ass3: Add answer to question about inhomogenity

superconductivity_assignment3_kvkempen.tex

@@ -71,6 +71,35 @@ The average field inside the cylinder is given by the following self-consisting
 Plugging in the values for \ce{Nb3Sn}, $B_E = \SI{1}{\tesla}$, and $\phi_0 = \SI{2.0678}{\weber}$, $B$ is found as $B = \SI{.986}{\tesla} \approx B_E$ by intersection.
 %https://www.wolframalpha.com/input?i=B+%3D+1+-+%282.0678*10%5E-15%29%2F%288*pi*%28124*10%5E-9%29%5E2%29+*+ln%282.0678*10%5E-15%2F%284*exp%281%29*%283.65*10%5E-9%29%5E2+*+B%29%29
 
+To investigate the inhomogenity of the field inside the cylinder, we look at the gradient $\nabla B$ inside the material.
+As we assume a vortex lattice that fully fills the cross section of the cylinder,
+and we assume that the fields due to each vortex die out quickly enough to not overlap,
+it suffices to calculate the gradient over just one vortex.
+These assumptions coincide with slide 15 of lecture 4, from which I took figure \ref{fig:lec4-vortexlattice}
+
+\begin{figure}[H]
+	\centering
+	\label{fig:lec4-vortexlattice}
+	\includegraphics[width=.4\textwidth]{lec4-vortexlattice.png}
+	\caption{The vortices are arranged in a lattice to maximize their distance, as this lowers their repulsive interaction and thus the energy.}
+\end{figure}
+
+On slide 19 from the same week, we find an expression $B(r)$ for the field at distance $r$ from the vortex core as
+\[
+	B = \frac{\phi_0}{2\pi\lambda^2} K_0(r/\lambda) = B_0 K_0(r/\lambda),
+\]
+where $K_0$ is the modified Bessel function of the second kind.
+For small $r$ (i.e. $r << \lambda$), we can approximate this and find that
+\[
+	K_0 \propto - \ln{(r/\lambda)},
+\]
+and notice a singularity at $r = 0$.
+For the gradient we thus find
+\[
+	\nabla B \propto \pfrac{K_0}{r}(r/\lambda) \propto \pfrac{-\ln{(r/\lambda)}}{r} = -1/r.
+\]
+The size of the supercurrent density has the same relation, $J_S \propto 1/r$.
+
 \section{Superconducting wire}
 
 \section{Fine type-II superconducting wire}