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\usepackage{sw20jrep}

%TCIDATA{TCIstyle=Report/report.lat,jrep,report}

%TCIDATA{Created=Tue Nov 26 09:32:02 1996}
%TCIDATA{LastRevised=Sun Dec 08 14:31:13 1996}
%TCIDATA{Language=American English}

\input tcilatex
\pagestyle{myheadings}
\markright{{\sf Time Distribution of Emitted Radiation    }}
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\begin{document}


\section{14.3\quad Time Distribution of Emitted Radiation.}

\subsection{14.3.1}

\quad \thinspace \textbf{a. }The distribution of emitted radiation can be
described by using the angular variables and the time, or by using the
angular variables and the frequency. Following Jackson, we give first the
distribution in time in this section, then the distribution in frequency in
Sect.14.5. The latter is probably approached more easily by working in
frequency (Fourier) space from the beginning, while the former follows
naturally from the space-time formulation of the previous sections.

\smallskip\ 

\textbf{b}. The radiative fields come exclusively from the terms
proportional to $\mathbf{\dot{\beta}=\dot{v}/}c$ in the Li\'{e}nard-Wiechert
fields, which behave as $R^{-1},$ i.e. 
\[
\mathbf{E}=\frac{e}{c}\left[ \frac{\mathbf{R\times }\left( \mathbf{r\times 
\dot{\beta}}\right) }{s^{3}}\right] =\frac{e}{cR}\frac{\mathbf{n\times }%
\left( \left( \mathbf{n-\beta }\right) \times \mathbf{\dot{\beta}}\right) }{%
\left( 1-\mathbf{\beta \cdot n}\right) ^{3}}\,,\quad \quad \quad \mathbf{B}=%
\mathbf{n\times E} 
\]
with $\mathbf{r}=\mathbf{R-\beta }R.$ (See sketch on page1.2), $\mathbf{n=R/}%
R,$ $s=R-\mathbf{\beta \cdot R.}$

\smallskip\ 

\textbf{c. }The Poynting vector $\mathbf{S=(}c/4\pi )\mathbf{E\times B}$ is
in the direction of $\mathbf{R}$ and has magnitude $\mathbf{(}c/4\pi )%
\mathbf{E}^2.$

\smallskip\ 

\textbf{d. }The power radiated per unit solid angle is $\mathbf{(}c/4\pi )R^2%
\mathbf{E}^2.$ Often one wishes to compute the energy radiated over a given
portion of the path $\mathbf{x}^{\prime }\left( t^{\prime }\right) $. This
leads to consider 
\[
\frac{dP\left( t^{\prime }\right) }{d\Omega }=\frac c{4\pi }R^2\left| 
\mathbf{E}\right| ^2\frac{dt}{dt^{\prime }}=\frac c{4\pi }R^2\left| \mathbf{E%
}\right| ^2\left( 1-\mathbf{\beta \cdot n}\right) 
\]
(from $t=t^{\prime }+R/c).$

\newpage

\subsection{14.3.2}

\quad \thinspace \textbf{a. }The general formula for the radiated power, as
a function of the time and angles as seen in the observer's system from the
instantaneous position of the particle, is then 
\begin{equation}
\frac{dP\left( t^{\prime }\right) }{d\Omega }=\frac{e^2}{4\pi c}\frac{%
\left\{ \mathbf{n\times }\left[ \left( \mathbf{n-\beta }\right) \times 
\mathbf{\dot{\beta}}\right] \right\} ^2}{\left( 1-\mathbf{n\cdot \beta }%
\right) ^5}\,.  \label{1}
\end{equation}

\smallskip\ 

\textbf{b}. We consider in detail two particular cases: $\mathbf{\dot{\beta}}
$ parallel to $\mathbf{\beta }$ (as in linear harmonic motion) and $\mathbf{%
\dot{\beta}}$ perpendicular to $\mathbf{\beta }$ (as in uniform circular
motion). In every case we let $\mathbf{\beta }$ be along the $z$ axis (polar
axis). Compare the non-relativistic case of Sect.14.2, where it is
convenient to take $\mathbf{\dot{\beta}}$ along the polar axis (note that $%
\Theta $ is the angle between $\mathbf{\dot{\beta}}$ and $\mathbf{R}$, while 
$\vartheta \;$is the angle between $\mathbf{\beta }$ and $\mathbf{R}$.

