How to Find the Electric Field on the XY-Plane from a Line Charge?

[bowlbase]

Homework Statement

A wire at extends for -L to L on the z-axis with charge $\lambda$. Find the field at points on the xy-plane

Homework Equations

$E(r)=k\int\frac{\rho}{r^2}dq$
$k=\frac{1}{4\pi ε_o}$

The Attempt at a Solution

First time I've looked for field on a plane so I wasn't sure if I'm doing this correctly.

$dq=\lambda dz=2L\lambda$
I made a right triangle with one side L and the other two x and y.
$r^2=L^2+x^2+y^2$ where only x and y change so I have dxdy.

So, the final integral

$k(2L\lambda)\int\int\frac{1}{L^2+x^2+y^2}dxdy$ with limits 0→∞.

I expect at large x,y that the field should be very small and small x,y it should be large. This integral satisfies both of those conditions.

The method I saw our professor do for another, somewhat similar, problem was exceedingly more complicated than this.

 

[CompuChip]

Hmm, the question is to find the field at some point (x, y, 0). So you should fix x and y, and add up all the contributions from the charges on the z-axis.
Therefore I would expect something like
$$E(x, y, 0) = k \int_{-L}^L \frac{\rho}{r^2} \, dz$$
(where did your $\rho$ go? Should there not be something with $\lambda$ in there)?

Try drawing such a point, and calculating $r$ and $\rho \, dz$ from the geometry of the problem rather than by trying to copy your notes.

 

[bowlbase]

$\rho$ is the charge density $\lambda$
I was trying to use $\rho$ in the general sense but I sort of got ahead of myself putting it in there.

$E(r)=k\int\frac{1}{r^2}dq$

$dq=\lambda dz$

so, $E(r)=k\int\frac{\lambda}{r^2}dz$


As soon as I walked away from the computer I knew I was wrong. My whole way of thinking was stuck in previous problems I've done.

X and Y are my points of interest and Z is my variable (for the line charge not on the plane). So if I want to figure out what the field is at X and Y I need to count up all the little pieces of charge at distance 'r' from them to the line.

I have $\int_{-L}^{L} \frac{\lambda}{z^2+x^2+y^2}dz$