Electric Potential ANSWERS TO QUESTIONS

Transcription

1 5 Elctic Potntil CHAPTE UTNE 5 Potntil Dinc nd Elctic Potntil 5 Potntil Dinc in Uniom Elctic ild 5 Elctic Potntil nd Potntil Eny Du to Point Chs 5 tinin th lu o th Elctic ild om th Elctic Potntil 55 Elctic Potntil Du to Continuous Ch Distiutions 56 Elctic Potntil Du to Chd Conducto 57 Th Millikn il Dop Expimnt 58 Appliction o Elctosttistics ANSWES T UESTNS 5 Whn on oct B with lctic ch is immsd in th lctic ild o noth ch o chs A, th systm posssss lctic potntil ny Th ny cn msud y sin how much wok th ild dos on th ch B s it mos to nc loction W choos not to isuli A s ct on B s n ction-t--distnc, ut s th sult o twostp pocss: Ch A cts lctic potntil thouhout th suoundin spc Thn th potntil cts on B to inct th systm with ny 5 Th potntil ny incss Whn n outsid nt mks it mo in th diction o th ild, th ch mos to ion o low lctic potntil Thn th poduct o its nti ch with low num o olts is hih num o ouls Kp in mind tht nti ch ls n lctic oc in th opposit diction to th ild, whil th potntil is th wok don on th ch to mo it in ild p unit ch 5 To mo lik chs toth om n ininit sption, t which th potntil ny o th systm o two chs is o, quis wok to don on th systm y n outsid nt Hnc ny is stod, nd potntil ny is positi As chs with opposit sins mo toth om n ininit sption, ny is lsd, nd th potntil ny o th st o chs coms nti 5 Th ch cn mod lon ny pth plll to th y- pln, nmly ppndicul to th ild 55 Th lctic ild lwys points in th diction o th tst chn in lctic potntil This is implid y th ltionships Ex x, E y nd E y 56 () Th quipotntil sucs nstin coxil cylinds ound n ininit lin o ch () Th quipotntil sucs nstin concntic sphs ound uniomly chd sph 57 th w potntil dinc twn two points on th conducto, th lctons in th conducto would mo until th potntil dinc dispps 5

2 5 Elctic Potntil 58 No Th uniomly chd sph, whth hollow o solid mtl, is n quipotntil olum Sinc th is no lctic ild, this mns tht th is no chn in lcticl potntil Th potntil t y point insid is th sm s th lu o th potntil t th suc 59 ninitly wy om lin o ch, th lin will not look lik point n ct, without ny distinuishin tus, it is not possil to tll th distnc om n ininitly lon lin o ch Anoth wy o sttin th nsw: Th potntil would di to ininity t ny init distnc, i it w o ininitly wy 5 Th smll sph will n th solution to th xmpl d to, qution stts tht ch will h th sm tio o ch to dius, q n this cs, th ch dnsity is suc ch q dnsity,, so th smll-dius sph will h th t ch dnsity π 5 Th min cto is th dius o th dom n otn olookd spct is lso th humidity o th i di i hs l dilctic kdown stnth, sultin in hih ttinl lctic potntil oth oundd octs ny, th mximum potntil miht ducd 5 Th intns otn oscilltin lctic ilds ound hih olt lins is l nouh to ioni th i suoundin th cls Whn th molculs cptu thi lctons, thy ls tht ny in th om o liht 5 A shp point in chd conducto would imply l lctic ild in tht ion An lctic disch could most sily tk plc t tht shp point 5 Us conducti ox to shild th quipmnt Any sty lctic ild will cus chs on th out suc o th conducto to n nd cncl th sty ild insid th olum it ncloss 55 No ch stys on th inn sph in quiliium th w ny, it would ct n lctic ild in th wi to push mo ch to th out sph All o th ch is on th out sph Tho, o ch is on th inn sph nd µ C is on th out sph 56 Th oundin wi cn touchd qully wll to ny point on th sph Elctons will din wy into th ound nd th sph will lt positily chd Th ound, wi, nd sph ll conductin Thy toth om n quipotntil olum t o olts duin th contct How clos th oundin wi is to th nti ch, lctons h no diiculty in moin within th mtl thouh th oundin wi to ound Th ound cn ct s n ininit souc o sink o lctons n this cs, it is n lcton sink SUTNS T PBEMS Sction 5 Potntil Dinc nd Elctic Potntil P5 nd N A 9 C W, so W 96 C J C 5 MJ

