hitt Phy2049: Magnetism 6/10/2011 Magnetic Field Units Force Between Two Parallel Currents Force Between Two Anti-Parallel Currents

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1 6/0/0 Phy049: Magsm Last lectue: t-avat s and Ampee s law: Magc eld due t a staght we Cuent lps (whle bts)and slends Tday: emnde and aaday s law. htt Tw lng staght wes pece the plane f the pape at vetces f an equlateal tangle as shwn. They each cay A but n the ppste ectn. The we n the left has the cuent cmng ut f the pape whle the we n the ght caes the cuent gng nt the pape. The c feld at the thd vetex (P) has the magntude and ectn (th s up): () 0 T, east () 7 T, west () 5T, nth (4) 6 T, suth (5) nne f these 4 cm X Magc eld Unts m the expessn f fce n a cuent-cayng we: max / I L Unts: ewtns/a m Tesla (I unt) Anthe unt: gauss 0-4 Tesla me sample c feld stengths: Eath: 0.5 gauss 0.5 x 0-4 T 5 x 0-5 T Galaxy: 0-6 gauss 0-0 T a : gauss tng elect: T upecnductng : 0 40 T Pulse : 00 T eutn sta: T Maga: 0 T ce etween Tw Paallel Cuents ce n I fm I 0I 0II IL I π π H ce twads I ce n I fm I 0I 0II I L I π π H ce twads I I I I Magc fces cause attactn between tw paallel cuents I ce etween Tw Ant-Paallel Cuents ce n I fm I 0I 0II I L I π π H ce away fm I ce n I fm I 0I 0II I L I π π H ce away fm I I I Magc fces epel tw antpaallel cuents I I Paallel Cuents (cnt.) Lk at them edge n t see felds me clealy Antpaallel: epel Paallel: attact

2 6/0/0 Cente f Ccula Cuent Lp aus and cuent : fnd feld at cente f lp 0 Dectn: H # (see pctue) If tuns clse tgethe Cuent Lp Example 500 A, 5 cm, 0 7 ( )( ) π.6t Challenge: Hw much heat s pduced? eld f lend mula fund fm Ampee s law cuent n tuns / mete 0n ~ cnstant nsde slend ~ ze utsde slend Mst accuate when L>> Example: 00A, n 0 tuns/cm n 000 tuns / m 7 ( π )( )( ) T eld at Cente f Patal Lp uppse lp cves angle φ 0 φ π Use example whee φ π (half ccle) Defne ectn nt page as pstve 0 π 0 π π π 0 4 Patal Lps (cnt.) te n pblems when yu have t evaluate a feld at a pnt fm seveal patal lps Only lp pats cntbute, pptnal t angle (pevus slde) taght sectns amed at pnt cntbute nthng e caeful abut sgns, e.g.n (b) felds patally cancel, wheeas n (a) and (c) they add ( z) π z The c feld f a c ple. Cnsde the c feld geneated by a we cl f aus whch caes a cuent. The c feld at a pnt P n the z-axs s gven by: ( + z ) / Hee z s the stance between P and the cl cente. pnts fa fm the lp ( z ) we can use the appxmatn: z π A Hee s the c pz π z π z ple mment f the lp. In vect fm: ( z) π z The lp geneates a c feld that has the same fm as the feld geneated by a ba. (9 4)

