ECE 307: Electricity and Magnetism Fall 2012

Transcription

1 C 307: lecriciy an Magneism Fall 01 Insrucor:.D. Williams, ssisan Professor lecrical an Compuer ngineering Uniersiy of labama in Hunsille 40 Opics uiling, Hunsille, l Phone: (5) , john.williams@uah.eu Course maerial pose on UH ngel course managemen websie Texbook: M.N.O. aiku, lemens of lecromagneics 5 h e. Oxfor Uniersiy Press, 009. Opional Reaing: H.M. hey, Di Gra Curl an all ha: an informal ex on ecor calculus, 4 h e. Noron Press, 005. ll figures aken from primary exbook unless oherwise cie.

2 Topics Coere Faraay s aw Chaper 9: Maxwell s quaions Transformer an Moional lecromoie Forces Displacemen Curren Magneizaion in Maerials Maxwell s quaions in Final Form Time arying Poenials (Opional) Time Harmonic Fiels (Opional) Homework: 3, 7, 9, 1, 13, 1, 18, 1,, 30, 33 8/17/01 ll figures aken from primary exbook unless oherwise cie.

3 Faraay s aw (1) We hae inrouce seeral mehos of examining magneic fiels in erms of forces, energy, an inucances. Magneic fiels appear o be a irec resul of charge moing hrough a sysem an emonsrae exremely similar fiel soluions for mulipoles, an bounary coniion problems. o is i no logical o aemp o moel a magneic fiel in erms of an elecric one? This is he quesion aske by Michael Faraay an oseph Henry in The resul is Faraay s aw for inuce Inuce elecromoie force () (in ols) in any close circui is equal o he ime rae of change of magneic flux by he circui N where, as before, is he flux linkage, is he magneic flux, N is he number of urns in he inucor, an represens a ime ineral. The negaie sign shows ha he inuce olage acs o oppose he flux proucing i. The saemen in blue aboe is known as enz s aw: he inuce olage acs o oppose he flux proucing i. xamples of generae elecric fiels: elecric generaors, baeries, hermocouples, fuel cells, phooolaic cells, ransformers. 8/17/01 3

4 Faraay s aw () To elaborae on, les consier a baery circui. The elecrochemical acion wihin he baery resuls an in prouce elecric fiel, f cuminae charges a he erminals proie an elecrosaic fiel e ha also exis ha couneracs he generae poenial The oal generae in he beween he wo open erminals in he baery is herefore Noe he following imporan facs n elecrosaic fiel canno mainain a seay curren in a close circui since l 0 IR n -prouce fiel is nonconseraie l P N f e f e f l 0 l xcep in elecrosaics, olage an poenial ifferences are usually no equialen 8/17/01 4 P N P N e f l l IR

5 For a single circui of 1 urn Transformer an Moional lecromoie Forces (1) l The ariaion of flux wih ime may be cause by hree ways 1. Haing a saionary loop in a ime-arying fiel. Haing a ime-arying loop in a saic fiel 3. Haing a ime-arying loop in a ime-arying fiel saionary loop in a ime-arying fiel l One of Maxwell s for ime arying fiels 8/17/01 5

6 Transformer an Moional lecromoie Forces () ime-arying loop in a saic fiel F Q u F F by okes' s Theorem in a moional fiel Fm u Q l u l m m m u Il Il ul m m u l ome care mus be use when applying his equaion 1. The inegral of presene is zero in he porion of he loop where u=0. Thus l is aken along he porion of he lop ha is cuing he fiel where u is no equal o zero. The irecion of he inuce fiel is he same as ha of m. The limis of he inegral are selece in he irecion opposie of he inuce curren, hereby saisfying enz s aw 8/17/01

7 Transformer an Moional lecromoie Forces (3) ime-arying loop in a ime-arying fiel m l u u l One of Maxwell s for ime arying fiels 8/17/01 7

8 Transformer an Moional lecromoie Forces: xample1 Conucing elemen is saionary an he magneic fiel aries wih ime m ssume he bar is hel saionary a y =0.08 m an = 4cos(10 )a z mwb/m ssume he lengh beween he wo conucing rails he bar slies along is 0.0 m (4)( xy(4)(10 3 8/17/ sin(10 )(10 )(10 (4)(10 ) )sin(10 )sin(10 3 )(10 (0.004 cos(10 ) xy ) )sin(10 ) )) aˆ z

