\documentclass[12pt]{article}
\usepackage{epsfig}
\topmargin=-0.5cm
\oddsidemargin=-0.5cm
\evensidemargin=-0.5cm
\textheight=23cm
\textwidth=17cm
\begin{document}
\newcommand{\bold}[1]{\mbox{\boldmath $#1$}}
\newcommand{\ems}{{\cal E}}
\begin{bf}
\noindent
Institutt for fysikk, NTNU

\noindent
TFY4155/FY1003: Elektrisitet og magnetisme

\noindent
V{\aa}r 2007

\end{bf}

\begin{center}
\begin{large}
{\O}ving 12
\end{large}
\end{center}

\noindent
Veiledning: Uke 13

\noindent
Innleveringsfrist: Onsdag 11. april

\vspace{0.7cm}
\noindent
{\em Exercise 1}

\vspace{0.2cm}
\noindent
An AC voltage source $\ems(t) = V_0 \, \cos\omega t$ is connected to a circuit
consisting of a capacitance $C$ and a resistance $R$ in parallel:

%%FIGURE
\vspace{0.5cm}
\centerline{\epsfxsize=3.0in \epsfbox{ov12_fig4.eps}}
\vspace{0.5cm}
%%END FIGURE

\vspace{0.3cm}
\noindent
Use Kirchhoff's rules to find the currents $I(t)$, $I_C(t)$ and $I_R(t)$.
Show that the total current delivered by the voltage source may be written
$$I(t) = I_0 \, \cos(\omega t - \alpha)$$
with amplitude
$$I_0 = \frac{V_0}{R} \, \sqrt{1 + (\omega RC)^2}$$
and phase constant
$$\alpha = -\arctan(\omega RC)$$
Hint: Use $\cos(a \pm b) = \cos a \, \cos b \mp \sin a \, \sin b$.

\vspace{0.2cm}
\noindent
Assume that the voltage source has amplitude $V_0 = 1.0$ V and
frequency $f = 1.0$ MHz, and that $R = 10 \, \Omega$ and $C = 16$ nF.
Find numerical values for $I_0$ and $\alpha$. Sketch $\ems(t)$, $I(t)$,
$I_R(t)$ and $I_C(t)$ for $t$ between 0 and $1/f$.

\vspace{0.2cm}
\noindent
With the given values of $V_0$, $R$ and $C$, sketch the functions
$I_0(\omega)$, $\alpha(\omega)$ and the {\em impedance} of the circuit,
$Z(\omega) = V_0/I_0(\omega)$
for angular frequencies between $10^5$ and $10^9$ s$^{-1}$.
Hint: Use a logarithmic scale along the $\omega$ axis.


\vspace{0.7cm}
\noindent
{\em Exercise 2}

\vspace{0.2cm}
\noindent
Particles with mass $m$, charge $q$, and velocity $\bold{v}$ enter a region
with "crossed" electric and magnetic field, $\bold{E}$ and $\bold{B}$,
as shown in the figure. $\bold{E}$ is directed downwards, and
$\bold{B}$ is directed into the paper plane. In the region with extent
$L$ we assume that the fields are homogeneous. Outside this region,
$\bold{E}=\bold{B}=0$.

%%FIGURE
\vspace{0.5cm}
\centerline{\epsfxsize=4.0in \epsfbox{ov12_fig1.eps}}
\vspace{0.5cm}
%%END FIGURE

\noindent
You keep the electric field strength $E$ constant throughout the experiment.
First, you set $B=0$ and register that the particles are deflected and hit
the detector in a distance $y$ above the center line (which is dashed). Next,
you repeat the experiment, but this time you adjust the value of $B$ until
the particles are no longer deflected.

\noindent
Show that the ratio between the charge and the mass of the particles is proportional
to the deflection (when $B=0$), i.e.,
$$\frac{q}{m} = - a y,$$
and find $a$ expressed in terms of $E$, $B$, $D$, and $L$.

\vspace{0.5cm}
\noindent
Given information: $\bold{F} = q \left(\bold{E} + \bold{v} \times \bold{B} \right)$ \hspace{0.3cm} 
(the Lorentz force)

\vspace{0.5cm}
\noindent
{\small In this manner, J. J. Thomson analyzed so called cathode rays in 1897, and
showed that these rays consisted of a particular type of particles with negative charge.
These were electrons that were emitted from the metal of the cathode. Thomson was the
first one to determine the ratio $e/m_e$. He found the same value for this ratio, independent
of what kind of metal he used in the cathode. Thus, he could conclude that the observed particles
had to be a {\em fundamental} ingredient of nature.}


\newpage
\noindent
{\em Exercise 3}

%%FIGURE
\vspace{0.5cm}
\centerline{\epsfxsize=3.0in \epsfbox{ov12_fig2.eps}}
\vspace{0.5cm}
%%END FIGURE

\vspace{0.2cm}
\noindent
The figure shows a mass spectrometer. An ion source emits charged particles.
Two slits, $S_1$ and $S_2$, make sure that a well {\em collimated} beam of
particles enter into the region with the magnetic field $\bold{B}$ (which
is directed into the paper plane). Between $S_1$ and $S_2$, we have a voltage
difference $V$ which accelerates the ions. The particle velocity at $S_2$ is much larger
than at $S_1$, so we may put $v=0$ at $S_1$. The ions are deflected through
an angle 180$^\circ$ by the magnetic field and are detected on a photographic plate $P$.

\noindent
The spectrometer is supposed to be used to separate the bromine isotopes $^{79}$Br and $^{81}$Br.
The source emits these isotopes as ions with charge $-e$. You  may assume that
the isotopes have atomic masses 79$m_p$ and 81$m_p$, respectively.

\noindent
On the photographic plate, we want a separation of at least 1.0 cm between the points
where the two isotopes strike. At the same time, we must make sure that both isotopes
really do hit the plate, which has a width $L=250$ cm, measured from where the ions enter
into the magnetic field. With these two conditions, what is the upper and lower limit
for the strength of the magnetic field $B$, when the
accelerating voltage drop $V$ is 400 V?

\vspace{0.3cm}
\noindent
Given information: $e = 1.6 \cdot 10^{-19}$ C, $m_p = 1.67 \cdot 10^{-27}$ kg \hspace{1.5cm}

\noindent
{\small [One of the answers is: $B_{\rm min} = 21$ mT]}


\end{document}
