Fundamentals of Magnetism Albrecht Jander Oregon State University Part I: H, M, B, χ, µ Part II: M(H) Magnetic Field from Current in a Wire H dl I r R c dl Ampere’s Circuital Law: Biot-Savart Law: × Jean-Baptiste Biot = = (1774 –1862) c 퐼4푑푙⃗ 푅� c � 퐻 ∙ 푑푙⃗ 퐼 푑퐻 2 휋 푅 = [A/m] 퐼 André-Marie Ampère 2휋휋 Félix Savart (1775–1836) 퐻 (1791 – 1841) Magnetic Field from Current Loop H r I Use Biot-Savart Law: In the center: × = = [A/m] 퐼4푑푙⃗ 푅� 푑퐻 2 퐼 휋 푅 퐻 2휋 Magnetic Field in a Long Solenoid H I JS I N turns L Inside, field is uniform with: = [A/m] = [A/m] 푁퐼 푆 퐻 퐿 퐻 퐽 Magnetic Field from “Small” Loop (Dipole) H r I Magnetic Field from Small Loop (Dipole) H R θ Compare to Electric dipole: For R >> r = 2 cos + sin ( ) [A/m] 4 퐼퐼 퐻 3 풓� 휃 휽� 휃 Magnetic (dipole)휋 moment:푅 = [Am2] 푚 퐼퐼⃗ Magnetic Field of a Dipole H R θ [A·m] For R >> r 푞푚 = 2 cos + sin ( ) [A/m] 4 푞푚푑 Magnetic퐻 (dipole) moment:3 푟̂ 휃 휃� 휃 휋푅 = [Am2] 푚 푞푚푑⃗ Magnetic Poles and Dipoles Electric charge Magnetic charge, or “pole” + q + qm = 4 푞 [A·m] 퐷 2 휋푅 Magnetic field from a magnetic “monopole” 푞푚 = [A/m] 4 푞푚 퐻 2 휋푅 Magnetic Field of a Dipole + qm d ̶ qm [A·m] 푞푚 Magnetic Field of a Dipole H R θ [A·m] For R >> r 푞푚 = 2 cos + sin ( ) [A/m] 4 푞푚푑 Magnetic퐻 (dipole) moment:3 푟̂ 휃 휃� 휃 휋푅 = [Am2] 푚 푞푚푑⃗ Orbital Magnetic Moment •Electrons orbiting a nucleus are like a circulating current producing e- a magnetic field. •For an electron with charge qe orbiting at a radius R with frequency f, the Orbital Magnetic Moment is 2 2 m = IA = −qe f πR [Am ] • It also has an Orbital Angular Momentum L = mevR = me 2πRf R • Note: m − q = e = gyromagnetic ratio L 2me Orbital Moment is Quantized Bohr model of the atom λdB •DeBroglie wavelength is: e- h R λdB = mev •Bohr Model: orbit must be integer number of wavelengths h h 2πR = NλdB = N = N mev me 2πRf • Thus the orbital magnetic moment is quantized: 2 h qe qe m = −qe f πR = N = N 2π 2me 2me • Magnetic moment restricted to multiples of Bohr Magneton qe −24 2 µB = = 9.2742×10 [Am ] Louis de Broglie (1892-1987) 2me Spin • Spin is a property of subatomic particles (just like charge or mass.) • A particle with spin has a magnetic dipole moment and angular momentum. Pauli and Bohr contemplate • Spin