From 1a6cb9891fa1b6cb306f7c02c92385202f0bc839 Mon Sep 17 00:00:00 2001 From: Kees van Kempen Date: Thu, 17 Mar 2022 15:09:18 +0100 Subject: [PATCH] ass3: Solve 9b, add sub letters, add todo --- superconductivity_assignment3_kvkempen.tex | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/superconductivity_assignment3_kvkempen.tex b/superconductivity_assignment3_kvkempen.tex index f2263e0..90f6f01 100755 --- a/superconductivity_assignment3_kvkempen.tex +++ b/superconductivity_assignment3_kvkempen.tex @@ -102,6 +102,7 @@ For the gradient we thus find The size of the supercurrent density has the same relation, $J_S \propto 1/r$. \section{Superconducting wire} +\textbf{(a)} The voltage $U = \SI{1.5e-5}{\volt}$ across the wire of length $\ell = \SI{.08}{\meter}$ induces a current $J_t$. % through the resistive wire with unknown resistivity $\rho$ according to Ohm's law. Due to the presence of the magnetic field $B = \SI{5}{\tesla}$, if the vortices move with velocity $v_L$, a Lorentz force $f_L$ per vortex acts on the vortices. This results in a power input $P_L = f_Lv_L = J_tBv_L$ per vortex. @@ -112,6 +113,18 @@ Equating these expressions and rewriting yields v_L = \frac{U}{B\ell} = \SI{3.75e5}{\meter\per\second}. % https://www.wolframalpha.com/input?i=1.5*10%5E-5+%2F+%285*+.08%29 \] +\todo{Direction?} + +\textbf{(b)} +The vortices are aranged in a lattice with separation $r_{sep} = \sqrt{\frac{\Phi_0}{B}}$. +They move along the wire with velocity $v_L$ as determined above. +The expected frequency is then given by their velocity over the separation, as that is the period of the changing fields due to the vortices: +\[ + f = \frac{v_L}{r_{sep}} = \frac{U}{B\ell}\sqrt{\frac{B}{\Phi_0}} = \frac{U}{\ell\sqrt{B\Phi_0}} = \SI{1.84}{\kilo\hertz}, + % https://www.wolframalpha.com/input?i=1.5*10%5E-5+%2F+%28.08%29+%2Fsqrt%285+*+2.067*10%5E%28-15%29%29 +\] +where we used that $\Phi_0 = \SI{2.067e-15}{\volt\second}$. +This is very close to what is written in the assignment, but not precisely the same, so maybe I used a different value for $\Phi_0$. \section{Fine type-II superconducting wire}