7.2.3 Inductance. Neumann Formula for the Mutual Inductance. Important Things about Mutual Inductance

Transcription

1 7..3 Inductnce Neumnn Fomul o the Mutul Inductnce Two loops o wie t est. A stedy cuent I ound loop ome psses though loop Φ Φ d ( A ) d A dl I dl A I dl ˆ Φ d nd dl ˆ Φ [ d ] I M I The constnt o popotionlity: mutul inductnce o the two loops. I dl dl Φ dl dl M Neumnn omul It involves doule line integl --- one integtion ound loop, the othe ound loop. Impotnt Things out Mutul Inductnce M dl dl It is not vey useul o pcticl clcultion, ut it evels two impotnt etues.. M is puely geometicl quntity, hving to do with the size, shpe, nd eltive position.. M M, so we cn dop the suscipts nd cll them M. Whteve the shpes nd positions o the loops, the lux though when we un cuent I ound is identicl to the lux though when we send the sme cuent I ound. Advntge o M M, see the ollowing exmples. 3 NΦ M I ( 7 n N A Exmple A cicul coil with coss-sectionl e o 4 cm hs tuns. It is plced t the cente o long solenoid tht hs 5 tuns/cm nd coss-sectionl e o cm, s shown elow. The xis o the coil coincides with the xis o the solenoid. Wht is thei mutul inductnce? olution: Φ A n I A )(5)()(.4) 7.54 uf Notice tht lthough M M, it would hve een much diicult to ind Φ ecuse the ield due to the coil is quite nonuniom. 4

2 el-inductnce It is convenient to expess the induced em in tems o cuent the thn the mgnetic lux though it. The mgnetic lux is diectly popotionl to the cuent lowing though it. N Φ I whee is constnt o popotionlity clled the selinductnce o coil. The I unit o sel-inductnce is the heny (H). The sel-inductnce o cicuit depends on its size nd its shpe. The sel-induced em in coil due to chnges in I tkes the om E dt 5 Exmple 7. Find the sel-inductnce o tooidl coil with ectngul coss section (inne dius, oute dius, height h), which cies totl n tuns. ol: mgnetic ield NI Inside tooidl π s NΦ NI hni h ds nd Φ ln( / ) I πs π hn π ln( ) 6 Exmple A coxil cle consists o n inne wie o dius tht cies cuent I upwd, nd n oute cylindicl conducto o dius tht cies the sme cuent downwd. Find the sel-inductnce o coxil cle o length. Ignoe the mgnetic lux within the inne wie. olution: I I, dφ da dx πx πx I I Φ dx ln I πx π ln π Hint: The diection o the mgnetic ield. Cicuits How does the cuent ise nd ll s unction o time in cicuit contining n inducto nd esisto in seies? em I dt αt et + β α I Ie Ie dt αt e : α E : E β β E t : I β E t I ( e ) Hint: Wht hppens when consides the inne lux? 7 8 ise αt The quntity τ/ is clled the time constnt.

3 Decy Cicuits I dt αt et I Ie α Ie dt αt e : α E t : I t E t E τ I e e αt The quntity τ/ is clled the time constnt Enegy in Mgnetic Field Inductnce (like cpcitnce) is n intinsiclly positive quntity. enz s lw dicttes tht the em is in such diection s to oppose ny chnge in cuent ck em. It tkes cetin mount o enegy to stt cuent lowing in cicuit. Wht we e concened with e the wok you must do ginst the ck em to get the cuent going. Is this ixed mount? Is it ecovele? Yes, you get it ck when the cuent is tuned o. It epesents enegy ltent in the cicuit o it cn e egd s enegy stoed in the mgnetic ield. Enegy toed in n Inducto The ttey tht estlishes the cuent in n inducto hs to do wok ginst the opposing induced em. The enegy supplied y the ttey is stoed in the inducto. In Kichho s loop ule, we otin di E i + dt di ie i + i dt du ie i +, whee U I dt powe supplied y the ttey powe dissipted in the esisto enegy chnge te The owe The wok done on unit chge, ginst the ck em, in one tip ound the cicuit is E. the wok done y you ginst the em The totl wok done on pe unit time is The totl wok is dw d( EQ) EI I dt dt dt W I I dt I Depends only on the geomety o the loop (in the om o ) nd the inl cuent I. in the inducto

