Chapter 28 Magnetic Induction

Transcription

1 Chapr 8 Magnic nducion Concpual Probls [SSM] (a) Th agnic quaor is a lin on h surfac of Earh on which Earh s agnic fild is horizonal. A h agnic quaor, how would you orin a fla sh of papr so as o cra h axiu agniud of agnic flux hrough i? (b) How abou h iniu agniud of agnic flux? Drin h Concp (a) Orin h sh so h noral o h sh is boh horizonal and prpndicular o h local angn o h agnic quaor. (b) Orin h sh of papr so h noral o h sh is prpndicular o h dircion of h noral dscribd in h answr o Par (a). A on of Earh s agnic pols, how would you orin a fla sh of papr so as o cra h axiu agniud of agnic flux hrough i? Drin h Concp (a) Orin h sh so h noral o h sh is vrical. (b) Any orinaion as long as h papr s plan is prpndicular o Earh s surfac a ha locaion. 3 [SSM] Show ha h following cobinaion of S unis is quivaln o h vol: T s. Drin h Concp Bcaus a vol is a joul pr coulob, w can show ha T h S unis ar quivaln o a vol by aking a sris of subsiuions and s siplificaions ha rducs hs unis o a joul pr coulob. Th unis of a sla ar N : A T s N A s N A s Subsiu h unis of an apr (C/s), rplac N wih J, and siplify o obain: T s J C s s J C 663

2 664 Chapr 8 Finally, bcaus a joul pr coulob is a vol: T s V 4 Show ha h following cobinaion of S unis is quivaln o h oh: Wb A s. Drin h Concp Bcaus a wbr is a nwon r pr apr, w can show ha h S unis Wb ar quivaln o an oh by aking a sris of A s subsiuions and siplificaions ha rducs hs unis o a vol pr apr. Bcaus a wbr is a N : A Wb A s N A A s J A s Subsiu h unis of an apr and siplify o obain: Wb A s J C A s s J A C Finally, bcaus a joul pr coulob is a vol: Wb A s V A Ω 5 [SSM] A currn is inducd in a conducing loop ha lis in a horizonal plan and h inducd currn is clockwis whn viwd fro abov. Which of h following sans could b ru? (a) A consan agnic fild is dircd vrically downward. (b) A consan agnic fild is dircd vrically upward. (c) A agnic fild whos agniud is incrasing is dircd vrically downward. (d) A agnic fild whos agniud is dcrasing is dircd vrically downward. () A agnic fild whos agniud is dcrasing is dircd vrically upward. Drin h Concp W know ha h agnic flux (in his cas h agnic fild bcaus h ara of h conducing loop is consan and is orinaion is fixd) us b changing so h only issus ar whhr h fild is incrasing or dcrasing and in which dircion. Bcaus h dircion of h agnic fild associad wih h clockwis currn is vrically downward, h changing fild ha is rsponsibl for i us b ihr incrasing vrically upward (no includd in h lis of possibl answrs) or a dcrasing fild dircd ino h pag. (d) is corrc.

3 Magnic nducion Giv h dircion of h inducd currn in h circui, shown on h righ in Figur 8-37, whn h rsisanc in h circui on h lf is suddnly (a) incrasd and (b) dcrasd. Explain your answr. Drin h Concp Th inducd f and inducd currn in h circui on h righ ar in such a dircion as o oppos h chang ha producs h (nz s aw). W can drin h dircion of h inducd currn in h circui. No ha whn is consan, B in h circui o h righ poins ou of h papr. (a) f incrass, dcrass and B in h circui o h righ dcrass. nz s law lls us ha h inducd currn is counrclockwis. (b) f dcrass, incrass and B in h circui o h righ incrass. nz s law lls us ha h inducd currn is clockwis. 7 [SSM] Th plans of h wo circular loops in Figur 8-38, ar paralll. As viwd fro h lf, a counrclockwis currn xiss in loop A. f h agniud of h currn in loop A is incrasing, wha is h dircion of h currn inducd in loop B? Do h loops arac or rpl ach ohr? Explain your answr. Drin h Concp Clockwis as viwd fro h lf. Th loops rpl ach ohr. 8 A bar agn ovs wih consan vlociy along h axis of a loop, as shown in Figur 8-39, (a) Mak a graph of h agnic flux hrough h loop as a funcion of i. ndica on h graph whn h agn is halfway hrough h loop by dsignaing his i. Choos h dircion of h noral o h fla surfac boundd by h fla surfac o b o h righ. (b) Mak a graph of h inducd currn in h loop as a funcion of i. Choos h posiiv dircion for h currn o b clockwis as viwd fro h lf. Drin h Concp W know ha, as h agn ovs o h righ, h flux hrough h loop firs incrass unil h agn is half way hrough h loop and hn dcrass. Bcaus h flux firs incrass and hn dcrass, h currn will chang dircions, having is axiu valus whn h flux is changing os rapidly. (a) and (b) Th following graph shows h flux and h inducd currn as a funcion of i as h bar agn passs hrough h coil. Whn h cnr of h agn passs hrough h plan of h coil dφ / and h currn is zro.

4 666 Chapr 8 flux currn i 9 A bar agn is ound on h nd of a coild spring and is oscillaing in sipl haronic oion along h axis of a loop, as shown in Figur 8-4. Th agn is in is quilibriu posiion whn is idpoin is in h plan of h loop. (a) Mak a graph of h agnic flux hrough h loop as a funcion of i. ndica whn h agn is halfway hrough h loop by dsignaing hs is and. (b) Mak a graph of h inducd currn in h loop as a funcion of i, choosing h currn o b posiiv whn i is clockwis as viwd fro abov. Drin h Concp Bcaus h agn ovs wih sipl haronic oion, h flux and h inducd currn will vary sinusoidally. Th currn will b a axiu whrvr h flux is changing os rapidly and will b zro whrvr h flux is onarily consan. (a), (b) Th following graph shows h flux, φ, and h inducd currn (proporional o dφ /) in h loop as a funcion of i. flux currn i Ti

5 Magnic nducion 667 A pndulu is fabricad fro a hin, fla pic of aluinu. A h boo of is arc, i passs bwn h pols of a srong prann agn. n Figur 8-4a, h al sh is coninuous, whras in Figur 8-4b, hr ar slos in h sh. Whn rlasd fro h sa angl, h pndulu ha has slos swings back and forh any is, bu h pndulu ha dos no hav slos sops swinging afr no or han on copl oscillaion. Explain why. Drin h Concp n h configuraion shown in (a), nrgy is dissipad by ddy currns fro h f inducd by h pndulu ovn. n h configuraion shown in (b), h slis inhibi h ddy currns and h braking ffc is graly rducd. A bar agn is droppd insid a long vrical ub. f h ub is ad of al, h agn quickly approachs a rinal spd, bu if h ub is ad of cardboard, h agn falls wih consan acclraion. Explain why h agn falls diffrnly in h al ub han i dos in h cardboard ub. Drin h Concp Th agnic fild of h falling agn ss up ddy currns in h al ub. Th ddy currns sablish a agnic fild ha xrs a forc on h agn opposing is oion; hus h agn is slowd down. f h ub is ad of a nonconducing arial, hr ar no ddy currns. A sall squar wir loop lis in h plan of his pag, and a consan agnic fild is dircd ino h pag. Th loop is oving o h righ, which is h +x dircion. Find h dircion of h inducd currn, if any, in h loop if (a) h agnic fild is unifor, (b) h agnic fild srngh incrass as x incrass, and (c) h agnic fild srngh dcrass as x incrass. Drin h Concp Th dircion of h inducd currn is in such a dircion as o oppos, or nd o oppos, h chang ha producs i (nz s aw). (a) Bcaus h applid fild is consan and unifor, hr is no chang in flux hrough h loop and, in accord wih Faraday s law, no inducd currn in h loop. (b) h posiiv noral dircion on h fla surfac boundd by h loop b ino h pag. Bcaus h srngh of h applid fild incrass o h righ, h flux hrough h loop incrass as i ovs o h righ. n accord wih nz s law, h dircion of h inducd currn will b such ha h flux hrough h loop du o is agnic fild will b opposi in sign h chang in flux of h applid fild. Thus, on h fla surfac boundd by h loop h agnic fild du o h inducd currn is ou of h pag. Using h righ-hand rul, h inducd currn us b counrclockwis.

