646 CHATE 24 ELECTIC OTENTIAL Electric otentil Energ The chnge U in the electric potentil energ U of point chrge s the chrge moves from n initil point i to finl point f in n electric fiel is U U f U i W, (24-1) where W is the work one b the electrosttic force (ue to the eternl electric fiel) on the point chrge uring the move from i to f. If the potentil energ is efine to be zero t infinit, the electric potentil energ U of the point chrge t prticulr point is U W. (24-2) Here W is the work one b the electrosttic force on the point chrge s the chrge moves from infinit to the prticulr point. Electric otentil Difference n Electric otentil We efine the potentil ifference V between two points i n f in n electric fiel s (24-7) where q is the chrge of prticle on which work W is one b the electric fiel s the prticle moves from point i to point f. The potentil t point is efine s V W (24-8) q. Here W is the work one on the prticle b the electric fiel s the prticle moves in from infinit to the point. The SI unit of potentil is the volt: 1 volt 1 joule per coulomb. otentil n potentil ifference cn lso be written in terms of the electric potentil energ U of prticle of chrge q in n electric fiel: (24-5) (24-6) Equipotentil Surfces The points on n equipotentil surfce ll hve the sme electric potentil. The work one on test chrge in moving it from one such surfce to nother is inepenent of the loctions of the initil n finl points on these surfces n of the pth tht joins the points. The electric fiel E : is lws irecte perpeniculrl to corresponing equipotentil surfces. Fining V from two points i n f is V V f V i W q, V V f V i U f q U i q U q. E : V U q, The electric potentil ifference between V f V i f E : s :, (24-18) where the integrl is tken over n pth connecting the points. If the integrtion is ifficult long n prticulr pth, we cn choose ifferent pth long which the integrtion might be esier. If we choose V i,we hve,for the potentil t prticulr point, V f E : s :. (24-19) i i otentil Due to oint Chrges The electric potentil ue to single point chrge t istnce r from tht point chrge is V 1 q (24-26) 4 r, where V hs the sme sign s q.the potentil ue to collection of point chrges is V n V i 1 (24-27) 4 n q i. r i otentil Due to n Electric Dipole At istnce r from n electric ipole with ipole moment mgnitue p q, the electric potentil of the ipole is V 1 p cos (24-3) 4 r 2 for r ; the ngle u is efine in Fig. 24-1. otentil Due to Continuous Chrge Distribution For continuous istribution of chrge, Eq. 24-27 becomes V 1 4, in which the integrl is tken over the entire istribution. (24-32) Clculting from V The component of in n irection is the negtive of the rte t which the potentil chnges with istnce in tht irection: E s V (24-4) s. The,, n z components of m be foun from E : i 1 E : q r E V (24-41) ; E V ; E z V z. When E : is uniform, Eq. 24-4 reuces to E V (24-42) s, where s is perpeniculr to the equipotentil surfces. The electric fiel is zero prllel to n equipotentil surfce. Electric otentil Energ of Sstem of oint Chrges The electric potentil energ of sstem of point chrges is equl to the work neee to ssemble the sstem with the chrges initill t rest n infinitel istnt from ech other. For two chrges t seprtion r, U W 1 q 1 q 2. 4 r (24-43) otentil of Chrge Conuctor An ecess chrge plce on conuctor will, in the equilibrium stte, be locte entirel on the outer surfce of the conuctor. The chrge will istribute itself so tht the following occur: (1) The entire conuctor, incluing interior points, is t uniform potentil. (2) At ever internl point, the electric fiel ue to the chrge cncels the eternl electric fiel tht otherwise woul hve been there. (3) The net electric fiel t ever point on the surfce is perpeniculr to the surfce. i 1 E :
QUESTIONS 647 AT 3 1 In Fig. 24-21, eight prticles 4q 2q +q form squre, with istnce between jcent prticles. Wht is the electric potentil t point t the center of the squre if the electric potentil is zero t +5q 5q infinit? 2 Figure 24-22 shows three sets of cross sections of equipotentil surfces; ll three cover the sme size region of spce. nk the q 2q +4q rrngements ccoring to the mgnitue of the electric fiel present Fig. 24-21 Question 1. in the region, gretest first. In which is the electric fiel i- recte own the pge? 2 V 4 6 8 1 14 V 12 1 (1) (2) (3) 2q +6q (1) +12q +q (3) Fig. 24-22 Question 2. +3q 4q (2) 6q 2q (4) Fig. 24-23 Questions 3 n 9. 1 V 3 Figure 24-23 shows four pirs of chrge prticles. For ech pir, let V t infinit n consier V net t points on the is. For which pirs is there point t which V net between the prticles n to the right of the prticles? (c) At such point is E : net ue to the prticles equl to zero? () For ech pir, re there off-is points (other thn t infinit) where V net? 4 Figure 24-24 gives the electric V potentil V s function of. nk the five regions ccoring to the mgnitue of the component of the electric fiel within them, 1 2 3 4 5 gretest first. Wht is the irection of the fiel long the is in region 2 n (c) region 4? 