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Spherical Charge Distribution - Electric Field Intensity

  1. Oct 9, 2014 #1
    1. The problem statement, all variables and given/known data
    A spherical charge distribution is given by [itex] p = p_0 (1- \frac{r^2}{a^2}), r\leq a[/itex] and [itex] p = 0, r \gt a [/itex], where a is the radius of the sphere.

    Find the electric field intensity inside the charge distribution.

    Well I thought I found the answer until I looked at the back of the book....

    2. Relevant equations
    [itex] \oint \vec E \cdot d \vec A = \frac{q_{inside}}{\epsilon_0}[/itex]

    3. The attempt at a solution
    [itex] \oint \vec E \cdot d \vec A = \frac{q_{inside}}{\epsilon_0}[/itex]

    [itex] E 4\pi r^2 = \frac{4\pi r^3 p}{3 \epsilon_0}[/itex]

    [itex] E = \frac{p_0 r}{3\epsilon_0}(1 - \frac{r^2}{a^2}) [/itex]

    but my book has as the answer:
    [itex] E = \frac{p_0 r}{\epsilon_0}(\frac{1}{3} - \frac{r^2}{5a^2}) [/itex]

    I have no idea where the extra factor came from... Did I do something wrong or is it a mis-print?
     
  2. jcsd
  3. Oct 9, 2014 #2

    gneill

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    Staff: Mentor

    The book's answer is correct.

    Since the charge density is changing with radius you'll have to do the integration over the interior volume (within the Gaussian surface) to find the total charge. You can't just multiply the volume by the fixed value of charge density at the Gaussian surface.
     
  4. Oct 10, 2014 #3
    got the right answer now - thanks for the help
     
  5. Oct 10, 2014 #4

    jtbell

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    Staff: Mentor

    Just as an aside, if you happen to be looking at this thread again, to put a "rho" (##\rho##) in your LaTeX equations, for charge density, type "\rho". (And similarly for other Greek letters. For capital Greek letters, capitalize the word, e.g. "\Sigma".)
     
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