Current transfer at the wheel-collector contact of a superconducting Barlow wheel and its control of the electromagnetic torque

In 1948 Heisenberg and von Laue asked whether the Barlow wheel - the classic radial-current homopolar motor - would rotate if made of a superconducting material, and showed within the London phenomenological theory that the answer is decided entirely at the contact: a resistance-free supercurrent contact transports the electromagnetic torque into the collector, so the wheel feels no torque, while a contact that forces normal current delivers the classical torque (thin wheel) or a parametrically suppressed torque (thick wheel). Because London phenomenology cannot decide which transfer mode a real contact realizes, the paper called for a completed microscopic electron theory of superconductivity and a theory of the contact itself. Those general theories now exist (BCS theory; the Josephson effect; Andreev-reflection microconstriction theory), but no prior work applies them to this system. The surviving open problem is to determine how current crosses the wheel-collector contact (supercurrent transfer versus Andreev-mediated normal transfer), under what physical conditions resistance-free supercurrent transfer at a moving contact is possible, and how the transfer mode controls the observable torque - a question now technically revived by fully superconducting homopolar DC machine designs that must collect current from superconducting armature circuits.

Background

The Barlow wheel, built by Peter Barlow in 1822, is the oldest homopolar motor (a motor whose torque comes from current flowing across a magnetic field in one fixed direction, rather than from alternating currents) and a long-standing lecture demonstration: a conducting wheel with its axle vertical and its rim dipping into a mercury trough, placed between the poles of a magnet. When a current $J$ is fed through the axle and leaves at the rim, it flows radially through the wheel, and the Lorentz force on this radial current in the axial magnetic field $H$ produces the torque

$D=-\frac{JHR^{2}}{2c}$

(in Gaussian units, used throughout the source; $R$ is the wheel radius and $c$ the speed of light), which sets the wheel spinning. In 1947 the Göttingen physicist R. W. Pohl asked Werner Heisenberg and Max von Laue what would happen if the wheel were made of a superconducting material - a material that, below a critical temperature, carries current with exactly zero resistance. Their answer appeared as 'Das Barlowsche Rad aus supraleitendem Material' (The Barlow wheel made of superconducting material), Zeitschrift für Physik 124, 514 (1948), dedicated to Richard Becker's sixtieth birthday.

The tool they used was the phenomenological theory of superconductivity of F. and H. London (1935). In that theory the zero-resistance current (the supercurrent) flows only in a thin surface layer of thickness $c\sqrt{\lambda}$ (the penetration depth, with $\lambda$ the London parameter), into which the magnetic field penetrates before being expelled from the bulk. A key structural result, developed by Becker, Heller and Sauter (1933), by the Londons, and by von Laue, and given a model interpretation by Heisenberg (1947), is that electromagnetic forces on the supercurrent are transmitted elastically inside the superconductor: the London stress tensor

$T_{ik}=\frac{\lambda}{2}\left(2\mathfrak{i}_i\mathfrak{i}_k-\delta_{ik}\mathfrak{i}^{2}\right),$

with $\mathfrak{i}$ the supercurrent density, has a divergence that cancels the Lorentz force density pointwise in the static case. Magnetic forces are therefore carried along the supercurrent streamlines like tension in a rope and appear as mechanical force only where the supercurrent terminates - typically at a boundary between superconducting and normal (non-superconducting) regions.

Applying this to the wheel, Heisenberg and von Laue found that everything is decided at the contact. If the current crosses the wheel-collector interface as supercurrent - a resistance-free transfer into a superconducting collector - the force is transported across the contact into the collector and the wheel feels no net torque: it does not rotate. If instead the supercurrent terminates in a normal region at the rim, the force is delivered to the wheel: for a thin wheel (thickness $d\ll c\sqrt{\lambda}$, too thin to form shielding layers) the full classical torque above reappears, while for a wheel thick enough to form shielding layers of thickness $c\sqrt{\lambda}$ ($d\gg c\sqrt{\lambda}$) the torque on the wheel is parametrically suppressed. The suppression factor, quoted by the paper as a parametric estimate, reads

$\frac{D'}{D}=\frac{2c\sqrt{\lambda}}{R}\cdot\frac{H'}{H}\approx\frac{4c\sqrt{\lambda}}{\pi d}\ll 1,$

where $H'$ is the magnetic field enhanced at the rim by the Meissner field distortion. In the paper's model of an ellipsoidal wheel cross-section with small axis $d$ and large axis $R$ (from its Eq. (2), in the limit $R\gg d$), $H'=H\,\frac{2R}{\pi d}\left(1+\frac{\pi d}{3R}+\cdots\right)$, so that $H'/H\approx 2R/(\pi d)$, which yields the quoted approximation. The paper also noted a possible asymmetry for very narrow contacts, in which electrons leaving the superconductor individually need not return the condensate stress to the wheel, so the torque could even depend on the current direction.

