Ampère's circuital law

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, displacement current is the term ${\mathbf{J}}_D={ϵ}_0{\partial }_t\mathbf{E}$ in the Ampère-Maxwell equation. It has the units of current density, but need not describe electric charge crossing the region. In Maxwell equations in matter, the corresponding term is ${\partial }_t\mathbf{D}$, with $\mathbf{D}$ the electric displacement field.

Consider a slowly charging parallel-plate capacitor with vacuum gap, plate area $A$, and charge $Q(t)$ on its positive plate. Neglect fringing and assume the charging timescale is long compared with the light-travel time across the apparatus. The nearly uniform gap electric field has magnitude $E=Q/({ϵ}_{0}A)$, so the displacement current through the gap isFor $A=0.010{\mathrm{m}}^2$ and charging electric current $I=1.0\mathrm{mA}$, this gives $dE/dt\simeq 1.13\times 10^{10}\mathrm{V}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ and $J_D=0.10A/{\mathrm{m}}^2$. No conducting electric current crosses the gap. A spanning surface cutting the wire and another passing between the plates nevertheless give the same magnetic circulation when the full Ampère-Maxwell equation is used. ^[ref].

The local consistency check is short. Without displacement current, taking the divergence of $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$ would force $\nabla \cdot \mathbf{J}=0$, allowing no local accumulation of electric charge. With displacement current, the divergence of a curl is zero and Gauss's law giveThis is precisely the charge continuity equation. Electric charge conservation fixes the required divergence of the added term; it does not by itself exclude other divergence-free additions. The specific Maxwell equations also yield the electromagnetic wave equation, linking this consistency correction to propagation at the speed of light.

The time derivative of the electric displacement field enters the macroscopic Ampère-Maxwell equation alongside free electric current density. In vacuum, ${\mathbf{J}}_D={ϵ}_0{\partial }_t\mathbf{E}$. It is not transport of free charges through the vacuum gap of a capacitor; its surface integral nevertheless supplies the term needed for compatibility with the continuity equation.

For a circular air-filled parallel-plate capacitor with voltage phasor $V_0{e}^{-i\omega t}$, uniform gap $d$ and radius $a$, the Ampère-Maxwell equation on a circular loop of radius $s<a$ gives the displayed magnetic field. The enclosed displacement current is proportional to $s^2$, whereas the circulation is $2\pi s{B}_{\varphi }$. Consequently its amplitude grows linearly from zero on the axis. This assumes negligible fringing fields and a quasistatic approximation $\omega a/c\ll 1$.

If a thin capacitor interior is modeled with axially uniform axisymmetric fields, Maxwell's equations give $r^{-1}(r{E}_z^{\prime }{)}^{\prime }+(\omega /c{)}^2{E}_z=0$. The solution regular at the axis is proportional to $J_0(\omega r/c)$. Normalizing to a central sinusoidal voltage gives $E_z=-(V_0/L)J_0(\omega r/c)\sin \omega t$ in this ideal interior model. It can change sign across the radius when propagation matters. Feed, edge and radiation boundary conditions are needed for a full driven-device solution.

For a uniform ohmic material under harmonic forcing, the magnitudes obey $|J_{\mathrm{cond}}|/|J_{\mathrm{disp}}|=\sigma /(\omega ϵ)$. A large ratio supports a local conduction-dominated approximation; it does not remove the displacement term from Maxwell's exact continuity relation.

A spanning surface that cuts a charging wire carries free current $I$. A surface passing through the capacitor gap instead carries displacement flux $\int {\partial }_t\mathbf{D}\cdot d\mathbf{A}=\dot{Q}=I$. Their common boundary has the same magnetic circulation only when both terms are included in Ampère's law.