Bf: the positive charges in the moving bar will flow downward

Transcription

1 31 Fdy s Lw CHAPTE OUTLNE 311 Fdy s Lw of nduction 31 Motionl mf 313 Ln s Lw 314 nducd mf nd Elctic Filds 315 Gntos nd Motos 316 Eddy Cunts 317 Mxwll s Equtions ANSWES TO QUESTONS Q311 Mgntic flux msus th flow of th mgntic fild though givn of loop vn though th fild dos not ctully flow By chnging th si of th loop, o th ointtion of th loop nd th fild, on cn chng th mgntic flux though th loop, but th mgntic fild will not chng Q31 Th mgntic flux is Φ B BA cosθ Thfo th flux is mximum whn B is ppndicul to th loop of wi nd o whn th is no componnt of mgntic fild ppndicul to th loop Th flux is o whn th loop is tund so tht th fild lis in th pln of its Q313 Th foc on positiv chgs in th b is F v Bf f th b is moving to th lft, positiv chg will mov downwd nd ccumult t th bottom nd of th b, so tht n lctic fild will b stblishd upwd Q314 No Th mgntic foc cts within th b, but hs no influnc on th fowd motion of th b q Q315 By th mgntic foc lw F v Bf: th positiv chgs in th moving b will flow downwd nd thfo clockwis in th cicuit f th b is moving to th lft, th positiv chg in th b will flow upwd nd thfo countclockwis in th cicuit Q316 W igno mchnicl fiction btwn th b nd th ils Moving th conducting b though th mgntic fild will foc chgs to mov ound th cicuit to constitut clockwis cunt Th downwd cunt in th b fls mgntic foc to th lft Thn countblncing pplid foc to th ight is quid to mintin th motion Q317 A cunt could b st up in th bclt by moving th bclt though th mgntic fild, o if th fild pidly chngd Q318 Moving mgnt insid th hol of th doughnut-shpd tooid will not chng th mgntic flux though ny tun of wi in th tooid, nd thus not induc ny cunt q 13

2 14 Fdy s Lw Q319 As wt flls, it gins spd nd kintic ngy t thn pushs ginst tubin blds, tnsfing its ngy to th oto coils of lg AC gnto Th oto of th gnto tuns within stong mgntic fild Bcus th oto is spinning, th mgntic flux though its tuns chngs in tim s Φ B BA cosω t Gntd in th oto is n inducd mf of ε Nd Φ B This inducd mf is th voltg diving th cunt in ou lctic pow lins Q3110 Ys Eddy cunts will b inducd ound th cicumfnc of th copp tub so s to fight th chnging mgntic flux by th flling mgnt f b mgnt is doppd with its noth pol downwds, ing of countclockwis cunt will suound its ppoching bottom nd nd ing of clockwis cunt will suound th cding south pol t its top nd Th mgntic filds ctd by ths loops of cunt will xt focs on th mgnt to slow th fll of th mgnt quit significntly Som of th oiginl gvittionl ngy of th mgnt will pp s intnl ngy in th wlls of th tub Q3111 Ys Th inducd ddy cunts on th sufc of th luminum will slow th dscnt of th luminum t my fll vy slowly Q311 Th mximum inducd mf will incs, incsing th tminl voltg of th gnto Q3113 Th incsing countclockwis cunt in th solnoid coil poducs n upwd mgntic fild tht incss pidly Th incsing upwd flux of this fild though th ing inducs n mf to poduc clockwis cunt in th ing Th mgntic fild of th solnoid hs dilly outwd componnt t ch point on th ing This fild componnt xts upwd foc on th cunt in th ing th Th whol ing fls totl upwd foc lg thn its wight FG Q3113 Q3114 Oscillting cunt in th solnoid poducs n lwys-chnging mgntic fild Vticl flux though th ing, ltntly incsing nd dcsing, poducs cunt in it with diction tht is ltntly clockwis nd countclockwis Th cunt though th ing s sistnc poducs intnl ngy t th t Q3115 () Th south pol of th mgnt poducs n upwd mgntic fild tht incss s th mgnt ppochs Th loop opposs chng by mking its own downwd mgntic fild; it cis cunt clockwis, which gos to th lft though th sisto Th noth pol of th mgnt poducs n upwd mgntic fild Th loop ss dcsing upwd flux s th mgnt flls wy, nd tis to mk n upwd mgntic fild of its own by cying cunt countclockwis, to th ight in th sisto

