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ass3: Fix task 8 with some Koen intervention

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Kees van Kempen
2022-03-17 14:29:43 +01:00
parent d9adf54595
commit cf406fc4da
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superconductivity_assignment3_kvkempen.tex
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@ -70,12 +70,13 @@ 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
This is in the range as provided in the assignment ($B = \SI[separate-uncertainty]{.981\pm.019}{\tesla}$).
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}
These assumptions coincide with slide 15 of lecture 4, from which I took figure \ref{fig:lec4-vortexlattice}.
\begin{figure}[H]
\centering
@ -96,7 +97,7 @@ For small $r$ (i.e. $r << \lambda$), we can approximate this and find that
and notice a singularity at $r = 0$.
For the gradient we thus find
\[
\nabla B \propto \pfrac{K_0}{r}(r/\lambda) \propto \pfrac{-\ln{(r/\lambda)}}{r} = -1/r.
\nabla B \propto \pfrac{K_0}{r}(r/\lambda) \propto \pfrac{-\ln{(r/\lambda)}}{r} = -\lambda/r.
\]
The size of the supercurrent density has the same relation, $J_S \propto 1/r$.