Equipment: Cow magnet and long copper tube and long aluminum tube, or cow magnet slug

Transcription

1 Mon., 3/30 Tues., 3/31 We., 4/1 Thus., 4/2 Fi., 4/ ,.7 Faaay & Emf & Inuctance ,.6 Enegy, Diff. fom, Supeconuctos Quiz Ch 22, Lab 10 Faaay s Law 23.1,2,7 Ampee- Maxwell, E&M Pulse RE27 HW21:RQ.12, 15,17; P.20, 23, 26 RE28 Equipment: Cow magnet an long coppe tube an long aluminum tube, o cow magnet slug Faaay s law: The inuce emf fo a close loop is equal to the ate of change of the magnetic flux on the aea enclose by the loop, emf Φ mag =. t Whee emf = l an Φ mag = E NC A n ˆ A. (Tm 2 = Webe) The book points out that, since 0 = l, we coul just as well wite emf = E l E C Review: julie s Lec W12_3, all questions; W13_1 Questions 1-3 Diffeent books ub slightly iffeent things Faaay s Law, but they ae often sloppy about specifying exactly what they mean. Ou book seems to be esticting the efinition to just the emf that aises because of a change in magnetic fiel. Look at Clicke questions 2 & 1. Relationship between Faaay s law an motional emf: Let s look again at the cicuit in a unifom magnetic fiel with a moving ba.

2 F L Recall that we efine emf as (non-coulombic) wok pe unit chage, so emf =. q Peviously, we foun the motional emf is vl by consieing the magnetic foce qv on the electons in the moving ba. Faaay s Law can be genealize to inclue motional emf as well as that ue to changing. Fo my own amusement (afte Pucell): Φ Φ = + t t A A A A A f i = A + = A + ( v l ) = A + = A + = A ( v ) A = ( ) v A A A = ( v l ) = A ( v ) A + ( l l ) (whee l is how much the boe has move uing.) Faaay s Law elates the fist tem in the integal to the non-coulombic electic fiel: A = E l = emf wheeas v A = v l = emf ( ) ( ) motional so, = emf + emfmotional = emf Φ l Fo moe geneal consumption: While this can be one in tems of the integal fo magnetic flux, the same esult can be moe easily obtaine by consie the change in flux though a iffeentially small patch of aea, ove which is elatively constant, = ( A) = A ( ) + ( A) = emf + L = emf + Lv = emf emf A x Monay, Mach. 30,

3 Some texts call this Faaay s Law. Whateve we call it, the fact of the matte is that, eithe way you magnetic flux changes, emf is inuce: Φ = emf. Φ While we eive this just consieing a iffeential patch of aea (ove which is unifom an along which v is constant) the final esult hols fo a much lage aea (ove which is not unifom an along which v is not constant.) The justification is that a geneal aea can be constucte out of a quilt of such iffeential patches. Rotation changes A pepenicula, an flux inuces emf. The geneal esult even hols when the aea o fiel is changing iection (say, via otation). Recall the otating coil that we consiee in a pevious chapte. Suppose the coil stats pepenicula to the magnetic fiel an otates at a ate ω. The angle of the coil is θ = ωt. The magnetic flux though the loop is: ( ωt) Φ( t) mag = A( t) = hw cos, whee A is the aea pepenicula to magnetic fiel. The inuce emf is: emf = mag = ωhwsin( ωt). Summay: An emf aoun a loop can esult fom a changing magnetic flux in two ways: the magnetic fiel can change the aea of the loop can change (in magnitue o oientation) The magnetic flux is Φ mag = A, so the emf is: emf = mag = ( A)= A A +. Monay, Mach. 30,

4 Thee ae some subtle iffeences between the two tems. D. If the magnetic fiel is changing, thee is a non-coulomb electic fiel E NC that culs in the egion of changing flux. That means that emf = E NC l 0. So the foce iving a cuent aoun a loop is electic. DA. If the aea is changing because pat of the loop is moving though a magnetic fiel, thee is a magnetic foce which ives electons aoun the loop. The non-coulomb foce in this case is the magnetic foce. In both cases, the size of the emf is equal to the ate of change of the magnetic flux. Some situations can be seen as one o the othe effect epening on the efeence fame but thee is not always one fame that woks fo all pats of the cicuit, so that moe geneally only woks locally. Coils: Suppose thee is a coil with N tuns, instea of a single loop. If the tuns ae all close togethe, the inuce electic fiel E NC is appoximately the same fo each one. The emf between the ens of the coil is the integal of E NC along the entie length: emf = E NC l = N N tuns one tun E NC l = N ( emf one tun). If the magnetic flux though one loop is Φ mag, the emf can also be witten as: Seveal ways to change the Magnetic Flux: mag emf = N. Execise Come up with ways to change the magnetic flux though a coil using eithe a secon coil o a pemanent magnet All of the following will esult in an inuce emf in the coil 2 on the ight. 1. Change the cuent in coil 1 I Move coil 1 (with cuent though it) v Move coil 2 (with cuent though coil 1) Monay, Mach. 30,

