Electric Potential. chapter

Transcription

1 chpte 25 Electic Potentil 25.1 Electic Potentil n Potentil Diffeence 25.2 Potentil Diffeence in Unifom Electic Fiel 25.3 Electic Potentil n Potentil Enegy Due to Point Chges 25.4 Otining the Vlue of the Electic Fiel fom the Electic Potentil 25.5 Electic Potentil Due to Continuous Chge Distiutions 25.6 Electic Potentil Due to Chge Conucto 25.7 The Millikn Oil-Dop Epeiment 25.8 Applictions of Electosttics In Chpte 23, we linke ou new stuy of electomgnetism to ou elie stuies of foce. Now we mke new link to ou elie investigtions into enegy. The concept of potentil enegy ws intouce in Chpte 7 in connection with such consevtive foces s the gvittionl foce n the elstic foce eete Pocesses occuing uing thunestoms cuse lge iffeences in y sping. By using the lw of consevtion electic potentil etween thuneclou n the goun. The esult of this of enegy, we coul solve vious polems in potentil iffeence is n electicl ischge tht we cll lightning, such s this isply ove Tucson, Aizon. (Keith Kent/Photo Reseches, Inc.) mechnics tht wee insolule with n ppoch using foces. The concept of potentil enegy is lso of get vlue in the stuy of electicity. Becuse the electosttic foce is consevtive, electosttic phenomen cn e conveniently escie in tems of n electic potentil enegy. This ie enles us to efine quntity known s electic potentil. Becuse the electic potentil t ny point in n electic fiel is scl quntity, we cn use it to escie electosttic phenomen moe simply thn if we wee to ely only on the electic fiel n electic foces. The concept of electic potentil is of get pcticl vlue in the opetion of electic cicuits n evices tht we will stuy in lte chptes. 710

2 25.1 Electic Potentil n Potentil Diffeence Electic Potentil n Potentil Diffeence When test chge q 0 is plce in n electic fiel E cete y some souce chge istiution, the electic foce cting on the test chge is q 0 E. The foce F e 5 q 0 E is consevtive ecuse the foce etween chges escie y Coulom s lw is consevtive. When the test chge is move in the fiel y some etenl gent, the wok one y the fiel on the chge is equl to the negtive of the wok one y the etenl gent cusing the isplcement. This sitution is nlogous to tht of lifting n oject with mss in gvittionl fiel: the wok one y the etenl gent is mgh, n the wok one y the gvittionl foce is 2mgh. When nlyzing electic n mgnetic fiels, it is common pctice to use the nottion s to epesent n infinitesiml isplcement vecto tht is oiente tngent to pth though spce. This pth my e stight o cuve, n n integl pefome long this pth is clle eithe pth integl o line integl (the two tems e synonymous). Fo n infinitesiml isplcement s of point chge q 0 immese in n electic fiel, the wok one within the chge fiel system y the electic fiel on the chge is W int 5 F e? s 5 q 0 E? s. As this mount of wok is one y the fiel, the potentil enegy of the chge fiel system is chnge y n mount U 52W int 52q 0 E? s. Fo finite isplcement of the chge fom point to point, the chnge in potentil enegy of the system DU 5 U 2 U is DU 52q 0 3 E? s (25.1) The integtion is pefome long the pth tht q 0 follows s it moves fom to. Becuse the foce q 0 E is consevtive, this line integl oes not epen on the pth tken fom to. Fo given position of the test chge in the fiel, the chge fiel system hs potentil enegy U eltive to the configution of the system tht is efine s U 5 0. Diviing the potentil enegy y the test chge gives physicl quntity tht epens only on the souce chge istiution n hs vlue t evey point in n electic fiel. This quntity is clle the electic potentil (o simply the potentil) V : V 5 U q 0 (25.2) Becuse potentil enegy is scl quntity, electic potentil lso is scl quntity. As escie y Eqution 25.1, if the test chge is move etween two positions n in n electic fiel, the chge fiel system epeiences chnge in potentil enegy. The potentil iffeence DV 5 V 2 V etween two points n in n electic fiel is efine s the chnge in potentil enegy of the system when test chge q 0 is move etween the points ivie y the test chge: Chnge in electic potentil enegy of system Pitfll Pevention 25.1 Potentil n Potentil Enegy The potentil is chcteistic of the fiel only, inepenent of chge test pticle tht my e plce in the fiel. Potentil enegy is chcteistic of the chge-fiel system ue to n intection etween the fiel n chge pticle plce in the fiel. DV ; DU 52 q 3 0 E? s (25.3) In this efinition, the infinitesiml isplcement s is intepete s the isplcement etween two points in spce the thn the isplcement of point chge s in Eqution Just s with potentil enegy, only iffeences in electic potentil e meningful. We often tke the vlue of the electic potentil to e zeo t some convenient point in n electic fiel. Potentil iffeence shoul not e confuse with iffeence in potentil enegy. The potentil iffeence etween n eists solely ecuse of souce chge n epens on the souce chge istiution (consie points n without the pesence of the test chge). Fo potentil enegy to eist, we must hve Potentil iffeence etween two points

3 712 CHAPTER 25 Electic Potentil Pitfll Pevention 25.2 Voltge A viety of phses e use to escie the potentil iffeence etween two points, the most common eing voltge, ising fom the unit fo potentil. A voltge pplie to evice, such s television, o coss evice is the sme s the potentil iffeence coss the evice. Despite popul lnguge, voltge is not something tht moves though evice. Pitfll Pevention 25.3 The Electon Volt The electon volt is unit of enegy, NOT of potentil. The enegy of ny system my e epesse in ev, ut this unit is most convenient fo esciing the emission n soption of visile light fom toms. Enegies of nucle pocesses e often epesse in MeV. E Figue 25.1 (Quick Quiz 25.1) Two points in n electic fiel. system of two o moe chges. The potentil enegy elongs to the system n chnges only if chge is move eltive to the est of the system. If n etenl gent moves test chge fom to without chnging the kinetic enegy of the test chge, the gent pefoms wok tht chnges the potentil enegy of the system: W 5 DU. Imgine n ity chge q locte in n electic fiel. Fom Eqution 25.3, the wok one y n etenl gent in moving chge q though n electic fiel t constnt velocity is W 5 q DV (25.4) Becuse electic potentil is mesue of potentil enegy pe unit chge, the I unit of oth electic potentil n potentil iffeence is joules pe coulom, which is efine s volt (V): 1 V ; 1 J/C Tht is, 1 J of wok must e one to move 1-C chge though potentil iffeence of 1 V. Eqution 25.3 shows tht potentil iffeence lso hs units of electic fiel times istnce. It follows tht the I unit of electic fiel (N/C) cn lso e epesse in volts pe mete: 1 N/C 5 1 V/m Theefoe, we cn intepet the electic fiel s mesue of the te of chnge of the electic potentil with espect to position. A unit of enegy commonly use in tomic n nucle physics is the electon volt (ev), which is efine s the enegy chge fiel system gins o loses when chge of mgnitue e (tht is, n electon o poton) is move though potentil iffeence of 1 V. Becuse 1 V 5 1 J/C n the funmentl chge is equl to C, the electon volt is elte to the joule s follows: 1 ev C? V J (25.5) Fo instnce, n electon in the em of typicl entl -y mchine my hve spee of m/s. This spee coespons to kinetic enegy J (using eltivistic clcultions s iscusse in Chpte 39), which is equivlent to ev. uch n electon hs to e ccelete fom est though potentil iffeence of 67 kv to ech this spee. Quick Quiz 25.1 In Figue 25.1, two points n e locte within egion in which thee is n electic fiel. (i) How woul you escie the potentil iffeence DV 5 V 2 V? () It is positive. () It is negtive. (c) It is zeo. (ii) A negtive chge is plce t n then move to. How woul you escie the chnge in potentil enegy of the chge fiel system fo this pocess? Choose fom the sme possiilities Potentil Diffeence in Unifom Electic Fiel Potentil iffeence etween two points in unifom electic fiel Equtions 25.1 n 25.3 hol in ll electic fiels, whethe unifom o vying, ut they cn e simplifie fo unifom fiel. Fist, consie unifom electic fiel iecte long the negtive y is s shown in Active Figue Let s clculte the potentil iffeence etween two points n septe y istnce, whee the isplcement s points fom tow n is pllel to the fiel lines. Eqution 25.3 gives V 2 V 5DV52 3 E? s 52 3 E s 1cos E s Becuse E is constnt, it cn e emove fom the integl sign, which gives DV 52E 3 s 52E (25.6)

4 25.2 Potentil Diffeence in Unifom Electic Fiel 713 When positive test chge moves fom point to point, the electic potentil enegy of the chge fiel system eceses. When n oject with mss moves fom point to point, the gvittionl potentil enegy of the oject fiel system eceses. ACTIVE FIGURE 25.2 () When the electic fiel E is iecte ownw, point is t lowe electic potentil thn point. () An oject of mss m moving ownw in gvittionl fiel g. E q 0 g m The negtive sign inictes tht the electic potentil t point is lowe thn t point ; tht is, V, V. Electic fiel lines lwys point in the iection of ecesing electic potentil s shown in Active Figue Now suppose test chge q 0 moves fom to. We cn clculte the chnge in the potentil enegy of the chge fiel system fom Equtions 25.3 n 25.6: DU 5 q 0 DV 5 2q 0 E (25.7) This esult shows tht if q 0 is positive, then DU is negtive. Theefoe, in system consisting of positive chge n n electic fiel, the electic potentil enegy of the system eceses when the chge moves in the iection of the fiel. Equivlently, n electic fiel oes wok on positive chge when the chge moves in the iection of the electic fiel. Tht is nlogous to the wok one y the gvittionl fiel on flling oject s shown in Active Figue If positive test chge is elese fom est in this electic fiel, it epeiences n electic foce q 0 E in the iection of E (ownw in Active Fig. 25.2). Theefoe, it cceletes ownw, gining kinetic enegy. As the chge pticle gins kinetic enegy, the potentil enegy of the chge fiel system eceses y n equl mount. This equivlence shoul not e supising; it is simply consevtion of mechnicl enegy in n isolte system s intouce in Chpte 8. The compison etween system of n electic fiel with positive test chge n gvittionl fiel with test mss in Active Figue 25.2 is useful fo conceptulizing electicl ehvio. The electicl sitution, howeve, hs one fetue tht the gvittionl sitution oes not: the test chge cn e negtive. If q 0 is negtive, then DU in Eqution 25.7 is positive n the sitution is evese. A system consisting of negtive chge n n electic fiel gins electic potentil enegy when the chge moves in the iection of the fiel. If negtive chge is elese fom est in n electic fiel, it cceletes in iection opposite the iection of the fiel. Fo the negtive chge to move in the iection of the fiel, n etenl gent must pply foce n o positive wok on the chge. Now consie the moe genel cse of chge pticle tht moves etween n in unifom electic fiel such tht the vecto s is not pllel to the fiel lines s shown in Figue In this cse, Eqution 25.3 gives DV 52 3 E? s 52E? 3 s 52E? s (25.8) whee gin E ws emove fom the integl ecuse it is constnt. The chnge in potentil enegy of the chge fiel system is Point is t lowe electic potentil thn point. u s Chnge in potentil etween two points in unifom electic fiel E Points n e t the sme electic potentil. Figue 25.3 A unifom electic fiel iecte long the positive is. DU 5 q 0 DV 52q 0 E? s (25.9)