3 Chpt Q3116 () Th btty mks countclockwis cunt 1 in th pimy coil, so its mgntic fild B 1 is to th ight nd incsing just ft th switch is closd Th scondy coil will oppos th chng with lftwd fild B, which coms fom n inducd clockwis cunt tht gos to th ight in th sisto (c) At stdy stt th pimy mgntic fild is unchnging, so no mf is inducd in th scondy Th pimy s fild is to th ight nd dcsing s th switch is opnd Th scondy coil opposs this dcs by mking its own fild to th ight, cying countclockwis cunt to th lft in th sisto FG Q3116 Q3117 Th motionl mf btwn th wingtips cnnot b usd to un light bulb To connct th light, n xt insultd wi would hv to b un out long ch wing, mking contct with th wing tip Th wings with th xt wis nd th bulb constitut singl-loop cicuit As th pln flis though unifom mgntic fild, th mgntic flux though this loop is constnt nd o mf is gntd On th oth hnd, if th mgntic fild is not unifom, lg loop towd though it will gnt pulss of positiv nd ngtiv mf This phnomnon hs bn dmonsttd with cbl unld fom th Spc Shuttl Q3118 No, thy do not Spcificlly, Guss s lw in mgntism pohibits mgntic monopols f mgntic monopols xistd, thn th mgntic fild lins would not hv to b closd loops, but could bgin o tmint on mgntic monopol, s thy cn in Guss s lw in lctosttics Q3119 () A cunt is inducd by th chnging mgntic flux though th ing of th tub, poducd by th high fquncy ltnting cunt in th coil (c) (d) Th high fquncy implis gt t of chng in th mgntic fild, fo lg inducd voltg Th sistnc of on cubic cntimt in th bulk sht mtl is low, so th t of poduction of intnl ngy is low At th sm, th cunt stts out cowdd into smll with high sistnc, so th tmptu iss pidly, nd th dgs mlt togth Th dgs must b in contct to llow th inducd mf to ct n lctic cunt ound th cicumfnc of th tub Additionlly, (duh) th two dgs must b in contct to b wldd t ll, just s you cn t glu two pics of pp togth without putting thm in contct with ch oth

4 16 Fdy s Lw SOLUTONS TO POBLEMS Sction 311 Sction 313 Fdy s Lw of nduction Ln s Lw ΦB NBA P311 ε ΦB BA P31 ε ε 160 mv nd f 500 mv 4 f f jf K JF loop 50 T 0500 T m 1 N s 1 V C 100 s 1 T C m 1 N m ε 160 mv 00 Ω F 0800 ma θ θ θ P313 ε π π KJ N BA cos cos f cos i 6 cos180 cos 0 NB T m 000 s ε +98 mv j f K J F K J dφb P314 () ε A db AB τ mx t τ j f m T ε 00 s 379 mv (c) At t 0 ε 8 0 mv P315 Noting unit convsions fom F q v B nd U qv, th inducd voltg is F K J + f f j F 1 N s K JF 1 V C K J 1 T C m N m θ ε N d BA 0 BA i cos T 0 00 m cos0 N s ε 300V 160 A 00 Ω P316 ε N d Φ B NBA 0 NBA NB π π ε ε f j f j s 300V

5 f dba P317 ε µ na d 3 0 V Chpt () 4 ing A ε B ing µ 0 ing 0 1 µ T (c) Coil s fild points downwd, nd is incsing, so B ing points upwd FG P317 f dba P318 ε µ na d µ nπ 0 0 () n ing ε µ π 0 µ 0 µ nπ B (c) Th coil s fild points downwd, nd is incsing, so B ing points upwd P319 () dφ B B d A µ h+ w 0 π x Ldx L dx L h+ w : Φ B µ 0 µ 0 F ln π x π h h Φ L µ 0L F h+ w µ NM π K JO L F QP 0L h+ w NM π K JO d ln ln h h QP 4π 10 7 TmA j100 mf F ln π K J b g 100 d B d ε ε 10 0 As 4 80 µ V Th long wi poducs mgntic flux into th pg though th ctngl, shown by th fist hnd in th figu to th ight As th mgntic flux incss, th ctngl poducs its own mgntic fild out of th pg, which it dos by cying countclockwis cunt (scond hnd in th figu) K J FG P318 FG P319

6 18 Fdy s Lw P3110 Φ B µ 0 n A b g solnoid Φ B d solnoidj j j b g b g f ε N d Nµ 0n π ε π 10 TmA m π m 600 Ascos 10t ε 14 cos 10t mv f P3111 Fo countclockwis tip ound th lft-hnd loop, with B At j f d At cos PQ 0 nd fo th ight-hnd loop, f d At + PQ 3 0 wh PQ 1 is th upwd cunt in QP d PQ i + PQ 3 f Thus, A nd A PQ 5 A 6PQ A + PQ 0 3 j b g f A PQ upwd, nd sinc Ω m m Ω Ts j0650 mf PQ 83 µ A upwd Ω Φ F P311 ε H G B N db K J b + b A N t A b g b g At t 500 s, ε Ts π m 61 8 mv 160t 0 0 f j P3113 B µ n µ n 30 0 A 1 Φ B Φ B µ 0 f 160tj 30 0 Af1 160tj Af 1 60f BdA n 30 0 A 1 da µ n π 0 Φ B 160 t ε N d Nµ n π f j j f b g ε 50 4π 10 N A 400 m 30 0 A π m 1 60 s ε g g t 160t 68 mv f countclockwis FG P3111 FG P3113

