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Electromagnetism - Lecture 12

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Electromagnetism - Lecture 12 Electromagnetism - Lecture 12 Ferromagnetism & Superconductivity • Ferromagnetism • Hysteresis & Permanent Magnets • Ferromagnetic Surfaces • Toroid with Ferromagnetic Core • Superconductivity • The Meissner Effect 1 Ferromagnetism Ferromagnetism occurs when the spins of conduction electrons in metals spontaneously align Caused by an exchange interaction U / S1:S2 The spontaneous alignment breaks down above a critical temperature, the Curie point T = TC For T > TC the metal is paramagnetic: C χM = TC = λC BL = λM T − TC where BL is the local field due to the spin-spin interactions For T < TC the metal is ferromagnetic with a large magnetization 2 Magnetic Saturation When all the spins are aligned the magnetization is saturated 2 −2λNeµB =kT MS = 2NeµBe 0 < T < TC Examples of ferromagnetic materials: Iron (Fe) TC = 1043K BS = 1:7T Cobalt (Co) TC = 1288K BS = 1:4T Nickel (Ni) TC = 627K BS = 0:5T The relative permeability µr = B/µ0H of ferromagnets is very 3 5 large and has a wide range of values µr = 10 − 10 3 Notes: Diagrams: 4 Magnetic Domains and Hysteresis The direction of the spontaneous M in a ferromagnet is random A macroscopic sample contains many magnetic domains, in each of which M points in a different direction. They are separated by domain walls A macroscopic ferromagnet can be unmagnetised if H = 0 Applying an external field H defines a preferred direction for M ) Domain walls move to favour the direction of H ) Electron spins rotate into alignment with H A hysteresis curve shows B = µ0(H + M) as a function of H When H is removed this can leave a permanent magnet The movement of domain walls is not completely reversible 5 Notes: Diagrams: 6 Ferromagnetic Surfaces Surfaces jj to M Surfaces ? to M Hjj is continuous B? is continuous 4 4 Bjj decreases by µr ≈ 10 H? increases by µr ≈ 10 Magnetization current JM = M × n^ No magnetization current No flux through surface Large flux through surface Example of a bar magnet with a magnetic dipole field outside it. For discussion - why is the direction of Hjj at the middle of the surface of a bar magnet opposite to M? 7 Notes: Diagrams: 8 Toroid with Ferromagnetic Core Use Amp`ere's law round a circular path at the centre of the core: H:dl = J:dS IL ZA H(2πR) = n(2πR)I H = nI B = µrµ0H If a small gap of length d is made in the core: Hcore(2πR − d) + Hgapd = n(2πR)I From boundary condition on B? at edges of gap: Bgap Bgap = Bcore = µrµ0Hcore Hgap = = µrHcore µ0 As a result of the gap Hcore is reduced but Hgap is large! (2πR)nI µr(2πR)nI Hcore = Hgap = 2πR + (µr − 1)d 2πR + (µr − 1)d 9 Notes: Diagrams: 10 Energy Stored in Toroid Magnetic energy density: dU 1 M = B:H dτ 2 Energy stored in ferromagnetic core (without gap): 2 (2πR)πa 2 2 2 2 U = BH = µ µ0π Ra n I M 2 r As a result of the gap the energy stored in the ferromagnetic core is reduced because Hcore is reduced: (2πR − d) 2 2 U = µ µ0 πa H core r 2 core ... but a lot of energy is stored in the gap! d 2 2 2 d 2 2 U = µ0 πa H = µ µ0 πa H gap 2 gap r 2 core 11 Notes: Diagrams: 12 Superconductivity Superconductivity occurs when conduction electrons in metals with wavenumber spin = k " and -k # form a Cooper pair Superconductivity breaks down above a critical temperature TC and above a critical magnetic field strength HC Type II superconductors have two transition temperatures Examples of Superconductors: Type I Metals (Al,Pb,Sn,Zn...) TC a few K BC up to 1T Type II Metal Alloys (NbTi) TC ≈ 10K BC = 15T Type II Ceramics (YBa2Cu3O7) TC ≈ 100K BC up to 300T 13 Properties of Superconductors • Perfect Conductivity No resisitivity ρ ! 0, σ ! 1 and no electric field E = 0 • Persistent Currents Any current density J is allowed J will continue to flow for ever! • Perfect Diamagnetism χM = −1, µr = 0 and no magnetic field B = 0 There is no magnetic flux inside a superconductor • Surface Magnetic Fields Can only have H tangential to surface A non-zero Hjj is associated with surface currents 14 The Meissner Effect A bar magnet levitates above the surface of a superconductor Understood using method of images: To satisfy the boundary condition H tangential to surface, the dipole field of the bar magnet has to be combined with the dipole field of an image bar magnet an equal distance behind the surface The relative orientation of the image magnet is not obvious! The lowest energy has the dipole moments parallel (not antiparallel) Force between bar magnet and image magnet is repulsive The image bar magnet is equivalent to the effect of physical surface currents that create Hjj at the superconducting surface 15 Notes: Diagrams: 16.
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