dualFarField.tex

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% Copyright © 2015 Peeter Joot.  All Rights Reserved.
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%\generatetitle{Duality transformation of the far field fields.}
\section{Duality transformation of the far field fields.}
\index{duality transformation}
\index{far field}
%\label{chap:dualFarField}
We've seen that the far field electric and magnetic fields associated with a magnetic vector potential were
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\begin{subequations}
\label{eqn:dualFarField:20}
\begin{equation}\label{eqn:dualFarField:40}
\BE = -j \omega \Proj_\T \BA,
\end{equation}
\begin{equation}\label{eqn:dualFarField:60}
\BH = \inv{\eta} \kcap \cross \BE.
\end{equation}
\end{subequations}
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What does \( \BH \) look like in terms of \( \BA \)?  Expanding the rejection of the radial component answers that
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\begin{equation}\label{eqn:dualFarField:140}
\BH
= -\frac{j \omega}{\eta} \kcap \cross \lr{ \BA - \lr{\BA \cdot \kcap} \kcap }.
\end{equation}
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The \( \kcap \) crossed terms are killed, leaving
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\begin{equation}\label{eqn:dualFarField:160}
\BH
= -\frac{j \omega}{\eta} \kcap \cross \BA.
\end{equation}
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It's worth a quick note that the duality transformation for this, referring to \citep{balanis2005antenna} \texttabref{3.2}, is
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\begin{subequations}
\label{eqn:dualFarField:80}
\begin{equation}\label{eqn:dualFarField:100}
\BH = -j \omega \Proj_\T \BF
\end{equation}
\begin{equation}\label{eqn:dualFarField:120}
\BE = \eta \kcap \cross \BH = j \omega \eta \kcap \cross \BF.
\end{equation}
\end{subequations}
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These show explicitly that neither the electric or magnetic far field have any radial component, matching with intuition for transverse propagation of the fields.
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