A Question. What is Magnetism?

Transcription

1 A Question What is Magnetism?

2 Basic Concepts: Magnetostatics J. M. D. Coey School of Physics and CRANN, Trinity College Dublin Ireland. 1. Introduction 2. Magnets 3. Fields 4. Forces and energy 5. Units and dimensions Comments and corrections please:

3 This series of three lectures covers basic concepts in magnetism; Firstly magnetic moment, magnetization and the two magnetic fields are presented. Internal and external fields are distinguished. The main characteristics of ferromagnetic materials are briefly introduced. Magnetic energy and forces are discussed. SI units are explained, and dimensions are given for magnetic, electrical and other physical properties. Then the electronic origin of paramagnetism of non-interacting electrons is calculated in the localized and delocalized limits. The multi-electron atom is analysed, and the influence of the local crystalline environment on its paramagnetism is explained. Assumed is an elementary knowledge of solid state physics, electromagnetism and quantum mechanics.

4 Books Some useful books include: J. M. D. Coey; Magnetism and Magnetic Magnetic Materials. Cambridge University Press (2010) 614 pp An up to date, comprehensive general text on magnetism. Indispensable! Stephen Blundell Magnetism in Condensed Matter, Oxford 2001 A good treatment of the basics. D. C. Jilles An Introduction to Magnetism and Magnetic Magnetic Materials, Magnetic Sensors and Magnetometers, CRC Press 480 pp : J. D. Jackson Classical Electrodynamics 3 rd ed, Wiley, New York 1998 A classic textbook, written in SI units. G. Bertotti, Hysteresis Academic Press, San Diego 2000 A monograph on magnetostatics A. Rosencwaig Ferrohydrodynamics, Dover, Mineola 1997 A good account of magnetic energy and forces L.D. Landau and E. M. Lifschitz Electrodynamics of Continuous Media Pergammon, Oxford 1989 The definitive text

5 614 pages. Published March 2010 Available from Amazon.co.uk ~ Introduction 2 Magnetostatics 3 Magnetism of the electron 4 The many-electron atom 5 Ferromagnetism 6 Antiferromagnetism and other magnetic order 7 Micromagnetism 8 Nanoscale magnetism 9 Magnetic resonance 10 Experimental methods 11 Magnetic materials 12 Soft magnets 13 Hard magnets 14 Spin electronics and magnetic recording 15 Other topics Appendices, conversion tables.

6 1. Introduction

7 Magnets and magnetization 0.03 A m A m A m 2 3 mm 10 mm 25 mm m is the magnetic (dipole) moment of the magnet. It is proportional to volume magnetic moment m = MV magnetization Magnetization is the intrinsic property of the material; Magnetic moment is a property of a particular magnet. volume Suppose they are made of Nd 2 Fe 14 B (M 1.1 MA m -1 ) What are the moments?

8 Magnetic moment - a vector Each magnet creates a field around it. This acts on any material in the vicinity but strongly with another magnet. The magnets attract or repel depending on their mutual orientation Nd 2 Fe 14 B Weak repulsion Weak attraction Strong attraction Strong repulsion tetragonal easy axis

9 Units What do the units mean? m A m 2 m M A m -1 1A 1 A m 2 10,000 turns 10,000 A Permanent magnets win over electromagnets at small sizes m = IA area of the loop m = nia number of turns Ampère,1821. A current loop or coil is equivalent to a magnet m C! Right-hand corkscrew

10 Magnetic field H Oersted s discovery I H δl r The relation between electric current and magnetic field was discovered by Hans-Christian Øersted, I H C Right-hand corkscrew Hdl = I Ampère s law Earth s field 40 Am -1 H = I/2πr If I = 1 A, r = 1 mm H = 159 A m -1

11 Magnets and currents Ampere and Arago s insight A magnetic moment is equivalent to a current loop. Provided the current flows in a plane m = IA j m = x M In general: m = ½ r j(r) d 3 r Where j is the current density; I = j.a So m = ½ r Idl = I da = m long solenoid with n turns. m = nia Polar vector Axial vector Space inversion - j m

12 Magnetization curves - Hysteresis loop spontaneous magnetization M remanence coercivity virgin curve initial susceptibility H major loop The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to a magnetic field. It reflects the arrangement of the magnetization in ferromagnetic domains. A broad loop like this is typical of a hard or permanent magnet.

