Elctcal Engy and apactanc 57. Bcaus lctc ocs a consvatv, th kntc ngy gand s qual to th dcas n lctcal potntal ngy, o + + 4 4 KE PE q( ).. so th coct choc s (a).. Fom consvaton o ngy, KE + PE KE + PE, o mv mv + q q o v q( ) v + m 5.6 5 4 ( 6. m s) + (.. ) 7 6. 6 kg Thus, th coct answ s choc. 9 5. 78 m s 4. In a unom lctc ld, th chang n lctc potntal s E ( x), gvng Ex x x x ( ) 5 and t s sn that th coct choc s (d). ( 9 ) (. m. m) 5 m 5 N 5. Wth th gvn spccatons, th capactanc o ths paalll plat capacto wll b ( ). 8. 85 N m. cm d. m κ 8. 85 F 88. 5 F 88. 5 pf and th coct choc s (a). 4 x m cm 6. Th total potntal at a pont du to a st o pont chags q s kq, wh s th dstanc om th pont o ntst to th locaton o th chag q. Not that n ths cas, th pont at th cnt o th ccl s qudstant om th 4 pont chags locatd on th m o th ccl. Not also that q + q + q4 ( +. 5.. 5) µ, so w hav kq kq kq kq4 k cnt + + + q + q + q ( q ) ( q + ) + k 4 k q 4 4. 5 o th total potntal at th cnt o th ccl s just that du to th st chag alon, and th coct answ s choc. 7. In a ss combnaton o capactos, th quvalnt capactanc s always lss than any ndvdual capactanc n th combnaton, manng that choc (a) s als. lso, o a ss combnaton o capactos, th magntud o th chag s th sam on all plats o capactos n th combnaton, makng both chocs (d) and () als. Th potntal dnc acoss th capactanc s Q, wh Q s th common chag on ach capacto n th combnaton. Thus, th lagst potntal dnc (voltag) appas acoss th capacto wth th last capactanc, makng choc th coct answ. 8. Kpng th capacto connctd to th batty mans that th potntal dnc btwn th plats s kpt at a constant valu qual to th voltag o th batty. Snc th capactanc o a paalll plat capacto s κ d, doublng th plat spaaton d, whl holdng oth chaactstcs o th capacto constant, mans th capactanc wll b dcasd by a acto o.
58 hapt 6 Th ngy stod n a capacto may b xpssd as U, so whn th potntal dnc s hld constant whl th capactanc s dcasd by a acto o, th stod ngy dcass by a acto o, makng (c) th coct choc o ths quston. 9. Whn th batty s dsconnctd, th s no long a path o chags to us n movng onto o o o th plats o th capacto. Ths mans that th chag Q s constant. Th capactanc o a paalll plat capacto s κ d and th dlctc constant s κ whn th capacto s a lld. Whn a dlctc wth dlctc constant κ s nstd btwn th plats, th capactanc s doubld ( ). Thus, wth Q constant, th potntal dnc btwn th plats, Q, s dcasd by a acto o, manng that choc (a) s a tu statmnt. Th lctc ld btwn th plats o a paalll plat capacto s E d and dcass whn dcass, makng choc () als and lavng (a) as th only coct choc o ths quston.. Onc th capacto s dsconnctd om th batty, th s no path o chags to mov onto o o o th plats, so th chags on th plats a constant, and choc () can b lmnatd. Th capactanc o a paalll plat capacto s κ d, so th capactanc dcass whn th plat spaaton d s ncasd. Wth Q constant and dcasng, th ngy stod n th capacto, U Q ncass, makng choc (a) als and choc tu. Th potntal dnc btwn th plats, Q Q d κ, ncass and th lctc ld btwn th plats, E d Q κ, s constant. Ths mans that both chocs (c) and (d) a als and lavs choc as th only coct spons.. apactancs connctd n paalll all hav th sam potntal dnc acoss thm and th quvalnt capactanc, + q + + L, s lag than th capactanc o any on o th capactos n th combnaton. Thus, choc (c) s a tu statmnt. Th chag on a capacto s Q ( ), so wth constant, but th capactancs dnt, th capactos all sto dnt chags that a popotonal to th capactancs, makng chocs (a),, (d), and () all als. Tho, (c) s th only coct answ.. Fo a ss combnaton o capactos, th magntud o th chag s th sam on all plats o capactos n th combnaton. lso, th quvalnt capactanc s always lss than any ndvdual capactanc n th combnaton. Tho, choc (a) s tu whl chocs and (c) a both als. Th potntal dnc acoss a capacto s Q, so wth Q constant, capactos havng dnt capactancs wll hav dnt potntal dncs acoss thm, wth th lagst potntal dnc bng acoss th capacto wth th smallst capactanc. Ths mans that choc (d) s als and choc () s tu. Thus, both chocs (a) and () a tu statmnts. NSWERS TO EEN NUMBERED ONEPTUL QUESTIONS. hangng th aa wll chang th capactanc and maxmum chag but not th maxmum voltag. Th quston dos not allow you to ncas th plat spaaton. You can ncas th maxmum opatng voltag by nstng a matal wth hgh dlctc stngth btwn th plats. 4. Elctc potntal s a masu o th potntal ngy p unt chag. Elctcal potntal ngy, PE Q, gvs th ngy o th total chag Q. 6. shap pont on a chagd conducto would poduc a lag lctc ld n th gon na th pont. n lctc dschag could most asly tak plac at th pont.
