Maxwell Equations

Maxwell’s four equations describe electrodynamics. Two constrain field divergence; two govern time evolution through field curls.

Fields and sources

Symbol	Meaning	SI unit
$\mathbf E$	Electric field	V/m
$\mathbf B$	Magnetic flux density	T
$\mathbf D$	Electric displacement	C/m²
$\mathbf H$	Magnetic field intensity	A/m
$\rho$	Charge density	C/m³
$\mathbf J$	Current density	A/m²

The divergence $\nabla\cdot$ measures the net outward flux near a point. The curl $\nabla\times$ measures local circulation. A time derivative $\partial_t$ measures change at a fixed position.

The four equations in vacuum

In vacuum, $\varepsilon_0$ is the vacuum permittivity and $\mu_0$ is the vacuum permeability. Charges and currents, if present, act as sources:

\nabla\cdot\mathbf E=\frac{\rho}{\varepsilon_0}.

Gauss’s law: positive charge produces outward electric flux; negative charge produces inward flux.

\nabla\cdot\mathbf B=0.

Gauss’s law for magnetism: magnetic field lines have no isolated sources or sinks in classical Maxwell theory.

\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}.

Faraday’s law: a changing magnetic flux induces a circulating electric field. The minus sign encodes Lenz’s law.

\nabla\times\mathbf B=\mu_0\mathbf J+ \mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t}.

Ampère–Maxwell law: electric current and a changing electric field produce magnetic circulation. Together with Faraday’s law, this permits waves in empty space.

Inside matter

Matter responds through electric polarization $\mathbf P$ (dipole moment per volume) and magnetization $\mathbf M$ (magnetic moment per volume). Define

\mathbf D=\varepsilon_0\mathbf E+\mathbf P, \qquad \mathbf B=\mu_0(\mathbf H+\mathbf M).

The macroscopic equations then separate free sources from bound material response:

\begin{aligned} \nabla\cdot\mathbf D&=\rho_{\mathrm f}, & \nabla\cdot\mathbf B&=0,\\ \nabla\times\mathbf E&=-\partial_t\mathbf B, & \nabla\times\mathbf H&=\mathbf J_{\mathrm f}+\partial_t\mathbf D. \end{aligned}

Bound material response is included in $\mathbf D$ and $\mathbf H$ ; $\rho_{\mathrm f}$ and $\mathbf J_{\mathrm f}$ describe free sources. See MIT’s material-response notes for details.

To close the system, specify constitutive relations. For a linear, isotropic, nondispersive medium at rest,

\mathbf D=\varepsilon\mathbf E,\qquad \mathbf B=\mu\mathbf H,\qquad \mathbf J_{\mathrm f}=\sigma\mathbf E+\mathbf J_{\mathrm s}.

Here $\varepsilon=\varepsilon_0\varepsilon_r$ , $\mu=\mu_0\mu_r$ , $\sigma$ is electric conductivity, and $\mathbf J_{\mathrm s}$ is an imposed source current. In a lossless, nonmagnetic dielectric, the refractive index satisfies $n^2=\varepsilon_r$ . Anisotropy makes the constitutive coefficients tensors; dispersion gives the response a time history.

Two evolution equations

For time-independent scalar $\varepsilon$ and $\mu$ , substituting the constitutive relations into the curl equations gives

\boxed{\frac{\partial\mathbf H}{\partial t} =-\frac{1}{\mu}\nabla\times\mathbf E}

\boxed{\frac{\partial\mathbf E}{\partial t} =\frac{1}{\varepsilon} \left(\nabla\times\mathbf H-\sigma\mathbf E-\mathbf J_{\mathrm s}\right)}.

The basic Yee scheme advances these equations. Gauss’s laws still constrain the initial fields and boundaries, and current must obey charge conservation:

\partial_t\rho_{\mathrm f}+\nabla\cdot\mathbf J_{\mathrm f}=0.

Consistent constraints persist because the divergence of a curl is zero. Discrete operators must preserve that identity.

Waves as a consistency check

In homogeneous, source-free, lossless material, differentiating once more in time yields

\frac{\partial^2\mathbf E}{\partial t^2} =\frac{1}{\mu\varepsilon}\nabla^2\mathbf E, \qquad v=\frac{1}{\sqrt{\mu\varepsilon}}.

In vacuum, $c_0=1/\sqrt{\mu_0\varepsilon_0}$ .

Continue to Yee-update.