#### Boundary Conditions
emdm.Me.bc

Discontinuities between different media, deduced from

Maxwell's equations (in matter), integral form

\begin{align} (i)~~ &\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\ (ii)~~ &\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\ (iii)~~ &\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\ (iv)~~ &\oint_{\cal P} {\bf H} \cdot d{\bf l} = I_{f_{enc}} + \frac{d}{dt} \int_{\cal S} {\bf D} \cdot d{\bf a}. \tag{Max_mat_int}\label{Max_mat_int} \end{align}

Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials: \({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so

- Gr (7.60)

\[ D^{\perp}_1 - D^{\perp}_2 = \sigma_f \tag{Ddisc}\label{Ddisc} \]

Same reasoning applied to \((ii)\) gives Bdisc

\[ B^{\perp}_1 - B^{\perp}_2 = 0 \]

For \((iii)\): Amperian loop straddling surface: \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\). Limit of small loop: flux vanishes, therefore

\[ {\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0 \]

Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\). No volume current can contribute, but a surface current can. Can write \(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\) and thus (as we got before in Hdisc)

\[ {\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n} \tag{Hdisc}\label{Hdisc} \]

These are the general boundary conditions for electrodynamics.

In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:

- Gr (7.64)

\begin{align} (i)~~ &\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = \sigma_f, \nonumber \\ (ii)~~ &B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\ (iii)~~ &{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\ (iv)~~ &\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}. \tag{disc_lm}\label{disc_lm} \end{align}

If there is no free charge and no free current at boundary:

- Gr (7.64)

\begin{align} (i)~~ &\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\ (ii)~~ &B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\ (iii)~~ &{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\ (iv)~~ &\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = 0. \tag{disc_nfc}\label{disc_nfc} \end{align}

These are basis of theory of reflection and refraction.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42