5 4. Rotate coil 1 v Rotate coil Move the magnet elative to the coil (inclues moving coil towa magnet) S 6. Rotate the magnet S 7. Rotate the coil S N N N v Clicke Questions 22.2 (about magnet oppe own tube) This is one of the situations escibe moving the magnet elative to the coils / tube walls. Demo: op a magnet own a coppe tube (not feomagnetic) vey slow compae to fee fall! Each coss section of the pipe can be consiee a loop. Thee will be inuce cuents aoun the pipe. These in tun pouce magnetic fiels, so it s like having two magnets inteact. You will explain the slowing in tems of foces in Pob (c). Inuctance: Faaay s Law tells us how an electic fiel an associate emf ae elate to a changing magnetic fiel. Of couse, a changing magnetic fiel is cause by a changing cuent, so we shoul be able to iectly elate the emf to its oot cause the changing cuent. We ll o that fo the simple case of a solenoi an then genealize. Monay, Mach. 30,

6 Suppose you have a cicuit with a loop o loops of wie (even the simplest cicuit makes one loop!). When cuent flows, thee is a magnetic fiel pouce that passes though the loop(s) which is popotional to the cuent. Theefoe, thee is a magnetic flux that is popotional to the cuent. Call the popotionality the inuctance (o self-inuctance ) L, so: Φ mag = LI. When the cuent changes thee will be an inuce emf accoing to Faaay s law: I emf in = = L. The negative sign inicates that the inuce emf will point against the changing cuent (upsteam if the cuent ties to incease, ownsteam if it ties to ecease), Consie the example of a single loop of wie with esistance R connecte to a vaiable powe supply. What iection is the emf? I (inceasing) vaiable powe supply If the powe supply is tune up an the cuent flowing CW inceases, the change in the magnetic fiel is ownwa. Since is upwa, the change in magnetic flux will inuce an emf that will act like a backwa battey which opposes the change in cuent. Recall the simulation I showe you last time this comes fom the electic fiel that points opposite to the chages acceleations. Example: inuctance of a solenoi with a aius R, length, an N tuns (tightly woun) The magnetic fiel insie the solenoi is: = µ 0 NI, along the axis of the solenoi. The magnetic flux is (appoximately) the same fo each tun of the solenoi, so: Φ mag = NΦ mag = N[ A]= N µ 0NI solenoi one tun Theefoe, self-inuctance of the solenoi is: ( πr 2 ) = µ 0N 2 πr I. Monay, Mach. 30,

7 L = µ 0 N 2 πr 2. We ll look at two examples of inuctos in cicuits. The symbol fo an inucto in a schematic cicuit iagam is a coil (because that s what they usually ae). RL Cicuit: a battey is connecte to a esisto an inucto in seies at time t = 0 R emf battey I L The loop ule in the CW iection (with the conventional cuent I) gives: V battey + V esisto + V inucto = 0 emf battey IR L I = 0. The secon tem is negative because the electic potential ops acoss a esisto in the iection the cuent moves. The thi tem is negative because the emf of the inucto is in the opposite iection as that of the battey. The solution with the initial conition that I( t = 0)= 0 is (check by substituting it back into the iff. eq.): Featues of the solution: I( t)= emf battey 1 e R The inucto is most impotant at ealy times when the cuent is changing apily. As the cuent changes moe slowly, the inucto becomes less impotant. Afte a lage enough time, the cuent is essentially constant an epens only on the battey an esisto: I emf battey R. The time constant fo the cicuit is L R, which etemines the length of the tansient behavio. All cicuits have some inuctance, but it is often vey small so the cuent comes to its final value quickly. LC Cicuit: a cicuit with just a capacito an an inucto (iealization because thee is always some esistance in a eal cicuit) R t L. Monay, Mach. 30,

8 C I L Suppose the uppe plate of the capacito has an initial chage of +Q i at t = 0. The loop ule in the CW iection (with the initial conventional cuent I) gives: The cuent is elate to the chage by: V capacito + V inucto = 0 1 I Q L C = 0. I = Q, (taking Q to measue the chage on the capacito) because the cuent is lage in the iection shown as the capacito ischages (as Q eceases). This gives: 1 C Q+ L 2 Q = Q = 1 Q. 2 LC The solution (confim by substituting it back in the iff. eq.) is: so: Q()= t Q i cos I( t)= Q = Q i 1 LC t LC sin 1 LC t, The cuent oscillates back an foth with a peio: T = 2π LC. In a eal cicuit with esistance, the cuent will also ecease ove time because enegy is issipate as heat. Monay, Mach. 30,