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Homework Help: 2-Dimensional Line charge, around a ring.

  1. Sep 21, 2008 #1
    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 [tex]\lambda=\lambda_{0}\sin(\theta)[/tex], where [tex]\lambda_{0}[/tex] is the max charge along the circumference.

    Picture1-3.gif

    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:

    [tex]d\vec{E_{r}}=d\vec{E_{x}}+d\vec{E_{y}}[/tex]

    [tex]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}}[/tex]

    [tex]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}}[/tex]

    When attempting to get dL into terms of [tex]d\theta[/tex] 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 [tex]r\;cos(\theta)[/tex] and [tex]r\;sin(\theta)[/tex] 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).
     
  2. jcsd
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