Inductance and Energy of B Maxwell s Equations Mon Potential Formulation HW8

Transcription

1 Wed. Fi Inductnce nd Enegy f Mxwell s Equtins Mn Ptentil Fmultin HW8 Whee we ve been Sttiny Chges pducing nd intecting vi Electic Fields Stedy Cuents pducing nd intecting vi Mgnetic Fields Whee we e ging Vying cuents nd chge distibutins pducing nd intecting with vying Electic nd Mgnetic Fields A step clse t

2 Genel Expessin f Chges whee E q q 4ππ F Q q qq 4ππ u 3 c v u + u nd intectin v V Q q F Q q Q E q + V q q q 4ππ F Q q u c v u 3 c c v u + u u 3 c v u + u + V c c v u + u Electic Depends n bseve s peceptin f suce chge s velcity nd cceletin Mgnetic Als depends n bseve s peceptin f ecipient chge s velcity 4_dite_field, kink.py

3 Mtinl Emf v Mve cnducting b css mgnetic field Mbile electns mve in espnse t mgnetic fce F mg ( e) v F Electn suplus ccumultes t ne end, deficiency t the elect Resulting electic field nd fce F + + F mg elect ( e)e gws until Felect + Fmg 0 e E + e v ( ) ( ) 0 Emf mg E v In tems f Vltge nd Emf: Emf mg Emf V mg F mg d E d V q vyˆ dy E dy V vl EL V Emf mg

4 Geneliztin f Flux Rule S(t) dl v d S(t+) d v dl Using vect identity () A ( C ) ( A ) C d ( v dl ) ( v) dl ( v ) dl Thus chnge in mgnetic flux thugh the lp dφ dl ( v ) te f chnge in mgnetic flux thugh the lp Φ F ( v ) mg dl dl q Φ Emf Wning: u deivtin used tht the chnging, d/, cespnded t mving chge, vdl. Nt pplicble when tht s nt the cse. th be pdxes (We will lte extend this esning t discuss sttiny chges but chnging fields) mg

5 Fdy s Lw F mg q dl Emf Φ t d vl v Which dives chges und the lp, vi mgnetic fce Fm the pespective f smene iding the lp Φ t Fm this pespective t we must see chges mve und the lp, thee must be fce ut mgnetic is defined s chge fce pptinl t chge s velcity; fm this pespective, thee is n v, s we cn t cll it mgnetic, hve t cll it electic. F elect q dl Emf

6 Fdy s Lw F mg q dl Emf Φ t d vl v Which dives chges und the lp, vi mgnetic fce Fm the pespective f smene iding the lp Φ t In mst genel cse Emf + Fm this pespective t we must see chges mve und the lp, thee must be fce d dφ Full time deivtive

7 Fdy s Lw F elect dl q qe dl d q t E dl d E d d E ( ) ciculting electic field is ccmpnied by time vying mgnetic field Fm the pespective f smene iding the lp th e pduced by time vying cuent nd chge distibutins

8 Oh, Inductin, let me cunt the wys induced emf in the cil n the ight I Chnge the cuent in cil v Mve cil (with cuent thugh it) Cme up with sme me

9 Inductin f the flling mgnet Nt mgnet Cppe pipe Why des the mgnet fll s slwly? Cppe pipe mgnet

10 Inductin f the flling mgnet Nt mgnet Cppe pipe Why des the mgnet fll s slwly? E dφ Emf Mens Emf s diectin is by left hnd ule und e cntining flux Cppe pipe mgnet 3_Fdy_Mgnet.py T the ight s dwnwd flux inceses T the left s dwnwd flux deceses Thus dives chges und pipe nd s tnsfes enegy. These chges in mtin pduce field which exets fce n mving chges in mgnets.

11 induced emf in the cil n the ight Lenz s Lw I Chnge the cuent in cil v Mve cil (with cuent thugh it) v Mve cil (with cuent thugh cil ) ntue bhs chnge in flux Induced Emf ( culed Electic field) dives cuent tht pduce mgnetic field which ptly cuntes the chnge in mgnetic flux. T the left s dwnwd flux deceses Cppe pipe mgnet Rtte cil (with cuent) T the ight s dwnwd flux inceses S N Dem! v Thus dives chges und pipe nd s tnsfes enegy. These chges in mtin pduce field which exets fce n mving chges in mgnets.

12 Using Fdy s Lw dφ E Φ Emf E dl e Exmple: vey lng slenid f dius with sinusidlly vying cuent such tht cs( ωt)zˆ. A cicul lp f dius / nd esistnce R is inseted. Wht is the cuent induced und the lp? ẑ I Dem!

