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In SI units, with microscopic charge density ${\rho }_e$ and current density $\mathbf{J}$, Maxwell's equations can be writtenThe first relates electric flux to charge. The second states that the net magnetic flux through a closed surface is zero. Faraday's law relates electric circulation to changing magnetic flux. The Ampère-Maxwell equation includes both conduction current and displacement current; the latter is essential to time-dependent fields and local charge conservation.

For fixed contours and spanning surfaces, the integral curl equations are $\oint \mathbf{E}\cdot d\mathbf{l}=-d{\Phi }_B/dt$ and $\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{\mathrm{enc}}+{\mu }_0{\epsilon }_{0}d{\Phi }_E/dt$. For a moving material contour the first must include the motional force term; its total emf is $\oint (\mathbf{E}+{\mathbf{v}}_{\mathrm{loop}}\times \mathbf{B})\cdot d\mathbf{l}$. This prevents confusing a changing circuit area with a partial time derivative of a static magnetic field.

Taking the divergence of the Ampère-Maxwell equation and using electric Gauss's law gives ${\partial }_t{\rho }_e+\nabla \cdot \mathbf{J}=0$. In a source-free vacuum, combining the two curl equations gives ${\nabla }^2\mathbf{E}-{\mu }_0{\epsilon }_0{\partial }_t^2\mathbf{E}=0$, and similarly for $\mathbf{B}$, predicting electromagnetic waves of speed $c=1/\sqrt{{\mu }_0{\epsilon }_0}$. The Lorentz force law then specifies how these fields act on a charge. Macroscopic matter descriptions may instead use $\mathbf{D}$ and $\mathbf{H}$ with free sources and constitutive relations; those should not be silently mixed with the microscopic-source equations above.

Here $\mathbf{H}$ is magnetic field intensity, $\mathbf{J}$ free electric current density, and $\mathbf{D}$ electric displacement. In vacuum, $\mathbf{B}={\mu }_0\mathbf{H}$ and $\mathbf{D}={ϵ}_0\mathbf{E}$. Integrating the equation gives Ampère's circuital law with displacement current; in a steady state the displacement term vanishes.

Take $z$ upward and define $V={\varphi }_{\mathrm{upper}}-{\varphi }_{\mathrm{lower}}=V_0\sin \omega t$. Neglect fringing and use the low-frequency thin-capacitor approximation. ThenInside the gap, a circular Ampère loop encloses only displacement current:Because $\hat{\mathbf{z}}\times \hat{\mathbf{\varphi }}=-\hat{\mathbf{r}}$, the Poynting vector isIt points inward while the electrical energy is increasing and outward while it decreases. The net power entering the curved side isThis is the leading quasistatic energy balance. The time-averaged field energies areFor the axisymmetric high-frequency capacitor field when $\omega a/c$ is not small, induction and radial propagation prevent a spatially uniform $E_z$. In an ideal axially uniform axisymmetric interior model, Faraday's and Ampère's equations giveThe solution regular at the axis and normalized to the central voltage is approximatelyAt $\omega =\pi c/a$, the radial profile is $J_0(\pi r/a)$: its first zero is at $r\simeq 0.765a$, and the edge has opposite phase, $J_0(\pi )\simeq -0.304$. This is an illustrative thin-disc interior profile; the exact driven high-frequency field also depends on the feed and edge/radiation boundary conditions. Continuing the uniform quasistatic field would be inconsistent.

For circular plates of radius $a$, separation $d\ll a$, and a thin axial resistor $R$, the quasistatic capacitance is $C={\epsilon }_0\pi a^2/d$. During discharge, $I=V/R=-C\dot{V}$. A circular Ampere-Maxwell law contour between the plates encloses wire current $I$ and opposing displacement current $-I{r}^2/a^2$, giving $H=I(1-r^2/a^2)/(2\pi r)$. The electric field has magnitude $V/d$, so the Poynting vector points radially inward with magnitude $V^2(1-r^2/a^2)/(2\pi rdR)$. Its flux through the cylindrical side is the displayed inward power. The field inside that cylinder releases $V^2{r}^2/(R{a}^2)$ per unit time; adding the incoming flux gives the resistor's full Joule heating rate $V^2/R$. End fringing, wire radius and magnetic-energy corrections are neglected.

An approximation that keeps slow electromagnetic time variation while treating propagation across characteristic size $L$ as rapid compared with the variation time $1/\omega$. The dimensionless parameter $\omega L/c$ measures that separation. Which terms of the Maxwell equations can be dropped still depends on the physical scaling; small propagation delay alone does not justify deleting both displacement current and electromagnetic induction. For a slowly driven parallel-plate capacitor, a nearly uniform gap electric field can coexist with the magnetic field generated by its changing electric flux.

A quasistatic electromagnetic approximation neglects propagation delay over a vacuum device of characteristic size $L$ when $\omega L/c\ll 1$. It does not automatically discard every time derivative: an Ampere-Maxwell law calculation in a capacitor gap must retain displacement current. In a medium use the relevant propagation and response scales. The dominant approximately static electric field can be used to calculate a smaller induced magnetic field, provided the corrections to the dominant field remain small. Near a resonance or for a large distributed circuit, the size criterion alone may not justify an instantaneous lumped model.

For a rigid frame with constant angular velocity $\mathbf{\Omega }$, add the Coriolis force and centrifugal force to the inviscid magnetohydrodynamic momentum equation. A constant magnetic permeability and solenoidal magnetic-field constraint split the Lorentz force density into magnetic pressure and magnetic tension. This nonrelativistic conducting-fluid approximation neglects the displacement current, net electric-charge force and viscosity. Differential rotation belongs to the fluid's velocity field, not to a spatially varying rigid-frame rotation vector. A time-dependent frame requires the additional Euler force.

With immobile ions, negligible collisions and electron density $n_e$, the plasma frequency is ${\omega }_p^2=n_e{e}^2/(m_e{ϵ}_0)$. Combining inertial electron current with vacuum displacement current gives ${\omega }^2={\omega }_p^2+c^2{k}^2$. Transverse fields are evanescent below ${\omega }_p$ and propagate above it.