( ) v2. Equipment: Inductance: Function Generator, nestable solenoids, oscilloscope. VPython. Intro: Faraday s Law as Correlation, not Causation

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1 Thus., 3/6 Fi., 3/7 Mn., 3/30 Tues., 3/3 Quiz Ch, Lab 9 Ampee s Law.-,0 Int t Faaday s & Lenz s R6.3-4,.7 Faaday & mf & Inductance R7 Lab 9 Wite-up HW:RQ., 5,7; P.0, 3, 6 quipment: Inductance: Functin Geneat, nestable slenids, scillscpe. VPythn Int: Faaday s Law as Celatin, nt Causatin Lk up the maste fce equatin fm the back f Giffiths Cntextualizing Faaday s law Statinay Chages. In the fist chapte f this bk, we met the electic inteactin f tw statinay chages: = qq F 4 3, f cnvenience sake, πε q we bke this int tw facts, q and ( ) = 4 3. That allwed us t πε fcus n the electic field f a suce chage withut having t wy abut the the chage. Mving Chages. A few chaptes late, we cnsideed tw mving chages. The easiest way t tansitin fm statinay t mving is t fist analyze the situatin in the fame f ne f the tw chages (whee it appeas t be statinay), and then tansfm int a fame in which it s mving. What yu get has tw distinct qq tems: v F = [ ( )( )] + [ ( )( )] c v ˆ ˆ ˆ 3 c v c v c v 0( ( cˆ v )) c Whee = ( tnw) ( tthen), that is, the sepaatin between the pbe chage and whee the suce chage was back when it emitted the fields that the pbe is expeiencing nw. The fist tem in culy backets depends nly n the suce s velcity, while the secnd tem depends n the velcity f the test chage t. Identifying the fist as an electic inteactin and the secnd as a magnetic inteactin, we have v v v F = q ) + q v B ( ) ( Ding sme ephasing, in tems f cuent lcatins f bth chages gives ( ) ( ) v c q = ˆ 3 0 ( ( ˆ ) ) v c µ ( ) ( ) v c q B ( ˆ = v ) 3/ 4 π ( ( ˆ ) ) v c

2 Acceleating Chages. S, we stated with statinay chages, then went f mving chages; the next natual step is acceleating chages. Nw yu get qq v F = ( c v )( cˆ v ) + (( cˆ v ) a) + ˆ c v ˆ c v + 3 0( ( cˆ v) ) c (Giffiths 0.67) [ ] [ [( )( ) (( c v ) a ) ] ˆ Again, the fist clump f tems des nt depend n the test chage s velcity, and the secnd des. S the fist set is the electic inteactin and the secnd is the magnetic inteactin. Ntice that the electic inteactin includes a tem pinting in a diectin that is distinctly nt alng (since it s gt an a in it). Futheme, the fist tem pints fm whee the suce chage wuld nw be if it wee ging at a cnstant speed, but nt fm whee it actually is (since its acceleating). S thee ae explicit and implicit dependences n the acceleatin f the chage. Ntice that we have ne tem f the electic field that pints adially (u gd, ld Culmbic field) but we als have a tem that pints in sme diectin dependent n the acceleatin f the suces. This is the Faaday field. Hee s what it lks like f just ne acceleating chage. Genealize t cuents Nw, if we have whle cuents f chages, then these elatins get ephased, and mst geneally in tems f cuent densities and time vaying cuent densities (instead f velcities and acceleatins). It is the latte, time vaying cuent densities that ae espnsible f the Faaday lectic Field, a.k.a., the nn-culmbic field. Hee s what it lks like f a cuent f acceleating chages. (the cuent is mving and acceleating t the ight). Fiday, Mach 7, 009

3 Ntice that this electic field ppses the change in cuent. This is a vey geneal chaacteistic f the Faaday lectic Field. Cntadicting bk The bk, alng with mst intducty texts, slips and says that the nn-culmbic electic field is pduced by the time vaying magnetic field. That may well have been hw Faaday intepeted what he bseved, but ve a centuy has passed and we ve leaned a few things bth this electic field and a time vaying magnetic field ae pduced by the same thing time vaying cuent densities. Faaday s Law is undeniably cect, but it des nt epesent a causal elatin, it epesents a celatin between tw effects f a shaed cause. Getting Familia with Faaday s lectic Field The simplest example f seeing the Faaday effect is a slenid with a changing cuent. The magnetic field pints alng the axis inside the slenid and is appximately ze utside as shwn belw. Let s say a cunte clckwise ciculating cuent inceases, the efeencing the illustatin f an acceleated cuent s field, we shuld get a clckwise electic field. Inductin. Meanwhile, if the cunte clckwise cuent is inceasing, then the magnetic field it pduces is als inceasing. S, celated t the culed electic field is the inceasing magnetic field. The cmmn language is t say that the changing magnetic field induces the culed electic field, and this pcess is called magnetic inductin. Unftunately, this is histic, but Fiday, Mach 7, 009 3