6 668 Chapr 8 (c) h posiiv noral dircion on h fla surfac boundd by h loop b ino h pag. Bcaus h srngh of h applid fild incrass o h righ, h flux hrough h loop dcrass as i ovs o h righ. n accord wih nz s law, h dircion of h inducd currn will b such ha h flux hrough h loop du o is agnic fild will b opposi in sign h chang in flux of h applid fild. Thus, on h fla surfac boundd by h loop h agnic fild du o h inducd currn is ino h pag. Using h righ-hand rul, h inducd currn us b clockwis. 3 f h currn in an inducor doubls, h nrgy sord in h inducor will (a) rain h sa, (b) doubl, (c) quadrupl, (d) halv. Drin h Concp Th agnic nrgy sord in an inducor is givn by U. Doubling quadrupls U. ) (c is corrc. 4 Two solnoids ar qual in lngh and radius, and h cors of boh ar idnical cylindrs of iron. Howvr, solnoid A has hr is h nubr of urns pr uni lngh as solnoid B. (a) Which solnoid has h largr slfinducanc? (b) Wha is h raio of h slf-inducanc of solnoid A o h slfinducanc of solnoid B? Drin h Concp Th slf-inducanc of a coil is givn by μ n A, whr n is h nubr of urns pr uni lngh and is h lngh of h coil. (a) Bcaus h wo solnoids ar qual in lngh and radius and hav idnical cors, hir slf-inducancs ar proporional o h squar of hir nubr of urns pr uni lngh. Hnc A has h largr slf-inducanc. (b) Th slf-inducancs of h wo coils ar givn by: Divid h slf-inducanc of coil A by h slf-inducanc of coil B and siplify o obain: A μna AA and μ n A B B B A B A μna AA A na AA A B μ nb AB B nb AB B or, bcaus h coils hav h sa lnghs and radii (hnc, h sa cross-scional aras), A B n n A B n n A B

7 Magnic nducion 669 f n incrass by a facor of 3, will dcras by h sa facor, bcaus h inducors ar ad fro h sa lngh of wir. Hnc: 5 [SSM] Tru or fals: 3n n B A B B 9 (a) Th inducd f in a circui is qual o h ngaiv of h agnic flux hrough h circui. (b) Thr can b a non-zro inducd f a an insan whn h flux hrough h circui is qual o zro. (c) Th slf inducanc of a solnoid is proporional o h ra of chang of h currn in h solnoid. (d) Th agnic nrgy dnsiy a so poin in spac is proporional o h squar of h agniud of h agnic fild a ha poin. () Th inducanc of a solnoid is proporional o h currn in i. (a) Fals. Th inducd f in a circui is qual o h ra of chang of h agnic flux hrough h circui. (b) Tru. Th ra of chang of h agnic flux can b non-zro whn h flux hrough h circui is onarily zro (c) Fals. Th slf inducanc of a solnoid is drind by is lngh, crossscional ara, nubr of urns pr uni lngh, and h prabiliy of h ar in is cor. (d) Tru. Th agnic nrgy dnsiy a so poin in spac is givn by B Equaion 8-: u. μ () Fals. Th inducanc of a solnoid is drind by is lngh, cross-scional ara, nubr of urns pr uni lngh, and h prabiliy of h ar in is cor. Esiaion and Approxiaion 6 Your basball aas, having jus sudid his chapr, ar concrnd abou gnraing nough volag o shock h whil swinging aluinu bas a fas balls. Esia h axiu possibl oional f asurd bwn h nds of an aluinu basball ba during a swing. Do you hink your a should swich o woodn bas o avoid lcrocuion? Picur h Probl Th ba is swung in Earh s agnic fild. W ll assu ha h bar swings such ha h axiu linar vlociy of h ba occurs a

8 67 Chapr 8 an angl such ha i is oving prpndicular o Earh s fild (i.. whn h ba is alignd norh-souh and oving as-ws). Th inducd f in h ba is givn by vb. A ba is roughly long, and a os is cnr is probably oving a 75 ph, or abou 33 /s. Earh s agnic fild is abou.3 G. Th f inducd in h ba is givn by: vb Undr h condiions rsuling in a axiu inducd f oulind abov: T 33 4 G ( /s).3 G ( ) V Bcaus V is so low, hr is no dangr of bing shockd and no rason o swich o woodn bas. 7 Copar h nrgy dnsiy sord in Earh s lcric fild nar is surfac o ha sord in Earh s agnic fild nar is surfac. Picur h Probl W can copar h nrgy dnsiy sord in Earh s lcric fild o ha of Earh's agnic fild by finding hir raio. W ll ak Earh s agnic fild o b.3 G and is lcric fild o b V/. Th nrgy dnsiy in an lcric fild E is givn by: u E Th nrgy dnsiy in a agnic fild B is givn by: B u μ Exprss h raio of u o u o obain: u u B μ E B μ E Subsiu nurical valus and valua u / u : or T.3 G 4 u G 8.9 u 7 ( 4π N/A )( C / N )( V/) ( 8 3 ) u u 3

9 Magnic nducion 67 8 A physics achr dos h following f donsraion. Sh has wo sudns hold a long wir conncd o a volr. Th wir is hld slack, so ha i sags wih a larg arc in i. Whn sh says sar, h sudns bgin roaing h wir as if hy wr playing jup rop. Th sudns sand 3. apar, and h sag in h wir is abou.5. Th oional f fro h jup rop is hn asurd on h volr. (a) Esia a rasonabl valu for h axiu angular spd ha h sudns can roa h wir. (b) Fro his, sia h axiu oional f in h wir. HNT: Wha fild is involvd in craing h inducd f? Picur h Probl W can us Faraday s law o rla h oional f in h wir o h angular spd wih which h sudns urn h jup rop. Assu ha Earh s agnic fild is.3 G. (a) ss unlikly ha h sudns could urn h jup rop wir fasr han 5. rv/s. (b) Th agnic flux φ hrough h roaing circular loop of wir varis sinusoidally wih i according o: Bcaus h avrag valu of h cosin funcion, ovr on rvoluion, is ½, h avrag ra a which h flux changs hrough h circular loop is: Fro Faraday s law, h agniud of h avrag oional f in h loop is: φ BAsin dφ BA cos φ BA d π r av d π r φ B B Subsiu nurical valus and valua :.5 π T.3G 4 G ( 3.4 rad/s).8v 9 (a) Esia h axiu possibl oional f bwn h wingips of a ypical corcial airlinr in fligh. (b) Esia h agniud of h lcric fild bwn h wingips. Picur h Probl Th oional f bwn h wingips of an airlinr is givn by vb. Assu a spd, rlaiv o Earh s agnic fild, of 5 i/h or abou /s and a wingspan of 7. Assu ha Earh s agnic fild is.3 G.

