Electric field due to a straight rod

[Physicslearner500039]

Homework Statement

A thin straight rod of length 2a carrying a uniformly distributed charge q is located in vacuum. Find the magnitude of the electric field strength as a function of the distance r from the rod's center along the straight line (a) perpendicular to the rod and passing through its center;
Investigate the obtained expressions at r >> a.

Relevant Equations

NA

The net electric field is
$2dE \cos\theta$
$dE = \lambda dx/(4\pi\epsilon (x^2 +r^2)) \\ 2dE \cos\theta = 2r\lambda dx /(4\pi\epsilon (x^2 +r^2)^\frac 3 2) \\ E_{net} = 2\lambda r /(4\pi\epsilon) \int_0^a dx /( (x^2 +r^2)^\frac 3 2) \\ E_{net} = 2\lambda r /(4\pi\epsilon) [\frac x {r^2(\sqrt{r^2 + x^2})}]_0^a \\ E_{net} = \lambda a/(2\pi\epsilon r \sqrt{r^2 + x^2})$
What mistake i am doing the answer does not tally with factor of 2.

 

[etotheipi]

Looks okay to me, just remember you should have $a$ instead of $x$ in that final line and also that $q = \lambda \times 2a$.