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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 8}\\%
% \vspace {3mm} (March 18, 2011) \\
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Download the Special Relativity and Maxwell Equations
script and read up to section~1.1.4 (included). Due
March 25.

\begin{description}

\item{(23)} A particle of mass $m$ and electric charge $q$ moves under 
the influence of a constant magnetic field of the form
${\bf B}({\bf r})\!=\!B_0\hat{\bf z}$. Obtain the most general 
solution for the velocity ${\bf v}(t)$ using Newton's second
law of motion in combination with the Lorentz force
$$ {\bf F}=m{\dot{\bf v}}=\frac{q}{c}{\bf v}\times{\bf B}\;.  $$
Homework, due March~28 before class (10 points). \bigskip

\item{(24)} SI units (due March 28 before class, 4 points).
\begin{enumerate}
\item How is one second [s] defined?
\item How is one meter [m] defined?
\item How is one kilogram [kg] defined?
\item How is one Newton [N] defined?
\item How is one Ampere [A] defined?
\item How is one Coulomb [C] defined?
\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 $\mu_0$?
\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 $\epsilon_0$?
\end{enumerate}

\item{(25)} Consider the 2D rotation
$$ \pmatrix{x'^{\,1}\cr x'^{\,4}\cr} = \pmatrix{\cos(\phi)& \sin(\phi)
   \cr - \sin(\phi)&\cos(\phi)\cr}\, \pmatrix{x^1\cr x^4\cr} $$
and substitute $\phi = i\,\zeta$, $x^4=i\,x^0$, $x'^{\,4}=i\,x'^{\,0}$. 
Write out the equations for $x'^{\,1}$ and~$x'^{\,0}$. Due in class 
(4~points).

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