# Two line charges lie in the XY plane

• weresquid
In summary, the problem involves two line charges with a linear charge density of 1 nC/m each, extending from (-2,2,0) to (2,2,0) and (-2,-2,0) to (2,-2,0) on the z-axis. The goal is to calculate the electric field in both magnitude and direction at all points on the z-axis. This can be done by setting up integrals and using the given formula (provided in the attached pdf) for each line charge and summing up their total electric fields. The main concern is the lack of z components in the given formula and how to relate it to the present problem.
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.

#### Attachments

• Line Charges.pdf
162.9 KB · Views: 257
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?

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

weresquid said:
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.)

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.

## 1. What is the formula for calculating the electric field between two line charges in the XY plane?

The formula for calculating the electric field between two line charges in the XY plane is: E = (k * λ1 * λ2 / r) * (cosθ1 + cosθ2), where k is the Coulomb constant, λ1 and λ2 are the line charges, r is the distance between the two charges, and θ1 and θ2 are the angles between the line joining the two charges and the XY plane.

## 2. How does the distance between the two line charges affect the strength of the electric field?

The strength of the electric field between two line charges in the XY plane is inversely proportional to the distance between the charges. This means that as the distance between the charges increases, the electric field strength decreases.

## 3. Can the direction of the electric field between the two line charges change?

Yes, the direction of the electric field between two line charges in the XY plane can change depending on the orientation of the charges. If the two line charges have the same sign, the electric field will point away from both charges. If the two line charges have opposite signs, the electric field will point towards the positive charge and away from the negative charge.

## 4. How do the magnitudes of the two line charges affect the electric field between them?

The magnitudes of the two line charges have a direct effect on the strength of the electric field between them. As the magnitudes of the charges increase, the electric field strength also increases.

## 5. Can the electric field between the two line charges be negative?

Yes, the electric field between two line charges in the XY plane can be negative. This occurs when the two line charges have the same sign and the electric field points towards the charges instead of away from them. This can happen if the distance between the charges is small enough or if the magnitudes of the charges are large enough.

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