3 6/0/0 Chapte 0 Inductn and Inductance In ths chapte we wll study the fllwng tpcs: -aaday s law f nductn -Lenz s ule -Electc feld nduced by a changng c feld Electc cuent wthut a battey -Inductance and mutual nductance - L ccuts -Enegy sted n a c feld (0 ) (0 ) aaday's expements In a sees f expements Mchael aaday n England and Jseph Heny n the U wee able t geneate electc cuents wthut the use f battees elw we descbe sme f these expements that helped fmulate whats s knwn as "aaday's law f nductn" The ccut shwn n the fgue cnssts f a we lp cnnected t a senstve ammete (knwn as a "galvanmete"). If we appach the lp wth a pemanent we see a cuent beng egsteed by the galvanmete. The esults can be summazed as fllws:. A cuent appeas nly f thee s elatve mtn between the and the lp. aste mtn esults n a lage cuent. If we evese the ectn f mtn the platy f the, the cuent eveses sgn and flws n the ppste ectn. The cuent geneated s knwn as " nduced cuent "; the emf that appeas s knwn as " nduced emf"; the whle effect s called " nductn" (0 ) lp lp In the fgue we shw a secnd type f expement n whch cuent s nduced n lp when the swtch n lp s ethe clsed pened. When the cuent n lp s cnstant n nduced cuent s bseved n lp. The cnclusn s that the c feld n an nductn expement can be geneated ethe by a pemanent by an electc cuent n a cl. aaday summazed the esults f hs expements n what s knwn as " aaday's law f nductn" An emf s nduced n a lp when the numbe f c feld lnes that pass thugh the lp s changng aaday's law s nt an explanatn f nductn but meely a descptn f f what nductn s. It s ne f the fu " Maxwell's equatns f electsm" all f whch ae statements f expemental esults. We have aleady encunteed Gauss' law f the electc feld, and Ampee's law (n ts ncmplete fm) da da ˆn φ Magc lux The c flux thugh a suface that bdes a lp s detemned as fllws:. we vde the suface that has the lp as ts bde nt aea elements f aea da.. each element we calculate the c flux thugh t: dacsφ Hee φ s the angle between the nmal nˆ and the c feld vects at the pstn f the element.. We ntegate all the tems. csφ da da I c flux unt : T m knwn as the Webe (symbl Wb) We can expess aaday's law f nductn n the flwng fm: The magntude f the emf E nduced n a cnductve lp s equal t ate at whch the c flux thugh the lp changes wth tme E (0 4) dacsφ da da ˆn φ Methds f changng thugh a lp. Change the magntude f wthn the lp. Change ethe the ttal aea f the cl the ptn f the aea wthn the c feld. Change the angle φ between and nˆ E Lenz's ule We nw cncentate n the negatve sgn n the equatn that expesses aaday's law. The ectn f the flw f nduced cuent n a lp s acuately pected by what s knwn as Lenz's ule. lp φ ˆn An Example. Pblem 0- A csφ ab cs ( ωt) E ωab sn ( ωt) ω π f E π fab sn ( π t) (0 5) An nduced cuent has a ectn such that the c feld due t the nduced cuent ppses the change n the c flux that nduces the cuent Lenz's ule can be mplemented usng ne f tw methds:. Oppstn t ple mvement In the fgue we shw a ba appachng a lp. The nduced cuent flws n the ectn ncated because ths cuent geneates an nduced c feld that has the feld lnes pntng fm left t ght. The lp s equvalent t a whse nth ple haces the cespnng nth ple f the ba appachng the lp. The lp epels the appachng and thus ppses the change n whch geneated the nduced cuent. (0 6)

