%begin preamble
\documentclass[11pt,titlepage]{article}
\usepackage[margin=1in,noheadfoot]{geometry}
\usepackage[parfill]{parskip}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{wrapfig}

\newcommand{\e}[1]{\times 10^{#1}}
\newcommand{\q}{\mathrm}
\newcommand{\where}[1][where]{\qquad\mbox{#1}\qquad}
\newcommand{\bs}[1]{\boldsymbol{#1}}
\newcommand{\bh}[1]{\hat{\boldsymbol{#1}}}
\newcommand{\bv}[1]{\vec{\boldsymbol{#1}}}
\newcommand{\bt}[1]{\tilde{\boldsymbol{#1}}}
\newcommand{\flow}[1]{\mathcal{E}_{#1}}
\newcommand{\flux}[1]{\Phi_{#1}}
\newcommand{\dslash}{d \hspace{-0.5ex}\rule[1.2ex]{0.6ex}{.1ex}}
\def\rcurs{{\mbox{$\resizebox{.16in}{.08in}{\includegraphics{ScriptR}}$}\!\!}}
\def\brcurs{{\mbox{$\resizebox{.16in}{.08in}{\includegraphics{BoldR}}$}\!\!}}
\def\hrcurs{{\mbox{$\hat \brcurs$}\!}}
%end preamble

\begin{document}
\begin{center}
  {\bf Arizona State University, Physics 311 \\
    Problem Set \#11, Rev. A, due Wednesday, 2014-04-30 }
\end{center}

\vspace{.3 in}

{\bf 1.} In a {\bf rail gun}, a conductive projectile of mass $m$ bridges the
gap between two conducting rods of radius $a$ and length $l$, separated by a
distance $d$.  Each rail is connected to one terminal of a low-inductance
capacitor $C$, with charge $Q$, initially at voltage $V_0$.  At $t=0$, the
projectile is placed touching both rails at the end near the capacitor,
completing the circuit.  The projectile is accelerated by the magnetic force
on the current $I$ passing through it.  The goal of this problem is to
calculate the velocity $v$ of projectile as it reaches the far end of the
rails.  Assume that the total resistance $R$ is dominated by the contact
resistance between the projectile and the rails, and that the inductance is
dominated by the rails (a single turn coil).

\quad a) Calculate the magnetic force on the projectile as a function of
current $I$ in the rails.  Calculate the power expended in accelerating the
projectile.  Model the lost power with an effective voltage drop $V_b = P/ I$
(back EMF) in the circuit.

\quad b) Calculate the inductance as a function of $x$, the position of the
projectile along the rail.  Model the rail as an LRC circuit to calculate
$dI/dt$ as a function of $Q$, $I$, $x$, and $v$.

\quad c) Numerically integrate $dI/dt$ of part (b) and $F=ma$ to determine the
position $x(t)$ of the projectile and charge $Q(t)$ in the capacitor as a
function of time.  Choose reasonable physical values for $V_0$, $C$, $R$, $m$,
$d$, and $a$.

A simple, yet effective, way to do this is in a spreadsheet with columns for
$t$, $Q$, $I$, $x$, and $v$.  Put the initial values in the first row.
Calculate each subsequent row at step $\Delta t$ forward in time using values
from the previous row.  For example, $Q_{i+i}= I_i \Delta t$ and $v_{i+1} =
F(I_i)\Delta t/m$.  Copy the cells (formulas) from one row to the next.

\quad d) Calculate the ratios of kinetic energy of the projectile versus
energy dissipated in the resistor, energy remaining in the capacitor, and
inductive energy, when the ciruit is broken as the it leaves the rails.  What
happens to the inductive energy?

\quad e) Bonus: Tune the capacitance as a function of the other parameters to
maximize the fraction of kinetic energy transferred to the projectile.

2. A {\bf quarter shrinker} can contract a quarter of initial diameter 24~mm
and thickness 1~mm to almost half its radius.  Of course mass is conserved, so
the quarter will be twice as thick.  The quarter is placed in an 11 turn
(33~mm long) solenoid of 3~mm diameter wire, with a winding diameter of 30~mm.
The solenoid is connected to a low-inductance high-voltage capacitor $C$.
Assume that the inductance $L$ and resistance $R$ is dominated by the coil,
and that the resistivity of Cu is $\rho=1.678~\mu\Omega\,$cm.

\quad a) Calculate $R$ and $L$ of the coil.  Neglecting the mutual inductance
of the quarter, calculate the current $I(t)$ through the coil as a function of
time.

\quad b) Calculate the induced current density in the the quarter as a
function of time.  Using the Lorentz force law, calculate the contractive
force on the quarter.  What do you think will happen?  See
\href{http://www.capturedlightning.com/frames/shrinkergallery.html}
{http://capturedlightning.com} for the answer!

Also, Griffiths 3ed[4ed] Ch.~7, \#%
14[14], %p305[316] drop bar magnet in conducting tube \\
20[22], %p327[327] M of large and small loop \\
30[32], %p321[332] dipoles mutual inductance \\
31[34], %p324[336] displacement current ideal \\
42[44], %p334[346] Meissner effect \\
48[50], %p336[348] betatron \\
53[57], %p338[350] transformer \\
54[58]. %p338[350] transformer \\

\end{document}
