Physics 202, Lecture 14 Today s Topics Sources of the Magnetic Field (Ch. 30) Review: iot-savart Law, Ampere s Law Displacement Current: Ampere-Maxwell Law Magnetism in Matter Maxwell s Equations (prelude) Faraday s Law of nduction (Ch. 31) Homework #6: due 10/22,10 PM. Optional reading quiz: due 10/19, 7 PM Sources of E and Fields: An overview Sources of electric fields: Static electric charges: Coulomb's Law/Gauss s Law Change of field (magnetic flux): Faraday's Law (today and next lectures) Sources of magnetic fields: Electric current: iot-savart Law/Ampere s Law Change of E field (electric flux): Ampere-Maxwell Law (today) summarized in Maxwell s Equations (later today) 1
Magnetic Force: Magnetic Forces and Fields F = d l Magnetic Field: iot-savart Law: = µ 0 4 Ampere s Law: # " d l " ˆr r 2 id s " = µ 0 enclosed θ r dl P R µ 0 = 4πx10-7 T m/a: permeability of free space Direction of integration along path: use right-hand rule Magnetic Fields: Examples (iot-savart) Current loop, distance x on loop axis (radius R): x = µ 0 R 2 2(x 2 + R 2 ) 3/2 center = µ 0 2R Center of arc (radius R, angle θ): Straight wire: finite length µ 0 = (cos 1 # cos 2) 4" a infinite wire: µ 0 = 2 a center = µ 0 4" R θ 1 θ 2 2
Magnetic Fields: Examples (Ampere s Law) nfinite wire: nside wire, radius R: = µ 0 2r Axial field inside toroid (N turns) = µ 0 N 2"r field inside long solenoid (L>>R) (n turns/length) Uniform field = µ 0n = µ 0 r 2 R 2 R x x x x x x x x x x x x x x x x x x x x x x x r x x x x x x x x x x x x x L R Question y A current flows in the +y direction in an infinite wire; a current also flows in loop. What is F x, the net force on the loop in the x-direction? F left F top X F right (a) F x < 0 (b) F x = 0 (c) F x > 0 F bottom x Recall: net force on a current loop in a uniform -field is zero -- but the -field of an infinite wire is not uniform Forces cancel on the top and bottom of the loop. Forces do not cancel on the left and right sides of the loop. The left segment is in a larger magnetic field than the right: F left > F right 3
Magnetic Field Generated by Varying E Eield General Form of Ampere s Law Ampere-Maxwell Law: any closed path d s d# = µ 0 + µ 0 " E 0 E " # Eid A "$ = µ 0 ( + d ) Changing electric field also generates a field Example: charging capacitor Text: problem 30.37 displacement d " # 0 current d E Magnetism in Matter The field produced along the axis of a circular loop (radius R) by a current is: µ 0 µ ẑ typical dipole behaviour 3 2"z µ is the magnetic moment = # area and z >> R Materials are composed of particles that have magnetic moments -- (negatively charged electrons circling around the positively charged nucleus). orbital angular momentum spin angular momentum (quantum mechanics) µ = e J = 9.27 10 "24 2m e T ohr magneton 4
Magnetization Apply external field 0. Field is changed within materials by these magnetic moments. Magnetization:total magnetic moment per unit volume The field in the material is Define H (magnetic field strength): Magnetic susceptibility: M = 0 + µ 0 M Text examples: Ch 30, #41,43 µ total V = µ 0 ( H + M ) M = H ( H = 0 µ 0 ) = µ H = µ 0 (1+ ) H permeability µ ( = m µ 0 ) Magnetic Materials Materials are classified by magnetic susceptibilities: Paramagnetic (aluminum, tungsten, oxygen, ) Atomic magnetic dipoles line up with the field, increasing it. Only small effects due to thermal randomization: χ ~ +10-5 Diamagnetic (gold, copper, water, as well as superconductors) Applied field induces an opposing field; usually very weak χ~ -10-5 Ferromagnetic (iron, cobalt, nickel, ) Dipoles prefer to line up with the applied field (similar to paramagnetic), but tend to all line up the same way due to collective effects: very strong enhancements χ ~ +10 +3-10 +5 Magnetic susceptibility temperature dependent (above range of typical values given at T=20 C) 5
Ferromagnets Dipoles tend to strongly align over small patches domains (even w/o external magnetic field). With external field, the domains align to produce a large net magnetization. Soft ferromagnets Domains re-randomize when magnetic field is removed Magnetic Domains Hard ferromagnets Domains persist even when the field is removed Permanent magnets Domains may be aligned in a different direction in a new external field Domains may be re-randomized by sudden physical shock f temperature is raised above Curie point (770 C for iron), domains will also randomize (like a paramagnet) Applied field aligns almost all the dipoles and the domains. Magnetization is then saturated : no further increase. Magnetic Domains Hard ferromagnets: Domains can persist even when the field is removed Permanent magnets Domains may be aligned in different directions by changing the applied field.. A memory effect that requires a large reverse field to significantly change the magnetization of the object: hysteresis. 6
Maxwell s Equations: Prelude Gauss s Law: Electric Fields Gauss s Law: Magnetic Fields Ampere-Maxwell: "# What about Eid A = q enclosed " 0 " id A = 0 id s d" = µ 0 enclosed + µ 0 E 0 " Eid s? E = " Eid A Recall definition of flux: M = " id A The Path ntegral of the E field Recall in electrostatics: Coulomb force conservative " Eid s = 0 define electrostatic potential: V b V a = Eid s b " a ut in the presence of moving charges ( fields): ds new contribution to E field, with = "# E NC id s " 0 nduced emf (Faraday) 7
Emf and Change of Magnetic Flux Also: battery-less flashlight Faraday s Law of nduction Emf induced in a circuit is proportional to the time rate of change of magnetic flux through the circuit. = " d# Circuit : any closed path (does not have to be a real conducting circuit) = " id A θ A Direction: opposes change in magnetic flux (Lenz s Law) 8
Exercises: Determine Direction of nduced Emf ndicate the direction of emf in the following cases: + + + + + + + + + + + + + + + + + + + + + + + + increases decreases decreases increases path outside decreases or Methods to Change Electric Flux d$ " = # = # uniform d( Acos ) Change of Φ emf To change Φ : Change emf produced by an induced E field Change A motional emf Change θ motional emf Combination of above Example: Ch. 31, #1 9