13 Another Question What is Magnetostatics? Maxwell s Equations D = ρ, B = 0, E = B/ t, H = j + D/ t. Electromagnetism with no timedependence In magnetostatics, we have only magnetic material and circulating currents in conductors, all in a steady state. The fields are produced by the magnets & the currents j = 0 B = 0 H = j.

14 2. Magnetization

15 Magnetic Moment and Magnetization The magnetic moment m is the elementary quantity in solid state magnetism. Define a local moment density - magnetization M(r, t) which fluctuates wildly on a subnanometer and a sub-nanosecond scale. Define a mesoscopic average magnetization The continuous medium approximation δm = MδV M can be the spontaneous magnetization M s within a ferromagnetic domain A macroscopic average magnetization is the domain average M = δm/δv M (r) M = Σ i M i V i / Σ i V i M s atoms The mesoscopic average magnetization

16 Magnetization and current density The magnetization of a solid is somehow related to a magnetization current density J m that produces it. Since the magnetization is created by bound currents, s J m.da = 0 over any surface. Using Stokes theorem M.dl = s ( M).dA and choosing a path of integration outside the magnetized body, we obtain s M.dA = 0, so we can identify J m J m = M We don t know the details of the magnetization currents, but we can measure the mesoscopic average magnetization and the spontaneous magnetization of a sample.

17 B and H fields in free space; permeability of free space µ 0 When illustrating Ampère s Law we labelled the magnetic field created by the current, measured in Am -1 as H. This is the magnetic field strength Maxwell s equations have another field, the magnetic flux density, labelled B, in the equation. B = 0. It is a different quantity with different units. Whenever H interacts with matter, to generate a force, an energy or an emf, a constant µ 0, the permeability of free space is involved. In free space, the relation between B and H is simple. They are numerically proportional to each other B-field Units Tesla, T B = µ 0 H Permeability of free space Units TmA -1 H-field Units A m -1 In practice you can never mix them up. The differ by almost a million! ( ) Tesla µ 0 depends on the definition of the Amp. It is precisely 4π 10-7 T m A -1

18 Field due to electric currents We need a differential form of Ampère s Law; The Biot-Savart Law δl I δb j B C Right-hand corkscrew (for the vector product)

19 Field due to a small current loop (equivalent to a magnetic moment) A B A =4δBsinε δl r I ε m B sinε=δl/2r m = I(δl) 2 A C So at a general point C, in spherical coordinates: msinθ m θ mcosθ B

20 Field due to a magnetic moment m The Earth s magnetic field is roughly that of a geocentric dipole tan I = B r /B θ = 2cotθ dr/rdθ = 2cotθ Solutions are r = c sin 2 θ m Equivalent forms

21 Magnetic field due to a moment m; Scaling m θ P r Why does magnetism lend itself to miniaturization? A H A = 2Ma 3 /4πr 3 ; H = (m/4πr 3 ){2cosθe r - sinθe θ } Just like the field of an electric dipole If a = 0.01m, r = 2a, M = 1 MA m -1 H A = 2M/16π = 40 ka m -1 A a m What is the average magnetization of the Earth? Magnet-generated fields just depend on M. They are scale-independent

22 3. Magnetic Fields

23 Magnetic flux density - B Now we discuss the fundamental field in magnetism. Magnetic poles, analogous to electric charges, do not exist. This truth is expressed in Maxwell s equation.b = 0. This means that the lines of the B-field always form complete loops; they never start or finish on magnetic charges, the way the electric E-field lines can start and finish on +ve and -ve electric charges. The same can be written in integral form over any closed surface S S B.dA = 0 (Gauss s law). The flux of B across a surface is Φ = B.dA. Units are Webers (Wb). The net flux across any closed surface is zero. B is known as the flux density; units are Teslas. (T = Wb m -2 ) Flux quantum Φ 0 = Wb (Tiny)

