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Physics 202, Lecture 14 Today s Topics Sources of the Magnetic Field (Ch 27) Review: The Ampere s Law Applications And Exercises of ampere s Law Straight line, Solenoid, Toroid Magnetism in Matter Ampere Review: Ampere s Law ! ! s Law: "! B• dl = µ0 I I any closed path It applies to any closed path It applies to any static B field It is practically useful in symmetric cases dl Ampere s Law can be derived from Biot-Savart Law Key point: Derivation relies on the fact that the current has no divergence(is a continuous flow does not increase or decrease at any point). This works for current loops or infinite currents. Solenoid The B field inside an ideal solenoid: Ideal: Infinitely long and tightly wound "! ! ! B• dl = µ 0 I any closed path Bl = µ 0 NI B = µ 0 nI n=N/L B field independent of where you are inside the solenoid ideal solenoid segment 3 at ∞ Solenoid Ideal vs. Real The B field inside an ideal solenoid is: B = µ0nI n=N/L ideal solenoid Compare Solenoid and Bar Magnet N S B Loose Solenoid Bar Magnet Tight Solenoid I Toroid The B field inside a toroid "! ! ! B• dl = µ 0 I any closed path B2! r = µ 0 NI µ0 NI B= 2πr b<r<c: B=0 outside the Torus because the Ampere’s loop does not enclose any net current Review: Electric Dipole Moments Electric dipole moment p. + +q τ = p×E U = -p • E -q - ∑F = 0 p= qd Dielectric material contains electric dipoles In an external field E0 , the dipoles line up at atomic level Eind is always opposite to E0 E=E0/κ<E0, C=κC0 (dielectric constant κ>1) More on Magnetic Dipole Moments Magnetic dipole moment µ. Note: B produced (at the center) is always in same direction as µ definition of magnetic moment Macroscopic µ=IA Microscopic From orbiting electrons (depends on their angular momentum) and quantum mechanical spin (special type of angular momentum) I=q/T=qv/2πr µ=IA =1/2 qvr = (q/2m)L 2 µ 0 IR Bz=! = 3 2z µ0µ B= 3 2z B Magnetism in Matter µ ∑F = 0 τ = µ ×B U = -µ • B µ in B Field External B and dipole B in the same direction! Total magnetic dipole moment µ and the magnetic field If all the individual dipoles line up the field can be very strong. Induced by applying an external field Configuration like a solenoid with constant field Depends on details of the material. Define the magnetization M where Bind=µ0M Magnetism in Matter Induced field Bind in response to an external B0: Bind = χB0 èthe net field inside: B = B0+Bind =(1+χ) B0 = (µm/µ0)B0 = KmB0 µm: magnetic permeability , χ: magnetic susceptibility, K: relative permeability Classification of Magnetic Matter Type Direction of Binduced Strength of Binduced χ=µm/µ0-1 Contributing Elements Ferromagnetic (e.g. Fe, Co, Ni…) Same as B0 Strong >>0 (~103) Domain of Magnetic Dipole Paramagnetic (e.g. Al, Ca,…) Same as B0 Weak >0 (~10-5) µatoms Diamagnetic (e.g. Cu, Au,…) Opposite Weak <0 (~-10-5) Magnetic Quadruple Superconductor Opposite -1 Quantum eff. =-B0 Permanent Magnetic Moments (domains) Inside Ferromagnetic Material polarized domains No external B field, permanent mag. moments exist, but oriented randomly à no induced B field B0 applied , permanent magnetic moments line up in the direction of B0 à strong induced B field Note: Inductance with a ferromagnetic core: B= µm/µ0B0 >>B0 Meissner Effect Certain superconductors (type I) exhibit perfect diamagnetism in superconducting state: no magnetic field allowed inside (Meissner Effect)
