From cf406fc4dab8c1a7d70f6a3151f18a79be56f207 Mon Sep 17 00:00:00 2001 From: Kees van Kempen Date: Thu, 17 Mar 2022 14:29:43 +0100 Subject: [PATCH] ass3: Fix task 8 with some Koen intervention --- superconductivity_assignment3_kvkempen.tex | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/superconductivity_assignment3_kvkempen.tex b/superconductivity_assignment3_kvkempen.tex index f40baca..25f1a41 100755 --- a/superconductivity_assignment3_kvkempen.tex +++ b/superconductivity_assignment3_kvkempen.tex @@ -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, $J_S \propto 1/r$.