5 714 CHAPTER 25 Electic Potentil 9 V 8 V 7 V 6 V Figue 25.4 (Quick Quiz 25.2) Fou equipotentil sufces. Finlly, we conclue fom Eqution 25.8 tht ll points in plne pepenicul to unifom electic fiel e t the sme electic potentil. We cn see tht in Figue 25.3, whee the potentil iffeence V 2 V is equl to the potentil iffeence V 2 V. (Pove this fct to youself y woking out two ot poucts fo E? s : one fo s, whee the ngle u etween E n s is ity s shown in Figue 25.3, n one fo s, whee u 5 0.) Theefoe, V 5 V. The nme equipotentil sufce is given to ny sufce consisting of continuous istiution of points hving the sme electic potentil. The equipotentil sufces ssocite with unifom electic fiel consist of fmily of pllel plnes tht e ll pepenicul to the fiel. Equipotentil sufces ssocite with fiels hving othe symmeties e escie in lte sections. Quick Quiz 25.2 The lele points in Figue 25.4 e on seies of equipotentil sufces ssocite with n electic fiel. Rnk (fom getest to lest) the wok one y the electic fiel on positively chge pticle tht moves fom to, fom to, fom to, n fom to. Emple 25.1 The Electic Fiel Between Two Pllel Pltes of Opposite Chge A ttey hs specifie potentil iffeence DV etween its teminls n estlishes tht potentil iffeence etween conuctos ttche to the teminls. A 12-V ttey is connecte etween two pllel pltes s shown in Figue The seption etween the pltes is cm, n we ssume the electic fiel etween the pltes to e unifom. (This ssumption is esonle if the plte seption is smll eltive to the plte imensions n we o not consie loctions ne the plte eges.) Fin the mgnitue of the electic fiel etween the pltes. B A V = 12 V OLUTION Conceptulize In Chptes 23 n 24, we investigte the unifom electic fiel etween pllel pltes. The new fetue to this polem is tht the electic fiel is elte to the new concept of electic potentil. Ctegoize The electic fiel is evlute fom eltionship etween fiel n potentil given in this section, so we ctegoize this emple s sustitution polem. Use Eqution 25.6 to evlute the mgnitue of the electic fiel etween the pltes: E 5 0 V B 2 V A 0 5 Figue 25.5 (Emple 25.1) A 12-V ttey connecte to two pllel pltes. The electic fiel etween the pltes hs mgnitue given y the potentil iffeence DV ivie y the plte seption. 12 V m V/m The configution of pltes in Figue 25.5 is clle pllel-plte cpcito n is emine in gete etil in Chpte 26. Emple 25.2 Motion of Poton in Unifom Electic Fiel A poton is elese fom est t point in unifom electic fiel tht hs mgnitue of V/m (Fig. 25.6). The poton unegoes isplcement of mgnitue m to point in the iection of E. Fin the spee of the poton fte completing the isplcement. OLUTION Conceptulize Visulize the poton in Figue 25.6 moving ownw though the potentil iffeence. The sitution is nlogous to n oject flling though gvittionl fiel. Figue 25.6 (Emple 25.2) A poton cceletes fom to in the iection of the electic fiel. v 0 v E

6 25.3 Electic Potentil n Potentil Enegy Due to Point Chges cont. Ctegoize The system of the poton n the two pltes in Figue 25.6 oes not intect with the envionment, so we moel it s n isolte system. Anlyze Use Eqution 25.6 to fin the potentil iffeence etween points n : Wite the ppopite euction of Eqution 8.2, the consevtion of enegy eqution, fo the isolte system of the chge n the electic fiel: DV 5 2E 5 2( V/m)(0.50 m) V DK 1 DU 5 0 ustitute the chnges in enegy fo oth tems: 1 1 2mv e DV 5 0 olve fo the finl spee of the poton: ustitute numeicl vlues: v 5 Å 22e DV m C V2 v 5 Å kg m/s Finlize Becuse DV is negtive fo the fiel, DU is lso negtive fo the poton fiel system. The negtive vlue of DU mens the potentil enegy of the system eceses s the poton moves in the iection of the electic fiel. As the poton cceletes in the iection of the fiel, it gins kinetic enegy while the electic potentil enegy of the system eceses t the sme time. Figue 25.6 is oiente so tht the poton moves ownw. The poton s motion is nlogous to tht of n oject flling in gvittionl fiel. Although the gvittionl fiel is lwys ownw t the sufce of the Eth, n electic fiel cn e in ny iection, epening on the oienttion of the pltes ceting the fiel. Theefoe, Figue 25.6 coul e otte 908 o 1808 n the poton coul move hoizontlly o upw in the electic fiel! 25.3 Electic Potentil n Potentil Enegy Due to Point Chges ˆ u s As iscusse in ection 23.4, n isolte positive point chge q pouces n electic fiel iecte illy outw fom the chge. To fin the electic potentil t point locte istnce fom the chge, let s egin with the genel epession fo potentil iffeence, V 2 V E? s whee n e the two ity points shown in Figue At ny point in spce, the electic fiel ue to the point chge is E 5 1k e q/ 2 2 ^ (Eq. 23.9), whee ^ is unit vecto iecte illy outw fom the chge. The quntity E? s cn e epesse s E? s 5 k q e 2 ^? s Becuse the mgnitue of ^ is 1, the ot pouct ^? s 5 s cos u, whee u is the ngle etween ^ n s. Futhemoe, s cos u is the pojection of s onto ^; theefoe, s cos u 5. Tht is, ny isplcement s long the pth fom point to point pouces chnge in the mgnitue of, the position vecto of the point eltive to the chge ceting the fiel. Mking these sustitutions, we q The two she cicles epesent intesections of spheicl equipotentil sufces with the pge. Figue 25.7 The potentil iffeence etween points n ue to point chge q epens only on the initil n finl il coointes n.

7 716 CHAPTER 25 Electic Potentil fin tht E? s 5 1k e q/ 2 2 ; hence, the epession fo the potentil iffeence ecomes V 2 V 52k e q 3 5 k q 2 e ` V 2 V 5 k e q c (25.10) Pitfll Pevention 25.4 imil Eqution Wning Do not confuse Eqution fo the electic potentil of point chge with Eqution 23.9 fo the electic fiel of point chge. Potentil is popotionl to 1/, whees the mgnitue of the fiel is popotionl to 1/ 2. The effect of chge on the spce suouning it cn e escie in two wys. The chge sets up vecto electic fiel E, which is elte to the foce epeience y test chge plce in the fiel. It lso sets up scl potentil V, which is elte to the potentil enegy of the two-chge system when test chge is plce in the fiel. Eqution shows us tht the integl of E? s is inepenent of the pth etween points n. Multiplying y chge q 0 tht moves etween points n, we see tht the integl of q 0 E? s is lso inepenent of pth. This ltte integl, which is the wok one y the electic foce on the chge q 0, shows tht the electic foce is consevtive (see ection 7.7). We efine fiel tht is elte to consevtive foce s consevtive fiel. Theefoe, Eqution tells us tht the electic fiel of fie point chge q is consevtive. Futhemoe, Eqution epesses the impotnt esult tht the potentil iffeence etween ny two points n in fiel cete y point chge epens only on the il coointes n. It is customy to choose the efeence of electic potentil fo point chge to e V 5 0 t 5 `. With this efeence choice, the electic potentil ue to point chge t ny istnce fom the chge is V 5 k q e (25.11) Figue 25.8 shows plot of the electic potentil on the veticl is fo positive chge locte in the y plne. Consie the following nlogy to gvittionl potentil. Imgine tying to oll mle tow the top of hill shpe like the sufce in Figue Pushing the mle up the hill is nlogous to pushing one positively chge oject tow nothe positively chge oject. imilly, the electic potentil gph of the egion suouning negtive chge is nlogous to hole with espect to ny ppoching positively chge ojects. A chge oject must e infinitely istnt fom nothe chge efoe the sufce in Figue 25.8 is flt n hs n electic potentil of zeo. We otin the electic potentil esulting fom two o moe point chges y pplying the supeposition pinciple. Tht is, the totl electic potentil t some point P ue to sevel point chges is the sum of the potentils ue to the iniviul chges. Fo goup of point chges, we cn wite the totl electic potentil t P s Electic potentil ue to sevel point chges V 5 k e i q i i (25.12) The e-own cuve shows the 1/ ntue of the electic potentil s given y Eqution Figue 25.8 () The electic potentil in the plne oun single positive chge is plotte on the veticl is. (The electic potentil function fo negtive chge woul look like hole inste of hill.) () The electic potentil in the plne contining ipole. Electic potentil (V) 1 0 y Electic potentil (V)

8 25.3 Electic Potentil n Potentil Enegy Due to Point Chges 717 The potentil enegy of the pi of chges is given y k e q 1 q 2 / 12. A potentil k e q 2 / 12 eists t point P ue to chge q 2. ACTIVE FIGURE 25.9 () Two point chges septe y istnce 12. () Chge q 1 is emove. 12 q 2 12 q 2 q 1 P q V k 2 e 12 whee the potentil is gin tken to e zeo t infinity n i is the istnce fom the point P to the chge q i. Notice tht the sum in Eqution is n lgeic sum of scls the thn vecto sum (which we use to clculte the electic fiel of goup of chges in Eq ). Theefoe, it is often much esie to evlute V thn E. The electic potentil oun ipole is illustte in Figue Notice the steep slope of the potentil etween the chges, epesenting egion of stong electic fiel, which cn lso e seen in the electic fiel line sketch in Figue Now consie the potentil enegy of system of two chge pticles. If V 2 is the electic potentil t point P ue to chge q 2, the wok n etenl gent must o to ing secon chge q 1 fom infinity to P without cceletion is q 1 V 2. This wok epesents tnsfe of enegy into the system, n the enegy ppes in the system s potentil enegy U when the pticles e septe y istnce 12 (Active Fig. 25.9). Theefoe, the potentil enegy of the system cn e epesse s 1 U 5 k q 1q 2 e (25.13) 12 If the chges e of the sme sign, then U is positive. Positive wok must e one y n etenl gent on the system to ing the two chges ne ech othe (ecuse chges of the sme sign epel). If the chges e of opposite sign, then U is negtive. Negtive wok is one y n etenl gent ginst the ttctive foce etween the chges of opposite sign s they e ought ne ech othe; foce must e pplie opposite the isplcement to pevent q 1 fom cceleting tow q 2. In Active Figue 25.9, we hve emove the chge q 1. At the position this chge peviously occupie, point P, Equtions 25.2 n cn e use to efine potentil ue to chge q 2 s V 5 U/q 1 5 k e q 2 / 12. This epession is consistent with Eqution If the system consists of moe thn two chge pticles, we cn otin the totl potentil enegy of the system y clculting U fo evey pi of chges n summing the tems lgeiclly. Fo emple, the totl potentil enegy of the system of thee chges shown in Figue is U 5 k e q 1q 2 1 q 1q 3 1 q 2q 3 (25.14) Physiclly, this esult cn e intepete s follows. Imgine q 1 is fie t the position shown in Figue ut q 2 n q 3 e t infinity. The wok n etenl gent must o to ing q 2 fom infinity to its position ne q 1 is k e q 1 q 2 / 12, which is the fist tem in Eqution The lst two tems epesent the wok equie to ing q 3 1 The epession fo the electic potentil enegy of system me up of two point chges, Eqution 25.13, is of the sme fom s the eqution fo the gvittionl potentil enegy of system me up of two point msses, 2Gm 1 m 2 / (see Chpte 13). The simility is not supising consieing tht oth epessions e eive fom n invesesque foce lw. The potentil enegy of this system of chges is given y Eqution q 1 13 q q 3 Figue Thee point chges e fie t the positions shown.

