Gauss's Law - A nonconducting spherical shell

[Edasaur]

1. Homework Statement

A nonconducting spherical shell of inner radius R1 and outer radius R2 contains a uniform volume charge density ρ throughout the shell. Use Gauss's law to derive an equation for the magnitude of the electric field at the following radial distances r from the center of the sphere. Your answers should be in terms of ρ, R_1, R_2, r, ε_0, and π.

a) R_1 < r < R_2
b) r > R_2

2. Homework Equations
∫E dA = Q_enc/ε_0


3. The Attempt at a Solution

For a), I tried using Gauss's law to find it and I arrived at:

E = [ρ(R_1)^3]/[3(ε_0)(r^2)]

For b), I also used Gauss's law to find:

[ρ(R_1)^3 + ρ(R_2)^3]/[3(ε_0)(r^2)]


I'm not quite sure what I'm doing wrong...

 

[vela]

Simply posting your answers is next to useless. Show your work.

 

[Edasaur]

Ok. Here's my work:

For a), my gaussian surface is between the two shells
For b), my gaussian surface is outside both shells

a) ∫E dA = Q_enc/ε_0
E(4πr^2) = [ρ((4/3)π(R_1)^3)]/[ε_0]
E = [ρ((4/3)π(R_1)^3]/[(ε_0)(4πr^2)]
E = [ρ(R_1)^3]/[3(ε_0)(r^2)]

b)∫E dA = Q_enc/ε_0
E(4πr^2) = [ρ((4/3)π(R_1)^3) + ((4/3)π(R_2)^3)]/[ε_0]
E = [ρ(R_1)^3 + ρ(R_2)^3]/[3(ε_0)(r^2)]

 

[vela]

Ok. Here's my work:

For a), my gaussian surface is between the two shells
For b), my gaussian surface is outside both shells

a) ∫E dA = Q_enc/ε_0
E(4πr^2) = [ρ((4/3)π(R_1)^3)]/[ε_0]
E = [ρ((4/3)π(R_1)^3]/[(ε_0)(4πr^2)]
E = [ρ(R_1)^3]/[3(ε_0)(r^2)]

Shouldn't the amount of charge inside the sphere vary with r? The expression you have on the righthand side is a constant that's equal to the amount of charge in a solid sphere of radius R₁ with charge density ρ. It's not applicable to this problem.

b)∫E dA = Q_enc/ε_0
E(4πr^2) = [ρ((4/3)π(R_1)^3) + ((4/3)π(R_2)^3)]/[ε_0]
E = [ρ(R_1)^3 + ρ(R_2)^3]/[3(ε_0)(r^2)]

I think if you figure out a), you'll see what's wrong here.

 

[Edasaur]

I just did a) again:
∫E dA = Q_enc/ε_0
E(4πr^2) = [ρ((4/3)π(r/R_1)^3)]/[ε_0]
E = (ρr)/(3(ε_0)(R_1)^3)

Does that seem right?

 

[vela]

When r=R₁, does it give the right answer?