A semi-infinite nonconducting rod (i.e. infinite in one direction only) has unif

Question

A semi-infinite nonconducting rod (i.e. infinite in one direction only) has uniform positive linear charge density lambda. Show that the electric field at point P makes an angle of 45 degrees with the rod and that this result is independent of the distance R. (HINT: Separately find the parallel and perpendicular (to the rod) components of the electric field at P, and then compare those components.)

rod
starts
here
_ +++++++++++++++++++++++++ =====> very long
| |
|
R |
|
| |
- P <== this point is a distance R from the end
of the rod

Explanation / Answer

Fix the end of the rod at the origin, and orient it along the x-axis. Let point (0, P) perpendicular to the end of the rod be given. Given any point (x, 0) on the rod of length dx, the separation vector from (0, P) to (x, 0) is (x, 0) - (0, P) = <x, -P> (I prefer to use <> for vectors and () for points, just a matter of personal preference though). This separation vector has magnitude ?[x

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