Phscs 1, Fall 01 6 Noember 01 Toda n Phscs 1: applcatons of nducton Generators, motors and back EMF Transformers Edd currents Vote toda! Hdropower generators on the Nagara Rer below the Falls. The ste of the world s frst commercal hdroelectrc generator, bult n 1881, les a few mles uprer on the US sde. (Nagara Falls Publc brar) 6 Noember 01 Phscs 1, Fall 01 1 Waterwheels, wndmlls and nducton Snce the thrd centur C, humans hae used gratdren flows of water, or wnd, to turn wheels. After Farada, nentors realed that rotatng a conductng loop about ts dameter n a magnetc feld generates current. So magnets and cols replaced the mllstones and pumps, and commercal electrc power was born. 6 Noember 01 Phscs 1, Fall 01 Magnetc AC generator Two ews of eternal torque rotatng loop about as: cost t 6 Noember 01 Phscs 1, Fall 01 3 (c) Unerst of Rochester 1
Phscs 1, Fall 01 6 Noember 01 Magnetc AC generator (contnued) cost d snt dt snt R At the nstant shown n the dagram, the loop s projected area s ncreasng, so the nduced EMF and current are flowng n the drecton that produces along +. 6 Noember 01 Phscs 1, Fall 01 4 Magnetc AC generator (contnued) Here s how one actuall gets the current out of the loop and nto an eternal crcut or power grd (from Gancol). Metal-brstled brushes make the (sldng) electrcal contacts. Usuall there are man turns to the loop, not just one. Wth N turns the EMF s N tmes larger than t would be wth a sngle turn. 6 Noember 01 Phscs 1, Fall 01 5 DC motor and back EMF The concept works n reerse, too. Run a DC current through the loop, and magnetc torque rotates the loop. t F3 6 Noember 01 Phscs 1, Fall 01 6 (c) Unerst of Rochester
Phscs 1, Fall 01 6 Noember 01 The loop gans angular momentum, so t wll oershoot the poston where = 0: perfect algnment of wth. t F3 6 Noember 01 Phscs 1, Fall 01 7 f the current drecton stas the same, the loop s poston wll now oscllate about the = 0 poston. t F3 6 Noember 01 Phscs 1, Fall 01 8 ut f, just after t passes that pont, the loop current s reersed, then the forces and torque reerse too, t t F 3 6 Noember 01 Phscs 1, Fall 01 9 (c) Unerst of Rochester 3
Phscs 1, Fall 01 6 Noember 01 and the loop keeps rotatng n the same drecton. So the trck s to swtch current polart just after passng through = 0. t 6 Noember 01 Phscs 1, Fall 01 10 Then eer par of postons 180 apart are the same ecept for segments and 4 beng swapped. t F 3 6 Noember 01 Phscs 1, Fall 01 11 Swappng the drecton of s generall done b hang the loop connected to ts power source wth metal-brstle brushes (as n the generator), n contact wth a splt crcular rng ( commutator ) t whch s fed to the loop and rotates wth t about ts as. As rotaton swaps the brushes from one half to the other of the commutator, the polart of the oltage on the loop swtches. 6 Noember 01 Phscs 1, Fall 01 1 (c) Unerst of Rochester 4
Phscs 1, Fall 01 6 Noember 01 That s not all there s to t, though: Snce the motor s loop rotates about ts dameter n a magnetc feld, there s an addtonal component of current and EMF for whch we haen t et accounted. the one that s there when the same components are functonng as an AC generator. Ths addtonal EMF opposes the EMF whch dres the rotaton. Thus the name back EMF. Fgures 7-4 and 7-3 n Gancol 6 Noember 01 Phscs 1, Fall 01 13 For eample: compare pages 6 (motor) and 3-4 (generator): the dre and generated currents alwas flow oppostel. t F3 6 Noember 01 Phscs 1, Fall 01 14 Suppose that the resultng motor turns at constant angular speed. Then, as we saw n the case of the generator, back snt back snt R When startng up ( near ero), the R total current s larger ( near ero). Ths s wh the lghts dm when powerful motors start up, but return to ther brghtness when the motor s up to speed. 6 Noember 01 Phscs 1, Fall 01 15 (c) Unerst of Rochester 5
Phscs 1, Fall 01 6 Noember 01 Transformers Consder two conductng cols, wth ther magnetc flu lnked b a ferromagnet as we consdered last lecture, and wth dfferent numbers of turns, NP and NSboth 1. Appl a snusodall-arng (.e. AC) oltage to the prmar sde. The flues n the two cols are related b NPP NSS dp ds NP NS dt dt (from Gancol) 6 Noember 01 Phscs 1, Fall 01 16 Transformers (contnued) ut the appled oltage and the flu derate are related b Farada s law: d V P P NP dt d S NS VS dt So f the loops are the same se, whch would requre P S, d VS VP dt NS NP (from Gancol) 6 Noember 01 Phscs 1, Fall 01 17 Transformers (contnued) Thus b lnkng dfferent numbers of loops, an AC oltage can be transformed nto a dfferent oltage, larger or smaller, wth ampltude gen b the rato of turns. Furthermore, snce energ s consered, P PVP SVS, and P P, or NSS NPP NSS NPP. (from Gancol) 6 Noember 01 Phscs 1, Fall 01 18 (c) Unerst of Rochester 6
Phscs 1, Fall 01 6 Noember 01 Edd currents Frst a smpler problem b wa of setup: A square conductng loop wth resstance R, on a sde, has two sdes parallel to a long wre carrng current. pull the loop awa from the wre at constant speed. How much current flows n the loop, and what force do eert? We know (5 October 01): 0 ˆ 0 ˆ n plane of loop r 6 Noember 01 Phscs 1, Fall 01 19 Edd currents (contnued) So the flu through the loop decreases as pull t awa: d 0 d 0 ln so a current must flow clockwse, as shown, to oppose the decrease. 6 Noember 01 Phscs 1, Fall 01 0 Edd currents (contnued) Use the chan rule to fnd : 1 d 1 d d R R dt R d dt 0 1 1 R 0 R 0 R Note that we could hae skpped a few steps b usng the fundamental theorem of calculus on last page. 6 Noember 01 Phscs 1, Fall 01 1 (c) Unerst of Rochester 7
Phscs 1, Fall 01 6 Noember 01 Edd currents (contnued) Magnetc forces cancel out on the segments to. The other two: 0 ˆ ˆ 3 0 ˆ R 0 F ˆ ˆ 3 0 ˆ 3 R 6 Noember 01 Phscs 1, Fall 01 F Dan Edd currents (contnued) s constant, so the net force on t s ero, and the force eert s FDan F 3 0 ˆ 1 1 R 3 0 ˆ R 0 Rˆ Rˆ R F Dan 6 Noember 01 Phscs 1, Fall 01 3 Edd currents (contnued) Whch means that the mechancal power eert s: dwdan PDan FDan dt 0 R R R Equal to the power dsspated n the loop, as seems approprate. F Dan 6 Noember 01 Phscs 1, Fall 01 4 (c) Unerst of Rochester 8
Phscs 1, Fall 01 6 Noember 01 Edd currents (contnued) Now, what f the loop s replaced wth a square sheet of conductor? Currents stll flow, but the can flow throughout the sheet, n the same general pattern as that of the square loop. Such orte-lke currents, nduced b flu changes through conductng sheets or olumes, are called edd currents. 6 Noember 01 Phscs 1, Fall 01 5 (c) Unerst of Rochester 9