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In a linear isotropic nondispersive medium with time-independent permittivity $ϵ$ and magnetic permeability $\mu$, the stored electromagnetic energy per volume is $u=ϵ|\mathbf{E}{|}^2/2+|\mathbf{B}{|}^2/(2\mu )$. For a short derivation, the Maxwell equations in matter and the divergence identity for a cross product give $\nabla \cdot (\mathbf{E}\times \mathbf{H})=-\mathbf{H}\cdot {\partial }_t\mathbf{B}-\mathbf{E}\cdot {\partial }_t\mathbf{D}-{\mathbf{J}}_f\cdot \mathbf{E}$. The constitutive relations turn the first two terms into $-{\partial }_{t}u$, proving the Poynting theorem. Choose $u=0$ when the electric field and magnetic field both vanish. Dispersive or time-dependent material response requires a different accounting of stored energy.

The Maxwell equations are the field equations of classical electromagnetism. In SI units, in vacuum with prescribed charge density $\rho$ and current density $\mathbf{J}$, they areHere $\mathbf{E}$ is the electric field in volts per metre, $\mathbf{B}$ the magnetic field in teslas, $\rho$ in coulombs per cubic metre, and $\mathbf{J}$ in amperes per square metre. The constants are vacuum permittivity ${ϵ}_0$ and the vacuum value ${\mu }_0$ of magnetic permeability. “Vacuum” specifies the field laws without a material response; “source-free” additionally means $\rho =0$ and $\mathbf{J}=0$. Matter described by averaged fields instead requires Maxwell equations in matter and constitutive relations.

The first equation, Gauss's law, makes electric charge a source of electric flux. Gauss's law for magnetism sets the net magnetic flux through a closed surface to zero. The two curl equations, Faraday's law and the Ampère-Maxwell equation, couple changes of the electric field and magnetic field. Appropriate initial conditions and boundary conditions select a physical solution; the two divergence equations constrain the initial fields. ^[ref].

Allow time dependence while keeping the constitutive relations linear, isotropic, nondispersive, and independent of time. The Maxwell equations in matter giveFor the Poynting vector $\mathcal{P}=\mathbf{E}\times \mathbf{H}$, the divergence identity yieldsThe first two terms are exact time derivatives. Taking the zero-field energy density as zero gives the electromagnetic energy densityConsequently the Poynting theorem is the continuity equationIn integral form,The left side is the rate of increase of stored electromagnetic energy plus outward Poynting flux. The right side accounts for work delivered to the free electric current; in an Ohmic medium ${\mathbf{J}}_f\cdot \mathbf{E}$ is the local rate of Joule heating.

Use the constitutive relations $\mathbf{D}=ϵ\mathbf{E}$ and $\mathbf{H}=\mathbf{B}/\mu$, where $\mathbf{D}$ is the electric displacement field, $\mathbf{H}$ the magnetic field strength, $ϵ$ the permittivity, and $\mu$ the magnetic permeability. Let ${\rho }_f$ be the charge density of free charge and ${\mathbf{J}}_f$ its current density. In this homogeneous medium, Gauss's law and the magnetostatic Ampère law are, respectively,Here $W$ is a volume with closed boundary $\partial W$, and $S$ is an oriented spanning surface whose boundary orientation follows the right-hand rule. Thus the electric field form is $\nabla \cdot \mathbf{E}={\rho }_f/ϵ$ and ${\oint }_{\partial W}\mathbf{E}\cdot \mathbf{n}dS=Q_f/ϵ$; the magnetic field form is $\nabla \times \mathbf{B}=\mu {\mathbf{J}}_f$ and ${\oint }_{\partial S}\mathbf{B}\cdot d\mathbf{\ell }=\mu I_f$.

The remaining static Maxwell equations areThese express Gauss's law for magnetism and the static limit of Faraday's law. The volume and circulation forms follow from the divergence theorem and Stokes theorem. The electric charge and electric current on the right of the macroscopic laws are free sources; material electric polarization and magnetization are already included in $\mathbf{D}$ and $\mathbf{H}$.

The source-free Maxwell equations in a linear medium areHere the constitutive relations are $\mathbf{D}=\epsilon \mathbf{E}$ and $\mathbf{B}=\mu \mathbf{H}$.

Substitution of the sinusoidal plane waves into the two divergence equations gives the transversality of an electromagnetic plane wave,Faraday's law and the Ampère-Maxwell equation give, respectively,Taking $\mathbf{k}\times$ the first equation and using transversality yieldsThus a nonzero plane electromagnetic wave in a linear medium must satisfyIn particular, ${\mathbf{E}}_0$, ${\mathbf{B}}_0$, and $\mathbf{k}$ are mutually orthogonal, with $|{\mathbf{B}}_0|=\sqrt{\mu \epsilon }|{\mathbf{E}}_0|$.

Take the interface to be $x=0$ and the plane of incidence to be the $xy$-plane. The incident, reflected, and transmitted wavevectors may be writtenThe phase matching at a planar wave interface requires one common angular frequency and one common tangential wavevector:Since $k_{\pm }=\omega \sqrt{{\epsilon }_{\pm }\mu }=n_{\pm }\omega /c$, this gives the Snell law for electromagnetic waves,

For the final polarization, take the signed electric amplitudes along $\hat{\mathbf{z}}$. The dielectric-interface boundary conditions require continuity of tangential $\mathbf{E}$ and $\mathbf{H}$. Sincetheir $z$- and $y$-components giveandEliminating $E_I$ yieldsUsing ${\theta }_R={\theta }_I$ and Snell's law,Therefore the reflected-to-transmitted transverse-electric amplitude ratio iswith the polarization directions chosen so that the displayed signed factor is nonnegative; otherwise its absolute value gives the magnitude ratio.

In the absence of free electric charge and electric current, the source-free Maxwell equations in a linear medium aretogether with the constitutive relations $\mathbf{D}=\epsilon \mathbf{E}$ and $\mathbf{B}=\mu \mathbf{H}$.

The constants specify isotropic, nondispersive linear response: $\kappa$ controls electric polarization, and $\chi$ controls magnetization. Their numerical relation to conventional electric susceptibility and magnetic susceptibility depends on the signs and field normalization in the action. In the rationalized units used here, the printed action gives the constitutive relationsThus its relative permittivity and magnetic permeability areHere conventional ${\chi }_m$ is defined by $\mathbf{M}={\chi }_m\mathbf{H}$, not by $\mathbf{M}={\chi }_m\mathbf{B}$. With the alternative repaired action that yields the PDF's target equation, ${\epsilon }_r$ is unchanged but ${\mu }_r=(1-\chi {)}^{-1}$ and ${\chi }_m=\chi /(1-\chi )$. In particular, a positive conventional electric susceptibility corresponds to negative $\kappa$ in the printed conventions; the problem does not specify signs for these constants.