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27
28 \begin{document}
29
30 % header
31 \begin{center}
32  {\LARGE {\bf Materials Physics I}\\}
33  \vspace{8pt}
34  Prof. B. Stritzker\\
35  WS 2007/08\\
36  \vspace{8pt}
37  {\Large\bf Tutorial 3 - proposed solutions}
38 \end{center}
39
40 \section{Drude theory of metallic conduction}
41 \begin{enumerate}
42  \item $U=IR \Rightarrow EL=jA\rho\frac{L}{A}
43              \Rightarrow E=j\rho$
44  \item \begin{itemize}
45         \item distance: $v\,dt$
46         \item number of electrons crossing $A$: $n(v\,dt)A$
47        \end{itemize}
48        $\Rightarrow$ $j=\frac{I}{A}=\frac{dQ/dt}{A}=\frac{-e\,n(v\,dt)A/dt}{A}
49                        =-nev$
50  \item \begin{itemize}
51         \item In the absence of an electric field, electrons are as likely
52               to be moving in any one direction as in any other.
53               The velocity averages to zero.
54               As expected, according to the above equation, there is no
55               net electric current density.
56         \item Since electrons emerge in a random direction
57               there will be no contribution from the thermal velocity
58               to the average electronic velocity.
59         \item $v_{average}=at=\frac{F}{m}\tau=-\frac{eE}{m}\tau$
60        \end{itemize}
61  \item \begin{itemize}
62         \item $j=\left(\frac{ne^2\tau}{m}\right)E$\\
63         \item $j=\sigma E \Rightarrow \sigma=\frac{ne^2\tau}{m}$
64        \end{itemize}
65  \item \begin{itemize}
66         \item Energy transfer: $\frac{m}{2}v_{drift}^2$,
67                                $\quad v_{drift}$:
68                                final drift velocity of the accelerated electron
69         \item $v_{drift}=-\frac{eE}{m}t_0$, $\quad t_0$:
70               free flight time (no collision) of the electron
71         \item $v_{average}=\frac{1}{t_0}\int_{0}^{t_0} v(t) dt
72                           =-\frac{eE}{m}\frac{1}{t_0}[\frac{t^2}{2}]_{0}^{t_0}
73                           =-\frac{eE}{m}\frac{t_0}{2}=:-\frac{eE}{m}\tau$,
74               $\qquad t_0=2\tau$, $v_{drift}=2v_{average}$
75               \includegraphics[width=12cm]{drude_v.eps}
76         \item Each of the $n$ electrons per unit volume
77               transfer the kinetic energy $\frac{1}{2}mv^2_{drift}$
78               once per $t_0$ to the lattice
79        \end{itemize}
80        \[
81        \Rightarrow \frac{P}{V}=\frac{E_{kin}}{Vt_0}
82                               =\frac{n\frac{1}{2}m\frac{e^2E^2}{m^2}t_0^2}{t_0}
83                               =n\frac{1}{2}\frac{e^2E^2}{m}2\tau
84                               =\sigma E^2=jE=j^2\rho=\frac{I^2}{A^2}\frac{A}{L}R
85                               =\frac{I^2R}{V}
86        \]
87        \[
88        \Rightarrow P=I^2R \textrm{ (Joule heating)}
89        \]
90 \end{enumerate}
91
92 \end{document}