7 Chpt *P3114 () Ech coil hs puls of voltg tnding to poduc countclockwis cunt s th pojctil ppochs, nd thn puls of clockwis voltg s th pojctil cds V d 150 m v 3 t s 65 ms 0 V 1 V t FG P3114 d θ P3115 ε θ NB N Bcos cos ε N Bcosθ j Lngth 4 N m 50 7 m 3 j f V s 136 m T T cos 30 0 f j f f f *P3116 () Suppos, fist, tht th cntl wi is long nd stight Th nclosd cunt of unknown mplitud cts cicul mgntic fild ound it, with th mgnitud of th fild givn by Amp s Lw t B ds µ 0 : B µ 0 mx sinω π t th loction of th ogowski coil, which w ssum is cntd on th wi This fild psss ppndicully though ch tun of th tooid, poducing flux BA µ 0mxAsin ω t π Th tooid hs π n tuns As th mgntic fild vis, th mf inducd in it is µ ε N d 0mxA d BA π n sinω tµ 0mxnAωcos ωt π This is n ltnting voltg with mplitud ε mx µ 0 naω mx Msuing th mplitud dtmins th si mx of th cntl cunt Ou ssumptions tht th cntl wi is long nd stight nd psss ppndicully though th cnt of th ogowski coil ll unncssy f th wi is not cntd, th coil will spond to stong mgntic filds on on sid, but to cospondingly wk filds on th opposit sid Th mf inducd in th coil is popotionl to th lin intgl of th mgntic fild ound th cicul xis of th tooid Amp s Lw sys tht this lin intgl dpnds only on th mount of cunt th coil ncloss t dos not dpnd on th shp o loction of th cunt within th coil, o on ny cunts outsid th coil

8 0 Fdy s Lw P3117 n tooid, ll th flux is confind to th insid of th tooid µ 0N 500µ 0 B π π Φ B Φ B ε ε 500µ BdA 0 mx sinω t π d 500µ b 0 mx F + sinω tln K J π F µ H G + ω ω π K J F N d Φ B b mx K J t 4 10 F + 4π N A j50 0 A fb377 d s gb mg f ln π 400 cm 04 Vfcosω t ln cos FG P3117 f 3 *P3118 Th upp loop hs π 0 05 m m Th inducd mf in it is cm KJ cosωt ε N d θ BA A db 3 cos 1 cos m b T s g V Th minus sign indicts tht it tnds to poduc countclockwis cunt, to mk its own mgntic fild out of th pg Similly, th inducd mf in th low loop is db ε NA cos θ π009 m f T s V V to poduc countclockwis cunt in th low loop, which bcoms clockwis cunt in th upp loop Th nt mf fo cunt in this sns ound th figu 8 is V V V t pushs cunt in this sns though sis sistnc π0 05 m f+ π0 09 m f 3 Ω m 64 Ω ε V Th cunt is 13 3 ma 64 Ω Sction 31 Sction 313 Motionl mf Ln s Lw P3119 () Fo mximum inducd mf, with positiv chg t th top of th ntnn, F q v Bf, so th uto must mov st j ff ε Bv T 10 m m cos s KJ 4 V

9 ε Bv P310 v 100 m s Chpt 31 1 FG P310 P311 () FB B B Whn ε nd ε Bv Bv B v w gt FB B f f f f N Th pplid foc is 300 N to th ight FG P311 P B v 600 W o P Fv 600 W P31 FB B nd ε Bv ε Bv so B v () F B v Fv B nd 0500 A 00 W fb g (c) Fo constnt foc, P F v 100 N 00 m s 00 W *P313 Modl th mgntic flux insid th mtllic tub s constnt s it shinks fom dius to dius : 50 T π B π B f j f F H G K J f 50 T 50 T T