5 Figure 24-25 shows three pths long which we cn move the positivel Fig. 24-24 Question 4. 3 chrge sphere A closer to positivel chrge sphere B, which 1 is hel fie in plce. Woul sphere A be move to higher or B + 2 + A lower electric potentil? Is the work one b our force n (c) b the Fig. 24-25 Question 5. electric fiel ue to B positive, negtive, or zero? () nk the pths ccoring to the work our force oes, gretest first. 3 5 6 Figure 24-26 shows four rrngements of chrge prticles, ll the sme istnce from the origin. nk the situtions ccoring to the net electric potentil t the origin, most positive first. Tke the potentil to be zero t infinit. +2q 9q 3q 2q (c) () Fig. 24-26 Question 6. 7 Figure 24-27 shows sstem of three chrge prticles. If ou move the prticle of chrge q from point A to point D, re the following quntities positive, negtive, or zero: the chnge in the electric potentil energ of the three-prticle sstem, the work one b the net electrosttic force on the prticle ou move (tht is, the net force ue to the other two prticles), n (c) the work one b our force? () Wht re the nswers to through (c) if, inste, the prticle is move from B to C? +q 2q 2q 2q q 4q +2q +2q 7q A +Q B C +Q D Fig. 24-27 Questions 7 n 8. 8 In the sitution of Question 7, is the work one b our force positive, negtive, or zero if the prticle is move from A to B, from A to C, n (c) from B to D? () nk those moves ccoring to the mgnitue of the work one b our force, gretest first. 9 Figure 24-23 shows four pirs of chrge prticles with ienticl seprtions. nk the pirs ccoring to their electric potentil energ (tht is, the energ of the two-prticle sstem), gretest (most positive) first. For ech pir, if the seprtion between the prticles is increse, Q + oes the potentil energ of the pir increse or ecrese? Q 4 (full ngle) 1 In Fig. 24-28, wht is the potentil t point ue to chrge Q t istnce from? Set V t infinit. In Fig. 24-28b, the sme chrge Q hs been spre uniforml over circulr rc of rius n centrl ngle 4. Q Wht is the potentil t point, the center of curvture of the rc? (c) In Fig. 24-28c, the sme chrge Q hs been spre uniforml over circle of rius. Wht is the potentil t point, the center of the circle? () nk the three situtions ccoring to the mgnitue of the electric fiel tht is set (c) up t, gretest first. Fig. 24-28 Question 1.
648 CHATE 24 ELECTIC OTENTIAL SSM Tutoring problem vilble (t instructor s iscretion) in WileLUS n WebAssign Worke-out solution vilble in Stuent Solutions Mnul WWW Worke-out solution is t Number of ots inictes level of problem ifficult ILW Interctive solution is t Aitionl informtion vilble in The Fling Circus of hsics n t flingcircusofphsics.com http://www.wile.com/college/hlli sec. 24-3 Electric otentil 1 SSM A prticulr 12 V cr btter cn sen totl chrge of 84 A h (mpere-hours) through circuit, from one terminl to the other. How mn coulombs of chrge oes this represent? (Hint: See Eq. 21-3.) If this entire chrge unergoes chnge in electric potentil of 12 V, how much energ is involve? 2 The electric potentil ifference between the groun n clou in prticulr thunerstorm is 1.2 1 9 V. In the unit electron-volts, wht is the mgnitue of the chnge in the electric potentil energ of n electron tht moves between the groun n the clou? 3 Much of the mteril mking up Sturn s rings is in the form of tin ust grins hving rii on the orer of 1 6 m. These grins re locte in region contining ilute ionize gs, n the pick up ecess electrons. As n pproimtion, suppose ech grin is sphericl, with rius 1. 1 6 m. How mn electrons woul one grin hve to pick up to hve potentil of 4 V on its surfce (tking V t infinit)? sec. 24-5 Clculting the otentil from the Fiel 4 Two lrge, prllel, conucting pltes re 12 cm prt n hve chrges of equl mgnitue n opposite sign on their fcing surfces. An electrosttic force of 3.9 1 15 N cts on n electron plce nwhere between the two pltes. (Neglect fringing.) Fin the electric fiel t the position of the electron. Wht is the potentil ifference between the pltes? 5 SSM An infinite nonconucting sheet hs surfce chrge ensit s.1 mc/m 2 on one sie. How fr prt re equipotentil surfces whose potentils iffer b 5 V? 6 When n electron moves from A to B long n electric fiel line in Fig. 24-29, the electric fiel oes 3.94 1 19 J of work on it. Wht re the electric potentil ifferences V B V A, V C V A, n (c) V C V B? Electric fiel line B A Equipotentils Fig. 24-29 roblem 6. 7 The electric fiel in region of spce hs the components E E z n E (4. N/C). oint A is on the is t 3. m, n point B is on the is t 4. m. Wht is the potentil ifference V B V A? 8 A grph of the component of the electric fiel s function of in region of spce is shown in Fig. 24-3. The scle of the verticl is is set b E s 2. N/C. The n z components of the electric C fiel re zero in this region. If the electric potentil t the origin is 1 V, wht is the electric potentil t 2. m, wht is the gretest positive vlue of the electric potentil for points on the is for which 6. m, n (c) for wht vlue of is the electric potentil zero? B 2 E (N/C) + 1 + q A q E s E s 1 2 3 4 5 6 (m) Fig. 24-3 roblem 8. 