Ampère-Maxwell equation

The Ampère-Maxwell equationrelates magnetic circulation to conduction and displacement current.

An Amperian loop is a closed curve used in the integral form of Ampère's circuital law or the Ampère-Maxwell equation.

In vacuum magnetostatics, the Ampère-Maxwell equation reduces to $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$. Applying Stokes theorem gives ${\oint }_C\mathbf{B}\cdot d\mathbf{\ell }={\mu }_0{I}_{\mathrm{enc}}$, where $I_{\mathrm{enc}}$ is the electric current crossing an oriented spanning surface. Time-dependent electric fields require the additional displacement current term in the Ampère-Maxwell equation.

In vacuum, the displacement-current density is ${ϵ}_0\partial E/\partial t$. Its contribution to magnetic circulation makes the Ampère-Maxwell equation compatible with changing electric charge.