9 718 CHAPTER 25 Electic Potentil fom infinity to its position ne q 1 n q 2. (The esult is inepenent of the oe in which the chges e tnspote.) Quick Quiz 25.3 In Active Figue 25.9, tke q 1 to e negtive souce chge n q 2 to e the test chge. (i) If q 2 is initilly positive n is chnge to chge of the sme mgnitue ut negtive, wht hppens to the potentil t the position of q 2 ue to q 1? () It inceses. () It eceses. (c) It emins the sme. (ii) When q 2 is chnge fom positive to negtive, wht hppens to the potentil enegy of the two-chge system? Choose fom the sme possiilities. Emple 25.3 The Electic Potentil Due to Two Point Chges As shown in Figue 25.11, chge q mc is locte t the oigin n chge q mc is locte t (0, 3.00) m. (A) Fin the totl electic potentil ue to these chges t the point P, whose coointes e (4.00, 0) m. OLUTION Conceptulize Recognize fist tht the 2.00-mC n mC chges e souce chges n set up n electic fiel s well s potentil t ll points in spce, incluing point P. Ctegoize The potentil is evlute using n eqution evelope in this chpte, so we ctegoize this emple s sustitution polem. y 6.00 mc 3.00 m 2.00 mc P 6.00 mc 3.00 m 2.00 mc 3.00 mc 4.00 m 4.00 m Figue (Emple 25.3) () The electic potentil t P ue to the two chges q 1 n q 2 is the lgeic sum of the potentils ue to the iniviul chges. () A thi chge q mc is ought fom infinity to point P. y Use Eqution fo the system of two souce chges: ustitute numeicl vlues: V P 5 k e q q 2 2 V P N? m 2 /C C 4.00 m V C 5.00 m (B) Fin the chnge in potentil enegy of the system of two chges plus thi chge q mc s the ltte chge moves fom infinity to point P (Fig ). OLUTION Assign U i 5 0 fo the system to the configution in which the chge q 3 is t infinity. Use Eqution 25.2 to evlute the potentil enegy fo the configution in which the chge is t P: U f 5 q 3 V P ustitute numeicl vlues to evlute DU: DU 5 U f 2 U i 5 q 3 V P ( C)( V) J Theefoe, ecuse the potentil enegy of the system hs ecese, n etenl gent hs to o positive wok to emove the chge q 3 fom point P ck to infinity. WHAT IF? You e woking though this emple with clssmte n she sys, Wit minute! In pt (B), we ignoe the potentil enegy ssocite with the pi of chges q 1 n q 2! How woul you espon?

10 25.4 Otining the Vlue of the Electic Fiel fom the Electic Potentil cont. Answe Given the sttement of the polem, it is not necessy to inclue this potentil enegy ecuse pt (B) sks fo the chnge in potentil enegy of the system s q 3 is ought in fom infinity. Becuse the configution of chges q 1 n q 2 oes not chnge in the pocess, thee is no DU ssocite with these chges. H pt (B) ske to fin the chnge in potentil enegy when ll thee chges stt out infinitely f pt n e then ought to the positions in Figue 25.11, howeve, you woul hve to clculte the chnge using Eqution Otining the Vlue of the Electic Fiel fom the Electic Potentil The electic fiel E n the electic potentil V e elte s shown in Eqution 25.3, which tells us how to fin DV if the electic fiel E is known. We now show how to clculte the vlue of the electic fiel if the electic potentil is known in cetin egion. Fom Eqution 25.3, we cn epess the potentil iffeence V etween two points istnce s pt s V 52E? s (25.15) If the electic fiel hs only one component E, then E? s 5 E. Theefoe, Eqution ecomes V 5 2E, o E 52 V (25.16) Tht is, the component of the electic fiel is equl to the negtive of the eivtive of the electic potentil with espect to. imil sttements cn e me out the y n z components. Eqution is the mthemticl sttement of the electic fiel eing mesue of the te of chnge with position of the electic potentil s mentione in ection Epeimentlly, electic potentil n position cn e mesue esily with voltmete ( evice fo mesuing potentil iffeence) n metestick. Consequently, n electic fiel cn e etemine y mesuing the electic potentil t sevel positions in the fiel n mking gph of the esults. Accoing to Eqution 25.16, the slope of gph of V vesus t given point povies the mgnitue of the electic fiel t tht point. When test chge unegoes isplcement s long n equipotentil sufce, then V 5 0 ecuse the potentil is constnt long n equipotentil sufce. Fom Eqution 25.15, we see tht V 52E? s 5 0; theefoe, E must e pepenicul to the isplcement long the equipotentil sufce. This esult shows tht the equipotentil sufces must lwys e pepenicul to the electic fiel lines pssing though them. As mentione t the en of ection 25.2, the equipotentil sufces ssocite with unifom electic fiel consist of fmily of plnes pepenicul to the fiel lines. Figue (pge 720) shows some epesenttive equipotentil sufces fo this sitution. If the chge istiution ceting n electic fiel hs spheicl symmety such tht the volume chge ensity epens only on the il istnce, the electic fiel is il. In this cse, E? s 5 E, n we cn epess V s V 5 2E. Theefoe, E 52 V (25.17) Fo emple, the electic potentil of point chge is V 5 k e q/. Becuse V is function of only, the potentil function hs spheicl symmety. Applying Eqution 25.17, we fin tht the mgnitue of the electic fiel ue to the point chge is E 5 k e q/ 2, fmili esult. Notice tht the potentil chnges only in the il iection, not in ny iection pepenicul to. Theefoe, V (like E ) is function

11 720 CHAPTER 25 Electic Potentil A spheiclly symmetic electic fiel pouce y point chge An electic fiel pouce y n electic ipole A unifom electic fiel pouce y n infinite sheet of chge q E c Figue Equipotentil sufces (the she lue lines e intesections of these sufces with the pge) n electic fiel lines. In ll cses, the equipotentil sufces e pepenicul to the electic fiel lines t evey point. Fining the electic fiel fom the potentil only of, which is gin consistent with the ie tht equipotentil sufces e pepenicul to fiel lines. In this cse, the equipotentil sufces e fmily of sphees concentic with the spheiclly symmetic chge istiution (Fig ). The equipotentil sufces fo n electic ipole e sketche in Figue 25.12c. In genel, the electic potentil is function of ll thee sptil coointes. If V() is given in tems of the Ctesin coointes, the electic fiel components E, E y, n E z cn eily e foun fom V(, y, z) s the ptil eivtives 2 E 52 'V ' E y 52 'V 'y E z 52 'V 'z (25.18) Quick Quiz 25.4 In cetin egion of spce, the electic potentil is zeo eveywhee long the is. (i) Fom this infomtion, you cn conclue tht the component of the electic fiel in this egion is () zeo, () in the positive iection, o (c) in the negtive iection. (ii) uppose the electic potentil is 12 V eveywhee long the is. Fom the sme choices, wht cn you conclue out the component of the electic fiel now? Emple 25.4 The Electic Potentil Due to Dipole An electic ipole consists of two chges of equl mgnitue n opposite sign septe y istnce 2 s shown in Figue The ipole is long the is n is centee t the oigin. (A) Clculte the electic potentil t point P on the y is. OLUTION Figue (Emple 25.4) An electic ipole locte on the is. q Conceptulize Compe this sitution to tht in pt (B) of Emple It is the sme sitution, ut hee we e seeking the electic potentil the thn the electic fiel. q y y P R 2 In vecto nottion, E is often witten in Ctesin coointe systems s whee = is clle the gient opeto. ' E 5 2=V 52i^ ' 1 j^ ' 'y 1 k^ ' 'z V

12 25.5 Electic Potentil Due to Continuous Chge Distiutions cont. Ctegoize Becuse the ipole consists of only two souce chges, the electic potentil cn e evlute y summing the potentils ue to the iniviul chges. Anlyze Use Eqution to fin the electic potentil t P ue to the two chges: V P 5 k e i q i q 5 k e i " 2 1 y 1 2q 2 " 2 1 y (B) Clculte the electic potentil t point R on the positive is. OLUTION Use Eqution to fin the electic potentil t R ue to the two chges: V R 5 k e i q i 5 k e 2q i 2 1 q k eq (C) Clculte V n E t point on the is f fom the ipole. OLUTION Fo point R f fom the ipole such tht.., neglect 2 in the enominto of the nswe to pt (B) n wite V in this limit: Use Eqution n this esult to clculte the component of the electic fiel t point on the is f fom the ipole: V R 5 lim.. 2 2k eq < 2 2k eq E 52 V k eq 2 5 2k e q k eq Finlize The potentils in pts (B) n (C) e negtive ecuse points on the positive is e close to the negtive chge thn to the positive chge. Fo the sme eson, the component of the electic fiel is negtive. WHAT IF? uppose you wnt to fin the electic fiel t point P on the y is. In pt (A), the electic potentil ws foun to e zeo fo ll vlues of y. Is the electic fiel zeo t ll points on the y is? Answe No. Tht thee is no chnge in the potentil long the y is tells us only tht the y component of the electic fiel is zeo. Look ck t Figue in Emple We showe thee tht the electic fiel of ipole on the y is hs only n component. We coul not fin the component in the cuent emple ecuse we o not hve n epession fo the potentil ne the y is s function of Electic Potentil Due to Continuous Chge Distiutions The electic potentil ue to continuous chge istiution cn e clculte using two iffeent methos. The fist metho is s follows. If the chge istiution is known, we consie the potentil ue to smll chge element q, teting this element s point chge (Fig ). Fom Eqution 25.11, the electic potentil V t some point P ue to the chge element q is V 5 k q e (25.19) whee is the istnce fom the chge element to point P. To otin the totl potentil t point P, we integte Eqution to inclue contiutions fom ll P 2 q q 1 q 3 Figue The electic potentil t point P ue to continuous chge istiution cn e clculte y iviing the chge istiution into elements of chge q n summing the electic potentil contiutions ove ll elements. Thee smple elements of chge e shown.

