A thin non-conducting rod of finite length L carries a total charge q, spread uniformly along it. We have to show that E at point P on the perpendicular bisector is given by

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As charge q is uniformly distributed over a length L, the linear charge density is

.

Looking at the direction of the electric field vectors at point P due to the symmetrically placed charge elements on the two sides of the rod, we note that the resultant electric field will be along the Y-axis.

The magnitude of the electric field at point P will be given by the integral

For calculating this integral, we make the substitution

We therefore have

.

And

Or