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The integral forms of Maxwell’s equations can be used to derive relations for the tangential and normal field components to the sources. These relations were mentioned in class. It’s a little late, but lets go over the derivation. This isn’t all review from first year electromagnetism since we are now using a magnetic source modifications of Maxwell’s equations.

The derivation below follows that of [1] closely, but I am trying it myself to ensure that I understand the assumptions.

The two infinitesimally thin pillboxes of fig. 1, and fig. 2 are used in the argument.

fig. 2: Pillboxes for tangential and normal field relations

fig. 1: Pillboxes for tangential and normal field relations

Maxwell’s equations with both magnetic and electric sources are

$$\begin{equation}\label{eqn:normalAndTangentialFields:20} \spacegrad \cross \boldsymbol{\mathcal{E}} = -\PD{t}{\boldsymbol{\mathcal{B}}} -\boldsymbol{\mathcal{M}} \end{equation}$$
$$\begin{equation}\label{eqn:normalAndTangentialFields:40} \spacegrad \cross \boldsymbol{\mathcal{H}} = \boldsymbol{\mathcal{J}} + \PD{t}{\boldsymbol{\mathcal{D}}} \end{equation}$$
$$\begin{equation}\label{eqn:normalAndTangentialFields:60} \spacegrad \cdot \boldsymbol{\mathcal{D}} = \rho_\textrm{e} \end{equation}$$
$$\begin{equation}\label{eqn:normalAndTangentialFields:80} \spacegrad \cdot \boldsymbol{\mathcal{B}} = \rho_\textrm{m}. \end{equation}$$

After application of Stokes’ and the divergence theorems Maxwell’s equations have the integral form

$$\begin{equation}\label{eqn:normalAndTangentialFields:100} \oint \boldsymbol{\mathcal{E}} \cdot d\Bl = -\int d\BA \cdot \lr{ \PD{t}{\boldsymbol{\mathcal{B}}} + \boldsymbol{\mathcal{M}} } \end{equation}$$
$$\begin{equation}\label{eqn:normalAndTangentialFields:120} \oint \boldsymbol{\mathcal{H}} \cdot d\Bl = \int d\BA \cdot \lr{ \PD{t}{\boldsymbol{\mathcal{D}}} + \boldsymbol{\mathcal{J}} } \end{equation}$$
$$\begin{equation}\label{eqn:normalAndTangentialFields:140} \int_{\partial V} \boldsymbol{\mathcal{D}} \cdot d\BA = \int_V \rho_\textrm{e}\,dV \end{equation}$$
$$\begin{equation}\label{eqn:normalAndTangentialFields:160} \int_{\partial V} \boldsymbol{\mathcal{B}} \cdot d\BA = \int_V \rho_\textrm{m}\,dV. \end{equation}$$

Maxwell-Faraday equation

First consider one of the loop integrals, like $\ref{eqn:normalAndTangentialFields:100}$. For an infinestismal loop, that integral is

$$\begin{equation}\label{eqn:normalAndTangentialFields:180} \begin{aligned} \oint \boldsymbol{\mathcal{E}} \cdot d\Bl &\approx \mathcal{E}^{(1)}_x \Delta x + \mathcal{E}^{(1)} \frac{\Delta y}{2} + \mathcal{E}^{(2)} \frac{\Delta y}{2} -\mathcal{E}^{(2)}_x \Delta x – \mathcal{E}^{(2)} \frac{\Delta y}{2} – \mathcal{E}^{(1)} \frac{\Delta y}{2} \\ &\approx \lr{ \mathcal{E}^{(1)}_x -\mathcal{E}^{(2)}_x } \Delta x + \inv{2} \PD{x}{\mathcal{E}^{(2)}} \Delta x \Delta y + \inv{2} \PD{x}{\mathcal{E}^{(1)}} \Delta x \Delta y. \end{aligned} \end{equation}$$

We let $\Delta y \rightarrow 0$ which kills off all but the first difference term.

The RHS of $\ref{eqn:normalAndTangentialFields:180}$ is approximately

$$\begin{equation}\label{eqn:normalAndTangentialFields:200} -\int d\BA \cdot \lr{ \PD{t}{\boldsymbol{\mathcal{B}}} + \boldsymbol{\mathcal{M}} } \approx – \Delta x \Delta y \lr{ \PD{t}{\mathcal{B}_z} + \mathcal{M}_z }. \end{equation}$$

If the magnetic field contribution is assumed to be small in comparison to the magnetic current (i.e. infinite magnetic conductance), and if a linear magnetic current source of the form is also assumed

$$\begin{equation}\label{eqn:normalAndTangentialFields:220} \boldsymbol{\mathcal{M}}_s = \lim_{\Delta y \rightarrow 0} \lr{\boldsymbol{\mathcal{M}} \cdot \zcap} \zcap \Delta y, \end{equation}$$

then the Maxwell-Faraday equation takes the form

$$\begin{equation}\label{eqn:normalAndTangentialFields:240} \lr{ \mathcal{E}^{(1)}_x -\mathcal{E}^{(2)}_x } \Delta x \approx – \Delta x \boldsymbol{\mathcal{M}}_s \cdot \zcap. \end{equation}$$

While $\boldsymbol{\mathcal{M}}$ may have components that are not normal to the interface, the surface current need only have a normal component, since only that component contributes to the surface integral.

