Electric field and potential outside and inside nonuniform spherical shell

[krickette]

Homework Statement

Find the electric field and potential inside and outside a hollow spherical shell (a \leq r \leq b) which carries a charge density proportional to the distance from the origin in the region, \rho=kr, for some constant k.

Homework Equations

The Attempt at a Solution

I think I've got the electric field inside the shell figured out, but I could be way off base. I have no idea how to do the rest...
help?

also, sorry, for some reason the <or= sign between a and r isn't working

 

[Curt Inman]

Remember Gauss' equation for the electric field: E=Q/(4pi e*r^2), where Q is the charge enclosed, pi=3.14159..., e* is a constant (the permittivity of free space), and r is the distance from the "center" of the charge distribution (analogous to center of mass). Inside the spherical shell (r<a<b), E=0 (no charge is enclosed). For a<r<b, the charge enclosed is obtained by integrating the charge density over the volume enclosed. In this case, the charge density is kr, and the differential element of volume is 4(pi)r^2 x incremental change in r (dr). This yields an enclosed charge of (pi)k(r^4-a^4), and substituting into Gauss' equation yields E=(k(r^4-a^4)/(4e*(r^2)). Outside the spherical shell, the enclosed charge is given by Q=(pi)k(b^4-a^4), and this yields E=(k(b^4-a^4))/(4e*(r^2)) for all r such that a<b<= r.