ass3: Solve 9b, add sub letters, add todo

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}