# 2-Dimensional Line charge, around a ring.

1. Sep 21, 2008

### MuckThatGuy

1. The problem statement, all variables and given/known data

This is a fairly easy problem, conceptually... but I can't seem to put the numbers together correctly.

A circle with a radius "r" has a linear charge along the circumference with a charge density of $$\lambda=\lambda_{0}\sin(\theta)$$, where $$\lambda_{0}$$ is the max charge along the circumference.

Find:
a.) The direction of the electric field at the center of the ring.
b.) The magnitude.

(They're obviously not looking for a numerical answer)

2. The attempt at a solution

a.) The direction is definitely down, however, I can't seem to 'show' this outside of just my own intuition.

My best attempt at getting everything together is as follows:

$$d\vec{E_{r}}=d\vec{E_{x}}+d\vec{E_{y}}$$

$$d\vec{E_{x}}=k\int\frac{dQ}{r^{2}}=k\int\frac{\lambda \;dL}{r^{2}}=k\int\frac{\lambda_{0}sin(\theta)\;dL}{r^{2}}$$

$$d\vec{E_{y}}=k\int\frac{dQ}{r^{2}}=k\int\frac{\lambda \;dL}{r^{2}}=k\int\frac{\lambda_{0}sin(\theta)\;dL}{r^{2}}$$

When attempting to get dL into terms of $$d\theta$$ is when problems arise. I know it has to be simple, but every method I've tried always ends up with a charge of 0 along both the x and y axis's (which is obviouslly wrong).

ETA: I'm not sure if it just suffices to change dL to the x and y components $$r\;cos(\theta)$$ and $$r\;sin(\theta)$$ respectively. It doesn't feel right to me, but it's the best I can come up with right now. (The book doesn't cover a problem like this).