13 722 CHAPTER 25 Electic Potentil Electic potentil ue to continuous chge istiution elements of the chge istiution. Becuse ech element is, in genel, iffeent istnce fom point P n k e is constnt, we cn epess V s V 5 k q e 3 (25.20) In effect, we hve eplce the sum in Eqution with n integl. In this epession fo V, the electic potentil is tken to e zeo when point P is infinitely f fom the chge istiution. The secon metho fo clculting the electic potentil is use if the electic fiel is ley known fom othe consietions such s Guss s lw. If the chge istiution hs sufficient symmety, we fist evlute E using Guss s lw n then sustitute the vlue otine into Eqution 25.3 to etemine the potentil iffeence DV etween ny two points. We then choose the electic potentil V to e zeo t some convenient point. Polem-olving ttegy CALCULATING ELECTRIC POTENTIAL The following poceue is ecommene fo solving polems tht involve the etemintion of n electic potentil ue to chge istiution. 1. Conceptulize. Think cefully out the iniviul chges o the chge istiution you hve in the polem n imgine wht type of potentil woul e cete. Appel to ny symmety in the ngement of chges to help you visulize the potentil. 2. Ctegoize. Ae you nlyzing goup of iniviul chges o continuous chge istiution? The nswe to this question will tell you how to pocee in the Anlyze step. 3. Anlyze. When woking polems involving electic potentil, ememe tht it is scl quntity, so thee e no components to consie. Theefoe, when using the supeposition pinciple to evlute the electic potentil t point, simply tke the lgeic sum of the potentils ue to ech chge. You must keep tck of signs, howeve. As with potentil enegy in mechnics, only chnges in electic potentil e significnt; hence, the point whee the potentil is set t zeo is ity. When eling with point chges o finite-size chge istiution, we usully efine V 5 0 to e t point infinitely f fom the chges. If the chge istiution itself etens to infinity, howeve, some othe ney point must e selecte s the efeence point. () If you e nlyzing goup of iniviul chges: Use the supeposition pinciple, which sttes tht when sevel point chges e pesent, the esultnt potentil t point P in spce is the lgeic sum of the iniviul potentils t P ue to the iniviul chges (Eq ). Emple 25.4 emonstte this poceue. () If you e nlyzing continuous chge istiution: Replce the sums fo evluting the totl potentil t some point P fom iniviul chges y integls (Eq ). The chge istiution is ivie into infinitesiml elements of chge q locte t istnce fom the point P. An element is then tete s point chge, so the potentil t P ue to the element is V 5 k e q/. The totl potentil t P is otine y integting ove the entie chge istiution. Fo mny polems, it is possile in pefoming the integtion to epess q n in tems of single vile. To simplify the integtion, give ceful consietion to the geomety involve in the polem. Emples 25.5 though 25.7 emonstte such poceue. To otin the potentil fom the electic fiel: Anothe metho use to otin the potentil is to stt with the efinition of the potentil iffeence given y Eqution If E is known o cn e otine esily (such s fom Guss s lw), the line integl of E? s cn e evlute. 4. Finlize. Check to see if you epession fo the potentil is consistent with the mentl epesenttion n eflects ny symmety you note peviously. Imgine vying pmetes such s the istnce of the osevtion point fom the chges o the ius of ny cicul ojects to see if the mthemticl esult chnges in esonle wy.

14 25.5 Electic Potentil Due to Continuous Chge Distiutions 723 Emple 25.5 Electic Potentil Due to Unifomly Chge Ring (A) Fin n epession fo the electic potentil t point P locte on the pepenicul centl is of unifomly chge ing of ius n totl chge Q. OLUTION Conceptulize tuy Figue 25.15, in which the ing is oiente so tht its plne is pepenicul to the is n its cente is t the oigin. Notice tht the symmety of the sitution mens tht ll the chges on the ing e the sme istnce fom point P. Ctegoize Becuse the ing consists of continuous istiution of chge the thn set of iscete chges, we must use the integtion technique epesente y Eqution in this emple. Anlyze We tke point P to e t istnce fom the cente of the ing s shown in Figue Use Eqution to epess V in tems of the geomety: V 5 k e 3 q 5 k e 3 q 2 2 Figue (Emple 25.5) A unifomly chge ing of ius lies in plne pepenicul to the is. All elements q of the ing e the sme istnce fom point P lying on the is. q " P Noting tht n e constnts, ing " in font of the integl sign n integte ove the ing: V 5 k e " q 5 k e Q " (25.21) (B) Fin n epession fo the mgnitue of the electic fiel t point P. OLUTION Fom symmety, notice tht long the is E cn hve only n component. Theefoe, pply Eqution to Eqution 25.21: E 52 V 52k eq /2 52k e Q /2 122 E 5 k e Q (25.22) /2 2 Finlize The only vile in the epessions fo V n E is. Tht is not supising ecuse ou clcultion is vli only fo points long the is, whee y n z e oth zeo. This esult fo the electic fiel gees with tht otine y iect integtion (see Emple 23.7). Emple 25.6 Electic Potentil Due to Unifomly Chge Disk A unifomly chge isk hs ius R n sufce chge ensity s. (A) Fin the electic potentil t point P long the pepenicul centl is of the isk. OLUTION Conceptulize If we consie the isk to e set of concentic ings, we cn use ou esult fom Emple 25.5 which gives the potentil ue to ing of ius n sum the contiutions of Figue (Emple 25.6) A unifomly chge isk of ius R lies in plne pepenicul to the is. The clcultion of the electic potentil t ny point P on the is is simplifie y iviing the isk into mny ings of ius n with, with e 2p. R 2 2 A 2p P continue

15 724 CHAPTER 25 Electic Potentil 25.6 cont. ll ings mking up the isk. Becuse point P is on the centl is of the isk, symmety gin tells us tht ll points in given ing e the sme istnce fom P. Ctegoize Becuse the isk is continuous, we evlute the potentil ue to continuous chge istiution the thn goup of iniviul chges. Anlyze Fin the mount of chge q on ing of ius n with s shown in Figue 25.16: Use this esult in Eqution in Emple 25.5 (with eplce y n Q eplce y q) to fin the potentil ue to the ing: q 5s A 5s12p 2 5 2ps V 5 k e q " k e2ps " To otin the totl potentil t P, integte this epession ove the limits 5 0 to 5 R, noting tht is constnt: V 5pk e s 3 R 0 2 " 2 1 5pk es R /2 2 This integl is of the common fom e u n u, whee n n u , n hs the vlue u n11 /(n 1 1). Use this esult to evlute the integl: V 5 2pk e s31r /2 2 4 (25.23) (B) Fin the component of the electic fiel t point P long the pepenicul centl is of the isk. OLUTION As in Emple 25.5, use Eqution to fin the electic fiel t ny il point: E 52 V 5 2pk es c1 2 1R /2 (25.24) 2 Finlize Compe Eqution with the esult of Emple The clcultion of V n E fo n ity point off the is is moe ifficult to pefom ecuse of the sence of symmety n we o not tet tht sitution in this ook. Emple 25.7 Electic Potentil Due to Finite Line of Chge A o of length, locte long the is hs totl chge Q n unifom line chge ensity l. Fin the electic potentil t point P locte on the y is istnce fom the oigin (Fig ). OLUTION Conceptulize The potentil t P ue to evey segment of chge on the o is positive ecuse evey segment cies positive chge. Notice tht we hve no symmety to ppel to hee, ut the simple geomety shoul mke the polem solvle. Ctegoize Becuse the o is continuous, we evlute the potentil ue to continuous chge istiution the thn goup of iniviul chges. Anlyze In Figue 25.17, the o lies long the is, is the length of one smll segment, n q is the chge on tht segment. Becuse the o hs chge pe unit length l, the chge q on the smll segment is q 5 l. P O y Figue (Emple 25.7) A unifom line chge of length, locte long the is. To clculte the electic potentil t P, the line chge is ivie into segments ech of length n ech cying chge q 5 l. q

16 25.6 Electic Potentil Due to Chge Conucto cont. Fin the potentil t P ue to one segment of the o: V 5 k e q 5 k e l " Fin the totl potentil t P y integting this epession ove the limits 5 0 to 5,: V 5 3, k e 0 l " Noting tht k e n l 5 Q/, e constnts n cn e emove fom the integl, evlute the integl with the help of Appeni B: V 5 k e l 3, 0 " k e Q, ln 1 1 " `, 0 Evlute the esult etween the limits: V 5 k Q e, 3ln 1, 1 " 2 1, ln 4 5 k Q e, ln, 1 " 2 1, 2 (25.25) Finlize If,,,, the potentil t P shoul ppoch tht of point chge ecuse the o is vey shot compe to the istnce fom the o to P. By using seies epnsion fo the ntul logithm fom Appeni B.5, it is esy to show tht Eqution ecomes V = k e Q/. WHAT IF? Wht if you wee ske to fin the electic fiel t point P? Woul tht e simple clcultion? Answe Clculting the electic fiel y mens of Eqution woul e little messy. Thee is no symmety to ppel to, n the integtion ove the line of chge woul epesent vecto ition of electic fiels t point P. Using Eqution 25.18, you coul fin E y y eplcing with y in Eqution n pefoming the iffeentition with espect to y. Becuse the chge o in Figue 25.6 Electic Potentil Due to Chge Conucto lies entiely to the ight of 5 0, the electic fiel t point P woul hve n component to the left if the o is chge positively. You cnnot use Eqution to fin the component of the fiel, howeve, ecuse the potentil ue to the o ws evlute t specific vlue of ( 5 0) the thn genel vlue of. You woul hve to fin the potentil s function of oth n y to e le to fin the n y components of the electic fiel using Eqution In ection 24.4, we foun tht when soli conucto in equiliium cies net chge, the chge esies on the conucto s oute sufce. Futhemoe, the electic fiel just outsie the conucto is pepenicul to the sufce n the fiel insie is zeo. We now genete nothe popety of chge conucto, elte to electic potentil. Consie two points n on the sufce of chge conucto s shown in Figue Along sufce pth connecting these points, E is lwys pepenicul to the isplcement s ; theefoe, E? s 5 0. Using this esult n Notice fom the spcing of the positive signs tht the sufce chge ensity is nonunifom. Pitfll Pevention 25.5 Potentil My Not Be Zeo The electic potentil insie the conucto is not necessily zeo in Figue 25.18, even though the electic fiel is zeo. Eqution shows tht zeo vlue of the fiel esults in no chnge in the potentil fom one point to nothe insie the conucto. Theefoe, the potentil eveywhee insie the conucto, incluing the sufce, hs the sme vlue, which my o my not e zeo, epening on whee the zeo of potentil is efine. E Figue An itily shpe conucto cying positive chge. When the conucto is in electosttic equiliium, ll the chge esies t the sufce, E 5 0 insie the conucto, n the iection of E immeitely outsie the conucto is pepenicul to the sufce. The electic potentil is constnt insie the conucto n is equl to the potentil t the sufce.