Elctcal Engy and apactanc 59 8. Th a ght dnt combnatons that us all th capactos n th ccut. Ths combnatons and th quvalnt capactancs a: ll th capactos n ss - q + + ll th capactos n paalll - + q + On capacto n ss wth a paalll combnaton o th oth two: q + +, q + +, q + + On capacto n paalll wth a ss combnaton o th oth two: q + +, q + +, q + +. Nothng happns to th chag th ws a dsconnctd. I th ws a connctd to ach oth, th chag apdly combns, lavng th capacto unchagd.. ll connctons o capactos a not smpl combnatons o ss and paalll ccuts. s an xampl o such a complx ccut, consd th ntwok o v capactos,,, 4, and 5 shown blow. Ths combnaton cannot b ducd to a smpl quvalnt by th tchnqus o combnng ss and paalll capactos. 4. Th matal o th dlctc may b abl to wthstand a lag lctc ld than a can wthstand bo bakng down to pass a spak btwn th capacto plats. PROBLEM SOLUTIONS 6. (a) Bcaus th lcton has a ngatv chag, t xpncs a oc n th dcton oppost to th ld and, whn lasd om st, wll mov n th ngatv x dcton. Th wok don on th lcton by th ld s 9 8 W F ( x) ( qe ) x. 6 75 N. m. 9 J x x ( ) Th chang n th lctc potntal ngy s th ngatv o th wok don on th patcl by th ld. Thus, PE W. 9 8 J contnud on nxt pag
6 hapt 6 (c) Snc th oulomb oc s a consvatv oc, consvaton o ngy gvs KE + PE, o KE m v PE PE, and v 8 PE. 9 J m 9. kg. 5 6 m s n th x dcton 6. (a) Th chang n th lctc potntal ngy s th ngatv o th wok don on th patcl by th ld. Thus, PE W qex x qey y + q( ) x + 5. 4 6 ( + ) 7 N. m + 5. 65 4 J Th chang n th lctcal potntal s th chang n lctc potntal ngy p unt chag, o PE + 4 5. 65 J + 5 6 q +5.4 6. Th wok don by th agnt movng th chag out o th cll s W W PE q nput ld + +. 6 9 J 9. 4 J PE 6.4 PE q( ) q( ), so q 7. 9 J. +6. J 9 5 J 6.5 E. 7 d. 5 m 6 N 6.6 Snc potntal dnc s wok p unt chag W, th wok don s q + 5 6 W q( ). 6 J 4. J 6 J 6.7 (a) E. d 5. m 5 N ( ) 9 5 4 F q E. 6. N. 8 N (c) W F s cosθ. 8 4 N 5.. 9 m cos 4. 8 7 J
Elctcal Engy and apactanc 6 6.8 (a) Usng consvaton o ngy, KE + PE, wth KE snc th patcl s stoppd, w hav PE KE m + ( 9 v. kg) ( 85 7 6. m s ) +. 7 J Th qud stoppng potntal s thn PE +. 7 q. 6 6 9 J.. k Bng mo massv than lctons, potons tavlng at th sam ntal spd wll hav mo ntal kntc ngy and qu a gat magntud stoppng potntal. (c) PE Snc KE m q stoppng q v, th ato o th stoppng potntal o a poton to q that o an lcton havng th sam ntal spd s p mp v ( + ) m m v ( ) p m 6.9 (a) Us consvaton o ngy ( KE + PE + PE ) ( KE + PE + PE ) s s + + o KE PE PE Thus, s ( KE ) snc th block s at st at both bgnnng and nd. ( PEs ) kxmax, wh x max s th maxmum sttch o th spng. max PE W QE x x kxmax QE xmax, gvng + max ( ) 6 4 QE 5. 4. 86 m k 78. N m t qulbum, ΣF Fs + F, o kxq + QE Tho, x QE x k q max. 8 cm 4. 6 m 4. 6 cm Th ampltud s th dstanc om th qulbum poston to ach o th tunng ponts at x and x 4. 6 cm, so 8 cm x.. max contnud on nxt pag
6 hapt 6 (c) Fom consvaton o ngy, KE PEs PE kxmax Q x, ths gvs max kx k max Q Q 6. Usng y v yt + ayt o th ull lght gvs v y t + a y t, o a Thn, usng v y v y t + + + + o k Q v + a ( y) o th upwad pat o th lght gvs y y y y y y t ( y) v v v. m s 4. s max a v t 4 4 y ( y ). 6 m ΣFy mg qe qe Fom Nwton s scond law, ay g + m m m. Equatng ths to th al qe y sult gvs ay g + m v, so th lctc ld stngth s t. Snc m y E g v q t. kg 6 5.. m s 4. s 9. 8 m s. 95 N 6. (a) ( ) Thus, y E max max. 6 m. 95 N 4. 4 4. k B kq kq B 9 9 ( 8. 99 N m ). 6. 5 m 9 9 ( 8. 99 N m ). 6. 75 m 5. 75 7. 9 7 7 7. 9 5. 75. 8 7 B + (c) Th ognal lcton wll b plld by th ngatvly chagd patcl whch suddnly appas at pont. Unlss th lcton s xd n plac, t wll mov n th oppost dcton, away om ponts and B, thby lowng th potntal dnc btwn ths ponts. 6. (a) t th ogn, th total potntal s ogn contnud on nxt pag kq kq + 9 ( 8. 99 N m ) 6 6 4. 5 4 +..5 m.8 m. 6
Elctcal Engy and apactanc 6 t pont B locatd at (. 5 cm, ), th ndd dstancs a x x y y B B + +. 5 cm. 5 cm. 95 cm and x x y y B B + + gvng B kq kq 9 + 8. 99 N m 6. (a) allng th. µ chag q,. 5 cm. 8 cm. 4 cm 6 4. 5 + 6. 4.95 m. 4 m. 6 k q q q k + + q + 8. 99 9 6 N m 8. 4. +. 6 m. m. 6 6 + +. 6. m. 67 6 Rplacng. by. 6 6 n pat (a) ylds. 6 ( ) 6.4 W q( ) q, and snc th 8. µ s nnt dstanc om oth chags. q q k + 9 8. 99 N m 6 6. 4. +. m (. ) + (.6 ) m. 5 6 ( ) 6 6 Thus, W 8.. 5 9. 8 J