The coordinate expression of $\ref{eqn:normalAndTangentialFields:240}$ can be written as

$$\begin{equation}\label{eqn:normalAndTangentialFields:260} – \boldsymbol{\mathcal{M}}_s \cdot \zcap = \lr{ \boldsymbol{\mathcal{E}}^{(1)} -\boldsymbol{\mathcal{E}}^{(2)} } \cdot \lr{ \ycap \cross \zcap } = \lr{ \lr{ \boldsymbol{\mathcal{E}}^{(1)} -\boldsymbol{\mathcal{E}}^{(2)} } \cross \ycap } \cdot \zcap. \end{equation}$$

This is satisfied when

$$\begin{equation}\label{eqn:normalAndTangentialFields:280} \boxed{ \lr{ \boldsymbol{\mathcal{E}}^{(1)} -\boldsymbol{\mathcal{E}}^{(2)} } \cross \ncap = – \boldsymbol{\mathcal{M}}_s, } \end{equation}$$

where $\ncap$ is the normal between the interfaces. I’d failed to understand when reading this derivation initially, how the $\boldsymbol{\mathcal{B}}$ contribution was killed off. i.e. If the vanishing area in the surface integral kills off the $\boldsymbol{\mathcal{B}}$ contribution, why do we have a $\boldsymbol{\mathcal{M}}$ contribution left. The key to this is understanding that this magnetic current is considered to be confined very closely to the surface getting larger as $\Delta y$ gets smaller.

Also note that the units of $\boldsymbol{\mathcal{M}}_s$ are volts/meter like the electric field (not volts/squared-meter like $\boldsymbol{\mathcal{M}}$.)

Ampere’s law

As above, assume a linear electric surface current density of the form

$$\begin{equation}\label{eqn:normalAndTangentialFields:300} \boldsymbol{\mathcal{J}}_s = \lim_{\Delta y \rightarrow 0} \lr{\boldsymbol{\mathcal{J}} \cdot \ncap} \ncap \Delta y, \end{equation}$$

in units of amperes/meter (not amperes/meter-squared like $\boldsymbol{\mathcal{J}}$.)

To apply the arguments above to Ampere’s law, only the sign needs to be adjusted

$$\begin{equation}\label{eqn:normalAndTangentialFields:290} \boxed{ \lr{ \boldsymbol{\mathcal{H}}^{(1)} -\boldsymbol{\mathcal{H}}^{(2)} } \cross \ncap = \boldsymbol{\mathcal{J}}_s. } \end{equation}$$

Gauss’s law

Using the cylindrical pillbox surface with radius $\Delta r$, height $\Delta y$, and top and bottom surface areas $\Delta A = \pi \lr{\Delta r}^2$, the LHS of Gauss’s law $\ref{eqn:normalAndTangentialFields:140}$ expands to

$$\begin{equation}\label{eqn:normalAndTangentialFields:320} \begin{aligned} \int_{\partial V} \boldsymbol{\mathcal{D}} \cdot d\BA &\approx \mathcal{D}^{(2)}_y \Delta A + \mathcal{D}^{(2)}_\rho 2 \pi \Delta r \frac{\Delta y}{2} + \mathcal{D}^{(1)}_\rho 2 \pi \Delta r \frac{\Delta y}{2} -\mathcal{D}^{(1)}_y \Delta A \\ &\approx \lr{ \mathcal{D}^{(2)}_y -\mathcal{D}^{(1)}_y } \Delta A. \end{aligned} \end{equation}$$

As with the Stokes integrals above it is assumed that the height is infinestimal with respect to the radial dimension. Letting that height $\Delta y \rightarrow 0$ kills off the radially directed contributions of the flux through the sidewalls.

The RHS expands to approximately

$$\begin{equation}\label{eqn:normalAndTangentialFields:340} \int_V \rho_\textrm{e}\,dV \approx \Delta A \Delta y \rho_\textrm{e}. \end{equation}$$

Define a highly localized surface current density (coulombs/meter-squared) as

$$\begin{equation}\label{eqn:normalAndTangentialFields:360} \sigma_\textrm{e} = \lim_{\Delta y \rightarrow 0} \Delta y \rho_\textrm{e}. \end{equation}$$

Equating $\ref{eqn:normalAndTangentialFields:340}$ with $\ref{eqn:normalAndTangentialFields:320}$ gives

$$\begin{equation}\label{eqn:normalAndTangentialFields:380} \lr{ \mathcal{D}^{(2)}_y -\mathcal{D}^{(1)}_y } \Delta A = \Delta A \sigma_\textrm{e}, \end{equation}$$

or

$$\begin{equation}\label{eqn:normalAndTangentialFields:400} \boxed{ \lr{ \boldsymbol{\mathcal{D}}^{(2)} – \boldsymbol{\mathcal{D}}^{(1)} } \cdot \ncap = \sigma_\textrm{e}. } \end{equation}$$

Gauss’s law for magnetism

The same argument can be applied to the magnetic flux. Define a highly localized magnetic surface current density (webers/meter-squared) as

$$\begin{equation}\label{eqn:normalAndTangentialFields:440} \sigma_\textrm{m} = \lim_{\Delta y \rightarrow 0} \Delta y \rho_\textrm{m}, \end{equation}$$

yielding the boundary relation

$$\begin{equation}\label{eqn:normalAndTangentialFields:420} \boxed{ \lr{ \boldsymbol{\mathcal{B}}^{(2)} – \boldsymbol{\mathcal{B}}^{(1)} } \cdot \ncap = \sigma_\textrm{m}. } \end{equation}$$

References

[1] Constantine A Balanis. Advanced engineering electromagnetics, volume 20, chapter Time-varying and time-harmonic electromagnetic fields. Wiley New York, 1989.