13 Using Fdy s Lw dφ E Φ Emf E dl e Execise: vey lng slenid, with dius nd n tuns pe unit length, cies time vying cuent, I(t). Wht s n expessin f the electic field ẑ distnce s fm xis? Recll tht inside slenid µ Inz. ˆ I

14 Using Fdy s Lw dφ E Φ Emf E dl e Exmple: A slwly vying ltenting cuent, I( t) I 0 cs( ωt), flws dwn lng, stight, thin wie nd etuns lng cxil cnducting tube f dius. In wht diectin must the electic field pint? ẑ I I E E Lenz lw sys in the diectin t dive cuent tht wuld ppse chnging flux, s dwn nd up s the cuent vies up nd dwn. ẑ Wht s the electic field? E dl d Clls f n Ampein lp E dl in E dl E d E dl + tp E dl + whee ut ( s ) z E( s ) z in ds z ut E dl + µ 0 I ˆ φ s <, πs 0 s >. nd I( t) I 0 cs( ωt) bttm E dl µ I ds z µ I πs ln z π sin

15 Using Fdy s Lw dφ E Φ Emf E dl e Exmple: A slwly vying ltenting cuent, I( t) I 0 cs( ωt), flws dwn lng, stight, thin wie nd etuns lng cxil cnducting tube f dius. In wht diectin must the electic field pint? I I ẑ E Wht s the electic field? ( E( s ) E( s )) z in E Lenz lw sys in the diectin t dive cuent tht wuld ppse chnging flux, s dwn nd up s the cuent vies up nd dwn. ẑ Right-hnd-side is independent f hw f E dl d ut f lp s ut is, s E is cnstnt utside. ut it ut µ I ln z shuld be 0 quite f wy, π sin s must be 0 eveywhee utside. E ( ) s in E( s in ) I ln t π s in µ ( ) Iω sin ωt ln π sin µ µ I cs ( ωt ) π ln s in

16 Inductnce Wht is flux thugh pth due t cuent fllwing pth? d µ dl I 4π µ dl I d 4π Φ Φ µ dl I d 4π Puely gemetic fct Equivilntly, cn ephse using pduct ules, use A t get sme esult M, Φ d I M, A Φ ( ) A d A dl µ Idl A 4π µ dl dl Φ M,I Φ I 4π Φ M,I µ dl µ dl dl d Symmetic between 4π 4π tw lps

17 Φ I Φ M, I Inductnce µ dl µ d 4π 4π dl dl As with Resistnce, smetimes it s esiest t d the gemetic integl, smetimes it s esiest t find flux, fct ut cuent, nd thus find M. Exmple: Cxil slenids f dii > nd windings pe length n nd n. ẑ Φ π ( ) N N n µ 0 n I l M, Φ µ ni ( π ) nl ( µ ( ) ) nn π l I Ovelpping vlumes Fdy s Lw: time vying cuent in ne slenid induces Emf nd dives cuent in the Emf d. Φ d ( M I ), Dem!

18 Φ I L Self Inductnce µ dl µ d 4π 4π dl dl Cuent pssing lng the lp is itself espnsible f flux thugh the lp dφ d Emf ( LI ) Time vying cuent lng ne segment f the lp pduces field nd Emf felt by the segments f the sme lp.. ẑ Exmple: single slenid Φ ( π )N N nl µ ni 0 ni µ ( )nl Φ µ ( π ) π L ( n l)i vlume

19 Enegy t Genete Cuent Cnside diving chges und slenid. Hw much wk wuld yu hve t d t get it ging? ẑ As yu ccelete it up t speed, self inductnce mens cunte fce is geneted, s yu must t lest pvide equl nd ppsite fce. F dl Emf q S pe unit chge, q Emf W W q Then the te t which wk is dne by the inductnce s emf is dq Emf P IEmf P O using the self-inductnce eltinship d S binging the cuent up t speed, yu I ( LI ) P must ppse this, nd invest enegy t d ( LI ) P P yu d ( LI ) W yu LI

20 Enegy t Genete Cuent Cnside diving chges und slenid. Hw much wk wuld yu hve t d t get it ging? ẑ W yu ( ) LI L µ n τ µ 0 Rephsing in tems f the cespnding field tht s geneted, W yu ( µ n τ ) I µ n n W yu µ τ Extplting t me genel cses, W τ µ d Whee is the enegy sted, field cuent? Neithe / bth enegy isn t substnce (n clic fluid ) t be sted sme whee. It s kinetic nd ptentil enegy, it s sted in the mtin f chges nd thei intectins situtin f cuent flwing nd field geneted. (Giffiths des me genel deivtin much like he did f the wk f geneting E field.)

21 Enegy t Genete Cuent W µ dτ Execise: Wk t tun n c-xil slenids f diffeent wie density, n, nd ppsite cuent, I. ẑ F n individul slenid µ 0nI 0 inside utside

22 Wed. Fi Inductnce nd Enegy f Mxwell s Equtins Mn Ptentil Fmultin HW8