10 67 Chapr 8 (a) Th oional f bwn h wingips is givn by: Subsiu nurical valus and valua : vb ( /s).3 G ( 7 ).5 V T G 4 (b) Th agniud of h lcric fild bwn h wingips is h raio of h ponial diffrnc bwn h and hir sparaion: Magnic Flux V E d.5 V 7 7 V/ A unifor agnic fild of agniud. T is in h +x dircion. A squar coil ha has 5.-c long sids has a singl urn and aks an angl θ wih h z axis, as shown in Figur 8-4. Find h agnic flux hrough h coil whn θ is (a) º, (b) 3º, (c) 6º, and (d) 9º. Picur h Probl Bcaus h surfac is a plan wih ara A and B is consan in agniud and dircion ovr h surfac and aks an angl θ wih h uni noral vcor, w can us φ BAcosθ o find h agnic flux hrough h coil. Th agnic flux hrough h coil is givn by: φ BAcosθ Subsiu for B and A o obain: φ T G 4 G 4 ( 5. ) cosθ ( 5. Wb) cosθ 4 (a) For θ : φ ( 5. Wb) 5. 4 (b) For θ 3 : φ ( 5. Wb) (c) For θ 6 : φ ( 5. Wb) cos Wb.5 Wb cos3 Wb.43Wb cos6 Wb.5Wb

11 Magnic nducion 673 (d) For θ 9 : 4 φ ( 5. Wb) cos9 [SSM] A circular coil has 5 urns and a radius of 5. c. is a h quaor, whr Earh s agnic fild is.7 G, norh. Th axis of h coil is h lin ha passs hrough h cnr of h coil and is prpndicular o h plan of h coil. Find h agnic flux hrough h coil whn h axis of h coil is (a) vrical, (b) horizonal wih h axis poining norh, (c) horizonal wih h axis poining as, and (d) horizonal wih h axis aking an angl of 3º wih norh. Picur h Probl Bcaus h coil dfins a plan wih ara A and B is consan in agniud and dircion ovr h surfac and aks an angl θ wih h uni noral vcor, w can us φ NBAcosθ o find h agnic flux hrough h coil. Th agnic flux hrough h coil is givn by: φ NBA cosθ NBπ r cosθ Subsiu for nurical valus o obain: ( 5. ) cosθ ( 3.7 μwb) cosθ T φ 5.7G π 4 G (a) Whn h plan of h coil is horizonal, θ 9 : (b) Whn h plan of h coil is vrical wih is axis poining norh, θ : (c) Whn h plan of h coil is vrical wih is axis poining as, θ 9 : (d) Whn h plan of h coil is vrical wih is axis aking an angl of 3 wih norh, θ 3 : φ ( 3.7 Wb) cos9 μ ( 3.7 μwb) cos 4μWb φ φ ( 3.7 Wb) cos9 μ ( 3.7 μwb) cos3 μwb φ A agnic fild of. T is prpndicular o h plan of a4 urn squar coil wih sids 5.-c long. (a) Find h agnic flux hrough h coil. (b) Find h agnic flux hrough h coil if h agnic fild aks an angl of 6º wih h noral o h plan of h coil.

12 674 Chapr 8 Picur h Probl Bcaus h squar coil dfins a plan wih ara A and B is consan in agniud and dircion ovr h surfac and aks an angl θ wih h uni noral vcor, w can us φ NBAcosθ o find h agnic flux hrough h coil. Th agnic flux hrough h coil is givn by: φ NBAcosθ Subsiu nurical valus for N, B, and A o obain: φ ( )( 5. ) 4.T ( 4. Wb) cosθ cosθ (a) For θ : φ ( 4. Wb) cos 4Wb (b) For θ 6 : φ ( 4. Wb) cos6 Wb 3 A unifor agnic fild B is prpndicular o h bas of a hisphr of radius. Calcula h agnic flux (in rs of B and ) hrough h sphrical surfac of h hisphr. Picur h Probl Noing ha h flux hrough h bas us also pnra h sphrical surfac (h ± in h answr blow), w can apply is dfiniion o xprss φ. Apply h dfiniion of agnic flux o obain: φ ± AB ± π B 4 Find h agnic flux hrough a 4-urn solnoid ha has a lngh qual o 5. c, has a radius qual o. c, and carris a currn of 3. A. Picur h Probl W can us φ NBAcosθ o xprss h agnic flux hrough h solnoid and B μn o rla h agnic fild in h solnoid o h currn in is coils. Assu ha h agnic fild in h solnoid is consan. Exprss h agnic flux hrough a coil wih N urns: Exprss h agnic fild insid a long solnoid: φ NBAcosθ B μn whr n is h nubr of urns pr uni lngh.

13 Magnic nducion 675 Subsiu o obain: φ Nμ nacosθ or, bcaus n N/ and θ, N μa N μπ r φ Subsiu nurical valus and valua φ : 7 ( 4) ( 4π N/A )( 3.A) π(. ) φ.5 758μWb 5 Find h agnic flux hrough a 8-urn solnoid ha has a lngh qual o 3. c, has a radius qual o. c, and carris a currn of. A. Picur h Probl W can us φ NBAcosθ o xprss h agnic flux hrough h solnoid and B μn o rla h agnic fild in h solnoid o h currn in is coils. Assu ha h agnic fild in h solnoid is consan. Exprss h agnic flux hrough a coil wih N urns: Exprss h agnic fild insid a long solnoid: Subsiu for B o obain: φ NBAcosθ B μn whr n is h nubr of urns pr uni lngh. φ NμnAcosθ or, bcaus n N/ and θ, N μa N μπ r φ Subsiu nurical valus and valua φ : φ 7 ( 8) ( 4π N/A )(.A) π (. ) Wb 6 A circular coil has 5. urns, has a radius 4. c, and is in a unifor agnic fild of 4. kg in h +x dircion. Find h flux hrough h coil whn h uni noral o h plan of h coil is (a)î, (b) ĵ, (c) ( i ˆ + ˆ j), (d) ˆk, and ().6î +.8ĵ.

14 676 Chapr 8 Picur h Probl W can apply h dfiniions of agn flux and of h do produc o find h flux for h givn uni vcors. Apply h dfiniion of agnic flux o h coil o obain: φ N B nda ˆ Bcaus B is consan: φ NB nˆ da N ( B nˆ ) S N( B nˆ ) π r Evalua B : B (.4T )iˆ S A Subsiu nurical valus and siplify o obain: φ ( 5.) [(.4T) ] (.4 ) π (.36T ) iˆ nˆ (a) Evalua φ for (b) Evalua φ for iˆ nˆ : φ (.36 T ) i i 3.Wb ˆj iˆ ˆj nˆ : φ (.36T ) ˆ ˆ (c) Evalua φ for n ˆ ( iˆ + ˆj ) : ( ) ( ˆ ˆ i + j) φ.36t.36t iˆ.3wb (d) Evalua φ for () Evalua φ for kˆ iˆ kˆ nˆ : φ (.36T ) nˆ.6ˆ i +.8 ˆj : φ (.36T ) iˆ (.6ˆi +.8 ˆ) 8Wb j 7 [SSM] A long solnoid has n urns pr uni lngh, has a radius, and carris a currn. A circular coil wih radius and wih N oal urns is coaxial wih h solnoid and quidisan fro is nds. (a) Find h agnic flux hrough h coil if >. (b) Find h agnic flux hrough h coil if <. Picur h Probl Th agnic fild ousid h solnoid is, o a good approxiaion, zro. Hnc, h flux hrough h coil is h flux in h cor of h solnoid. Th agnic fild insid h solnoid is unifor. Hnc, h flux hrough h circular coil is givn by h sa xprssion wih rplacing :

15 Magnic nducion 677 (a) Th flux hrough h larg circular loop ousid h solnoid is givn by: Subsiuing for B and A and siplifying yilds: (b) Th flux hrough h coil whn < is givn by: φ NBA ( μ n )( π ) μ nnπ φ N ( μ n )( π ) μ nnπ φ N 8 (a) Copu h agnic flux hrough h rcangular loop shown in Figur (b) Evalua your answr for a 5. c, b c, d. c, and A. Picur h Probl W can us h hin o s up h ln of ara da and xprss h flux dφ hrough i and hn carry ou h dails of h ingraion o xprss φ. (a) Th flux hrough h srip of ara da is givn by: Exprss B a a disanc x fro a long, sraigh wir: Subsiu o obain: d φ BdA whr da bdx. μ B 4π x μ π x μ dφ bdx π x μb π dx x ngra fro x d o x d + a: φ d + a μb π d dx x μb d + a ln π d (b) Subsiu nurical valus and valua φ : 7 ( 4π N/A )( A)(.) 7.c φ ln π.c.5 μwb 9 A long cylindrical conducor wih a radius and a lngh carris a currn. Find h agnic flux pr uni lngh hrough h ara indicad in Figur Picur h Probl Considr an ln of ara da dr whr r. W can us is dfiniion o xprss dφ hrough his ara in rs of B and Apr s law o xprss B as a funcion of. Th fac ha h currn is uniforly disribud

