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Split the microscopic sources into free and bound parts and defineSubstitution of $\rho ={\rho }_{\mathrm{f}}-\nabla \cdot P$ into Gauss's law gives $\nabla \cdot D={\rho }_{\mathrm{f}}$. Similarly, using $J=J_{\mathrm{f}}+{\partial }_{t}P+\nabla \times M$ in the Ampère-Maxwell equation gives $\nabla \times H=J_{\mathrm{f}}+{\partial }_{t}D$. The homogeneous Maxwell equations are unchanged. Thus

Maxwell equations areTaking the divergence of the Ampère-Maxwell equation and using Gauss's law givesThereforebecause $J$ vanishes on the boundary. Componentwise integration by parts also giveswith the boundary term again zero.

Varying $\bar{\psi }$ and $\psi$ independently givesand its adjointwhere $D_{\mu }\bar{\psi }={\partial }_{\mu }\bar{\psi }-ie{A}_{\mu }\bar{\psi }$. Varying $A_{\nu }$ and integrating the Pauli term by parts gives the modified Maxwell equations

Varying $A_{\nu }$ gives the Maxwell equationsOn the vacuum manifold, choose unitary gauge so that $\varphi =v$. Then $\mathbf{J}=-2e^2{v}^2\mathbf{A}$ and the London equation isFor a static configuration, $\nabla \times \mathbf{B}=\mathbf{J}$ and $\nabla \cdot \mathbf{B}=0$, soThe magnetic field therefore decays exponentially inside the condensate. This is the Meissner effect, with penetration depth equal to the inverse gauge-boson mass.

The first vacuum Maxwell equations equation is preserved for every vector field because part a givesOn two-forms in four dimensions, the mixed volume tensor $T_{ab}{}^{cd}$ represents twice the Hodge star operator. Part b therefore says that a conformal Killing field commutes with the Hodge star:It follows thatHence $\tilde{F}={\mathcal{L}}_{X}F$ satisfies both vacuum Maxwell equations whenever $F$ does.

Let $\hat{n}$ be the unit normal pointing from the minus medium to the plus medium. Integrating the Maxwell equations across an infinitesimal pillbox or loop gives the dielectric-interface boundary conditionsandThus tangential $E$ and $H$ are continuous, as are normal $D$ and $B$.

Because $D=ϵE$, the normal electric field generally jumps according toThe tangential displacement field generally jumps because ${ϵ}_{+}\ne {ϵ}_{-}$. Since the permeability is common, continuity of tangential $H$ is continuity of tangential $B$; together with normal-$B$ continuity, every component of $B$ is continuous. The same common $\mu$ also makes the normal component of $H$ continuous.

Vacuum permittivity is the electromagnetic constant ${ϵ}_0$ appearing in Coulomb's law and the SI vacuum Maxwell equations.