# Potential Energy from an infinite line charge

by hover
Tags: charge, gauss, line, potential energy
 P: 344 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??? Thanks
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P: 26,148
hi hover!
 Quote by hover F= qE = (q*Q')/(2∏rε)
yes
 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)
no
W = ∫ F dr
P: 344
 Quote by tiny-tim hi hover! yes no … W = ∫ F dr
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!

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Thanks
P: 26,148
Potential Energy from an infinite line charge

 Quote by hover I knew something was fishy …
what's wrong with being fishy?
P: 344
 Quote by tiny-tim what's wrong with being fishy?
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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