16 678 Chapr 8 ovr h cross-scional ara of h conducor allows us o s up a proporion fro which w can obain as a funcion of r. Wih hs subsiuions in plac w can ingra dφ o obain φ /. Noing ha B is paralll o nˆ ovr h nir ara, xprss h flux dφ hrough an ara dr: Apply Apr s law o h currn conaind insid a cylindrical rgion of radius r < : Solving for B yilds: Using h fac ha h currn is uniforly disribud ovr h crossscional ara of h conducor, xprss is variaion wih disanc r fro h cnr of h conducor: Subsiu for C in quaion () and siplify o obain: d φ BdA Bdr () C B π rb μ d μc B () πr ( r) πr π () r C μ r μ πr π B r C r Subsiuing for B in quaion () yilds: μ φ π d rdr ngra dφ fro r o r o obain: μ μ φ rdr π 4π Divid boh sids of his quaion by o xprss h agnic flux pr uni lngh: nducd EMF and Faraday s aw φ μ 4π 3 Th flux hrough a loop is givn by φ..4, whr φ is in wbrs and is in sconds. (a) Find h inducd f as a funcion of i. (b) Find boh φ and a,. s, 4. s, and 6. s.

17 Magnic nducion 679 Picur h Probl Givn φ as a funcion of i, w can us Faraday s law o xprss as a funcion of i. (a) Apply Faraday s law o xprss h inducd f in h loop in rs of h ra of chang of h agnic flux: dφ d () [.(.4) ] Wb (..4) s (..4)V whr is in vols and is in sconds. (b) Evalua φ a : φ ( s).( ) (.4)( ) Evalua a : ( s) [.( ).4] V.4V Procd as abov o copl h abl o h righ: φ (s) (Wb) (V) Th flux hrough a loop is givn by φ..4, whr φ is in wbrs and is in sconds. (a) Skch graphs of agnic flux and inducd f as a funcion of i. (b) A wha i(s) is h flux iniu? Wha is h inducd f a ha (hos) i(s)? (c) A wha i(s) is h flux zro? Wha is (ar) h inducd f(s) a hos i(s)? Picur h Probl W can find h i a which h flux is a iniu by looking for h lows poin on h graph of vrsus and h f a his i by drining h valu of V a his i fro h graph. W can inrpr h graphs o find h is a which h flux is zro and h corrsponding valus of h f.

18 68 Chapr 8 (a) Th flux, φ, and h inducd f,, ar shown as funcions of in h following graph. Th solid curv rprsns φ, h dashd curv rprsns flux f.5 flux (Wb) f (V) (s) -. (b) frring o h graph, w s ha h flux is a iniu whn. s and ha V(. s). (c) Th flux is zro whn and 4. s. ().4 V and (4. s).4 V. 3 A solnoid ha has a lngh qual o 5. c, a radius qual o.8 c, and 4 urns is in a rgion whr a agnic fild of 6 G xiss and aks an angl of 5º wih h axis of h solnoid. (a) Find h agnic flux hrough h solnoid. (b) Find h agniud of h avrag f inducd in h solnoid if h agnic fild is rducd o zro in.4 s. Picur h Probl W can us is dfiniion o find h agnic flux hrough h solnoid and Faraday s law o find h f inducd in h solnoid whn h xrnal fild is rducd o zro in.4 s. (a) Exprss h agnic flux hrough h solnoid in rs of N, B, A, and θ : φ NBAcosθ NBπ cosθ Subsiu nurical valus and valua φ : ( 4)( 6. T) (.8 ) φ π 3. Wb 3.Wb cos5

19 Magnic nducion 68 (b) Apply Faraday s law o obain: Δφ Δ.V 3.Wb.4s 33 [SSM] A -urn circular coil has a diar of. c, a rsisanc of 5. Ω, and h wo nds of h coil ar conncd oghr. Th plan of h coil is prpndicular o a unifor agnic fild of agniud. T. Th dircion of h fild is rvrsd. (a) Find h oal charg ha passs hrough a cross scion of h wir. f h rvrsal aks. s, find (b) h avrag currn and (c) h avrag f during h rvrsal. Picur h Probl W can us h dfiniion of avrag currn o xprss h oal charg passing hrough h coil as a funcion of av. Bcaus h inducd currn is proporional o h inducd f and h inducd f, in urn, is givn by Faraday s law, w can xprss ΔQ as a funcion of h nubr of urns of h coil, h agnic fild, h rsisanc of h coil, and h ara of h coil. Knowing h rvrsal i, w can find h avrag currn fro is dfiniion and h avrag f fro Oh s law. (a) Exprss h oal charg ha passs hrough h coil in rs of h inducd currn: la h inducd currn o h inducd f: Using Faraday s law, xprss h inducd f in rs of φ : Subsiu in quaion () and siplify o obain: ΔQ av Δ () av Δφ Δ Δφ φ ΔQ Δ Δ Δ π NB d NBA 4 NBπd whr d is h diar of h coil.

20 68 Chapr 8 Subsiu nurical valus and valua ΔQ: ΔQ ( )(.T) π (. ) ( 5.Ω).57 C.6 C (b) Apply h dfiniion of avrag currn o obain: (c) Using Oh s law, rla h avrag f in h coil o h avrag currn: av av ΔQ.57 C.57 A Δ.s.6A av 68V (.57 A)( 5.Ω) 34 A h quaor, a -urn coil ha has a cross-scional ara of 3 c and a rsisanc of 5. Ω is alignd so ha is plan is prpndicular o Earh s agnic fild of.7 G. (a) f h coil is flippd ovr in.35 s, wha is h avrag inducd currn in i during h.35 s? (b) How uch charg flows hrough a cross scion of h coil wir during h.35 s? Picur h Probl (a) Bcaus h avrag inducd currn is proporional o h inducd f and h inducd f, in urn, is givn by Faraday s law, w can find av fro h chang in h agnic flux hrough h coil, h rsisanc of h coil, and h i rquird for h flipping of h coil. (b) Knowing h avrag currn, w can us is dfiniion o find h charg flowing in h coil. (a) Th avrag inducd currn is givn by: av Th inducd f in h coil is h ra a which h agnic flux is changing: Subsiuing for yilds: Δφ Δ av φ Δ NBA Δ NBA Δ Subsiu nurical valus and valua av : av T ( ).7 G 3 c 4 G ( 5. Ω)(.35 s) c 8 μa

21 Magnic nducion 683 (b) Th avrag currn is also givn by: Subsiu nurical valus and valua ΔQ: ΔQ av ΔQ avδ Δ ΔQ (.8 A)(.35 s) 8 μc 35 A currn ingraor asurs h currn as a funcion of i and ingras (adds) h currn o find h oal charg passing hrough i. (Bcaus dq/, h ingraor calculas h ingral of h currn or Q.) A circular coil ha has 3 urns and a radius qual o 5. c is conncd o such an insrun. Th oal rsisanc of h circui is. Ω. Th plan of h coil is originally alignd prpndicular o Earh s agnic fild a so poin. Whn h coil is road hrough 9º abou an axis ha is in h plan of h coil, a charg of 9.4 μc passs hrough h currn ingraor is asurd o b 9.4 μc. Calcula h agniud of Earh s agnic fild a ha poin. Picur h Probl W can us Faraday s law o xprss Earh s agnic fild a his locaion in rs of h inducd f and Oh s law o rla h inducd f o h charg ha passs hrough h currn ingraor. Using Faraday s law, xprss h inducd f in rs of h chang in h agnic flux as h coil is road hrough 9 : Solving for B yilds: B Δφ NBA NB r π Δ Δ Δ Δ Nπ r Using Oh s law, rla h inducd f o h inducd currn: ΔQ Δ whr ΔQ is h charg ha passs hrough h currn ingraor. Subsiu for and siplify o ΔQ Δ Q obain: Δ B Δ Nπ r Nπ r Subsiu nurical valus and valua B: B ( 9.4μC)(.Ω) ( 3) π (.5 ) 79.8 μ T

