# E&M fun (the derivation for electric field from a charged rod to a point charge)

(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! )

[ATTACH=CONFIG]92[/ATTACH]

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

Step 1:

Step 2:

Step 3:

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

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