second tutorial

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+\pdfoutput=0
+\documentclass[a4paper,11pt]{article}
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+\usepackage{amsmath}
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+
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+\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}