10 Fdy s Lw *P314 Obsv tht th homopol gnto hs no commutto nd poducs voltg constnt in tim: DC with no ippl n tim, th disk tuns by ngl dθ ω Th out bush slids ov distnc dθ Th dil lin to th out bush swps ov 1 1 da dθ ω Th mf gntd is ε N d BA da F ε ω H G 1 1fB cos 0 B K J (W could think of this s following fom th sult of Exmpl 314) Th mgnitud of th mf is b g f b go QP F H G FG P314 F π ε ω H G K J L B N M d v N s C m m v min 60 smin ε 4 1 V A f positiv chg q shown, tuning with th disk, fls mgntic foc qv B dilly outwd Thus th out contct is positiv K J *P315 Th spd of wvs on th wi is T v µ 67 N m kg 98 ms n th simplst stnding-wv vibtion stt, d NN 064 m λ λ 18 m v 98 ms nd f 33 H λ 18 m () Th chnging flux of mgntic fild though th cicuit contining th wi will div cunt to th lft in th wi s it movs up nd to th ight s it movs down Th mf will hv this sm fquncy of 33 H Th vticl coodint of th cnt of th wi is dscibd by x Acos ωt 15 cm cos π33t s ts vlocity is v f b g dx 15 cmfb 33 s gsinb 33 t sg cm πf s m s ts mximum spd is π π Th inducd mf is ε Bv, with mplitud f 3 3 ε mx Bv mx T 0 0 m m s V

11 P316 ε N d θ θ BA NB A cos cos 300 m 300 m sin m ε Tf f f cos s 11 V 011 A 100 Ω F H G K J 11 V Chpt 31 3 FG P316 Th flux is into th pg nd dcsing Th loop mks its own mgntic fild into th pg by cying clockwis cunt b gb g P317 ω 00 v s π d v 400 π d s 1 ε Bω 83 mv P318 () B B i nd B xt dcss; thfo, th xt xt inducd fild is B B i (to th ight) nd th 0 0 cunt in th sisto is dictd to th ight B B i xt fild B xt j incss; thfo, th inducd B i j is to th ight, nd th cunt in th sisto is dictd to th ight (c) B B k xt xt j into th pp nd B xt dcss; 0 B0 k j into th thfo, th inducd fild is B pp, nd th cunt in th sisto is dictd to th ight FG P318 (d) By th mgntic foc lw, FB q v B f Thfo, positiv chg will mov to th top of th b if B is into th pp

12 4 Fdy s Lw P319 () Th foc on th sid of th coil nting th fild (consisting of N wis) is f f F N LB N wb Th inducd mf in th coil is Φ f ε N d B N dbwx NBwv ε NBwv so th cunt is countclockwis Th foc on th lding sid of th coil is thn: F F H G K J N NBwv wb N B w v to th lft Onc th coil is ntily insid th fild, NBA constnt, Φ B so ε 0, 0, nd F 0 FG P319 (c) As th coil stts to lv th fild, th flux dcss t th t Bwv, so th mgnitud of th cunt is th sm s in pt (), but now th cunt is clockwis Thus, th foc xtd on th tiling sid of th coil is: F N B w v to th lft gin P3130 Look in th diction of b Th b mgnt cts fild into th pg, nd th fild incss Th loop will ct fild out of th pg by cying countclockwis cunt Thfo, cunt must flow fom b to though th sisto Hnc, V V will b ngtiv P3131 Nm th cunts s shown in th digm: Lft loop: + Bdv 11 0 ight loop: + Bdv At th junction: Thn, Bdv Bdv So, Bdv3 1 1 Bdv 1b1 + g F v3 v KJ upwd L O ms ms b T g mf b g Ωf b g Ωf NM P 500 Ωf100 Ωf Ωf150 Ωf Ωf150 ΩfQ Bd b FG P3131 P 145 µ A upwd

13 Chpt 31 5 Sction 314 P313 () nducd mf nd Elctic Filds db Φ B 600 t 800 t ε d π db 800 π 0050 At t 00 s, E π π b g b g b g Whn 6 00t 8 00t 0, t 133 s F qe N FG P313 P3133 db dφ 00600t ε B db π1 πe 1 At t 300 s, E π π 1 db F H G K J m F Tsjb300 sg 1 N s 1 T C m K J E N C ppndicul to 1 nd countclockwis FG P3133 dφ P3134 () E d πe B π j so E 987 mv m cos 100π t db Th E fild is lwys opposit to incsing B clockwis b g b g Sction 315 Gntos nd Motos b g f f f kv P3135 () εmx NABω π 7 54 f ε t NBAωsinωt NBAωsinθ ε is mximl whn sinθ 1 π o θ ± FG P3135 so th pln of coil is plll to B

