= Meissner effect and flux history
{c}
{title2=$\partial_t\mathbf B=0\text{ when }\mathbf E=0$}
Compare two ideal models for a stationary, solid, <simply connected> sample. In the <perfect conductor> flux-history model the interior <electric field> is constrained to zero from the moment perfect conduction starts. <Faraday's law of induction> then gives
$$
\partial_t\mathbf B=-\nabla\times\mathbf E=0.
$$
Consequently the interior <magnetic field> retains its value at that moment. Cooling in an already applied <magnetic field> freezes a nonzero interior value; starting perfect conduction in zero <magnetic field> and then applying a <magnetic field> preserves zero. The zero initial interior <magnetic field> often assumed in <perfect conductor> wave-reflection problems is one particular preparation, not a general prediction of expulsion. <University Physics Volume 3 by Samuel J. Ling and others/9.8 Superconductivity>[§9.8, Eq. (9.38) and the argument that zero resistivity conserves existing flux rather than expelling it]{ref}
\Image[/meissner-cooling-and-field-history.png]
{title=Two preparation histories distinguish the <Meissner state> from <perfect conductor> flux memory}
{description=Original schematic end states for a solid sample much thicker than its <London penetration depth>, not measured data or transition kinetics. Both rows end at the same low <temperature> and applied weak <magnetic field>. Blue arrows indicate interior <magnetic field>; orange edges represent screening <electric currents>. The near-zero interior in the <Meissner state> omits its finite penetration layer. The <perfect conductor> panel uses the ideal zero-interior-<electric field> constraint.}
The <Meissner state> instead selects the screened equilibrium after either preparation, provided that equilibrium is accessible and the applied <magnetic field> lies in the <Meissner state> regime. The first <London equation> alone illustrates the missing information: at fixed $\lambda_L$, combining it with <Faraday's law of induction> gives
$$
\partial_t\!\left(\nabla\times\mathbf J+\frac{\mathbf B}{\mu_0\lambda_L^2}\right)=0.
$$
This permits an arbitrary time-independent integration field. The second <London equation> imposes the additional zero value of that bracket in the vortex-free local model.<Lecture notes on the electrodynamics of metals and superconductors>[§2, pp. 12–13, Eqs. (64), (66)–(68): time-independent freedom and the second London equation]{ref} Establishing <superconductivity> during cooling is not the fixed-parameter, zero-<electric field> model: transient <electric fields> accompany changing <magnetic flux>, and $\lambda_L$ itself depends on <temperature>. The equilibrium <London equations> predict final spatial screening, not the cooling rate or the time-resolved expulsion pulse.
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