Understanding stable levitation of superconductors from intermediate electromagnetics A. Badía-Majósa͒ Departamento de Física de la Materia Condensada-I.C.M.A., C.P.S.U.Z., María de Luna 1, E-50018 Zaragoza, Spain ͑Received 15 February 2006; accepted 28 July 2006͒ Levitation experiments with superconductors in the Meissner state are hindered by low stability except for specifically designed configurations. In contrast, magnetic force experiments with strongly pinned superconductors and permanent magnets display high stability, allowing the demonstration of striking effects, such as lateral or inverted levitation. These facts are explained by using a variational theory. Illustrations based on calculated magnetic field lines for various configurations are presented. They provide a qualitative physical understanding of the stability features. © 2006 American Association of Physics Teachers. ͓DOI: 10.1119/1.2338548͔ I. INTRODUCTION A. Type-I superconductors Levitation experiments based on the repulsive ͑or attrac- We begin with the definition of the electric current density tive͒ force between permanent magnets and superconductors for conducting media are common. Almost everyone is fascinated and stimulated J = nqv, ͑1͒ by the observation of floating objects. When the setup in- cludes a high pinning ͑or high critical current͒ supercon- where n is the volume density of the charge carriers, q their ductor, the possibilities of lifting moderate weights and dis- effective charge, and v their velocity. If the underlying ma- playing lateral or inverted levitation are even more attractive. terial is such that charges can move without friction, New- Recent developments of vehicles capable of supporting sev- ton’s second law gives eral passengers1 have increased the interest in this topic. dJ nq2 1 As the number of levitation phenomena becomes larger, = E ϵ 2 E, ͑2͒ the difficulty of giving a reasonable explanation on how levi- dt m ␮0␭ tation works and giving a quantitative analysis has also in- where ␭ defines a characteristic length, called the London creased. It is simple to understand that a magnet floating penetration depth. Equation ͑2͒ leads to the property above a superconductor is related to flux expulsion and we 2 may use the standard image technique in magnetostatics to dJ d ␮0␭ E·J= ␮ ␭2 ·J= J2 . ͑3͒ make quantitative estimates. Specialized image models have 0 dt dtͩ 2 ͪ also been introduced, which allow us to understand attractive forces.2 However, such approximations are only useful for If this relation is included in Poynting’s theorem, it is appar- small displacements of tiny magnets close to the supercon- ent that the standard electromagnetic field energy is aug- ductor and do not include material parameters. mented by a new form of reversible storage, related to the As an alternative and complementary point of view to pre- kinetics of the moving charges. If we neglect the presence of vious work,3 I will give a more general theoretical frame- electrostatic charges, we have the energy conservation law work and a number of examples. The presentation is aimed d B2 ␭2 dU Ã B͉2dV ϵ =0, ͑4͒ ١͉ + at students who have had an intermediate course on electro- dV ͵ 3 ͵ magnetism and some background in classical mechanics. The dtͩ 2␮0 V 2␮0 ͪ dt R S main concepts involved are electromagnetic energy, thermo- dynamic reversibility and irreversibility, Lenz-Faraday’s law where VS denotes the superconducting volume. of induction, and the use of variational principles. The pre- Because energy is conserved, the system will settle in sentation emphasizes the peculiarities of levitation with some equilibrium configuration. If we use the field B as the type-I and type-II superconductors. In both cases, the limita- independent variable and let ␦ denote derivatives with re- tions imposed by Earnshaw’s theorem,4 which restricts levi- spect to it, we may express the equilibrium condition as the tation in electromagnetic systems, are avoided.5 minimization of B2 ␭2 Ã B͉2dV, ͑5͒ ١͉ + U = dV ͵ 3 2␮ ͵ 2␮ R 0 VS 0 II. BASIC SUPERCONDUCTIVITY AND which implies that VARIATIONAL PRINCIPLES ␦U =0. ͑6͒ ␦B Superconductivity is a complex phenomenon, whose na- ture combines electromagnetic, thermodynamic, and quan- We minimize the quantity in Eq. ͑5͒ to obtain the static con- tum effects. For our purposes, the governing equations may figuration for fixed sources ͑boundary conditions for the be found from relatively simple considerations. fields͒. Notice that this formulation reflects flux expulsion 1136 Am. J. Phys. 74 ͑12͒, December 2006 http://aapt.org/ajp © 2006 American Association of Physics Teachers 1136 ͑diamagnetism͒ in superconductors, because B2 is mini- mized. This expulsion is the Meissner