4 6/0/0 mtn. Oppstn t flux change Example a : a appaches the lp wth the nth ple facng the lp. As the ba appaches the lp the feld pnts twads the left and ts magntude nceases wth tme at the lcatn f the lp. Thus the magntude f the lp c flux als nceases. The nduced cuent flws n the cunteclckwse (CCW) ectn s that the nduced c feld ppses the feld. The feld. The nduced cuent s thus tyng t pevent fm nceasng. emembe t was the ncease n that geneated the nduced cuent n the fst place. mtn. Oppstn t flux change Example b : a mves away fm the lp wth nth ple facng the lp. As the ba mves away fm the lp the feld pnts twads the left and ts magntude deceases wth tme at the lcatn f the lp. Thus the magntude f the lp c flux als deceases. The nduced cuent flws n the clckwse (CW) ectn s that the nduced c feld adds t the feld. The feld +. The nduced cuent s thus tyng t pevent fm deceasng. emembe t was the decease n that geneated the nduced cuent n the fst place. (0 7) (0 8). Oppstn t flux change Example c : a appaches the lp wth suth ple facng the lp. mtn As the ba appaches the lp the feld pnts twads the ght and ts magntude nceases wth tme at the lcatn f the lp. Thus the magntude f the lp c flux als nceases. The nduced cuent flws n the clckwse (CW) ectn s that the nduced c feld ppses the feld. The feld. The nduced cuent s thus tyng t pevent fm nceasng. emembe t was the ncease n that geneated the nduced cuent n the fst place. mtn. Oppstn t flux change Example d : a mves away fm the lp wth suth ple facng the lp. As the ba mves away fm the lp the feld pnts twads the ght and ts magntude deceases wth tme at the lcatn f the lp. Thus the magntude f the lp c flux als deceases. The nduced cuent flws n the cunteclckwse (CCW) ectn s that the nduced c feld adds t the feld. The feld +. The nduced cuent s thus tyng t pevent fm deceasng. emembe t was the decease n that geneated the nduced cuent n the fst place. (0 9) (0 0) (0 ) Inductn and enegy tansfes y Lenz's ule, the nduced cuent always ppses the extenal agent that pduced the nduced cuent. Thus the extenal agent must always d wk n the lp-c feld system. Ths wk appeas as themal enegy that gets sspated n the esstance f the lp we. Lenz's ule s actually a ffeent fmulatn f the pncple f enegy cnsevatn Cnsde the lp f wh L shwn n the fgue. Pat f the lp s lcated n a egn whee a unfm c feld exsts. The lp s beng pulled utsde the c feld egn wth cnstant speed v. The c flux thugh the lp A Lx E The flux deceases wth tme dx E Lv L Lv (0 ) The ate at whch themal enegy s sspated n Lv L v Pth ( eqs.) The c fces n the we sdes ae shwn n the fgue. ces and cancel each the. Lv ce L Lsn 90 L L L v The ate at whch the extenal agent s pducng mechancal wk Pext L v v ( eqs.) If we cmpae equatns and we see that ndeed the mechancal wk dne by the extenal agent that mves the lp s cnveted nt themal enegy that appeas n the lp wes. 4

5 6/0/0 (0 ) Eddy cuents We eplace the we lp n the pevus example wth a sld cnductng plate and mve the plate ut f the c feld as shwn n the fgue. The mtn between the plate and nduces a cuent n the cnduct and we encunte an ppsng fce. Wth the plate the fee electns d nt fllw ne path as n the case f the lp. Instead the electns swl aund the plate. These cuents ae knwn as " eddy cuents". As n the case f the we lp the esult s that mechancal enegy that mves the plate s tansfmed nt themal enegy that heats up the plate. Induced electc felds Cnsde the cppe ng f aus shwn n the fgue. It s paced n a unfm c feld pntng nt the page, that ceases as functn f tme. The esultng change n c flux nduces a cuent n the cunteclck-wse (CCW) ectn. The pesence f the cuent n the cnductng ng mples that an nduced electc feld E must be pesent n de t set the electns n mtn. Usng the agument abve we can efmulate aaday's law as fllws: A changng c feld pduces an electc feld te : The nduced electc feld s geneated even n the absense f the cppe ng. (0 4) Cnsde the ccula clsed path f aus shwn n the fgue t the left. The pctue s the same as that n the pevus page except that the cppe ng has been emved. The path s nw an abstact lne. The emf alng the path s gven by the equatn: E E ds ( eqs.) The emf s als gven by aaday's law: E ( eqs.) If we cmpae eqs. wth eqs. we get: E ds E ds Eds cs0 E ds π E d π π d d π E π E (0 5) l L n la Inductance Cnsde a slend f length l that has lps f aea A each, and n wnngs pe unt length. A cuent l flws thugh the slend and geneates a unfm c feld n nsde the slend. The slend c flux A ( l ) The ttal numbe f tuns nl n A The esult we gt f the specal case f the slend s tue f any nduct. L. Hee L s a cnstant knwn as the nductance f the slend. The nductance depends n the gemety f the patcula nduct. Inductance f the slend the slend n la L n la (0 6) E L elf Inductn lp In the pctue t the ght we lp aleady have seen hw a change n the cuent f lp esults n a change n the flux thugh lp, and thus ceates an (0 7) nduced emf n lp If we change the cuent thugh an nduct ths causes a change n the c flux L thugh the nduct accng t the equatn: L Usng aaday's law we can detemned the esultng emf knwn as self nduced emf. E L I unt f L : the Heny (symbl: H) An nduct has nductance L H f a cuent change f A/s esults n a self-nduced emf f V. E ( t) t ( e ) L τ L ccuts Cnsde the ccut n the uppe fgue wth the swtch n the mddle pstn. At t 0 the swtch s thwn n pstn a and the equvelent ccut s shwn n the lwe fgue. It cntans a battey wth emf E, cnnected n sees t a esst and an nduct L (thus the name " L ccut"). Ou bjectve s t calculate the cuent as functn f tme t. We wte Kchhff's lp ule statng at pnt x and mvng aund the lp n the clckwse ectn. L + E 0 L + E The ntal cntn f ths pblem s: (0) 0. The slutn f the ffeental equatn that satsfes the ntal cntn s: E t L ( t) ( e ) The cnstant τ s knwn as the " tme cns tant" f the L ccut. 5

