The True Nature of Superconductivity — Epoche B2
Beyond Zero Resistance: The True Nature of Superconductivity Ask what a superconductor is and almost everyone answers the same way: a material that carries electricity with no resistance. That is true, and it is spectacular — a current started in a superconducting ring can circulate for years with no battery to sustain it. Yet zero resistance, taken by itself, does not pin down what a superconductor is . A hypothetical perfect conductor — an ordinary metal in which the electrons somehow stopped scattering — would also carry current without loss, and it would still not be a superconductor. The property that actually defines the superconducting state, and that betrays its quantum-mechanical origin, is a magnetic one: the Meissner effect [1] , the active expulsion of magnetic field from the bulk. This note builds that claim from the ground up — what zero resistance really implies, how a perfect conductor behaves in a magnetic field, and why the Meissner effect demands something more. What zero resistance does, and does not, say Resistance is captured by Ohm's law in its local form, $\mathbf{E} = \rho\,\mathbf{J}$, relating the electric field $\mathbf{E}$ inside a conductor to the current density $\mathbf{J}$ it drives, through the resistivity $\rho$. "Zero resistance" means $\rho \to 0$, so a steady current can flow with no electric field, $\mathbf{E}=0$. Microscopically, if the charge carriers (number density $n_s$, charge $-e$, mass $m$) feel no drag at all, Newton's second law for one of them is simply $m\,\mathrm{d}\mathbf{v}/\mathrm{d}t = -e\mathbf{E}$: the field accelerates the carriers rather than fixing their speed. Writing the supercurrent density as $\mathbf{J}_s = -n_s e\,\mathbf{v}$ and differentiating in time gives the first London equation [2] , $$ \frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m}\,\mathbf{E} \equiv \frac{1}{\Lambda}\,\mathbf{E}, \qquad \Lambda \equiv \frac{m}{n_s e^2}. $$ Here $\Lambda$ is a material constant set by the density and mass of the superconducting carriers; the smaller it is, the more current a given field builds up. The equation says something decisive: in the steady state $\partial \mathbf{J}_s/\partial t = 0$, so $\mathbf{E}=0$ even while a current flows. That is the entire content of zero resistance. But notice what is missing — this relation involves only the electric field. It says nothing yet about how the material treats a magnetic field, and that is exactly where the real physics hides. A perfect conductor only freezes the flux To bring in magnetism we take the curl of the first London equation and use Faraday's law of induction, $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$, which links a changing magnetic field $\mathbf{B}$ to a circulating electric field. Interchanging the space and time derivatives gives $$ \frac{\partial}{\partial t}\Big[\nabla\times(\Lambda\mathbf{J}_s) + \mathbf{B}\Big] = 0. $$ Read this carefully: it fixes only the rate of change of the bracketed quantity, forcing it to stay at whatever value it already had. So a perfect conductor freezes in the magnetic flux that happened to thread it at the moment its resistance vanished. Cool such a material in a field and the field stays trapped inside; switch the external field off afterwards and persistent currents keep the trapped flux alive. A perfect conductor, in short, remembers its history — a consequence of Lenz's law, that induced currents oppose any change of flux, and nothing more. The Meissner effect: expulsion, not memory Real superconductors do not behave this way. In 1933 Walther Meissner and Robert Ochsenfeld reported that a material cooled through its critical temperature $T_c$ while sitting in a magnetic field pushes the field out as it becomes superconducting, ending with $$ \mathbf{B} = 0 $$ throughout its interior — regardless of whether the field was applied before or after cooling. The final state carries no memory of how it was reached; it is a genuine equilibrium phase, not a frozen accident. To capture this, the brothers Fritz and Heinz London made a decisive move in 1935. Rather than keep the time-integrated relation above, which merely holds the bracket constant , they postulated that the bracket is identically zero : $$ \nabla\times(\Lambda\mathbf{J}_s) = -\mathbf{B}. $$ This is the second London equation . It is a genuinely new physical assumption, not derivable from $\rho=0$ alone; it is the mathematical statement of the Meissner effect. Everything that distinguishes superconductivity from mere perfect conduction follows from choosing $0$ here rather than "an arbitrary constant" [3] . How deep does the field reach? The penetration depth The second London equation, combined with magnetostatics, predicts exactly how a field dies away inside the material. Ampère's law (with no displacement current) reads $\nabla\times\mathbf{B} = \mu_0\mathbf{J}_s$, where $\mu_0$ is the permeability of free space. Take the curl of it and substitute the second London equation on the right: $\nabla\times(\nabla\times\mathbf{B}) = \mu_0\,\nabla\times\mathbf{J}_s = -\mu_0\mathbf{B}/\Lambda$. The vector identity $\nabla\times(\nabla\times\mathbf{B}) = \nabla(\nabla\cdot\mathbf{B}) - \nabla^2\mathbf{B}$ together with $\nabla\cdot\mathbf{B}=0$ turns the left-hand side into $-\nabla^2\mathbf{B}$, and the two minus signs cancel to give a screening equation for the field alone: $$ \nabla^2\mathbf{B} = \frac{\mathbf{B}}{\lambda_L^{2}}, \qquad \lambda_L \equiv \sqrt{\frac{m}{\mu_0 n_s e^2}}. $$ For a field $B_0$ applied parallel to a flat surface at $x=0$, with the superconductor filling $x>0$, this reduces to $\mathrm{d}^2B/\mathrm{d}x^2 = B/\lambda_L^{2}$, whose physically sensible solution decays exponentially into the bulk: $$ B(x) = B_0\,e^{-x/\lambda_L}. $$ The field does not stop dead at the surface; it soaks in a short way and is cancelled by supercurrents flowing in a thin surface layer. The length scale $\lambda_L$ — the London penetration dept