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% header
\begin{center}
 {\LARGE {\bf Materials Physics I}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 WS 2007/08\\
 \vspace{8pt}
 {\Large\bf Tutorial 2 - proposed solutions}
\end{center}

\section{Drude theory of metallic conduction}
\begin{enumerate}
 \item $U=IR \Rightarrow EL=jA\rho\frac{L}{A}
             \Rightarrow E=j\rho$
 \item \begin{itemize}
        \item distance: $v\,dt$
        \item number of electrons crossing $A$: $n(v\,dt)A$
       \end{itemize}
       $\Rightarrow$ $j=\frac{I}{A}=\frac{dQ/dt}{A}=\frac{-e\,n(v\,dt)A/dt}{A}
                       =-nev$
 \item \begin{itemize}
        \item In the absence of an electric field, electrons are as likely
	      to be moving in any one direction as in any other.
	      The velocity averages to zero.
	      As expected, according to the above equation, there is no
	      net electric current density.
        \item Since electrons emerge in a random direction
	      there will be no contribution from the thermal velocity
	      to the average electronic velocity.
        \item $v_{average}=at=\frac{F}{m}\tau=-\frac{eE}{m}\tau$
       \end{itemize}
 \item \begin{itemize}
	\item $j=\left(\frac{ne^2\tau}{m}\right)E$\\
	\item $j=\sigma E \Rightarrow \sigma=\frac{ne^2\tau}{m}$
       \end{itemize}
 \item \begin{itemize}
        \item Energy transfer: $\frac{m}{2}v_{drift}^2$,
                               $\quad v_{drift}$:
		               final drift velocity of the accelerated electron
	\item $v_{drift}=-\frac{eE}{m}t_0$, $\quad t_0$:
	      free flight time (no collision) of the electron
        \item $v_{average}=\frac{1}{t_0}\int_{0}^{t_0} v(t) dt
	                  =-\frac{eE}{m}\frac{1}{t_0}[\frac{t^2}{2}]_{0}^{t_0}
			  =-\frac{eE}{m}\frac{t_0}{2}=:-\frac{eE}{m}\tau$,
              $\qquad t_0=2\tau$, $v_{drift}=2v_{average}$
	      \includegraphics[width=12cm]{drude_v.eps}
	\item Each of the $n$ electrons per unit volume
	      transfer the kinetic energy $\frac{1}{2}mv^2_{drift}$
	      once per $t_0$ to the lattice
       \end{itemize}
       \[
       \Rightarrow \frac{P}{V}=\frac{E_{kin}}{Vt_0}
                              =\frac{n\frac{1}{2}m\frac{e^2E^2}{m^2}t_0^2}{t_0}
			      =n\frac{1}{2}\frac{e^2E^2}{m}2\tau
			      =\sigma E^2=jE=j^2\rho=\frac{I^2}{A^2}\frac{A}{L}R
			      =\frac{I^2R}{V}
       \]
       \[
       \Rightarrow P=I^2R \textrm{ (Joule heating)}
       \]
\end{enumerate}

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