23 ELECTRIC PTENTIAL whee d l is n infinitesiml displcement long the pticle s pth nd f is the ngle etween F nd d l t ech point long the pth. econd, if the foce F is consevtive, s we defined the tem in ection 7.3, the wok done y F cn lwys e expessed in tems of potentil enegy U. When the pticle moves fom point whee the potentil enegy is U to point whee it is U, the chnge in potentil enegy is DU 5 U 2 U nd the wok done y the foce is W 23. Electic Potentil Enegy 78 LEARNING GAL W 5 U 2 U 52 U 2 U 2 52DU (wok done y consevtive foce) (23.2) By studying this chpte, you will len: How to clculte the electic potentil enegy of collection of chges. The mening nd significnce of electic potentil. How to clculte the electic potentil tht collection of chges poduces t point in spce. How to use euipotentil sufces to visulize how the electic potentil vies in spce. How to use electic potentil to clculte the electic field.?in one type of welding, electic chge flows etween the welding tool nd the metl pieces tht e to e joined togethe. This poduces glowing c whose high tempetue fuses the pieces togethe. Why must the tool e held close to the pieces eing welded? When W is positive, U is gete thn U, DU is negtive, nd the potentil enegy deceses. Tht s wht hppens when sell flls fom high point () to lowe point () unde the influence of the eth s gvity; the foce of gvity does positive wok, nd the gvittionl potentil enegy deceses (Fig. 23.). When tossed ll is moving upwd, the gvittionl foce does negtive wok duing the scent, nd the potentil enegy inceses. Thid, the wokenegy theoem sys tht the chnge in kinetic enegy DK 5 K 2 K duing ny displcement is eul to the totl wok done on the pticle. If the only wok done on the pticle is done y consevtive foces, then E. (23.2) gives the totl wok, nd K 2 K 52U 2 U 2. We usully wite this s K U 5 K U (23.3) Tht is, the totl mechnicl enegy (kinetic plus potentil) is conseved unde these cicumstnces. 23. The wok done on sell moving in unifom gvittionl field. ject moving in unifom gvittionl field w 5 mg h The wok done y the gvittionl foce is the sme fo ny pth fom to : W 52DU 5 mgh 780 This chpte is out enegy ssocited with electicl intections. Evey time you tun on light, CD plye, o n electic pplince, you e mking use of electicl enegy, n indispensle ingedient of ou technologicl society. In Chptes 6 nd 7 we intoduced the concepts of wok nd enegy in the context of mechnics; now we ll comine these concepts with wht we ve lened out electic chge, electic foces, nd electic fields. Just s the enegy concept mde it possile to solve some kinds of mechnics polems vey simply, using enegy ides mkes it esie to solve viety of polems in electicity. When chged pticle moves in n electic field, the field exets foce tht cn do wok on the pticle. This wok cn lwys e expessed in tems of electic potentil enegy. Just s gvittionl potentil enegy depends on the height of mss ove the eth s sufce, electic potentil enegy depends on the position of the chged pticle in the electic field. We ll descie electic potentil enegy using new concept clled electic potentil, o simply potentil. In cicuits, diffeence in potentil fom one point to nothe is often clled voltge. The concepts of potentil nd voltge e cucil to undestnding how electic cicuits wok nd hve eully impotnt pplictions to electon ems used in cnce diothepy, high-enegy pticle cceletos, nd mny othe devices. 23. Electic Potentil Enegy The concepts of wok, potentil enegy, nd consevtion of enegy poved to e extemely useful in ou study of mechnics. In this section we ll show tht these concepts e just s useful fo undestnding nd nlyzing electicl intections. Let s egin y eviewing thee essentil points fom Chptes 6 nd 7. Fist, when foce F cts on pticle tht moves fom point to point, the wok done y the foce is given y line integl: W W 5 3 F # dl 5 3 F cos f dl (wok done y foce) (23.) Electic Potentil Enegy in Unifom Field Let s look t n electicl exmple of these sic concepts. In Fig. 23.2 pi of chged pllel metl pltes sets up unifom, downwd electic field with mgnitude E. The field exets downwd foce with mgnitude F 5 E on positive test chge. As the chge moves downwd distnce d fom point to point, the foce on the test chge is constnt nd independent of its loction. o the wok done y the electic field is the poduct of the foce mgnitude nd the component of displcement in the (downwd) diection of the foce: W 