Electricity and Magnetism on lines of charge

[goodam]

Homework Statement

Two infinitely-long lines of charge run parallel to the z-axis. One has a positive uniform charge per unit length, [lambda]>0, and goes through the x y plane at x=0, y=d/2. The other has a negative uniform charge per unit length, -[lambda], and goes through x=0, y=-d/2. Nothing changes with the z coordinate; the state of affairs in any plane parallel to the x y plane is the same in the x y plane.

Homework Equations

This is where I need help.

The Attempt at a Solution

My thought is the charge will be 0, but I cannot prove this without an equation or some work.

 

[dynamicsolo]

What is the question the problem is asking about?

 

[Moetasim]

I would first clarify the question and make sure I understand the setup correctly. I would ask for more information, such as the distance between the two lines of charge (d) and the values of [lambda].

Assuming that the distance between the two lines is d and [lambda] is equal in magnitude for both lines, the electric field at any point in the x-y plane can be calculated using the superposition principle, which states that the total electric field at a point is the vector sum of the electric fields produced by each individual charge.

Using this principle, the total electric field at a point in the x-y plane would be the sum of the electric fields produced by the positive line of charge and the negative line of charge. Since the lines are parallel and the distance between them is constant, the electric field produced by each line will also be parallel and in opposite directions.

Therefore, the electric field at any point in the x-y plane will be cancelled out and the net electric field will be zero. This means that the charge in the x-y plane will also be zero and the state of affairs in any plane parallel to the x-y plane will be the same as in the x-y plane.

To prove this, I would use the equation for the electric field produced by a line of charge, which is E = [lambda]/(2πε0r), where [lambda] is the charge per unit length, ε0 is the permittivity of free space, and r is the distance from the line of charge.

By plugging in the given values, we can see that the electric fields produced by the positive and negative lines of charge will have equal magnitudes but opposite directions, resulting in a net electric field of zero.

In conclusion, the state of affairs in any plane parallel to the x-y plane will be the same as in the x-y plane and the net charge in the x-y plane will be zero.