tutorial 3 + solution

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+\pdfoutput=0
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+\usepackage[T1]{fontenc}
+\usepackage{amsmath}
+\usepackage{ae}
+\usepackage{aecompl}
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+\usepackage{color}
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+
+\begin{document}
+
+% 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 Obviously 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}
+
+\end{document}