Potential Energy from an infinite line charge

In summary: PIn summary, the conversation discusses a question about finding the potential energy given to a point charge from an infinite line charge. The conversation includes the use of Gauss' Law and the Work Formula to find the solution. After a mistake is pointed out, the final equation for potential energy is found to be dependent on the distance between the starting and final positions of the point charge.
  • #1
hover
343
0

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
 
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  • #2
hi hover! :smile:
hover said:
F= qE = (q*Q')/(2∏rε)

yes :smile:
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 :redface:

W = ∫ F dr :wink:
 
  • #3
tiny-tim said:
hi hover! :smile:


yes :smile:


no :redface:

W = ∫ F dr :wink:

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:
 
  • #4
hover said:
I knew something was fishy …

what's wrong with being fishy? :confused:
 
  • #5
tiny-tim said:
what's wrong with being fishy? :confused:

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!
 

1. What is the formula for calculating the potential energy from an infinite line charge?

The formula for calculating the potential energy from an infinite line charge is U = λQ/4πε0r, where U is the potential energy, λ is the linear charge density, Q is the charge on the object, ε0 is the permittivity of free space, and r is the distance from the line charge.

2. How does the potential energy from an infinite line charge differ from that of a point charge?

The potential energy from an infinite line charge is directly proportional to the distance from the line charge, while the potential energy from a point charge is inversely proportional to the distance squared. This means that the potential energy from an infinite line charge decreases at a slower rate as the distance increases compared to a point charge.

3. Can the potential energy from an infinite line charge ever be negative?

No, the potential energy from an infinite line charge will always be positive. This is because the linear charge density and charge on the object will also always be positive, and the distance from the line charge cannot be negative.

4. How does the potential energy from an infinite line charge change as the linear charge density or charge on the object is increased?

The potential energy from an infinite line charge will increase proportionally as the linear charge density or charge on the object is increased. This is because both of these factors are directly proportional to the potential energy in the formula U = λQ/4πε0r.

5. Can the potential energy from an infinite line charge ever be zero?

Yes, the potential energy from an infinite line charge can be zero if the linear charge density or charge on the object is zero. This means that there is no electric field present and therefore no potential energy associated with the line charge.

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