From: hackbard Date: Mon, 5 May 2008 14:41:35 +0000 (+0200) Subject: second tutorial X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=b8f5baaff6d109fe71cea75dda7359935f254153;p=lectures%2Flatex.git second tutorial --- diff --git a/solid_state_physics/tutorial/2_02.tex b/solid_state_physics/tutorial/2_02.tex new file mode 100644 index 0000000..5f280f0 --- /dev/null +++ b/solid_state_physics/tutorial/2_02.tex @@ -0,0 +1,99 @@ +\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})} +\renewcommand{\labelenumii}{\arabic{enumii})} +\renewcommand{\labelenumiii}{\roman{enumiii})} + +\begin{document} + +% header +\begin{center} + {\LARGE {\bf Materials Physics II}\\} + \vspace{8pt} + Prof. B. Stritzker\\ + SS 2008\\ + \vspace{8pt} + {\Large\bf Tutorial 2} +\end{center} + +\section{Critical current in the surface region of a type 1 superconductor} +There is an exponential decay of the current in the surface region of +a superconductor. +For a cylindric wire the equation +\[ + j_s(r)=j_s(R)\exp\left(\frac{-(R-r)}{\lambda}\right) +\] +is given. +$R$ is the radius of the wire, $r$ is the distance from the cylinder axis +and $\lambda$ is the London penetration depth. + +\begin{enumerate} + \item Derive an expression for the critical current density at the + surface of the wire with subject to the critical current $I_c$ + of the wire. Assume, that the penetration depth $\lambda$ is much + smaller than the radius $R$ of the cylinder. + {\bf Hint:} + Use the relation $I_c=\int_0^R dr \int_0^{2\pi} d\phi \, j_c(r)$ + and integration by parts. + \item Calculate $j_c(R,T=0K)$ for a wire of Sn with a radius of 1 mm at $T=0K$. + The critical current and penetration depth at $T=0K$ are + $I_c=75\, A$ and $\lambda =300\cdot 10^{-10}\, m$. +\end{enumerate} + +\section{Penetration of the magnetic field into a type 1 superconductor} +In the following, the behaviour of the magnetic field ${\bf B}_s({\bf r})$ +in the surface layer of a superconductor is calculated. +In the superconductor the magnetic field is given by +\[ + {\bf B}_s=\mu_0 \left({\bf H}_a + {\bf M}_s\right) +\] +in which ${\bf H}_a$ is the strength of the applied magnetic field and +${\bf M}_s$ is the magnetization of the superconductor. + +\begin{enumerate} + \item Set up the differential equation for ${\bf B}_s$. + {\bf Hint:} + Use the appropriate Maxwell equation to connect the magnetic field + to the generating current densities. + What is the value of the current density responsible for the external + magnetic field inside the superconductor? + Apply the second London equation + $\nabla \times {\bf j}_s=-{\bf B}_s/\Lambda$ + relating the supercurrent to the magnetic field. + \item Consider a superconducting half space. The interface of the + superconductor ($x>0$) and the vacuum ($x<0$) is located at $x=0$. + A magnetic field ${\bf B}_a=\mu_0 H_a {\bf e}_z$ + parallel to the surface is applied. + Calculate and sketch the decay of + ${\bf B}_s=B_{s_z}(x) {\bf e}_x$ + in the superconductor. + Introduce the London penetration depth $\lambda=\sqrt{\Lambda/\mu_0}$. + \item Out of this, calculate the screening current density ${\bf j}_s$. + What is the direction of the current? + Calculate the value of ${\bf j}_s$ at the interface? +\end{enumerate} + +\end{document}