Question

A positive electric charge Qis spread out uniformly along a wire of length L, with one end of the wire located at x=0and the other at x=L. We are going to find the electric field Elocated at x=Dalong the same line as the wire. We will once more use the idea of superposition: the total electric field at the point x=Dis due to the sum of the small electric fields produced by each piece of the charged wire.

1. If the little piece of wire is located at a position xit will produce a piece of the electric field Eat the point x=D. Call this dE. What is the magnitude of dEin terms of Q, L, x, and dx?

2. Now get the total field Eby adding up all the dEfrom the different parts of the wire.

Answer 1

1) The charge Q is linearly distributed on a wire of length L. Linear charge density can be given as:

Charge on a small part of wire of length dx can be given as:

Formula for electric field:

where r = distance between charge and the point

If the charge is located at x and point is at D:

r = x-D

2) Total electric field = submission of all small electric fields because of small parts of wire

Sumbission of small continuous parts: Integration