24 The B-field Sources of B electric currents in conductors moving charges magnetic moments time-varying electric fields. (Not in magnetostatics) In a steady state: Maxwell s equation B = µ 0 I/2πr e x e y e z / x / y / z (Stokes theorem; ) I B x B Y B Z Field at center of current loop

25 Forces between conductors; Definition of the Amp F = q(e + v x B) Lorentz expression. Note the Lorentz force does no work on the charge Gives dimensions of B and E. If E = 0 the force on a straight wire arrying a current I in a uniform field B is F = BIl The force between two parallel wires each carrying one ampere is precisely N m -1. (Definition of the Amp) The field at a distance 1 m from a wire carrying a current of 1 A is 0.2 µτ [T is equivalent to Vsm -1 ] B = µ 0 I 1 /2πr Force per meter = µ 0 I 1 I 2 /2πr

26 The range of magnitude of B Human Brain Human Heart The tesla is a very large unit Largest continuous field acheived in a lab was 45 T Interstellar Space Interplanetary Space Earth's Field at the Surface Solenoid Permanent Magnet Superconducting Magnet Hybrid Magnet Pulse Magnet Explosive Flux Compression 1E-15 1E-12 1E-9 1E-6 1E E6 1E9 1E12 1E15 pt µt T Largest continuous laboratory field 45 T (Tallahassee) MT Neutron Star TT Magnetar

27 Typical values of B Electromagnet 1 T Human brain 1 ft Magnetar 1012 T Permanent magnets 0.5 T Earth 50 µt Helmholtz coils 0.01 T Electromagnet 1 T Superconducting magnet 10 T

28 Sources of uniform magnetic fields Helmholtz coils Long solenoid B =µ 0 ni B = (4/5) 3/2 µ 0 NI/a Halbach cylinder r 2 r 1 B =µ 0 M ln(r 2 /r 1 )

29 The H-field. Ampère s law for the field is free space is x B = µ 0 ( j c + j m ) but j m cannot be measured! Only the conduction current j c is accessible. We showed on slide 16 that J m = M Hence (B/µ 0 M) = µ 0 j c We can retain Ampère s law is a usable form provided we define H = B/µ 0 M Then x H = µ 0 j c And B = µ 0 (H + M)

30 The H-field. The H-field plays a critical role in condensed matter. The state of a solid reflects the local value of H. Hysteresis loops are plotted as M(H) Unlike B, H is not solenoidal. It has sources and sinks in a magnetic material wherever the magnetization is nonuniform..h = -.M The sources of H (magnetic charge, q mag ) are distributed in the bulk with charge density -.M at the surface with surface charge density M.e n Coulomb approach to calculate H (in the absence of currents) Imagine H due to a distribution of magnetic charges q m (units Am) so that H = q m r/4πr 3 [just like electrostatics]

31 Potentials for B and H It is convenient to derive a field from a potential, by taking a spatial derivative. For example E = - ϕ e (r) where ϕ e (r) is the electric potential. Any constant ϕ 0 can be added. For B, we know from Maxwell s equations that.b = 0. There is a vector identity. X 0. Hence, we can derive B(r) from a vector potential A(r) (units Tm), B(r) = A(r) The gradient of any scalar f can be added to A (a gauge transformation) This is because of another vector identity.f 0. Generally, H(r) cannot be derived from a potential. It satisfies Maxwell s equation H = j c + D/ t. In a static situation, when there are no conduction currents present, H = 0, and H(r) = - ϕ m (r) In these special conditions, it is possible to derive H(r) from a magnetic scalar potential ϕ m (units A). We can imagine that H is derived from a distribution of magnetic charges ± q m.

32 Relation between B and H in a material The general relation between B, H and M is B = µ 0 (H + M) i.e. H = B/µ 0 - M H M B We call the H-field due to a magnet; stray field outside the magnet demagnetizing field, H d, inside the magnet

33 Boundary conditions Conditions on the fields It follows from Gauss s law S B.dA = 0 that the perpendicular component of B is continuous. (B 1 - B 2 ).e n = 0 It follows from from Ampère s law H.dl = I = 0 that the parallel component of H is continuous. since there are no conduction currents on the surface. (H 1 - H 2 ) x e n = 0 Conditions on the potentials Since S B.dA = A.dl (Stoke s theorem) (A 1 - A 2 ) x e n = 0 The scalar potential is continuous ϕ m1 = ϕ m2

34 Boundary conditions in linear, isotropic homogeneous media In LIH media, B = µ 0 µ r H Hence B 1 e n = B 2 e n H 1 e n = µ r2 /µ r1 H 2 e n So field lies ~ perpendicular to the surface of soft iron but parallel to the surface of a superconductor. Diamagnets produce weakly repulsive images. Paramagnets produce weakly attractive images. Soft ferromagnetic mirror Superconducting mirror

35 Field calculations Volume integration Integrate over volume distribution of M Sum over fields produced by each magnetic dipole element Md 3 r. Using Gives (Last term takes care of divergences at the origin)

36 Field calculations Ampèrian approach Consider bulk and surface current distributions j m = x M and j ms = M x e n Biot-Savart law gives For uniform M, the Bulk term is zero since x M = 0

37 Field calculations Coulombian approach Consider bulk and surface magnetic charge distributions ρ m = -.M and ρ ms = M.e n H field of a small charged volume element V is So δh = (ρ m r/4πr 3 ) δv For a uniform magnetic distribution the first term is zero..m = 0

38 Hysteresis loop; permanent magnet (hard ferromagnet) spontaneous magnetization M remanence coercivity virgin curve initial susceptibility H major loop A broad M(H) loop

39 Hysteresis loop; temporary magnet (soft ferromagnet) M s M Slope here is the initial susceptibility χ i > 1 H Susceptibility is defined as χ = M/H Applied field A narrow M(H) loop

40 Paramagnets and diamagnets; antiferromagnets. Only a few elements and alloys are ferromagnetic. (See the magnetic periodic table). The atomic moments in a ferromagnet order spontaneosly parallel to eachother. Most have no spontaneous magnetization, and they show only a very weak response to a magnetic field. Here χ << 1 χ is M paramagnet diamagnet H A few elements and oxides are antiferromagnetic. The atomic moments order spontaneously antiparallel to eachother. Ordered T < T c Disordered T > T c

41 1 H 1.00 Magnetic Periodic Table 2 He Li s 0 11 Na s 0 4 Be s 0 12 Mg s 0 Atomic Number Typical ionic change Antiferromagnetic T N (K) 66 Dy f Atomic symbol Atomic weight Ferromagnetic T C (K) 5 B Al p 6 6 C Si N O P S 9 F Ne Cl 18 Ar K s 0 37 Rb s 0 20 Ca s 0 38 Sr s 0 21 Sc d 0 39 Y d 0 22 Ti d 0 40 Zr d 0 23 V d 2 41 Nb d 0 24 Cr d Mo d 1 25 Mn d Tc Fe d Ru d 5 27 Co d Rh d 6 28 Ni d Pd d 8 29 Cu d 9 47 Ag d Zn d Cd d Ga d In d Ge 33 As 34 Se Sn d Sb Te Br I Kr Xe Cs s 0 56 Ba s 0 57 La f 0 72 Hf d 0 73 Ta d 0 74 W d 0 75 Re d 3 76 Os d 5 77 Ir d 5 78 Pt d 8 79 Au d Hg d Tl d Pb d Bi Po At Rn Fr Ra s 0 89 Ac f0 59 Pr 61 Pm 62 Sm f 2 58 Ce f Th f 0 91 Pa f 0 60 Nd f U f 2 93 Np f f Pu Eu f Am Gd f Cm Tb f Bk Dy f Cf Ho f Es Er f Fm Tm f Md Yb f No Lu f Lr 260 Nonmetal Diamagnet Ferromagnet T C > 290K Metal Paramagnet Antiferromagnet with T N > 290K Radioactive BOLD Magnetic atom Antiferromagnet/Ferromagnet with T N /T C < 290 K

42 Susceptibilities of the elements

43 Magnetic fields - Internal and applied fields In ALL these materials, the H-field acting inside the material is not the one you apply. These are not the same. If they were, any applied field would instantly saturate the magnetization of a ferromagnet when χ > 1. M iron Consider a thin film of iron. M s Field applied to film Field applied to film Hʹ substrate Internal field H = H + H d Demagnetizing field External field M s M s = 1.72 MAm -1

44 Demagnetizing field in a material - H d The demagnetizing field depends on the shape of the sample and the direction of magnetization. For simple uniformly-magnetized shapes (ellipsoids of revolution) the demagnetizing field is related to the magnetization by a proportionality factor N known as the demagnetizing factor. The value of N can never exceed 1, nor can it be less than 0. H d = - N M More generally, this is a tensor relation. N is then a 3 x 3 matrix, with trace 1. That is N x + N y + N z = 1 Note that the internal field H is always less than the applied field H since H = H - N M

45 Demagnetizing factor N for special shapes. Shapes N = 0 N = 1/3 N = 1/2 N = 1 H H H

46 The shape barrier N < 0.1 Daniel Bernouilli 1743 Shen Kwa 1060 S N T. Mishima 1931 N = 0.5 New icon for permanent magnets! Gowind Knight 1760

47 The shape barrier ovecome! H d = - N M working point M s M (Am -1 ) M r H d H c H d H (Am -1 )

48 Demagnetizing factors N I for a general ellipsoid. Calculate from analytial formulae N x + N y + N z = 1

49 Other shapes cannot be unoformly magnetized.. When measuring the magnetization of a sample H is always taken as the independent variable, M = M(H).

50 Paramagnetic and ferromagnetic responses Susceptibility of linear, isotropic and homogeneous (LIH) materials M = χ H χ is external susceptibility (no units) It follows that from H = H + H d that 1/χ = 1/χ - N Typical paramagnets and diamagnets: χ χ (10-5 to 10-3 ) Paramagnets close to the Curie point and ferromagnets: χ >> χ χ diverges as T T C but χ never exceeds 1/N. M M M H M s /3 Ms /3 H H H' H

51 Demagnetizing correction to M(H) and B(H) loops. N = 0 N = ½ N = 1 The B(H) loop is deduced from M(H) using B = µ 0 (H + M).

52 Permeability In LIH media B =µ H defines the permeability µ Units: TA -1 m Relative permeability is defined as µ r = µ / µ 0 B = µ 0 (H + M) gives µ r = 1 + χ

53 4. Energy and forces

54 Energy of ferromagnetic bodies Torque and potential energy of a dipole in a field, assumed to be constant. Force Γ = m x B Γ = mb sinθ. ε = -m.b ε = -mb cosθ. B θ m In a non-uniform field, f = - ε m f = m. B Magnetostatic (dipole-dipole) forces are long-ranged, but weak. They determine the magnetic microstructure (domains). ½µ 0 H 2 is the energy density associated with a magnetic field H M 1 MA m -1, µ 0 H d 1 T, hence µ 0 H d M 106 J m -3 Products B.H, B.M, µ 0 H 2, µ 0 M 2 are all energies per unit volume. Magnetic forces do no work on moving charges f = q(v x B) [Lorentz force] No potential energy associated with the magnetic force.

55 Reciprocity theorem The interaction of a pair of dipoles, ε p, can be considered as the energy of m 1 in the field B 21 created by m 2 at r 1 or vice versa. ε p = -m 1.B 21 = -m 2.B 12 So ε p = -(1/2)(m 1.B 21 + m 2.B 12 ) Extending to magnetization distributions: ε = -µ 0 M 1.H 2 d 3 r = -µ 0 M 2.H 1 d 3 r

56 Self Energy The interaction of the body with the field it creates. H d. Consider the energy to bring a small moment δm into position within the magnetized body H loc =H d +(1/3)M δε = - µ 0 δm.h loc Integration over the whole sample gives The magnetostatic self energy is defined as Or equivalently, using B = µ 0 (H + M) and B.H d d 3 r=0 For a uniformly magnetized ellipsoid

57 Energy associated with a field General expression for the energy associated with a magnetic field distribution Aim to maximize energy associated with the field created around the magnet, from previous slide: Can rewrite as: where we want to maximize the integral on the left. Energy product: twice the energy stored in the stray field of the magnet -µ 0 i B.H d d 3 r

58 Work done by an external field Elemental work δw to produce a flux change δφ is I δφ Ampere: H.dl = I So δw = δφ H.dl So in general: δw = δb.h.d 3 r H = H + H d B = µ 0 (H + M) Subtract the term associated with the H-field in empty space, to give the work done on the body by the external field; gives

59 Thermodynamics First law: du = H x dx + dq (U,Q, F, G are in units of Jm -3 ) dq = TdS Four thermodynamic potentials; U(X,S) M ΔG E(H X,S) F(X,T) = U - TS df = HdX - SdT ΔF G(H X,T) = F- H X X dg = -XdH - SdT Magnetic work is HδB or µ 0 H δm df = µ 0 H dm - SdT dg = -µ 0 MdH - SdT S = -( G/ T) H µ 0 M = -( G/ H ) T Maxwell relations H ( S/ H ) T = - µ 0 ( M/ T) H etc.

60 Magnetostatic forces Force density on a magnetized body at constant temperature F m = - G Kelvin force General expression, for when M is dependent on H is V =1/d d is the density

61 Anisotropy - shape θ The energy of the system increases when the magnetization deviates by an angle θ from its easy axis. E = K sin 2 θ The anisotropy may be due to shape. The energy of the magnet in the demagnetizing field is -(1/2)µ 0 mh d For shape anisotropy K sh = (1/4)µ 0 M 2 (1-3N ) For example, if M = 1 MA m -1, N = 1, K sh = 630 kj m -3 e.g. if a thin film is to have its magnetization perpendicular to the plane, there must be a stronger intrinsic magnetocrystalline anisotropy K 1.

62 Polarization and anisotropy field Engineers often quote the polarization of a magnetic material, rather than its magnetization. The relation is simple; J = µ 0 M. For example, the polarization of iron is 2.15 T; Its magnetization is 1.71 MA m -1 The defining relation is then: B = µ 0 H + J The anisotropy of whatever origin may be represented by an effective magnetic field, the anisotropy field H a Ha = 2K/µ0 Ms The coercivity can never exceed the anisotropy field. In practice it is rarely possible to obtain coercivity greater than about 25 % of H a. Coercivity depends on the microsctructure, and the ability to impede fprmation of reversed domains Sometimes anisotropy field is quoted in Tesla. B a = µ 0 H a.

63 Some expressions involving B F = q(e + v B) Force on a charged particle q F = B ll Force on current-carrying wire E = - dφ/dt Faraday s law of electromagnetic induction E = -m.b Energy of a magnetic moment F = m.b Γ = m x B Force on a magnetic moment Torque on a magnetic moment

64 5. Units

65 A note on units: Magnetism is an experimental science, and it is important to be able to calculate numerical values of the physical quantities involved. There is a strong case to use SI consistently Ø SI units relate to the practical units of electricity measured on the multimeter and the oscilloscope Ø It is possible to check the dimensions of any expression by inspection. Ø They are almost universally used in teaching Ø Units of B, H, Φ or dφ/dt have been introduced. BUT Most literature still uses cgs units, You need to understand them too.

66 SI / cgs conversions: m M σ H B Φ dφ/dt χ SI units B = μ 0 (H + M) A m 2 A m -1 (10-3 emu cc -1 ) A m 2 kg -1 (1 emu g -1 ) A m -1 (4π/ Oe) Tesla (10000 G) Weber (Tm 2 ) (10 8 Mw) V (10 8 Mw s -1 ) - (4π cgs) cgs units B = H + 4πM emu emu cc -1 (1 k A m -1 ) emu g -1 (1 A m 2 kg -1 ) Oersted (1000/4π 80 A m -1 ) Gauss (10-4 T) Maxwell (G cm 2 ) (10-8 Wb) Mw s -1 (10 nv) - (1/4π SI)

67 Mechanical Quantity Symbol Unit m l t i θ Area A m Volume V m Velocity v ms Acceleration a ms Density d kg m Energy ε J Momentum p kg m s Angular momentum L kg m 2 s Moment of inertia I kg m Force f N Force density F Nm Power P W Pressure P Pa Stress σ Nm Elastic modulus K Nm Frequency f s Diffusion coefficient D m 2 s Viscosity (dynamic) η Nsm Viscosity ν m 2 s Planck s constant Js

68 Electrical Quantity Symbol Unit m l t i θ Current I A Current density j Am Charge q C Potential V V Electromotive force E V Capacitance C F Resistance R Resistivity ϱ m Conductivity σ Sm Dipole moment p Cm Electric polarization P Cm Electric field E Vm Electric displacement D Cm Electric flux C Permittivity ε Fm Thermopower S VK Mobility µ m 2 V 1 s

69 Magnetic Quantity Symbol Unit m l t i θ Magnetic moment m Am Magnetization M Am Specific moment σ Am 2 kg Magnetic field strength H Am Magnetic flux Wb Magnetic flux density B T Inductance L H Susceptibility (M/H) χ Permeability (B/H) µ Hm Magnetic polarization J T Magnetomotive force F A Magnetic charge q m Am Energy product (BH) Jm Anisotropy energy K Jm Exchange stiffness A Jm Hall coefficient R H m 3 C Scalar potential ϕ A Vector potential A Tm Permeance P m Tm 2 A Reluctance R m AT 1 m

70 Thermal Quantity Symbol Unit m l t i θ Enthalpy H J Entropy S JK Specific heat C JK 1 kg Heat capacity c JK Thermal conductivity κ Wm 1 K Sommerfeld coefficient γ Jmol 1 K Boltzmann s constant k B JK (1) Kinetic energy of a body: ε = 1 2 mv2 [ε] = [1, 2, 2, 0, 0] [m] = [1, 0, 0, 0, 0] [v 2 ] = 2[0, 1, 1, 0, 0] [1, 2, 2, 0, 0] (2) Lorentz force on a moving charge; f = qv B [f ] = [1, 1, 2, 0, 0] [q] = [0, 0, 1, 1, 0] [v] = [0, 1, 1, 0, 0] [1, 0, 2, 1, 0] [B] = [1, 1, 2, 0, 0] (3) Domain wall energy γ w = AK (γ w is an energy per unit area) [γ w ] = [εa 1 ] [ AK] = 1/2[AK] = [1, 2, 2, 0, 0] [ A] = 1 [1, 1, 2, 0, 0] 2 [ 1,1, 2, 0, 0] [ K] = 1 [1, 1, 2, 0, 0] - [0, 0, 0, 0 ] 2 [1, 0, 2, 0, 0] = [1, 0, 2, 0, 0] (4) Magnetohydrodynamic force on a moving conductor F = B B

71 = (4) Magnetohydrodynamic force on a moving conductor F = σ v B B (F is a force per unit volume) [F ] = [FV 1 ] [σ ] = [ 1, 3, 3, 2, 0] = [1, 1, 2, 0, 0] [v] = [0, 1, 1, 0, 0] [0, 3, 0, 0, 0] [B 2 2[1, 0, 2, 1, 0] ] = [1, 2, 2, 0, 0] [1, 2, 2, 0, 0] (5) Flux density in a solid B = µ 0 (H + M) (note that quantities added or subtracted in a bracket must have the same dimensions) [B] = [1, 0, 2, 1, 0] [µ 0 ] = [1, 1, 2, 2, 0] [0, 1, 0, 1, 0] [M], [H ] = [1, 0, 2, 1, 0] (6) Maxwell s equation H = j + dd/dt. [ H] = [Hr 1 ] [j] = [0, 2, 0, 1, 0] [dd/dt] = [Dt 1 ] = [0, 1, 0, 1, 0] = [0, 2, 1, 1, 0] [0,1,0,0,0] [0, 0, 1, 0, 0] = [0, 2, 0, 1, 0] = [0, 2, 0, 1, 0] (7) Ohm s Law V = IR = [1, 2, 3, 1, 0] [0, 0, 0, 1, 0] + [1, 2, 3, 2, 0] = [1, 2, 3, 1, 0] (8) Faraday s Law E = / t = [1, 2, 3, 1, 0] [1, 2, 2, 1, 0] [0, 0, 1, 0, 0] = [1, 2, 3, 1, 0]