HW1 Solutions (u Tus, Apr 8) 1. T&M 8.P.4 A unior 1.-T agntic il is in th +z irction. A conucting ro o lngth 15 c lis paralll to th y axis an oscillats in th x irction with isplacnt givn by x (. c) cos 1 πt whr 1πt has units o ra/s. (a) Fin an xprssion or potntial irnc btwn th ns o th ro as a unction o ti. (b) What is th axiu potntial irnc btwn th ns o th ro? Pictur th Probl Th ro is xcuting sipl haronic otion in th xy plan, i.., in a plan prpnicular to th agntic il. Th inuc in th ro is a consqunc o its otion in this agntic il an is givn by vb. Bcaus w r givn th position o th oscillator as a unction o ti, w can irntiat this xprssion to obtain v. (a) Th potntial irnc btwn th ns o th ro is givn by: x vb B x Evaluat x/: [(.c) cos1 t] Substitut nurical valus an valuat : 1 (.c)( 1s ) ( 7.54 /s) sin1 t # 1 sin1 t ( 1. T)(.15)( 7.54 /s) sin1 t #( 1.4V) sin t (b) Th axiu potntial irnc btwn th ns th ro is th aplitu o th xprssion or riv in Part (a): ax 1.4 V
. T&M 8.P.4 A unior agntic il is stablish prpnicular to th plan o a loop o raius 5 c, rsistanc.4 ohs, an ngligibl slinuctanc. Th agnitu o th il is incrasing at a rat o 4 T/s. Fin (a) th inuc in th loop, (b) th inuc currnt in th loop, an (c) (c) th rat o Joul hating in th loop. Pictur th Probl (a) W can in th agnitu o th inuc by applying Faraay s law to th loop. (b) an (c) Th application o Oh s law will yil th inuc currnt in th loop an w can in th rat o Joul hating using P R. (a) Apply Faraay s law to xprss th inuc in th loop in trs o th rat o chang o th agntic il: Substitut nurical valus an valuat : B R # ( AB) A (.5 ) ( 4. T/s).314 V.314 V B (b) Using Oh s law, rlat th inuc currnt to th inuc voltag an th rsistanc o th loop an valuat : (c) Exprss th rat at which powr is issipat in a conuctor in trs o th inuc currnt an th rsistanc o th loop an valuat P: R.314 V.4.785 A.7854 A P R (.7854 A) (.4 ).47 µw
3. T&M 8.P.47 Th currnt in a 8. H coil is qual to 3. A an is incrasing at A/s. (a) Fin th agntic lux through th coil. (b) Fin th inuc in th coil. Pictur th Probl W can us L to in th agntic lux through th coil. W can apply Faraay s law to in th inuc in th coil. (a) Th agntic lux through L th coil is th prouct o th slinuctanc o th coil an th currnt it is carrying: Whn th currnt is 3. A: ( 8.H)( 3.A) 4. Wb (b) Us Faraay s law to rlat ε, L L, an : Substitut nurical valus an valuat ε : ( 8. H)( A/s) 1.6kV 4. T&M 8.P.54 A coil with a sl-inuctanc o H an a rsistanc o 1 ohs is connct across a 4 V battry o ngligibl intrnal rsistanc. (a) What is th inal currnt? (b) How uch nrgy is stor in th inuctor whn th inal currnt is attain? Pictur th Probl Th currnt in an LR circuit, as a unction o ti, is givn 1 t, whr ε /R an τ L/R. Th nrgy stor in th inuctor by ( ) unr stay-stat conitions is stor in its agntic il an is givn by
U L. 1 (a) Th inal currnt is th quotint o th o th battry an th rsistanc o th coil: R 4.V 1.. A (b) Th nrgy stor in th inuctor is: U L 1 4.J 1 (.H)(.A) 5. T&M 8.P.57 A long cylinrical wir has a raius qual to.6 c an carris a currnt o 75 A uniorly istribut ovr its cross-sctional ara. Fin th agntic nrgy pr unit lngth within th wir. Pictur th Probl Consir a cylinrical annulus o thicknss r at a raius r < a. W can us its inition to xprss th total agntic nrgy U insi th cylinrical annulus an ivi both sis o this xprssion by th lngth o th wir to xprss th agntic nrgy pr unit lngth U'. ntgration o this xprssion will giv us th agntic nrgy pr unit lngth within th wir.
