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

by brandon_1892
Tags: ampere's law complex
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Feb5-13, 05:31 PM
P: 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?
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Feb5-13, 11:17 PM
elfmotat's Avatar
P: 260
If you wanted to get hand-wavy, I suppose you could view current as the residue of the magnetic field.

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