From a7f0bd01ad41cd6b3508f30a4ff9b7cbb2f829c7 Mon Sep 17 00:00:00 2001 From: hackbard Date: Thu, 22 Nov 2007 15:08:52 +0100 Subject: [PATCH] tutorial 3 + solution --- solid_state_physics/tutorial/1_03.tex | 111 +++++++++++++++++++++++++ solid_state_physics/tutorial/1_03s.tex | 71 ++++++++++++++++ 2 files changed, 182 insertions(+) create mode 100644 solid_state_physics/tutorial/1_03.tex create mode 100644 solid_state_physics/tutorial/1_03s.tex diff --git a/solid_state_physics/tutorial/1_03.tex b/solid_state_physics/tutorial/1_03.tex new file mode 100644 index 0000000..a6efba3 --- /dev/null +++ b/solid_state_physics/tutorial/1_03.tex @@ -0,0 +1,111 @@ +\pdfoutput=0 +\documentclass[a4paper,11pt]{article} +\usepackage[activate]{pdfcprot} +\usepackage{verbatim} +\usepackage{a4} +\usepackage{a4wide} +\usepackage[german]{babel} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{ae} +\usepackage{aecompl} +\usepackage[dvips]{graphicx} +\graphicspath{{./img/}} +\usepackage{color} +\usepackage{pstricks} +\usepackage{pst-node} +\usepackage{rotating} + +\setlength{\headheight}{0mm} \setlength{\headsep}{0mm} +\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} +\setlength{\oddsidemargin}{-10mm} +\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm} +\setlength{\textheight}{26cm} \setlength{\headsep}{0cm} + +\renewcommand{\labelenumi}{(\alph{enumi})} + +\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} diff --git a/solid_state_physics/tutorial/1_03s.tex b/solid_state_physics/tutorial/1_03s.tex new file mode 100644 index 0000000..1174731 --- /dev/null +++ b/solid_state_physics/tutorial/1_03s.tex @@ -0,0 +1,71 @@ +\pdfoutput=0 +\documentclass[a4paper,11pt]{article} +\usepackage[activate]{pdfcprot} +\usepackage{verbatim} +\usepackage{a4} +\usepackage{a4wide} +\usepackage[german]{babel} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{ae} +\usepackage{aecompl} +\usepackage[dvips]{graphicx} +\graphicspath{{./img/}} +\usepackage{color} +\usepackage{pstricks} +\usepackage{pst-node} +\usepackage{rotating} + +\setlength{\headheight}{0mm} \setlength{\headsep}{0mm} +\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} +\setlength{\oddsidemargin}{-10mm} +\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm} +\setlength{\textheight}{26cm} \setlength{\headsep}{0cm} + +\renewcommand{\labelenumi}{(\alph{enumi})} + +\begin{document} + +% 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 distance: $v\,dt$\\ + number of electrons crossing $A$: $n(v\,dt)A$\\ + $\Rightarrow$ $j=\frac{I}{A}=\frac{dQ/dt}{A}=\frac{-e\,n(v\,dt)A/dt}{A} + =-nev$ + \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 Energy transfer: $\frac{m}{2}v_{drift}^2$, + $\qquad v_{drift}$: + end drift velocity of the accelerated electron\\ + $v_{drift} \ne v_{average}$ + + +\end{enumerate} + +\end{document} -- 2.20.1