Electromagnetism - Lecture 12. Ferromagnetism & Superconductivity

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1 Electromagnetism - Lecture 12 Ferromagnetism & Superconductivity Ferromagnetism Hysteresis & Permanent Magnets Ferromagnetic Surfaces Toroid with Ferromagnetic Core Superconductivity The Meissner Effect 1

2 Ferromagnetism Ferromagnetism occurs when the spins of conduction electrons in metals spontaneously align Caused by an exchange interaction U S 1.S 2 The spontaneous alignment breaks down above a critical temperature, the Curie point T = T C For T > T C the metal is paramagnetic: χ M = C T T C T C = λc B L = λm where B L is the local field due to the spin-spin interactions For T < T C the metal is ferromagnetic with a large magnetization 2

3 Magnetic Saturation When all the spins are aligned the magnetization is saturated M S = 2N e µ B e 2λN eµ 2 B /kt 0 < T < T C Examples of ferromagnetic materials: Iron (Fe) T C = 1043K B S = 1.7T Cobalt (Co) T C = 1288K B S = 1.4T Nickel (Ni) T C = 627K B S = 0.5T The relative permeability µ r = B/µ 0 H of ferromagnets is very large and has a wide range of values µ r =

4 Notes: Diagrams: 4

5 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 = µ 0 (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

6 Notes: Diagrams: 6

7 Ferromagnetic Surfaces Surfaces to M Surfaces to M H is continuous B is continuous B decreases by µ r 10 4 H increases by µ r 10 4 Magnetization current J M = M ˆn 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 H at the middle of the surface of a bar magnet opposite to M? 7

8 Notes: Diagrams: 8

9 Toroid with Ferromagnetic Core Use Ampère s law round a circular path at the centre of the core: H.dl = J.dS L H(2πR) = n(2πr)i H = ni B = µ r µ 0 H If a small gap of length d is made in the core: H core (2πR d) + H gap d = n(2πr)i From boundary condition on B at edges of gap: B gap = B core = µ r µ 0 H core A H gap = B gap µ 0 = µ r H core As a result of the gap H core is reduced but H gap is large! H core = (2πR)nI 2πR + (µ r 1)d H gap = µ r (2πR)nI 2πR + (µ r 1)d 9

10 Notes: Diagrams: 10

11 Energy Stored in Toroid Magnetic energy density: du M dτ = 1 2 B.H Energy stored in ferromagnetic core (without gap): U M = (2πR)πa2 2 BH = µ r µ 0 π 2 Ra 2 n 2 I 2 As a result of the gap the energy stored in the ferromagnetic core is reduced because H core is reduced: (2πR d) U core = µ r µ 0 πa 2 Hcore but a lot of energy is stored in the gap! U gap = µ 0 d 2 πa2 H 2 gap = µ 2 rµ 0 d 2 πa2 H 2 core 11

12 Notes: Diagrams: 12

13 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 T C and above a critical magnetic field strength H C Type II superconductors have two transition temperatures Examples of Superconductors: Type I Metals (Al,Pb,Sn,Zn...) T C a few K B C up to 1T Type II Metal Alloys (NbTi) T C 10K B C = 15T Type II Ceramics (YBa 2 Cu 3 O 7 ) T C 100K B C up to 300T 13

14 Properties of Superconductors Perfect Conductivity No resisitivity ρ 0, σ and no electric field E = 0 Persistent Currents Any current density J is allowed J will continue to flow for ever! Perfect Diamagnetism χ M = 1, µ r = 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 H is associated with surface currents 14

15 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 H at the superconducting surface 15

16 Notes: Diagrams: 16

Electromagnetism - Lecture 10. Magnetic Materials

Electromagnetism - Lecture 10. Magnetic Materials Electromagnetism - Lecture 10 Magnetic Materials Magnetization Vector M Magnetic Field Vectors B and H Magnetic Susceptibility & Relative Permeability Diamagnetism Paramagnetism Effects of Magnetic Materials