9 An infinite nonconucting sheet hs surfce chrge ensit s 5.8 pc/m 2. How much work is one b the electric fiel ue to the sheet if prticle of chrge q 1.6 1 19 C is move from the sheet to point t istnce 3.56 cm from the sheet? If the electric potentil V is efine to be zero on the sheet, wht is V t? 1 Two uniforml chrge, infinite, nonconucting plnes re prllel to z plne n positione t 5 cm n 5 cm. The chrge ensities on the plnes re 5nC/m 2 n 25 nc/m 2, respectivel. Wht is the mgnitue of the potentil ifference between the origin n the point on the is t 8 cm? (Hint: Use Guss lw.) 11 A nonconucting sphere hs rius 2.31 cm n uniforml istribute chrge q 3.5 fc.tke the electric potentil t the sphere s center to be V. Wht is V t ril istnce r 1.45 cm n r.(hint: See Section 23-9.) sec. 24-7 otentil Due to Group of oint Chrges 12 As spce shuttle moves through the ilute ionize gs of Erth s ionosphere, the shuttle s potentil is tpicll chnge b 1. V uring one revolution.assuming the shuttle is sphere of rius 1 m, estimte the mount of chrge it collects. 13 Wht re the chrge n the chrge ensit on the surfce of conucting sphere of rius.15 m whose potentil is 2 V (with V t infinit)? 14 Consier point chrge q 1. mc, point A t istnce 1 2. m from q, n point B t istnce 2 1. m. If A n B re imetricll opposite ech other, s in Fig. 24-31, wht is the elec- B 2 Fig. 24-31 roblem 14. 1 A
OBLEMS 649 AT 3 tric potentil ifference V A V B? Wht is tht electric potentil ifference if A n B re locte s in Fig.24-31b? 15 SSM ILW A sphericl rop of wter crring chrge of 3 pc hs potentil of 5 V t its surfce (with V t infinit). Wht is the rius of the rop? If two such rops of the sme chrge n rius combine to form single sphericl rop, wht is the potentil t the surfce of the new rop? 16 Figure 24-32 shows rectngulr rr of chrge +2q 1 +4q 2 3q 1 prticles fie in plce, with istnce 39. cm n the chrges shown s integer multiples of q 1 3.4 pc n q 2 6. pc. With V t infinit, wht is the net electric poten- q 1 +4q 2 +2q 1 til t the rectngle s center? (Hint: Thoughtful emintion cn reuce Fig. 24-32 roblem 16. the clcultion.) 17 In Fig. 24-33, wht is the net electric potentil t point ue to the four prticles if V t infinit, q 5. fc, n 4. cm? + +q +q + 18 Two chrge prticles re shown in Fig. 24-34. rticle 1, with chrge q 1, is fie in plce t istnce. rticle 2, with chrge q 2, cn be move long the is. Figure 24-34b gives the net electric potentil V t the origin ue to the two prticles s function of the coorinte of prticle 2.The scle of the is is set b s 16. cm. The plot hs n smptote of V 5.76 1 7 V s :.Wht is q 2 in terms of e? 1 2 q q Fig. 24-33 roblem 17. V (1 7 V) 4 1 Fig. 24-34 roblem 18. s (cm) electric potentil efine to be V t infinit, wht re the finite positive n negtive vlues of t which the net electric potentil on the is is zero? 2 Two prticles, of chrges q 1 n q 2, re seprte b istnce in Fig. 24-35. The net electric fiel ue to the prticles is zero t /4. With V t infinit, locte (in terms of ) n point on the is (other thn t infinit) t which the electric potentil ue to the two prticles is zero. sec. 24-8 otentil Due to n Electric Dipole 21 ILW The mmoni molecule NH 3 hs permnent electric ipole moment equl to 1.47 D, where 1 D 1 ebe unit 3.34 1 3 C m. Clculte the electric potentil ue to n mmoni molecule t point 52. nm w long the is of the ipole. (Set V t infinit.) 22 In Fig. 24-36, prticle of elementr chrge e is initill t coorinte z 2 nm on the ipole is (here z is) through n electric ipole, on the positive sie of the ipole. (The origin of z is t the center of the ipole.) The prticle is then move long circulr pth roun the ipole center until it is t coorinte z 2 nm, on the negtive sie of the ipole is. Figure 24-36b gives the work W one b the force moving the prticle versus the ngle u tht loctes the prticle reltive to the positive irection of the z is.the scle of the verticl is is set b W s 4. 1 3 J.Wht is the mgnitue of the ipole moment? +e + z θ W (1 3 J) Fig. 24-36 roblem 22. sec. 24-9 otentil Due to Continuous Chrge Distribution 23 Figure 24-37 shows nonconucting ro of length L 6. cm n uniform liner chrge ensit l 3.68 pc/m. Assume tht the electric potentil is efine to be V t infinit. Wht is V t point t istnce 8. cm long the ro s perpeniculr bisector? Figure 24-37b shows n ienticl ro ecept tht one hlf is now negtivel chrge. Both hlves hve liner chrge ensit of mgnitue 3.68 pc/m.with V t infinit, wht is V t? W s θ 19 In Fig. 24-35, prticles with the chrges q 1 5e n q 2 15e re fie in plce with seprtion of 24. cm.with q q 1 2 Fig. 24-35 roblems 19, 2, n 97. +++++++++++++ +++++++ L/2 L/2 L/2 L/2 Fig. 24-37 roblem 23.
