Question

A very long straight uniformly charged thread carries a charge λ per unit length. Find the magnitude and direction of the electric field strength at a point which is at a distance y from the thread and lies on the perpendicular passing through one of the thread's ends.

Answer 1

The problem is reduced to finding E_x and E_y viz. the projections of vector E in Fig, where it is assumed that λ > 0.

Let us start E_x. The contribution to E_x from the charge element of the segment dx is 

Let us reduce this expression to the form convenient for integration. In our case,  dx = rdα/cos α, r = y/cos α. Then

Integrating this expression over α between 0 and π/2, we find

In order to find the projection E it is sufficient to recall that differs from dE in that sin α in (1) is simply replaced by cos α. 

This gives

We have obtained an interesting result :

E_x = E_y in dependently of y, 

i.e. vector E oriented at the angle of 45° to the rod. The modulus of  is