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Cauchy's Integral Formula and Ampere's Law - Any Connection?

  1. Feb 5, 2013 #1
    I know how, by Amp[itex]\grave{e}[/itex]re's Law for the loop integral of a magnetic field, $$\oint_C \textbf B \cdot d \textbf s = \mu_0 I$$ and this is zero when there is no current enclosed in the loop; there is a nonzero result when there is a current.

    With Cauchy's Integral Theorem, if a function is analytic on and in the loop integrated over, $$\oint_C f(z) dz = 0$$; otherwise there may be a nonzero result.

    Since essentially the magnetic field approaches infinity right in the current, I thought maybe Ampere's Law is an instance of Cauchy's Integral Formula/Theorem; when there is no current enclosed, the magnetic field has no such center, so all of it would be analytic.

    Is this an actual connection, then?
  2. jcsd
  3. Feb 5, 2013 #2
    If you wanted to get hand-wavy, I suppose you could view current as the residue of the magnetic field.
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