17 726 CHAPTER 25 Electic Potentil c k e Q R R V E k e Q k e Q 2 Eqution 25.3, we conclue tht the potentil iffeence etween n is necessily zeo: V 2 V E? s 5 0 This esult pplies to ny two points on the sufce. Theefoe, V is constnt eveywhee on the sufce of chge conucto in equiliium. Tht is, the sufce of ny chge conucto in electosttic equiliium is n equipotentil sufce: evey point on the sufce of chge conucto in equiliium is t the sme electic potentil. Futhemoe, ecuse the electic fiel is zeo insie the conucto, the electic potentil is constnt eveywhee insie the conucto n equl to its vlue t the sufce. R Figue () The ecess chge on conucting sphee of ius R is unifomly istiute on its sufce. () Electic potentil vesus istnce fom the cente of the chge conucting sphee. (c) Electic fiel mgnitue vesus istnce fom the cente of the chge conucting sphee. Becuse of the constnt vlue of the potentil, no wok is equie to move test chge fom the inteio of chge conucto to its sufce. Consie soli metl conucting sphee of ius R n totl positive chge Q s shown in Figue As etemine in pt (A) of Emple 24.3, the electic fiel outsie the sphee is k e Q / 2 n points illy outw. Becuse the fiel outsie spheiclly symmetic chge istiution is ienticl to tht of point chge, we epect the potentil to lso e tht of point chge, k e Q /. At the sufce of the conucting sphee in Figue 25.19, the potentil must e k e Q /R. Becuse the entie sphee must e t the sme potentil, the potentil t ny point within the sphee must lso e k e Q /R. Figue is plot of the electic potentil s function of, n Figue 25.19c shows how the electic fiel vies with. When net chge is plce on spheicl conucto, the sufce chge ensity is unifom s inicte in Figue If the conucto is nonspheicl s in Figue 25.18, howeve, the sufce chge ensity is high whee the ius of cuvtue is smll (s note in ection 24.4) n low whee the ius of cuvtue is lge. Becuse the electic fiel immeitely outsie the conucto is popotionl to the sufce chge ensity, the electic fiel is lge ne conve points hving smll ii of cuvtue n eches vey high vlues t shp points. In Emple 25.8, the eltionship etween electic fiel n ius of cuvtue is eploe mthemticlly. Emple 25.8 Two Connecte Chge phees Two spheicl conuctos of ii 1 n 2 e septe y istnce much gete thn the ius of eithe sphee. The sphees e connecte y conucting wie s shown in Figue The chges on the sphees in equiliium e q 1 n q 2, espectively, n they e unifomly chge. Fin the tio of the mgnitues of the electic fiels t the sufces of the sphees. q 1 1 OLUTION Conceptulize Imgine the sphees e much fthe pt thn shown in Figue Becuse they e so f pt, the fiel of one oes not ffect the chge istiution on the othe. The conucting wie etween them ensues tht oth sphees hve the sme electic potentil. Ctegoize Becuse the sphees e so f pt, we moel the chge istiution on them s spheiclly symmetic, n we cn moel the fiel n potentil outsie the sphees to e tht ue to point chges. q 2 Figue (Emple 25.8) Two chge spheicl conuctos connecte y conucting wie. The sphees e t the sme electic potentil V. 2

18 25.6 Electic Potentil Due to Chge Conucto cont. Anlyze et the electic potentils t the sufces of the sphees equl to ech othe: V 5 k e q k e q 2 2 olve fo the tio of chges on the sphees: (1) q 1 q Wite epessions fo the mgnitues of the electic fiels t the sufces of the sphees: Evlute the tio of these two fiels: E 1 5 k e q n E 2 5 k e q E 1 5 q E 2 q ustitute fo the tio of chges fom Eqution (1): (2) E E Finlize The fiel is stonge in the vicinity of the smlle sphee even though the electic potentils t the sufces of oth sphees e the sme. If 2 0, then E 2 `, veifying the sttement ove tht the electic fiel is vey lge t shp points. A Cvity Within Conucto uppose conucto of ity shpe contins cvity s shown in Figue Let s ssume no chges e insie the cvity. In this cse, the electic fiel insie the cvity must e zeo egless of the chge istiution on the outsie sufce of the conucto s we mentione in ection Futhemoe, the fiel in the cvity is zeo even if n electic fiel eists outsie the conucto. To pove this point, ememe tht evey point on the conucto is t the sme electic potentil; theefoe, ny two points n on the cvity s sufce must e t the sme potentil. Now imgine fiel E eists in the cvity n evlute the potentil iffeence V 2 V efine y Eqution 25.3: V 2 V E? s Becuse V 2 V 5 0, the integl of E? s must e zeo fo ll pths etween ny two points n on the conucto. The only wy tht cn e tue fo ll pths is if E is zeo eveywhee in the cvity. Theefoe, cvity suoune y conucting wlls is fiel-fee egion s long s no chges e insie the cvity. The electic fiel in the cvity is zeo egless of the chge on the conucto. Figue A conucto in electosttic equiliium contining cvity. Coon Dischge A phenomenon known s coon ischge is often oseve ne conucto such s high-voltge powe line. When the electic fiel in the vicinity of the conucto is sufficiently stong, electons esulting fom nom ioniztions of i molecules ne the conucto ccelete wy fom thei pent molecules. These pily moving electons cn ionize itionl molecules ne the conucto, ceting moe fee electons. The oseve glow (o coon ischge) esults fom the ecomintion of these fee electons with the ionize i molecules. If conucto hs n iegul shpe, the electic fiel cn e vey high ne shp points o eges of the conucto; consequently, the ioniztion pocess n coon ischge e most likely to occu oun such points. Coon ischge is use in the electicl tnsmission inusty to locte oken o fulty components. Fo emple, oken insulto on tnsmission towe hs shp eges whee coon ischge is likely to occu. imilly, coon ischge will occu t the shp en of oken conucto stn. Osevtion of these ischges is ifficult ecuse the visile ition emitte is wek n

19 728 CHAPTER 25 Electic Potentil ACTIVE FIGURE chemtic wing of the Millikn oil-op pptus. Oil oplets Pinhole q v Telescope with scle in eyepiece most of the ition is in the ultviolet. (We will iscuss ultviolet ition n othe potions of the electomgnetic spectum in ection 34.7.) Even use of titionl ultviolet cmes is of little help ecuse the ition fom the coon ischge is ovewhelme y ultviolet ition fom the un. Newly evelope ul- spectum evices comine now-n ultviolet cme with visilelight cme to show ylight view of the coon ischge in the ctul loction on the tnsmission towe o cle. The ultviolet pt of the cme is esigne to opete in wvelength nge in which ition fom the un is vey wek. With the electic fiel off, the oplet flls t teminl velocity v T une the influence of the gvittionl n g foces. v T v T mg qe F D F D q mg When the electic fiel is tune on, the oplet moves upw t teminl velocity v T une the influence of the electic, gvittionl, n g foces. E Figue The foces cting on negtively chge oil oplet in the Millikn epeiment The Millikn Oil-Dop Epeiment Roet Millikn pefome illint set of epeiments fom 1909 to 1913 in which he mesue e, the mgnitue of the elementy chge on n electon, n emonstte the quntize ntue of this chge. His pptus, igmme in Active Figue 25.22, contins two pllel metllic pltes. Oil oplets fom n tomize e llowe to pss though smll hole in the uppe plte. Millikn use -ys to ionize the i in the chme so tht fee electons woul hee to the oil ops, giving them negtive chge. A hoizontlly iecte light em is use to illuminte the oil oplets, which e viewe though telescope whose long is is pepenicul to the light em. When viewe in this mnne, the oplets ppe s shining sts ginst k ckgoun n the te t which iniviul ops fll cn e etemine. Let s ssume single op hving mss m n cying chge q is eing viewe n its chge is negtive. If no electic fiel is pesent etween the pltes, the two foces cting on the chge e the gvittionl foce mg cting ownw 3 n viscous g foce F D cting upw s inicte in Figue The g foce is popotionl to the op s spee s iscusse in ection 6.4. When the op eches its teminl spee v T the two foces lnce ech othe (mg 5 F D ). Now suppose ttey connecte to the pltes sets up n electic fiel etween the pltes such tht the uppe plte is t the highe electic potentil. In this cse, thi foce qe cts on the chge op. Becuse q is negtive n E is iecte ownw, this electic foce is iecte upw s shown in Figue If this upw foce is stong enough, the op moves upw n the g foce F D cts ownw. When the upw electic foce qe lnces the sum of the gvittionl foce n the ownw g foce F D, the op eches new teminl spee v9 T in the upw iection. With the fiel tune on, op moves slowly upw, typiclly t tes of huneths of centimete pe secon. The te of fll in the sence of fiel is comple. Hence, one cn follow single oplet fo hous, ltentely ising n flling, y simply tuning the electic fiel on n off. 3 Thee is lso uoynt foce on the oil op ue to the suouning i. This foce cn e incopote s coection in the gvittionl foce mg on the op, so we will not consie it in ou nlysis.

20 25.8 Applictions of Electosttics 729 Afte ecoing mesuements on thousns of oplets, Millikn n his cowokes foun tht ll oplets, to within out 1% pecision, h chge equl to some intege multiple of the elementy chge e: q 5 ne n 5 0, 21, 22, 23,... whee e C. Millikn s epeiment yiels conclusive evience tht chge is quntize. Fo this wok, he ws we the Noel Pize in Physics in Applictions of Electosttics The pcticl ppliction of electosttics is epesente y such evices s lightning os n electosttic pecipittos n y such pocesses s eogphy n the pinting of utomoiles. cientific evices se on the pinciples of electosttics inclue electosttic genetos, the fiel-ion micoscope, n ion-ive ocket engines. Detils of two evices e given elow. The Vn e Gff Geneto Epeimentl esults show tht when chge conucto is plce in contct with the insie of hollow conucto, ll the chge on the chge conucto is tnsfee to the hollow conucto. In pinciple, the chge on the hollow conucto n its electic potentil cn e incese without limit y epetition of the pocess. In 1929, Roet J. Vn e Gff ( ) use this pinciple to esign n uil n electosttic geneto, n schemtic epesenttion of it is given in Figue This type of geneto ws once use etensively in nucle physics esech. Chge is elivee continuously to high-potentil electoe y mens of moving elt of insulting mteil. The high-voltge electoe is hollow metl ome mounte on n insulting column. The elt is chge t point y mens of coon ischge etween com-like metllic neeles n goune gi. The neeles e mintine t positive electic potentil of typiclly 10 4 V. The positive chge on the moving elt is tnsfee to the ome y secon com of neeles t point. Becuse the electic fiel insie the ome is negligile, the positive chge on the elt is esily tnsfee to the conucto egless of its potentil. In pctice, it is possile to incese the electic potentil of the ome until electicl ischge occus though the i. Becuse the ekown electic fiel in i is out V/m, sphee 1.00 m in ius cn e ise to mimum potentil of V. The potentil cn e incese futhe y incesing the ome s ius n plcing the entie system in contine fille with high-pessue gs. Vn e Gff genetos cn pouce potentil iffeences s lge s 20 million volts. Potons ccelete though such lge potentil iffeences eceive enough enegy to initite nucle ections etween themselves n vious tget nuclei. mlle genetos e often seen in science clssooms n museums. If peson insulte fom the goun touches the sphee of Vn e Gff geneto, his o he oy cn e ought to high electic potentil. The peson s hi cquies net positive chge, n ech stn is epelle y ll the othes s in the opening photogph of Chpte 23. The Electosttic Pecipitto One impotnt ppliction of electicl ischge in gses is the electosttic pecipitto. This evice emoves pticulte mtte fom comustion gses, theey eucing i pollution. Pecipittos e especilly useful in col-uning powe plnts n inustil opetions tht genete lge quntities of smoke. Cuent systems e le to eliminte moe thn 99% of the sh fom smoke. Figue (pge 730) shows schemtic igm of n electosttic pecipitto. A high potentil iffeence (typiclly 40 to 100 kv) is mintine etween wie Metl ome P Goun Belt Insulto The chge is eposite on the elt t point n tnsfee to the hollow conucto t point. Figue chemtic igm of Vn e Gff geneto. Chge is tnsfee to the metl ome t the top y mens of moving elt.

