(First off, I know Mr. F probably has this derivation on the course notes, but I figured it would be great practice to write it out, and also people might be more likely to look at it if it is typed out.)
We just started Electricity & Magnetism in Physics C, and right off the bat the difficulty is enormous. Now I understand why college engineers always complain about electrostatics. At the moment we are derriving the electric field on a point charge, from uniformly charged objects. We've done a rod, a ring, and a disc, perpendicular to a point charge. They are quite difficult, because they involve differentials and require you to manipulate the givens to a integrable point. I figured for my blog post this week i would go through the steps and derrive the electric field on a point charge from a uniformly charged rod.
Here is the picture I drew in photoshop to represent the problem I will be derriving (this picture is much better if you keep in mind that I figured my way around an advanced photoshop that I've never used! )
Also, let it be known that the rod as a uniform linear charge density of
Since the rod is equal on either side of the y axis, the x components of the electric field will cancel, so we only need to worry about the y components. Also, I left it out, but the angle between r and y is and the the equal angle that would be on the the other side of y if i had made another vector to the end of the rod would be . Finally, the far left side of the rod is and the far right side of the rod is
And there you have it! (Jeez that equation editor takes so much time to use...)
If your problem using a rod that has an infinite length... you can back up a few steps before you finally solve your last integral in terms of and and plug in and for the angles, because as , aka