3 7 q P5 K q 77 5 Chpt 5 5 q 6 9 C P5 () Eny o th poton-ild systm is consd s th poton mos om hih to low potntil, which cn dind o this polm s moin om down to Ki + Ui + E mch K + U + q + m p + J 6 C 67 K J k p C 9 7 p 5 5 m s () Th lcton will in spd in moin th oth wy, om i to : K + U + E K + U i i mch + + m + q 9 9 k + 6 C J C 69 6 m s P5 W K q k m s 6 C om which, 5 Sction 5 Potntil Dinc in Uniom Elctic ild P55 () W ollow th pth om (, ) to ( cm, ) to ( cm, 5 cm) U (wok don) U (wok om oiin to ( cm, )) (wok om ( cm, ) to ( cm, 5 cm)) Not tht th lst tm is qul to cus th oc is ppndicul to th displcmnt 6 x U qe x C 5 m m 6 J () U 6 J 5 JC 5 6 q C 5 P56 E d 5 JC 6 67 NC 67 MNC m

4 5 Elctic Potntil P57 U m i P 8 9 k ms 7 ms 6 J U q : Th oiin is t hihst potntil P58 () Ed 5 9 m m 59 () m q : 55 6 m s P59 E ds E ds E ds Ecos8 dy Ecos 9 dx B B B A A B C B A A C A G P59 *P5 Assum th opposit Thn t som point A on som quipotntil suc th lctic ild hs nono componnt E p in th pln o th suc t tst ch stt om point A nd mo B som distnc on th suc in th diction o th ild componnt Thn E ds is nono A Th lctic potntil chs coss th suc nd it is not n quipotntil suc Th contdiction shows tht ou ssumption is ls, tht E p, nd tht th ild is ppndicul to th quipotntil suc P5 () Aitily choos t Thn t oth points Ex nd U Ex Btwn th ndpoints o th motion, K+ U + U K+ U + U s i s E kxmx Exmx so xmx k G P5 () At quiliium, x s + o kx E So th quiliium position is t x E k continud on nxt p

5 Chpt 5 55 (c) Th lock s qution o motion is kx E m d x x + dt E t x x k, o x x E +, k so th qution o motion coms: E + k x + K J + E m d x E k d x k, o k dt dt H G m K J x This is th qution o simpl hmonic motion x ω x with ω Th piod o th motion is thn (d) K+ U + U + E K+ U + U s i mch s k m π T ω + + µ kmx + kx Ex E µ km xmx k mx mx mx m π k P5 o th nti motion, y yi yit+ yt t i + yt so y my: m qe mi t E m i q t o th upwd liht: + y y ymx E dy+ k 6 5 C y t K J mh G K J y yi yd ii i i + K Jymx nd y mx i t t y m i m i y it q t K J q t K J mx H G K J ms 98 ms ms s P k s nd E i q t P5 Aitily tk t th initil point Thn t distnc d downild, wh is th od lnth, Ed nd U λ Ed () K + U K+ U i + µ λed 6 λed Cm NC m µ k m ms i () Th sm

6 56 Elctic Potntil P5 Aitily tk t point P Thn (om Eqution 58) th potntil t th oiinl position o th ch is Es E cosθ At th inl point, E Suppos th tl is ictionlss: K + U K+ U i qe cosθ m qe 6 qe cos θ C N C 5 m cos 6 m k ms Sction 5 Elctic Potntil nd Potntil Eny Du to Point Chs P55 () Th potntil t cm is k q () Th potntil t cm is k q N m C 6 C 7 m N m C 6 C 7 79 Thus, th dinc in potntil twn th two points is 79 m (c) Th ppoch is th sm s o xcpt th ch is 6 9 C This chns th sin o ch nsw, with its mnitud minin th sm Tht is, th potntil t cm is 7 Th potntil t cm is , so 79 8 P56 () Sinc th chs qul nd plcd symmticlly, () Sinc qe, E (c) k q 899 N m 5 5 k 9 C 6 C 8 m G P56 P57 () E π π E 5 m 6 m () π 6 m m C 6 m µ C