64 hapt 6 6.5 (a) k q 8. 99 9 9 9 N m 5...75 m.75 m PE k q q 8. 99 9 N m 9 9 ( 5. )..5 m. 85 7 J Th ngatv sgn mans that postv wok must b don to spaat th chags (that s, bng thm up to a stat o zo potntal ngy). 6.6 Th potntal at dstanc. m om a chag Q + 9. 9 s kq 9 9 ( 8. 99 N m ) 9.. m +7 Thus, th wok qud to cay a chag q. 9 om nnty to ths locaton s W + 9 7 q. 7 8. 9 J 6.7 Th Pythagoan thom gvs th dstanc om th mdpont o th bas to th chag at th apx o th tangl as ( 4. ) (. ) 5 5 cm cm cm m Thn, th potntal at th mdpont o th bas s k q, o 8. 99 9 9 N m 7.. m 4.. k + 9 9 ( 7. ) + + ( 7. ). m 5 m
Elctcal Engy and apactanc 65 6.8 Outsd th sphcal chag dstbuton, th potntal s th sam as o a pont chag at th cnt o th sph, Thus, kq, wh Q. 9 PE q kq and om consvaton o ngy, ( KE) ( PE ), o k Q mv kq. Ths gvs v m, o v 9 N m 9 9 8. 99 (. ). 6 9. kg. m. m v 7. 5 6 m s 6.9 (a) Whn th chag conguaton conssts o only th two potons ( q and q n th sktch), th potntal ngy o th conguaton s PE a kqq 9 9 ( 8. 99 N m ). 6 5 6. m o PE a. 84 4 J Whn th alpha patcl ( q n th sktch) s addd to th conguaton, th a th dstnct pas o patcls, ach o whch posssss potntal ngy. Th total potntal ngy o th conguaton s now PE b k q q k q q k q q k + + PEa + ( ) wh us has bn mad o th acts that q q q q and (. m ) + (. m ) 4. 4 m 4. 4 5 m. lso, not that th st tm n ths computaton s just th potntal ngy computd n pat (a). Thus, PE b 4k PEa + ( ) 9 4 8. 99 N m 9. 6 4. 84 J + 5 4. 4 m. 55 J contnud on nxt pag
66 hapt 6 (c) I w stat wth th th-patcl systm o pat and allow th alpha patcl to scap to nnty [thby tunng us to th two-patcl systm o pat (a)], th chang n lctc potntal ngy wll b 4 PE PE PE. 84 J. 55 J. 7 J a b (d) onsvaton o ngy, KE + PE, gvs th spd o th alpha patcl at nnty n th stuaton o pat (c) as m v PE, o α α v α ( PE ). α ( J) 7 7 m 6. 64 kg 8. 8 6 m s () Whn, statng wth th th-patcl systm, th two potons a both allowd to scap to nnty, th wll b no manng pas o patcls and hnc no manng potntal ngy. Thus, PE PEb PEb, and consvaton o ngy gvs th chang n kntc ngy as KE PE + PE b. Snc th potons a dntcal patcls, ths ncas n kntc ngy s splt qually btwn thm gvng KE m v PE poton p p b PEb o v p m p. 55 J 7. 4 m s 7.67 kg 6. (a) I a poton and an alpha patcl, ntally at st 4. m apat, a lasd and allowd to cd to nnty, th nal spds o th two patcls wll d bcaus o th dnc n th masss o th patcls. Thus, attmptng to solv o th nal spds by us o consvaton o ngy alon lads to a stuaton o havng on quaton wth two unknowns, and dos not pmt a soluton. In th stuaton dscbd n pat (a) abov, on can obtan a scond quaton wth th two unknown nal spds by usng consvaton o lna momntum. Thn, on would hav two quatons whch could b solvd smultanously both unknowns. contnud on nxt pag
Elctcal Engy and apactanc 67 (c) Fom consvaton o ngy: mα v + mpv α p kqα qp + 9 9 9 kqα qp 8. 99 N m.. 6 o mα vα + mpvp 5 4. m ( ) yldng m α v α m p v + p. J [] Fom consvaton o lna momntum, m α v α + m p v p o vα m v p m α p [] Substtutng Equaton [] nto Equaton [] gvs m α m m p α p mp p v + v. J o m m p α + mp p v. J and. J. J 7 7 7 α.67 6.64 +.67 kg v p mp m + mp. 5 ( ) 7 m s Thn, Equaton [] gvs th nal spd o th alpha patcl as v α m v p. 67 p m 6. 64 α 7 7 kg 7 6 kg (. 5 m s ). 64 m s kq 6. so kq ( 8. 99 9 N m )( 8. 9 ) 7. 9 m Fo, 5., and 5.,. 79 m,.44 m, and.88 m Th ad a nvsly popotonal to th potntal. 6. By dnton, th wok qud to mov a chag om on pont to any oth pont on an qupotntal suac s zo. Fom th dnton o wok, W ( F cosθ ) s, th wok s zo only s o F cosθ. Th dsplacmnt s cannot b assumd to b zo n all cass. Thus, on must qu that F cosθ. Th oc F s gvn by F qe and nth th chag q no th ld stngth E can b assumd to b zo n all cass. Tho, th only way th wok can b zo n all cass s cosθ. But cosθ, thn θ 9 o th oc (and hnc th lctc ld) must b ppndcula to th dsplacmnt s (whch s tangnt to th suac). That s, th ld must b ppndcula to th qupotntal suac at all ponts on that suac.
