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Electrodynamics                                                                      Nov 14, 2018

Tutorial-VII

Potentials of a moving point charge:- The Leinard Wiechert potentials for a moving point charge are given by 
\begin{eqnarray} \phi(\vec{r}, t) &=& \frac{e}{4\pi\epsilon_0} \frac{1}{s}\label{eq01}\\ \vec{A}(\vec{r}, t) &=& \frac{\mu_0}{4\pi} \frac{q\vec{v}}{s}\label{eq02}. \end{eqnarray} where \begin{eqnarray} s &=& R- \frac{\vec{v}\cdot\vec{R}}{c} \label{eq03}\\ \vec{R} &=& \vec{r}- \vec{r}{'} , \qquad R = |\vec{r}- \vec{r}{'}|.\label{eq04}\\ t{'} &=& r- \frac{R}{c} = t-\frac{|\vec{r}-\vec{r}{'}|}{c} \label{eq05} \end{eqnarray} Here \(\vec{r}{'}\) and \(t'\) are retarded position vector and retarded times. To compute the fields we will need the partial derivatives of potentials w.r.t. \(\vec{r}\) and \(t\), In the following we use the notation \begin{equation} \pp[F]{t} = \pp[F]{t}\big|_{\vec{r}} \qquad \nabla F = \nabla F \big|_t. \end{equation} here \(\nabla F= (\pp[F]{x}, \pp[F]{y},\pp[F]{z})\). We will also use the notation \begin{equation} \nabla_1 F =\nabla F \Big|_{t'} = \Big(\pp[F]{x}\big|_{t'}, \pp[F]{y}\big|_{t'},\pp[F]{z}\big|_{t'}\Big). \end{equation} \(F\) may depend on \(\vec{r}\) and \(t\) implicitly through the position \(\vec{r}{'}\) at retarded time, \(t'\). We want to compute the electric and magnetic fields at a point \(\vec{r},t\) in terms of velocity and acceleration of the particle. The potentials depend on the retarded position \(\vec{r}{'}\) and retarded time \(t'\) which in turn depend on the field point \(\vec{r},t\).

1. Prove the following identities which will be required for calculation of fields. \begin{eqnarray}\label{eq06} \pp[R]{t'} &=& - \frac{\vec{R}.\vec{v}}{R},\\\label{eq07} \pp[f]{t} &=& \frac{R}{s}\pp[f]{t'}\\\label{eq08} \nabla f &=& \nabla_1 f - \frac{\vec{R}}{cs}\pp[f]{t'} \end{eqnarray} where \(\nabla_1 f= \nabla f|_{t'}\). {Rc}\Big)}\label{eq10} {R} \nabla t'
2. Since \(t,t',\vec{r},\vec{r}{'}\) appear in the potentials implicitly through \(s\), compute the derivatives \(\nabla_1s\) and \(\pp[s]{t'}\) of \(s\).
3. We have the electric field given by \begin{equation} \vec{E}=-\nabla \phi -\pp[\vec{A}]{t} \end{equation} Using the expressions for the potentials from \eqRef{eq01} and \eqRef{eq02}, verifty that \begin{eqnarray} \frac{4\pi\epsilon_0}{e} \vec{E} &=& \frac{1}{s^2}\nabla_1 s -\frac{1}{cs^3} \big(\vec{R}- \frac{R\vec{v}}{c}\big)\pp[s]{t'} - \frac{R}{c^2s^2} \dot{\vec{v}} \end{eqnarray}
4. Substituting for \(\nabla_1s\)%, from \eqRef{E18}, show that \begin{eqnarray} \frac{4\pi\epsilon_0}{e} \vec{E}&=& \dot{\vec{v}} \\&=& \frac{1}{s^2R} \Big(\vec{R}-\frac{R\vec{v}}{c}\Big) -\frac{1}{cs^3} \Big(\vec{R}- \frac{R\vec{v}}{c}\big)\Big\{- \frac{\vec{R}\cdot\vec{v}}{R} + \frac{v^2}{c} - \frac{\vec{R}\cdot\dot{\vec{v}}}{c} \Big\} - \frac{R}{c^2s^2} \dot{\vec{v}} \nonumber\\\ \end{eqnarray}
5. Now collect all terms which are independent of \(\dot{\vec{v}}\). Show that these can be rearranged to give \begin{eqnarray} \frac{1}{s^3}\Big(\vec{R}-\frac{R\vec{v}}{c}\Big)\Big[1 - \frac{v^2}{c^2}\Big] \label{eq24} \end{eqnarray}
6. Verify that the terms having \(\dot{v}\) can be rearranged to get \begin{eqnarray} \frac{\vec{R}\cdot\vec{v}}{c}\big) \Big]= \frac{1}{c^2s^3} \vec{R} \times\Big[(\vec{R}-\frac{R\vec{v}}{c}) \times \dot{\vec{v}} \Big]\label{eq25} \end{eqnarray}
7. Finally collect all the terms and prove \begin{equation} \vec{E} = \frac{e}{4\pi\epsilon_0}\left( \frac{1}{s^3}\Big(\vec{R}-\frac{R\vec{v}}{c}\Big)\Big[1 - \frac{v^2}{c^2}\Big] + \frac{1}{c^2s^3} \vec{R} \times\Big[(\vec{R}-\frac{R\vec{v}}{c}) \times \dot{\vec{v}} \Big] \right) \end{equation}
8. Write a translation dictionary for notation used here and that used in the book .

References : Jack Vanderlinde, Classical Electromagnetic Theory, 2nd edn(2004)
Springer India Pvt Ltd, New Delhi.