14 6 Fdy s Lw P3136 Fo th ltnto, ω Φ B F b H G K J F H G 3000 v ming K J π d 1 min 1 v 60 s 314 d s j b g jb g f ε N d d cos 314t sin 314t Tm s Tm s f f () ε 19 6 V sin 314t ε mx 19 6 V P3137 B µ n 4π 10 TmA 00 m 15 0 A T 0 j j f j Fo th smll coil, Φ B NBA NBAcosωt NB π cosωt dφb Thus, ε NBπ ωsinωt 3 f j b g 1 j b g f b g ε T π m 4 00πs sin 4 00πt 8 6 mv sin 4 00πt P3138 As th mgnt otts, th flux though th coil vis sinusoidlly in tim with Φ B 0 t t 0 Choosing th flux s positiv whn th fild psss fom lft to ight though th of th coil, th flux t ny tim my b wittn s ΦB Φ mx sinω t so th inducd mf is givn by / mx t/t (ωt/π ) dφ ε B ωφ mx cos ωt FG P3138 ε ω Φ Th cunt in th coil is thn mx cosωt mx cos ωt *P ma 10 V M To nly th ctul cicuit, w modl it s 10 V 850 ma 118 Ω ε bck () Th loop ul givs + 10 V 0 85 A 11 8 Ω bck 0 f ε ε bck V 110 (c) Th sisto is th dvic chnging lcticl wok input into intnl ngy: f f P 085 A 118 Ω 853 W With no motion, th moto dos not function s gnto, nd ε bck 0 Thn 10 V 11 8 Ω 0 10 A c c P 10 A 11 8 Ω 1 kw c f c f f

15 1 P3140 () ε BAω B π ω π ε mx 130 T 050 m 400 π d s ε F mx H G K J f f b g mx π 160 V π ε επ θ BAω d sinθdθ π ε ε Chpt 31 7 t Figu 1 t (c) (d) Th mximum nd vg ε would min unchngd S Figu 1 t th ight Figu FG P3140 () S Figu t th ight f j f j f P3141 () Φ B BA cosθ BA cos ωt T m cos π 60 0 t 8 00 mt m cos 377t dφ ε B (c) ε (d) P () P Fv 30 Vfsin377tf 30 Afsin377tf 910 Wfsin 377tf f f P τω so τ 4 1 mn m sin 377t ω Sction 316 Eddy Cunts P314 Th cunt in th mgnt cts n upwd mgntic fild, so th N nd S pols on th solnoid co shown coctly On th il in font of th bk, th upwd flux of B incss s th coil ppochs, so cunt is inducd h to ct downwd mgntic fild This is clockwis cunt, so th S pol on th il is shown coctly On th il bhind th bk, th upwd mgntic flux is dcsing Th inducd cunt in th il will poduc upwd mgntic fild by bing countclockwis s th pictu coctly shows

16 8 Fdy s Lw P3143 () At tminl spd, Mg F wb B ε wb Bwv T wb F H G K J F H G K J B w v T o v Mg T B ω (c) Th mf is dictly popotionl to v T, but th cunt is invsly popotionl to A lg mns smll cunt t givn spd, so th loop must tvl fst to gt FB mg FG P3143 At givn spd, th cunt is dictly popotionl to th mgntic fild But th foc is popotionl to th poduct of th cunt nd th fild Fo smll B, th spd must incs to compnst fo both th smll B nd lso th cunt, so vt B Sction 317 Mxwll s Equtions P3144 F m qe+ q v B so E+ v B m 19 j i j k wh v B j i j 4 00 j i j i j P3145 F m qe+ qv B j ms E+ v B m 19 i j k wh v B f j fk j 80 0 j+ 60 0k j+ 60 0k j+ k ms j k ms 9 j j Additionl Poblms f F j H G K J 1 f + NM j QP f f j L 3 O f NM m j cos QP T j s j t s j V jcos π53t s j P3146 ε N d θ π BA N db cos cos 0 ε L π O d m mt 3 0 mt sin π 53t s ε π π π ε

17 Chpt 31 9 P3147 () Doubling th numb of tuns Amplitud doubls: piod unchngd Doubling th ngul vlocity doubls th mplitud: cuts th piod in hlf (c) Doubling th ngul vlocity whil ducing th numb of tuns to on hlf th oiginl vlu FG P3147 Amplitud unchngd: cuts th piod in hlf F 150 T 500 T f j jf K J s B P3148 ε N BA cosθ N π cos m 1 t () ε 0875 V Ω 43 8 A f f P ε 0875 V 438 A 383 W P3149 n th loop on th lft, th inducd mf is f b g dφ ε A db B π 0100 m 100 T s πv nd it ttmpts to poduc countclockwis cunt in this loop n th loop on th ight, th inducd mf is f b g FG P3149 dφ ε B π m 100 T s 5πV 0875 V nd it ttmpts to poduc clockwis cunt Assum tht 1 flows down though th 600-Ω sisto, flows down though th 500-Ω sisto, nd tht 3 flows up though th 300-Ω sisto Fom Kichhoff s junction ul: (1) Using th loop ul on th lft loop: π () 1 3 Using th loop ul on th ight loop: π (3) Solving ths th qutions simultnously, A, 0860 A, nd A 3

