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\centerline{\bf Electrodynamics A (PHY 5346) Fall 2016 
Classwork and Homework} \bigskip

Every exercise counts 10 points unless stated differently.
\bigskip

{\bf Set 1:}

\begin{enumerate}

\item Exercise E.2: A non-relativistic distance measurement.
Homework, due 9/13/2016 before class.

\item Exercise E.4: Non-relativistic Galilei transformation.
Homework, due 9/13/2016 before class.

\item Exercise E.5: $x_{\alpha}x^{\alpha}$ contractions.
Homework, due 9/13/2016 before class.

\item Time and relativistic distance measurements.
Classwork, due 9/1/2016 in class.

A Cesium clock counts 2,757,789,531,312 cycles (starting at 0). Find
the elapsed time in seconds (you may round to the nearest integer
number)?

An observer $O_1$ is located in an inertial system and flashes at
times $1,\,2,\,3\,\dots\,[s]$ light signals towards another observer
$O_2$, who reflects them back with a mirror. Assume that the returned
signals are received by $O_1$ at the following times: 

$ (1)~1.002,\,2.002,\,3.002\,\dots\, [s]$, 

$ (2)~1.002,\,2.004,\,3.006\,\dots\, [s]$, 

$ (3)~1.002,\,2.004,\,3.008\,\dots\, [s]$. \\
Determine for each case whether the data are consistent with assuming 
that $O_2$ is also in an inertial system. If this is the case, write 
down the equation for the distance of $O_2$ as function of the time 
as seen by $O_1$. Approximate the speed of light by $300,000\,[km/s]$,
but perform all calculations with a precision of at least four digits.
\smallskip \medskip % Solution in the next appendix.

\end{enumerate}
\bigskip

{\bf Set 2:}

\begin{enumerate}

\setcounter{enumi}{4}

\item Exercise E.6: Time in Minkowski space.
Homework, due 9/15/2016 before class.

\item Exercise E.8: Euclidean and hyperbolic rotations.
Homework, due 9/15/2016 before class.

\end{enumerate}
\bigskip \eject

{\bf Set 3:}

\begin{enumerate}

\setcounter{enumi}{6}

\item Exercise E.10 without part (5): Lorentz group Lie matrix 
generator. Classwork, 5 points, due 9/15/2016 in class.

\item Exercise E.10 part (5). Homework, 5 points, 
due 9/22/2016 before class.

\item Exercise E.11: A rotation and a boost generator.
Homework, due 9/22/2016 before class.

\item Exercise E.12: Four-dimensional Levi-Civita tensor.
Homework, due 9/22/2016 before class.

\item Exercise E.14: Addition theorem for transverse velocity
components. Homework, due 9/22/2016 before class.

\end{enumerate}
%\bigskip

{\bf Set 4:}

\begin{enumerate}

\setcounter{enumi}{11}

\item Exercise E.17: End of spacetrip. Homework, due 9/29/2016 before class.

\item Exercise E.19: Redshift. Homework, due 9/29/2016 before class.

\item Exercise E.21: Relativistic energy-momentum conservation. \hfil
                     \newline Homework, due 9/29/2016 before class.

\item Electromagnetic field tensor, due  9/27/2016 in class.

\noindent
(A) Compare $\partial_{\alpha}F^{\alpha\beta}=(4\pi/c)\,J^{\beta}$ 
for $\beta=0$ with the inhomogeneous Maxwell equations 
$$ \nabla\cdot\vec{E} = 4\pi \rho = \frac{4\pi}{c}\,J^0
   ,~~~\nabla \times \vec{B} - {1\over c}{\partial\vec{E}
   \over\partial t}={4\pi\over c}{\vec J} $$
to identify $F^{00}$, $F^{10}$, $F^{20}$ and $F^{30}$ in terms of
$\vec{E}$ and $\vec{B}$. Continue with $\beta=1$, then $\beta=2$. 
Do you need $\beta=3$ too?

