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

Discussion in 'Homework Help' started by StasKO, Dec 25, 2013.

  1. StasKO

    Thread Starter Member

    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=-\nablaψ. 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!!