may be thought of as conceptually the “spin” of a tippy-top arising from a spinning sphere of charge. (However, note that neutrons also have spin but no charge!) Electron Spin • When measured in a particular direction, the measured angular momentum of an electron is L = ± z 2 • When measured in a particular direction, the measured magnetic moment of an electron is Compare: qe mz = sz = ±µB Spin Orbital me mz = ±µB mz = NµB • We say “spin up” and “spin down” m − q m − q = e = e • Note: m − q L me L 2me = e L me m − q g = 2 = g e g =1 L 2me 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 Walter Gerlach Silberatoms". Zeitschrift für Physik A Hadrons and Nuclei 9. (1889-1979) Stern-Gerlach Experiment Quantum Numbers of Electron Orbitals •Electrons bound to a nucleus move in orbits identified by quantum numbers (from solution of Schrödinger's equation): n 1… principal quantum number, identifies the “shell” l 0..n-1 angular momentum q. #, type of orbital, e.g. s,p,d,f ml -l..l magnetic quantum number ms +½ or -½ spin quantum number For example, the electron orbiting the Spectroscopic notation hydrogen nucleus (in the ground state) has: l=0 s l=1 p n=1, l=0, m =0, m =+½ l s l=2 d l=3 f In spectroscopic notation: 1s1 n l # of electrons in orbital 1s22s2 Spin and Orbital Magnetic Moment • Total orbital magnetic moment (sum over all electron orbitals) mtot _ orbital = µB ∑ ml • Total spin magnetic moment mtot _ spin = 2µB ∑ ms full orbitals: no net moment E.g. Iron: 1s22s22p63s23p63d64s2 How to fill remaining d orbitals (l=2) 6 electrons for 10 spots: ml=-2 ml=-1 ml=0 ml=+1 ml=+2 Hund’s rules: ms=+½ Maximize Σms ms=-½ Then maximize Σml m = 2µ m = 4µ tot _ orbital B tot _ spin B Magnetization, M Atomic magnetic moment: What is H here? [Am2] M Object푚 magnetic푎푎푣푚 moment: 2 . = [Am ] Magnetic moment per unit volume: Atomic푚표표표 density:∑ 푚푎푎표푚 . [1/m3] = [A/m] = 푁 푚표표표 푑 푀 ≝ 푚푎푎표푚푑 푣푣푙 푣푣푙 Equivalent Loop Currents Atomic magnetic moment: 퐼 What is H here? [Am2] = I M Object푚푎푎 magnetic푣푚 moment:퐼⃗ 2 . = [Am ] Magnetic moment per unit volume: Atomic푚표표표 density:∑ 푚푎푎표푚 . [1/m3] = [A/m] = 푁 푚표표표 푑 푀 ≝ 푚푎푎표푚푑 푣푣푙 푣푣푙 Equivalent Surface Current Atomic magnetic moment: 퐼 What is H here? = I [Am2] M 