4 inaccuate language (much like heat flw ) it s the changing cuent density that causes bth magnetic and electic effects. It is smetimes called a nn-culmb electic field NC. That said, the celatin between the fields is fa simple t expess than is the causatin between the cuents and the fields. Diectin f the Culy lectic Field: The ight hand ule is simila t the ne used t detemine the diectin f the magnetic field pduced by a cuent: With yu ight thumb pinted in the diectin f db dt, yu finges will cul in the diectin f NC. This is the ppsite f the diectin f the change in the cuent density that pduces the change in magnetic field. T use this ule, it is useful t knw that the diectin f the change in the magnetic field f a small time B is the same diectin as db dt. xamples: () B is ut and inceasing B B i The diectin f the change in the magnetic field B = B f B i is utwad as shwn belw. Side View (ut t ight) B B f The diectin f d B dt is inwad (because f the minus sign), s NC culs clckwise. B i B f Fiday, Mach 7, 009 4

5 NC B d B dt () B is in and deceasing (xecise f students) The diectin f the change in the magnetic field B = B f B i is utwad as shwn belw. Side View (ut t ight) B f B B i The diectin f d B dt is inwad (because f the minus sign), s NC culs clckwise. Faaday s Law: Magnetic flux is defined as: Φ = B da mag, We aleady encunteed this in Ch ; hweve then we wee thinking f the flux ut thugh a clsed suface (like a ball), and that summed t 0. Nw we e just cnsideing the flux thugh an pen suface (like sheet f pape). The induced emf f a clsed lp is defined as: emf = dl. Faaday s law: F NC The induced emf f a clsed lp is equal t the ate f change f the magnetic flux n the aea enclsed by the lp, emf F Φmag =. t Ntatin: the subscipt F indicates that we e just talking abut the emf due t Faaday s effect and the A indicates that we e hlding Aea cnstant. We ll late see that hlding B cnstant A Fiday, Mach 7, 009 5

6 and allwing A t vay als gives ise t an emf what we ve aleady cme t call mtinal emf. The minus sign is t emind yu hw t detemine the diectin f NC fm db dt. This is a new law, which cannt be deived fm what we have dne peviusly. Mind, t the extent that all magnetic phenmenn can be deived fm electic nes and the ppe elativistic tansfmatins, this can be easned (I ve seen a numbe f papes d s); hweve, while that is, n sme philsphical level, cmfting, pactically, it s t much f a pain t d. xample: A lp aund a slenid Suppse the slenid has a adius and B f the slenid is ut and inceasing at a cnstant ate db dt. Cnside a lp f adius centeed n the slenids axis. By symmety, the nn-culmb electic field NC must be the same size eveywhee n the lp, s: emf = NC d l = NC ( π) The flux n the suface suunded by the lp depends n the size f the lp: () > magnetic flux inside whle slenid cntibutes, s: Φ mag = B ( π ) B In this case: which falls ff as /. emf = dφ mag dt NC ( π)= db dt NC = db dt ( π ) ( ) () < nly magnetic flux inside lp cntibutes, s: ( ) Φ mag = B π, Fiday, Mach 7, 009 6

7 In this case: which inceases as. Tw Slenids. Retuning t emf F A ( ) NC ( π)= db π dt ( ) NC = db dt, Φmag =, If we have just ne lp, it has aea A, and flux BA. If we have t tw lps, it has aea A, and flux BA, N lps means flux NBA, and cespnding Φmag emf F = N t A While the emf aund ne lp may be t small t measue. The emf alng a cil f 500 lps, i.e., alng a slenid, may be quite measuable. Dem: One slenid nested inside anthe. Lp utside slenid What abut a lp that des nt suund the slenid with a changing cuent? Thee ae tw ways t explain why the emf f the lp is ze: The magnetic field utside the slenid is appximately ze, s thee is n change in the magnetic flux as the cuent changes. The length f the acs is pptinal t and the size f the nn-culmb electic field NC is pptinal t / (utside the slenid). Ntice that NC pints alng d l f ne ac and ppsite t d l f the the ac. NC is pependicula t the staight segments. The shape f the lp desn t matte (any lp can be appximated as acs and adial segments). F example, the eadings n ammetes vltmetes just depends n whethe nt the lp is aund the egin whee the magnetic flux is changing. Fiday, Mach 7, 009 7

