EM M97 1
From QED
A spherical shell is uniformly polarized. The electric polarization per unit volume is , and the inner and outer radii of the shell are a and b. Find the electric potential in each of the three regions: r>b, a<r<b, and r<a.
The effective charge density is given by:
Where ρ is the free space charge density and P is the polarization. Since there is no free space charge density, and the polarization is constant except on boundaries, we can write the effective surface charge density on the spheres as:
Where is the polarization on the outside sphere, and is the unit vector in the radial direction. So:
Where we have chosen the θ = 0 direction to be in the direction, so that θ is the angle between . The general solution for the Laplace equation in spherical coordinates with azimuthal symmetry is:
Where Pl is the l'th Legendre polynomial. Obviously our solution should use l=1:
So it is just a matter of matching the boundary conditions. If we label the regions I (r < a), II (a < r < b) and III (r > b), it will make things easier. Since we must have reasonable behavior as :
We now find boundary conditions at r=a . Continuity:
And derivative:
And for r=b , Continuity:
And derivative:
Collecting all of our equations:
Solving the first two together for C :
Solving the last two for B :
And plugging onto the continuity equations:
Plugging in:
This page was recovered in October 2009 from the Plasmagicians page on Prelim_M97_EM1 dated 22:52, 18 December 2005.