21 730 CHAPTER 25 Electic Potentil The high negtive electic potentil mintine on the centl wie cetes coon ischge in the vicinity of the wie. Insulto Bttey Clen i out Dity i in Dit out Weight Photogphs coutesy of Btemn Engineeing c Figue () chemtic igm of n electosttic pecipitto. Compe the i pollution when the electosttic pecipitto is () opeting n (c) tune off. unning own the cente of uct n the wlls of the uct, which e goune. The wie is mintine t negtive electic potentil with espect to the wlls, so the electic fiel is iecte tow the wie. The vlues of the fiel ne the wie ecome high enough to cuse coon ischge oun the wie; the i ne the wie contins positive ions, electons, n such negtive ions s O 2 2. The i to e clene entes the uct n moves ne the wie. As the electons n negtive ions cete y the ischge e ccelete tow the oute wll y the electic fiel, the it pticles in the i ecome chge y collisions n ion cptue. Becuse most of the chge it pticles e negtive, they too e wn to the uct wlls y the electic fiel. When the uct is peioiclly shken, the pticles ek loose n e collecte t the ottom. In ition to eucing the level of pticulte mtte in the tmosphee (compe Figs n c), the electosttic pecipitto ecoves vlule mteils in the fom of metl oies. Definitions The potentil iffeence DV etween points n in n electic fiel E is efine s DV ; DU 52 q 3 E? s (25.3) 0 whee DU is given y Eqution 25.1 elow. The electic potentil V 5 U/q 0 is scl quntity n hs the units of joules pe coulom, whee 1 J/C ; 1 V. ummy An equipotentil sufce is one on which ll points e t the sme electic potentil. Equipotentil sufces e pepenicul to electic fiel lines. continue

22 Ojective Questions 731 Concepts n Pinciples When positive test chge q 0 is move etween points n in n electic fiel E, the chnge in the potentil enegy of the chge fiel system is DU 52q 0 3 E? s (25.1) The potentil iffeence etween two points n septe y istnce in unifom electic fiel E, whee s is vecto tht points fom tow n is pllel to E, is DV 52E 3 s 52E (25.6) If we efine V 5 0 t 5 `, the electic potentil ue to point chge t ny istnce fom the chge is V 5 k q e (25.11) The electic potentil ssocite with goup of point chges is otine y summing the potentils ue to the iniviul chges. The potentil enegy ssocite with pi of point chges septe y istnce 12 is U 5 k e q 1q 2 12 (25.13) We otin the potentil enegy of istiution of point chges y summing tems like Eqution ove ll pis of pticles. If the electic potentil is known s function of coointes, y, n z, we cn otin the components of the electic fiel y tking the negtive eivtive of the electic potentil with espect to the coointes. Fo emple, the component of the electic fiel is E 52 V (25.16) The electic potentil ue to continuous chge istiution is V 5 k q e 3 (25.20) Evey point on the sufce of chge conucto in electosttic equiliium is t the sme electic potentil. The potentil is constnt eveywhee insie the conucto n equl to its vlue t the sufce. Ojective Questions enotes nswe ville in tuent olutions Mnul/tuy Guie 1. A poton is elese fom est t the oigin in unifom electic fiel in the positive iection with mgnitue 850 N/C. Wht is the chnge in the electic potentil enegy of the poton fiel system when the poton tvels to m? () J () J (c) J () J (e) J 2. An electon in n -y mchine is ccelete though potentil iffeence of V efoe it hits the tget. Wht is the kinetic enegy of the electon in electon volts? () ev () ev (c) ev () ev (e) ev 3. A helium nucleus (chge 5 2e, mss kg) tveling t m/s entes n electic fiel, tveling fom point, t potentil of V, to point, t V. Wht is its spee t point? () m/s () m/s (c) m/s () m/s (e) m/s 4. The electic potentil t m is 120 V, n the electic potentil t m is 190 V. Wht is the component of the electic fiel in this egion, ssuming the fiel is unifom? () 140 N/C () 2140 N/C (c) 35.0 N/C () N/C (e) 75.0 N/C 5. Rnk the potentil enegies of the fou systems of pticles shown in Figue OQ25.5 fom lgest to smllest. Inclue equlities if ppopite. Q Q 2Q 2 c Q Q Q Figue OQ25.5 Q 2Q 2 6. In cetin egion of spce, unifom electic fiel is in the iection. A pticle with negtive chge is cie fom cm to cm. (i) Does the electic potentil enegy of the chge fiel system () incese,

23 732 CHAPTER 25 Electic Potentil () emin constnt, (c) ecese, o () chnge unpeictly? (ii) Hs the pticle move to position whee the electic potentil is () highe thn efoe, () unchnge, (c) lowe thn efoe, o () unpeictle? 7. Rnk the electic potentils A t the fou points shown in Figue OQ25.7 fom lgest to smllest. C 8. Consie the equipotentil sufces shown in Figue In this egion of spce, wht is the ppoimte iection of the elec- D tic fiel? () It is out of the Q pge. () It is into the pge. (c) It is tow the top of Figue OQ25.7 the pge. () It is tow the ottom of the pge. (e) The fiel is zeo. 2Q 9. Rnk the electic potentil enegies of the systems of chges shown in Figue OQ25.9 fom lgest to smllest. Inicte equlities if ppopite. Q Q Q c Q Q Q Q Q Q Q Figue OQ25.9 Q Q Q Q 10. Fou pticles e positione on the im of cicle. The chges on the pticles e mc, mc, mc, n mc. If the electic potentil t the cente of the cicle ue to the mc chge lone is V, wht is the totl electic potentil t the cente ue to the fou chges? () V () V (c) 0 () V (e) V 11. In cetin egion of spce, the electic fiel is zeo. Fom this fct, wht cn you conclue out the electic potentil in this egion? () It is zeo. () It oes not vy with B position. (c) It is positive. () It is negtive. (e) None of those nswes is necessily tue. 12. A pticle with chge nc is on the is t the point with coointe 5 0. A secon pticle, with chge nc, is on the is t m. (i) Is the point t finite istnce whee the electic fiel is zeo () to the left of 5 0, () etween 5 0 n m, o (c) to the ight of m? (ii) Is the electic potentil zeo t this point? () No; it is positive. () Yes. (c) No; it is negtive. (iii) Is thee point t finite istnce whee the electic potentil is zeo? () Yes; it is to the left of 5 0. () Yes; it is etween 5 0 n m. (c) Yes; it is to the ight of m. () No. 13. A filment unning long the is fom the oigin to cm cies electic chge with unifom ensity. At the point P with coointes ( cm, y cm), this filment cetes electic potentil 100 V. Now we nothe filment long the y is, unning fom the oigin to y cm, cying the sme mount of chge with the sme unifom ensity. At the sme point P, is the electic potentil cete y the pi of filments () gete thn 200 V, () 200 V, (c) 100 V, () etween 0 n 200 V, o (e) 0? 14. In iffeent epeimentl tils, n electon, poton, o ouly chge oygen tom (O 22 ), is fie within vcuum tue. The pticle s tjectoy cies it though point whee the electic potentil is 40.0 V n then though point t iffeent potentil. Rnk ech of the following cses ccoing to the chnge in kinetic enegy of the pticle ove this pt of its flight fom the lgest incese to the lgest ecese in kinetic enegy. In you nking, isply ny cses of equlity. () An electon moves fom 40.0 V to 60.0 V. () An electon moves fom 40.0 V to 20.0 V. (c) A poton moves fom 40.0 V to 20.0 V. () A poton moves fom 40.0 V to 10.0 V. (e) An O 22 ion moves fom 40.0 V to 60.0 V. 15. (i) A metllic sphee A of ius 1.00 cm is sevel centimetes wy fom metllic spheicl shell B of ius 2.00 cm. Chge 450 nc is plce on A, with no chge on B o nywhee ney. Net, the two ojects e joine y long, thin, metllic wie (s shown in Fig ), n finlly the wie is emove. How is the chge she etween A n B? () 0 on A, 450 nc on B () 90.0 nc on A n 360 nc on B, with equl sufce chge ensities (c) 150 nc on A n 300 nc on B () 225 nc on A n 225 nc on B (e) 450 nc on A n 0 on B (ii) A metllic sphee A of ius 1 cm with chge 450 nc hngs on n insulting the insie n unchge thin metllic spheicl shell B of ius 2 cm. Net, A is me tempoily to touch the inne sufce of B. How is the chge then she etween them? Choose fom the sme possiilities. Anol Aons, the only physics teche yet to hve his pictue on the cove of Time mgzine, suggeste the ie fo this question. Conceptul Questions 1. Distinguish etween electic potentil n electic potentil enegy. enotes nswe ville in tuent olutions Mnul/tuy Guie 2. Descie the motion of poton () fte it is elese fom est in unifom electic fiel. Descie the chnges (if

24 Polems 733 ny) in () its kinetic enegy n (c) the electic potentil enegy of the poton fiel system. 3. When chge pticles e septe y n infinite istnce, the electic potentil enegy of the pi is zeo. When the pticles e ought close, the electic potentil enegy of pi with the sme sign is positive, whees the electic potentil enegy of pi with opposite signs is negtive. Give physicl eplntion of this sttement. 4. Descie the equipotentil sufces fo () n infinite line of chge n () unifomly chge sphee. 5. Wht etemines the mimum electic potentil to which the ome of Vn e Gff geneto cn e ise? 6. tuy Figue 23.3 n the ccompnying tet iscussion of chging y inuction. When the gouning wie is touche to the ightmost point on the sphee in Figue 23.3c, electons e ine wy fom the sphee to leve the sphee positively chge. uppose the gouning wie is touche to the leftmost point on the sphee inste. () Will electons still in wy, moving close to the negtively chge o s they o so? () Wht kin of chge, if ny, emins on the sphee? Polems The polems foun in this chpte my e ssigne online in Enhnce WeAssign 1. enotes stightfow polem; 2. enotes intemeite polem; 3. enotes chllenging polem 1. full solution ville in the tuent olutions Mnul/tuy Guie 1. enotes polems most often ssigne in Enhnce WeAssign; these povie stuents with tgete feeck n eithe Mste It tutoil o Wtch It solution vieo. enotes sking fo quntittive n conceptul esoning enotes symolic esoning polem enotes Mste It tutoil ville in Enhnce WeAssign enotes guie polem she enotes pie polems tht evelop esoning with symols n numeicl vlues ection 25.1 Electic Potentil n Potentil Diffeence ection 25.2 Potentil Diffeence in Unifom Electic Fiel 1. () Clculte the spee of poton tht is ccelete fom est though n electic potentil iffeence of 120 V. () Clculte the spee of n electon tht is ccelete though the sme electic potentil iffeence. 2. How much wok is one (y ttey, geneto, o some othe souce of potentil iffeence) in moving Avogo s nume of electons fom n initil point whee the electic potentil is 9.00 V to point whee the electic potentil is V? (The potentil in ech cse is mesue eltive to common efeence point.) 3. A unifom electic fiel of mgnitue 325 V/m is iecte in the negtive y iection in Figue P25.3. The coointes of point e (20.200, ) m, n those of point e (0.400, 0.500) m. Clculte the electic potentil iffeence V 2 V using the she-line pth. 4. () Fin the electic potentil iffeence DV e equie to Figue P25.3 stop n electon (clle stopping potentil ) moving with n initil spee of m/s. () Woul poton tveling t the sme spee equie gete o lesse mgnitue of electic potentil iffeence? Eplin. (c) Fin symolic epession fo the tio of the poton stopping potentil n the electon stopping potentil, DV p /DV e. 5. An electon moving pllel to the is hs n initil spee of m/s t the oigin. Its spee is euce to m/s t the point cm. () Clculte y E the electic potentil iffeence etween the oigin n tht point. () Which point is t the highe potentil? 6. tting with the efinition of wok, pove tht t evey point on n equipotentil sufce, the sufce must e pepenicul to the electic fiel thee. 7. A pticle hving chge q mc n mss m kg is connecte to sting tht is L m long n tie to the pivot point P in Figue P25.7. The pticle, sting, n pivot point ll lie on fictionless, hoizontl tle. The pticle is elese fom est when the sting mkes n ngle u with unifom electic fiel of mgnitue E V/m. Detemine the spee of the pticle when the sting is pllel to the electic fiel. P m v 0 q u L Top view Figue P Review. A lock hving mss m n chge 1Q is connecte to n insulting sping hving foce constnt k. The lock lies on fictionless, insulting, hoizontl tck, n the system is immese in unifom electic fiel of mgnitue E iecte s shown in Figue P25.8 (pge 734). The lock is elese fom est when the sping is unstetche (t 5 0). We wish to show tht the ensuing motion of the lock is simple hmonic. () Consie the system of the lock, the sping, n the electic fiel. Is this system isolte o nonisolte? () Wht kins of potentil enegy v E

