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Magnetic Dipole Moment Magnetic Dipole Moment 2 R2 I 1 2 p far from coil: µ0 π far from dipole: E = Bz = 3 z 3 4π z 4πε0 z µ0 2µ p = sq Bz = 3 magnetic 4π z dipole moment: µ = πR2 I = AI µ - vector in the direction of B Twisting of a Magnetic Dipole The magnetic dipole moment µ acts like a compass needle! In the presence of external magnetic field a current-carrying loop rotates to align the magnetic dipole moment µ along the field B. The Magnetic Field of a Bar Magnet How does the magnetic field around a bar magnet look like? N S Magnets and Matter How do magnets interact with each other? Magnets interact with iron or steel, nickel, cobalt. Does it interact with charged tape? Does it work through matter? Does superposition principle hold? Similarities with E-field: • can repel or attract • superposition • works through matter Differences with E-field: • B-field only interacts with some objects • curly pattern • only closed field lines Magnetic Field of Earth The magnetic field of the earth has a pattern that looks like that of a bar magnet Horizontal component of magnetic field depends on latitude Maine: ~1.5.10-5 T Indiana: ~2.10-5 T Florida: ~3.10-5 T Can use magnetic field of Earth as a reference to determine unknown field. Magnetic Monopoles An electric dipole consists of two opposite charges – monopoles Break magnet: S S N N There are no magnetic monopoles! The Atomic Structure of Magnets The magnetic field of a current loop and the magnetic field of a bar magnet look the same. Electrons µ 2µ B = 0 , µ = π R2 I atom 4π z3 What is the direction? One loop: What is the average current I? e N S current=charge/second: I = t 2π R ev T = I = v 2πR ev 1 µ = πR2 = eRv 2πR 2 Magnetic Dipole Moment 1 Magnetic dipole moment of 1 atom: µ = eRv 2 Method 1: use quantized angular momentum Orbital angular momentum: L = Rmv 1 1 e 1 e µ = eRv = Rmv = L 2 2 m 2 m Quantum mechanics: L is quantized: L = n, = 1.05×10−34 J⋅s 1 e If n=1: µ = L = 0.9 × 10−23 A ⋅ m2 per atom 2 m Magnetic Dipole Moment 1 Magnetic dipole moment of 1 atom: µ = eRv 2 Method 2: estimate speed of electron dp Momentum principle: = F dt net Circular motion: p = p = const dp v = ω p = mv = Fnet dt R ω – angular speed 2 2 mv 1 e ω = v / R = 2 R 4πε0 R 1 e2 v = ≈ 1.6 ×106 m/s µ ≈ 1.3 × 10−23 A ⋅ m2 /atom 4πε0 Rm Magnetic Dipole Moment Magnetic dipole moment of 1 atom: µ ≈ 10−23 A ⋅m2/atom Mass of a magnet: m~5g 6.1023 atoms Assume magnet is made of iron: 1 mole – 56 g number of atoms = 5g/56g . 6.1023 ~ 6.1022 22 −23 2 µmagnet ≈ 6 × 10 ⋅10 = 0.6 A ⋅ m Modern Theory of Magnets 1. Orbital motion There is no ‘motion’, but a distribution Spherically symmetric cloud (s-orbital) has no µ Only non spherically symmetric orbitals (p, d, f) contribute to µ There is more than 1 electron in an atom Modern Theory of Magnets Alignment of atomic dipole moments: ferromagnetic materials: most materials iron, cobalt, nickel Modern Theory of Magnets 2. Spin Electron acts like spinning charge - contributes to µ Electron spin contribution to µ is of the same order as one due to orbital momentum Neutrons and proton in nucleus also have spin but their µ‘s are much smaller than for electron 1 e same angular momentum: µ ≈ 2 m NMR, MRI – use nuclear µ Nuclear Magnetic Resonance Proton spin Magnet N S Felix Bloch Edward Purcell (1905 -1983) (1912-1997) B field Magnetic Resonance Imaging B Modern Theory of Magnets Magnetic domains Very pure iron – no residual magnetism spontaneously disorders Hitting or heating can also demagnetize Why are there Multiple Domains? Magnetic domains Iron Inside a Coil Multiplier effect: Bnet = Bcoil + Biron Bnet > Bcoil Electromagnet: Magnetic Field of a Solenoid Step 1: Cut up the distribution into pieces Step 2: Contribution of one piece origin: center of the solenoid µ 2π R2 I B = 0 one loop: z 2 3/2 4π R2 + (d − z) ( ) B Number of loops per meter: N/L Number of loops in Δz: (N/L) Δz µ 2π R2 I N Field due to Δz: ΔB = 0 Δz z 2 3/2 4π (R2 + (d − z) ) L Magnetic Field of a Solenoid Step 3: Add up the contribution of all the pieces µ 2π R2 I N dB = 0 dz z 2 3/2 4π (R2 + (d − z) ) L 2 L /2 µ0 2π R NI dz Bz = 3/2 4π L ∫ 2 2 − L /2 (R + (d − z) ) B Magnetic field of a solenoid: ⎡ ⎤ µ0 2π NI d + L / 2 d − L / 2 Bz = ⎢ − ⎥ 4π L ⎢ 2 2 2 2 ⎥ ⎣ (d + L / 2) + R (d − L / 2) + R ⎦ Magnetic Field of a Solenoid ⎡ ⎤ µ0 2π NI d + L / 2 d − L / 2 Bz = ⎢ − ⎥ 4π L ⎢ 2 2 2 2 ⎥ ⎣ (d + L / 2) + R (d − L / 2) + R ⎦ Special case: R<<L, center of the solenoid: µ 2π NI ⎡ L / 2 −L / 2 ⎤ µ 2π NI 0 ⎢ ⎥ 0 Bz ≈ − = [2] 4π L ⎢ 2 2 ⎥ 4π L ⎣ (L / 2) (L / 2) ⎦ µ NI B ≈ 0 in the middle of a long solenoid z L Compact Muon Solenoid Detector Particle Physics experiment at CERN Largest superconducting magnet in the world B = 4 Tesla Diameter: 15 m Length: 21.5 m Total weight: 12500 Tons STAR Time Projection Chamber .
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