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Outline of proof that a magnetic field calculated by the Biot-Savart law will always satisfy Gauss's law for magnetism and Ampere's law. We start with the Biot-Savart law:
Plugging in the well-known relation
and using the product rule for curls, as well as the fact that J does not depend on the unprimed coordinates, this equation can be rewritten as
Since the divergence of a curl is always zero, this establishes Gauss's law for magnetism. Next, taking the curl of both sides, using the formula for the curl of a curl (see the article Curl (mathematics)), and again using the fact that J does not depend on the unprimed coordinates, we eventually get the result
Finally, plugging in the relations
(where δ is the Dirac delta function), using the fact that the divergence of J is zero (due to the assumption of magnetostatics), and performing an integration by parts, the result turns out to be
i.e. Ampere's law.
thus the ampere's circuital law is prooved
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