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Missing a Factor of 2 in a Poynting Vector Verification

  1. Dec 16, 2015 #1
    The question reads:

    A fat wire, radius a, carries a constant current I, uniformly distributed over its cross section. A narrow gap in the wire, of width w << a, forms a parallel-plate capacitor.

    I have drawn a red box at the bottom of the page where the Poynting theorem is supposedly verified - however it seems to be a factor of 2 out. I am happy that the derivation for uem is correct, it must be something to do with the very last line. Could someone please point me in the right direction, excuse the pun.

    Poynting.png
     
    Last edited: Dec 16, 2015
  2. jcsd
  3. Dec 16, 2015 #2
    You should be careful about taking the divergence in cylindrical coordinates.
     
  4. Dec 16, 2015 #3
    Is this a hint? I know there are some steps are missing, does a factor of two comes out when the divergence of s is taken in cylindrical coordinates?
     
  5. Dec 16, 2015 #4
    Yes, it was a hint. The divergence of a vector field with only a radial component is given by [tex] \nabla \cdot (A_s\hat{s})=\frac{1}{s}\frac{\partial}{\partial s}(sA_s). [/tex]

    So in your example ## A_s =s## and the divergence becomes

    [tex] \nabla \cdot (s \hat{s})=\frac{1}{s}\frac{\partial}{\partial s}(s^2)=2. [/tex]
     
  6. Dec 16, 2015 #5
    Thanks so much, this answers my question perfectly.
     
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