Physics 11b Lecture #11


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1 Physics 11b Lectue #11 Mgnetic Fields Souces of the Mgnetic Field S&J Chpte 9, 3
2 Wht We Did Lst Time Mgnetic fields e simil to electic fields Only diffeence: no single mgnetic pole Loentz foce Moving chge in mgnetic field is deflected Unifom field Cycloton motion Cycloton fequency depends only on q/m nd Nonunifom field cn tp chged pticles Vn Allen belt F = qv Loentz foce on cuent Stight wie: F = IL v ω = = q m
3 Tody s Gols Wok out the Loentz foce on bityshped cuent Don t woy Esie thn you think Clculte toque on cuent loop in field Do you emembe wht ws toque? Compe this with the toque on electic/mgnetic dipole A little efeshe on dipole nd enegy Discuss Hll effect Stt ceting mgnetic field iotsvt Lw
4 Mgnetic Foce on Cuent Cuent I flows in wie in unifom field Totl foce on cuent is F = IL Wht if the wie is cuved? Cut it into smll pieces (s usul) I L df = Ids Integte long the wie F = I ds b df ds b is inside the integl It my not be constnt long the wie!
5 Unifom Field If field is unifom, the integl simplifies b ( b ) F I d I d I = s = s = L L is the vecto fom to b Net foce on cuent due to unifom field is identicl fo ny pth with the sme end points We cn use the esiest pth (stight line) fo ou clcultion Coolly: net foce on closed loop cuent fom unifom field is zeo I I L b
6 Simple Exmple Cuent I flows on Dshped wie in unifom field Wht s the foce on the ound side? Foce on the stight side is esy F ( ˆ) ( ˆ) ˆ = IL = I y z = Ix Foce on the ound side must be the sme size nd opposite diection F = Ixˆ y x Shpe elly doesn t mtte
7 Toque Cuent I flows ound squeshped wie in unifom field y Foces on the 4 sides e b c d F ˆ = F = I y b Totl foce is zeo (of couse) b b c c d d F + F + F + F = ut the foces en t completely blnced They will otte the wie F = F = c d Tht s toque (textbook Chpte 11) τ = = F I d z b F y c F x x
8 Toque Toque τ is defined s τ = F fo ech foce F vecto fom xis to the line of foce When the wie is pllel to τ = xˆ ( Fyˆ) = Fzˆ = I zˆ If the wie is t n ngle τ = sin θxˆ ( Fyˆ) = I sinθzˆ Define vecto A pependicul to the wie nd A = τ = IA F y y F θ A F F x x
9 Whee Ae We? A cuent loop in n unifom field eceives no net foce but finite toque given by τ = IA A A is the e vecto pependicul to the loop Note the diections of A nd τ They e elted to the diections of the cuent nd ottion by the ighthnd ule τ This is how n electic moto woks A coil (= wie loop) on otting xle in mgnetic field
10 E&M Dipoles Electic nd mgnetic dipoles e quite simil + S N In unifom E o field, they eceive no net foce ut thee cn be toques E F = qe F = qm + S N τ = F= q E= p E Electic dipole moment: p q τ = F= q m = µ Mgnetic dipole moment: µ q m
11 Cuent Loop nd Dipole Mgnetic dipole is suspiciously simil to cuent loop No net foce fom, but toque τ = µ Fo cuent loop: τ = IA τ A A cuent loop esponds to field like mgnetic dipole µ = IA S N F = q m µ qm
12 Cuent Loop nd Dipole Remembe: thee is no such thing s mgnetic chge All mgnets e dipoles ut how cn we build dipole if we don t hve +q m nd q m? Wht if ll the mgnets e mde of tiny cuent loops? Afte ll, thee e electic chges eveywhee Mgnetic phenomen cn be explined by electic chges in motion We ll come bck to this (textbook 3.8) S N µ q m µ = IA
13 E&M Dipoles nd Enegy Conside electic dipole in n unifom E field Unstble = enegy high + E + F = qe p q + Stble = enegy Low The potentil enegy t n bity ngle θ is U = qecosθ = pe Exctly the sme pplies to mgnetic dipole U = µ E θ + θ S N
14 Hll Effect Cuent I flows in piece of conducto inside mgnetic field I q v d Loentz foce F ct on the cies ut they cn t flow out of the conducto Sufce chge ppes F = qv v d E q FE = qe d Edwin Hll ( ) Potentil diffeence V H between top nd bottom is clled the Hll voltge
15 Hll Voltge Fom lectue #8 I = nqv A Since F = F E I E = vd = nqa d F = qv v d E q FE = qe d d The conducto hs height d, thickness t Id I V = H Ed = nqa = R I V H H = nqt t R H is the Hll coefficient of this conducto Hll effect cn be used to detemine the sign nd the density of the chge cies in (semi)conducto R H 1 nq
16 Ceting Mgnetic Field Electic field is ceted by chge distibutions Cn t do this with mgnetic field No mgnetic chge! Must do with mgnetic dipoles, which we believe e mde of cuent loops Mgnetic fields e ceted by electic cuents Discoveed in 1819 by Hns Østed I Hns Østed ( )
17 iotsvt Lw Coulomb s Lw descibes the E field due to infinitesimlly smll chge 1 dqˆ de = 4πε field due to cuent in shot piece of wie is given by the iotsvt Lw µ Ids ˆ d = I 4π Stikingly simil except fo: 1 µ ε dqˆ Id ˆ s dq ds d de
18 iotsvt Lw d = µ ˆ Ids 4π Constnt µ is the pemebility of fee spce µ = π This is exct. Will explin why T m A The Id s ˆ pt is inteesting d is pependicul to both ds nd Size of d depends on the ngle between ds nd Ids Ids sin θ θ I ds d =
19 StightLine Cuent Cuent I flows in n infinitelylong stight wie field t distnce fom the wie? µ Ids ˆ µ Idxsinθ d= = zˆ 4π 4π Geomety tells us = + x sinθ = µ I Idx d = 3 4 π ( + x ) This cn be integted, but not esy ds d z θ y x
20 StightLine Cuent Esie to integte if switch fom dx to dθ dθ x = dx = tnθ sin θ Also = sinθ d = = µ Idxsinθ 4π µ I sinθdθ 4π µ I π µ I sin π µ I = θdθ = [ cosθ] = 4π 4π π I ds d z θ y field goes ound the wie nd decese s 1/ x
21 Summy Loentz foce on cuent Simplifies to F = IL if the field is unifom Toque on cuent loop in field Cuent loop looks like mgnetic dipole Hll effect H V = RH I t F = I ds Mgnetic fields e ceted by electic cuent µ Ids ˆ iotsvt Lw d = 4π µ I field by n infinite stightline cuent = π R H b 1 nq τ = IA µ = IA
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