22 684 Chapr 8 Moional EMF 36 A 3.-c long rod ovs sadily a 8. /s in a plan ha is prpndicular o a agnic fild of 5 G. Th vlociy of h rod is prpndicular o is lngh. Find (a) h agnic forc on an lcron in h rod, (b) h lcrosaic fild in h rod, and (c) h ponial diffrnc bwn h nds of h rod. Picur h Probl W can apply h quaion for h forc on a chargd paricl oving in a agnic fild o find h agnic forc acing on an lcron in h rod. W can us E v B o find E andv E, whr is h lngh of h rod, o find h ponial diffrnc bwn is nds. (a) la h agnic forc on an lcron in h rod o h spd of h rod, h lcronic charg, and h agnic fild in which h rod is oving: F q v B and F qvbsinθ Subsiu nurical valus and valua F: F 9 (.6 C)( 8. /s) (.5T) 6.4 N sin9 (b) Exprss h lcrosaic fild E in h rod in rs of h agnic fild B : Subsiu nurical valus and valua B: E v B and E vbsinθ whr θ is h angl bwn v and B. E ( 8./s)(.5T) sin9.4v/.4v/ (c) la h ponial diffrnc bwn h nds of h rod o is lngh and h lcric fild E: V E Subsiu nurical valus and valua V: V (.4 V/)(.3 ).V 37 A 3.-c long rod ovs in a plan ha is prpndicular o a agnic fild of 5 G. Th vlociy of h rod is prpndicular o is lngh. Find h spd of h rod if h ponial diffrnc bwn h nds is 6. V.

23 Magnic nducion 685 Picur h Probl W can us E v B o rla h spd of h rod o h lcric fild in h rod and agnic fild in which i is oving and V E o rla h lcric fild in h rod o h ponial diffrnc bwn is nds. Exprss h lcrosaic fild E in h rod in rs of h agnic fild B and solv for v: la h ponial diffrnc bwn h nds of h rod o is lngh and h lcric fild E and solv for E: Subsiu for E o obain: E v B and E v whr θ is Bsinθ h angl bwn v and B. V E v V B sinθ V E Subsiu nurical valus and v 6.V.5T valua v: ( )(.3) 4/s 38 n Figur 8-45, l h agnic fild srngh b.8 T, h rod spd b /s, h rod lngh b c, and h rsisanc of h rsisor b. Ω. (Th rsisanc of h rod and rails ar ngligibl.) Find (a) h inducd f in h circui, (b) h inducd currn in h circui (including dircion), and (c) h forc ndd o ov h rod wih consan spd (assuing ngligibl fricion). Find (d) h powr dlivrd by h forc found in Par (c) and () h ra of Joul haing in h rsisor. Picur h Probl Bcaus h spd of h rod is consan, an xrnal forc us ac on h rod o counr h agnic forc acing on h inducd currn. W can us h oional-f quaion vb o valua h inducd f, Oh s law o find h currn in h circui, Nwon s nd law o find h forc ndd o ov h rod wih consan spd, and P Fv o find h powr inpu by h forc. (a) la h inducd f in h circui o h spd of h rod, h agnic fild, and h lngh of h rod: vb.6v ( /s)(.8t)(.).6v

24 686 Chapr 8 (b) Using Oh s law, rla h currn in h circui o h inducd f and h rsisanc of h circui: (c) Bcaus h rod is oving wih consan spd in a sraigh lin, h n forc acing on i us b zro. Apply Nwon s nd law o rla F o h agnic forc F : Solving for F and subsiuing for F yilds: Subsiu nurical valus and valua F:.6V.8A.Ω No ha, bcaus h rod is oving o h righ, h flux in h rgion dfind by h rod, h rails, and h rsisor is incrasing. Hnc, in accord wih nz s law, h currn us b counrclockwis if is agnic fild is o counr his incras in flux. F x F F F F F B (.8T)(.8A)(.).3N.8 N (d) Exprss h powr inpu by h forc in rs of h forc and h vlociy of h rod: () Th ra of Joul ha producion is givn by: P Fv P (.8 N)( /s).3w (.8A) (.Ω).3W 39 A -c by 5.-c rcangular loop (Figur 8-46) ha has a rsisanc qual o.5 Ω ovs a a consan spd of.4 c/s hrough a rgion ha has a unifor.7-t agnic fild dircd ou of h pag as shown. Th fron of h loop nrs h fild rgion a i. (a) Graph h flux hrough h loop as a funcion of i. (b) Graph h inducd f and h currn in h loop as funcions of i. Nglc any slf-inducanc of h loop and consruc your graphs o includ h inrval 6 s.

25 Magnic nducion 687 Picur h Probl W ll nd o drin how long i aks for h loop o coplly nr h rgion in which hr is a agnic fild, how long i is in h rgion, and how long i aks o lav h rgion. Onc w know hs is, w can us is dfiniion o xprss h agnic flux as a funcion of i. W can us Faraday s law o find h inducd f as a funcion of i. (a) Find h i rquird for h loop o nr h rgion whr hr is a unifor agnic fild: sid of loop v c.4 c/s 4.7s ing w rprsn h wih of h loop, xprss and valua φ for < < 4.7s : φ NBA NBwv (.7T)(.5 )(.4 /s) (.4 Wb/s) Find h i during which h loop is fully in h rgion whr hr is a unifor agnic fild: sid of loop v c.4 c/s 4.7s i.., h loop will bgin o xi h rgion whn 8.33 s. Exprss φ for 4.7s < < 8.33s : φ NBA NB w (.7T)(.)(.5 ) 8.5 Wb Th lf-nd of h loop will xi h fild whn.5 s. Exprss φ for 8.33s < <.5s : φ + b whr is h slop of h lin and b is h φ -inrcp. For 8.33 s and φ 8.5 Wb:.5 Wb ( 8.33s) + b For.5 s and φ : (.5s) + b 8 () () Solv quaions () and () siulanously o obain: Th loop will b coplly ou of h agnic fild whn >.5 s and: φ (.4 Wb/s) 5.5Wb + φ

26 688 Chapr 8 Th following graph of φ () was plod using a spradsh progra Magnic flux, Wb , s (b) Using Faraday s law, rla h inducd f o h agnic flux: During h inrval < < 4.7s : During h inrval 4.7s < < 8.33s : dφ d d [(.4 Wb/s) ].4 V [ 8.5 Wb] During h inrval 8.33s < <.5s : d.4 V [(.4 Wb/s) + 5.5Wb] For >.5 s: Th currn in ach of hs inrvals is givn by Oh s law:

27 Magnic nducion 689 Th following graph of () was plod using a spradsh progra. f, V , s Th following graph of () was plod using a spradsh progra , A , s 4 A unifor.-t agnic fild is in h +z dircion. A conducing rod of lngh 5 c lis paralll o h y axis and oscillas in h x dircion wih displacn givn by x (. c) cos (π), whr π has unis of rad/s. (a) Find an xprssion for h ponial diffrnc bwn h nds h rod as a funcion of i? (b) Wha is h axiu ponial diffrnc bwn h nds h rod? Picur h Probl Th rod is xcuing sipl haronic oion in h xy plan, i.., in a plan prpndicular o h agnic fild. Th f inducd in h