65 CHATE 24 ELECTIC OTENTIAL 24 In Fig. 24-38, plstic ro hving uniforml istribute chrge Q 25.6 pc hs been bent into circulr rc of rius 3.71 cm n centrl ngle f 12.With V t infinit, wht is the electric potentil t, the center of curvture of the ro? 25 A plstic ro hs been bent into circle of rius 8.2 cm. It hs chrge Q 1 4.2 pc uniforml istribute long onequrter of its circumference n chrge Q 2 6Q 1 uniforml istribute long the rest of the circumference (Fig. 24-39). With V t infinit, wht is the electric potentil t the center C of the circle n point, on the centrl is of the circle t istnce D 6.71 cm from the center? Q 2 C 26 Figure 24-4 shows thin ro with uniform chrge ensit of 2. mc/m. Evlute the electric potentil t point if D L/4.. D o L 27 In Fig. 24-41, three thin plstic ros form qurter-circles with common center of curvture t the origin. Fig. 24-4 roblem 26. The uniform chrges on the ros re Q 1 3 nc, Q 2 3.Q 1, n Q 3 8.Q 1.Wht is the net electric potentil t the origin ue to the ros? D Q 1 Fig. 24-39 roblem 25. φ Q Fig. 24-38 roblem 24. 29 In Fig. 24-43, wht is the net electric potentil t the origin ue to the circulr rc of chrge Q 1 7.21 pc n the two prticles of chrges Q 2 4.Q 1 n Q 3 2.Q 1? The rc s center of curvture is t the origin n its rius is 2. m; the ngle inicte is u 2.. 3 Q 1 Q 3 2. θ Fig. 24-43 roblem 29. The smiling fce of Fig. 24-44 consists of three items: 1. thin ro of chrge 3. mc tht forms full circle of rius 6. cm; 2. secon thin ro of chrge 2. mc tht forms circulr rc of rius 4. cm, subtening n ngle of 9 bout the center of the full circle; 3. n electric ipole with ipole moment tht is perpeniculr to ril line n hs mgnitue 1.28 1 21 C m. Wht is the net electric potentil t the center? Q 2 (cm) 4. Q 2 2. Q 3 1. Q 1 (cm) Fig. 24-41 roblem 27. 28 Figure 24-42 shows thin plstic ro of length L 12. cm n uniform positive chrge Q 56.1 fc ling on n is.with V t infinit, fin the electric potentil t point 1 on the is, t istnce 2.5 cm from one en of the ro. D 1 2 ++++++++++ L Fig. 24-42 roblems 28, 33, 38, n 4. Fig. 24-44 roblem 3. 31 SSM WWW A plstic isk of rius 64. cm is chrge on one sie with uniform surfce chrge ensit s 7.73 fc/m 2, D n then three qurnts of the isk re remove. The remining qurnt is shown in Fig. 24-45. With V t infinit, wht is the potentil ue to the remining qurnt t point, which is on the Fig. 24-45 roblem 31. centrl is of the originl isk t istnce D 25.9 cm from the originl center? 32 A nonuniform liner chrge istribution given b l b, where b is constnt, is locte long n is from to.2 m. If b 2 nc/m 2 n V t infinit, wht is the electric potentil t the origin n the point.15 m on the is? 33 The thin plstic ro shown in Fig. 24-42 hs length L 12. cm n nonuniform liner chrge ensit l c, where c 28.9
OBLEMS 651 AT 3 pc/m 2.With V t infinit, fin the electric potentil t point 1 on the is, t istnce 3. cm from one en. V (V) V (V) the kinetic energ of the prticle t the instnt it hs move 4 cm if Q 2 mc n Q 2 mc? 42 Wht is the electric potentil energ of two electrons seprte b 2. nm? If the seprtion increses, oes the potentil energ increse or ecrese? +q q 43 SSM ILW WWW How much work is + require to set up the rrngement of Fig. 24-47 if q 2.3 pc, 64. cm, n the prticles re initill infinitel fr prt n t rest? sec. 24-1 Clculting the Fiel from the otentil 34 Two lrge prllel metl pltes re 1.5 cm prt n hve chrges of equl mgnitues but opposite signs on their fcing surfces. Tke the potentil of the negtive plte to be zero. If the potentil hlfw between the pltes is then 5. V, wht is the electric fiel in the region between the pltes? 35 The electric potentil t points in n plne is given b V (2. V/m 2 ) 2 (3. V/m 2 ) 2. In unit-vector nottion, wht is the 44 In Fig. 24-48, seven chrge prticles re electric fiel t the point (3. m, 2. m)? fie in plce to form squre with n ege + length of 4. cm. How much work must we o q +q 36 The electric potentil V in the spce between two flt prllel to bring prticle of chrge 6e initill t pltes 1 n 2 is given (in volts) b V 15 2, where (in meters) Fig. 24-47 rest from n infinite istnce to the center of is the perpeniculr istnce from plte 1. At 1.3 cm, wht roblem 43. the squre? is the mgnitue of the electric fiel n is the fiel irecte towr or w from plte 1? 37 SSM Wht is the mgnitue of the electric fiel t the point (3.î 2.ĵ 4.kˆ ) m if the electric potentil is given b V 2.z 2 2e, where V is in volts n,, n z re in meters? e 3e 38 Figure 24-42 shows thin plstic ro of length L 13.5 cm n uniform chrge 43.6 fc. In terms of istnce, fin n epression for the electric potentil t point 1. Net, substitute +2e vrible for n fin n epression for the mgnitue of the +3e component E of the electric fiel t 1. (c) Wht is the irection of +3e +e E reltive to the positive irection of the is? () Wht is the vlue of E t 1 for 6.2 cm? (e) From the smmetr in Fig. 24-48 roblem 44. Fig. 24-42, etermine E t 1. 