\smallskip\ 

\textbf{c. }In \textsl{linear motion}, $\mathbf{\beta \times \dot{\beta}}=0$
and (\ref{1}) reduces to

\begin{equation}
\frac{dP\left( t^{\prime }\right) }{d\Omega }=\frac{e^{2}\dot{v}^{2}}{4\pi
c^{3}}\frac{\sin ^{2}\vartheta }{\left( 1-\beta \cos \vartheta \right) ^{5}}%
\,.  \label{2}
\end{equation}
This agrees with the Larmor result of 2.1.c for $\beta \ll 1$, since in this
case $\vartheta =\Theta .$ But the denominator in (\ref{2}) causes a strong
enhancement of the power radiated in the near forward direction for $\beta $
approaching $1.$ The peak intensity occurs for 
\begin{equation}
\cos \vartheta _{\max }=\frac{1}{3\beta }\left[ \sqrt{1+15\beta ^{2}}%
-1\right] .  \label{2'}
\end{equation}
For $\beta \rightarrow 1,$ one can use $\beta ^{2}=1-\dfrac{1}{\gamma ^{2}}$
to obtain $\vartheta _{\max }\simeq \dfrac{1}{2\gamma }\,.$

\smallskip\ 

\textbf{d}. The forward peaking within an angle such that $2\vartheta \gamma
\sim 1$ is a general characteristic of radiation by relativistic particles.

\newpage 

\subsection{14.3.3}

\quad \thinspace \textbf{a. }In the relativistic limit $\beta \rightarrow 1,$
$\gamma \rightarrow \infty ,$ $\vartheta $ small, we have $1-\beta \cos
\vartheta \simeq \dfrac{1}{2\gamma ^{2}}+\dfrac{\vartheta ^{2}}{2}$ and 
\begin{equation}
\frac{dP\left( t^{\prime }\right) }{d\Omega }\approx \frac{8}{\pi }\frac{%
e^{2}\dot{v}^{2}}{c^{3}}\gamma ^{8}\frac{\left( \gamma \vartheta \right) ^{2}%
}{\left( 1+\gamma ^{2}\vartheta ^{2}\right) ^{5}}\,.  \label{3}
\end{equation}
This peaks at $\gamma \vartheta =1/2,$ in agreement with eq. (\ref{2'}), and
has a $\left\langle \Delta \vartheta ^{2}\right\rangle \simeq \gamma
^{-2}=1-\beta ^{2}.$

\ 

\textbf{b. }The total radiated power, for any $\beta $, is obtained by
integrating (\ref{2}): 
\begin{equation}
P_{lin}\left( t^{\prime }\right) =\frac{2}{3}\frac{e^{2}}{c^{3}}\,\dot{v}%
^{2}\gamma ^{6}\,.  \label{4}
\end{equation}
Note that this is the power instantaneously radiated by the charge at
(laboratory) time $t^{\prime }.$ If we were to compute the power crossing a
sphere of radius $R$ as a function of $t,$ we would have a hard time because
the emitting charge is not sitting still at the center of the sphere.

\ 

\textbf{c. }Here are angular distribution plots for $\beta =0.05$ (the
classical ``donut'', slightly deformed) and $\beta =0.5$ (in cutout
perspective, to show the ``rabbit ears'').

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\newpage

Here are angular distribution plots for $\beta =0.9$ and $\beta =0.99,$ both
in perspective (left) and in cut-out front view to show the ``rabbit ears''.%
\vspace{-0.5in}

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\vspace{-0.7in} \enlargethispage*{1in}

