What is the Electric Field at P(0,0,4)?

[LeeroyJenkins]

Homework Statement

Infinite unknown line charges of 5nC/m lie along the (positive and negative) x and y axes in free space. Find E at P(0,0,4)

Homework Equations

E = (pl/2(pi)(eo)p)ap
pi = 3.14...
eo - permitivity of free space 8.854 x 10^-12
pl = line charge density
ap = unit vector

The Attempt at a Solution

really just need to know how to solve for the unit vector pl then I can do the rest.
I was thinking along the lines of ap = (z-4) az/ sqrt((z-4)^2) then plug in z... not really sure. Any help would be appreciated.

The answer is 45az but how?

 

[LowlyPion]

Why the unit vector pl?

Through symmetry (You're on the z-axis) isn't the set up equivalent to just a single line charge with density 10 taken from the radius of 4?

 

[nlightner]

To solve for the electric field at P(0,0,4), we can use the superposition principle, which states that the total electric field at a point is the vector sum of the electric fields produced by individual charges.

First, we need to determine the direction of the electric field at P(0,0,4). Since the line charges are along the x and y axes, the electric field will be in the z direction, perpendicular to the plane formed by the x and y axes. Therefore, the unit vector ap will be in the z direction, which is given by az.

Next, we can calculate the electric field produced by each line charge at P(0,0,4) using the given equation E = (pl/2(pi)(eo)p)ap.

For the line charge along the positive x-axis, pl = 5nC/m, p = 4m, and the distance between the charge and P is 4m. Substituting these values into the equation, we get E = (5nC/m x 4m)/(2(pi)(8.854 x 10^-12)(4m)) az = 45az.

Similarly, for the line charge along the negative x-axis, pl = -5nC/m, p = -4m, and the distance between the charge and P is also 4m. Substituting these values into the equation, we get E = (-5nC/m x -4m)/(2(pi)(8.854 x 10^-12)(4m)) az = 45az.

Adding these two electric fields together (since they are in opposite directions), we get a total electric field of 45az at P(0,0,4).