8 Fiday, Mach 7, 009 8

9 What fllws ae backgund ntes f the instuct. (Nte: Giffiths illustates the equivalence f the tw appaches: using Faaday s law with etaded ptentials and using Culmb s law with elativistic tansfmatins f the simple case f cnstant velcity. The deivatin f q n 0.68 uses V and A (and thus Faaday s) and etaded ptential while the deivatin f the same elatin, nw dubbed q n.9 uses just Culmb s and elativistic tansfmatins.) Pi t this chapte, we d seen that an electic field is geneated by chages. F a simple, statinay chage, Culmb s law gives: = 0 Whee the subscipt stands f bsevatin lcatin. If we have a distibutin f pint chages, the field is fund by the Supepsitin Pinciple: q ˆ, ( ) qi ρ q = ˆ i ˆ i 0 i 0 q q dvl whee ρ is the chage density, which is a functin f psitin. Late, we nted that the electic field due t a mving chage is a little diffeent. One way t find this is t tansfm fm the fame in which the chage is statinay t ne in which it is mving at a cnstant speed v. Then, the electic field f a pint chage sensed by the bseve is = 0 ( v c) ( vsin θ c) ( ) 3 q whee q is the angle between v and. ˆ (Giffiths 3 d d..9) Nte: ne might be able t genealize this f an acceleating chage withut appealing t Geneal Relativity by successively tansfming t fames in which the chage is mving at incementally diffeent velcities. If we have a distibutin f pint chages, each mving at its wn velcity, I imagine that this wuld lk like ( vq c) ( v sin θ c) ( ) ( ) ρ q = ˆ qdvl 3 0 q q q Whethe the chage is mving nt, Gauss s law hlds: Fiday, Mach 7, 009 9

10 n ˆ da= q inside ε 0 div( ) = = ρ ε 0 Inteestingly, Giffith s als aives at.9 by a diffeent ut (labeling the equatin 0.68). The tw main featues f this the ut ae that it elates the electic field nw t the chage distibutin back then, in paticula, back when the field was emitted that, taveling at the speed f light, gets t us nw: t then = t nw /c. But in this case, Culmb s law is nt enugh, a new law / a new mechanism f geneating electic field is equied: Faaday s Law. One way f phasing this law is dj( q, t then ) Faaday( tnw) = dvl 4 (fm Giffiths 3 d 0.7,.8,.9) πε c dt O, putting it in slightly me familia fm: Faaday Faaday Faaday ( t ( t ( t nw nw nw ) = q ) = t c B( tnw) ) = t c J q (, t ) (, t ) dj(, t ) q dj q then q then dvl dt q + c dt q then dvl ˆ F a pint chage suce, athe than a cntinuus cuent, this is Faaday ( t nw ( c v)( a c v) + ( c v + a) qc ) = / v 3 ( c v) c F cmpaisn, the Clmbic tem is Cul ( t nw ) = qc ( c v ) 3 [( c v ) v ( c v + a) ] S the ttal electic field f a mving pint chage is thei sum: ( t nw ( c v) [( c v) v ( c v )( cˆ v ) + (( c v ) a )] qc ) = ˆ 3 Which educes t.9 / 0.68 if thee is n acceleatin. (0.63) (0.6) (0.65, w/.64) This is a ae causal phasing f Faaday s Law that an electic field is geneated by a changing cuent density. In essence, the ttal electic field depends nt just n the chage distibutin (Culmb s Law), but hw the flw f chages is changing (Faaday s Law). Recall that cuent is the suce f magnetic field. S this elatinship between electic field and cuent implies ne between electic field and time vaying magnetic field. That is the much Fiday, Mach 7, 009 0

11 me cmmn way f phasing Faaday s law, but, fm a philsphical stand pint, it s wth nting that the elatinship between the electic and time vaying magnetic fields is a celatin while the ne between electic field and time vaying cuent is a causatin time vaying magnetic field des nt cause an electic field, athe, time vaying cuent causes bth electic and time vaying magnetic fields. It s als inteesting t nte that, fm the pespective f any ne f the suce chages, it itself is nt mving, s it itself is exeting a fce n anthe chage nly via a culmbic electic field. Fiday, Mach 7, 009