The paper's conclusion is that the London phenomenology cannot decide which transfer mode a real contact realizes. Written before any microscopic theory of superconductivity existed, it states that what is needed is a completed electron theory of superconductivity and a theory of the contact itself, capable of deciding under what physical conditions a resistance-free supercurrent transfer is possible, when the contact is forced normal by local current density or geometry, and how these conditions control the experimentally observed torque.

Both named general theories have since been supplied. The microscopic electron theory arrived as BCS theory (Bardeen, Cooper and Schrieffer, 1957), in which the supercurrent is carried by a condensate of bound electron pairs (Cooper pairs). The theory of the contact arrived with the Josephson effect (1962) - the passage of supercurrent through a weak contact between two superconductors - and with the mechanism of Andreev reflection, the microscopic process by which current crosses from a normal metal into a superconductor: an electron incident on the interface is retro-reflected as a hole, depositing a Cooper pair in the condensate, with the conversion between normal current and supercurrent taking place over roughly the Cooper-pair size near the contact; this mechanism underlies the standard theory of metallic normal-superconductor microconstrictions - very small-area contacts between a normal metal and a superconductor (Blonder, Tinkham and Klapwijk, 1982; see also Asano's 2021 monograph on Andreev reflection in superconducting junctions). Together these theories answer, in general form, how current crosses a contact and when such transfer is resistance-free (below a critical current density, above which the contact region is driven normal). Engineering practice bears on the feasibility side of the contact question: superconducting homopolar machines with liquid-metal current collectors have been analyzed and operated with stationary superconducting windings, and designs for fully superconducting homopolar DC machines - which would have to collect current directly from superconducting armature circuits (the current-carrying rotating part) through brushes - have been proposed.

What has never been done is to apply this machinery to the actual question of 1948: how the microscopic current-transfer mode at the wheel-collector contact controls the torque of a superconducting Barlow wheel, and whether the resistance-free supercurrent-transfer regime can be realized in a rotating contact.

Problem Statement

Determine, within microscopic superconductivity theory, how electrical current crosses the contact between a superconducting Barlow wheel (radius $R$, thickness $d$, axial feed current $J$ drained at the rim) and its current collector in an axial magnetic field $H$:

(1) the microscopic mode of transfer at the interface - resistance-free supercurrent transfer versus normal-current transfer mediated by Andreev reflection - as a function of the physical conditions at the contact: contact geometry and area, local current density relative to the critical (pair-breaking) current density, temperature, materials, and the geometry-enhanced field $H'$ at the rim;

(2) the consequence of the transfer mode for the mechanical torque on the wheel - consistent with the London-phenomenology limits of Heisenberg and von Laue (zero net torque on the wheel for ideal supercurrent transfer into a superconducting collector; the classical torque $D=-JHR^{2}/(2c)$ for a thin wheel, too thin to form shielding layers, whose supercurrent terminates in a normal region at the rim; a parametrically suppressed torque, by a small factor of order penetration depth over wheel dimension involving the rim-enhanced field $H'$, for a wheel thick enough to form shielding layers, $d\gg c\sqrt{\lambda}$) - and giving the torque for realistic contact regimes between these limits;

(3) whether a resistance-free supercurrent transfer is realizable at the collector contact of an actually rotating superconducting wheel - Pohl's original question of whether the wheel would still function - and what torque the wheel experiences in the realized transfer regime.

The physical conditions listed in (1) delimit the class of contact regimes over which the transfer-mode criterion must hold, rather than a separate sub-question for each condition; and the "realistic contact regimes between these limits" in (2) are contacts that are neither ideal resistance-free supercurrent contacts (which give zero net torque) nor fully normal (which give the classical or parametrically suppressed London-phenomenology torque above), so that the torque must be given for the crossover between the limiting regimes.

This is the surviving core of the 1948 Heisenberg-von Laue question: the general microscopic electron theory and the general theory of contact current transfer now exist, but their application to this system - the contact-mode-to-torque map and its experimental realization - is absent from the literature.

References

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