9 Conucor moes a a elociy u = 0a y m/s in consan magneic fiel =4a z mwb/m ssume he lengh beween he wo conucing rails he bar slies along is 0.0 m m Transformer an Moional lecromoie Forces: xample l 0aˆ 0.08x 0.08x m y u u 0.004aˆ z l xaˆ x 8/17/01 9

10 Transformer an Moional lecromoie Forces: xample 3 Conucor moes a a elociy u = 0a y m/s in ime arying magneic fiel =4cos(10 -y)a z mwb/m ssume he lengh beween he wo conucing rails he bar slies along is 0.0 m 0(10 8/17/01 10 l 3 3 0aˆ y (10 )(4)cos(10 y) aˆ z (10 )(4)cos(10 3 (10 )(4) 10 sin(10 y) aˆ ˆ z xyaz 3 )(4) cos(10 y) x 3 3 (10 )(4) cos(10 y) x 10 (4) cos(10 3 )(4) cos(10 y) x 3 3 (10 )(4) cos(10 y) x 10 (4) cos(10 3 )(4) cos(10 y) x 3 3 (10 )(4) 8(10 ) cos(10 y) x 10 0(10 0( x cos(10 40 cos(10 y) 4000x cos(10 y) 40 cos(10 ) ) u y) aˆ ) x ) x l z xyaˆ xaˆ (4) cos(10 x z ) x

11 Displacemen Curren (1) es now examine ime epenen fiels from he perspecie on mpere s aw. H H 0 0 H H 0 D D H This ecor ieniy for he cross prouc is mahemaically ali. Howeer, i requires ha he coninuiy eqn. equals zero, which is no ali from an elecrosaics sanpoin! D Thus, les a an aiional curren ensiy erm o balance he elecrosaic fiel requiremen D We can now efine he isplacemen curren ensiy as he ime eriaie of he isplacemen ecor noher of Maxwell s for ime arying fiels This one relaes Magneic Fiel Inensiy o conucion an isplacemen curren ensiies 8/17/01 11

12 Displacemen Curren () Using our unersaning of conucion an isplacemen curren ensiy. e s es his heory on he simple case of a capaciie elemen in a simple elecronic circui. D H D I H l I H l H l 1 I enc enc I 0 D ase on he equaion for isplacemen curren ensiy, we can efine he isplacemen curren in a circui as shown mpere s circui law o a close pah proies he following eqn. for curren on he firs sie of he capaciie elemen Howeer surface is he opposie sie of he capacior an has no conucion curren allowing for no enclose curren a surface Q I If =0 on he secon surface hen mus be generae on he secon surface o creae a ime isplace curren equal o curren on surface 1 8/17/01 1

13 Displacemen Curren (3) how ha I enc on surface 1 an Q/ on surface of he capacior are boh equal o C(/) D D I from surface 1 I c Q s D C C 8/17/01 13

14 Maxwell s Time Depenen quaions I was ames Clark Maxwell ha pu all of his ogeher an reuce elecromagneic fiel heory o 4 simple equaions. I was only hrough his clarificaion ha he iscoery of elecromagneic waes were iscoere an he heory of ligh was eelope. The equaions Maxwell is creie wih o compleely escribe any elecromagneic fiel (eiher saically or ynamically) are wrien as: Differenial Form Inegral Form Remarks D 0 H D D l H l Gauss s aw Nonexisence of he Magneic Monopole Faraay s aw mpere s Circui aw 8/17/ D

15 8/17/01 15 few oher key equaions ha are rouinely use are lise oer he nex couple of slies Maxwell s Time Depenen quaions () 0 ˆ ˆ ˆ 0 ˆ n s n n n a a D D K a H H a Coninuiy quaion Compaibiliy quaions ounary Coniions for Perfec Conucor ounary Coniions quilibrium quaions 0 m m D H D m = free magneic ensiy 0 0 H 0 n 0 orenz Force aw u Q F

16 Maxwell s Time Depenen quaions: Ieniy Map 8/17/01 1

17 8/17/01 17 Time arying Poenials poenials for Coniion orenz pply g choo by coniions fiel ecor he imi yiels ieniy ecor he pplying D H aw Circui s mpere pplying 0 : : sin : : 1 1 : ' aw s Faraay pplying from of Definiion R R poenials Fiel 0 : ' : 4 4 :

18 8/17/01 18 Wae quaion u c n c u yiels space free In Refracie inex pee of he wae in a meium pee of ligh in a acuum