18 30 Fdy s Lw P3150 Th mf inducd btwn th nds of th moving b is f fb g ε Bv 50 T m 8 00 m s 7 00 V Th lft-hnd loop contins dcsing flux wy fom you, so th inducd cunt in it will b clockwis, to poduc its own fild dictd wy fom you Lt 1 psnt th cunt flowing upwd though th 00-Ω sisto Th ight-hnd loop will cy countclockwis cunt Lt 3 b th upwd cunt in th 500-Ω sisto () Kichhoff s loop ul thn givs: V 00 Ω 0 nd V f Ωf A A (c) Th totl pow dissiptd in th sistos of th cicuit is b g f f P ε1 + ε3 ε V 350 A A 343 W Mthod 1: Th cunt in th sliding conducto is downwd with vlu 350 A A 490 A Th mgntic fild xts foc of Fm B 4 90 A f0 350 m f 50 T f 4 9 N dictd towd th ight on this conducto An outsid gnt must thn xt foc of 49 N to th lft to kp th b moving Mthod : Th gnt moving th b must supply th pow ccoding to P F v Fv cos0 Th foc quid is thn: P 34 3 W F v 800 m s 49 N P3151 Suppos w wp twnty tuns of wi into flt compct cicul coil of dimt 3 cm Suppos w us b mgnt to poduc fild 10 3 T though th coil in on diction long its xis Suppos w thn flip th mgnt to vs th flux in 10 1 s Th vg inducd mf is thn K J Φ θ ε N N BA B cos cos180 cos 0 NB π t f j b g F jf 3 ε 0 10 T π m ~ s 4 V K J

19 P315 ε + ε inducd nd ε inducd d BAf F m dv Bd dv Bd Bd bε + ε m m dv Bd ε Bvdf m To solv th diffntil qution, lt u ε Bvd du Bd dv 1 du Bd Bd m u so u u0 du u t 0 Bd m f ntgting fom t 0 to t t, ln u Bd u m t 0 o u Bm u0 Sinc v 0 whn t 0, u 0 ε f nd u ε Bvd Thfo, ε Bvd ε ε v 1 Bd inducd Bm Bm j g Chpt FG P315 *P3153 Th nclosd flux is Φ B BA Bπ Th pticl movs ccoding to F m : qvbsin90 mv qb mv B m v Thn Φ B π qb () ΦBq B v π m Tm j30 10 C j 0 6 Tf π 10 kg j 5 ms Engy fo th pticl-lctic fild systm is consvd in th fiing pocss: U i K f : q V 1 mv V mv q kg j m sj C j 15 V

20 3 Fdy s Lw *P3154 () Consid n nnulus of dius, wih d, hight b, nd sistivity ρ Aound its cicumfnc, voltg is inducd ccoding to ε N d ω π + π ω ω B t B mxbcos g mx sin t Th sistnc ound th loop is ρ ρ π A bd x b g ε Bmxπ ω sinωt bd Bmxbωdsinωt Th ddy cunt in th ing is d sistnc ρ π ρ b b 3 Bmxπ bω dsin ωt Th instntnous pow is dp i ε d ρ 1 1 Th tim vg of th function sin ωt cos ωt is so th tim-vgd pow dlivd to th nnulus is B b d dp mx π 3 ω 4ρ Th pow dlivd to th disk is P dp F KJ 4 4 Bmxπbω πbmx bω P 0 4ρ 4 16ρ Whn B mx gts two tims lg, B mx B mx 4 0 g g πω b 3 d ρ nd P gt 4 tims lg (c) Whn f nd ω π f doubl, ω nd P gt 4 tims lg (d) Whn doubls, 4 nd P bcom 4 16 tims lg ε B A P3155 b15 0 µ T g0 00 mf so q 0500 Ω 10 µ C dq P3156 () ε wh ε N d ΦB N so dq d ΦB nd th chg though th cicuit will b Q N bφ Φ1g L F H G NM K JO QP Q 4 00 Ωf Cj NA 4 100f m j Q N BA cos 0 BA cos π BAN so B Φ Φ1 050 T