푛� [A/m] 푎푎푣푚 J Object푚 magnetic moment:퐼⃗ S 2 . = [Am ] Equivalent surface current: Atomic푚표표표 density:∑ 푚푎푎표푚 = × [A/m] [1/m3] = 푁 퐽⃗푆 푀 푛� 푑 푣푣푙 Magnetic Pole Model Atomic magnetic moment: What is H here? 2 = [Am ] M Object푚푎푎 magnetic푣푚 푞 푚moment:푑⃗ 2 . = [Am ] Magnetic moment per unit volume: Atomic푚표표표 density:∑ 푚푎푎표푚 [A/m] = [1/m3] 푚 푁 푀 ≝ 푑 푣푣푙 푣푣푙 Equivalent Surface Pole Density Surface charge density: [Am] = [m2] 푞푚 Atomic magnetic moment: 휌푚푚 2 푎푟푎푎 = [Am ] M Object푚푎푎 magnetic푣푚 푞 푚moment:푑⃗ 2 . = [Am ] Magnetic pole density: Atomic푚표표표 density:∑ 푚푎푎표푚 = [A/m] [1/m3] = 푁 휌푚푚 푀 ∙ 푛� 푑 푣푣푙 Example: NdFeB Cylinder 3 cm A 2 cm M JS 6 M =10 [A/m] 6 [A/m] JS =10 -6 [m3] [A] vol 14·10 I = 20000 -4 [m2] 2 A 7·10 m =≃ 14 [Am ] m = IA = 14 [Am2] ≃ Example: NdFeB Cylinder 3 cm + + ++ + + ρ + + ++ + + + ms 2 cm M 6 [A/m] 6 M =10 ρms =10 [A/m] -6 [m3] -4 [m2] vol 14·10 A 7·10 [Am] 2 q = 700 m =≃ 14 [Am ] m≃ -2 [m] d =2·10 [Am2] m = qmd = 14 Example: NdFeB Cylinder + + + +++ + + + + + + + M Result is the same external to magnetic material. Hint: we are “far” away from the dipoles!!! Example: NdFeB Cylinder + + + +++ + + + + + + + M Result is different inside the magnetic material. The Constitutive Relation = ( + ) 푩 흁ퟎ 푯 푴 B = Magnetic flux density [ Tesla ] H = Magnetic field, “Magnetizing force” [A/m] M = Magnetization [A/m] 2 µ0 = Magnetic constant, “Permeability of free space” [Tesla-m/A] [Henry/m] [N/A ] Note, in vacuum, M is zero: = 푩 흁ퟎ푯 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 2×10−7 newton per meter of length. I 1 = = I 2 흁ퟎ푰ퟏ 푩 흁ퟎ푯 F = × ퟐ흅풓 B 푭 푰ퟐ푳 푩 Maxwell’s Equations (Magnetostatics) Differential form Integral form = = 훁 ∙ 푩 ퟎ � 푩 ∙ 풅풔 ퟎ × = = 훁 푯 푱⃗ � 푯 ∙ 풅풍⃗ 푰 With: = ( + ) ퟎ 푩 흁= 푯 ( 푴 + ) = = [A/m2] 훁 ∙ 푩 흁ퟎ 훁 ∙ 푯 훁 ∙ 푴 훁 ∙ 푯 −훁 ∙ 푴 흆풎 Magnetic charge density i.e. magnetic charge/volume Example: NdFeB Cylinder + + ++ + + + + ++ + + + M B H = ( + ) 푩 흁ퟎ 푯 푴 Boundary Conditions B1┴ B2┴ H1|| H2|| The perpendicular component of B The transverse component of H is is continuous across a boundary continuous across a boundary (unless there is a true current on the boundary) Demagnetizing Fields + + ++ + + + + ++ + + + M Hd 푯풅 ∝ 푴 = 푯풅 −푵푴 Demagnetizing Factor