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1 Maxwell equations

1.1 Maxwell time domain equations

\[\begin{array} [c]{lll}-\nabla \times \overline{\mathcal{E}} & = & \frac{\partial \overline{\mathcal{B}}}{\partial t}+\overline{\mathcal{M}}^{i}\\ & & \\ \nabla \times \overline{\mathcal{H}} & = & \frac{\partial \overline{\mathcal{D}}}{\partial t}+\overline{\mathcal{J}}^{c}+\overline{\mathcal{J}}^{i}\end{array} \]

\[\begin{array} [c]{lll}\nabla \cdot \overline{\mathcal{B}} & = & 0\\ & & \\ \nabla \cdot \overline{\mathcal{D}} & = & q \end{array} \]

1.1.1 equation of continuity

\[ \nabla \cdot \overline{\mathcal{J}}=-\frac{\partial q_{v}}{\partial t} \]

1.2 Maxwell integral form of time domain equations

\[\begin{array} [c]{lll}\oint \overline{\mathcal{E}}\cdot d\mathbf{l} & = & -\frac d{dt}\iint \overline{\mathcal{B}}\cdot d\mathbf{s}\\ & & \\ \oint \overline{\mathcal{H}}\cdot d\mathbf{l} & = & \frac d{dt}\iint \overline{\mathcal{D}}\cdot d\mathbf{s}+\iint \overline{\mathcal{J}}\cdot d\mathbf{s}\\ & & \\ \iint \overline{\mathcal{B}}\cdot d\mathbf{s} & = & 0\\ & & \\ \iint \overline{\mathcal{D}}\cdot d\mathbf{s} & = & \iiint q_{v}d_{\tau }\end{array} \]

1.3 Relation of field to circuit quantities

\[\begin{array} [c]{lll}v\left ( \text{voltage in volts}\right ) & = & \int \overline{\mathcal{E}}\cdot d\mathbf{l}\\ & & \\ i\left ( \text{current in amp}\right ) & = & \iint \overline{\mathcal{J}}\cdot d\mathbf{s}\\ & & \\ q\left ( \text{chanrge in coulombs}\right ) & = & \iiint q_{v}d_{\tau }\\ & & \\ \psi \left ( \text{magnetic flux in weber}\right ) & = & \iint \overline{\mathcal{B}}\cdot d\mathbf{s}\\ & & \\ \psi ^{e}\left ( \text{electric flux in coulombs}\right ) & = & \iint \overline{\mathcal{D}}\cdot d\mathbf{s}\\ & & \\ u\left ( \text{magnetomotive force in amp}\right ) & = & \int \overline{\mathcal{H}}\cdot d\mathbf{s}\end{array} \]

1.4 relations of a complex domain to time domain

\[ \overline{\mathcal{A}}=\sqrt{2}Re\left ( \mathbf{A}e^{j\omega t}\right ) \]

1.5 Maxwell equations in complex form

\[\begin{array} [c]{lll}-\nabla \times \mathbf{E} & = & j\omega \widehat{\mu }\left ( \omega \right ) \mathbf{H}+\mathbf{M}^{i}=\widehat{z}\left ( \omega \right ) \mathbf{H}+\mathbf{M}^{i}\\ & & \\ \nabla \times \mathbf{B} & = & j\omega \widehat{\epsilon }\left ( \omega \right ) \mathbf{E}+\mathbf{J}^{c}=j\omega \widehat{\epsilon }\left ( \omega \right ) \mathbf{E}+ \widehat{\sigma }\left ( \omega \right ) \mathbf{E}=\left ( j\omega \widehat{\epsilon }\left ( \omega \right ) +\widehat{\sigma }\left ( \omega \right ) \right ) \mathbf{E}=\widehat{y}\left ( \omega \right ) \mathbf{E}\end{array} \]

in free space\[\begin{array} [c]{c}\widehat{y}\left ( \omega \right ) =j\omega \epsilon _{o}\\ \widehat{z}\left ( \omega \right ) =j\omega \mu _{o}\end{array} \]

for all frequencies and all field intensities.

for non-magnetic metals

\[\begin{array} [c]{c}\widehat{y}\left ( \omega \right ) =\sigma +j\omega \epsilon _{o}\\ \widehat{z}\left ( \omega \right ) =j\omega \mu _{o}\end{array} \]

in ferromagnetic metals\[\begin{array} [c]{c}\widehat{y}\left ( \omega \right ) =\sigma +j\omega \widehat{\epsilon }\\ \widehat{z}\left ( \omega \right ) =j\omega \widehat{\mu }\end{array} \]

in good dielectric (nonmagnetic dielectric)\[\begin{array} [c]{c}\widehat{y}\left ( \omega \right ) =j\omega \widehat{\epsilon }\\ \widehat{z}\left ( \omega \right ) =j\omega \mu _{o}\end{array} \]

where\[ \widehat{\epsilon }\left ( \omega \right ) =\epsilon ^{^{\prime }}-j\epsilon ^{^{\prime \prime }}=\left | \widehat{\epsilon }\right | e^{-j\delta } \]

where \(\epsilon ^{^{\prime }}\) called a-c capacitivity, \(\epsilon ^{^{\prime \prime }}\) called dielectric loss factor, \(\delta \) called dielectric loss angle.

and\[ \widehat{\mu }\left ( \omega \right ) =\mu ^{^{\prime }}-j\mu ^{^{\prime \prime }}=\left | \widehat{\mu }\right | e^{-j\delta _{m}} \]

where \(\mu \)\(^{^{\prime }}\) called a-c inductitvity, \(\mu \)\(^{^{\prime \prime }}\) called magnetic loss factor, \(\delta _{m}\) called magnetic loss angle.

2 some relations

\[ k=k^{^{\prime }}-jk^{^{\prime \prime }} \]

where \(K\) is the wave number\[ k=\sqrt{-\widehat{z}\widehat{y}} \]

and\[ \eta =\mathcal{R+}j\mathcal{X} \]

where \(\eta \) is the intrinisc impedence. for air\[\begin{array} [c]{c}k=\omega \sqrt{\mu \epsilon }\\ \eta =\sqrt{\frac \mu \epsilon }\end{array} \]

speed of light\[ c=\frac 1{\sqrt{\epsilon _{o}\mu _{o}}}=3\times 10^{8}m/s \]

wave impedence, is the ratio of components of \(\mathbf{E}\) to components of \(\mathbf{H}\)

interinsic wave length \(\lambda \)=\(\frac{2\pi }k\)