What is the formula for the electric field above a plane?

[likephysics]

Homework Statement

This is problem 47(chapter 21) in the textbook - Physics for engineers and scientists (Giancoli)
Uniformly charged wire has length L, where point 0 is the mid point. Show that the field at P, perpendicular distance x from 0 is

E= (lambda/2*pi*epsilon_0) *(L/x*sqrt(L^2+4x^2)

Homework Equations

The Attempt at a Solution

I tried solving it, I got E = - (lambda/2pi*epsilon_0)*[1/sqrt(L^2+4x^2)-1/2x)
Is something wrong with my integration?

My attempt is correct until

E= lamda/r^2*4pi*epsilon_0 * integration (cos theta)dl

After this, I use cos theta = x/r (r is the hypotenuse = sqrt (L^2+4x^2))

I am trying to do this instead of taking r=x cos theta.

 

[Delphi51]

Looks like something wrong with the "(cos theta)dl."
I'll use A instead of theta. And z in place of your l, running from -L/2 to L/2.
I get some constants times integral of cos(A)/(x^2 + z^2)*dz
Since tan(A) = z/x, I can use z = x*tan(A) to simplify the integral.
And dz = x*sec^2(A) dA
After the dust settles on this change of variable, I get integral of cos(A)dA.

Not the difference from your integral: I have dA where you have dl
I end up with the given answer.

 

[likephysics]

If I use r=cos theta/x, then i get the correct result.
I am trying to get there without that substitution and just cos theta = x/sqrt(x^2+(l/2)^2)
huh. Doesn't work if I use the integral but does work if I take l/2 as y initially. The whole term becomes (x^2+y^2) and then apply limit to get answer.