4 Enegy Density o the Mgnetic Field We hve expessed the totl enegy stoed in the inducto in tems o the cuent nd we know the mgnetic ield is popotionl to the cuent. Cn we expess the totl mgnetic enegy in tems o the -ield? Yes. et s conside the cse o solenoid. U u n A I ( ni) A A (The enegy density o mgnetic ield in ee spce) Although this eltion hs een otined om specil cse, the expession is vlid o ny mgnetic ield. 3 Genelized Totl Enegy Thee is nice wy to wite the totl mgnetic enegy W. Φ d ( A) d A dl I : suce ounded y : peimete o the loop ( ) W I Φ I I A d l A I dl genelize to the volume cuent W ( ) dl ( ) dτ, whee AI AJ J W ( ) dτ [ ( )] dτ A J A 4 Genelized Totl Enegy II oduct ule 6, ( A ) ( A) A ( ) A ( ) ( A) ( A ) W [ ( )] dτ [ ( )] dτ A A divegence theoem dτ ( ) d A ll spce ( ) d A W τ d ll spce 5 Electic nd Mgnetic Field Enegy Electic ield enegy enegy density Welec ( ) d ρ τ E dτ, ue E Mgnetic ield enegy Wmg ( ), A J dτ dτ u Mgnetic ields themselves do no wok. Whee does the enegy come om? A chnging mgnetic ield induces n electic ield which cn do wok. 6

5 U U E E 4 J/m Exmple The ekdown electic ield stength o i is 3x 6 /m. A vey lge mgnetic ield stength is T. compe the enegy densities o the ield. olution: (.5)(8.85 J/m )(3 Mgnetic ields e n eective mens o stoing enegy without ekdown o the i. Howeve, it is diicult to poduce such lge ields ove lge egions. 6 ) 7 Exmple Use the expession o the enegy density o the mgnetic ield to clculte the sel-inductnce o tooid with ectngul coss section. olution: NI π h( NI ) du d h(πd) d h( NI ) hn I U d ln( ) I N h ln( ) π Cn we use the concept o mgnetic lux to deive the selinductnce? 8 Exmple 7.3 A long coxil cle cies cuent I (the cuent lows down the suce o the inne cylinde, dius, nd ck long the oute cylinde, dius ) s shown in the Figue. Find the mgnetic enegy stoed in section o length l. ol: I mgnetic ield φˆ π s I enegy density u 8π s mgnetic enegy sel-inductnce I I W udτ lπsds lln( ) 8π s l W I ln( ) π Mxwell s Equtions 7.3. Electodynmics eoe Mxwell E ρ (Guss's lw) (no nme) electomgnetic theoy ove centuy go E (Fdy's lw) t J (Ampee's lw) A tl inconsistency in Ampee s lw ( ) J Ampee s lw is incoect o the nonstedy cuent.

6 The Electic nd Mgnetic Fields Two distinct kinds o electic ields: E (in sttic cse): ttiuted to electic chges, using Coulom s lw. E (in nonstedy cse): ssocited with chnging mgnetic ield, using Fdy s lw. Two distinct kinds o mgnetic ields: (in sttic cse): ttiuted to electic cuents, using Ampee s lw. (in nonstedy cse): ssocited with chnging electic ield, using? Anothe Inconsistency o Ampee s w How do we detemine the enclosed cuent I enc? d l I enc * The simplest suce---the wie punctue this suce so I enc I Ampee s lw is ok. * A ollon-shped suce---no cuent psses though this suce. so I enc Ampee s lw is not vlid! Fo nonstedy cuent, the cuent enclosed y loop is n ill-deined. How Mxwell Fixed Ampee s w Applying the continuity eqution nd Guss s lw, the oending tem cn e ewitten: ρ ( E) J ( ) A new cuent E J J+ kills o the ext divegence ( ) ( ) ( + E J J ) Electic Anlogy o Fdy s w Mxwell s tem cues the deect in Ampee s lw, nd moeove, it hs cetin esthetic ppel. Fdy s lw A chnging mgnetic ield induces electic ield. A chnging electic ield induces mgnetic ield. When E is constnt (electosttic+mgnetosttic), we will hve J. E plys cucil ole in the EM wve popgtion. 3 Mxwell clled this ext tem the displcement cuent. J E d misleding nme, nothing to do with cuent 4

