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Gauss's Law - A nonconducting spherical shell

  1. Oct 12, 2012 #1
    1. The problem statement, all variables and given/known data

    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. Relevant 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...
     
  2. jcsd
  3. Oct 12, 2012 #2

    vela

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    Simply posting your answers is next to useless. Show your work.
     
  4. Oct 12, 2012 #3
    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)]
     
  5. Oct 12, 2012 #4

    vela

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    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.

    I think if you figure out a), you'll see what's wrong here.
     
  6. Oct 13, 2012 #5
    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?
     
  7. Oct 13, 2012 #6

    vela

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