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Electrodynamics MOC

Maxwell’s equations in materials

By introducing the auxiliary [[Electric displacement|-field]] and H-field and separating free charge and current from those arising from electric polarization and magnetization, Maxwell’s equations may become

  1. Gauß’s law for diëlectrics
  2. Gauß’s law for magnetic flux
  3. Faraday’s law of induction
  4. Ampère’s law for magnets

Differential form

\begin{align*} \vab{\nabla} \cdot \vab D = \rho_{f} \end{align*}

Erroneous nesting of equation structures2. $$ \begin{align*} \vab\nabla \cdot \vab B = 0 \end{align*} $$ 3. $$ \begin{align*} \vab\nabla \times \vab E = -\frac{\partial \vab B}{\partial t} \end{align*} $$ 4. $$ \begin{align*} \vab{\nabla} \times \vab H = \vab J_{f} + \frac{ \partial \vab D }{ \partial t } \end{align*}

Integral form

\begin{align*} \oiint_{\partial\Omega} \vab D \cdot d\vab a = \iiint_{\Omega} \rho_{f} , d\tau’ \end{align*}

Erroneous nesting of equation structures2. $$ \begin{align*} \oiint_{\partial \Omega}{\vab B \cdot d\vab a} = 0 \end{align*} $$ 3. $$ \begin{align*} \mathcal{E} =\oint_{\partial \Sigma}{\vab E \cdot d \vab \ell} &= -\frac{\partial \Phi_B}{\partial t} \\\\ &= -\frac{\partial}{\partial t} \iint_\Sigma{\vab B \cdot d \vab a} \end{align*} $$ 4. $$ \begin{align*} \oint_{\partial\Sigma} \vab H \cdot d\vab r &= \mu_{0}\left( I_{f,\Sigma} + \frac{ \partial \Phi_{D,\Sigma} }{ \partial t } \right) \\ &= \left( \iint_{\Sigma} \vab J_{f} \cdot d\vab a + \frac{d}{dt} \iint_{\Sigma} \vab D \cdot d\vab a \right) \end{align*}

Sources

Noting the expressions for bound charge density and bound current density, as well as current due to changes in electric polarization density, we have

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