TimeHarmonic

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This section is used the describe an electromagnetic field problem in the frequency domain. This means that the electromagnetic field depends harmonically on time with an angular frequency \omega. The relevant fields are replaced by complex phasors \VField{E}(\pvec{x}), \VField{H}(\pvec{x}), etc., related to the actual fields by

\begin{eqnarray*} \VField{E}(\pvec{x}, t) & = & \Re \left (\VField{E}(\pvec{x}) e^{-i \omega t} \right ),\\ \VField{H}(\pvec{x}, t) & = & \Re \left (\VField{H}(\pvec{x}) e^{-i \omega t} \right ). \end{eqnarray*}

This allows to write Maxwell’s equations as

\begin{eqnarray*} \curl \VField{E} & = & i \omega \left( \mu \VField{H} +\tilde{\mu} \VField{E} \right), \\ \curl \VField{H} & = & - i \omega \left( \varepsilon \VField{E}+\tilde{\varepsilon} \VField{H} \right) +\VField{J}, \\ \divo \left( \mu \VField{H} + \tilde{\mu} \VField{E} \right) & = & 0, \\ \divo \left( \varepsilon \VField{E}+\tilde{\varepsilon} \VField{H} \right) & = & \SField{\rho}. \end{eqnarray*}

When splitting the electric current into the ohmic current \VField{J}^{\mathrm{(Ohm)}} and the impressed current \VField{J}^{\mathrm{(imp)}}, that is

\begin{eqnarray*} \VField{J} & = & \VField{J}^{\mathrm{(Ohm)}}+\VField{J}^{\mathrm{(imp)}} = \sigma \VField{E}+\VField{J}^{\mathrm{(imp)}}, \end{eqnarray*}

the second equation reads as

\begin{eqnarray*} \curl \VField{H} & = & - i \omega \left( \TField{\varepsilon}_{\mathrm{C}} \VField{E}+\tilde{\varepsilon} \VField{H} \right)+\VField{J}^{\mathrm{(imp)}}, \end{eqnarray*}

with the complex permittivity tensor

\begin{eqnarray*} \TField{\varepsilon}_{\mathrm{C}} = \TField{\varepsilon}+\frac{i}{\omega}\sigma. \end{eqnarray*}

In the context of time-harmonic problems the index ‘\mathrm{C}’ will be dropped and it should be kept in mind that the permittivity tensor is allowed to have complex-valued entries.

From this one obtains independent equations of second order for the electric and magnetic fields:

\begin{align*} {} & \curl \mu^{-1} \curl \VField{E} \\ {} & \phantom{xxx} +i \omega \tilde{\varepsilon} \mu^{-1} \curl \VField{E}-i \omega \curl \mu^{-1} \tilde{\mu} \VField{E} \\ {} & \phantom{xxx} - \omega^2 \left ( \TField{\varepsilon} - \tilde{\varepsilon} \mu^{-1} \tilde{\mu} \right) \VField{E} = i \omega \VField{J}^{\mathrm{(imp)}}, \end{align*}

and

\begin{align*} {} & \curl \varepsilon^{-1} \curl \VField{H} \\ {} & \phantom{xxx} -i \omega \tilde{\mu} \varepsilon^{-1} \curl \VField{H}+i \omega \curl \varepsilon^{-1} \tilde{\varepsilon} \VField{H} \\ {} & \phantom{xxx} - \omega^2 \left( \mu - \tilde{\mu} \varepsilon^{-1} \tilde{\varepsilon} \right) \VField{H} \\ {} & \phantom{xxxxxx} = \curl \varepsilon^{-1} \VField{J}^{\mathrm{(imp)}}-i \omega \tilde{\mu} \varepsilon^{-1} \VField{J}^{\mathrm{(imp)}}. \end{align*}