7 P58 () E x kq kq + x x coms E k x + q q + x x Chpt 5 57 Diidin y k, qx q x x x + Tho E whn x ± m (Not tht th positi oot dos not cospond to physiclly lid sitution) K J () kq kq + x x + q q o k x x Ain solin o x, qx q x o x whn x 667 m nd q x q x o x < x m P59 k q i i i M N 87 7 M q P5 () U 5 C C 8 99 m C π 5 m P G P J Th minus sin mns it tks 86 7 much l sption J to pull th two chs pt om 5 cm to () + π π C 8 99 m C C 8 99 m C 75 m 75 m

8 58 Elctic Potntil P5 U q + q + q q U U q q q + + π 6 9 C 8 99 N m C m 5 m 895 J P5 () kq kq kq + H G K J 9 6 G 8 99 N m C C m + 5 m k 6 () U q C J C 9 65 J 6 m + 5 m G P5 P5 U U + U + U + U U + U + U + U + U + U + U k k k U s s s U k k + 5 G P5 s s An ltnt wy to t th tm + dionl pis is to coni tht th sid pis nd c P5 Ech ch cts qul potntil t th cnt Th totl potntil is: M N P k q kq 5 M P 5 P55 () Ech ch sptly cts positi potntil ywh Th totl potntil poducd y th th chs toth is thn th sum o th positi tms Th is no point loctd t init distnc om th chs, t which this totl potntil is o () kq kq kq +

9 Chpt 5 59 P56 Consid th two sphs s systm () Constion o momntum: m i+ m i k q q By constion o ny, d nd mkqq m m + m + d m k 8 m s m 7 k kqq + m o m k q q m + m + + kqq m m + d m k N m C C C k 8 k 8 m m 8 ms 55 ms K J () th sphs mtl, lctons will mo ound on thm with nliil ny loss to plc th cnts o xcss ch on th insids o th sphs Thn ust o thy touch, th cti distnc twn chs will lss thn + nd th sphs will lly moin st thn clcultd in () P57 Consid th two sphs s systm () Constion o momntum: m i+ m i o m m k q q By constion o ny, d nd kqq + k q q m + m + + kqq m m + d m mkqq m m + m + d m m mkqq m m + m + d H G K J () th sphs mtl, lctons will mo ound on thm with nliil ny loss to plc th cnts o xcss ch on th insids o th sphs Thn ust o thy touch, th cti distnc twn chs will lss thn + nd th sphs will lly moin st thn clcultd in ()

10 6 Elctic Potntil *P58 () n n mpty unis, th -nc ch cn plcd t its loction with no ny instmnt At distnc o cm, it cts potntil 9 9 kq 899 N m C C 5 k m To plc th -nc ch th w must put in ny U q C 5 5 J 9 5 Nxt, to in up th -nc ch quis ny U + U q + q q C 8 99 N m C J 5 J Th totl ny o th th chs is U + U + U 5 J C C + m 8 m () Th th ixd chs ct this potntil t th loction wh th outh is lsd: N m C + C m Eny o th systm o ou chd octs is consd s th outh ch lis wy: m + q m q K J + i K J J 9 + C k + k 6 ms *P59 Th oiinl lcticl potntil ny is U q q kq d n th inl coniution w h mchnicl quiliium Th spin nd lctosttic ocs on ch ch k d + q kq kq Thn k n th inl coniution th totl potntil d 8d d ny is kq kx q q kq kq + + Th missin ny must h com intnl 8d d 9 d ny, s th systm is isoltd: kq kq + E int d 9d kq Eint 5 9 d

11 k k k k P5 () x x + x + x k k x + G x x k x + G H + K J Chpt 5 6 () k k k k y y y+ y k y y+ y k y y + G P5() G P5() 9 9 P5 k k 8 99 so N m C 8 C 7 m o, 5, nd 5, 7 m, m, nd 88 m Th dii insly popotionl to th potntil P5 Usin constion o ny o th lph pticl-nuclus systm, w h K + U Ki + Ui But U kq α q old i nd i Thus, U i i Also K ( t tunin point), so U K o min kq q i α old mαα min 9 9 kq α qold 8 99 N m C 79 6 C m 7 7 α α 6 6 k m s 7 m 7 m

