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%TCIDATA{Created=Mon Nov 25 15:26:43 1996}
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\markright{{\sf Radiation by Moving Charges    }}
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\begin{document}


\section{14.1\quad Radiation by Moving Charges.}

\subsection{14.1.1}

\quad \thinspace \textbf{a. }Although the approach used in Ch.9, starting
from 
\begin{equation}
A^{\beta }\left( k_{0},\,\mathbf{x}\right) =\frac{4\pi }{c}\int d^{3}\mathbf{%
x}^{\prime }D(k_{0},\,\mathbf{x}-\mathbf{x}^{\prime })J^{\beta }\left(
k_{0},\,\mathbf{x}^{\prime }\right)  \label{1}
\end{equation}
is appropriate to treat any problem with assigned sources $J^{\beta },$ it
is sometimes more convenient to approach the problem entirely in space-time,
using 
\begin{equation}
A^{\beta }\left( \mathsf{x}\right) =\frac{4\pi }{c}\int d^{4}x^{\prime }D(%
\mathsf{x}-\mathsf{x}^{\prime })\,J^{\beta }\left( \mathsf{x}^{\prime
}\right) \,.  \label{2}
\end{equation}

\smallskip\ 

\textbf{b}. This is especially true when a microscopic approach is taken.
Then the current density consists in fact of discrete particles and it may
be advantageous to construct the total field by adding the fields of the
individual particles. Accordingly, we are led to consider the current due to
a single particle moving with 4-velocity $u^{\alpha }.$ This is written in
the manifestly covariant form 
\begin{equation}
J^{\alpha }\left( \mathsf{x}^{\prime }\right) =ec\int d\tau \,u^{\alpha
}\left( \tau \right) \delta ^{\left( 4\right) }\left( \mathsf{x}^{\prime }-%
\mathsf{x}^{\prime }\left( \tau \right) \right)  \label{3}
\end{equation}
where $\mathsf{x}^{\prime }\left( \tau \right) $ is the particle path in
parametric form (see 12.3.1 for an example). Using $d\tau =dt^{\prime
}/\gamma $ and $u^{\alpha }=\gamma \left( c,\mathbf{v}\right) ,$ we can put (%
\ref{3}) in the more usual form 
\begin{eqnarray}
\mathbf{J}\left( \mathbf{x}^{\prime },\,t^{\prime }\right) &=&e\mathbf{v}%
\left( t^{\prime }\right) \delta ^{\left( 3\right) }\left( \mathbf{x}%
^{\prime }-\mathbf{x}^{\prime }\left( t^{\prime }\right) \right)  \label{4}
\\
\mathbf{J}_{0}\left( \mathbf{x}^{\prime },\,t^{\prime }\right) &=&c\rho
\left( t^{\prime }\right) =ec\delta ^{\left( 3\right) }\left( \mathbf{x}%
^{\prime }-\mathbf{x}^{\prime }\left( t^{\prime }\right) \right) .  \nonumber
\end{eqnarray}
[We use $\int cd\tau \gamma \mathbf{v}\delta \left( x^{0\prime }-x^{0\prime
}\left( \tau \right) \right) =c\gamma \mathbf{v/}\left| dx^{0\prime }/d\tau
\right| =\mathbf{v].}$

\smallskip\ 

\textbf{c. }To obtain explicit forms for $A^{\beta }\left( x\right) ,$ we
insert (12.11.5) in (\ref{2}). This gives the manifestly covariant form (see
Jackson) 
\begin{equation}
A^{\alpha }\left( \mathsf{x}\right) =\left. \frac{eu^{\alpha }\left( \tau
\right) }{\mathsf{u}\cdot \left[ \mathsf{x}-\mathsf{x}^{\prime }\left( \tau
\right) \right] }\right| _{ret},  \label{5}
\end{equation}
where $ret\,$ stands for ``tretartded'' and means: $\tau $ is such that $%
x^{0}=x^{0\prime }\left( \tau \right) +\left| \mathbf{x}-\mathbf{x}^{\prime
}\left( \tau \right) \right| .$