Exprss th agntic nrgy within th cylinrical annulus: Divi both sis o th quation by to xprss th agntic nrgy pr unit lngth U' : Us Apr s law to xprss th agntic il insi th wir at a istanc r < a ro its cntr: U B V µ B µ r r B µ annulus r r B U' rr (1) µ µ C rb µ C B r whr C is th currnt insi th cylinr o raius r. Bcaus th currnt is uniorly istribut ovr th crosssctional ara o th wir: C r r a C a Substitut or C to obtain: Substituting or B in quation (1) an sipliying yils: ntgrat U' ro r to r a: µ r B a ' µ r $ % a µ 3 U ' & # rr r r 4 µ 4 a U' a 4 µ 3 µ a r r 4 4 4 a # 4 a 4 µ 16 Rarks: Not that th agntic nrgy pr unit lngth is inpnnt o th raius o th cylinr an pns only on th total currnt. 6. T&M 8.P.61 n th circuit givn in th igur blow, lt E 1 V, R 3. ohs, an L.6 H. Th switch is clos at ti t. (a) At ti t.5 s, in th rat at which th battry supplis nrgy. (b) Fin th rat o joul hating at this ti. (c) Fin th rat at
which nrgy is bing stor in th inuctor at this ti. Pictur th Probl W can in th currnt using ( 1 t ), whr ε /R,an τ L/R, an its rat o chang by irntiating this xprssion with rspct to ti. Exprss th pnnc o th currnt on an τ : Evaluating an τ yils: ( t) ( 1 t ) R 1.V 3. 4. A an L R.6H 3.Ù.s t Substitut or an τ to obtain: ( t) ( 4.A)( 1.s ) Exprss /: t. s 1 ( 4.A)( )( 5.s ) t. s (. A/s) (a) Th rat at which th battry supplis nrgy is givn by: P t.s Substituting or an ε yils: P( t) ( 4.A)( 1 )( 1. V) t ( 48.W)( 1.s ) Th rat at which th battry supplis nrgy at t.5 s is: P.s (.5 s) ( 48.W)( 1.5 ) 44.1W (b) Th rat o Joul hating is: Substitut or an R an sipliy to obtain: P J R ( ) 1 P J # 4. A ( ) 1 48. W t. s ( ) $ % t. s ( ) ( 3. )
Th rat o Joul hating at t.5 s is: P J.5 s.s (.5 s) ( 48. W)( 1 ) 4.4 W (c) Us th xprssion or th agntic nrgy stor in an inuctor to xprss th rat at which nrgy is bing stor: U L 1 [ L ] L Substitut or L,, an / to obtain: U L t.s (.6 H)( 4.A)( 1 )(. A/s) t.s t.s ( 48. W)( 1 ) t.s Evaluat this xprssion or t.5 s: U L $ # t.5 s.5 s.s.5 s.s ( 48.W)( 1 ) 3.6 W Rarks: Not that, to a goo approxiation, U L / P P J. 7. T&M 8.P.68 A circuit consists o a coil that has a sl-inuctanc qual to 5. H an an intrnal rsistanc qual to 15. Ω, an ial 1. V battry, an an opn switch--all connct in sris. At t th switch is clos. Fin th ti whn th rat at which nrgy is issipat in th coil quals th rat at which agntic nrgy is stor in th coil. Pictur th Probl th currnt is initially zro in an LR circuit, its valu at so latr ti t is givn by ( 1 t ), whr ε /R an τ L/R is th ti constant or th circuit. W can in th ti at which th powr issipation in th rsistor quals th rat at which agntic nrgy is stor in th inuctor
by quating xprssions or ths rats an using th xprssion or an its rat o chang. Exprss th rat at which agntic nrgy is stor in th inuctor: Exprss th rat at which powr is issipat in th rsistor: Equat ths xprssions to obtain: Exprss th currnt an its rat o chang: Substitut or / in quation (1) an sipliy to obtain: Solving or t an substituting or τ yils: Substitut nurical valus an valuat t: 1 [ L ] U L P R R L 1 an ( ) t L (1) t ( 1 ) t ' t ( & ' t ( # ( ' ) ( $ 1 % ( or t t 1 L t ln 1 ln R t 1 t 1 t 5. H 15. ln 1 31µs ( 1 % & # ' $ 8. T&M 8.P.7 Prior to about 196, agntic il strngth was asur by ans o a rotating coil gausstr. This vic us a sall loop o any turns rotating on an axis prpnicular to th agntic il at airly high sp an connct to an ac volttr by ans o slip rings lik thos shown in th igur blow. Th snsing coil or a rotating coil gausstr has 4
