Physics 202, Lecture 14 Tody s Topics Sources of the Mgnetic Field (Ch. 28) Biot-Svrt Lw Ampere s Lw Mgnetism in Mtter Mxwell s Equtions Homework #7: due Tues 3/11 t 11 PM (4th problem optionl) Mgnetic Force: Mgnetic Forces nd Fields F = Id l B Mgnetic Field: Biot-Svrt Lw: B = µ 0 4 Ampere s Lw: # " Id l " ˆr r 2 Bid s " = µ 0 I enclosed θ r dl P R I I µ 0 = 4πx10-7 T m/a: permebility of free spce Direction of integrtion long pth: use right-hnd rule 1
B Field of Stright Wire, length L (Text exmple 28-13) B field t point P is: B = µ 0 I (cos 1 # cos 2) 4" When L=infinity (text exmple 28-11): B = µ 0I 2 θ 1 θ 2 (return to this lter with Ampere s Lw) Text: 28.43 Question y A current I flows in the +y direction in n infinite wire; current I lso flows in loop. Wht is F x, the net force on the loop in the x-direction? I F left F top X I F right () F x < 0 (b) F x = 0 (c) F x > 0 F bottom x 2
B Field of Circulr Current Loop on Axis Use Biot-Svrt Lw (text exmple 28-12) µ0 IR 2 Bx = 2( x 2 + R 2 ) 3 / 2 Bcenter = µ0 I 2R See lso: center of rc (text exmple 28-13) Text: 28.37 B of Circulr Current Loop: Field Lines B 3
Mgnetic Fields (Biot-Svrt): Summry Current loop, distnce x on loop xis (rdius R): B x = µ 0 IR 2 B 2(x 2 + R 2 ) 3/2 center = µ I 0 2R Center of rc (rdius R, ngle θ): Stright wire: finite length µ I B = (cos 1 # cos 4" infinite wire: 0 2 µ 0I B = 2 ) B center = µ 0 I 4" R θ 1 θ 2 Ampere s Lw: Ampere s Lw I " ny closed pth Bid l = µ 0 I encl dl pplies to ny closed pth, ny sttic B field useful for prcticl purposes only for situtions with high symmetry 4
Ampere s Lw: B-field of Stright Wire Use symmetry: I Choose loop to be circle of rdius R centered on the wire in plne to wire. Why? Mgnitude of B is constnt (function of R only) Direction of B is prllel to the pth. # B id l " = # Brd = 2"rB = µ 0 I encl = µ 0 I B = µ 0 I 2r Text exmple: 28-6 B Field Inside Long Wire Totl current I flows through wire of rdius into the screen s shown. Wht is the B field inside the wire? By symmetry -- tke the pth to be circle of rdius r: " Bid l = B2r Current pssing through circle: I encl = r 2 Bid l = B2r = µ o I encl 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 2 I " B in = µ Ir 0 2 2 5
B Field of Long Wire Inside the wire: (r < ) B = µ 0 I 2 r 2 Outside the wire: ( r > ) B B = µ 0 I 2 r r Text: 28.31 Ampere s Lw: Toroid Toroid: N turns with current I. B θ =0 outside toroid (Consider integrting B on circle outside toroid: net current zero) B θ inside: consider circle of rdius r, centered t the center of the toroid. Bid l = B2r = µ o I encl = " µ o NI B = µ 0 NI 2r x x x x x x x x r x x x x x x x x B 6
B Field of Solenoid Inside solenoid: source of uniform B field Solenoid: current I flows through wire wrpped n turns per unit length on cylinder of rdius nd length L. L If << L, the B field is pproximtely contined within the solenoid, in the xil direction, nd of constnt mgnitude. In this limit, cn clculte the field using Ampere's Lw Ampere s Lw: Solenoid The B field inside n idel solenoid is: B = µ 0nI n=n/l idel solenoid segment 3 t 7
Solenoid: Field Lines Right-hnd rule: direction of field lines north pole south pole Like br mgnet (except it cn be turned on nd off) Text: 28.68 Mgnetic Fields (Ampere s Lw): Summry Infinite wire: Inside wire, rdius R: B = µ 0 I 2r Axil field inside toroid (N turns) B = µ 0 NI 2"r B field inside long solenoid (L>>R) (n turns/length) Uniform field B = µ ni 0 B = µ 0Ir 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 8
Mgnetism in Mtter The B field produced long the xis of circulr loop (rdius R) by current I is: B µ 0 µ ẑ typicl dipole behviour 3 2"z µ is the mgnetic moment = I # re nd z >> R Mterils re composed of prticles tht hve mgnetic moments -- (negtively chrged electrons circling round the positively chrged nucleus). orbitl ngulr momentum spin ngulr momentum (quntum mechnics) µ B = q e J = 9.27 10 "24 2m e T Bohr mgneton Mgnetiztion Apply externl B field B 0. Field is chnged within mterils by these mgnetic moments. Mgnetiztion:totl mgnetic moment per unit volume The B field in the mteril is Define H (mgnetic field strength): Mgnetic susceptibility: M B = B 0 + µ 0 M µ totl V B = µ 0 ( H + M ) M = H ( H = B 0 µ 0 ) B = µ H = µ 0 (1 + ) H permebility µ ( = m µ 0 ) 9
Mgnetic Mterils Mterils re clssified by mgnetic susceptibilities: Prmgnetic (luminum, tungsten, oxygen, ) Atomic mgnetic dipoles line up with the field, incresing it. Only smll effects due to therml rndomiztion: χ ~ +10-5 Dimgnetic (gold, copper, wter, s well s superconductors) Applied field induces n opposing field; usully very wek χ~ -10-5 Ferromgnetic (iron, coblt, nickel, ) Dipoles prefer to line up with the pplied field (similr to prmgnetic), but tend to ll line up the sme wy due to collective effects: very strong enhncements χ ~ +10 +3-10 +5 Mgnetic susceptibility temperture dependent (bove rnge of typicl vlues t T=20 C) Ferromgnets Dipoles tend to strongly lign over smll ptches domins (even w/o externl mgnetic field). With externl field, the domins lign to produce lrge net mgnetiztion. Soft ferromgnets Domins re-rndomize when mgnetic field is removed Mgnetic Domins Hrd ferromgnets Domins persist even when the field is removed Permnent mgnets Domins my be ligned in different direction in new externl field Domins my be re-rndomized by sudden physicl shock If temperture is rised bove Curie point (770 C for iron), domins will lso rndomize (like prmgnet) 10
Applied field ligns lmost ll the dipoles nd the domins. Mgnetiztion is then sturted : no further increse. Mgnetic Domins Hrd ferromgnets: Domins cn persist even when the field is removed Permnent mgnets Domins my be ligned in different directions by chnging the pplied field.. A memory effect tht requires lrge reverse field to significntly chnge the mgnetiztion of the object: hysteresis. Mxwell s Equtions: Sttics " " " " Eid A = q encl 0 Eid l = 0 Bid A = 0 Bid l = µ 0 I encl Guss s Lw: Electric Fields Conservtive nture of electrosttic force Guss s Lw: Mgnetic Fields Ampere s Lw 11
Mxwell s Equtions: Generl Cse Preview: " Eid A = q encl Bid A = 0 " 0 Eid l = d " Bid A dt " Bid l d " = µ 0 I encl + µ 0 " 0 dt (time-dependent fields) Guss s Lw: Electric Fields Frdy s Lw (next week) Guss s Lw: Mgnetic Fields Eid A Ampere-Mxwell (lter in sem.) (interdependence of E nd B: crucil for wht s next) 12