2 and now there is a boundary condition at r=R1.

now i do understand the whole proccess but i cant understand 2 assumptions that he made:

1) in the region r->R2 (outer most region) the symmetry of the field is spherical
2) the surface charge distribution is uniform on r=R2 but non-uniform (depends on theta) on r=R1.

i guess the second assumption is a consequence of the first one but i cant see why the first assumption is true and how it is possible that on a perfect conductor one lip will have a non uniform charge dist. and on its second lip the charge is distributed uniformly??

and another question regarding the former problem (with only the inner shell):
could i have solved it by using gaussian surface around the shell instead of laplace equation??

thank you for your help!!