25 734 CHAPTER 25 Electic Potentil eist within this system? (c) Cll the initil configution of the system tht eisting just s the lock is elese fom est. The finl configution is when the lock momentily comes to est gin. Wht is the vlue of when the lock comes to est momentily? () At some vlue of we will cll 5 0, the lock hs zeo net foce on it. Wht nlysis moel escies the pticle in this sitution? (e) Wht is the vlue of 0? (f) Define new coointe system 9 such tht how tht 9 stisfies iffeentil eqution fo simple hmonic motion. (g) Fin the peio of the simple hmonic motion. (h) How oes the peio epen on the electic fiel mgnitue? ection 25.3 Electic Potentil n Potentil Enegy Due to Chge Pticles Note: Unless stte othewise, ssume the efeence level of potentil is V 5 0 t 5 `. 10. () Clculte the electic potentil cm fom n electon. () Wht is the electic potentil iffeence etween two points tht e cm n cm fom n electon? (c) How woul the nswes chnge if the electon wee eplce with poton? 11. Two point chges e on the y is. A 4.50-mC chge is locte t y cm, n mC chge is locte t y cm. Fin the totl electic potentil t () the oigin n () the point whose coointes e (1.50 cm, 0). 12. The two chges in Figue A P25.12 e septe y cm. Fin the electic potentil t () point A n () point B, which is hlfwy 60.0 etween the chges. B 13. Thee positive chges 15.0 nc 27.0 nc e locte t the cones of n equiltel tingle s in Figue P25.12 Figue P Fin n epession fo Q the electic potentil t the cente of the tingle. 14. Two chge pticles, Q nc n Q nc, e septe y 35.0 cm. () Wht is the electic potentil enegy of the pi? Eplin the significnce of the lgek m, Q 0 Figue P An insulting o hving line chge ensity l mc/m n line mss ensity m kg/m is elese fom est in unifom electic fiel E V/m iecte pepenicul to the o (Fig. P25.9). () Dete- E E mine the spee of the o fte it hs l, m tvele 2.00 m. () Wht If? How oes Figue P25.9 you nswe to pt () chnge if the electic fiel is not pepenicul to the o? Eplin. Q E 2Q Figue P25.13 ic sign of you nswe. () Wht is the electic potentil t point miwy etween the chge pticles? 15. Given two pticles with 2.00-mC chges s shown in Figue P25.15 n pticle with chge q C t the oigin, () wht is the net foce eete y the two 2.00-mC chges on the test chge q? () Wht is the electic fiel t the oigin ue to the two 2.00-mC pticles? (c) Wht is the electic potentil t the oigin ue to the two 2.00-mC pticles? 2.00 mc m y q 2.00 mc m Figue P The two chges in Figue P25.16 e septe y istnce cm, n Q nc. Fin () the electic potentil t A, () the electic potentil t B, n (c) the electic potentil iffeence etween B n A. 17. Fou point chges ech hving chge Q e locte Q 2Q t the cones of sque hving sies of length. Figue P25.16 Fin epessions fo () the totl electic potentil t the cente of the sque ue to the fou chges n () the wok equie to ing fifth chge q fom infinity to the cente of the sque. 18. At cetin istnce fom chge pticle, the mgnitue of the electic fiel is 500 V/m n the electic potentil is kv. () Wht is the istnce to the pticle? () Wht is the mgnitue of the chge? 19. A pticle with chge 1q is t the oigin. A pticle with chge 22q is t m on the is. () Fo wht finite vlue(s) of is the electic fiel zeo? () Fo wht finite vlue(s) of is the electic potentil zeo? 20. The thee chge pticles in Figue P25.20 e t the vetices of n isosceles tingle (whee cm). Tking q mc, clculte the electic potentil t point A, the mipoint of the se. 21. Two pticles ech with chge A mc e locte on the is. q q One is t m, n the othe is t m. () Detemine the Figue P25.20 electic potentil on the y is t y m. () Clculte the chnge in electic potentil enegy of the system s thi chge pticle of mc is ought fom infinitely f wy to position on the y is t y m. 22. how tht the mount of wok equie to ssemle fou ienticl chge pticles of mgnitue Q t the cones of sque of sie s is 5.41k e Q 2 /s. A q 2 B

26 Polems Fou ienticl chge pticles (q mc) e locte on the cones of ectngle s shown in Figue P The imensions of the ectngle e L cm n W cm. Clculte the chnge in electic potentil enegy of the system s the pticle t the lowe left q q cone in Figue P25.23 is ought to this position fom infinitely f wy. Assume the othe thee pticles in Figue P25.23 emin fie in position. 24. Two chge pticles of equl mgnitue e locte long the y is equl istnces ove n elow the is s shown in Figue P () Plot gph of the electic potentil t points long the is ove the intevl 23,, 3. You shoul plot the potentil in units of k e Q/. () Let the chge of the pticle locte t y 5 2 e negtive. Plot the potentil long the y is ove the intevl 24, y, 4. q 25. Five pticles with equl negtive chges 2q e plce symmeticlly oun cicle of ius R. Clculte the electic potentil t the cente of the cicle. 26. Thee pticles with equl positive chges q e t the cones of n q equiltel tingle of sie s shown in Figue P () At wht point, if ny, in the plne of the pticles is the electic potentil zeo? () Wht is the electic potentil t the position q q of one of the pticles ue to the othe two pticles in the tingle? Figue P Review. Two insulting sphees hve ii cm n cm, msses kg n kg, n unifomly istiute chges mc n 3.00 mc. They e elese fom est when thei centes e septe y 1.00 m. () How fst will ech e moving when they collie? () Wht If? If the sphees wee conuctos, woul the spees e gete o less thn those clculte in pt ()? Eplin. 28. Review. Two insulting sphees hve ii 1 n 2, msses m 1 n m 2, n unifomly istiute chges 2q 1 n q 2. They e elese fom est when thei centes e septe y istnce. () How fst is ech moving when they collie? () Wht If? If the sphees wee conuctos, woul thei spees e gete o less thn those clculte in pt ()? Eplin. 29. How much wok is equie to ssemle eight ienticl chge pticles, ech of mgnitue q, t the cones of cue of sie s? 30. Review. A light, unstesse sping hs length. Two ienticl pticles, ech with chge q, e connecte to the opposite ens of the sping. The pticles e hel sttiony istnce pt n then elese t the sme moment. The system then oscilltes on fictionless, hoizontl tle. The sping hs it of intenl kinetic fiction, y W L q Figue P25.23 y Q Q Figue P25.24 so the oscilltion is mpe. The pticles eventully stop viting when the istnce etween them is 3. Assume the system of the sping n two chge pticles is isolte. Fin the incese in intenl enegy tht ppes in the sping uing the oscilltions. 31. In 1911, Enest Ruthefo n his ssistnts Geige n Msen conucte n epeiment in which they scttee lph pticles (nuclei of helium toms) fom thin sheets of gol. An lph pticle, hving chge 12e n mss kg, is pouct of cetin ioctive ecys. The esults of the epeiment le Ruthefo to the ie tht most of n tom s mss is in vey smll nucleus, with electons in oit oun it. (This is the plnety moel of the tom, which we ll stuy in Chpte 42.) Assume n lph pticle, initilly vey f fom sttiony gol nucleus, is fie with velocity of m/s iectly tow the nucleus (chge 179e). Wht is the smllest istnce etween the lph pticle n the nucleus efoe the lph pticle eveses iection? Assume the gol nucleus emins sttiony. 32. Fou ienticl pticles, ech hving chge q n mss m, e elese fom est t the vetices of sque of sie L. How fst is ech pticle moving when thei istnce fom the cente of the sque oules? ection 25.4 Otining the Vlue of the Electic Fiel fom the Electic Potentil 33. The potentil in egion etween 5 0 n m is V 5 1, whee V n V/m. Detemine () the potentil t 5 0, 3.00 m, n 6.00 m n () the mgnitue n iection of the electic fiel t 5 0, 3.00 m, n 6.00 m. 34. Figue P25.34 epesents V (V) gph of the electic potentil in egion of 20 spce vesus position, whee the electic fiel 10 is pllel to the is. (cm) Dw gph of the component of the electic fiel Figue P25.34 vesus in this egion. 35. Ove cetin egion of spce, the electic potentil is V y 1 2yz 2. () Fin the epessions fo the, y, n z components of the electic fiel ove this egion. () Wht is the mgnitue of the fiel t the point P tht hs coointes (1.00, 0, 22.00) m? 36. An electic fiel in egion of spce is pllel to the is. The electic potentil vies with position s shown in Figue P Gph the component of the electic fiel vesus position in this egion of spce. V (V) (cm) Figue P25.36

