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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 8}\\%
\vspace {3mm} (February 21, 2009) \\
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Read chapter 7.3 of Griffith (due Monday February 23). \medskip

Also, {\bf bring a calculator} on Monday.

\begin{description}

\item{(34)} SI units (due February 23 in class).
\begin{enumerate}
\item How is one second [s] defined (1~point -- done)?
\item How is one meter [m] defined (1~point -- done)?
\item How is one kilogram [kg] defined (1~point -- done)?
\item How is one Newton [N] defined (1~point -- done)?
\item How is one Ampere [A] defined (1~point)?
\item How is one Coulomb [C] defined (1~point)?
\item From the Biot-Savart law it follows that the force exerted by
one (infinitely long) wrire on a section of length $l$ of a parallel 
wire is given by
$$ F = \frac{\mu_0}{2\pi}\,\frac{I_1\,I_2}{d}\,l\ ,$$
where $\mu_0$ is the {\it permeability of vacuum}. What is the (exact) 
value of $\mu_0$ (1~point)?
\item Using the above units, Coulomb's law read
$$ F(r) = \frac{\kappa_0\,Q_1\,Q_2}{r^2}\,,~~~ 
   \kappa_0 = \frac{1}{4\pi\,\epsilon_0} $$
where $\kappa_0$ is called {\it Coulomb constant} and $\epsilon_0$ {\it 
permittivity of vacuum}. They can, for instance, be estimated by force 
measurements. In SI units $\epsilon_0=8.85418782\times 10^{-12}$. What 
are the physical dimensions of $\epsilon_0$ (1~point)?
\end{enumerate}

\item{(35)} EM waves in vacuum (due February 23 in class).
\begin{enumerate}
\item Use $\epsilon_{ijk}$ notation and Einstein's summation
convention to show 
$$\nabla\times(\nabla\times\vec{A}) = \nabla\,(\nabla\cdot\vec{A})
  - \nabla^2\vec{A}~~~\rm (3~points)\ .$$
\item Use SI units and write down Maxwell's equations in vacuum 
      (2 points).
\item Calculate $\nabla\times(\nabla\times\vec{E})$ to derive
$$ \nabla^2\vec{E} = \left(\frac{1}{v} \frac{\partial\ }{\partial t}
   \right)^2 \vec{E}~~{\rm with}~~v = 
   \frac{1}{\sqrt{\mu_0\epsilon_0}} ~~~\rm (3~points)\ .$$
\item Calculate $v=1/\sqrt{\epsilon_0\,\mu_0}$ and make an observation 
      (2~points). 
\end{enumerate}

\item{(36)} Homework, due February 27 before class (4 points). Repeat 
the previous considerations for $\nabla\times(\nabla\times\vec{B})$.

\end{description}

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Read chapter 9.2.1 of Griffith.
