Semicircular Line Charge Electric Field

[jegues]

Homework Statement

See figure attached.

Homework Equations

The Attempt at a Solution

The question I have is with regards to, $dE_{x}. dE_{y}, dE_{z}, dE_{h}$.

First,

$dE_{z} = dEcos \alpha \quad \text{(form of z component of spherical coordinates)}$

$dE_{x} = -dE_{h}cos\phi = -dEcos\phi sin\alpha \quad \text{(form of x component of spherical coordinates)}$

$dE_{y} = -dE_{h}sin\phi = -dEsin\phi sin\alpha \quad \text{(form of y component of spherical coordinates)}$

Is there something to take away from this? He does this problem as a line integral in each direction respectively, but can I do it in one line integral? If so, how would I set that up?

Edit:

Here's my attempt at it. (See 2nd figure attached)

I didn't end up getting to the actual answer in the textbook, but it appears as though I'm pretty close. The answer they give is,

$\frac{p_{l}a}{2 \epsilon_{0}(z^{2} + a^{2})^{\frac{3}{2}}} \left( \frac{-a}{\pi}\hat{i} + \frac{z}{2} \hat{j} \right)$

My y-component disapeers as it is supposed to, and if I pull out a factor of, $2\pi$ from my vector, I get part of the form we're looking for in the answer. My x-component is missing a factor of a, and my z-component a factor a z and I have a root(2) sitting in the denominator.

So the final answer I obtained, trying to get it as close to theirs as possible,

$\frac{p_{l}a}{2\sqrt{2} \epsilon_{0}(a^{2} + z^{2})} (\frac{-\hat{i}}{\pi} + \frac{\hat{j}}{2})$

I don't know if I am totally wrong, or if I just screwed up one portion of the problem.

Any ideas? Does anyone see any problems?

 

[jegues]

Bump, still looking for some help!

 

[Spinnor]

You are on the right track, but I would have written down separate expressions for Ex and Ez. You have the magnitude right but not the right parts. See the attached.