Physics 202, Lecture 14

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1 Physics 202, Lecture 14 Today s Topics Sources of the Magnetic Field (Ch. 27) Review of Biot-Savart Law Ampere s Law Magnetism in Matter

2 Magnetic Fields (Biot-Savart): Summary Current loop, distance on loop ais (radius R): B = µ 0 IR 2 B 2( 2 + R 2 ) 3/2 center = µ I 0 2R Center of arc (radius R, angle θ): Straight wire: finite length B = infinite wire: µ 0I (cosθ1 cosθ 4πa µ 0I B = 2πa 2 ) B center = µ 0 Iθ 4π R θ 1 θ 2

3 Ampere s Law: Ampere s Law I any closed path Bid l = µ 0 I encl dl applies to any closed path, any static B field useful for practical purposes only for situations with high symmetry

4 Ampere s Law: B-field of Straight Wire Use symmetry: I Choose loop to be circle of radius R centered on the wire in a plane to wire. Why? Magnitude of B is constant (function of R only) Direction of B is parallel to the path. B idl = Brdθ = 2πrB = µ 0 I encl B = µ 0 I 2πr = µ 0 I

5 B Field Inside a Long Wire Total current I flows through wire of radius a into the screen as shown. What is the B field inside the wire? By symmetry -- take the path to be a circle of radius r: Bid l = B2πr Current passing through circle: I encl = r 2 Bid l = B2πr = µ o I encl r a a 2 I B in = µ 0Ir 2πa 2

6 B Field of a Long Wire Inside the wire: (r < a) B = µ 0 I 2π r a 2 a Outside the wire: ( r > a ) B B = µ 0 I 2π r r

7 Ampere s Law: Toroid Toroid: N turns with current I. B θ =0 outside toroid! (Consider integrating B on circle outside toroid: net current zero) B θ inside: consider circle of radius r, centered at the center of the toroid. Bidl = B2πr = µ o I encl = µ o NI B = µ 0NI 2πr r B

8 B Field of a Solenoid Inside a solenoid: source of uniform B field Solenoid: current I flows through a wire wrapped n turns per unit length on a cylinder of radius a and length L. L a If a << L, the B field is approimately contained within the solenoid, in the aial direction, and of constant magnitude. In this limit, can calculate the field using Ampere's Law!

9 Ampere s Law: Solenoid The B field inside an ideal solenoid is: B = µ 0nI n=n/l ideal solenoid segment 3 at

10 Magnetism in Matter The B field produced along the ais of a circular loop (radius R) by a current I is: B µ 0 µ 2πz ẑ 3 typical dipole behaviour µ is the magnetic moment = I 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 B = q e J = m e T Bohr magneton

11 Magnetization Apply eternal B field B 0. Field is changed within materials by these magnetic moments. Magnetization:total magnetic moment per unit volume The B field in the material is Define H (magnetic field strength): Magnetic susceptibility: M B = B 0 + µ 0M M = χ H µ total V B = µ 0 ( H + M ) ( H = B 0 µ 0 ) B = µ H = µ 0 (1 + χ) H permeability µ ( = κ m µ ) 0

12 Magnetic Materials Materials are classified by magnetic susceptibilities: Paramagnetic (aluminum, tungsten, oygen, ) Atomic magnetic dipoles line up with the field, increasing it. Only small effects due to thermal randomization: χ ~ Diamagnetic (gold, copper, water, as well as superconductors) Applied field induces an opposing field; usually very weak χ~ 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 ~ Magnetic susceptibility temperature dependent (above range of typical values at T=20 C)

Ferromagnets Dipoles tend to strongly align over small patches domains (even w/o eternal magnetic field). With eternal field, the domains align to produce a large net magnetization.

13 Ferromagnets Dipoles tend to strongly align over small patches domains (even w/o eternal magnetic field). With eternal 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 eternal field Domains may be re-randomized by sudden physical shock If temperature is raised above Curie point (770 C for iron), domains will also randomize (like a paramagnet)

Physics 202, Lecture 14

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)