On the dynamics of the Meissner and the Becker-London effects

TL;DR

This work demonstrates that the dynamical evolution of the Meissner and Becker-London effects can be accurately captured by solving Maxwell equations augmented with the London current, even when the superconducting fraction $f({\bf x},t)$ is time- and space-dependent. By analyzing simple plate and cylinder geometries under cool-first and field-first protocols, the authors reveal rapid initial field propagation followed by slow relaxation toward the appropriate screening profiles, with the dynamics governed by Proca-like equations that include $\lambda_{\rm eff}(t)$ and $B_{\rm eff}(t)$. A key result is the energy accounting: part of the external-work power is stored in the superconducting state, part radiates away, and part converts to Joule heating, with the relation $W_B=\Delta F$ emerging in the nucleation context and latent-heat considerations aligning with thermodynamics. The analysis reconciles Hirsch's criticisms with the conventional theory, clarifying how phase-nucleation, front propagation, and momentum exchange with the lattice yield reversible dynamics in slow processes and establishing a framework for transient electromagnetic phenomena in superconductors. Overall, the study highlights the richness of superconducting dynamics within the standard London electrodynamics and provides a quantitative bridge between microscopic pictures and macroscopic electromagnetic responses.

Abstract

It is generally accepted that the most fundamental property of a superconductor is that it exhibits the Meissner effect. Of similar importance is the Becker-London effect, i.e. generation of magnetic field inside a rotating superconductor. Hirsch has recently pointed out that, within the conventional theory of superconductivity, the question about how these effects are generated dynamically has not even been asked yet. Here we fill in this gap in the literature by a detailed study of the evolution of the electromagnetic field for both of these effects. To this end, we solve the Maxwell equations supplemented by the simplest conventional constitutive equation for a superconductor, namely the London equation. We demonstrate that, contrary to the expectations of Hirsch, the conventional theory does correctly describe the dynamics of both, the Meissner and the Becker-London effect. We find that the dynamics of the studied processes is quite rich and interesting even at this level of description.

Figures (14)

• Figure 1: Cut through the plate (shaded region) by the plane $yz$. Shown is the orientation of the relevant fields at an intermediate stage of the process leading to the Meissner effect, as obtained from the numerical solution. The crosses (dots) indicate vectors oriented into (out of) the paper plane.
• Figure 2: Temporal evolution of the magnetic field profile in the plate (for $\lambda/L=0.1$ and $c\tau/L=20$) for the cool-first protocol. The dots correspond to \ref{['eq:meissner_plate']} with $B_{\rm eff}=B_0$. The inset shows the field profiles for times $\lambda/c<t<L/c$, when the field in the middle of the plate is still identically zero.
• Figure 3: Temporal evolution of the Poynting vector in close vicinity of the plate (for $\lambda/L=0.1$ and several values of $\tau$) for the cool-first protocol. The lines are analytical estimates in the large-$\tau$ limit, described in the text. The Poynting vector is proportional to $\cosh^{-4}(t/\tau)$ and is measured in units of $cB_0^2/\mu_0$. The inset shows the scaling of the total radiated energy $W_{\rm rad}$ (in units of $B_0^2L/\mu_0$) with $\tau$.
• Figure 4: Temporal evolution of the magnetic field profile in the sample (for $\lambda/L=0.1$ and $c\tau/L=200$) for the field-first protocol. The dots correspond to \ref{['eq:meissner_plate']} with $\lambda$ replaced by $\lambda_{\rm eff}(t)$, see \ref{['eq:lambda_eff']}.
• Figure 5: Spatial profile of the relevant fields in the sample (for $\lambda/L=0.1$, $c\tau/L=200$ and at very short time $ct/L=0.5$) for the field-first protocol. The Maxwell displacement current is defined as $\mu_0 j_{\rm M}=c^{-2}\partial E/\partial t$. The magnetic field, electric field, and the current densities are measured in units of $B_0$, $cB_0$, and $B_0/(\mu_0 L)$, respectively.