Potential Energy from an infinite line charge

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

    This isn't a real HW problem for me but just a question I asked myself and I am slightly confused by the solution I get. Here is the situation. You have an infinite line charge and a point charge q. Find the potential energy given to the point charge from the infinite line charge.

    2. Relevant equations

    Gauss' Law
    Work Formula

    3. The attempt at a solution

    Here is my solution.

    ∫E*dS = Q/ε

    Q=∫Q'*dL where Q' is charge per length integrated from 0 to L

    Q = (Q')L

    ∫E*dS = E*2∏rL

    E*2∏rL = (Q')L/ε

    E = Q'/(2∏rε)

    We know that F = qE so

    F= qE = (q*Q')/(2∏rε)

    Work done on a point particle to move it from the line charge to a distance r would be

    W = F*r = (q*Q')/(2∏ε)

    So my final answer is

    Potential Energy = (q*Q')/(2∏ε)

    My math certainly leads up to this answer but I am finding it slightly difficult to accept. I just feel that the potential energy should depend on the distance from the line charge to the point charge but this equation says otherwise. Am I doing something wrong or is my math right???

  2. jcsd
  3. tiny-tim

    tiny-tim 26,016
    Science Advisor
    Homework Helper

    hi hover! :smile:
    yes :smile:
    no :redface:

    W = ∫ F dr :wink:
  4. THAT makes more sense! I knew something was fishy when the potential had no dependence on the distance. The other equation I used can only be used if the force doesn't depend on the distance r which isn't the case here. Since F(dot)dr is equal to F_r*dr, the new equation would then be

    Potential energy = ((q*Q')/(2∏ε))*ln(b/a)

    where a is the starting position and b is the final position radially. The only staring position particle q can't have is where a = 0. I think this is the correct equation.

    If there is something I still missed, let me know but otherwise, thanks for helping me find my mistake!:biggrin:
  5. tiny-tim

    tiny-tim 26,016
    Science Advisor
    Homework Helper

    what's wrong with being fishy? :confused:
  6. Ah yes, I see how this relates to your avatar! :P

    There is nothing wrong with being fishy as long as I can catch the fishies!... err I mean fishiness!
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