What is the total force on a line charge due to another? Hw question

AI Thread Summary
The discussion revolves around calculating the total force on one line charge due to another parallel line charge with the same linear charge density. The key challenge is setting up the integral for the electric field generated by the first line charge, which is not constant along the length of the wire. Participants emphasize the need for two integrals: one for the electric field from the first wire and another to sum the contributions from the second wire. The correct approach involves considering the radial component of the electric field and its dependence on the geometry of the setup. Overall, the conversation highlights the importance of properly accounting for the electric field variations in the calculation.
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Homework Statement


linecharges.jpg

Two line charges of the same length L are parallel to each other and located in the xy plane. They each have the same linear charge density λ=constant. Find the total force on II due
to I.

Homework Equations


F =QE
Q = \lambdaL
E = ? This is where I'm stuck.

The Attempt at a Solution


This got me a 0 points on the homework so I know its very wrong. I really just need some clues as to how to set up the integral, or if this is just some kind of trick question, what makes it tricky. I've got tunnel vision at this point so any help will be greatly appreciated.

d\vec{E} = \frac{\lambda dl \hat{r}}{4\pi\epsilon a^{2}}

\vec{E} = \frac{1}{4\pi\epsilon}\int\frac{\lambda dl\hat{r}}{a^{2}}

=\frac{L\lambda}{a^{2}4\pi\epsilon}

\vec{F} = \frac{(L\lambda)^{2}}{a^{2}4\pi\epsilon}
 
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Use Gauss's law to find the electric field generated by an infinite line charge.

EDIT: Nevermind, just read they weren't infinite.

You will have 2 integrals. One for the electric field from wire 1 (and your setup for the E-field from wire doesn't look right at all, you need some z-dependence in it). You will also need a 2nd that runs over the 2nd wire summing up \lambda E_x(y_2) dy_2 to get the net force in the x-direction. This is because the electric field isn't constant over the length of the wire, so you can't just say F = E*Q.
 
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Wouldn't the radial component be E_{x}, & depend only on y and x, since this is a strictly 2-dimensional problem?

Using the hyperphysics site and the setup you see in the picture, I got:

dE_{x} = \frac{\lambda xdy}{4\pi\epsilon r^{2}r}

E_{x} = \frac{\lambda}{x4\pi\epsilon}\frac{L}{\sqrt{x^{2}+L^{2}}}

Aside from that, thank you so much for the input, you broke my tunnel vision! I can see what to do now in principle and its very helpful. :smile:
 
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