state. The more con- ventional differential statement B+␭2͑١Ã١ÃB͒=0 ͑the London equation6͒ follows from the zero derivative condi- tion Eq. ͑6͒. B. Type-II superconductors For certain superconducting materials, we need to relax the assumption that the electric current flow is completely free of losses. In the hard type-II superconductors only low ͑undercritical͒ currents are lossless. As a first approximation, we may assume lossless behavior for JഛJc ͑Jc is the critical current for the material͒, and Ohm’s law for JϾJc as for normal metals ͓here E=␳sc͑J−Jc͔͒. The nature of such re- sponse of the charge carriers, including the interpretation of the intrinsic parameter Jc, may be found in Ref. 7. In brief, Fig. 1. ͕E,J͖ graph ͑conduction law͒ for a hard type-II superconductor, these materials can hold an internal magnetic flux as long as according to Bean’s model ͑Ref. 8͒. Vertical lines correspond to infinite the field gradients ͑J͒ remain below a threshold. Above this resistivity above Jc. value, the flux is unpinned and the underlying currents flow dissipatively. I now show that the behavior of these materials also fol- lows a variational law. Recall that the dynamical equations of 2 ͑⌬B͒ ␳⌬t 2 Ã B͉ dV. ͑11͒ ١͉ + a single particle under conservative forces may be obtained Fn ϵ dV ͵ 3 2␮ ͵ 2␮ by a minimum action principle. That is the minimization of R 0 Vmetal 0 ͐Ldt implies that If we take variations with respect to the variable B, we ob- Lץ Lץ d = , ͑7͒ tain ␦Fn /␦B=0, and xץ ͪ ˙xץ dtͩ where L=mx˙2 /2−V is the Lagrangian. Note that nonconser- ⌬B ␳ Ã B ͑12͒ ١ Ã ١−= vative forces may not be treated variationally, so that New- ⌬t ͩ ␮ ͪ ton’s second law is equivalent to 0 L as expected. This equation in terms of increments is just theץ Lץ d = + Fncons. ͑8͒ .١ÃE͒−=tץ/Bץ͑ x time-discretized version of Faraday’s lawץ ͪ ˙xץ dtͩ The application of the previous ideas to superconducting Nevertheless, if we assume that Fncons is a viscous drag media follows naturally. We have to use a suitable form for force ͑Fncons=−m␥x˙͒ with a friction constant ␥,Eq.͑8͒ the energy loss that is related to overcritical current flow. The ⌬t ˆ may be obtained from minimizing ͐0 Ldt: simplest theory that accounts for losses in such conditions was proposed by Bean8 in the context of magnetic hysteresis. Lˆ In terms of a conduction law, Bean’s model is equivalent toץ ˆLץ d = . ͑9͒ -x the E͑J͒ relation sketched in Fig. 1. Note that the multivalץ ͪ ˙xץ dtͩ ued graph corresponds to the overdamped limit of the physi- We have defined Lˆ ϵL+͑m␥x˙2 /2͒t, and assumed that ⌬x˙ cal properties mentioned previously: nondissipative current Ӷx˙ for increments within the time interval ͓0,⌬t͔. Then Eq. flow is allowed for current densities below a critical value Jc, x as required. and induced electric fields relate to the current density flowץ/Vץ− ˙leads to mx¨ =−m␥x 9͒͑ The result ͑9͒ is a quasistationary variational principle, by an infinite slope resistivity. Obviously, the vertical rela- which may be applied in a time discretized description of the tion is just an idealization of real cases in which there is a system ͑redefine ͓0,⌬t͔ and iterate͒. very high slope ␳sc. In terms of Eq. ͑10͒, Bean’s law can be Generalization is possible if we recall that for a single written as a quasistationary principle in which the modified particle, m␥x˙2 is the energy loss per unit time. For instance, Lagrangian is ˆ = + t, with the dissipation function L L D the eddy-current problem in normal ͑ohmic͒ metals may be solved iteratively by minimizing 0 if J Ͻ Jc ⌬t = ͑13͒ ˆ D ͭϱ if J Ͼ Jc. ͮ Sn ϵ dVdt, ͑10͒ ͵ ͵ 3 0 R L The variational principle admits a simple treatment of this ˆ 2 with ϵB /2␮0 +͑E·J/2͒t as the modified Lagrangian den- singular behavior. We may minimize the first term in Eq. L sity for the magnetostatic field. Recall that E·J corresponds ͑11͒ and replace the second one by the cutoff condition J to the energy dissipation per unit time and volume. Then, if ഛJc. we assume ⌬EӶE and use the stationary relations E=␳J By using a time discretization in layers of step size ␦t, that 9 ١ÃB=␮0J in the time interval ͓0,⌬t͔, the minimization is, tn =n␦t, we obtain the following iterative model. We and functional in Eq. ͑10͒ becomes minimize the quantity 1137 Am. J. Phys., Vol. 74, No. 12, December 2006 A. Badía-Majós 1137 Fig. 2. Magnetic field lines around a horizontal magnetic dipole over a superconducting tape, for three positions of the magnet. Induced currents within the tape flow perpendicularly to the plot ͑outward within the small segment below the magnet, and inward for the rest͒. To the right, we plot the magnetostatic potential energy Um =−m·BS for small horizontal displace- ments of the magnet around each position.