21 Chpt ε P3157 () ε Bv 0360 V 0900 A FB B 0108 N (c) Sinc th mgntic flux BA is in ffct dcsing, th inducd cunt flow though is fom b to Point b is t high potntil FG P3157 (d) No Mgntic flux will incs though loop to th lft of b H countclockwis cunt will flow to poduc upwd mgntic fild Th cunt in is still fom b to P3158 ε Bv t distnc fom wi ε µ 0 π F H G K J v v FG P3158 f j f P3159 () At tim t, th flux though th loop is Φ B BAcosθ + bt π cos 0 π + bt At t 0, Φ B π d B d+ bt ε π πb Φ f (c) ε π b (d) P ε π b F j KJ π b π 4 b f 1 d F P3160 ε π π H G K J NBA db K () Q Cε Cπ K B into th pp is dcsing; thfo, cunt will ttmpt to countct this Positiv chg will go to upp plt (c) Th chnging mgntic fild though th nclosd inducs n lctic fild, suounding th B-fild, nd this pushs on chgs in th wi

22 34 Fdy s Lw P3161 Th flux though th coil is Φ B BA BA cosθ BA cosω t Th inducd mf is b g ω ε N d Φ NBA d cos B t NBAωsinωt f jb g () εmx NBAω T m 30 0 d s 36 0 V dφ B Φ B ε d, thus N mx ε mx 36 0 V V Wb s N 600 (c) At t s, ω t 150 d nd ε ε sin 150 d 360 V sin 150 d 359 V mx f f f (d) Th toqu on th coil t ny tim is F ε ε τ µ ω H G ω K J F H G mx B NA B NAB t K J f sin sinω t ε mx 36 0 Vf ω b30 0 d s g10 0 Ωf Whn ε ε mx, sin ω t 1 00 nd τ 43 N m P316 () ΦB W us ε N, with N 1 t Tking m to b th dius of th wsh, nd h 0500 m, Φ B BA BA AB B π 1 1 F 0 0 b g F f f Th tim fo th wsh to dop distnc h (fom st) is: µ µ µ Thfo, ε 0 h 0 h g 0 gh h+ h+ h h+ nd ε f f f 7 3 4π 10 TmA j m j10 0 A f b0 500 m mg µ µ h h+ KJ µ 0 1 h+ 1 K J µ 0 π π h+ h g j f 980 ms 0500 m 97 4 nv Sinc th mgntic flux going though th wsh (into th pln of th pp) is dcsing in tim, cunt will fom in th wsh so s to oppos tht dcs Thfo, th cunt will flow in clockwis diction P3163 Find n xpssion fo th flux though ctngul swpt out by th b in tim t Th mgntic fild t distnc x fom wi is v vt B µ 0 nd Φ B BdA π x Thfo, Φ B µ 0vt π + dx x dφb µ 0v Thn, ε ln 1 + π wh vt is th distnc th b hs movd in tim t F K J FG P3163

23 Chpt P3164 Th mgntic fild t distnc x fom long wi is B µ 0 Find n xpssion fo th flux π x though th loop dφ B µ + w 0 µ 0 dx µ 0 w dxf so Φ B F + π x ln 1 x K J π π Thfo, dφb µ 0v ε π w + w f nd ε µ 0v π w + w f 3 j Tm P3165 W givn Φ B 600 t 180 t nd dφ ε B 18 0t t Mximum E occus whn d 36 0 t which givs t 100 s ε Thfo, th mximum cunt (t t 100 s ) is f V 600 A 300 Ω P3166 Fo th suspndd mss, M: F Mg T M Fo th sliding b, m: F T B m, wh B v Mg m+ M v dv α βv b f o g t 0 0 ε Bv dv Mg B v m+ M M+ m Mg wh α M+ m nd β B M+ m f f 1 β t Mg 1 B t M+ m j f α Thfo, th vlocity vis with tim s v β B P3167 () ε N d Φ NA db NA B bµ 0ng wh A of coil N numb of tuns in coil nd n numb of tuns p unit lngth in solnoid b g f b g j b g j f b g 119 Vfcosb10πtg 119 Vf cos b10π tg Thfo, ε Nµ An d 0 4sin 10π t Nµ 0An 480π cos 10πt 7 3 ε 40 4π 10 π m π cos 10πt ε V nd P V 1 1 Fom cos θ + cos θ th vg vlu of cos θ is 1, so P V 800 Ω f f 800 Ω 88 5 mw

