1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Can someone help me understand resultant cross product vector on half loop of current

  1. Nov 24, 2008 #1
    I can't seem to integrate this correctly. I only need the half loop integration, as i have the correct integration for the infinite lines.

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

    We start with the top half of a half-loop of radius R, centered at the origin, with infinite line segments travelling in the +/- x directions.

    [tex]B=\frac{a}{r^3} (-\vec{k} )[/tex]

    2. Relevant equations

    dF=I (dL x B)

    3. The attempt at a solution

    My attempt at a solution ends up with me getting confused over how to determine the resultant vector by cross product.

    =I Rd\theta (-cos\theta(-\vec{\i})-sin\theta(-\vec{\j})) \times \frac {a}{r^3} (-\vec{k} )
    now, [tex]d\theta = \pi [/tex], and
    I integrate [tex]\int_{-\pi /2}^{\pi/2}(-cos\theta(-\vec{i})-sin\theta(-\vec{j}) [/tex]

    which results in

    [tex]\int_{-\pi /2}^{\pi/2}(-cos\theta(-\vec{i})-\int_{-\pi /2}^{\pi/2}(sin\theta(-\vec{j}) [/tex]

    [tex]\left[(-sin\theta(-\vec{i}) \right]_{-\pi /2}^{\pi/2}+\left[(cos\theta(-\vec{j})\right]_{-\pi /2}^{\pi/2} [/tex]
    yielding 2+0=2

    I then integrate [tex]2\pi IRa \int_{-R}^{R} \frac{dx}{x^3}[/tex]

    but my final result ends up being

    [tex]=2IR\pi a [\frac{-1}{2R^2}-\frac{-1}{2R^2}][/tex] which is zero and I need it to be one.

    Also, I believe the correct answer is
    [tex]\frac{2Ia}{R^2}(-\vec{j})[/tex] which does not have the RΠ factor which I end up with...
    Can someone please help me comprehend this better, as it has been many years since I have had calculus and haven't been able to find any examples of this type of integration.

    Thank you very much!
    Last edited: Nov 24, 2008
  2. jcsd
  3. Nov 24, 2008 #2
    here are some pics

    here are the pictures of my problem and, seemingly, my result.

    Attached Files:

    Last edited: Nov 24, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook