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Fundamentals of Magnetism Albrecht Jander Oregon State University Part I: H, M, B, Ο, µ Part II: M(H) Magnetic Field from Current in a Wire H dl I c dl Ampereβs Circuital Law: c André-Marie Ampère (1775β1836) r R Biot-Savart Law: οΏ½ π» β ππβ = πΌ ππ» = c π»= πΌ [A/m] 2ππ πΌπΌπβ × π οΏ½ 4π π 2 Jean-Baptiste Biot (1774 β1862) Félix Savart (1791 β 1841) Magnetic Field from Current Loop H r I Use Biot-Savart Law: ππ» = πΌπΌπβ × π οΏ½ 4π π 2 In the center: π»= πΌ [A/m] 2π Magnetic Field in a Long Solenoid H I JS I N turns L Inside, field is uniform with: π»= ππ [A/m] πΏ π» = π½π [A/m] Magnetic Field from βSmallβ Loop (Dipole) H r I Magnetic Field from Small Loop (Dipole) H R ΞΈ Compare to Electric dipole: For R >> r πΌπΌ οΏ½ sin(π) π»= 2ποΏ½ cos π + π½ 3 4ππ Magnetic (dipole) moment: π = πΌπ΄β [Am2] [A/m] Magnetic Field of a Dipole H R ΞΈ ππ For R >> r ππ π π»= 2πΜ cos π + ποΏ½sin(π) 3 4ππ Magnetic (dipole) moment: π = ππ πβ [A/m] [Am2] [AΞm] Magnetic Poles and Dipoles Electric charge Magnetic charge, or βpoleβ + qm + q π π·= 4ππ 2 Magnetic field from a magnetic βmonopoleβ ππ π»= 4ππ 2 [A/m] ππ [AΞm] Magnetic Field of a Dipole + qm d ΜΆ qm ππ [AΞm] Magnetic Field of a Dipole H R ΞΈ ππ For R >> r ππ π π»= 2πΜ cos π + ποΏ½sin(π) 3 4ππ Magnetic (dipole) moment: π = ππ πβ [A/m] [Am2] [AΞm] Orbital Magnetic Moment e- β’Electrons orbiting a nucleus are like a circulating current producing a magnetic field. β’For an electron with charge qe orbiting at a radius R with frequency f, the Orbital Magnetic Moment is m = IA = βqe f ΟR 2 β’ It also has an Orbital Angular Momentum L = me vR = me 2ΟRf R β’ Note: m β qe = = gyromagnetic ratio L 2me [Am2] Orbital Moment is Quantized Bohr model of the atom Ξ»dB R β’DeBroglie wavelength is: e- Ξ»dB = h me v β’Bohr Model: orbit must be integer number of wavelengths 2ΟR = NΞ»dB = N h h =N me v me 2ΟRf β’ Thus the orbital magnetic moment is quantized: m = β qe f ΟR 2 = N ο¨q h qe =N e 2Ο 2me 2me β’ Magnetic moment restricted to multiples of Bohr Magneton ο¨q e µB = = 9.2742 ×10 β 24 [ Am 2 ] 2me Louis de Broglie (1892-1987) Spin β’ Spin is a property of subatomic particles (just like charge or mass.) β’ A particle with spin has a magnetic dipole moment and angular momentum. β’ Spin may be thought of conceptually as arising from a spinning sphere of charge. (However, note that neutrons also have spin but no charge!) Pauli and Bohr contemplate the βspinβ of a tippy-top Electron Spin β’ When measured in a particular direction, the measured angular momentum of an electron is Lz = ± ο¨ 2 β’ When measured in a particular direction, the measured magnetic moment of an electron is ο¨q mz = s z e = ± µ B me β’ We say βspin upβ and βspin downβ β’ Note: m β qe = L me Compare: Spin Orbital mz = ± µ B m z = Nµ B m β qe = L me m β qe = L 2me g=2 β qe m =g L 2me g =1 Stern-Gerlach Experiment - 1922 Demonstrated that magnetic moment is quantized with ±µB Otto Stern (1888-1969) Walther Gerlach, Otto Stern (1922). "Das magnetische Moment des Silberatoms". Zeitschrift für Physik A Hadrons and Nuclei 9. Walter Gerlach (1889-1979) Stern-Gerlach Experiment Quantum Numbers of Electron Orbitals β’Electrons bound to a nucleus move in orbits identified by quantum numbers (from solution of Schrödinger's equation): n l ml ms 1β¦ principal quantum number, identifies the βshellβ 0..n-1 angular momentum q. #, type of orbital, e.g. s,p,d,f -l..l magnetic quantum number +½ or -½ spin quantum number For example, the electron orbiting the hydrogen nucleus (in the ground state) has: n=1, l=0, ml=0, ms=+½ In spectroscopic notation: 1s1 Spectroscopic notation l=0 s l=1 p l=2 d l=3 f n l 1s22s2 # of electrons in orbital Spin and Orbital Magnetic Moment β’ Total orbital magnetic moment (sum over all electron orbitals) mtot _ orbital = µ B β ml β’ Total spin magnetic moment mtot _ spin = 2 µ B β ms full orbitals: no net moment E.g. Iron: 1s22s22p63s23p63d64s2 How to fill remaining d orbitals (l=2) 6 electrons for 10 spots: ms=+½ ml=-2 ml=-1 ml=0 ml=+1 ml=+2 ο© ο© ο© ο© ο© οͺ ms=-½ mtot _ orbital = 2 µ B mtot _ spin = 4 µ B Hundβs rules: Maximize Ξ£ms Then maximize Ξ£ml Magnetization, M Atomic magnetic moment: πππππ What is H here? [Am2] M Object magnetic moment: ππππ. = βπππππ Atomic density: π π= π£π£π£ [1/m3] [Am2] Magnetic moment per unit volume: ππππ. πβ = πππππ π π£π£π£ [A/m] Equivalent Loop Currents πΌ Atomic magnetic moment: πππππ = Iπ΄β What is H here? M [Am2] Object magnetic moment: ππππ. = βπππππ Atomic density: π π= π£π£π£ [1/m3] [Am2] Magnetic moment per unit volume: ππππ. πβ = πππππ π π£π£π£ [A/m] Equivalent Surface Current πΌ Atomic magnetic moment: πππππ = Iπ΄β M [Am2] ποΏ½ JS Object magnetic moment: ππππ. = βπππππ Atomic density: π π= π£π£π£ [1/m3] What is H here? [A/m] [Am2] Equivalent surface current: π½βπ = π × ποΏ½ [A/m] Magnetic Pole Model Atomic magnetic moment: πππππ = ππ πβ What is H here? M [Am2] Object magnetic moment: ππππ. = βπππππ Atomic density: π π= π£π£π£ [1/m3] [Am2] Magnetic moment per unit volume: π πβ π£π£π£ [A/m] Equivalent Surface Pole Density Surface charge density: πππ Atomic magnetic moment: πππππ = ππ πβ M [Am2] ππ = ππππ Object magnetic moment: ππππ. = βπππππ Atomic density: π π= π£π£π£ [1/m3] [Am2] Magnetic pole density: πππ = π β ποΏ½ [A/m] [Am] [m2] Example: NdFeB Cylinder 3 cm A 2 cm M =106 M [A/m] vol β 14·10-6 m = 14 JS [Am2] JS =106 [m3] [A/m] I = 20000 A β 7·10-4 [A] [m2] m = IA = 14 [Am2] Example: NdFeB Cylinder 3 cm Οms + ++ + + ++ ++++ + + 2 cm M =106 M vol β 14·10-6 m = 14 Οms =106 [A/m] [Am2] [m3] [A/m] A β 7·10-4 qm = 700 [m2] [Am] d =2·10-2 [m] m = qmd = 14 [Am2] Example: NdFeB Cylinder + ++ + + ++ ++++ + + M Result is the same external to magnetic material. Hint: we are βfarβ away from the dipoles!!! Example: NdFeB Cylinder + ++ + + ++ ++++ + + M Result is different inside the magnetic material. The Constitutive Relation π© = ππ (π― + π΄) B = Magnetic flux density [ Tesla ] H = Magnetic field, βMagnetizing forceβ M = Magnetization [A/m] [A/m] µ0 = Magnetic constant, βPermeability of free spaceβ [Tesla-m/A] [Henry/m] [N/A2] Note, in vacuum, M is zero: π© = ππ π― What is µo ? µ0 comes from the SI definition of the Ampere: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2×10β7 newton per meter of length. I1 I2 F B ππ π°π π© = ππ π― = ππ π π = π°π π³ × π© Maxwellβs Equations (Magnetostatics) Differential form πβπ©=π With: π × π― = π±β π© = ππ (π― + π΄) π β π© = ππ (π β π― + π β π΄) Integral form οΏ½ π© β π π = π οΏ½ π― β π πβ = π° π β π― = βπ β π΄ = ππ [A/m2] Magnetic charge density i.e. magnetic charge/volume Example: NdFeB Cylinder + ++ + + ++ ++++ + + M B π© = ππ (π― + π΄) H Boundary Conditions B1β΄ B2β΄ The perpendicular component of B is continuous across a