12 6 Elctic Potntil P5 Usin constion o ny k kq w h: + m which is: o k m Thus, 76 6 m s N m C 6 C C 9 k m m wh i i q q q q q q q q q q q q k P 6 6 kq P P 96 J kqq i P5 U, summd o ll pis o i, U U U P55 Ech ch mos o on its dionl lin All chs h qul spds K + U K+ U kq kq kq kq + + m H G K J kq m + i 8 kq m G P5 P56 A cu hs ds nd 6 cs Consquntly, th d pis sptd y s, 6 c dionl pis sptd y s nd intio dionl pis sptd s U kq + + s P kq 8 s Sction 5 tinin th lu o th Elctic ild om th Elctic Potntil P57 + x + 7 m x () At x, At x m, () At x 6 m, d E 7 m 7 N C in th + xdiction dx

13 P58 () o < k d E d Chpt 5 6 () o k d k E d P59 5x x y+ y Elut E t,, E E E x y K J k 5+ 6xy x x y + 5 y N C x y E E + E + E P5 () EA > EB sinc E s () EB s 6 cm NC down (c) Th iu is shown to th iht, with smpl ild lins sktchd in P5 E E y y y y M N M k ln + + y y P P k y k y M + y + + y P y + y P G P5 Sction 55 Elctic Potntil Du to Continuous Ch Distiutions P5 k + k k k 55 5

14 6 Elctic Potntil P5 () α λ xp C H G K J m m C m dq λdx () k k kα xdx k d d+ x α ln + d K J P kdq P5 k αxdx t x + x Thn x, nd dx d k α G P5 d kα d d kα + kα ln + + k H K + α M N M kα x x k K J + K J + + M P x ln α K J + k α + + ln M P + kα + + M k + α ln + + P55 d P56 d π dq All its o ch t th sm distnc om So π kdq + x P P P P M N K J + H G K J + K J C 899 N m C 5 M m π wh dq σda σπd d πσk π kσ x + x + + x P P G P56

15 dq P57 k k ll ch smicicl λdx ds dx k k x + λ + λ x kλ kλln x + + k x π λ ln k ln + k λπ + k ln k λπ + ln Chpt 5 65 Sction 56 Elctic Potntil Du to Chd Conducto P58 Sustitutin in lus into kq N m Cq 75 m Sustitutin q 5 7 C, 7 5 C N 9 6 C 56 lctons P59 () E ; kq M () E kq MN C wy kq M (c) E kq kq 67 M 9 MN C wy

16 66 Elctic Potntil *P55 () Both sphs must t th sm potntil ccodin to kq wh lso q + q C Thn q q q q q 6 + q kq 6 6 C C + 6 cm cm 6 kq C C C N m C 9 C 9 6 C on th smll sph m 5 5 () utsid th l sph, kq 5 E 5 6 m 5 6 m wy utsid th smll sph, E 67 m wy m Th smll sph cis lss ch ut cts much ston lctic ild thn th l sph Sction 57 Th Millikn il Dop Expimnt Sction 58 Appliction o Elctosttistics P55 () E mx k k mx H G K J H G K J 6 m 6 k mx Emx 5 5 () k mx E mx o k ST mx mx UW mx 6 Emx 5 75 µ C k P55 kq nd E kq Sinc E, () () E 5 6 m nd 6 m q µ C k

17 Chpt 5 67 Additionl Polms P55 U q k qq J 5M P55 () To mk spk 5 mm lon in dy i twn lt mtl plts quis potntil dinc 6 ~ Ed m 5 m 5 () Th o you skin is phps 5 m, so modl you ody s sph with this suc ts dius is in y 5 m π, 5 m W qui tht you t th potntil ound in pt (): kq 5 5 m J q N m C C k q 5 8 C ~ C 7 6 K J J N m P555 () U kqq J () U kqq (c) U kqq k P556 om Exmpl 55, th potntil ctd y th in t th lcton s sttin point is i x k k πλ x + i + i whil t th cnt, it is + m + i d i d i m π kλ i m π m s π k λ om constion o ny, xi G 7 +