turns an an ara o 1.4 c. Th coil rotats at 18 rp. th agntic il strngth is.45 T, in th axiu inuc in th coil an th orintation o th coil rlativ to th il or which this axiu inuc occurs. Pictur th Probl W can apply Faraay s law an th inition o agntic lux to riv an xprssion or th inuc in th rotating coil gausstr. Us Faraay s law to xprss th # inuc : Using th inition o agntic lux, rlat th agntic lux through th loop to its angular vlocity: Substitut or ( t) to obtain: an sipliy ( t ) NBA t cos # # NBA NBA sin whr NBA ax [ NBAcos t] (# sin t) t ax sin t Substitut nurical valus an valuat ε ax : ( ) 18 rv ax ( 4) (.45 T) 1.4 1 #4 $ % & in ra rv 1in 6s ' ( ).475V Th axiu inuc occurs at th instant th noral to th plan o th coil is prpnicular to th agntic il B. At this instant, is zro, but is a axiu. 9. T&M 8.P.78 Th rctangular coil in th igur has 8 turns, is 5 c wi an 3 c long, an is locat in a agntic il B.14 T irct
out o th pag as shown. Only hal o th coil in th rgion o th agntic il. Th rsistanc o th coil is 4 Ω. Fin th agnitu an irction o th inuc currnt i th coil is oving with a vlocity o. /s to th right, up th pag, to th lt, an own th pag. Pictur th Probl Th currnt quals th inuc ivi by th rsistanc. W can calculat th inuc in th circuit as th coil ovs by calculating th rat o chang o th lux through th coil. Th lux is proportional to th ara o th coil in th agntic il. W can in th irction o th currnt ro Lnz s law. (a) an (c) Exprss th agnitu o th inuc currnt: (1) R Using Faraay s law, xprss th agnitu o th inuc : Whn th coil is oving to th right (or to th lt), th lux os not chang (until th coil lavs th rgion o agntic il). Thus: R
(b) Ltting x rprsnt th lngth o th si o th rctangular coil that is in th agntic il, xprss th agntic lux through th coil: NBwx Coput th rat o chang o th lux whn th coil is oving up or own: ( 8)(.14T)(.5)(. /s) x NBw 5.6V Substitut in quation (1) to obtain: 5.6V 4.3A clockwis () Whn th coil is oving ownwar, th outwar lux crass an th inuc currnt will b in such a irction as to prouc outwar lux. Th agnitu o th currnt is th sa as in Part (b) an.3a countrclockwis. 1. T&M 3.P.19 Thr is a currnt o 1 A in a rsistor that is connct in sris with a paralll-plat capacitor. Th plats o th capacitor hav an ara o.5, an no ilctric xists btwn th plats. (a) What is th isplacnt currnt btwn th plats? (b) What is th rat o chang o th lctric il strngth btwn th plats? (c) Fin th valu o th lin intgral B l, whr th intgration path is a 1-c-raius circl that lis in a plan that is paralll with th plats an is copltly within th rgion btwn th. Pictur th Probl W can us th consrvation o charg to in, th initions o th isplacnt currnt an lctric lux to in E/, an Apr s law to valuat B aroun th givn path. C
(a) Fro consrvation o charg w know that: 1A (b) Exprss th isplacnt currnt : Substituting or E/ yils: Substitut nurical valus an valuat E/: (c) Apply Apr s law to a circular path o raius r btwn th plats an paralll to thir suracs to obtain: [ EA] E E C A ( & 8.85 1 '.3 1 B µ 1 ) 1 nclos 1A C N V s % # $ E A (.5 ) Assuing that th isplacnt currnt is uniorly istribut an ltting A rprsnt th ara o th circular plats yils: nclos r A nclos r A Substitut or nclos to obtain: Substitut nurical valus an valuat B : µ r # B C A C # 7 ( 4% 1 N / A )% (.1) ( 1A) $ B.79 µ T C.5