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Homework Help: Odd Gauss's Law question

  1. Jan 27, 2010 #1
    1. The problem statement, all variables and given/known data

    Calculate E inside and outside an infinite cylinder of uniform volume charge density using the differential form of Gauss's law.

    2. Relevant equations

    [tex]\nabla[/tex] E = [tex]\frac{p}{e0}[/tex]

    p = charge density

    Divergence in cylindrical polars:


    3. The attempt at a solution

    I'm aware this is much easier using the integral form. I have no problem with calculating E field of various symmetrical shapes using the integral form. However I specifically have to use the differential form. I've never seen an example of this and have looked for quite a while, and I'm not really sure what I'm doing at all.

    All I can think is that by symmetry, differential of E in terms of theta and z are zero, but this still leaves an awkward derivative of E in terms of r, and I'm not sure what to do at that point.
  2. jcsd
  3. Jan 27, 2010 #2


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    You have to use the awkward derivative, sorry. I am going to change symbols and use r for the radial coordinate and ρ for the volume charge density which is constant in this case. You get

    [tex]\frac{d(E_r r)}{dr}=\frac{\rho \; r}{\epsilon_0}[/tex]

    Can you solve this differential equation?
  4. Jan 27, 2010 #3
    That's pretty much what I did, but I didn't think to multiply through by r. After that the integration is quite straight forward, and by varying the limits accordingly I get the same result as by the integral form for E field both inside and outside the cylinder.

    Thanks a lot!
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