18 68 Elctic Potntil *P557 Th plts ct uniom lctic ild to th iht in th pictu, with mnitud d d Assum th ll swins smll distnc x to th iht t mos to plc wh th olt ctd y th plts is low y Ex d x ts ound connction mintins it t y llowin x kq x ch q to low om ound onto th ll, wh + q Thn th ll d kd x ls lctic oc qe to th iht o quiliium this must lncd y th kd x hoiontl componnt o stin tnsion ccodin to Tcosθ m Tsinθ kd tnθ x x kd m o smll x Thn kd m H G K J is lss thn this lu, th only quiliium position o th ll is hnin stiht down xcds this lu th ll will swin o to on plt o th oth P558 () Tk th oiin t th point wh w will ind th potntil n in, o width dx, hs ch dx nd, ccodin to Exmpl 55, cts potntil h kdx d h x + Th whol stck o ins cts potntil d ll ch k d+ h+ d+ h + K ln ln h d+ d + d + h d h kdx k x+ x + h x h H + d + d () A disk o thicknss dx hs ch dx nd ch-p- dx h π Accodin to h Exmpl 56, it cts potntil dx d k π x + x hh π K nttin, d h k k x x + dxxdx x x x x hh K M h + + H + + ln K + d+ h d M N k h d h d h d d dh h d h d h ln M G d+ d + G H P d P P559 W dq wh kq Tho, W k

19 Chpt σ 6 Cm P56 Th positi plt y itsl cts ild E kn C wy 8 85 C N m om th + plt Th nti plt y itsl cts th sm si ild nd twn th plts it is in th sm diction Toth th plts ct uniom ild 7 kn C in th spc twn () () Tk t th nti plt Th potntil t th positi plt is thn cm 7 kn C dx Th potntil dinc twn th plts is 7 NC m 88 K J + m + q m q i 9 7 q 6 C 88 m 78 J K J (c) 6 km s i d ii 5 6 ms + m (d) + x x 9 ms 7 6 () m 67 k 9 m s 6 5 N () E q N C 7 kn C B P56 () B A E ds nd th ild t distnc om uniomly A chd od (wh > dius o chd od) is λ k E λ π n this cs, th ild twn th cntl wi nd th coxil cylind is dictd ppndicul to th lin o ch so tht k B A d k H G λ λ ln, o kλ ln H G G P56 continud on nxt p

20 7 Elctic Potntil () om pt (), whn th out cylind is considd to t o potntil, th potntil t distnc om th xis is k H G λ ln K J Th ild t is in y E k λ H G But, om pt (), kλ ln Tho, E P56 () om Polm 6, H G ln E ln K J W qui ust outsid th cntl wi m ln 85 H G m o m H G 85m ln kλ W sol y homin in on th quid lu m m m H G ln Thus, to th siniicnt ius, mm () At, 5 k E H G K J ln 85 m m 85 m 9 k m P56 E d λ ln π λ π d

21 Chpt 5 7 *P56 Tk th illusttion psntd with th polm s n initil pictu No xtnl hoiontl ocs ct on th st o ou lls, so its cnt o mss stys ixd t th loction o th cnt o th squ As th chd lls nd swin out nd wy om ch oth, lls nd mo up with qul y-componnts o locity Th mximum-kinticny point is illusttd Systm ny is consd: + CM + kq kq + m + m + m + m G P56 kq m kq m k q k q P565 o th in ch distiution, x, y, + wh x+ + y + Th suc on which x, y, is in y kq nd x y, o This is: x+ + y + x + y H G K J H G K J [] 8 which my wittn in th om: x y x y Th nl qution o sph o dius cntd t x, y, is: x x + y y + [] o x + y + + x x+ y y+ + x + y + Compin qutions [] nd [], it is sn tht th quipotntil suc o which is indd sph nd tht: Thus, x 8 x ; y ; ; x + y + K J 6, y, nd 9 9 Th quipotntil suc is tho sph cntd t K J,,, hin dius