68 hapt 6 ( + ), whch gvs 6. Fom consvaton o ngy, KE + PE KE PE kqq + mα + v o kqq k 79 m v m v α α N m 8. 99 ( 58 ). 6 7 7 6. 64 kg. m s 9 9 ( ) ( ). 74 4 m 6.4 (a) Th dstanc om any on o th cons o th squa to th pont at th cnt s on hal th lngth o th dagonal o th squa, o dagonal a + a a a Snc th chags hav qual magntuds and a all th sam dstanc om th cnt o th squa, thy mak qual contbutons to th total potntal. Thus, total kq kq 4sngl 4 4 chag a 4 k Q a Th wok qud to cay chag q om nnty to th pont at th cnt o th squa s qual to th ncas n th lctc potntal ngy o th chag, o W PE PE q q k Q k cnt total 4 a 4 qq a 6.5 (a) d 8. 85 N m 6 (. m ). 8 8 m F Q E d max max max 8 6 (. F) (. N )( 8 m ) 7 Q 7. µ 6.6 (a) 9.. µ F Q ( ) (. µ F )(. ) 6. µ 6.7 (a) Th capactanc o ths a lld dlctc constant, κ. paalll plat capacto s k d 4 ( N m )( m ). 8. 85.. 5 m. 6 F. 6 pf contnud on nxt pag
Elctcal Engy and apactanc 69 Q ( ). 6 F.. 6 6. 6. p (c). E 8. m 8. N d.5 m 6 6 6.8 (a) Q ( ) 4. F. 48. 48. µ 6 6 Q ( ) 4. F. 5 6. 6. µ. 6.9 (a) E. 4 m. k m dctd towad th ngatv plat d.8 m d 4 ( 8. 85 N m ) 7. 6 m.8 m. 74 F. 74 pf (c) Q ( ). 74 F. 7. 47 74. 7 p on on plat and 74. 7 p on th oth plat. 6., so d d ( 8. 85 N m ). m 5 6. F 9 Å d. m. Å m. 9 m 6. (a) ssumng th capacto s a-lld ( κ ), th capactanc s d ( 8. 85 N m )(. m ) 5. 9. m F 9 Q ( ) 5. 9 F 6.. 54 (c) 6. E. m. N d. m Q. 54 (d) σ. m () 9. 77 8 m Incasng th dstanc spaatng th plats dcass th capactanc, th chag stod, and th lctc ld stngth btwn th plats. Ths mans that all o th pvous answs wll b dcasd.
