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Magnetic field of infinite wire

  1. Feb 4, 2015 #1
    1. The problem statement, all variables and given/known data
    An infinite wire carries current I.

    upload_2015-2-4_18-24-50.png

    I hope the picture works! The vertical arrow shows the direction of current in the wire. The green arrow has length a and ends at point P. What is the magnitude of the magnetic field at point P?
    2. Relevant equations
    dB = (μ/4π) (I dl×r )/r3
    Where the r on the numerator is a position vector.

    3. The attempt at a solution
    The horizontal part of the wire is irrelevant, as l is in the same direction as r at that point.

    r is the distance of P from the vertical wire, I think. I called the direction of the green arrow the positive x direction, the direction of current and of dl is the y direction. I have to find the magnitude of the field, so I have to integrate the expression for dB. dl can be replaced with dy as they are in the same direction.

    I, μ and 4π are constants and can be taken out of an integral. The magnitude of the cross product on the numerator is given by r dy sinφ, where r now cancels with the r in the denominator.
    I have to do (μI / 4π) ∫ (dy sinφ) / r2
    So the distance r to the wire will be given by Pythagoras' theorem, so r = (x2+y2)1/2.
    My major problem is, I don't really know which angle φ is. It's one of two! Is sinφ given by x/(x2+y2) or should y be on the numerator instead?

    Anyway, I carried on assuming x is on the numerator. Then the integral becomes:
    ∫ x dy / (x2+y2)3/2

    So do a substitution of y=xtanφ. And then how I carry on really depends on what angle φ is!
    Thanks for any help :)
     
    Last edited: Feb 4, 2015
  2. jcsd
  3. Feb 4, 2015 #2
    Oh, and all black lines indicate a part of the wire, which does turn a corner.
     
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