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In physics, Ampère's law, discovered by André-Marie Ampère, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. It is the magnetic equivalent of Faraday's law of induction.

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where

$oint\_C$ is the closed line integral around contour (closed curve) $C$.

$mathbf$ is the magnetic flux density in teslas,

$dmathbf$ is an infinitesimal element (differential) of the contour $C$,

$mathbf$ is the current density (in amperes per square meter) through the surface S enclosed by contour C

$d\; mathbf\; !$ is a differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S

$I\_\; !$ is the current enclosed by the curve $C$, or strictly, the current that penetrates surface $S$,

$mu\_0\; =\; 4\; pi\; imes\; 10^$ is the permeability of free space (in henries per meter),

where

$imes,$ is the cross product operator

$mathbf\; =\; mu\; mathbf$

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$oint\_C\; mathbf\; cdot\; dmathbf\; =\; iint\_S\; mathbf\; cdot\; d\; mathbf\; +$

$mathbf\; =\; varepsilon\; mathbf$

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Maxwell's equations

Biot-Savart law

Faraday's law of induction

Gauss's law

Electric current

Vector calculus

Stokes' theorem