6 6/0/0 E t L ( t) ( e ) Hee τ t ( ) The vltage acss the esst V E e. t The vltage acss the nduct VL L Ee The slutn gves 0 at t 0 as equed by the ntal cntn. The slutn gves ( ) E / The ccut tme cnstant τ L / tells us hw fast the cuent appaches ts temnal value. ( t τ ) ( 0.6 )( E / ) ( t τ ) ( )( E / ) ( t 5 τ ) ( 0.99 )( E / ) If we wat nly a few tme cnstants the cuent, f all pactcal pupses has eached ts temnal value ( E / ). (0 9) L U Enegy sted n a c feld We have seen that enegy can be sted n the electc feld f a capact. In a smla fashn enegy can be sted n the c feld f an nduct. Cnsde the ccut shwn n the fgue. Kchhff's lp ules gves: E + + The tem E descbes the ate at whch the batte delves enegy t the ccut L If we multply bth sdes f the equatn we get: E L The tem s the ate at whch themal enegy s pduced n the esst Usng enegy cnsevatn we cnclude that the tem L s the ate at whch enegy s sted n the nduct. du L du L We ntegate bth sdes f ths equatn: U L L L (0 0) l u Enegy densty f a c feld Cnsde the slend f length l and lp aea A that has n wnngs pe unt length. The slend caes a cuent that geneates a unfm c feld n nsde the slend. The c feld utsde the slend s appxmately ze. The enegy U sted by the nduct s equal t L Ths enegy s sted n the empty space whee the c feld s pesent U We defne as enegy densty u whee V s the vlume nsde V n A n n the slend. The densty u l Al n Al Ths esult, even thugh t was deved f the specal case f a unfm c feld, hlds tue n geneal. (0 ) Mutual Inductn Cnsde tw nducts whch ae placed clse enugh s that the c feld f ne can nfluence the the. In fg.a we have a cuent n nduct. That ceates a c feld n the vcnty f nduct. As a esult, we have a c flux M thugh nduct. If cuent vaes wth tme, then we have a tme vayng flux thugh nduct and theefe an nduced emf acss t. E M E M M s a cnstant that depends n the gemety f the tw nducts as well as the elatve pstn. (0 ) (0 ) E M (0 4) E M E M In fg.b we have a cuent n nduct. That ceates a c feld n the vcnty f nduct. As a esult, we have a c flux M thugh nduct. If cuent vaes wth tme, then we have a tme vayng flux thugh nduct and theefe an nduced emf acss t. E M M s a cnstant that depends n the gemety f the tw nducts as well as the elatve pstn. It can be shwn that the cnstants M and M ae equal. M M M The cnstant M s knwn as the " mutual nductance" between the tw cls. Mutual nductance s a cnstant that depends n the gemety f the tw nducts as well as the elatve pstn. The I unt f M : the Heny (H) The expessns f the nduced emfs acss the tw nducts becme: M M E E M M 6

7 6/0/0 Electns ae gng aund a ccle n a cunteclckwse ectn as shwn. At the cente f the ccle they pduce a c feld that s: Lng paallel wes cay equal cuents nt ut f the page. ank accng t the magntude f the c feld at the cente f the squae, hghest fst. (Paenthess means the same magntude). e A. nt the page. ut f the page C. t the left D. t the ght E. ze. C,D, (A,). A,, (C,D)., A, C, D 4. D, (A, ), C Lng, staght, paallel wes cay equal cuents nt ut f page. ank accng t the magntude f the fce n the cental we.. d, c, a, b. a, b, c, d. b, c, d, a 4. c, a, b, d 5. b, d, c, a 7