45 ILW A prticle of chrge q is fie t point, n secon 39 An electron is plce in n plne where the electric potentil epens on n s shown in Fig. 24-46 (the potentil prticle of mss m n the sme chrge q is initill hel istnce r oes not epen on z). The scle of the verticl is is set b V s 1 from.the secon prticle is then relese. Determine its spee when it is istnce r 5 V. In unit-vector nottion, wht is the electric force on the 2 from. Let q 3.1 mc, m 2 mg, r 1.9 mm, n r electron? 2 2.5 mm. 46 A chrge of 9. nc is uniforml istribute roun thin plstic ring ling in z plne with the ring center t the origin. A V s V s 6. pc point chrge is locte on the is t 3. m. For ring rius of 1.5 m, how much work must n eternl force o on the point chrge to move it to the origin?.2 47 Wht is the escpe spee for n electron initill t rest.2.4.4 on the surfce of sphere with rius of 1. cm n uniforml istribute chrge of 1.6 1 15 C? Tht is, wht initil spee must the electron hve in orer to rech n infinite istnce from the V s V s sphere n hve zero kinetic energ when it gets there? (m) (m) 48 A thin, sphericl, conucting shell of rius is mounte Fig. 24-46 roblem 39. on n isolting support n chrge to potentil of 125 V. An electron is then fire irectl towr the center of the shell, from point t istnce r from the center of the shell (r ). Wht 4 The thin plstic ro of length L 1. cm in Fig. 24-42 hs initil spee v is neee for the electron to just rech the shell nonuniform liner chrge ensit l c, where c 49.9 pc/m 2. before reversing irection? With V t infinit, fin the electric potentil t point 2 on 49 Two electrons re fie the is t D 3.56 cm. Fin the electric fiel component 2. cm prt. Another electron is q 1 Q E t 2. (c) Wh cnnot the fiel component E t 2 be foun using the result of? between the two. Wht is its initil shot from infinit n stops miw + 2. + θ1 θ2 sec. 24-11 Electric otentil Energ of spee? Sstem of oint Chrges 5 In Fig. 24-49, how much work q 2 41 A prticle of chrge 7.5 mc is relese from rest t the must we o to bring prticle, of point 6 cm on n is. The prticle begins to move ue to chrge Q 16e n initill t rest, the presence of chrge Q tht remins fie t the origin. Wht is long the she line from infinit to Fig. 24-49 roblem 5.
652 CHATE 24 ELECTIC OTENTIAL the inicte point ner two fie prticles of chrges q 1 4e n q 2 q 1 /2? Distnce 1.4 cm, u 1 43, n u 2 6. 51 In the rectngle of Fig. 24-5, the sies hve lengths 5. cm n 15 cm, q 1 5. mc, n q 2 2. mc.with V t infinit, wht is the electric potentil t corner A n corner B? (c) How much work is require to move chrge q 3 3. mc from B to A long igonl of the rectngle? () Does this work increse or ecrese the electric potentil energ of the threechrge sstem? Is more, less, or the sme work require if q 3 is move long pth tht is (e) insie the rectngle but not on igonl n (f) outsie the rectngle? 52 Figure 24-51 shows n electron moving long n electric ipole is towr the negtive sie of the ipole. The ipole is fie in plce. The electron ws initill ver fr from the ipole, with kinetic energ 1 ev. Figure 24-51b gives the kinetic energ K of the electron versus its istnce r from the ipole center. The scle of the horizontl is is set b r s.1 m. Wht is the mgnitue of the ipole moment? q 1 B A + q 2 Fig. 24-5 roblem 51. 55 An electron is projecte with n initil spee of 3.2 1 5 m/s irectl towr proton tht is fie in plce. If the electron is initill gret istnce from the proton, t wht istnce from the proton is the spee of the electron instntneousl equl to twice the initil vlue? 56 Figure 24-53 shows three prticles on n is. rticle 1 (with chrge of 5. mc) n prticle 2 (with chrge of 3. mc) re fie in plce with seprtion 4. cm. rticle 3 cn be move long the is to the right of prticle 2. Figure 24-53b gives the electric potentil energ U of the three-prticle sstem s function of the coorinte of prticle 3. The scle of the verticl is is set b U s 5. J. Wht is the chrge of prticle 3? 1 2 3 U (J) U s U s Fig. 24-53 roblem 56. 1 2 (cm) K (ev) 1 5 + e r (m) Fig. 24-51 roblem 52. 53 Two tin metl spheres A n B, mss m A 5. g n m B 1. g, hve equl positive chrge q 5. mc. The spheres re connecte b mssless nonconucting string of length 1. m, which is much greter thn the rii of the spheres. Wht is the electric potentil energ of the sstem? Suppose ou cut the string. At tht instnt, wht is the ccelertion of ech sphere? (c) A long time fter ou cut the string, wht is the spee of ech sphere? 54 A positron (chrge e, mss equl to the electron mss) is moving t 1. 