Question

4 wires with uniform linear density of charge λ, of length 2L c / u form a square with center at the origin of the XY plane, find the electrical force exerted by these 4 wires on a load Q> 0 at a height h from the center of the square

a) raise all the integrals on the side parallel to the Y axis and located in the 4th and 1st quadrants

b) Find the electric force exerted by these 4 wires on a load Q> 0 at a height h from the center of the square

Answer 1

The electric field of a finite line charge is

where

(If the wire is infinite then and we get back the infinite wire result.)

In this case the angle can be determined by dot product. The location of charge Q is

The location of the end of wire, say, in the first quadrant is

Thus, the angles required are given by

The perpendicular distance d to the charge is

So the electric field of only one wire is

Due to symmetry of the problem only z components will remain because all others will cancel. Hence the net field is

and hence the force exerted is