7 hapt 6 6. ΣF mg y T cos 5. mg o T cos 5. ΣF x qe T sn 5. mg tan 5. o E mg tan 5. q mgd tan 5. Ed q ( 5 6 kg )( 9. 8 m s )(. 4 m) tan 5... k 9. 6. (a) apactos n a ss combnaton sto th sam chag, Q q ( ), wh q s th quvalnt capactanc and s th potntal dnc mantand acoss th ss combnaton. Th quvalnt capactanc o th gvn ss combnaton s +, o q +, gvng q. 5 µ F 6. 5 µ F q. 79 µf. 5 µ F + 6. 5 µ F so th chag stod on ach capacto n th ss combnaton s Q q ( ) (. 79 µ F )( 6. ). 7 µ Whn connctd n paalll, ach capacto has th sam potntal dnc, 6., mantand acoss t. Th chag stod on ach capacto s thn Fo. 5 µ F: Q ( ) (. 5 µ F )( 6. ) 5. µ Fo 6. 5 µ F: Q ( ) ( 6. 5 µ F )( 6. ) 7. 5 µ 6.4 (a) Whn connctd n ss, th quvalnt capactanc s +, o q q + 4. µ F 8. 5 µ F 4. µ F + 8. 5 µ F. 8 µ F Whn connctd n paalll, th quvalnt capactanc s q + 4. µ F + 8. 5 µ F. 7 µ F
Elctcal Engy and apactanc 7 6.5 (a) Fst, w plac th paalll combnaton btwn ponts b and c by ts quvalnt capactanc, bc. µ F + 6. µ F 8. µ F. Thn, w hav th capactos n ss btwn ponts a and d. Th quvalnt capactanc o ths ccut s tho + + 8. µ q ab bc cd F 8. µ F gvng q. 67 µ F Th chag stod on ach capacto n th ss combnaton s Qab Qbc Qcd q ( ad ) (. 67 µ F )( 9. ) 4. µ Thn, not that ccut s: bc Qbc 4. µ.. Th chag on ach capacto n th ognal 8. µ F bc On th 8. µ F btwn a and b: Q8 Q 4. µ ab On th 8. µ F btwn c and d: Q8 Q 4. µ cd On th. µ F btwn b and c: Q ( bc ) (. µ F )(. ) 6. µ On th 6. µ F btwn b and c: Q6 6 ( bc ) ( 6. µ F )(. ) 8. µ (c) Not that ab Qab ab 4. µ 8. µ F., and that cd Qcd cd 4. µ 8. µ F.. W al ound that bc., so w conclud that th potntal dnc acoss ach capacto n th ccut s 8 6 8. 6.6 paalll + 9. pf 9. pf [] + ss + ss Thus, usng Equaton [], ss +. pf ( 9. pf ) 9. pf +. pf, whch ducs to 9. pf 8. pf, o 6. pf. pf ( ) Tho, th 6. pf and, om Equaton [],. pf o. pf and 6. pf. W conclud that th two capactancs a. pf and 6. pf.
7 hapt 6 6.7 (a) Th quvalnt capactanc o th ss combnaton n th upp banch s upp F + 6 F +. µ. µ 6. µ F. mf 6. mf o upp. µ F. mf 4. mf Lkws, th quvalnt capactanc o th ss combnaton n th low banch s + + low. µ F 4. µ F 4. µ F o low. µ F 9. Ths two quvalnt capactancs a connctd n paalll wth ach oth, so th quvalnt capactanc o th nt ccut s +. µ F +. µ F. µ F q upp low Not that th sam potntal dnc, qual to th potntal dnc o th batty, xsts acoss both th upp and low banchs. Th chag stod on ach capacto n th ss combnaton n th upp banch s Q Q6 Qupp upp ( ) (. µ F )( 9. ) 8 µ and th chag stod on ach capacto n th ss combnaton n th low banch s Q Q6 Qlow low ( ) (. µ F )( 9. ) µ (c) Th potntal dnc acoss ach o th capactos n th ccut s: Q µ. µ F Q 8 µ. µ F 6. 6. 4 6 Q4 µ 4. µ F 4 Q6 8 µ 6. µ F 6..
Elctcal Engy and apactanc 7 6.8 (a) Th quvalnt capactanc o th ss combnaton n th ghtmost banch o th ccut s ght 4 F + 8 F +. µ. µ 4. µ F o ght 6. µ F Fgu P6.8 Th quvalnt capactanc o th th capactos now connctd n paalll wth ach oth and wth th batty s q 4. µ F +. µ F + 6. µ F. µ F Dagam (c) Th total chag stod n ths ccut s Q. µf 6. total q (d) o Q total 4 µ Th chags on th th capactos shown n Dagam a: Q4 4 ( ) ( 4. µ F )( 6. ) 44 µ Dagam Q ( ) (. µ F )( 6. ) 7 µ Qght ght ( ) ( 6. µ F )( 6. ) 6 µ Ys. Q + Q + Q Q 4 ght total as t should. () Th chag on ach capacto n th ss combnaton n th ghtmost banch o th ognal ccut (Fgu P6.8) s Q Q Q 4 8 ght 6 µ () (g) 4 Q4 6 µ 9. 4. µ F 8 4 Q8 6 µ 8 7. Not that 8 + 4 6. as t should. 8. µ F
74 hapt 6 6.9 Fgu Fgu Fgu Th ccut may b ducd n stps as shown abov. 4. µ F 4. 96. µ Usng Fgu, Q ac Thn, n Fgu, Qac 96. µ ( ) 6. ab 6. µ F ab 4. 6. and bc ac ab 8. Fnally, usng Fgu, Q ( ) ab (. µ F )( 6. ) 6. µ Q 5 ( 5. µ F )( ) ab 8. µ, Q 8 ( 8. µ F )( ) bc 64. µ and Q 4 ( 4. µ F )( ) bc. µ 6.4 Fom Q, th ntal chag o ach capacto s Q (. µ F )(. ) µ and Q x x t th capactos a connctd n paalll, th potntal dnc acoss ach s., and th total chag o Q Q + Q x µ s dvdd btwn th two capactos as Q. µ F.. µ and Qx Q Q µ. µ 9. µ Thus, x Qx 9. µ.. µ F
