Lecture 5: Classical Model of a SuperconductorSuperconductor: Classical ModelSuperconducting Infinite SlabFields and Currents for |y|< aSuperconducting Sphere: Bulk Approximation R >> lCurrent along a cylinder: bulk superconductorField along a cylinder: bulk superconductorField along a hollow cylinderMultiply Connected SuperconductorField along a hollow cylinderFlux trapped in a hollow cylinderSuperconducting CircuitsDC Flux TransformerSuperconducting Memory0 to 1 StorageMagnetic Monopole DetectorMagnetic Monopole DetectorInductance measurementExperimentTemperature dependent lTwo Fluid Model for wttr<<1, T< TcTwo Fluid ModelComplex wavenumberMassachusetts Institute of Technology 6.763 2003 Lecture 5Lecture 5: Classical Model of a SuperconductorOutline1. First and Second London Equations2. Examples• Superconducting Slab• Bulk Sphere3. Non-simply connected superconductors• Hollow cylinder• Superconducting circuitso DC flux transformero Superconducting memory loopo Magnetic monopole detector4. Two Fluid ModelSeptember 18, 2003Massachusetts Institute of Technology 6.763 2003 Lecture 5Superconductor: Classical Modelfirst London Equationsecond London Equationpenetration depthWhen combined with Maxwell’s equation in the MQS limitMassachusetts Institute of Technology 6.763 2003 Lecture 5Superconducting Infinite SlabLetTherefore,and2axBoundary Conditions demandMassachusetts Institute of Technology 6.763 2003 Lecture 5Fields and Currents for |y|< aThin film limitλλBulk limitMassachusetts Institute of Technology 6.763 2003 Lecture 5Superconducting Sphere: Bulk Approximation R >> λMassachusetts Institute of Technology 6.763 2003 Lecture 5Current along a cylinder: bulk superconductorThe fields from Ampere’s lawInside: Outside:Therefore,Massachusetts Institute of Technology 6.763 2003 Lecture 5Field along a cylinder: bulk superconductorMassachusetts Institute of Technology 6.763 2003 Lecture 5Field along a hollow cylinderSolution 1or Solution 2?Massachusetts Institute of Technology 6.763 2003 Lecture 5Multiply Connected SuperconductorFirst LondonMaxwellThereforeandFor a contour within the bulk where J = 0, flux remains constantMassachusetts Institute of Technology 6.763 2003 Lecture 5Field along a hollow cylinderZero Field Initially SolutionFinite Field Initially SolutionMassachusetts Institute of Technology 6.763 2003 Lecture 5Flux trapped in a hollow cylinderMassachusetts Institute of Technology 6.763 2003 Lecture 5Superconducting CircuitsA generalization to any closed superconducting circuit is that the total flux linkage in a circuit remains constant.Then if a circuit has N elements that can contain flux,Sources of Flux linkageSelf-inductanceMutual inductance External fluxMassachusetts Institute of Technology 6.763 2003 Lecture 5DC Flux TransformerIf the B field is measured of the transported fluxB can be amplifiedFlux can be transportedMassachusetts Institute of Technology 6.763 2003 Lecture 5Superconducting MemoryMassachusetts Institute of Technology 6.763 2003 Lecture 50 to 1 Storage01Massachusetts Institute of Technology 6.763 2003 Lecture 5Magnetic Monopole DetectorMaxwell’s Equations with Monopole density ρmThe signs insure electric and magnetic charge conservation.Massachusetts Institute of Technology 6.763 2003 Lecture 5Magnetic Monopole DetectorTake the line integralTotal of Flux and magnetic charge is conserved.Massachusetts Institute of Technology 6.763 2003 Lecture 5Inductance measurementFrom the measurement of the inductance, the penetration depth can determined.IRFor a normal metalAnd For a superconductor,andMassachusetts Institute of Technology 6.763 2003 Lecture 5ExperimentThe penetration depth λ is temperature dependent !Massachusetts Institute of Technology 6.763 2003 Lecture 5Temperature dependent λA good guess to let depend on temperature for T< Tcn n(T)n/ntotTMassachusetts Institute of Technology 6.763 2003 Lecture 5Two Fluid Model for ωτtr<<1, T< TcJtot= Js(T) + Jn(T)Λ(T) 1/σ0(T)Massachusetts Institute of Technology 6.763 2003 Lecture 5Two Fluid ModelConstitutive relations for two fluid modelMaxwellGivesMassachusetts Institute of Technology 6.763 2003 Lecture 5Complex wavenumberFor a sinusoidal drive,2axFor a slab in a uniform fieldThe smaller length determines the length