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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.

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.