\documentclass[12pt]{article}
\usepackage{wrapfig,graphicx}
\usepackage[colorlinks]{hyperref}
%\newcolumntype{d}[0]{D{.}{.}{-4}}
\setlength{\topmargin}{0.02in}
\setlength{\parskip}{4.0mm}
\setlength{\textheight}{9in}
\setlength{\parindent}{0pt}
\newcommand{\ttmdump}[1]{#1}
\begin{document}
Not knowing anything in this area, I searched for a way to calculate energy loss suffered by a charge on encountering a cavity in an otherwise smooth vacuum chamber. I found \href{http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4433.pdf}{a note by Bob Palmer}\cite{bp}. His equation (2.11) for energy lost is exactly the geometry under consideration:
\[U=(0.85)\frac{Q^2}{4\pi\epsilon_0}\sqrt{\frac{g}{2w}}\,\frac{1}{a}\sim(1.6)
\frac{Q^2}{4\pi\epsilon_0}\,\frac{1}{a}\]
Namely, a smooth beam pipe of radius $a$ interrupted by a gap of width $g$ extending to a radius $\gg a$. (See Fig.) $w$ is the length of the wake at the wall; it's some combination of bunch length, $\gamma$, and $a$.
\begin{wrapfigure}{l}{6cm}\vspace{-1cm}
\includegraphics[width=6cm]{bp}
\end{wrapfigure}
I'm using $a=25$\,mm, $g=60$\,mm, bunch length $\sim$ 8\,mm to 30\,mm.
For the nominal bunch charge of $Q=16\,$pC, we find \[1.5\times 10^{-10}\,\mbox{joule.}\] And at $650\,$MHz, this is \[P=0.1\,\mbox{Watt.}\]
To make the connection to conventional notation, note that the ``loss factor'' is \[k=\frac{1.6}{4\pi\epsilon_0}\,\frac{1}{a}=50\,\Omega\ \frac{c}{a}=0.6\,\mbox{V/pC}\]
and this is a fairly typical value; see slide 5 of \href{http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-wp-016-ch23-CNg.pdf}{talk by Karl Bane}\cite{kb}. One sees from that talk that one can reduce this an order of magnitude by tapering and shielding. I do not think it is worth it as 100\,mW will not cause any problems. Comments?
\begin{thebibliography}{99}
\bibitem{bp}Bob Palmer: {\em A Qualitative Study of Wake
Fields for Very Short Bunches}, SLAC-PUB-4433 (1987).
\bibitem{kb} Karl Bane: {\em Numerical Calculations of the NLC Damping Ring Impedances} Presented at the Broadband Impedance Measurements \&
Modeling Workshop, Feb 28 - Mar 2, 2000, Stanford.
\end{thebibliography}
\end{document}