The Conditions for Biot-Savart Law to Ampere's Law

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Main Question or Discussion Point

Mathematically, what conditions must a B-field that obeys the Biot-Savart Law satisfy before it will obey Ampere's Law?

Additionally, what conditions must the B-field obey in order to satisfy Faraday's Law?
 

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vanhees71
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vanhees71
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Of course, you can do that, but that leads to very cumbersome and not very physically intuitive non-local equations. The local representation is given by Jefimenko's equation rather than by lumping field parts into the sources. It's also clear from relativistic covariance, that the electromagnetic field builds one "unit" in the sense that the electric and magnetic components together are the components of the electromagnetic field-strength (or Faraday) tensor ##F_{\mu \nu}## and the charge density and current density together are components of a four-vector field ##j^{\mu}=(c \rho,\vec{j})##.
 
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Both laws have applications in aerodynamics for calculating the velocity induced by vortex lines by using the magnetic induction current formula B=µH
 

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