Chapte 31 Faaday s Law Change oving --> cuent --> agnetic field (static cuent --> static agnetic field) The souce of agnetic fields is cuent. The souce of electic fields is chage (electic onopole). Altenating agnetic field --> change the flux --> ef Changing agnetic flux --> induced ef --> induced cuents Motion of a conducto in a agnetic field --> otional ef electic flux: EA --> agnetic flux: E da E da Unit of agnetic flux: webe (T ) <--> Gauss s Law: <--> Gauss s Law: da E da Q enclosed ε 31.1 Faaday s Law of Induction Expeientalists, Faaday and Heny et al., found that a ef is induced in the cicuit if the agnetic flux though the suface bounded by the cicuit is changed. The ways fo changing the agnetic flux: 1. the cuents that geneate the agnetic field incease o decease. the peanent agnet oves fowad o backwad 3. the cicuit is otating in a static agnetic field 4. the cicuit is oving in a nonunifo agnetic field 5. the aea of the cicuit is inceasing o deceasing Faaday s law: ε d d ( A) ( A ) d ε cosθ 1
Exaple: A unifo agnetic field akes an angle of 3 o with the axis of a cicula coil of 3 tuns and a adius of 4 c. The agnitude of the agnetic field inceases at a ate of 85 T / s while its diection eains fixed. Find the agnitude of the induced ef in the coil. A Nπ cosθ, ε d d N π cosθ Exaple: A agnetic field is pependicula to the plane of the page. is unifo thoughout a ciculaegion of adius. Outside this egion, equals zeo. The diection of eains fixed and the ate of change of is d/. What ae the agnitude and diection of the induced electic field in the plane of the page (a) a distance < fo the cente of the egion and a distance > fo the cente, whee. d < : π, E dl da --> d π E π C d > : π, E dl da --> d π E π C Copae with the Apee s law: dl µ I π µ I enc, enc Exaple: A sall coil of N tuns has its plane pependicula to a unifo static agnetic field. The coil is connected to a cuent integato. Find the chage passing though the coil if the coil is otated though 18 o about the axis. NAcosθ, dq d dθ I ε NAsinθ NAsinθ dq dθ --> π NA Q sin d θ θ NA
31. Motional EMF The ef induced in a conducto oving though a agnetic field is called otional ef. A conducting od oves in a agnetic field. qe qv E v V El vl ε lx ε lv I ε d l lv Phenoena of The Induced ef and Cuents: I --> ε? h f Loentz f pull v 1. You use the foce f pull to pull the wie.. You found that the wie is oving at a constant velocity v. 3. Thee ust be one opposite foce that cancels the pulling foce. 4. The foce geneated by the agnetic field and doing on the (oving) chage ay be Loentz foce f Loentz. 5. The electic field that induced the cuent I o dive the chage q ove ust be elated to the Loentz foce. 6. The chage is oving to the ight at a constant speed of v. The chage will be exeted with a Loentz foce of f qv qe that dive the cuent I to flow. 7. The diving electic potential of the induced cuent I is V El vl 3
Exaple: A od of ass and esistance slides on fictionless conducting ails with a sepaation distance of l in a egion of static unifo agnetic field. An extenal agent is pushing the od, aintaining its otion to the ight at constant speed v. At tie t, the agent abuptly stops pushing the od continuous fowad. The od is slowed down by the agnetic foce. Find the speed v of the od as a function of tie. F qv Il, dv l --> Diffeential Eq: v vl I ε l --> F a Il v Solve the diffeential eq: v v e l t The geneal equation fo otional ef: V vl ( v ) l --> dv ( v ) dl --> dv ( v ) In static agnetic field & without otation: dv v dl Othewise: ε dv ( ) ( v ) d dl dl Exaple: Motional ef induced in a otating ba. dv v dl ω ( ) ( )d l 1 ε ω d ω l 31.3 Lenz s Law Lenz s Law: The induced ef is in such a diection as to opposite, o tend to oppose, the change that poduces it. The wod change is a key wod in the expession of the Lenz s law. 4
When a agnetic flux though a suface changes, the agnetic field due to any induced cuent poduces a flux of its own though the sae suface and in opposite to the change. I Does the induced ef still exist when no cicula loop is placed? Exaple: Find the diection of the induced cuent in the loop. The agnetic flux changes by inceasing o deceasing the cuent: Exaple: A ectangula coil of N tuns, each of wih a and length b; whee N 8, a c, and b 3 c; is located in a agnetic field.8 T diected into the page, with only half of the coil in the egion of the agnetic field. The esistance of the coil is 3 Ω. Find the agnitude and diection of the induced cuent if the coil is oved with a speed of /s (a) to the ight, (b) up, and (c) down. (a) (b) NA Nax, up: >, down: < ε --> d Na ε I 5
31.4 Induced ef and Electic Fields Wok: qε Fs qeπ ε Eπ A π, ε π d d E ε π π π d d d d E dl π Exaple: A long solenoid of adius has n tuns of wie pe unit length and caies a tie-vaying cuent that vaies sinusoidally as I Iax cos( ωt), whee I ax is the axiu cuent and ω is the angula fequency of the ac cuent souce. (a) Deteine the agnitude of the induced electic field outside the solenoid at a distance > fo its long cental axis. (b) What is the agnitude of the induced electic field inside the solenoid, a distance fo its axis? in µ ni cos ax (a) d E dl (b) d πe ( ωt) d ( ) πe π µ ni cos( ωt) ( π µ ni cos( ωt) ) ax ax 31.5 Geneatos and Motos 6
Acosθ d dθ ε N NAsinθ NAω sinθ DC geneato: 31.6 Eddy Cuents A changing flux sets up ciculating cuents --> eddy cuents (Lenz s law) 1. in the tansfoe, eddy cuent which geneates Joule heating should be pevented. eddy cuent in induction cooke (induction oven) 3. eddy cuent is used foapid heating What is induction heating? Fig 1 Induction heating is a noncontact heating ethod Fig Heat enegy (E) poduced in an electic cicuit is equal to I _. Induction heating (Fig. 1) is a noncontact heating ethod; one in which an electically conductive ateial (typically a etal) is heated by an altenating agnetic field. Invisible lines of foce ae ceated by a wok coil when a cuent flows though it, the esult of which is an induced cuent in the conductive wokpiece. Heating esults due to the Joule effect and, to a lesse degee, agnetic hysteesis (i.e., powe loss othe than by eddy cuents in a agnetic ateial caused by evesals of the agnetic field). Joule s Law (Fig. ) states that the ate at which heat enegy is poduced in any pat of an electic cicuit is easued by the poduct of the squae of the cuent (I) ties the esistance () of that pat of the cicuit. ef: http://www.industialheating.co/cda/aticleinfoation/featues/np Featues Ite/,83,14816,.htl 4. eddy cuent is used to dap echanical vibation agnet coppe esistance 7