Electric field of a non-conducting shell

[motyapa]

Homework Statement

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

Homework Equations

[/B]
$$\varepsilon_0\int E \cdot dA = Qenc$$

The Attempt at a Solution

[/B]
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 I've done so far!

 

[ehild]

It is correct, nice work!