Two line charges lie in the XY plane

[weresquid]

Homework Statement

Line A extends from (2,2, 0) to (-2, 2, 0) and line B extends from (2, -2, 0) to (-2, -2, 0). Each has a linear charge density ρl = 1 nC / m. You want to calculate the magnitude and direction of the electric field due to the two line charges for all points on the z-axis.

a.) Sketch the charge distribution, and set up the integrals you need in order to
solve this problem. Indicate the vector R for each line charge on your plot,
and determine expressions for R and R³

b.) Prove that the magnitude of the electric field in the x and y directions is
zero.

Homework Equations

See pdf attachment.

The Attempt at a Solution

So far I've tried doing the integration technique shown in the attached pdf (this was a class example) and I am not quite sure if that's correct since there are no Z components. Also this problem is with two line charges and not one... so I guess I am basically having trouble finding out where to start? Any help is appreciated.

 

[haruspex]

What do you see as the differences between the class example and the present problem? What can you do to relate the one to the other?

 

[weresquid]

I'm guessing that I could do that formula for each line and then just sum up their total e-fields?

 

[haruspex]

I'm guessing that I could do that formula for each line and then just sum up their total e-fields?

Yes. And the other difference is? (You mentioned a concern regarding z components.)

 

[Nickie]

I would start by setting up the problem and identifying the given information. We have two line charges in the XY plane with a linear charge density of ρl = 1 nC/m. The lines extend from (2,2,0) to (-2,2,0) and (2,-2,0) to (-2,-2,0). We are interested in calculating the electric field at all points on the z-axis.

To solve this problem, we will need to use Coulomb's law, which states that the electric field at a point due to a line charge is given by E = (ρl/2πε) * (1/R), where R is the distance from the point to the line charge and ε is the permittivity of free space.

a.) To visualize the charge distribution, we can sketch the two lines on the XY plane. We can also label the vector R for each line charge, which will help us in setting up the integrals. For line A, R will be from the point (0,0,z) to the point (x,y,0). For line B, R will be from the point (0,0,z) to the point (x,y,0). We can also determine the expressions for R and R3, which are simply the distance from the point to the line charge and its cubed value, respectively.

To solve for the electric field at any point on the z-axis, we will need to integrate over the entire length of the line charges. The integral for line A will be from x = -2 to 2 and the integral for line B will also be from x = -2 to 2. The integrals will have the form of ∫(ρl/2πε) * (1/R) * dx. We will need to use the expressions for R and R3 that we determined earlier in the integrals.

b.) To prove that the electric field in the x and y directions is zero, we can use symmetry arguments. Since the two line charges are symmetric with respect to the y-axis, the electric fields due to each line charge will cancel out in the x direction. Similarly, since the two line charges are symmetric with respect to the x-axis, the electric fields due to each line charge will cancel out in the y direction. Therefore, the magnitude of the electric field in the x and y directions is zero.