Perfect Conductivity Lecture 2OutlinePersistent CurrentsCharging up a superconducting loopParts of a Physical TheoryLRC CircuitSimpler Circuits and Time ConstantsReducing the Circuit to a simpler form ?Order of time constantsMoral of time constantsDistributed SystemsDistributied systems con’tQuasistatic LimitTime ConstantsOrder of time constantsMagnetoQuasiStaticsMQS: Magnetic Diffusion EquationMassachusetts Institute of Technology 6.763 2003 Lecture 2Perfect Conductivity Lecture 2Terry P. OrlandoDept. of Electrical Engineering MITSeptember 9, 2003Massachusetts Institute of Technology 6.763 2003 Lecture 2Outline1. Persistent Currents2. Parts of a Physical Theory3. Circuits and Time Constants4. Distributive Systems and Time constantsA.QuasistaticsB.MagnetoQuasiStatics (MQS)Massachusetts Institute of Technology 6.763 2003 Lecture 2Persistent CurrentsIf the field is turned off, thenIf the loop is made out of a superconductor,Experimentally the dc resistivity of a superconductor is at least as small as . The superconducting state is “truly” zero dc resistance.LRMassachusetts Institute of Technology 6.763 2003 Lecture 2Perfect Conductivity: t << τRL= L/R, sytem looks like R is zeroSuperconductivity: for all time, R is zeroMassachusetts Institute of Technology 6.763 2003 Lecture 2Charging up a superconducting loopRLISuperconducting materialThis Persistent Mode is the basis of MRI magnets, SMES, flux memory….Massachusetts Institute of Technology 6.763 2003 Lecture 2Parts of a Physical Theory1. Governing Laws:Maxwell’s Equations, Newton’s equations, 2. Constitutive Laws:Models of the systemlike ohm’s law,3. Summary Relations:Transfer functions, Dispersion relationsMassachusetts Institute of Technology 6.763 2003 Lecture 2LRC CircuitiiCvCRLCiL.vLiR.vRv+-RjωL1jωC1. Governing Equations Current conservation: i=iC+ iLiL =iREnergy Conservation v = vC= vR+ vLMassachusetts Institute of Technology 6.763 2003 Lecture 22. Constitutive RelationsMassachusetts Institute of Technology 6.763 2003 Lecture 23. Summary RelationRjωL1jωCMassachusetts Institute of Technology 6.763 2003 Lecture 2Simpler Circuits and Time ConstantsRLLCRCEnergy stored in capacitorEnergy stored in inductorResonant transfer of energy between L and CMassachusetts Institute of Technology 6.763 2003 Lecture 2Reducing the Circuit to a simpler form ?RjωL1jωC???RLLCRCMassachusetts Institute of Technology 6.763 2003 Lecture 2Order of time constantsω1/τLR 1/τRC1/τLCω1/τRC 1/τLR1/τLCaa bbτLR> τRCLow RRLRLaabbτLR< τRCHigh RRLow frequency circuitRCLow frequency circuitCMassachusetts Institute of Technology 6.763 2003 Lecture 2Moral of time constantsIf you know what frequency range you want to study or what physics dominates the problem, then you can solve a simpler problem.Useful, especially in more complex situations.Massachusetts Institute of Technology 6.763 2003 Lecture 2Distributed Systems1. Governing Equations: Maxwell’s EquationsFaraday’s LawAmpere’s LawGauss LawGauss’ Magnetic LawConservations lawsCharge conservation Also Poynting’sMassachusetts Institute of Technology 6.763 2003 Lecture 2Distributied systems con’t2. Constitutive RelationsLocal in space, linear time invariantOhm’s Law3. Summary relationsComplex: Search first for first order in time approximationMassachusetts Institute of Technology 6.763 2003 Lecture 2Quasistatic LimitSpeed of lightWavelength of E&M waveLength scale of systemFrequency(angular)If the dimensions of a structure are much less than thewavelength of an electromagnetic field interacting with it, the coupling between the associated electric and magnetic fields is weak and a quasistatic approximation is appropriate.Massachusetts Institute of Technology 6.763 2003 Lecture 2Time ConstantsElectromagnetic coupling timeCharge relaxation timeMagnetic diffusion timewhdσ0,µ0,εivMassachusetts Institute of Technology 6.763 2003 Lecture 2Order of time constantsω1/τm 1/τc1/τemω1/τc 1/τm1/τemaa bbτm > τc τm < τeHigh conductivityLowconductivityEQSMQSRLLow frequency circuitLow frequency circuitRaaRLbbRCCMassachusetts Institute of Technology 6.763 2003 Lecture 2MagnetoQuasiStaticswhdσ0,µ0,εivSolve firstSolve for E once B is foundBoundary conditions:Massachusetts Institute of Technology 6.763 2003 Lecture 2MQS: Magnetic Diffusion EquationFor a metal B = µ0H, D = ε0 E and J = σ0E, so thatwhdσ0,µ0,εivMagnetic Diffusion