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Simple Radiating Systems

Let us start by writing the integral equation for the vector potential $\mbox{\boldmath$A$}(\mbox{\boldmath$x$})$ where we presume that we've already transformed the IWE into the IHE. We will choose to use the outgoing wave Green's function to make it clear that the field we are looking for is the one that the source is emitting, not one that it is absorbing.


\begin{displaymath}
\mbox{\boldmath$A$}(\mbox{\boldmath$x$}) = +\mu_0 \int \fra...
...$x$}'\vert}
\mbox{\boldmath$J$}(\mbox{\boldmath$x$}') d^3x'.
\end{displaymath} (11.68)

There is no inhomogeneous term if there are no boundaries with a priori known boundary conditions.

Note that a more general solution would be one that allowed for absorption of incoming waves as well as the emission of outgoing waves, but that this would require knowing something about the sources outside the domain considered to be infinite. We will talk about this later (scattering theory and the optical theorem).

From $\mbox{\boldmath$A$}(\mbox{\boldmath$x$})$ we can easily find $\mbox{\boldmath$B$}$ or $\mbox{\boldmath$H$}$:

\begin{displaymath}
\mbox{\boldmath$B$}= \mu_0 \mbox{\boldmath$B$}= \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}
\end{displaymath} (11.69)

(by definition). Outside of the source, though (where the currents are all zero) Ampere's law tells us that:
\begin{displaymath}
\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$H$}= -i\omega \mbox{\boldmath$D$}
\end{displaymath} (11.70)

or
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle -i\omega\mu_0\epsilon_0 \mbox{\boldmath$E$}$ (11.71)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle -i\frac{\omega}{c^2} \mbox{\boldmath$E$}= i\frac{k}{c} \mbox{\boldmath$E$}$ (11.72)

or
\begin{displaymath}
\mbox{\boldmath$E$}= i \frac{c}{k}\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$}
\end{displaymath} (11.73)

Doing the integral above can be quite difficult in the general case. However, we'll find that for most reasonable, physical situations we will be able to employ certain approximations that will enable us to obtain a systematic hierarchy of descriptions that converge to the correct answer as accurately as you like, at the same time they increase our physical insight into the radiative processes.



Subsections
next up previous contents
Next: The Zones Up: Radiation Previous: Green's Function for the   Contents
Robert G. Brown 2007-12-28