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\newpage

\subsection{14.3.4}

\quad \thinspace \textbf{a. }For a charge in \textsl{uniform circular motion}
we use 
\[
\left[ \mathbf{n\times }\left( (\mathbf{n-\beta })\times \mathbf{\dot{\beta}}%
\right) \right] ^{2}=\dot{\beta}^{2}\left[ \left( 1-\beta \cos \vartheta
\right) ^{2}-\frac{\sin ^{2}\vartheta \cos ^{2}\varphi }{\gamma ^{2}}\right] 
\]
and get 
\begin{equation}
\frac{dP\left( t^{\prime }\right) }{d\Omega }=\frac{e^{2}}{4\pi c^{3}}\frac{%
\dot{v}^{2}}{\left( 1-\beta \cos \vartheta \right) ^{3}}\left[ 1-\frac{\sin
^{2}\vartheta \cos ^{2}\varphi }{\gamma ^{2}\left( 1-\beta \cos \vartheta
\right) ^{2}}\right] \,.  \label{5}
\end{equation}

\smallskip\ 

\textbf{b.} Plots of this are given on next page. In the relativistic limit, 
$\gamma \gg 1,$ we can use again $\sin \vartheta \rightarrow \vartheta ,$ $%
1-\beta \cos \vartheta \rightarrow \left( 1+\gamma ^{2}\vartheta ^{2}\right)
/2\gamma ^{2}\,,$ as in 2.3a.

\smallskip\ 

\textbf{c. }The total radiated power, by an integration of (\ref{5}) is 
\begin{equation}
P_{circ}\left( t^{\prime }\right) =\frac{2}{3}\frac{e^{2}}{c^{3}}\,\dot{v}%
^{2}\gamma ^{4}\,.  \label{6}
\end{equation}

\smallskip\ 

\textbf{d. }The comparison of (\ref{4}) and (\ref{6}) is very instructive.
For equal values of $\dot{v}$, (\ref{4}) is larger by a factor $\gamma ^{2}.$
A more significant comparison, however, is made for equal values of $d%
\mathbf{p/}dt,$ with $\mathbf{p}=m\gamma \mathbf{v.}$%
\begin{eqnarray*}
\text{For }\mathbf{\dot{v}\,}/\!/\,\mathbf{v,\quad }\frac{d\mathbf{p}}{dt}
&=&m\left( \gamma \mathbf{\dot{v}+}\gamma ^{3}\frac{v^{2}}{c^{2}}\mathbf{%
\dot{v}}\right) =m\dot{v}\gamma ^{3} \\
\text{For }\mathbf{\dot{v}}_{\perp }\mathbf{v,\quad }\frac{d\mathbf{p}}{dt}
&=&m\gamma \dot{v}\,.
\end{eqnarray*}
Hence: 
\[
P_{lin}\left( t^{\prime }\right) =\frac{2}{3}\frac{e^{2}}{c^{3}}\frac{1}{%
m^{2}}\left( \frac{dp}{dt^{\prime }}\right) ^{2} 
\]
while: 
\[
P_{circ}\left( t^{\prime }\right) =\frac{2}{3}\frac{e^{2}}{c^{3}}\frac{%
\gamma ^{2}}{m^{2}}\left( \frac{dp}{dt^{\prime }}\right) ^{2}\,. 
\]
$P_{circ}$ is larger by $\gamma ^{2}.$

\smallskip\ 

\textbf{e. }The difference between ``longitudinal'' and ``transverse''
accelerated behavior of relativistic particles discussed above should be
well known from elementary relativistic mechanics. One sometimes speaks of a
``longitudinal mass'' $m\gamma ^{3}$ and a ``transverse mass'' $m\gamma .$

\newpage Here are angular distribution plots of the synchrotron radiation
emitted by a particle at a given instant, eq.(\ref{5}). To accommodate the
way Maple works best, the orbit is in the $xz$ plane with $\mathbf{v}$ along 
$x,$ and what is actually plotted is (by a suitable redefinition of angles) 
\[
\frac{1}{\left( 1+\beta \sin \phi \cos \theta \right) ^{3}}\left( 1-\frac{%
\left( 1-\beta ^{2}\right) \cos ^{2}\phi }{\left( 1+\beta \sin \phi \cos
\theta \right) ^{2}}\right) 
\]
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\newpage

Here are cross-sectional plots in the plane of the orbit ($\phi =0,$ dotted
line) and in the plane containing $\mathbf{v}$ and the perpendicular to the
orbit ($\phi =\pi /2,$ solid line). The top panel shows plots for $v=c/20$
(dotted line), the middle panel for $v=$ $c/2,$ the bottom panel is for $%
v=0.9c$. Note the different scales.

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\end{document}