1 7 m/s in the positive irection of n is when, t, it encounters n electric fiel irecte long the is. The electric potentil V ssocite with the fiel is given in Fig. 24-52. The scle of the verticl is is set b V s 5. V. Does the positron emerge from the fiel t (which mens its motion is reverse) or t.5 m (which mens its motion is not reverse)? Wht is its spee when it emerges? r s 57 SSM Ienticl 5 mc chrges re fie on n is t 3. m.a prticle of chrge q 15 mc is then relese from rest t point on the positive prt of the is. Due to the smmetr of the sitution, the prticle moves long the is n hs kinetic energ 1.2 J s it psses through the point, 4. m. Wht is the kinetic energ of the prticle s it psses through the origin? At wht negtive vlue of will the prticle momentril stop? 58 roton in well. Figure 24-54 shows electric potentil V long n is.the scle of the verticl is is set b V s 1. V. A proton is to be relese t 3.5 cm with initil kinetic energ 4. ev. If it is initill moving in the negtive irection of the is, oes it rech turning point (if so, wht is the coorinte of tht point) or oes it escpe from the plotte region (if so, wht is its spee t )? If it is initill moving in the positive irection of the is, oes it rech turning point (if so, wht is the coorinte of tht point) or oes it escpe from the plotte region (if so, wht is its spee t 6. cm)? Wht re the (c) mgnitue F n () irection (positive or negtive irection of the is) of the electric force on the proton if the proton moves just to the left of 3. cm? Wht re (e) F n (f) the irection if the proton moves just to the right of 5. cm? V (V) V s V (V) V s 1 2 3 4 5 6 7 Fig. 24-54 roblem 58. (cm) 2 5 (cm) Fig. 24-52 roblem 54. 59 In Fig. 24-55, chrge prticle (either n electron or proton) is moving rightwr between two prllel chrge pltes seprte b istnce 2. mm. The plte potentils re V 1 7. V n V 2 5. V. The prticle is slowing from n initil
OBLEMS 653 AT 3 spee of 9. km/s t the left plte. Is the prticle n electron or proton? Wht is its spee just s it reches plte 2? 6 In Fig. 24-56, we move n electron from n infinite istnce to point t istnce 8. cm from tin chrge bll. The move requires work W 2.16 1 13 J b us. Wht is the chrge Q on the bll? In Fig. 24-56b, the bll hs been slice up n the slices spre out so tht n equl mount of chrge is t the hour positions on circulr clock fce of rius 8. cm. Now the electron is brought from n infinite istnce to the center of the circle. With tht ition of the electron to the sstem of 12 chrge prticles, wht is the chnge in the electric potentil energ of the sstem? e e Q V 1 V 2 Fig. 24-55 roblem 59. 61 Suppose N electrons cn be plce in either of two configurtions. In configurtion 1, the re ll plce on the circumference of nrrow ring of rius n re uniforml istribute so tht the istnce between jcent electrons is the sme everwhere. In configurtion 2, N 1 electrons re uniforml istribute on the ring n one electron is plce in the center of the ring. Wht is the smllest vlue of N for which the secon configurtion is less energetic thn the first? For tht vlue of N, consier n one circumference electron cll it e. How mn other circumference electrons re closer to e thn the centrl electron is? sec. 24-12 otentil of Chrge Isolte Conuctor 62 Sphere 1 with rius 1 hs positive chrge q. Sphere 2 with rius 2. 1 is fr from sphere 1 n initill unchrge.after the seprte spheres re connecte with wire thin enough to retin onl negligible chrge, is potentil V 1 of sphere 1 greter thn, less thn, or equl to potentil V 2 of sphere 2? Wht frction of q ens up on sphere 1 n (c) sphere 2? () Wht is the rtio s 1 /s 2 of the surfce chrge ensities of the spheres? 63 SSM WWW Two metl spheres, ech of rius 3. cm, hve center-to-center seprtion of 2. m. Sphere 1 hs chrge 1. 1 8 C; sphere 2 hs chrge 3. 1 8 C. Assume tht the seprtion is lrge enough for us to s tht the chrge on ech sphere is uniforml istribute (the spheres o not ffect ech other). With V t infinit, clculte the potentil t the point hlfw between the centers n the potentil on the surfce of sphere 1 n (c) sphere 2. 64 A hollow metl sphere hs potentil of 4 V with respect to groun (efine to be t V ) n chrge of 5. 1 9 C. Fin the electric potentil t the center of the sphere. Fig. 24-56 roblem 6. 65 SSM Wht is the ecess chrge on conucting sphere of rius r.15 m if the potentil of the sphere is 15 V n V t infinit? 66 Two isolte, concentric, conucting sphericl shells hve rii 1.5 m n 2 1. m, uniform chrges q 1 2. mc n q 2 1. mc, n negligible thicknesses. Wht is the mgnitue of the electric fiel E t ril istnce r 4. m, r.7 m, n (c) r.2 m? With V t infinit, wht is V t () r 4. m, (e) r 1. m, (f) r.7 m, (g) r.5 m, (h) r.2 m, n (i) r? ( j) Sketch E(r) n V(r). 67 A metl sphere of rius 15 cm hs net chrge of 3. 1 8 C. Wht is the electric fiel t the sphere s surfce? If V t infinit, wht is the electric potentil t the sphere s surfce? (c) At wht istnce from the sphere s surfce hs the electric potentil ecrese b 5 V? Aitionl roblems 68 Here re the chrges n coorintes of two point chrges locte in n plne: q 1 3. 1 6 C, 3.5 cm,.5 cm n q 2 4. 1 6 C, 2. cm, 1.5 cm. How much work must be one to locte these chrges t their given positions, strting from infinite seprtion? 69 SSM A long, soli, conucting cliner hs rius of 2. cm. The electric fiel t the surfce of the cliner is 16 N/C, irecte rill outwr. Let A, B, n C be points tht re 1. cm, 2. cm, n 5. cm, respectivel, from the centrl is of the cliner. Wht re the mgnitue of the electric fiel t C n the electric potentil ifferences V B V C n (c) V A V B? 7 The chocolte crumb mster. This stor begins with roblem 6 in Chpter 23. From the nswer to prt of tht problem, fin n epression for the electric potentil s function of the ril istnce r from the center of the pipe. (The electric potentil is zero on the groune pipe wll.) For the tpicl volume chrge ensit r 1.1 1 3 C/m 3, wht is the ifference in the electric potentil between the pipe s center n its insie wll? (The stor continues with roblem 6 in Chpter 25.) 