of the wire. Assume, that the penetration depth $\lambda$ is much
smaller than the radius $R$ of the cylinder.
{\bf Hint:}
- Use the relation $I_c=\int_0^R dr \int_0^{2\pi} d\phi \, j_c(r)$
+ 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 radius of 1 mm at $T=0K$.
+ \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.
-In the superconductor the magnetic field is given by
+Inside the superconductor the magnetic field is given by
\[
{\bf B}_s=\mu_0 \left({\bf H}_a + {\bf M}_s\right)
\]
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 ${\bf B}_a=\mu_0 H_a {\bf e}_z$
Calculate and sketch the decay of
${\bf B}_s=B_{s_z}(x) {\bf e}_x$
in the superconductor.
- Introduce the London penetration depth $\lambda=\sqrt{\Lambda/\mu_0}$.
+ 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?