\newpage

\subsection{14.1.2}

\quad \quad \textbf{a. }The potentials (\ref{5}) are known as the
Li\'{e}nard-Wiechert potentials. The 3-vector form of these potentials is
actually more useful and transparent. It can be derived from (\ref{5}) (see
Jackson) or more directly by using (from (\ref{2}), (\ref{4}), (12.11.6)) 
\begin{equation}
\mathbf{A}=\frac{e}{c}\int \frac{dx^{\prime \,0}}{R\left( t^{\prime }\right) 
}\delta \left( x^{0}-x^{\prime \,0}-R\left( t^{\prime }\right) \right) 
\mathbf{v}\left( t^{\prime }\right) ,\quad \quad \phi =e\int \frac{%
dx^{\prime \,0}}{R\left( t^{\prime }\right) }\delta \left( x^{0}-x^{\prime
\,0}-R\left( t^{\prime }\right) \right)  \label{6}
\end{equation}
with $R\left( t^{\prime }\right) =\left| \mathbf{x-x}^{\prime }\left(
t^{\prime }\right) \right| .$ The result is

\begin{equation}
\mathbf{A}\left( \mathbf{x,\,}t\right) =\left[ \frac{e\mathbf{\beta }}{%
\left( 1-\beta _{/\!/}\right) R}\right] _{ret}\quad \quad \quad \phi \left( 
\mathbf{x,\,}t\right) =\left[ \frac e{\left( 1-\beta _{/\!/}\right)
R}\right] _{ret}\,.  \label{7}
\end{equation}
Here $\mathbf{\beta }=\mathbf{v/}c$ and $\beta _{/\!/}$ is the component of $%
\mathbf{\beta }$ in the line-of-sight direction, i.e. in the direction of $%
\mathbf{x-x}^{\prime }\left( t^{\prime }\right) .$ The time $t$ appears
impilcitly through the condition, indicated by the subscript ``ret'', that $%
t^{\prime }$ must satisfy the $\delta $-function in (\ref{6}), i.e., 
\begin{equation}
t^{\prime }=t-R\left( t^{\prime }\right) /c\,.  \label{8}
\end{equation}

\smallskip\ 

\textbf{b.} In this formulation it appears clearly that the fields are not
determined at all by the present position of the particle, but by its
position at the earlier time $t^{\prime }$ that satisfies (\ref{8}). To
avoid confusion, it must be understood that both $t$ and $t^{\prime }$ are
measured in the frame of the observer, which usually coincides with the
laboratory frame. See sketch.

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\newpage

\subsection{14.1.3}

\quad \quad \textbf{a. }As we know, the potentials are interesting, but the
physical effects are due to the fields. The necessary differentiations to
obtain $F^{\mu \nu }$ can be done in the manifestly covariant form, leading
to Jackson's (14.11), which is rather useless as such. The 3-space forms
(14.13) and (14.14) are much more instructive. Their derivation is involved,
no matter how one proceeds. The 3-space procedure followed in the old
Jackson and given in full detail in Appendix is probably still the easiest.
The result is most easily expressed in terms of the vector $\mathbf{r}=%
\mathbf{R-\beta }R,$ which is the oriented segment $\overrightarrow{PO}$ of
the sketch on previous page, and of $s=OQ=R\left( 1-\beta _{/\!/}\right) =%
\mathbf{r\cdot R/}R;$%
\begin{eqnarray}
\mathbf{E(x,}\,t) &=&e\frac{\mathbf{r}}{\gamma ^2s^3}+\frac ec\frac{\mathbf{%
R\times }\left( \mathbf{r\times \dot{\beta}}\right) }{s^3}  \label{9} \\
\mathbf{B(x,}\,t) &=&\frac{\mathbf{R}}R\times \mathbf{E\,.}  \nonumber
\end{eqnarray}

\smallskip\ 

\textbf{b.} Note also that eqns. (\ref{7}) can be written as 
\begin{equation}
\mathbf{A(x,}\,t)=\frac{e\mathbf{\beta }}s\quad \quad \quad \phi =\frac es\,.
\label{10}
\end{equation}
In this deceptively simple notation they have the same form as for static
fields (this is Panofsky and Phillips' notation).

\smallskip\ 

\textbf{c.} If the particle is moving with uniform velocity, $P$ is the
present position of the particle (i.e. its position at time $t)$ and $r$ is
its present distance from the observer. It is a straightforward exercise to
show that in this case the fields (\ref{9}) reduce to those found in
Sect.11.10. See Jackson..

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