24 36 Fdy s Lw f f θ θ *P3168 () ε N d θ ω BA d B B d 1 1 cos 1 cos 0 B 05 T 05 m d s 015 V 015 V clockwis Th sign indicts tht th inducd mf poducs clockwis cunt, to mk its own mgntic fild into th pg f Th c PQ hs lngth At this instnt θ ωt d s 0 5 s 05 d θ 05 d f05 m f 05 m Th lngth of th cicuit is 0 5 m m m 1 5 m its sistnc is 15 m5 b Ω m g 65 Ω Th cunt is 015 V 0000 A clockwis 65 Ω *P3169 Suppos th fild is vticlly down Whn n lcton is moving wy fom you th foc on it is in th diction givn by qv B s wy down c b g lft ight Thfo, th lctons cicult clockwis FG P3169 () As th downwd fild incss, n mf is inducd to poduc som cunt tht in tun poducs n upwd fild This cunt is dictd countclockwis, cid by ngtiv lctons moving clockwis Thfo th oiginl lcton motion spds up mv At th cicumfnc, w hv Fc mc : qvbc sin90 mv q B c Th incsing mgntic fild B v in th nclosd by th obit poducs tngntil lctic fild ccoding to d db E ds Bv A Ebπg π v E db v An lcton fls tngntil foc ccoding to Ft mt: qe m dv Thn q db v m dv q Bv mv qbc nd B B P3170 Th inducd mf is ε Bv wh B µ 0, vf vi + gt 980 m s j t, nd π y j 1 yf yi gt 0800 m 490 m s t 7 4 4π 10 TmA j00 Af ε 0300m f980 m s j jt t V π m 4 90 m s t t j j0300 f f At t 0300 s, ε V 98 3 µ V v c

25 P3171 Th mgntic fild poducd by th cunt in th stight wi is ppndicul to th pln of th coil t ll points within th coil Th mgnitud of th fild is B µ 0 Thus, th flux linkg is π h+ w µ 0NL d µ 0NmxL h+ w NΦ B ln sinbωt φg π π h h F K J + Chpt Finlly, th inducd mf is FG P3171 µ 0NmxLω F w ε ln 1 + ω φ π K J cosb t + g h 7 1 4π 10 j100f50 0f0 00 m f00π s j F 500 cm ε ln 1 + ω φ π K J cosb t + g 500 cm ε 87 1 mvfcosb00πt + φg Th tm sinbωt + φg in th xpssion fo th cunt in th stight wi dos not chng ppcibly whn ω t chngs by 010 d o lss Thus, th cunt dos not chng ppcibly duing tim intvl < s 1 00π s j j j qul to th distnc to W dfin citicl lngth, c t ms s m which fild chngs could b popgtd duing n intvl of s This lngth is so much lg thn ny dimnsion of th coil o its distnc fom th wi tht, lthough w consid th stight wi to b infinitly long, w cn lso sfly igno th fild popgtion ffcts in th vicinity of th coil Moov, th phs ngl cn b considd to b constnt long th wi in th vicinity of th coil f th fquncy w w much lg, sy, 00π 10 5 s 1, th cosponding citicl lngth would b only 48 cm n this sitution popgtion ffcts would b impotnt nd th bov xpssion fo ε would qui modifiction As ul of thumb w cn consid fild popgtion ffcts fo cicuits of lbotoy si to b ngligibl fo fquncis, f ω π, tht lss thn bout 106 H P317 Φ B BA cosθ dφ B ω BA sin θ ; sinθ τ Bsinθ sin θ θ FG P317

26 38 Fdy s Lw ANSWES TO EVEN POBLEMS P ma P3140 () 1 60 V ; 0; (c) no chng; (d) nd () s th solution P314 () s th solution; 3 79 mv; (c) 8 0 mv P314 both coct; s th solution j ms P µ s P i 1 76 j f b g P318 () µ n π countclockwis; P µ 0 π n ; (c) upwd 4 1 t P3148 () 43 8 A; 38 3 P mv cos10tf P3150 () 3 50 A up in Ω nd 1 40 A up in 5 Ω; 34 3 W ; (c) 4 9 N P mv P315 s th solution P3114 () s th solution; 65 m/s P3116 s th solution P ma countclockwis in th low loop nd clockwis in th upp loop P m s P31 () 500 ma; 00 W; (c) 00 W P V with th out contct positiv P ma clockwis P318 () to th ight; to th ight; (c) to th ight; (d) into th pp P3130 ngtiv; s th solution 4 P3154 () π Bmx b ω ; 4 tims lg; 16ρ (c) 4 tims lg; (d) 16 tims lg P3156 () s th solution; 0 50T P3158 s th solution P3160 () Cπ K; th upp plt; (c) s th solution P316 () 97 4 nv; clockwis P3164 P3166 µ 0v π Mg B w + w B t M m 1 + f f P313 () N downwd P3168 () 015 V to poduc clockwis cunt; ppndicul to 1 ; 1 33 s 00 ma clockwis b g b g Vfsin f P3134 () 987 mv m cos 100π t ; clockwis P3136 () t ; 19 6 V P3138 s th solution P t P317 s th solution ; 98 3 µ V