Field on a plane from line charge

1. Nov 20, 2013

bowlbase

1. The problem statement, all variables and given/known data
A wire at extends for -L to L on the z-axis with charge $\lambda$. Find the field at points on the xy-plane

2. Relevant equations

$E(r)=k\int\frac{\rho}{r^2}dq$
$k=\frac{1}{4\pi ε_o}$
3. 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.

2. Nov 20, 2013

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.

3. Nov 20, 2013

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