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

In summary, the problem involves two line charges with the same linear charge density and length, parallel to each other in the xy plane. The goal is to find the total force on one charge due to the other. The solution involves using Gauss's law and setting up two integrals to calculate the electric field and then summing them to find the net force. The electric field will vary with distance from the wire, making it necessary to integrate over the length of the wire.
  • #1
ProtonHarvest
4
0

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 = [tex]\lambda[/tex]L
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.

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

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

=[tex]\frac{L\lambda}{a^{2}4\pi\epsilon}[/tex]

[tex] \vec{F} = \frac{(L\lambda)^{2}}{a^{2}4\pi\epsilon}[/tex]
 
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  • #2
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 [itex]\lambda E_x(y_2) dy_2[/itex] 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.
 
Last edited:
  • #3
Wouldn't the radial component be [tex]E_{x}[/tex], & 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:

[tex]dE_{x} = \frac{\lambda xdy}{4\pi\epsilon r^{2}r}[/tex]

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

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:
 

1. What is a line charge?

A line charge is a distribution of electric charge that is confined to a one-dimensional line or curve.

2. What is the total force on a line charge due to another?

The total force on a line charge due to another is the sum of all the individual forces between each pair of charges along the line. This can be calculated using Coulomb's Law, which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

3. How is the direction of the force determined?

The direction of the force on a line charge due to another is determined by the relative positions of the two charges. If the charges are of the same sign, the force will be repulsive and if they are of opposite signs, the force will be attractive.

4. What factors affect the total force on a line charge?

The total force on a line charge is influenced by the magnitude and sign of the charges, as well as the distance between them. In addition, the medium in which the charges are located can also affect the force due to its dielectric constant.

5. How is the total force on a line charge calculated?

The total force on a line charge can be calculated by summing up all the individual forces between each pair of charges along the line. Alternatively, it can also be calculated by integrating the electric field along the line charge using the equation F = qE, where q is the charge and E is the electric field strength.

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