# Electric field of a non-conducting shell

1. Apr 13, 2015

### motyapa

1. The problem statement, all variables and given/known data

Not sure if I'm doing this problem correctly (no answer key for these practice problems). I just want to check with people that know this material well enough.

A hollow spherical non-conducting shell of inner radius a and outer radius b carries charge density p = C/r^2 in the region a =< r =< b. Find the electric field in the following regions

r < a
a < r < b
r > b

2. Relevant equations

$$\varepsilon_0\int E \cdot dA = Qenc$$

3. The attempt at a solution

for r < a

Qenc = 0 so E = 0

for a < r < b

$$Qenc = \int _a^r pdV$$

Volume of a sphere with radius r $$4/3 \pi r^3$$

so then $$dV = 4\pi r^2 dr$$

which means $$Qenc = \int_a^r C/r^2 4\pi r^2 dr$$ or $$\int_a^r4C \pi dr$$

Solving I get $$Qenc = 4\pi C (r-a)$$

Now that I have Qenc I can use

$$\varepsilon_0\int E \cdot dA = Qenc$$

using a gaussian surface of a sphere with radius r, I do

$$\varepsilon_0EA = 4\pi C (r-a)$$

A = 4\pi r^2 so that leaves me with

$$E = C(r-a)/r^2\varepsilon_0$$

for r > b

I used a similar process except I did

$$Qenc = \int _a^b pdV$$

making $$Qenc = 4\pi C (b-a)$$

so then $$E = C(b-a)/r^2\varepsilon_0$$

while my answers make sense to me, I'd like to make sure I'm not making any mistakes because this question is harder than anything ive done so far!

2. Apr 14, 2015

### ehild

It is correct, nice work!