# Gauss' law electrostatics problem involving charge densities

1. Nov 19, 2013

### chipotleaway

1. The problem statement, all variables and given/known data
A nonconducting spherical shell has a thickness $b-a$, where b is the outer radius and a the inner radius has a volume charge density $\rho=\frac{A}{r}, r\in[a,b]$. If there is a charge +q located at the center, what must $A$ be in order for the electric field to be uniform in the shell?

3. The attempt at a solution
The electric field for any $r\in[a,b]$ must be equal to the field at $a$, whiuch is $E_1=k\frac{q}{a^2}$. The field at any point in the shell is $E_2=k(\frac{q}{r^2}+\frac{4A\pi}{3r}(r^3-a^3)\frac{1}{r^2})$. I equated the two expressions and tried to solve for A but the expression I'm getting is not in agreeance with the the solution. Is this approach correct?

Here's a diagram of the problem (right)

2. Nov 19, 2013

### vanhees71

The approach is correct, but you should reconsider the charge inside your Gaußian surface. It's not uniform, and thus you have to really evaluate the volme integral!

3. Nov 19, 2013

### chipotleaway

Got it! Thanks :p

4. Nov 20, 2013

### rude man

5. Nov 20, 2013

### Joncat

A non-conducting spherical shell carries a non-uniform charge density ρ=ρ0r1/r. Determine the electric field in the regions:
A) 0<r<r1
B)r1<r<r0
C)r>r0

r1 is radius to inside of shell. r0 is radius to outside of shell.