Potential Energy from an infinite line charge

[hover]

Homework Statement

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.

Homework Equations

Gauss' Law
Work Formula

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

 

[tiny-tim]

hi 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 ​

 

[hover]

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!

 

[tiny-tim]

I knew something was fishy …

what's wrong with being fishy?

 

[hover]

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!