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Semicircular Line Charge Electric Field

  1. Sep 15, 2011 #1
    1. The problem statement, all variables and given/known data

    See figure attached.

    2. Relevant equations



    3. The attempt at a solution

    The question I have is with regards to, [itex]dE_{x}. dE_{y}, dE_{z}, dE_{h}[/itex].

    First,

    [itex]dE_{z} = dEcos \alpha \quad \text{(form of z component of spherical coordinates)} [/itex]

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

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

    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,

    [itex]\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)[/itex]

    My y-component disapeers as it is supposed to, and if I pull out a factor of, [itex]2\pi[/itex] 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,

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

    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?
     

    Attached Files:

    Last edited: Sep 15, 2011
  2. jcsd
  3. Sep 18, 2011 #2
    Bump, still looking for some help!
     
  4. Sep 18, 2011 #3
    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.
     

    Attached Files:

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