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Lorentz force on current-carrying wire

  1. Apr 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Given: a quarter-circle of wire (radius r=0.75m) in a uniform 1.7T magnetic field carrying a current of 3.5A (see diagram)
    physproblem.png
    Find: The force on the wire.

    Note: the connecting wires delivering the current to the quarter circle are parallel to the magnetic field and experience no force.


    2. Relevant equations
    dF = I dS x B


    3. The attempt at a solution
    I know of two ways to solve this.

    The first is to use the fact that the force is path-independent and use the F = I L x B formulation; we're not supposed to do that, as this is supposed to be a calculus problem.

    The second is the way my instructor suggested to the class, which is to argue by symmetry that the force is directed at a 45 degree angle outward from the origin. I can then treat it like a scalar integration problem and use the fact that the magnitude of dS is r d(theta). It's really a quite simple problem this way, but it only works because the problem is particularly simple.

    Having done vector calculus, I feel like there ought to be a third way to attack the thing; it should be possible to parameterize the curve and do something like a line integral. The advantage would be that the same approach would work for a messier problem, where e.g. the magnetic field was nonuniform so the direction wasn't plainly obvious. But I'm not quite sure where to begin. I've got the following parameterization:
    x(t) = r sin t
    y(t) = r cos t
    z(t)=0
    0 <= t <= pi/2
    but I'm not sure where to go from there. Any ideas?
     
  2. jcsd
  3. Apr 13, 2009 #2

    Redbelly98

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    I haven't thought this through completely, but it seems the arc-length formula would be useful:

    dS = (dx2 + dy2)1/2 = (1 + (dy/dx)2)1/2 dx

    And then express dy and dx in terms of t and dt.
     
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