7 The Displcement Cuent How the displcement cuent esolves the pdox o the chging cpcito. The electic ield etween the two cpcito pltes is σ Q E the chge on the plte A the e o the plte E Q I J A A J J+ J tot d JJ, J d t the lt suce J, J d J t the lloon-shped suce Mxwell s Equtions Mxwell s equtions in the tditionl wy. E ρ (Guss's lw) (no nme) E (Fdy's lw) t (Ampee's lw with J+ Mxwell's coection) oentz oce lw Continuity eqution F q( E+ v ) ρ J t 6 Mxwell s Equtions (II) Anothe expession o the Mxwell s equtions. ρ E E+ J The ields (E nd ) on the let nd the souces (ρ nd J) on the ight. Mxwell s equtions tell you how souces poduce ields; ecipoclly, the oentz oce lw tells you how ields ect souces. A nonline eedck Mgnetic Chge I thee is mgnetic chge ρ m nd the coesponding cuent o the mgnetic cuent J m, the Mxwell s equtions ed ρ e E E+ Jm ρ J m e oth chges would e conseved: ρ e Je, nd Jm ρm A symmetic etween E nd E - E Q: Hs ny one eve ound the mgnetic chge? No. 8

8 7.3.5 Mxwell s Equtions in Mtte When woking with mteils tht e suject to electic nd mgnetic poliztion, thee is moe convenient wy to wite the Mxwell s equtions. ttic cse: An electic poliztion poduces ound chge: ρ A mgnetic poliztion esults in ound cuent: J M Nonsttic cse: Any chnge in the electic poliztion involves low o ound chge. dσ d J ˆ p whee σ n d dt d poliztion cuent (nothing to do with the ound cuent). 9 oliztion nd ound Cuents ound cuent J : mgnetiztion o the mteil involving the spin nd oitl motion o electons. oliztion cuent J p : the line motion o chge when the electic poliztion chnges. Now ρ ρ + ρ ρ J J + J + Jp J + M+ t Guss s lw: E ( ρ ) ( E+ ) ρ Ampee s lw: ( J + M+ ) + ( M) J + ( E+ ) 3 Mxwell s Equtions in Mtte In tems o ee chges nd cuents, Mxwell s equtions ed D ρ E+ D H J The constitutive eltions: χee M χ H m Dieentil om D ρ oundy Conditions (I) Integl om D d ρ ove ny enclosed suce. d D d ρ D D σ D D σ o D ( ) E+ + χe E E H M ( + χ m) H H we thin Gussin pillox d σ 3 3

9 Dieentil om E + D oundy Conditions (II) Integl om E dl d o ny suce ounded y the + d H l J D closed loop. E dl d E l E l d E E H J d vey thin Ampein loop stddling the suce // // H dl J + d D H l H l I + D d ( H H ) l I I enc 33 enc K ( nˆ l) ( K nˆ) l H H ( K nˆ) enc oundy Conditions in ine Medi D D σ E E H H ( K nˆ ) // // In cse o line medi, D nd H cn e expess in tems o E nd. // // E E σ E E K nˆ I thee is no ee chge o ee cuent t the intece, then // // E E E E 34 Homewok o Chp.7 (pt II) o. 3, 4, 4, 53, 57 35