Electric field of a line of charge and direction

[Ahmed A]

Homework Statement

A straight nonconducting plastic wire 8.50 cm long carries a charge density of +175 nC/m distributed uniformly along its length. It is lying on a horizontal table top. Find the magnitude and direction of the electric field this wire produces at a point 6.00 cm directly above its midpoint.

Homework Equations

E = \frac{1}{4πε_0} * \frac{Q}{r^2}
λ = 175 nC/m
a = 4.25 cm
y = 6.00 cm

The Attempt at a Solution

I made a diagram with the rod along the x-axis, with the y-axis intersecting it at the center. The endpoints of the rod are ±a where a=4.25cm.

dE = \frac{1}{4πε_0} * \frac{λdx}{x^2+y^2}
dE_y = dE sin(θ)
sin(θ) = \frac{y}{\sqrt{x^2+y^2}}

dE_y = \frac{λy}{4πε_0} * \frac{dx}{(x^2+y^2)^\frac{3}{2}}
E_y = \frac{λy}{4πε_0} * \int^{a}_{-a} \frac{dx}{(x^2+y^2)^\frac{3}{2}}
E_y = \frac{λa}{2πε_{0}y\sqrt{x^2+y^2}}

The result I got after plugging all the numbers in is E = 3.03 * 10⁴ N/C but other sources online say the correct answer is 4.28 * 10⁴ N/C. What did I do wrong?

 

[gneill]

I don't see a problem with what you've done. (Although you did forget to replace the 'x²' within the radical of the final line with 'a²' -- no doubt a minor oversight).

Your answer looks okay to me.