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Curl H outside a long, thin wire of constant current

  1. May 12, 2013 #1

    rude man

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    1. The problem statement, all variables and given/known data
    Per Maxwell's equation and my H. H. Skilling EM textbook, curl H = 0 in the absence of current density.

    But consider a long, thin wire along the z axis carrying time-invariant current I. By Ampere's law, at a point (x,0) outside the wire, H = I/2πx j . But curl H computed by Cramer's rule from the curl determinant is not zero.

    2. Relevant equations

    curl H = J + ∂D /∂t
    2πxH = I

    3. The attempt at a solution

    Since I is time-invariant there is no time-varying electric field so ∂D/∂t = 0.

    Since H = H j = I/2πx j at (x,0),
    curl H = ∂H/∂x k - ∂H/∂z i
    but for a long wire, ∂H/∂z = 0 so we wind up with
    curl H = dH/dx k
    = -I/2π x2 k ≠ 0.

    So how is the paradox resolved, please? If you're familiar with the concept of little "paddle wheels" (Skilling uses) indicating curl, even there it seems curl H ≠ 0 outside the wire.

    All glory and thanks to the resolver!
     
    Last edited: May 12, 2013
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  3. May 12, 2013 #2

    TSny

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    If you want to work in Cartesian coordinates, then first work out the general expression for the components of H in terms of x and y. Since you are going to take derivatives, you don't want to let y = 0 until after taking the derivatives.
     

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    Last edited: May 12, 2013
  4. May 12, 2013 #3

    rude man

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    Thanks, T. Will explore! I had a faint idea I shouldn't ignore the x component of H but never paid it enough attention.
     
  5. May 12, 2013 #4

    rude man

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    I switched to cylindrical coordinates, and lo & behold ... :smile:

    I won't bother with the cartesian but obviously am now a Believer! Thanks again, will get to sleep a bit faster now ...
     
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