28 69 Chapr 8 rod is a consqunc of is oion in his agnic fild and is givn by vb. Bcaus w r givn h posiion of h oscillaor as a funcion of i, w can diffrnia his xprssion o obain v. (a) Th ponial diffrnc bwn h nds of h rod is givn by: dx vb B dx d Evalua dx/: [(.c) cosπ ] Subsiu nurical valus and valua : (.c)( s ) ( 7.54 /s) sinπ π sinπ (.T)(.5)( 7.54 /s) sinπ (.4V) sin π (b) Th axiu ponial diffrnc bwn h nds h rod is h apliud of h xprssion for drivd in Par (a): ax.4 V 4 [SSM] n Figur 8-47, h rod has a ass and a rsisanc. Th rails ar horizonal, fricionlss and hav ngligibl rsisancs. Th disanc bwn h rails is. An idal bary ha has an f is conncd bwn poins a and b so ha h currn in h rod is downward. Th rod rlasd fro rs a. (a) Driv an xprssion for h forc on h rod as a funcion of h spd. (b) Show ha h spd of h rod approachs a rinal spd and find an xprssion for h rinal spd. (c) Wha is h currn whn h rod is oving a is rinal spd? Picur h Probl (a) Th n forc acing on h rod is h agnic forc i xprincs as a consqunc of carrying a currn and bing in a agnic fild. Th n f ha drivs in his circui is h diffrnc bwn h f of h bary and h f inducd in h rod as a rsul of is oion. Applying a righhand rul o h rod rvals ha h dircion of his agnic forc is o h righ. Hnc h rod will acclra o h righ whn i is rlasd. (b) W can obain h quaion of oion of h rod by applying Nwon s nd law o rla is acclraion o, B,, and. (c) ing v v in h quaion for h currn in h circui will yild currn whn h rod is a is rinal spd.

29 Magnic nducion 69 (a) Exprss h agnic forc on h currn-carrying rod: F B Th currn in h rod is givn by: B v () Subsiuing for yilds: F B v B B ( B v) (b) ing h dircion of oion of h rod b h posiiv x dircion, apply F x a x o h rod: B ( B v) dv Solving for dv yilds: dv B ( B v) No ha as v incrass, B v, dv and h rod approachs is rinal spd v. S dv o obain: B ( B v ) v B (c) Subsiu v for v in quaion B () o obain: B 4 A unifor agnic fild is sablishd prpndicular o h plan of a loop ha has a radius qual o 5. c and a rsisanc qual o.4 Ω. Th agniud of h fild is incrasing a a ra of 4. T/s. Find (a) h agniud of h inducd f in h loop, (b) h inducd currn in h loop, and (c) h ra of Joul haing in h loop. Picur h Probl (a) W can find h agniud of h inducd f by applying Faraday s law o h loop. (b) and (c) Th applicaion of Oh s law will yild h inducd currn in h loop and w can find h ra of Joul haing using P. (a) Apply Faraday s law o xprss h inducd f in h loop in rs of h ra of chang of h agnic fild: φ d db π d ( AB) A db

30 69 Chapr 8 Subsiu nurical valus and valua : (b) Using Oh s law, rla h inducd currn o h inducd volag and h rsisanc of h loop and valua : (c) Exprss h ra a which powr is dissipad in a conducor in rs of h inducd currn and h rsisanc of h loop and valua P: π (.5 ) ( 4. T/s).34 V.785A.34 V.34V.7854A.4Ω P (.7854 A) (.4Ω).47 μw 43 n Figur 8-48, a conducing rod ha has a ass and a ngligibl rsisanc is fr o slid wihou fricion along wo paralll fricionlss rails ha hav ngligibl rsisancs sparad by a disanc and conncd by a rsisanc. Th rails ar aachd o a long inclind plan ha aks an angl θ wih h horizonal. Thr is a agnic fild dircd upward as shown. (a) Show ha hr is a rarding forc dircd up h inclin givn by F ( B v cos θ)/. (b) Show ha h rinal spd of h rod is v ( g sin θ)/ ( B cos θ). Picur h Probl Th fr-body diagra shows h forcs acing on h rod as i slids down h inclind plan. Th rarding forc is h coponn of F acing up h inclin, i.., in h x dircion. W can xprss F using h xprssion for h forc acing on a conducor oving in a agnic fild. cognizing ha only h horizonal coponn of h rod s vlociy v producs an inducd f, w can apply h xprssion for a oional f in conjuncion wih Oh s law o find h inducd currn in h rod. n Par (b) w can apply Nwon s nd law o obain an xprssion for dv/ and s his xprssion qual o zro o obain v. y θ F r n F r θ g r x

31 Magnic nducion 693 (a) Exprss h rarding forc acing on h rod: Exprss h inducd f du o h oion of h rod in h agnic fild: F F cosθ () whr F B and is h currn inducd in h rod as a consqunc of is oion in h agnic fild. B vcosθ Using Oh s law, rla h currn in h circui o h inducd f: B v cosθ Subsiu in quaion () o obain: (b) Apply Fx ax o h rod: Whn h rod rachs is rinal spd v, dv : F B v cosθ B cosθ B v cos θ B v dv g sinθ cos θ and dv B v g sinθ cos θ B v g sinθ cos θ Solv for v o obain: v gsinθ B cos θ 44 A conducing rod of lngh roas a consan angular spd abou on nd, in a plan prpndicular o a unifor agnic fild B (Figur 8-49). (a) Show ha h ponial diffrnc bwn h nds of h rod is θ. (b) h angl θ bwn h roaing rod and h dashd lin b givn byθ. Show ha h ara of h pi-shapd rgion swp ou by h rod during i is θ. (c) Copu h flux φ hrough his ara, and apply dφ / (Faraday s law) o show ha h oional f is givn by B.

32 694 Chapr 8 Picur h Probl W can us F qv B o xprss h agnic forc acing on h oving chargd body. Exprssing h f inducd in a sgn of h rod of lngh dr and ingraing his xprssion ovr h lngh of h rod will lad us o an xprssion for h inducd f. (a) Us h oional f quaion o xprss h f inducd in a sgn of h rod of lngh dr and a a disanc r fro h pivo: ngra his xprssion fro r o r o obain: (b) Using Faraday s law, rla h inducd f o h ra a which h flux changs: Exprss h ara da, for any valu of θ, bwn r and r + dr: d Brdv Brdr d B rdr B dφ da rθdr ngra fro r o r o obain: A θ rdr θ (c) Using is dfiniion, xprss h agnic flux hrough his ara: φ BA B θ Diffrnia φ wih rspc o i o obain: d dθ [ θ ] B B B 45 [SSM] A.-c by.5-c rcangular coil has 3 urns and roas in a rgion ha has a agnic fild of.4 T. (a) Wha is h axiu f gnrad whn h coil roas a 6 rv/s? (b) Wha us is angular spd b o gnra a axiu f of V? Picur h Probl W can us h rlaionship ax πnbaf o rla h axiu f gnrad o h ara of h coil, h nubr of urns of h coil, h agnic fild in which h coil is roaing, and h angular spd a which i roas.

33 Magnic nducion 695 (a) la h inducd f o h agnic fild in which h coil is roaing: NBA πnbaf () ax Subsiu nurical valus and valua ax : ( 3)(.4T)(. )(.5 )( 6s ) 4V ax π (b) Solv quaion () for f: f ax πnba Subsiu nurical valus and valua f: f π V ( 3)(.4T)(. )(.5 ) 486 rv/s 46 Th coil of Probl 45 roas a 6 rv/s in a agnic fild. f h axiu f gnrad by h coil is 4 V, wha is h agniud of h agnic fild? Picur h Probl W can us h rlaionship ax NBA o rla h axiu f gnrad o h ara of h coil, h nubr of urns of h coil, h agniud of h agnic fild in which h coil is roaing, and h angular spd a which i roas. la h inducd f o h agnic fild in which h coil is roaing: ax ax NBA B NA Subsiu nurical valus and valua B: B π 4V ( 3)(. )(.5 )( 6 rv/s).7t nducanc 47 Whn h currn in an 8.-H coil is qual o 3. A and is incrasing a A/s, find (a) h agnic flux hrough h coil and (b) h inducd f in h coil. Picur h Probl W can us φ o find h agnic flux hrough h

