Law

Ampère's law

\oint B \cdot d ℓ = μ_{0} I_{enc}

Statement

The line integral of B around any closed loop equals μ₀ times the current enclosed by that loop.

Symbols and units

$I_{enc}$	current enclosed by the loop	A
$d ℓ$	line element along the loop	m
$μ_{0}$	permeability of free space	4π×10⁻⁷ T·m/A
$B$	magnetic field	T

When it applies

Always true; useful for finding B directly only where symmetry allows it out of the integral.

Don't use when

Always true but only useful for finding B directly where symmetry lets it come out of the line integral — without that symmetry it gives the circulation but not the field itself.

Derivation

The magnetic analogue of Gauss's law: a closed loop replaces a closed surface, and symmetric current distributions allow B to be pulled out of the integral, exactly as symmetric charge distributions allow E to be pulled out in Gauss's law.

\oint B \cdot d ℓ = μ_{0} I_{enc}

Watch out for

Always true, but useful for finding B directly only where symmetry lets it come out of the integral — a straight wire, a solenoid, a toroid.

Connects to

Gauss's law