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Infinitely Long Wire Potential

  1. Sep 20, 2015 #1
    I am doing homework for my Intermediate Electricity & Magnetism class. We are currently working on Electric Potential. There are a few problems I have attempted but I have gotten stuck on. I think I keep getting stuck when I have to find what |r-r'| is for uniformly charged objects to some point at which I am finding the potential. Often the geometry gets very much involved and the integrals ugly, but I will post a more simple one (at least I think.) Thank you.

    1. The problem statement, all variables and given/known data

    Find the potential (using the general expression for V obtained from Coulomb’s law) a distance s from an infinitely long straight wire that carries a uniform line charge density λ. (Hint: assume wire is length L, integrate, take the limit L → ∞, express in terms of a reference length s0, and discard a large constant.) Compute the gradient of your potential to check that it yields the correct field.

    2. Relevant equations
    [tex] v = \frac{1}{4\pi\epsilon_{0}} \int \frac{\lambda}{|r-r'|} dl[/tex]

    3. The attempt at a solution
    I would've used Gauss Law to do this, but the prompt says to use Coulomb's.

    Following the instructions stated in the prompt, I first try to draw a diagram with a wire with a point s away from the wire in the perpendicular direction. I assume the wire has length L. I then set λ = q/L, and dl = dx.

    What I seem to be stuck on with a lot of these problems is finding |r-r'| though. For this one I thought it would be, [tex] \sqrt{s^2 + x^2}[/tex] because that is the magnitude of a hypotenuse of a triangle with one leg going to s, and one leg going to some distance x on the wire.

    Since I was told to assume the wire is of length L, I made my limits of integration - L/2 and L/2.

    This all gives me the integral

    [tex] v = \frac{q/L}{4\pi\epsilon_{0}} \int_{-L/2}^{L/2} \frac{\lambda}{\sqrt{s^2 + x^2}} dx[/tex]

    When I solve this integral I get

    [tex] = \frac{q/L}{4\pi\epsilon_{0}} (\ln({\sqrt{s^2 + \frac{L^2}{4}} + \frac{L}{2}}) - \ln({\sqrt{s^2 + \frac{L^2}{4}} + \frac{-L}{2}}))[/tex]

    When I take the limit at L -> infinity of this, I get 0. What did I do wrong/not consider?
     
  2. jcsd
  3. Sep 25, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Sep 25, 2015 #3
    I already handed in the homework and got the solutions. Was suppose to manipulate the last expression to get it in a form in which I could get a limit. Thanks for the bump though. :)
     
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