34 696 Chapr 8 coil. W can apply Faraday s law o find h inducd f in h coil. (a) Th agnic flux hrough φ h coil is h produc of h slfinducanc of h coil and h currn i is carrying: Whn h currn is 3. A: ( 8.H)( 3.A) 4. Wb (b) Us Faraday s law o rla, d, and d : φ Subsiu nurical valus and valua : ( 8.H)( A/s).6kV 48 A 3-urn solnoid has a radius qual o. c; a lngh qual o 5. c, and a -urn solnoid has a radius qual o 5. c and is also 5.-c long. Th wo solnoids ar coaxial, wih on nsd coplly insid h ohr. Wha is hir uual inducanc? Picur h Probl W can find h uual inducanc of h wo coaxial φ solnoids using M, μnn πr. Subsiu nurical valus and valua M, : M 7 3 ( 4π N/A ) (.5 ) π (. ).89H, [SSM] An insulad wir ha has a rsisanc of 8. Ω/ and a lngh of 9. will b usd o consruc a rsisor. Firs, h wir is bn in half and hn h doubld wir is wound on a cylindrical for ( Figur 8-5) o cra a 5.-c-long hlix ha has a diar qual o. c. Find boh h rsisanc and h inducanc of his wir-wound rsisor. Picur h Probl No ha h currn in h wo pars of h wir is in opposi dircions. Consqunly, h oal flux in h coil is zro. W can find h rsisanc of h wir-wound rsisor fro h lngh of wir usd and h rsisanc pr uni lngh. Bcaus h oal flux in h coil is zro:

35 Magnic nducion 697 Exprss h oal rsisanc of h wir: Subsiu nurical valus and valua : Ω 8. Ω 8. ( 9.) 6Ω 5 You ar givn a lngh of wir ha has radius a and ar old o wind i ino an inducor in h shap of a hlix ha has a circular cross scion of radius r. Th windings ar o b as clos oghr as possibl wihou ovrlapping. Show ha h slf-inducanc of his inducor is μ r a. Picur h Probl Th wir of lngh and radius a is shown in h diagra, as is h inducor consrucd wih his wir and whos inducanc is o b found. W can us h quaion for h slf-inducanc of a cylindrical inducor o driv an xprssion for. 4 l a a... r... d Th slf-inducanc of an inducor wih lngh d, cross-scional ara A, and nubr of urns pr uni lngh n is: Th nubr of urns N is givn by: Assuing ha a << r, h lngh of h wir is rlad o N and r: Solving for d yilds: n Ad () μ d N n a N d a d N a a d π r π r a ( π r) π r d

36 698 Chapr 8 Subsiu for d, A, and n in quaion a a π r () o obain: μ ( π ) μ r a 5 Using h rsul of Probl 5, calcula h slf-inducanc of an inducor wound fro c of wir ha has a diar of. ino a coil ha has a radius of.5 c. Picur h Probl W can subsiu nurical valus in h xprssion drivd in Probl 5o find h slf-inducanc of h inducor. r 4 Fro Probl 5 w hav: rd 4a μ Subsiu nurical valus and valua : 7 ( 4π N/A )(.5c)( c) 4(.5 ).6μH 5 n Figur 8-5, circui has a oal rsisanc of 3 Ω. Afr swich S is closd, h currn in circui incrass raching a valu of 5. A afr a long i. A charg of μc passs hrough h galvanor in circui during h i ha h currn in circui is incrasing. Wha is h uual inducanc bwn h wo coils? Picur h Probl W can apply Kirchhoff s loop rul o h galvanor circui o rla h ponial diffrnc across o h ponial diffrnc across. ngraion of his quaion ovr i will yild an quaion ha rlas h uual inducanc bwn h wo coils o h sady-sa currn in circui and h charg ha flows hrough h galvanor. Apply Kirchhoff s loop rul o h galvanor circui: ngra ach r fro o : d d M + or Md + d M and M d + d + Q Bcaus : M Q M Q

37 Subsiu nurical valus and valua M: M Magnic nducion ( 3Ω)(. C) 5.A. H 53 [SSM] Show ha h inducanc of a oroid of rcangular cross scion, as shown in Figur 8-5 is givn by μ o N H ln( b / a) whr N is h π oal nubr of urns, a is h insid radius, b is h ousid radius, and H is h high of h oroid. Picur h Probl W can us Apr s law o xprss h agnic fild insid h rcangular oroid and h dfiniion of agnic flux o xprss φ hrough h oroid. W can hn us h dfiniion of slf-inducanc of a solnoid o xprss. Using h dfiniion of h slfinducanc of a solnoid, xprss in rs of φ, N, and : Apply Apr s law o a closd pah of radius a < r < b: Exprss h flux in a srip of high H and wih dr : Subsiuing for B yilds: Nφ () C B d Bπ r μ or, bcaus C N, μn Bπ r μn B πr d φ BHdr dφ μnh dr πr C ngra dφ fro r a o r b o obain: φ μ NH b dr μnh b ln π r π a a Subsiu for φ in quaion () and siplify o obain: Magnic Enrgy μn H b ln π a 54 A coil ha has a slf-inducanc of. H and a rsisanc of. Ω is conncd o an idal 4.-V bary. (a) Wha is h sady-sa currn?

38 7 Chapr 8 (b) How uch nrgy is sord in h inducor whn h sady-sa currn is sablishd? Picur h Probl Th currn in an circui, as a funcion of i, is givn by ( f ), whr f / and /. Th nrgy sord in h inducor undr sady-sa condiions is sord in is agnic fild and is givn by U. f (a) Th final currn is h quoin of h f of h bary and h rsisanc of h coil: 4.V.Ω f.a (b) Th nrgy sord in h inducor is: U f 4.J (.H)(.A) 55 [SSM] n a plan lcroagnic wav, h agniuds of h lcric filds and agnic filds ar rlad by E cb, whr c μ is h spd of ligh. Show ha whn E cb h lcric and h agnic nrgy dnsiis ar qual. Picur h Probl W can xain h raio of u o u E wih E cb and c μ o show ha h lcric and agnic nrgy dnsiis ar qual. Exprss h raio of h nrgy B dnsiy in h agnic fild o h u μ B nrgy dnsiy in h lcric fild: u E μ E E Bcaus E cb: u u B E μ c B μ c Subsiuing for c and siplifying yilds: u u E μ μ u u E 56 A -urn solnoid has a cross-scional ara qual o 4. c and lngh qual o 3 c. Th solnoid carris a currn of 4. A. (a) Calcula h agnic nrgy sord in h solnoid using U, whr μ n A. (b) Divid your answr in Par (a) by h volu of h rgion insid h solnoid o find h agnic nrgy pr uni volu in h solnoid. (c) Chck your Par (b) rsul by copuing h agnic nrgy dnsiy fro μ B / μ whr B μ n.

39 Magnic nducion 7 Picur h Probl W can us n A o find h inducanc of h solnoid and B μ μ n o find h agnic fild insid i. (a) Exprss h agnic nrgy sord in h solnoid: la h inducanc of h solnoid o is dinsions and propris: U μ n A Subsiu for o obain: U μ n A Subsiu nurical valus and 7 valua U : U ( 4π N/A ) 4 ( 4. )(.3)( 4.A) 53.6 J 54J.3 (b) Th agnic nrgy pr uni 4 volu in h solnoid is: V A ( 4. )(.3) U U.45kJ/ J (c) Th agnic nrgy dnsiy in h solnoid is givn by: B u μ Subsiuing for B and siplifying yilds: ( μ n ) u μ μn N μ μ Subsiu nurical valus and valua u : u 7 ( 4π N/A )( )( 4. A) (.3 ).45 kj/ 3 57 A long cylindrical wir has a radius qual o. c and carris a currn of 8 A uniforly disribud ovr is cross-scional ara. Find h agnic nrgy pr uni lngh wihin h wir.

