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Integrate each conservative ideal magnetohydrodynamics law through a thin pillbox around the stationary shock. Mass conservation givesThe MHD momentum-flux tensor isIts normal and tangential components givewhere $\Pi =p+B^2/(2{\mu }_0)$ is total pressure. The solenoidal condition gives $[B_x]=0$, while steady Faraday's law gives continuity of tangential electric field,Finally, material energy flux plus the normal Poynting vector component giveswhere $h$ is specific enthalpy per unit mass.

Gravity is a bounded volume force, so its integral across a shock whose thickness tends to zero vanishes. Viscosity and resistivity inside the layer may produce entropy; consequently entropy flux need not be equal on both sides, although the second law requires nonnegative net entropy production.

Insert $\rho ={ϵ}_0\nabla \cdot E$ and $J={\mu }_0^{-1}\nabla \times B-{ϵ}_0{E}_t$ into the Lorentz force and use vector identities plus Faraday's law. Rearrangement gives ${\partial }_t{g}_i+{\partial }_j{\sigma }_{ij}=-f_i$. Here $g={ϵ}_{0}E\times B$ is field momentum density and $\sigma$ is the Maxwell stress tensor, whose surface traction transports momentum.

For a solenoidal vector field $\mathbf{B}$ and constant magnetic diffusivity $\eta$, the magnetic induction equation isIt combines Faraday's law with the moving-conductor relation for electrical conductivity, neglecting displacement current. With incompressible flow, its advective form is ${\partial }_t\mathbf{B}+(\mathbf{u}\cdot \nabla )\mathbf{B}=(\mathbf{B}\cdot \nabla )\mathbf{u}+\eta {\nabla }^2\mathbf{B}$.

For a circuit whose flux linkage $\Lambda$ is proportional to its current, the self-inductance is the coefficient $L$ in $\Lambda =LI$. Faraday's law gives the induced electromotive force $-L\dot{I}$ when $L$ is constant. In series with a resistance $R$ and a constant applied electromotive force, the current obeys $L\dot{I}+RI={\mathcal{E}}_0$, with time constant $L/R$.