Perfect Conductivity Lecture 2 Terry P. Orlando Dept. of Electrical Engineering MIT September 13, 2005 Massachusetts Institute of Technology 6.763 2005 Lecture 2 Outline 1. Persistent Currents 2. Parts of a Physical Theory 3. Circuits and Time Constants 4. Distributive Systems and Time constants A.Quasistatics B.MagnetoQuasiStatics (MQS) Massachusetts Institute of Technology 6.763 2005 Lecture 2 1Persistent Currents If the field is turned off, then as L R Massachusetts Institute of Technology 6.763 2005 Lecture 2 If the loop is made out of a superconductor, Experimentally the dc resistivity of a superconductor is at least as small . The superconducting state is “truly” zero dc resistance. Perfect Conductivity: t << τRL= L/R, sytem looks like R is zero Superconductivity: for all time, R is zero Massachusetts Institute of Technology 6.763 2005 Lecture 2 2Charging up a superconducting loop R LI Superconducting material This Persistent Mode is the basis of MRI magnets, SMES, flux memory…. Massachusetts Institute of Technology 6.763 2005 Lecture 2 Parts of a Physical Theory 1. Governing Laws:Maxwell’s Equations, Newton’s equations, 2. Constitutive Laws:Models of the system like ohm’s law, 3. Summary Relations: Transfer functions, Dispersion relations Massachusetts Institute of Technology 6.763 2005 Lecture 2 3LRC Circuit R jωL1 jωC 1. Governing Equations C + iL iL =iR Energy Conservation v = vC = vR + vL i iC vC R L C iL.vL iR.vR v + -Massachusetts Institute of Technology 6.763 2005 Lecture 2 Current conservation: i=i2. Constitutive Relations Massachusetts Institute of Technology 6.763 2005 Lecture 2 4R jωL1 jωC Massachusetts Institute of Technology 6.763 2005 Lecture 2 3. Summary Relation Simpler Circuits and Time Constants R L LC RC Energy stored Resonant transfer of Energy stored in in inductor energy between L and C capacitor Massachusetts Institute of Technology 6.763 2005 Lecture 2 5Reducing the Circuit to a simpler form ? R L LC RC R jωL1 jωC ? ? ? Massachusetts Institute of Technology 6.763 2005 Lecture 2 ω 1/τLR 1/τRC1/τLC ω 1/τRC 1/τLR1/τLC τLR > τRC Low R τLR < τRC High R R L Low frequency Low frequency R L R CRC aa a a bb b b Massachusetts Institute of Technology 6.763 2005 Lecture 2 Order of time constants circuit circuit 6Moral of time constants If 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 2005 Lecture 2 Distributed Systems Gauss Law Gauss’ Magnetic Law Conservations laws Charge conservation Also Poynting’s Massachusetts Institute of Technology 6.763 2005 Lecture 2 1. Governing Equations: Maxwell’s Equations Faraday’s Law Ampere’s Law 7Distributied systems con’t 2. Constitutive RelationsLocal in space, linear time invariant Ohm’s Law 3. Summary relations Complex: Search first for first order in time approximation Massachusetts Institute of Technology 6.763 2005 Lecture 2 Quasistatic Limit Length scale of system Speed of light Frequency (angular)Wavelength of E&M wave If the dimensions of a structure are much less than the wavelength 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 2005 Lecture 2 8Time Constants Electromagnetic coupling time Charge relaxation time Magnetic diffusion time Image removed for copyright reasons. Please see: Figure 2.9, page 34, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 2 Order of time constants abba ωω 1/τc 1/τem 1/τm1/τm 1/τem 1/τc High Lowτm > τc τm < τeconductivity conductivity EQSMQS R L R L a b RaLow Low frequency frequency C R b C circuit circuit Massachusetts Institute of Technology 6.763 2005 Lecture 2 9MagnetoQuasiStatics Solve for E once B is found Solve first Boundary conditions: Image removed for copyright reasons. Please see: Figure 2.9, page 34, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 2 MQS: Magnetic Diffusion Equation For a metal B = µ0 H, D = ε0 E and J = σ0E, so that Image removed for copyright reasons. Please see: Figure 2.9, page 34, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Magnetic Diffusion Equation Massachusetts Institute of Technology 6.763 2005 Lecture 2