How Do You Calculate Electric Field for a Non-Parallel Finite Line of Charge?

[Drpepperment]

Homework Statement

Last week I https://www.physicsforums.com/showthread.php?t=483819" which I believe didn't get answered because my description of the problem was a lot messier than it needs to be. Hopefully today, I will not make the same mistake.

In physics books, when calculating the electric field due to a finite line of charge, the problem is usually setup as shown in the top half of my figure. The point of interest / evaluation is placed on the Y axis, and the two endpoints of the line are placed right on the x axis.

What I would like to know is how to set up my integral to solve for this basic problem, if my line weren't parallel with the X axis. I know I could rotate everything by some angle theta (the angle between the line and the x-axis), do my calculations, then rotate everything back into place. But I would much rather discover the correct way to arrange my integrals to solve for this sort of problem directly, rather than altering the reference frame.

Homework Equations

Please see attached diagram below.

The Attempt at a Solution

[PLAIN]http://authman.net/problem.jpg


Thanks in advance for any direction that is bestowed..

 

[BigJDubs]

The electric field due to a finite line of charge can be calculated using the equation:E = (K*q)/(r^2)where K is Coulomb's constant, q is the charge along the line, and r is the distance from the point of evaluation to the charge on the line. In order to solve the problem when the line is not parallel to the x-axis, one must first determine the distance r between the point and each increment of charge along the line. This can be done by using the Pythagorean theorem, which states that the square of the distance between two points is equal to the sum of the squares of the differences in their x-coordinates and y-coordinates. In this case, the distance formula can be written as:r^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 where x_1 and y_1 are the coordinates of the point of evaluation and x_2 and y_2 are the coordinates of the charge along the line. Using this distance formula, the integral for calculating the electric field due to a finite line of charge can be written as:E = (K*q)/(r^2) = (K*q)/((x_1 - x_2)^2 + (y_1 - y_2)^2) Integrating over the entire length of the line gives: E = K*q*∫[(x_1 - x_2)^2 + (y_1 - y_2)^2]^(-1/2)dx This integral can be solved using standard integration techniques.