Elctcal Engy and apactanc 75 6.4 (a) Fom Q ( ), Q 5 ( 5. µ F )( 5. ). 5 µ. 5 m and Q 4 ( 4. µ F )( 5. ). µ. m Whn th two capactos a connctd n paalll, th quvalnt capactanc s q + 5. µ F + 4. µ F 65. µ F. Snc th ngatv plat o on was connctd to th postv plat o th oth, th total chag stod n th paalll combnaton s Q Q Q. µ. 5 µ 75 µ 4 5 Th potntal dnc acoss ach capacto o th paalll combnaton s Q 75 µ 65. µ F q. 5 and th nal chag stod n ach capacto s Q5 ( ) ( 5. µ F )(. 5 ) 88 µ and Q4 Q Q5 75 µ 88 µ 46 µ 6.4 (a) Th ognal ccut ducs to a sngl quvalnt capacto n th stps shown blow. s + + 5. µ F. µ F. µ F p s + + s (. µ F ) +. µ F 8.66 µ F p + (. µ F). µ F q +.66 F + 8 µ. µ F p p 6. 4 µ F contnud on nxt pag
76 hapt 6 Th total chag stod btwn ponts a and b s Q total q ( ) ab ( 6. 4 µ F )( 6. ) 6 µ Thn, lookng at th thd gu, obsv that th chags o th ss capactos o that gu a Qp Qp Qtotal 6 µ. Thus, th potntal dnc acoss th upp paalll combnaton shown n th scond gu s Qp 6 µ ( ) 4. 8 p 8.66 µ F p Fnally, th chag on s Q ( ) p (. µ F )( 4. 8 ) 8. 6 µ 6.4 Fom Q, th ntal chag o ach capacto s Q (. µ F )(. ). µ and Q (. µ F)( ) t th capactos a connctd n paalll, th potntal dnc acoss on s th sam as that acoss th oth. Ths gvs Q Q. µ F. µ F o Q Q [] Fom consvaton o chag, Q + Q Q + Q. µ. Thn, substtutng om Equaton [], ths bcoms Q + Q. µ, gvng Q µ Fnally, om Equaton [], Q µ 6.44 Rcognz that th 7. µ F and th 5. µ F o th cnt banch a connctd n ss. Th total capactanc o that banch s s +. 9 µ F 5. 7. Thn cognz that ths capacto, th 4. µ F capacto, and th 6. µ F capacto a all connctd n paalll btwn ponts a and b. Thus, th quvalnt capactanc btwn ponts a and b s q 4. µ F +. 9 µ F + 6. µ F. 9 µ F
Elctcal Engy and apactanc 77 Q 6 6.45 Engy stod ( ) ( 4. 5 F) (. ). 4 4 J 6.46 (a) Th quvalnt capactanc o a ss combnaton o and s q 8 F + 6 F +. µ. µ 6. µ F o q. µ F Whn ths ss combnaton s connctd to a.- batty, th total stod ngy s Total ngy stod 4. F 8 64 J 6 q ( ) ( )(. ). Th chag stod on ach o th two capactos n th ss combnaton s Q Q Qtotal q ( ) (. µ F )(. ) 44 µ. 44 4 and th ngy stod n ach o th ndvdual capactos s Engy stod n 4 Q (. 44 ) 6 8. F 5. 76 4 J and Engy stod n 4 Q (. 44 ) 6 6. F. 88 4 J 4 4 4 Engy stod n + Engy stod n 5. 76 J +. 88 J 8. 64 J, whch s th sam as th total stod ngy ound n pat (a). Ths must b tu th computd quvalnt capactanc s tuly quvalnt to th ognal combnaton. (c) I and had bn connctd n paalll ath than n ss, th quvalnt capactanc would hav bn q + 8. µ F + 6. µ F 54. µ F. I th total ngy stod q ( ) n ths paalll combnaton s to b th sam as was stod n th ognal ss combnaton, t s ncssay that. Total ngy stod 8 64 4 J q 54. 6 F 5. 66 Snc th two capactos n paalll hav th sam potntal dnc acoss thm, th ngy stod n th ndvdual capactos ( ) s dctly popotonal to th capactancs. Th lag capacto,, stos th most ngy n ths cas.
78 hapt 6 6.47 (a) Th ngy ntally stod n th capacto s Q ( Engy stod ) ( ) (. µf )( 6. ) 54. µ J Whn th capacto s dsconnctd om th batty, th stod chag bcoms solatd wth no way o th plats. Thus, th chag mans constant at th valu Q as long as th capacto mans dsconnctd. Snc th capactanc o a paalll plat capacto s κ d, whn th dstanc d spaatng th plats s doubld, th capactanc s dcasd by a acto o (..,. 5 µ F). Th stod ngy (wth Q unchangd) bcoms ( Engy stod) Q Q Q ( ) Engy stod 8 µ J (c) Whn th capacto s connctd to th batty, th potntal dnc btwn th plats s stablshd at th ognal valu o ( ) 6., whl th capactanc mans at. 5 µ F. Th ngy stod und ths condtons s ( Engy stod ) ( F). 5 µ ( 6. ) 7. µ J 6.48 Th ngy tansd to th wat s.. W Q 5 8 Thus, m s th mass o wat bold away, W m c( T ) + Lv bcoms. 5 7 J J. 5 7 J m 486. ( ) +. 6 6 J kg kg 7. 5 J gvng m.55 J kg 9. 79 kg 6.49 (a) Not that th chag on th plats mans constant at th ognal valu, Q, as th dlctc s nstd. Thus, th chang n th potntal dnc, Q, s du to a chang n capactanc alon. Th ato o th nal and ntal capactancs s κ d κ d and Q Q ( ) ( ) ( ) 85. 5.. 4 Thus, th dlctc constant o th nstd matal s κ. 4, and th matal s pobably nylon (s Tabl 6.). I th dlctc only patally lld th spac btwn th plats, lavng th manng spac a-lld, th quvalnt dlctc constant would b somwh btwn κ. (a) and κ. 4. Th sultng potntal dnc would thn l somwh btwn ( ) 85. and ( ) 5..