Demagnetizing Factors: Special Cases = + + = 푯풅 −푵푴 푵풙 푵풚 푵풛 ퟏ Sphere Thin Sheet Long Rod = = = = = ퟏ = 푵풙 푵풚 푵풛 푵풙 푵풚 ퟎ z ퟑ 풛 = = 푵 ퟏ x ퟏ 푵풙 푵=풚 y ퟐ 푵풛 ퟎ Demagnetizing Factors “Demag” fields in ellipsoids of revolution are uniform. = 푯풅 + −푵+푴 = 푵풙 푵풚 푵풛 ퟏ Example: Ba-Ferrite Sphere Barrium Ferrite: ~ [kA/m] 푴 ퟏퟏퟎ M H B = = = = ( + ) 풙 풚 풛 ퟏ 푵 푵 푵 푩 = 흁ퟎ(푯 푴[kA/m] ) = ퟑ ퟎ 풅 푩 = 흁. ퟏퟎퟎ [Tesla] 푯 =−푵푴 [kA/m] 푯풅 −ퟏퟎ 푩 ퟎ ퟏퟐퟏ Example: Ba-Ferrite Ellipsoid Barrium Ferrite: ~ [kA/m] 푴 ퟏퟏퟎ M H B = = . = ( + ) 푵풙 푵풚 ퟎ ퟒퟏ = ퟎ( . [kA/m] ) = 푩 흁 푯 푴 ퟎ 풅 푩 = 흁. ퟖퟐ ퟏ[Tesla] 푯 =−푵푴. [kA/m] 푯풅 −ퟏퟔ ퟏ 푩 ퟎ ퟏퟎퟒ Example: Ba-Ferrite Ellipsoid Barrium Ferrite: ~ [kA/m] 푴 ퟏퟏퟎ M H B = . = ( + ) 푵풛 ퟎ ퟏ = ퟎ( [kA/m] ) = 푩 흁 푯 푴 ퟎ 풅 푩 = 흁. ퟏퟑퟏ [Tesla] 푯 =−푵푴 [kA/m] 푯풅 −ퟏퟏ 푩 ퟎ ퟏퟔ Susceptibility and Permeability Ideal, linear soft magnetic material Ideal permanent magnet M M H H = Real, useful magnetic material Susceptibility [unitless] M 푴 흌푯 = ( + ) 푩 = 흁ퟎ(푯+ 흌)푯 H 푩 = 흁ퟎ ퟏ 흌 푯 Rel. Permeability 푩 흁ퟎ흁풓푯 [unitless] Example, Long Rod in a Long Solenoid I =1 A = 1000 = 999 N/L = 1000/m 휇휋 휒 = = 1 [kA/m] 푁퐼 = = = 퐻 퐿 999 [kA/m] 푁퐼 푀= 휒퐻 +휒 퐿 = (1000kA/m)= 1.25 [Tesla] or: 0 퐵 휇0 퐻 푀 휇 = = 1.25 [Tesla] Note: without the rod: = = 0.00125 [Tesla] 퐵 휇0휇휋퐻 퐵 휇0퐻 Example, Soft Magnetic Ball in a Field Hext M Hd B = = + = 푴 흌푯 흌 푯풆풙풆 푯풅 흌 푯풆풙풆 − 푵푴 = = < ! + + 흌 흌 푴 푯풆풙풆 흌풆풆풆 ퟑ‼ ퟏ 푵흌 ퟏ 푵흌 Example, Soft Magnetic Ball in a Field B = = + = 푴 흌푯 흌 푯풆풙풆 푯풅 흌 푯풆풙풆 − 푵푴 = = < ! + + 흌 흌 푴 푯풆풙풆 흌풆풆풆 ퟑ‼ ퟏ 푵흌 ퟏ 푵흌 Example, Thin Film in a Field Hext = = + = 푴 흌푯풊풊풔풊 흌 푯풆풙풆 푯풅 흌 푯풆풙풆 − 푴 = + 흌 푴 푯풆풙풆 ퟏ 흌 = = + ퟏ 푯풊풊풔풊 푯풆풙풆 푩풊풊풔풊 퐵Why?풆풙풆 ퟏ 흌 Torque and Zeeman Energy F + + + +++ + + Force: + + + + + = B M Torque: 퐹⃗ 푞푚퐵 = × F Energy: 풯 푚 퐵 = 퐸 −푚 ∙ 퐵 Self Energy and Shape Anisotropy M = Hd M 1 Hd 2 0 퐸 � 휇 푀 ∙ 퐻푑푣 Higher energy Lower energy 푣표표 “Shape Anisotropy” = ( ) E 2 Hard Hard 1 2 퐸 2휇0푀푚 푁푥 − 푁푌 푠푠푛 휃 Easy axis Easy Easy M θ 0 180 360 Hard axis Magnetic Circuits path length, l I = area,A N turns � 푯 ∙ 풅풍⃗ 푵푰 Assuming flux is uniform and contained inside permeable material: = = 흓 = 푩 흁ퟎ흁풓푯 푨 Magneto-motive force, “MMF”푵푰 흓흓 Reluctance, = 풍 Flux 흓 흁푨 Magnetic Circuits 흓ퟏ 흓ퟐ NI ퟑ 품품품 흓 흓품품품 흓 흓ퟒ 흓ퟏ Solve using electric circuit theory.