\noindent (B) The result of (A) translates into (see the lecture notes) 
$$ ({^*\!F}^{\alpha\beta}) = \left(\begin{matrix} 
         0      & ^*F^{01} & ^*F^{02} & ^*F^{03} \\
       ^*F^{10} &  0       & ^*F^{12} & ^*F^{13} \\
       ^*F^{20} & ^*F^{21} &  0       & ^*F^{23} \\
       ^*F^{30} & ^*F^{31} & ^*F^{32} &  0 \end{matrix}\right)\
    =\ \left(\begin{matrix} 
         0   & -B^x & -B^y & -B^z \\
         B^x &  0   &  E^z & -E^y \\
         B^y & -E^z &  0   &  E^x \\
         B^z &  E^y & -E^x &  0   \end{matrix}\right)\ . $$
Show that this form and $\partial_{\alpha}\,^*F^{\alpha\beta}=0$ 
imply the homogeneous Maxwell equations 
$$ \nabla\cdot \vec{B} = 0\,,~~~\nabla \times \vec{E} +
   {1\over c} {\partial \vec{B}\over \partial t} = 0\,. $$
Fix $\beta$ to 0, 1, 2, 3 and discuss each case separately.
\bigskip
\end{enumerate}

{\bf Set 5:}

\begin{enumerate}

\setcounter{enumi}{15}

\item Exercise E.18: Time dilation for a satellite.
                     Homework, due 10/6/2016 before class.

\item Exercise E.26: Homogeneous Maxwell equations.
                     Homework, due 10/6/2016 before class.

\item Exercise E.27: Lorentz transformations of electric and magnetic 
                     fields Homework, due 10/6/2016 before class.

\end{enumerate}

{\bf Set 6:}

\begin{enumerate}
\setcounter{enumi}{18}

\item Exercise E.31: Fields of a moving charge.
                     Homework, due 10/13/2016 before class.
\bigskip

\item Exercise: Divergence and Laplace operator.  
      Classwork, due 10/13/2016 in class.

\begin{enumerate}

\item Calculate $ \nabla\cdot\vec{r}$.
\bigskip

\item Calculate $\label{vecrfr}\nabla\cdot\,\vec{r}\,f(r)$ as
      scalar function of $r$.
\bigskip

\item Example: Calculate (b) for $f(r)=r^{n-1}$.
\bigskip

\item Calculate the divergence of the electric field $\vec{E}=q\,\hat{r}/r^2$
for $r\ne 0$.
\bigskip

\item Calculate $ \nabla^2\,(q/r) $ for $r\ne 0$.

\end{enumerate}

\end{enumerate}
\bigskip


{\bf Set 7:}

\begin{enumerate}

\setcounter{enumi}{20}

\item Exercise E.36: Electric and magnetic fields for an infinitely
                     long wire. Homework, due 10/20/2016 before class
                     or till 10/21/2016 5pm in David Clarke's mailbox.

\item Exercise E.37: Charge density from a potential. Homework, 
                     due 10/20/2016 before class
                     or till 10/21/2016 5pm in David Clarke's mailbox.

\item Exercise E.38: Symmetry of Dirichlet Green functions. Homework, 
                     due 10/20/2016 before class
                     or till 10/21/2016 5pm in David Clarke's mailbox.
\end{enumerate}
\bigskip \vfil\eject


{\bf Set 8:}

\begin{enumerate}

\setcounter{enumi}{23}

\item Exercise E.40: Method of images for a plane. Homework, 
                     due 10/27/2016 before class.

\item Exercise E.42: Potential from distinct BCs and half-spheres. 
                     Homework, due 10/27/2016 before class.

\item Exercise E.46: Potential in a rectangular box. Homework, 
                     due 10/27/2016 before class.

\end{enumerate}
\bigskip

{\bf Set 9:}

\begin{enumerate}

\setcounter{enumi}{26}

\item Exercise E.41: Potential over a plane. Due 11/3/2016 before class.

\item Exercise E.44: Angled plates. Homework, due 11/3/2016 
                     before class.

\item Exercise: Expansion in cylindrical coordinates. 
Due 11/3/2016 before class: \medskip

For simplicity we use dimensionless numbers in the following problem.

Consider a cylinder length $L=1$ and radius $\rho_0=1$. The potential 
on the mantel and bottom surface is zero and on the top surface it is 
given by 
$$  R(\rho)\ =\ J_0\left(x_{01}\,\frac{\rho}{\rho_0}\right)\,, $$
where $x_{01}$ is the first zero of the Bessel function $J_0(x)$.

\begin{enumerate}

\item Find the potential everywhere inside the cylinder.

\item Evaluate the potential numerically for $\rho=0$, $z=1/2$.

\item Evaluate the potential numerically for $\rho=1/2$, $z=1/2$.