71 SSM Strting from Eq. 24-3, erive n epression for the electric fiel ue to ipole t point on the ipole is. 72 The mgnitue E of n electric fiel epens on the ril istnce r ccoring to E A/r 4, where A is constnt with the unit volt cubic meter. As multiple of A, wht is the mgnitue of the electric potentil ifference between r 2. m n r 3. m? 73 If n isolte conucting sphere 1 cm in rius hs net chrge of 4. mc n if V t infinit, wht is the potentil on the surfce of the sphere? Cn this sitution ctull occur, given tht the ir roun the sphere unergoes electricl brekown when the fiel ecees 3. MV/m? q 3 74 Three prticles, chrge q 1 1 mc, q 2 2 mc, n q 3 3mC, re positione t the vertices of n isosceles tringle s shown in Fig. 24-57. If 1 cm n b 6. cm, how much work must n eternl gent o to echnge the positions of q 1 n q 3 n, inste, q 1 n q 2? 75 An electric fiel of pproimtel 1 V/m is often observe ner the surfce of Erth. If this were the fiel over the entire q 2 b Fig. 24-57 roblem 74. q 1
654 CHATE 24 ELECTIC OTENTIAL surfce, wht woul be the electric potentil of point on the surfce? (Set V t infinit.) 76 A Gussin sphere of rius 4. cm is centere on bll tht hs rius of 1. cm n uniform chrge istribution. The totl (net) electric flu through the surfce of the Gussin sphere is 5.6 1 4 N m 2 /C. Wht is the electric potentil 12. cm from the center of the bll? 77 In Millikn oil-rop eperiment (Section 22-8), uniform electric fiel of 1.92 1 5 N/C is mintine in the region between two pltes seprte b 1.5 cm. Fin the potentil ifference between the pltes. 78 Figure 24-58 shows three circulr, nonconucting rcs of rius 8.5 cm. The chrges on the rcs re q 1 4.52 pc, q 2 2.q 1, q 3 3.q 1. With V t infinit, wht is the net electric potentil of the rcs t the common center of curvture? q 1 45. 45. q 2 q 3 Fig. 24-58 roblem 78. 79 An electron is relese from rest on the is of n electric ipole tht hs chrge e n chrge seprtion 2 pm n tht is fie in plce.the relese point is on the positive sie of the ipole, t istnce 7. from the ipole center. Wht is the electron s spee when it reches point 5. from the ipole center? 8 Figure 24-59 shows ring of outer rius 13. cm, inner rius r.2, n uniform surfce chrge ensit s 6.2 pc/m 2. With V t infinit, fin the electric potentil t point on the centrl is of the ring, t istnce z 2. from the center of the ring. Fig. 24-59 roblem 8. z r 81 Electron in well. Figure 24-6 shows electric potentil V long n is. The scle of the verticl is is set b V s 8. V. An electron is to be relese t 4.5 cm with initil kinetic energ 3. ev. If it is initill moving in the negtive irection of the is, oes it rech turning point (if so, wht is the coorinte of tht point) or oes it escpe from the plotte region (if so, wht is its spee t )? If it is initill moving in the positive irection of the is, oes it rech turning point (if so, wht is the coorinte of σ tht point) or oes it escpe from the plotte region (if so, wht is its spee t 7. cm)? Wht re the (c) mgnitue F n () irection (positive or negtive irection of the is) of the electric force on the electron if the electron moves just to the left of 4. cm? Wht re (e) F n (f) the irection if it moves just to the right of 5. cm? V (V) V s 1 2 3 4 5 6 7 Fig. 24-6 roblem 81. (cm) 82 If Erth h uniform surfce chrge ensit of 1. electron/m 2 ( ver rtificil ssumption), wht woul its potentil be? (Set V t infinit.) Wht woul be the mgnitue n (c) irection (rill inwr or outwr) of the electric fiel ue to Erth just outsie its surfce? 83 In Fig. 24-61, point is t istnce 1 4. m from prticle 1 1 q 1 (q 1 2e) n istnce 2 2. m 2 from prticle 2 (q 2 2e), with both prticles fie in plce. With V t infinit, wht is V t? If we bring prticle of chrge q 3 2e from infinit Fig. 24-61 q 2 roblem 83. to, how much work o we o n (c) wht is the poten- til energ of the three-prticle stem? 84 A soli conucting sphere of rius 3. cm hs chrge of 3 nc istribute uniforml over its surfce. Let A be point 1. cm from the center of the sphere, S be point on the surfce of the sphere, n B be point 5. cm from the center of the sphere.wht re the electric potentil ifferences V S V B n V A V B? 85 In Fig. 24-62, we move prticle of chrge 2e in from infinit to the is.how much work o we o? Distnce D is 4. m. +2e D +e D +2e Fig. 24-62 roblem 85. 86 Figure 24-63 shows hemisphere with chrge of 4. mc is- tribute uniforml through its volume. The hemisphere lies on n plne the w hlf grpefruit might lie fce own on kitchen Fig. 24-63 roblem 86. tble. oint is locte on the plne, long ril line from the hemisphere s center of curvture, t ril istnce 15 cm. Wht is the electric potentil t point ue to the hemisphere? 87 SSM Three.12 C chrges form n equilterl tringle 1.7 m on sie. Using energ supplie t the rte of.83 kw, how mn s woul be require to move one of the chrges to the mipoint of the line joining the other two chrges?
