1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Electrostatic Energy of Sphere in Shell

  1. Sep 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Compute, in the following two ways, the electrostatic energy [itex]W[/itex] of the uniformly charged solid sphere of radius [itex]a[/itex] (charge density [itex]\rho[/itex]) that is surrounded concentrically by a uniformly charged thin
    spherical shell of radius [itex]b[/itex] (surface charge density [itex]\sigma[/itex]), where the charge densities satisfy [itex]\frac {4\pi a^3}{3}\rho + 4\pi^2 \sigma = 0[/itex] That is, the sum of all the charge is zero.
    (a) Compute [itex]W=\int \frac {\epsilon_0 E^2}{2}d\tau[/itex].
    (b) Compute [itex]W=\int \frac {\rho V}{2}d\tau[/itex].

    2. Relevant equations
    [itex]V = -\int E \cdot dl[/itex]

    3. The attempt at a solution
    For part (a) the electric field between the shell and the sphere is [itex]E = \frac {a^3 \rho}{3\epsilon_0 r^2}[/itex]. Plug it into the equation and integrate using spherical coordinates. I got [itex]W_1 = \frac {2\pi a^5 \rho^2}{9\epsilon_0}(1 - \frac {a}{b})[/itex]. Then the electric field inside the sphere is [itex]E=\frac {r\rho}{3\epsilon_0}[/itex]. Integrating over the sphere I got [itex]W_2 = \frac {2\pi a^5 \rho^2}{45 \epsilon_0}[/itex].

    In part (b) I found [itex]V[/itex] from [itex]E[/itex]. Integrating over the volume between the shell and sphere I got [itex]W_1 = \frac {2\pi a^3 \rho^2}{18\epsilon_0} b^2 - \frac {2\pi a^5 \rho^2}{6\epsilon_0} + \frac {2\pi a^5 \rho^2}{9\epsilon_0}(\frac {a}{b})[/itex]. Then integrating over the sphere I got [itex]W_2 = \frac {2\pi a^5 \rho^2}{45 \epsilon_0}[/itex].

    It seems that I went wrong finding the volume integral using [itex]\int \frac {\rho V}{2}d\tau[/itex]. My setup was
    [itex]W_1 = \frac {1}{2} \frac {a^3 \rho^2}{3 \epsilon_0}\int_{0}^{2\pi} \int_0 ^\pi \int_a ^b \left (\frac {1}{r} - \frac {1}{b} \right ) (r^2 \sin \theta dr d\theta d\phi) [/itex].
    Can anyone spot my mistake?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted