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Force acting on conducting contour

  1. Mar 12, 2016 #1
    1. The problem statement, all variables and given/known data
    We have contours C1 and C2 located in vacuum and there are constant currents in them, I1 and I2 respectively. Find the expression for magnetic force that acts on one very small element (dl) of the circle C2 and it's coming from one very small element of the contour C1. Sketch the contours with all the vectors that appear in the expression.

    2. Relevant equations
    F
    =IlxB

    3. The attempt at a solution

    This is how i tried to solve it.

    First, the formula for magnetic force is F=IlxB but, since i have to choose very small elements of the contours it means that i am only interested in dl and not the whole lenght of the contour l so force coming form the one small element of C1 and it's acting on one small element of C2 should be F=I2dl2xB where I2 is the current in C2, dl2 is the small element of C2 and B is magnetic induction vector of one element of C1 in the point where dl2 si located, which means that i should find B.


    Using Biot-Savart's law we have B=μ0I1dl1/(4πr2)

    where r is the distance from one element to another,

    now, since axb=absinα , α-angle between a and b

    then, the final solution is

    F=μ0I2I1dl1dl2/(4πr2)

    Now, i need to know if this is correct, then i'll do the sketch. So, i'd appreciate any help here.
     
    Last edited: Mar 12, 2016
  2. jcsd
  3. Mar 16, 2016 #2

    rude man

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    The Biot-Savart law gives B as a vector too, and in this case a differential one at that, so you should write
    dB = μ0I1dl1 x r/(4πr2)
    with r being the unit vector pointing from dl1 to dl2.

    Then you can write the element of force dF in terms of dB and dl2.
     
  4. Mar 21, 2016 #3

    Finally someone responded! :)

    Ok, i know, B is a vector too, but if we look only at intensity, is my final solution in my previous post correct?
     
  5. Mar 21, 2016 #4

    rude man

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    Finally, you responded (5 days after I responded) :-)
    Your final expression is correct only if the vector elements dl1 and dl2 have the same direction (or opposite direction). For example, if the two elements were at 90 degrees then dF = 0.

    You didn't include a picture of the two circles so maybe that would resolve the issue. And in any case you force is also an element of force so needs to be written as dF or dF.
     
  6. Mar 21, 2016 #5

    I understand what are you saying, it's because of the cross product. Thanks a lot for help!
     
  7. Mar 21, 2016 #6

    rude man

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    Right, and yer' welcome!
     
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