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% 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 $I_c=\int_0^R dr \int_0^{2\pi} d\phi \, j_c(r) r$
       and integration by parts.
 \item Calculate $j_c(R,T=0K)$ for a wire of Sn with a diameter 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.
Inside 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
       $\nabla \times {\bf j}_s=-{\bf B}_s/\Lambda$
       with $\lambda=\sqrt{\Lambda/\mu_0}$
       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 $\lambda$.
 \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}

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