27 736 CHAPTER 25 Electic Potentil 37. The electic potentil insie chge spheicl conucto of ius R is given y V 5 k e Q/R, n the potentil outsie is given y V 5 k e Q/. Using E 5 2V/, eive the electic fiel () insie n () outsie this chge istiution. 38. Figue P25.38 shows sevel equipotentil lines, ech lele y its potentil in volts. The istnce etween the lines of the sque gi epesents 1.00 cm. () Is the mgnitue of the fiel lge t A o t B? Eplin how you cn tell. () Eplin wht you cn etemine out E t B. (c) Repesent wht the electic fiel looks like y wing t lest eight fiel lines. 39. It is shown in Emple 25.7 tht the potentil t point P istnce ove one en of unifomly chge o of length, lying long the is is V 5 k Q e, ln, 1 " 2 1, 2 Use this esult to eive n epession fo the y component of the electic fiel t P. ection 25.5 Electic Potentil Due to Continuous Chge Distiutions 40. A unifomly chge insulting o of length 14.0 cm is ent into the shpe of semicicle s shown in Figue P The o hs totl chge of mc. Fin the electic potentil t O, the cente of the semicicle. 41. Consie ing of ius R with the totl chge Q spe unifomly ove its peimete. Wht is the potentil iffeence etween the point t the cente of the ing n point on its is istnce 2R fom the cente? 42. A o of length L (Fig. P25.42) lies long the is with its left en t the oigin. It hs nonunifom chge ensity l 5, whee is positive constnt. () Wht e the units of? () Clculte the electic potentil t A. 43. Fo the ngement escie in Polem 42, clculte the electic potentil t point B, which lies on the pepenicul isecto of the o istnce ove the is. 44. A wie hving unifom line chge ensity l is ent into the shpe shown in Figue P Fin the electic potentil t point O. 6 Numeicl vlues e in volts. A 8 Figue P25.38 y A B O Figue P25.40 B L Figue P25.42 Polems 42 n 43. 2R ection 25.6 Electic Potentil Due to Chge Conucto 45. How mny electons shoul e emove fom n initilly unchge spheicl conucto of ius m to pouce potentil of 7.50 kv t the sufce? 46. The electic fiel mgnitue on the sufce of n iegully shpe conucto vies fom 56.0 kn/c to 28.0 kn/c. Cn you evlute the electic potentil on the conucto? If so, fin its vlue. If not, eplin why not. 47. Electic chge cn ccumulte on n iplne in flight. You my hve oseve neele-shpe metl etensions on the wing tips n til of n iplne. Thei pupose is to llow chge to lek off efoe much of it ccumultes. The electic fiel oun the neele is much lge thn the fiel oun the oy of the iplne n cn ecome lge enough to pouce ielectic ekown of the i, ischging the iplne. To moel this pocess, ssume two chge spheicl conuctos e connecte y long conucting wie n 1.20-mC chge is plce on the comintion. One sphee, epesenting the oy of the iplne, hs ius of 6.00 cm; the othe, epesenting the tip of the neele, hs ius of 2.00 cm. () Wht is the electic potentil of ech sphee? () Wht is the electic fiel t the sufce of ech sphee? 48. A spheicl conucto hs ius of 14.0 cm n chge of 26.0 mc. Clculte the electic fiel n the electic potentil t () cm, () cm, n (c) cm fom the cente. ection 25.8 Applictions of Electosttics 49. Lightning cn e stuie with Vn e Gff geneto, which consists of spheicl ome on which chge is continuously eposite y moving elt. Chge cn e e until the electic fiel t the sufce of the ome ecomes equl to the ielectic stength of i. Any moe chge leks off in spks Figue P25.49 s shown in Figue P Assume the ome hs imete of 30.0 cm n is suoune y y i with ekown electic fiel of V/m. () Wht is the mimum potentil of the ome? () Wht is the mimum chge on the ome? Aitionl Polems O 50. Why is the following sitution impossile? In the Boh moel of the hyogen tom, n electon moves in cicul oit out poton. The moel sttes tht the electon cn eist only in cetin llowe oits oun the poton: those whose ius stisfies 5 n 2 ( nm), whee n 5 1, 2, R Figue P R E. R. Deggine/H. Amstong Roets

28 Polems 737 3,.... Fo one of the possile llowe sttes of the tom, the electic potentil enegy of the system is ev. 51. Review. Fom lge istnce wy, pticle of mss 2.00 g n chge 15.0 mc is fie t 21.0i^ m/s stight tow secon pticle, oiginlly sttiony ut fee to move, with mss 5.00 g n chge 8.50 mc. Both pticles e constine to move only long the is. () At the instnt of closest ppoch, oth pticles will e moving t the sme velocity. Fin this velocity. () Fin the istnce of closest ppoch. Afte the intection, the pticles will move f pt gin. At this time, fin the velocity of (c) the 2.00-g pticle n () the 5.00-g pticle. 52. Review. Fom lge istnce wy, pticle of mss m 1 n positive chge q 1 is fie t spee v in the positive iection stight tow secon pticle, oiginlly sttiony ut fee to move, with mss m 2 n positive chge q 2. Both pticles e constine to move only long the is. () At the instnt of closest ppoch, oth pticles will e moving t the sme velocity. Fin this velocity. () Fin the istnce of closest ppoch. Afte the intection, the pticles will move f pt gin. At this time, fin the velocity of (c) the pticle of mss m 1 n () the pticle of mss m The liqui-op moel of the tomic nucleus suggests high-enegy oscilltions of cetin nuclei cn split the nucleus into two unequl fgments plus few neutons. The fission poucts cquie kinetic enegy fom thei mutul Coulom epulsion. Assume the chge is istiute unifomly thoughout the volume of ech spheicl fgment n, immeitely efoe septing, ech fgment is t est n thei sufces e in contct. The electons suouning the nucleus cn e ignoe. Clculte the electic potentil enegy (in electon volts) of two spheicl fgments fom unium nucleus hving the following chges n ii: 38e n m, n 54e n m. 54. Review. In fi wethe, the electic fiel in the i t pticul loction immeitely ove the Eth s sufce is 120 N/C iecte ownw. () Wht is the sufce chge ensity on the goun? Is it positive o negtive? () Imgine the sufce chge ensity is unifom ove the plnet. Wht then is the chge of the whole sufce of the Eth? (c) Wht is the Eth s electic potentil ue to this chge? () Wht is the iffeence in potentil etween the he n the feet of peson 1.75 m tll? (Ignoe ny chges in the tmosphee.) (e) Imgine the Moon, with 27.3% of the ius of the Eth, h chge 27.3% s lge, with the sme sign. Fin the electic foce the Eth woul then eet on the Moon. (f) tte how the nswe to pt (e) compes with the gvittionl foce the Eth eets on the Moon. 55. The electic potentil immeitely outsie chge conucting sphee is 200 V, n 10.0 cm fthe fom the cente of the sphee the potentil is 150 V. Detemine () the ius of the sphee n () the chge on it. The electic potentil immeitely outsie nothe chge conucting sphee is 210 V, n 10.0 cm fthe fom the cente the mgnitue of the electic fiel is 400 V/m. Detemine (c) the ius of the sphee n () its chge on it. (e) Ae the nswes to pts (c) n () unique? 56. On y winte y, you scuff you lethe-sole shoes coss cpet n get shock when you eten the tip of one finge tow metl ookno. In k oom, you see spk pehps 5 mm long. Mke oe-of-mgnitue estimtes of () you electic potentil n () the chge on you oy efoe you touch the ookno. Eplin you esoning. 57. The electic potentil eveywhee on the y plne is V 5 36 " y " 2 1 1y whee V is in volts n n y e in metes. Detemine the position n chge on ech of the pticles tht cete this potentil. 58. () Use the ect esult fom Emple 25.4 to fin the electic potentil cete y the ipole escie in the emple t the point (3, 0). () Eplin how this nswe compes with the esult of the ppoimte epession tht is vli when is much gete thn. 59. Clculte the wok tht must e one on chges ought fom infinity to chge spheicl shell of ius R m to totl chge Q mc. 60. Clculte the wok tht must e one on chges ought fom infinity to chge spheicl shell of ius R to totl chge Q. 61. Fom Guss s lw, the electic fiel set up y unifom line of chge is l E 5 2pP 0 ^ whee ^ is unit vecto pointing illy wy fom the line n l is the line chge ensity long the line. Deive n epession fo the potentil iffeence etween 5 1 n Why is the following sitution Q impossile? You set up n pptus in you lotoy s follows. The is is R the symmety is of sttiony, unifomly chge ing of ius R m n chge Q mc (Fig. P25.62). You plce pticle with chge Q 5 Figue P mc n mss m kg t the cente of the ing n nge fo it to e constine to move only y long the is. When it is isplce slightly, the pticle is epelle y the ing n cceletes long the is. P The pticle moves fste thn you epecte n stikes the opposite wll of you lotoy t 40.0 m/s. 63. The thin, unifomly chge o shown in Figue P25.63 hs line chge ensity l. Fin n epession fo the electic potentil t P. L Q v Figue P25.63

29 738 CHAPTER 25 Electic Potentil 64. A unifomly chge filment lies long the is etween m n 5 1, m s shown in Figue P The totl chge on the filment is 1.60 nc. Clculte successive ppoimtions fo the electic potentil t the oigin y moeling the filment s () single chge pticle t m, () two nC chge pticles t m n m, n (c) fou nC chge pticles t m, m, m, n m. () Eplin how the esults compe with the potentil given y the ect epession P y V 5 k eq, ln, 1 Figue P Review. Two pllel pltes hving chges of equl mgnitue ut opposite sign e septe y 12.0 cm. Ech plte hs sufce chge ensity of 36.0 nc/m 2. A poton is elese fom est t the positive plte. Detemine () the mgnitue of the electic fiel etween the pltes fom the chge ensity, () the potentil iffeence etween the pltes, (c) the kinetic enegy of the poton when it eches the negtive plte, () the spee of the poton just efoe it stikes the negtive plte, (e) the cceletion of the poton, n (f) the foce on the poton. (g) Fom the foce, fin the mgnitue of the electic fiel. (h) How oes you vlue of the electic fiel compe with tht foun in pt ()? 66. A Geige Muelle tue is ition etecto tht consists of close, hollow, metl cyline (the cthoe) of inne ius n coil cylinicl wie (the noe) of ius (Fig. P25.66). The chge pe unit length on the noe is l, n the chge pe unit length on the cthoe is 2l. A gs fills the spce etween the electoes. When the tue is in use (Fig. P25.66) n high-enegy elementy pticle psses though this spce, it cn ionize n tom of the gs. The stong electic fiel mkes the esulting ion n electon ccelete in opposite iections. They stike othe molecules of the gs to ionize them, poucing n vlnche of electicl ischge. The pulse of electic cuent etween the wie n the cyline is counte y n etenl cicuit. () how tht the mgnitue of the electic potentil iffeence etween the wie n the cyline is DV 5 2k e l ln () how tht the mgnitue of the electic fiel in the spce etween cthoe n noe is DV E 5 ln 1 / 2 1 whee is the istnce fom the is of the noe to the point whee the fiel is to e clculte. 67. When n unchge conucting sphee of ius is plce t the oigin of n yz coointe system tht lies in n initilly unifom electic fiel E 5 E 0 k^, the esulting electic potentil is V(, y, z) 5 V 0 fo points insie the sphee n E 0 3 z V 1, y, z2 5 V 0 2 E 0 z y 2 1 z 2 2 3/2 fo points outsie the sphee, whee V 0 is the (constnt) electic potentil on the conucto. Use this eqution to etemine the, y, n z components of the esulting electic fiel () insie the sphee n () outsie the sphee. Chllenge Polems 68. A pticle with chge q is locte t 5 2R, n pticle with chge 22q is locte t the oigin. Pove tht the equipotentil sufce tht hs zeo potentil is sphee centee t (24R/3, 0, 0) n hving ius 5 2 3R. 69. () A unifomly chge cylinicl shell with no en cps hs totl chge Q, ius R, n length h. Detemine the electic potentil t point istnce fom the ight en of the cyline s shown in Figue P uggestion: Use the esult of Emple 25.5 y teting the cyline s collection of ing chges. () Wht If? Use the esult of Emple 25.6 to solve the sme polem fo soli cyline. h R Figue P As shown in Figue P25.70, two lge, pllel, veticl conucting pltes septe y istnce e chge so tht thei potentils e 1V 0 n 2V 0. A smll conucting ll of mss m n ius R (whee R,, ) hngs miwy etween the pltes. The the of length L suppoting the ll is conucting wie connecte to goun, so Anoe Cthoe l l Figue P25.66 Hnk Mogn/Photo Reseches, Inc. V 0 L V 0 Figue P25.70