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Uniformly charged disk and the E field some distance Z from the center

  1. Jan 23, 2013 #1
    1. The problem statement, all variables and given/known data

    Hi,

    I have a problem that describes a uniformly charged disk and the electric field a distance z from the center.

    I have found an equation that describes the E field at any point z already. Now I have to find out how the E field decreases as z increases-- as 1/r^2, I assume, but I am not sure.

    2. Relevant equations

    E = 2[itex]\pi[/itex]kσz(1-[itex]\frac{z}{\sqrt{z^{2}+R^{2}}}[/itex]) where R is the radius of the disk and z is the distance away from the disk.

    3. The attempt at a solution

    I know I have to do a binomial expansion of (R/z). I think I am having mathematical issues rather than physical issues.

    Before the expansion, I need to get some term (R/z). Does taking out a z^{2) from the denominator leave a z^{4} out front? Or a z^{3} out front? If I can figure this out then I know the rest.
     
  2. jcsd
  3. Jan 23, 2013 #2

    TSny

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    ##\sqrt{z^2+R^2} = \sqrt{z^2(1+(R/z)^2)}## and ##\sqrt{(a b)} = \sqrt{a}\sqrt{b}##
     
  4. Jan 23, 2013 #3

    TSny

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    I believe you must have a typo in this expression. The dimensions on the right do not match the dimensions of electric field.
     
  5. Jan 23, 2013 #4
    Good catch: The expression should be E = 2[itex]\pi[/itex]kσ(1-[itex]\frac{z}{\sqrt{z^{2}+R^{2}}}[/itex]).

    So, the denominator of the last term can be rewritten as z[itex]\sqrt{1-(R^{2}/z^{2})}[/itex]. After a binomial expansion, I got that the E field decreases as 1/z.
     
  6. Jan 26, 2013 #5

    TSny

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    I don't think that's correct. Check your binomial expansion simplification. Your original idea that the disk should behave as a point charge for large distances is right.
     
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