OBLEMS 655 AT 3 88 Two chrges q 2. mc re C fie istnce 2. cm prt (Fig. 24-64). With V t infinit, wht is the electric potentil t /2 point C? You bring thir + /2 /2 + chrge q 2. mc from infinit to q q C. How much work must ou o? (c) Wht is the potentil energ U of Fig. 24-64 roblem 88. the three-chrge configurtion when the thir chrge is in plce? 89 Initill two electrons re fie in plce with seprtion of 2. mm. How much work must we o to bring thir electron in from infinit to complete n equilterl tringle? 9 A prticle of positive chrge Q is fie t point. A secon prticle of mss m n negtive chrge q moves t constnt spee in circle of rius r 1, centere t. Derive n epression for the work W tht must be one b n eternl gent on the secon prticle to increse the rius of the circle of motion to r 2. 91 Two chrge, prllel, flt conucting surfces re spce 1. cm prt n prouce potentil ifference V 625 V between them. An electron is projecte from one surfce irectl towr the secon. Wht is the initil spee of the electron if it stops just t the secon surfce? +q 1 q 2 q 3 92 In Fig. 24-65, point is t the center of the rectngle. With V t infinit, q 1 5. fc, q 2 2. fc, q 3 3. fc, n 2.54 cm, wht is the net electric potentil t ue to the si chrge prticles? +q 3 q 2 +q 1 93 SSM A uniform chrge of Fig. 24-65 roblem 92. 16. mc is on thin circulr ring ling in n plne n centere on the origin. The ring s rius is 3. cm. If point A is t the origin n point B is on the z is t z 4. cm, wht is V B V A? 94 Consier point chrge q 1.5 1 8 C, n tke V t infinit. Wht re the shpe n imensions of n equipotentil surfce hving potentil of 3. V ue to q lone? Are surfces whose potentils iffer b constnt mount (1. V, s) evenl spce? 95 SSM A thick sphericl shell of chrge Q n uniform volume chrge ensit r is boune b rii r 1 n r 2 r 1.With V t infinit, fin the electric potentil V s function of istnce r from the center of the istribution, consiering regions r r 2, r 2 r r 1, n (c) r r 1. () Do these solutions gree with ech other t r r 2 n r r 1? (Hint: See Section 23-9.) 96 A chrge q is istribute uniforml throughout sphericl volume of rius. Let V t infinit. Wht re V t ril istnce r n the potentil ifference between points t r n the point t r? 97 Figure 24-35 shows two chrge prticles on n is. Sketch the electric fiel lines n the equipotentil surfces in the plne of the pge for q 1 q, q 2 2q n q 1 q, q 2 3q. 98 Wht is the electric potentil energ of the chrge configur- tion of Fig. 24-8? Use the numericl vlues provie in the ssocite smple problem. 99 Using Eq. 24-32, show tht the electric potentil t point on the centrl is of thin ring (of chrge q n rius ) n t istnce z from the ring is V 1 q 4 z 2. 2 From this result, erive n epression for the electric fiel mgnitue E t points on the ring s is; compre our result with the clcultion of E in Section 22-6. 1 An lph prticle (which hs two protons) is sent irectl towr trget nucleus contining 92 protons. The lph prticle hs n initil kinetic energ of.48 pj. Wht is the lest center-to-center istnce the lph prticle will be from the trget nucleus, ssuming the nucleus oes not move? 11 In the qurk moel of funmentl prticles, proton is compose of three qurks: two up qurks, ech hving chrge 2e/3, n one own qurk, hving chrge e/3. Suppose tht the three qurks re equiistnt from one nother. Tke tht seprtion istnce to be 1.32 1 15 m n clculte the electric potentil energ of the sstem of onl the two up qurks n ll three qurks. 12 A proton of kinetic energ 4.8 MeV trvels he-on towr le nucleus.assuming tht the proton oes not penetrte the nucleus n tht the onl force between proton n nucleus is the Coulomb force, clculte the smllest center-to-center seprtion p between proton n nucleus when the proton momentril stops. If the proton were replce with n lph prticle (which contins two protons) of the sme initil kinetic energ, the lph prticle woul stop t center-to-center seprtion. Wht is / p? 13 In Fig. 24-66, two prticles of 1 2 chrges q 1 n q 2 re fie to n is. If thir prticle, of chrge 1.5 6. mc, is brought from n infinite istnce to point, the three- Fig. 24-66 roblem 13. prticle sstem hs the sme electric potentil energ s the originl two-prticle sstem.wht is the chrge rtio q 1 /q 2? 14 A chrge of 1.5 1 8 C lies on n isolte metl sphere of rius 16. cm. With V t infinit, wht is the electric potentil t points on the sphere s surfce? 15 SSM A soli copper sphere whose rius is 1. cm hs ver thin surfce coting of nickel. Some of the nickel toms re rioctive, ech tom emitting n electron s it ecs. Hlf of these electrons enter the copper sphere, ech epositing 1 kev of energ there. The other hlf of the electrons escpe, ech crring w chrge e. The nickel coting hs n ctivit of 3.7 1 8 rioctive ecs per secon. The sphere is hung from long, nonconucting string n isolte from its surrounings. How long will it tke for the potentil of the sphere to increse b 1 V? How long will it tke for the temperture of the sphere to increse b 5. K ue to the energ eposite b the electrons? The het cpcit of the sphere is 14 J/K.