\end{enumerate}

\item Exercise: Classwork, due 11/3/2016 in class (5 points). \medskip

Use 
$$ L_-\ =\ \hat{x}\cdot\vec{L} - i\,\hat{y}\cdot\vec{L} $$
to find in spherical coordinates:
$$ L_-\ =\ e^{-i\,\phi}\,\left(-\frac{\partial\ }{\partial\theta} 
   + i\,\cot\theta\,\frac{\partial\ }{\partial\phi}\right)\,.  $$

\end{enumerate}
\bigskip

\vfil\eject

{\bf Set 10:}

\begin{enumerate}

\setcounter{enumi}{30}

\item Exercise E.39: Paul Trap. Due 11/10/2016 before class.
\bigskip

\item Exercise: Potential of two point charges.
Due 11/10/2016 before class: 

Two point charges are located on the $z$-axis: $q_1$  at $z_1$, 
$0<z_1<R$ and $q_2$ at $z_2>R$. The resulting potential is
$$ \Phi(\vec{x}\,)\ =\ \frac{q_1}{|\vec{x}-z_1\hat{z}|} + 
            \frac{q_2}{|\vec{x}-z_2\hat{z}|}\,. $$

\begin{enumerate}

\item Calculate $q_2$ and $z_2$ as functions of $q_1$, $z_1$ and $R$
from the requirement that the potential is zero at $z=R$ and $z=-R$.

\item Calculate the resulting potential on the surface of the 
sphere of radius $R$ with center at $\vec{x}=0$.

\end{enumerate}
\bigskip

\item Exercise E.49: Potential inside a hollow sphere with Dirichlet
BC. Due 11/10/2016 before class.

\end{enumerate}
\bigskip


{\bf Set 11:}

\begin{enumerate}

\setcounter{enumi}{33}

\item Exercise E.48: A symmetry relation for spherical harmonics. 
Due 11/17/2016 before class.
\smallskip

\item Exercise E.51: Dirichlet Green function of a cylinder.
Due 11/17/2016 before class.
\smallskip

\item Exercise E.52: Point charge at the center of a rectangular 
box.  Due 11/17/2016 before class.
\end{enumerate}
\bigskip

{\bf Set 12:}

\begin{enumerate}

\setcounter{enumi}{36}

\item Exercise: Classwork, due 11/15/2016 in class. \medskip

\begin{enumerate}
\item 10 points: Use Taylor series expansion to derive the Cartesian 
quadrupole expansion 
$$ \Phi(\vec{x}\,)\ =\ \int d^3x'\,\frac{\rho(\vec{x}^{\,'}\,)}
   {|\vec{x}-\vec{x}^{\,'}\,|}\ =\ {q\over r}+{\vec{p}\cdot\vec{x}\over 
   r^3}+{1\over 2} \sum_{i=1}^3 \sum_{j=1}^3 Q^{ij}\,{x^i\,x^j\over r^5}
   + ...\ .\  $$

\item Due 11/29/2016, 10 points: 
Derive the same result by expressing the spherical multipoles 
$q_{20}$, $q_{21}$ and $q_{22}$ in terms of Cartesian multipoles $Q^{ij}$.
Compare (E.56) of the lecture notes.
\end{enumerate}

\item Exercise E.55: Number of Cartesian multipoles.  
Due 11/29/2016 before class.
\smallskip

\item Exercise E.57: Quadrupole moment of an ellipsoid.  
Due 11/29/2016 before class.
\smallskip

\item Exercise E.58: Dielectric sphere.
Due 11/29/2016 before class.
\medskip
\eject \Large

\item Exercise E.34 (c): Classwork, 5 points. 

Due 11/22/2016 in class. 
\medskip

Use
$$ \vec{a}\times\vec{b}\ =\ \sum_{i=1}^3\sum_{j=1}^3\sum_{k=1}^3
   \epsilon^{ijk}\,\hat{x}^i\,\partial_j\,b^k$$
and
$$ \left(\vec{a}\times\vec{b}\,\right)^k\ =\ 
   \sum_{l=1}^3\sum_{m=1}^3\epsilon^{klm}\,\partial_l\,b^m $$ 
to derive
$$ \nabla\times\left(\nabla\times\vec{b}\,\right)\ =\ \nabla\,
   \left(\nabla\cdot\vec{b}\,\right) -\nabla^2\,\vec{b}\,.$$

\end{enumerate}
\bigskip


% \begin{figure}[tb] % From EMscript/figures_plt/minkowski_space_ex.
% \vspace{5pc}
% \centerline{\hbox{ \psfig{figure=figures/e1mink.eps,width=10cm} }}
% \caption{Minkowski space in which observer~A is at rest and
% flashes a light signal at observer~B, who moves with speed $4c/5$
% and flashes the signal back. \label{fig_exmink1} } \end{figure}
% \vfil

\end{document}
