Lecture 8: Perfect DiamagnetismDescription of Perfect DiamagetismMethods I and IIExample: Magnetized SphereBoundary ConditionsMagnetized SphereMagnetized Sphere in fieldComparison of MethodsMethods I and II: SummaryWhy Method IIMethod I: The EnergyMethod I: The CoenergyInterpretation of the CoenergyMethod IIExample: Energy of a Superconducting SphereMagnetic LevitationMagnetic vs. Gravitational ForcesMagnetic Levitation Equilibrium PointLevitating magnets and trainsMaglev TrainPrinciple of MaglevReal trains have wheels and high-voltage railsOur Approach to SuperconductivityMassachusetts Institute of Technology 6.763 2003 Lecture 8Lecture 8: Perfect DiamagnetismOutline1. Description of a Perfect Diamagnet• Method I and Method II• Examples 2. Energy and Coenergy in Methods I and II3. Levitating magnets and Maglev trainsSeptember 30, 2003Massachusetts Institute of Technology 6.763 2003 Lecture 8Description of Perfect DiamagetismSurface currents or internal induced magnetization?Massachusetts Institute of Technology 6.763 2003 Lecture 8Methods I and IIMethod IMethod IIB field is the same in both methods.Massachusetts Institute of Technology 6.763 2003 Lecture 8Example: Magnetized SphereFor this example: Laplace’s equationMassachusetts Institute of Technology 6.763 2003 Lecture 8Boundary ConditionsInside the sphere:Outside the sphere:Boundary Conditions:Therefore,Massachusetts Institute of Technology 6.763 2003 Lecture 8Magnetized SphereMassachusetts Institute of Technology 6.763 2003 Lecture 8Magnetized Sphere in fieldFor this to describe a superconductor (bulk limit), then B=0 inside.Therefore,So thatMassachusetts Institute of Technology 6.763 2003 Lecture 8Comparison of MethodsMethod IMethod IIInside a bulk superconducting sphere:B field is the same, but not H.Massachusetts Institute of Technology 6.763 2003 Lecture 8Methods I and II: SummaryMethod I Method IIMaxwell’s EquationsLondon EquationsMassachusetts Institute of Technology 6.763 2003 Lecture 8Why Method IIQuestion: The constitutive relations and London’s Equations have gotten much more difficult. So why do Method II?Answer: The Energy and Thermodynamics are easier, especially when there is no applied current.So we will find the energy stored in both methods.Poynting’s theorem is a result of Maxwell’s equation, so both methods giveMassachusetts Institute of Technology 6.763 2003 Lecture 8Method I: The EnergyCombining the constitutive relations with Poynting’s theorem,Power dW/dt in the E&M fieldThe energy stored in the electromagnetic field is So that the energy W is a function of D, B, and ΛJsI= vs. However, one rarely has control over these variables, but rather over their conjugates E, HI, and JsI.Massachusetts Institute of Technology 6.763 2003 Lecture 8Method I: The CoenergyThe coenergy is a function of E, HI, and JsI is defined byand with givesMQSEQS(The coenergy is the Free Energy at zero temperature.)Massachusetts Institute of Technology 6.763 2003 Lecture 8Interpretation of the CoenergyConsider the case where there are only magnetic fields:The energy and coenergy contain the same information.Massachusetts Institute of Technology 6.763 2003 Lecture 8Method IIIn the important case when of the MQS limit and Note that these two relations apply also in free space.Massachusetts Institute of Technology 6.763 2003 Lecture 8Example: Energy of a Superconducting SphereMassachusetts Institute of Technology 6.763 2003 Lecture 8Magnetic LevitationMassachusetts Institute of Technology 6.763 2003 Lecture 8Magnetic vs. Gravitational ForcesηMassachusetts Institute of Technology 6.763 2003 Lecture 8Magnetic Levitation Equilibrium PointForce mainly due to bending of flux lines here.Force is upwardsEquilibrium pointMassachusetts Institute of Technology 6.763 2003 Lecture 8Levitating magnets and trainsB = 1 TeslaArea = 0.5 cm2η0 = 1 cmForce = 1 NewtonEnough to lift magnet, but not a trainB = 2 TeslaArea = 100 cm2η0= 10 cmForce = 1,200 NewtonsEnough to lift a trainMassachusetts Institute of Technology 6.763 2003 Lecture 8Maglev TrainMagnets on the train are superconducting magnets; the rails are ohmic!Massachusetts Institute of Technology 6.763 2003 Lecture 8Principle of MaglevThe train travels at a velocity U, and the moving flux lines and the rails “see” a moving magnetic field at a frequency of ω ~ U/R. If this frequency is much larger than the inverse of the magnetic diffusion time, then the flux lines are “repelled” from the ohmic rails.From the previous numbers, U > 40 km/hr for levitation.Massachusetts Institute of Technology 6.763 2003 Lecture 8Real trains have wheels and high-voltage railsSynchronous motor action down the rails provides thrust to accelerate train to the needed velocity to levitate, and provides a source of energy to further accelerate the train and to overcome the losses due to drag from the wind.Massachusetts Institute of Technology 6.763 2003 Lecture 8Our Approach to SuperconductivitySuperconductor as a perfect conductor & perfect diamagnetMacroscopic Quantum Model Ψ(r)Supercurrent Equation Js(r)Ginzburg-LandauΨ(r) = | Ψ(r)|2e iθ(r,t)BCSType II SuperconductivityLarge-Scale ApplicationsJosephson EquationsSmall-Scale ApplicationsClassicalMacroscopic Quantum ModelMicroscopic