40 7 Chapr 8 Picur h Probl Considr a cylindrical annulus of hicknss dr a a radius r < a. W can us is dfiniion o xprss h oal agnic nrgy du insid h cylindrical annulus and divid boh sids of his xprssion by h lngh of h wir o xprss h agnic nrgy pr uni lngh du'. ngraion of his xprssion will giv us h agnic nrgy pr uni lngh wihin h wir. dr r a Exprss h agnic nrgy wihin h cylindrical annulus: du B V μ B πr dr μ B μ annulus πr dr Divid boh sids of h quaion by o xprss h agnic nrgy pr uni lngh du' : du' B πrdr () μ Us Apr s law o xprss h agnic fild insid h wir a a disanc r < a fro is cnr: μc π rb μc B πr whr C is h currn insid h cylindr of radius r. Bcaus h currn is uniforly disribud ovr h crossscional ara of h wir: C πr r C πa a Subsiu for C o obain: μr B πa

41 Magnic nducion 73 Subsiuing for B in quaion () and siplifying yilds: ngra du' fro r o r a: μr πa μ 3 du ' πrdr r dr 4 μ 4πa U' a 4 μ 3 μ a r dr 4 4 4πa 4πa 4 μ 6π arks: No ha h agnic nrgy pr uni lngh is indpndn of h radius of h cylindr and dpnds only on h oal currn. 58 A oroid ha has a an radius qual o 5. c and a circular loops wih radii qual o. c is wound wih a suprconducing wir. Th wir has a lngh qual o and carris a currn of 4 A. (a) Wha is h nubr of urns of h wir? (b) Wha is h agnic fild srngh and agnic nrgy dnsiy a h an radius? (c) Esia h oal nrgy sord in his oroid by assuing ha h nrgy dnsiy is uniforly disribud in h rgion insid h oroid. Picur h Probl W can find h nubr of urns on h coil fro h lngh of h suprconducing wir and h cross-scional radius of h coil. W can us B ( μn ) ( πr an ) o find h agnic fild a h an radius. W can find h nrgy dnsiy in h agnic fild fro u B ( μ ) and h oal nrgy sord in h oroid by uliplying u by h volu of h oroid. (a) Exprss h nubr of urns in rs of h lngh of h wir and lngh rquird pr urn πr: N πr Subsiu nurical valus and N π valua N: (. ) (b) B insid a ighly wound oroid or radius r is givn by: Subsiu nurical valus and valua h agnic fild a h an radius: μn B π r B 7 ( 4π N/A )( 7958)( 4A) π (.5 ).547 T.55T

42 74 Chapr 8 Th nrgy dnsiy in h agnic fild is givn by: B u μ Subsiu nurical valus and valua u : u (.547 T) 7 ( 4π N/A ).58MJ/ 3.58 MJ/ 3 (c) la h oal nrgy sord in h oroid o h nrgy dnsiy in is agnic fild and h volu of h oroid: Think of h oroid as a cylindr of radius r and high πr an o obain: U u V V oroid ( π ran ) π r ran π oroid r Subsiu for V oroid o obain: U π r ranu Subsiu nurical valus and valua U : Circuis 3 (. ) (.5 )(.58 MJ/ ) 5.9kJ U π 59 [SSM] A circui consiss of a coil ha has a rsisanc qual o 8. Ω and a slf-inducanc qual o 4. H, an opn swich and an idal -V bary all conncd in sris. A h swich is closd. Find h currn and is ra of chang a is (a), (b). s, (c).5 s, and (d). s. Picur h Probl W can find h currn using ( ) f whr f / and / and is ra of chang by diffrniaing his xprssion wih rspc o i. Exprss h dpndnc of h currn on f and : f ( )

43 Magnic nducion 75 Evaluaing f and yilds: V 8.Ω and 4.H 8.Ω f.5a.5s.5 s Subsiu for f and o obain: (.5A)( ) Exprss d/: d.5 s (.5A)( )( s ).5 s ( 5. ka/s) (a) Evalua and d/ a : ( ) (.5A)( ) and d ( 5. ka/s) 5.kA/s (b) Evaluaing and d/ a. s yilds: (c) Evalua and d/ a.5 s o obain: (d) Evaluaing and d/ a. s yilds:. s.5 s (. s) (.5A)( ) and d.5 s.7a ( 5. ka/s).5ka/s. s.5 s.5 s.5 s (.5 s) (.5A)( ) and d.5 s 7.9A ( 5. ka/s) 9. ka/s.5 s.5 s. s.5 s (. s) (.5A)( ).8A and

44 76 Chapr 8 d. s ( 5. ka/s) 3.38kA/s. s.5 s 6 n h circui shown in Figur 8-53, h hrow of h ak-bforbrak swich has bn a conac a for a long i and h currn in h. H coil is qual o. A. A h hrow is quickly ovd o conac b. Th oal rsisanc + r of h coil and h rsisor is. Ω. Find h currn whn (a).5 s, and (b) s. Picur h Probl W can find h currn using ( ), currn a i and /. whr is h Exprss h currn as a funcion of i: Evaluaing yilds: ( ) (.A).H.s.Ω. s Subsiu for o obain: ( ) ( ).A (a) Whn.5 s: (.5 s) (.A) 3.5A (b) Whn. s: (. s) ( A) (.A).5 s. s. s. s A 6 [SSM] n h circui shown in Figur 8-54, l. V, 3. Ω, and.6 H. Th swich, which was iniially opn, is closd a i. A i.5 s, find (a) h ra a which h bary supplis nrgy, (b) h ra of Joul haing in h rsisor, and (c) h ra a which nrgy is bing sord in h inducor. Picur h Probl W can find h currn using ( ), whr f /,and /, and is ra of chang by diffrniaing his xprssion wih rspc o i. Exprss h dpndnc of h currn on f and : ( ) ( ) f f

45 Magnic nducion 77 Evaluaing f and yilds:.v 3.Ω and.6h 3.Ω f 4.A.s.s Subsiu for f and o obain: ( ) ( 4.A)( ) Exprss d/: d. s ( 4.A)( )( 5.s ). s (. A/s) (a) Th ra a which h bary supplis nrgy is givn by: P.s Subsiuing for and yilds: P( ) ( 4.A)( )(. V).s ( 48.W)( ) Th ra a which h bary supplis nrgy a.5 s is: P.5.s (.5 s) ( 48.W)( ) 44.W (b) Th ra of Joul haing is: Subsiu for and and siplify o obain: Th ra of Joul haing a.5 s is: (c) Us h xprssion for h agnic nrgy sord in an inducor o xprss h ra a which nrgy is bing sord: P P J J P J du.s [( 4.A)( )] ( 3. Ω).s ( 48. W)( ).5 s.s (.5 s) ( 48. W)( ) d [ ] 4.4 W d

46 78 Chapr 8 Subsiu for,, and d/ o obain: du.s (.6 H)( 4.A)( )(. A/s).s.s ( 48. W)( ).s Evalua his xprssion for.5 s: du.5 s.5 s.s.5 s.s ( 48. W)( ) 3.6 W arks: No ha, o a good approxiaion, du / P P J. 6 How any i consans us laps bfor h currn in an circui (Figur 8-54) ha is iniially zro rachs (a) 9 prcn, (b) 99 prcn, and (c) 99.9 prcn of is sady-sa valu? Picur h Probl f h currn is iniially zro in an circui, is valu a so lar i is givn by ( ) f, whr f / and / is h i consan for h circui. W can find h nubr of i consans ha us laps bfor h currn rachs any givn fracion of is final valu by solving his quaion for /. Exprss h fracion of is final valu o which h currn has risn as a funcion of i: f ln f (a) Evalua / for / f.9: 9% ln (.9). 3 (b) Evalua / for / f.99: 99% ln (.99) 4. 6 (c) Evalua / for / f.999: 99.9% ln (.999) [SSM] A circui consiss of a 4.-H coil, a 5-Ω rsisor, a.-v idal bary and an opn swich all conncd in sris. Afr h swich is closd: (a) Wha is h iniial ra of incras of h currn? (b) Wha is h ra