22 7 Elctic Potntil P566 () om Guss s lw, E A (no ch within) E k q A B 899 E C k () k C q q A A 8 H G K J m 9 + q 5 B q 5 B At, 5 nsid, B K J K J m d K J At, so A +5 5 P567 om Exmpl 55, th potntil t th cnt o th in is k i nd th potntil t n ininit distnc om th in is Thus, th initil nd inl potntil nis o th point ch-in systm : K J U i k i G P567 nd U om constion o ny, K + U Ki + Ui o M k + + iin P568 k λdx x + k M kλln x+ x + kλln P M + + M P

23 *P569 () kq kq kq Chpt 5 7 () om th iu, o >>, cosθ Thn kq kp cosθ cosθ E kp cosθ n sphicl coodints, th θ componnt o th dint is K J θ Tho, o >> E E θ θ K J kp, kp nd E 9 nd E θ Eθ 9 kp sinθ G P569 Ths sults sonl o >> Thi dictions s shown in iu 5 (c) How, o, E This is unsonl, sinc is not much t thn i it is (c) x kpy + y nd E E x y kpxy x x + y 5 kp y x y x + y 5

24 7 Elctic Potntil P57 nsid th sph, Ex Ey E utsid, So Ex + + E x y x E x x y P + + E y y y E + E x + y + H 5 5 Ey E x y y E y x y E E E x y E x y 5 K J E E + E x y x + y Ex x x E E + x + y + H K 5 K J + + K K J P57 o n lmnt o which is in o dius nd width d, d dq σda Cπd nd d x Cπ k Cπ k + x + x ln + x + + x P kdq + x P57 du dq wh th potntil kq Th lmnt o ch in shll is dq ρ (olum lmnt) o dq ρπ d nd th ch q in sph o dius is q d H G π πρ ρ K J Sustitutin this into th xpssion o du, w h du kq dq H G K J π kρ d k d H G K J ρ π 6π H G ρ H G 6π 6π U du k ρ d k ρ 5 H G 5 5 But th totl ch, ρ π Tho, U 5 k

25 Chpt 5 75 *P57 () Th whol ch on th cu is 6 7 q Cm m C Diid up th cu into 6 o mo lmnts Th littl cu lld cts t P kq potntil Th oths in th m hoiontl ow hind it contiut kq m Th littl cus in th ows continin nd c dd kq m P nd th its in ow d mk potntil t P kq m P 9 7 d c 5 cm Nm C Th whol potntil t P is C 6 m mo sudiisions o th l cu, w t th sm nsw to ou diits G P57 w us P () A sph cntd t th sm point would ct potntil 9 7 kq Nm C 8988, l y C m ANSWES T EEN PBEMS P5 6 9 C P5 () k ; () 96 5 mj P5 5 P56 67 MN C P58 () 59 ; () 55 Mm s P5 s th solution P5 k P5 m s P56 () ; () ; (c) 5 k P58 () 8 m; () 667 m nd m P5 () 86 nj ; () P5 5kq P56 () 8 m s nd 55 m s ; () t P58 () 5 µ J; () 6 km s P5 s th solution P5 7 m P5 96 J kq P56 8 s

26 76 Elctic Potntil P58 () ; () k dilly outwd P56 () 88 ; () J ; (c) 6 km s ; (d) 9 Gm s towd th nti plt; P5 () l t A; () N C down; () 65 6 N towd th nti plt; (c) s th solution () 7 kn C towd th nti plt P5 55 k k + P5 α ln M + + M P56 π k σ x + x + P58 56 lctons P55 () 5 k ; () 5 M m wy om th l sph nd 67 M m wy om th smll sph P55 () µ C; () m P55 () ~ ; () ~ 6 P556 5 Mm s P558 () k h () k h G H dh h + C P P d+ h+ d+ h + ln G d+ d + d+ h d+ h + d d + ; d+ h+ d+ h + ln d+ d + P P56 () mm; () 9 k m P56 kq m 89 9 P566 () E A ; E B 5 outwd; E C outwd; K J K J () A 5 ; B 5 + C M 5 K J P568 kλ ln M P m dilly m dilly 5 ; 5 ; P57 Ex E x x + y + Ey E y x + y + E x y E E + x + y + 5 E insid P57 5 k 89 9 K J ; outsid nd