Ampère-Maxwell's law

Ampère-Maxwell’s law describes how magnetic fields are generated by electric currents and changing electric fields.

Differential form

$$\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\partial }{\partial t}\mathbf{E}$$

where
$\mathbf{B}(\mathbf{x},t)$	magnetic field (or magnetic flux density), measured in tesla $\text{T}$
$\mathbf{x}$	position vector in three-dimensional space, typically expressed as $\mathbf{x}=(x,y,z)$ and measured in meters $\text{m}$
$t$	time, measured in seconds $\text{s}$
$\nabla \times$	curl operator which measures the rotation of a field in space, and which applied to $\mathbf{B}$ is: $\nabla \times \mathbf{B}=\left(\frac{\partial }{\partial y}{B}_z-\frac{\partial }{\partial z}{B}_y\right)\mathbf{i}+\left(\frac{\partial }{\partial z}{B}_x-\frac{\partial }{\partial x}{B}_z\right)\mathbf{j}+\left(\frac{\partial }{\partial x}{B}_y-\frac{\partial }{\partial y}{B}_x\right)\mathbf{k},$ measured in amperes per meter $\frac{\text{A}}{\text{m}}$
${\mu }_0$	permeability of free space, measured in henry per meter $\frac{\text{H}}{\text{m}}$
$\mathbf{J}(\mathbf{x},t)$	current density vector field, measured in amperes per square meter $\frac{\text{A}}{{\text{m}}^2}$
${\epsilon }_0$	permittivity of free space, measured in farads per meter $\frac{\text{F}}{\text{m}}$
$\mathbf{E}(\mathbf{x},t)$	electric field, measured in volts per meter $\frac{\text{V}}{\text{m}}$

Integral form

$${\oint }_C\mathbf{B}\cdot d\mathbf{s}={\mu }_0{\int }_S\mathbf{J}\cdot d\mathbf{A}+{\mu }_0{\epsilon }_0\frac{d}{dt}{\int }_S\mathbf{E}\cdot d\mathbf{A}$$

where
$C$	closed loop (or contour) in three-dimensional space
${\oint }_C$	line integral around the closed path $C$ measuring the total circulation of the electric field along the loop
$d\mathbf{s}$	differential line element that represents an infinitesimally small segment along the loop $C$
$S$	surface whose boundary is the closed loop $C$
${\int }_S$	surface integral which sums the contribution of the field $\mathbf{B}$ over the entire surface $S$
$d\mathbf{A}$	differential surface element representing an infinitesimally small patch of the surface $S$