Elctcal Engy and apactanc 79 6.5 (a) Th capactanc o th capacto whl a-lld s d 4 ( 8. 85 N m )( 5. m ). 5 Th ognal chag stod on th plats s m. 48 F. 48 pf Q ( ). 48 F. 5 7 ( ) 7 p Snc dstlld wat s an nsulato, ntoducng t btwn th solatd capacto plats dos not allow th chag to chang. Thus, th nal chag s Q 7 p. t mmson dstlld wat ( κ 8 s Tabl 6. ), th nw capactanc s κ ( 8)(. 48 pf ) 8 pf and th nw potntal dnc s Q 7 p 8 pf. 4. (c) Th ngy stod n a capacto s: Engy stod Q. Thus, th chang n th stod ngy du to mmson n th dstlld wat s E Q Q Q 7 8 9 4. 57 J 45. 7 J 45. 7 nj 8 F. 48 F 6.5 (a) Th dlctc constant o Tlon s κ., so th capactanc s d ( ). 8. 85 N m 75 m κ 4 9 8. F 8. nf.4 m Fo Tlon, th dlctc stngth s E max 6. 6 m, so th maxmum voltag s 6 E d 6. m. 4 m max max ( ) max. 4. 4 k 6.5 Bo th capacto s olld, th capactanc o ths paalll plat capacto s w L κ κ d d wh s th suac aa o on sd o a ol stp. Thus, th qud lngth s L ( ) ( ) 8 d w 9. 5 F. 5 m.7) 8. 85 N m 7. m κ (. 4 m
8 hapt 6 m. kg 6.5 (a) 9. 9 ρ kg m 6 m Snc 4 π, th adus s 4π 4π 4π 4π, and th suac aa s 9. 9 m 4π 4π 6 4. 54 m κ d ( N m )( m ) 5. 8. 85 4. 54 9 m. F ( ) (c) Q ( ). F. and th numb o lctonc chags s n Q 4 4.. 6 9.6 5 6.54 Snc th capactos a n paalll, th quvalnt capactanc s + + q + + + + d d d d o q wh + + d 6.55 Snc th capactos a n ss, th quvalnt capactanc s gvn by d d d d + d + d + + + + q o q wh d d + d + d d 6.56 (a) Plas to th soluton o Poblm 6.7, wh th ollowng sults w obtand:. mf 6. mf. µ q F Q Q6 8 µ Q Q4 µ Th total ngy stod n th ull ccut s thn 6 ( Engy stod) total q ( ). F 9.. 5 J. 5 J. 5 mj. mf 9. 4. mf contnud on nxt pag
Elctcal Engy and apactanc 8 (c) Th ngy stod n ach ndvdual capacto s 6 Q ( ) Fo. µ F: ( Engy stod) 6. F Fo. µ F: Engy stod Fo 4. µ F: Engy stod Fo 6. µ F: Engy stod 4 6. 6 J. 6 mj 6 Q ( 8 ) 6. F 5. 4 J 5. 4 mj 6 Q ( ) 4 6 4. F 4. 8 J. 8 mj 6 Q ( 8 ) 6 6 6. F Th total ngy stod n th ndvdual capactos s 6. 7 J. 7 mj Engy stod.6 + 5. 4 +. 8 +. 7 mj. 5 mj Engy stod total Thus, th sums o th ngs stod n th ndvdual capactos quals th total ngy stod by th systm. 6.57 In th absnc o a dlctc, th capactanc o th paalll plat capacto s d Wth th dlctc nstd, t lls on-thd o th gap btwn th plats as shown n sktch (a) at th ght. W modl ths stuaton as consstng o a pa o capactos, and, connctd n ss as shown d d d k d k c n sktch at th ght. In alty, th low plat o and th upp plat o a on and th sam, consstng o th low suac o th dlctc shown n sktch (a). Th capactancs n th modl o sktch a gvn by: (a) d c κ κ d d and d d and th quvalnt capactanc o th ss combnaton s q d κ d d + + κ κ + d κ + d κ + κ κ κ and κ q + κ
8 hapt 6 6.58 Fo th paalll combnaton: p + whch gvs p [] Fo th ss combnaton: + o s s s s Thus, w hav p s s s s and quatng ths to Equaton [] abov gvs o + W wt ths sult as : + p p s s s p p s and us th quadatc omula to obtan ± 4 Thn, Equaton [] gvs 4 p p p s p p p s 6.59 Th chag stod on th capacto by th batty s Q ( ) ( ) Ths s also th total chag stod n th paalll combnaton whn ths chagd capacto s connctd n paalll wth an unchagd. -µ F capacto. Thus, acoss th paalll combnaton, Q p ( ) gvs ( + ). µ F. o 7... µ F and. (. µ F ) 4. 9 µ F 7. s th sultng voltag 6.6 (a) Th. -µ s locatd.5 m om pont P, so ts contbuton to th potntal at P s k q 9 8. 99 N m ( ) 6..5 m. 8 4 Th potntal at P du to th. -µ chag locatd.5 m away s k q 9 8. 99 N m ( )..5 m 6. 6 4 (c) Th total potntal at pont P s P + + 4 4. 8. 6. 8 (d) Th wok qud to mov a chag q. µ to pont P om nnty s 6 4 W q q.. 8 5. 4 J P ( )
