Lecture 8: Perfect Diamagnetism Outline 1. Description of a Perfect Diamagnet • Method I and Method II • Examples 2. Energy and Coenergy in Methods I and II 3. Levitating magnets and Maglev trains October 4, 2005 Massachusetts Institute of Technology 6.763 2005 Lecture 8 Description of Perfect Diamagetism Image removed for copyright reasons. Please see: Figure 2.17, page 50, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Surface currents or internal induced magnetization? Massachusetts Institute of Technology 6.763 2005 Lecture 8 1Methods I and II Image removed for copyright reasons. Please see: Figure 4.10, page 173, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Method I Method II B field is the same in both methods. Massachusetts Institute of Technology 6.763 2005 Lecture 8 Example: Magnetized Sphere Image removed for copyright reasons. Please see: Figure 4.11, page 174, 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 8 For this example: Laplace’s equation 2Boundary Conditions Inside the sphere: Boundary Conditions: Therefore, Magnetized Sphere Image removed for copyright reasons. Please see: Figure 4.11, page 174, 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 8 Outside the sphere: Massachusetts Institute of Technology 6.763 2005 Lecture 8 3Magnetized Sphere in field B=0 inside. Therefore, So that Image removed for copyright reasons. Please see: Figure 4.11, page 174, 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 8 For this to describe a superconductor (bulk limit), then Comparison of Methods Method I Method IIInside a bulk superconducting sphere: B field is the same, but not H. Massachusetts Institute of Technology 6.763 2005 Lecture 84Methods I and II: Summary Method I London Equations Maxwell’s Equations Massachusetts Institute of Technology 6.763 2005 Lecture 8 Method II Why Method II Question: 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 give Massachusetts Institute of Technology 6.763 2005 Lecture 8 5Method I: The Energy D, B, and ΛJs I= vs . but jugates E, HI, and Js I. Method I: The Coenergy E, HI, and Js I is defined by gives and with MQSEQS Massachusetts Institute of Technology 6.763 2005 Lecture 8 Combining the constitutive relations with Poynting’s theorem, Power dW/dt in the E&M field The energy stored in the electromagnetic field is So that the energy W is a function of However, one rarely has control over these variables, rather over their conMassachusetts Institute of Technology 6.763 2005 Lecture 8 The coenergy is a function of (The coenergy is the Free Energy at zero temperature.) 6Interpretation of the Coenergy Consider the case where there are only magnetic fields: Image removed for copyright reasons. Please see: Figure 4.12, page 184, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. The energy and coenergy contain the same information. Massachusetts Institute of Technology 6.763 2005 Lecture 8 Method II Massachusetts Institute of Technology 6.763 2005 Lecture 8 In the important case when of the MQS limit and Note that these two relations apply also in free space. 7Example: Energy of a Superconducting Sphere Image removed for copyright reasons. Please see: Figure 2.18, page 50, 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 8 Magnetic Levitation Image removed for copyright reasons. Please see: Figure 4.17, page 195, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Image removed for copyright reasons. Please see: Figure 4.18, page 195, 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 8 8Magnetic vs. Gravitational Forces Image removed for copyright reasons. Please see: Figure 4.17, page 195, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Image removed for copyright reasons. Please see: Figure 4.19, page 196, 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 8 Force is upwards Image removed for copyright reasons. Please see: Figure 4.20, page 198, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Image removed for copyright reasons. Please see: Figure 4.21, page 199, 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 8 Magnetic Levitation Equilibrium Point Equilibrium point 9Levitating magnets and trains Image removed for copyright reasons. Please see: Figure 4.17, page 195, from Orlando, T., and K. Delin. Image removed for copyright reasons. Foundations of Applied Superconductivity. Reading, MA: Please see: Figure 4.22, page 202, from Orlando, T., and K. Delin. Addison-Wesley, 1991. ISBN: 0201183234. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. ηB = 1 Tesla Area = 0.5 cm2 B = 2 Tesla 0 = 1 cm Area = 100 cm2 Force = 1 Newton η0 = 10 cm Force = 1,200 Newtons Enough to lift magnet, but not a train Enough to lift a train Massachusetts Institute of Technology 6.763 2005 Lecture 8 Maglev Train