boundary H1|| H2|| The transverse component of H is continuous across a boundary (unless there is a true current on the boundary) Demagnetizing Fields + ++ + + ++ ++++ + + M Hd π―π β π΄ π―π = βπ΅π΅ Demagnetizing Factor Demagnetizing Factors: Special Cases π―π = βπ΅π΅ π΅π + π΅π + π΅π = π Sphere Thin Sheet Long Rod π π΅π = π΅π = π΅π = π z x y π π΅π = π΅π = π π΅π = π π΅π = π΅π = π π΅π = π Demagnetizing Factors βDemagβ fields in ellipsoids of revolution are uniform. π―π = βπ΅π΅ π΅π + π΅π + π΅π = π Example: Ba-Ferrite Sphere Barrium Ferrite: π΄~πππ M [kA/m] B H π π΅π = π΅π = π΅π = π π―π = βπ΅π΅ π―π = βππ [kA/m] π© = ππ (π― + π΄) π© = ππ (πππ [kA/m] ) π© = π. πππ [Tesla] Example: Ba-Ferrite Ellipsoid Barrium Ferrite: π΄~πππ M [kA/m] B H π΅π = π΅π = π. ππ π―π = βπ΅π΅ π―π = βππ. π [kA/m] π© = ππ (π― + π΄) π© = ππ (ππ. π [kA/m] ) π© = π. πππ [Tesla] Example: Ba-Ferrite Ellipsoid Barrium Ferrite: π΄~πππ M [kA/m] B H π΅π = π. π π―π = βπ΅π΅ π―π = βππ π© = ππ (π― + π΄) [kA/m] π© = ππ (πππ π© = π. ππ [kA/m] ) [Tesla] Susceptibility and Permeability Ideal, linear soft magnetic material Ideal permanent magnet M M H H Real, useful magnetic material M π΄ = ππ Susceptibility [unitless] π© = ππ (π― + ππ―) H π© = ππ (π + π)π― π© = ππ ππ π― Rel. Permeability [unitless] Example, Long Rod in a Long Solenoid I =1 A N/L = 1000/m ππ = 1000 π = 999 ππ = 1 [kA/m] πΏ ππ = ππ = π = 999 [kA/m] πΏ π»= π or: π΅ = π0 π» + π = π0 (1000kA/m)= 1.25 [Tesla] π΅ = π0 ππ π»= 1.25 [Tesla] Note: without the rod: π΅ = π0 π»= 0.00125 [Tesla] Example, Soft Magnetic Ball in a Field Hext M B Hd π΄ = ππ = π π―πππ + π―π = π π―πππ β π΅π΅ π π΄= π―πππ π + π΅π ππππ π = < πβΌ! π + π΅π Example, Soft Magnetic Ball in a Field π΄ = ππ = π π―πππ + π―π = π π―πππ β π΅π΅ π π΄= π―πππ π + π΅π ππππ π = < πβΌ! π + π΅π B Example, Thin Film in a Field Hext π΄ = ππ―ππππππ = π π―πππ + π―π = π π―πππ β π΄ π π΄= π―πππ π+π π―ππππππ π = π―πππ π+π π©ππππππ = π΅πππ Why? Torque and Zeeman Energy F + ++ + + ++ ++++ + + M F Force: B πΉβ = ππ π΅ Torque: π― =π×π΅ Energy: πΈ = βπ β π΅ Self Energy and Shape Anisotropy πΈ= 1 2 οΏ½ π0 π π£ππ β π»ππ M Hd Hd M Higher energy Lower energy βShape Anisotropyβ Hard 0 Hard Easy 180 ΞΈ 360 M Hard axis E Easy 2 πΈ = 12π0 ππ ππ₯ β ππ π π π 2 (π) Easy axis Magnetic Circuits path length, l οΏ½ π― β π πβ = π΅π΅ I area,A N turns Assuming flux is uniform and contained inside permeable material: π΅π΅ = ππ Magneto-motive force, βMMFβ Flux π π© = ππ ππ π― = π¨ Reluctance, π π‘= ππ Magnetic Circuits π‘π π‘π π‘π NI π‘πππ π‘π π‘π Solve using electric circuit theory. ππππ Units Confusion CGS Units SI Units B [Gauss] B : [Tesla] H [Oersted] H : [A/m] M [emu/cc] M : [A/m] 4ΟM [Gauss] π© = π― + ππ π π© β π― [dyne/cm3] π΅π + π΅π + π΅π = ππ π© = ππ π― + π΄ π© β π― [J/m3] π΅π + π΅π + π΅π = π Books on Magnetism B.D. Cullity, Introduction to Magnetic Materials, Wiley-IEEE Press, 2010. (revised version with C.D. Graham). M. Coey, Magnetism and Magnetic Materials, Cambridge University Press, 2010. S. Chikazumi, Physics of Magnetism, John Wiley and Sons, 1984. R.C. OβHandley, Modern Magnetic Materials, John Wiley & Sons, 2000. R.L. Comstock, Introduction to Magnetism and Magnetic Recording, John Wiley & Sons, 1999. D. Jiles, Introduction to Magnetism and Magnetic Materials, CRC Press, 1998. Bozorth, Ferromagnetism,1951 (reprinted by IEEE Press, 1993.)
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