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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 3}
\end{center}

\section{Drude theory of metallic conduction}
{\bf Motivation:} In the following excercise we will reconsider once more the
Drude theory of metals.
We will end up with an expression for the electrical conductivity of a metal.
In addition we will deduce the expression of power loss
for current flowing in a wire.

{\bf Our understanding of condensed matter} is based on the notion of a solid
being composed of heavy, positively charged ions
and light, negatively charged valence electrons.
The ions consist of the nuclei and core electrons tightly bound to the nuclei
which thus do not contribute to the metallic conductivity.
The mobile valence electrons on the other hand are responsible for the
electrical and thermal conductivity of the metal.

{\bf The basic assumptions of the Drude model} are presented in the following.
Basically the theory is constructed by applying the kinetic theory of gases
to a metal, considered as a gas of free non-interacting valence electrons.
Briefly outlined, the models assumptions are mentioned:
\begin{itemize}
 \item Between collisions:\\
       Independent electron approximation
       $\rightarrow$ no electron-electron interaction\\
       Free electron approximation
       $\rightarrow$ no electron-ion interaction
 \item Electrons collide with the large heavy ions.
       Collisions are instantaneous events abruptly altering the velocity of
       an electron and randomly changing its direction.
 \item On average, electrons travel for a time $\tau$
       before its next collision.\\
       $\Rightarrow$ Probability of a collision for an electron in an
       infinitesimal time interval $dt$ is $dt/\tau$.
 \item Thermal equilibrium achieved by collisions only.\\
       $\Rightarrow$ Electron's speed after collision determined
                     according to local temperature.
\end{itemize}

Consider a wire of length $L$ and cross-sectional area $A$.
The wire has a resistance $R$.

\begin{enumerate}
 \item According to Ohm's law ($U=IR$) the current $I$ flowing in that wire
       is proportional to the potential drop $U$.
       The resistance depends on the shape of the wire ($R=\rho\frac{L}{A}$).
       Rewrite Ohm's law eliminating this dependence using
       the resitivity $\rho$ which is only characterized by the metal.
       {\bf Hint:} $U=EL$ is the potential drop along the wire
       ($E$: electric field)
       and $j=I/A$ is the current density.
 \item Find an expression for the current density if $n$ electrons
       per unit volume move with velocity $v$.
       {\bf Hint:} What distance the electrons travel in a time $dt$?
       How many electrons will cross an area $A$ perpendicular to the
       direction of flow in a time $dt$?
       Remember that the current $I$ is the derivative of charge $Q$
       with respect to time.
 \item What is the average velocity of the electrons in the absence
       of an electric field?
       What does this mean for the contribution of the
       thermal electronic velocity after a collsion
       to the average electronic velocity?
       Find an expression for the electric field dependent
       average electronic velocity.
 \item Rewrite the current density using the average electronic velocity
       and find an expression for the conductivity $\sigma=1/\rho$.
 \item The resistance is caused by collisions of the electrons
       with the lattice.
       Energy is not conserved in the collisions.
       Find an expression for the power loss in the considered wire.
\end{enumerate}

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