Elctcal Engy and apactanc 8 6.6 Th stags o th ducton o ths ccut a shown blow. Thus, q 6. 5 µ F 6.6 (a) Du to sphcal symmty, th chag on ach o th concntc sphcal shlls wll b unomly dstbutd ov that shll. Insd a sphcal suac havng a unom chag dstbuton, th lctc ld du to th chag on that suac s zo. Thus, n ths gon, th potntal du to th chag on that suac s constant and qual to th potntal at th suac. Outsd a sphcal suac havng a unom chag dstbuton, th potntal du kq to th chag on that suac s gvn by, wh s th dstanc om th cnt o that suac and q s th chag on that suac. In th gon btwn a pa o concntc sphcal shlls, wth th nn shll havng chag + Q and th out shll havng adus b and chag Q, th total lctc potntal s gvn by + du to nn shll du to out shll k Q k + Q b kq b Th potntal dnc btwn th two shlls s tho k Q k Q a b a b b b k Q b a ab Th capactanc o ths dvc s gvn by Q ab k b a Whn b >> a, thn b a b. Thus, n th lmt as b, th capactanc ound abov bcoms ab k b a k π 4 6.6 Th ngy stod n a chagd capacto s W a. Hnc, W J 4. 47 4. 47 k 6. F
84 hapt 6 6.64 Fom Q, th capactanc o th capacto wth a btwn th plats s Q 5 µ t th dlctc s nstd, th potntal dnc s hld to th ognal valu, but th chag changs to Q Q + µ 5 µ. Thus, th capactanc wth th dlctc slab n plac s Q 5 µ Th dlctc constant o th dlctc slab s tho µ κ 5 5 µ 5 5. 6.65 Th chags ntally stod on th capactos a Q ( ) ( 6. µ F )( 5 ). 5 µ and Q ( ) (. µ F )( 5 ) 5. µ Whn th capactos a connctd n paalll, wth th ngatv plat o on connctd to th postv plat o th oth, th nt stod chag s Q Q Q. 5 µ 5. µ. µ Th quvalnt capactanc o th paalll combnaton s q + 8. µ F. Thus, th nal potntal dnc acoss ach o th capactos s Q ( ). µ 5 8. µ F q and th nal chag on ach capacto s Q ( 6. µ F )( 5 ) 75 µ. 75 m and Q ( ) ( µ ) µ. F 5 5. 5 m 6.66 Th ngy qud to mlt th lad sampl s W m c ( T ) + L Pb ( + 6 6. kg 8 J kg ) 7.. 4. 5 J kg. 8 J Th ngy stod n a capacto s W, so th qud potntal dnc s W. 8 J 6 5. F
Elctcal Engy and apactanc 85 6.67 Whn xcss chag sds on a sphcal suac that s a movd om any oth chag, ths xcss chag s unomly dstbutd ov th sphcal suac, and th lctc potntal at th suac s th sam as all th xcss chag w concntatd at th cnt o th sphcal suac. In th gvn stuaton, w hav two chagd sphs, ntally solatd om ach oth, wth chags and potntals o: Q + 6. µ, kq R wh R. cm, Q 4. µ, and k Q R wth R 8. cm. Whn ths sphs a thn connctd by a long conductng thad, th chags a dstbutd ( yldng chags o Q and Q, spctvly) untl th two suacs com to a common potntal ( kq R kq R ). Whn qulbum s stablshd, w hav: Fom consvaton o chag: Q + Q Q + Q Q + Q +. µ [] Fom qual potntals: kq kq R R R Q R Q o Q. 5Q [] Substtutng Equaton [] nto [] gvs: +. µ Q. 8 µ. 5 Thn, Equaton [] gvs: Q. 5. 8 µ. µ 6.68 Th lctc ld btwn th plats s dctd downwad wth magntud E y d 5. 4. m N m Snc th gavtatonal oc xpncd by th lcton s nglgbl n compason to th lctcal oc actng on t, th vtcal acclaton s a y Fy qey 9 4. 6 5. N m m m 9. kg ( ) + 8. 78 5 m s (a) t th closst appoach to th bottom plat, v y. Thus, th vtcal dsplacmnt om pont O s ound om v v a y as y a y + y y y 5 ( 8. 78 m s ) 6 v sn θ 5. 6 m s sn 45. 89 mm Th mnmum dstanc abov th bottom plat s thn D d + y. mm. 89 mm. mm Th tm o th lcton to go om pont O to th upp plat s ound om y v yt + ayt as 6 +. m 5. 6 m sn 45 t + s 8 78 5. m t s Solvng o t gvs a postv soluton o t. 9 s. Th hozontal dsplacmnt om pont O at ths tm s 6 9 x v xt 5. 6 m s cos 45. s 4. 4 mm