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Net Force on a circular current carrying wire, from an infinite wire.

  1. Apr 26, 2014 #1
    1. The problem statement, all variables and given/known data

    -I've attached a picture of the problem-

    An infinitely long straight wire of steady current I1 is placed to the left of a circular wire of current I2 and radius a as shown. The center of the circular wire is distance d(≥ a) away from the straight wire. Let’s find the net magnetic (Lorentz) force acting on the entire circular wire as follows:

    (A) Express the magnetic field Bp at point P (due to the
    current I1) in given quantities (including its direction).

    (B) The Lorentz force due to the magnetic field Bp acting on a small current segment I2dℓ at P is given by

    dF = I2dℓ × Bp

    Express dF = (dFx, dFy, dFz) in component representation in given quantities.

    (C) By integrating your results from (B) show explicitly that the net Lorentz force for the
    entire circular wire is given by

    Fnet = µoI1I2[itex]\left(1-[itex]\frac{d}{\sqrt{d2-a2}}[/itex]\right)[itex][itex]\hat{y}[/itex]​


    2. Relevant equations

    Most given in question.

    Bp=[itex]\frac{-μ0I1}{2π(d+x)}[/itex]​


    3. The attempt at a solution

    The first two parts I got through pretty easily. Part (A) was just giving the equation of the B-field for an infinite wire. For part (B) I ended up with:

    dF=[itex]\left(\frac{-μ0I1I2acosødø}{2(d+acosø)}\right)[/itex][itex]\hat{x}[/itex]+[itex]\left(\frac{-μ0I1I2asinødø}{2(d+acosø)}\right)[/itex][itex]\hat{y}[/itex]​

    which I got from taking the cross product of I2dl and Bp, and am fairly certain is correct.

    Now, for part (C) I keep getting stuck, I see what I'm supposed to get for Fnet, but my answer always has a natural log, or becomes zero, and I don't know how else to approach the problem.
     

    Attached Files:

  2. jcsd
  3. Apr 26, 2014 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Hello bluecadeetthree. Welcome to PF!

    Yes, I think that's correct. (I tried to fix the latex, hope I got it the way you intended.)

    The integration of the y-component is straightforward. (The answer for this component should also be evident by symmetry.) The x-integration seems more difficult. I threw it into Mathematica and it popped out the answer. I also managed to do it by converting it to a contour integral in the complex plane. I'm not seeing an elementary way to do it, but I'm probably just not seeing it. I spent a few minutes searching integral tables on the internet but did not find it.

    The answer for the net force that you stated in part (C) appears to have the wrong direction.
     
  4. Apr 26, 2014 #3
    I was thinking the given net force had the wrong direction. I'll have to email my professor about that. Thanks for your help!
     
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