A point charge +Q is surrounded by a spherical shell of inner radius a and outer radius b. The spherical shell has charge density [tex]\alpha[/tex]r.
a) What is the electric field for r < a?
b) What is the electric field for a < r < b?
c) What is the electric field for r > b?
d) What is the electric potential for r < a?
e) What is the electric potential for a < r < b?
f) What is the electric potential for r > b?
The Attempt at a Solution
For practice, I'm trying to do the problem in two ways: finding the electric fields first and then integrating to get the potential, and finding the potential first and differentiating to get the fields. But it seems that I'm stuck with both ways:
I've already computed the electric fields (using Gauss' Law). Here they are:
r < a: Q / (4*Pi*Epsilon*r^2)
a < r < b: (Q + (1/2)*[tex]\alpha[/tex]*(r^2-a^2)) / (4*Pi*Epsilon*r^2)
r > b: Q + (1/2)*[tex]\alpha[/tex]*(b^2-a^2)) / (4*Pi*Epsilon*r^2)
To find the potentials, I thought I could simply integrate the fields from r to Infinity. This works in the first case (r < a), giving me a potential of Q / (4*Pi*Epsilon*r)
However, this is as far as it goes: the integral for the second case (a < r < b) doesn't converge if I use r and Infinity as bounds again. I assume I need to use different bounds, but I'm still a bit unclear on how exactly this works.
The second way, finding the potential first, is even harder. I assume it would be
k * Integral(dq/r), but I can't figure out how to express the differential element dq.
Thanks for any hints.