5 Fd 5 Ed (23.4) This wok is positive, since the foce is in the sme diection s the net displcement of the test chge. The y-component of the electic foce, F y 52 E, is constnt, nd thee is no x- o z-component. This is exctly nlogous to the gvittionl foce on mss m ne the eth s sufce; fo this foce, thee is constnt y-component F y 52mg nd the x- nd z-components e zeo. Becuse of this nlogy, we cn conclude tht the foce exeted on y the unifom electic field in Fig. 23.2 is consevtive, just s is the gvittionl foce. This mens tht the wok W done y the field is independent of the pth the pticle tkes fom to. We cn epesent this wok with potentil-enegy function U, just s we did fo gvittionl potentil enegy in ection 7.. The potentil enegy fo the gvittionl foce F y 52mg ws U 5 mgy; hence the potentil enegy fo the electic foce F y 52 E is U 5 Ey (23.5) When the test chge moves fom height y to height y, the wok done on the chge y the field is given y W 52DU 52 U 2 U 2 52 Ey 2 Ey 2 5 E y 2 y 2 (23.6) 23.2 The wok done on point chge moving in unifom electic field. Compe with Fig. 23.. Point chge moving in unifom electic field The wok done y the electic foce is the sme fo ny pth fom to : W 5 2DU 5 Ed y F 5 E y d
782 CHAPTER 23 Electic Potentil 23. Electic Potentil Enegy 783 23.3 A positive chge moving () in the diection of the electic field nd () in the diection opposite. 23.4 A negtive chge moving () in the diection of the electic field nd () in the diection opposite. Compe with Fig. 23.3. () Positive chge moves in the diection of E: Field does positive wok on chge. U deceses. y y y F 5 E y y () Positive chge moves opposite E: Field does negtive wok on chge. U inceses. y y y F 5 E When is gete thn (Fig. 23.3), the positive test chge moves downwd, in the sme diection s ; the displcement is in the sme diection s the foce F 5, so the field does positive wok nd U deceses. [In pticul, if y 2 y 5 d s in Fig. 23.2, E. (23.6) gives W 5 Ed, in geement with E. (23.4).] When y is less thn y (Fig. 23.3), the positive test chge moves upwd, in the opposite diection to 0 ; the displcement is opposite the foce, the field does negtive wok, nd U inceses. If the test chge is negtive, the potentil enegy inceses when it moves with the field nd deceses when it moves ginst the field (Fig. 23.4). Whethe the test chge is positive o negtive, the following genel ules pply: U inceses if the test chge moves in the diection opposite the electic foce F 5 (Figs. 23.3 nd 23.4); U deceses if moves in the sme diection s F 0 0 5 0 (Figs. 23.3 nd 23.4). This is the sme ehvio s fo gvittionl potentil enegy, which inceses if mss m moves upwd (opposite the diection of the gvittionl foce) nd deceses if m moves downwd (in the sme diection s the gvittionl foce). CAUTIN Electic potentil enegy The eltionship etween electic potentil enegy chnge nd motion in n electic field is n impotnt one tht we ll use often. It s lso eltionship tht tkes little effot to tuly undestnd. Tke the time to eview the peceding pgph thooughly nd to study Figs. 23.3 nd 23.4 cefully. Doing so now will help you temendously lte! Electic Potentil Enegy of Two Point Chges The ide of electic potentil enegy isn t esticted to the specil cse of unifom electic field. Indeed, we cn pply this concept to point chge in ny electic field cused y sttic chge distiution. Recll fom Chpte 2 tht () Negtive chge moves in the diection of E: Field does negtive wok on chge. U inceses. y F 5 E () Negtive chge moves opposite E: Field does positive wok on chge. U deceses. y we cn epesent ny chge distiution s collection of point chges. Theefoe it s useful to clculte the wok done on test chge moving in the electic field cused y single, sttiony point chge. We ll conside fist displcement long the dil line in Fig. 23.5, fom point to point. The foce on is given y Coulom s lw, nd its dil component is (23.7) If nd hve the sme sign ( o 2) the foce is epulsive nd F is positive; if the two chges hve opposite signs, the foce is ttctive nd F is negtive. The foce is not constnt duing the displcement, nd we hve to integte to clculte the wok W done on y this foce s moves fom to. We find W 5 3 F d 5 3 (23.8) d 5 2 The wok done y the electic foce fo this pticul pth depends only on the endpoints. In fct, the wok is the sme fo ll possile pths fom to. To pove this, we conside moe genel displcement (Fig. 23.6) in which nd do not lie on the sme dil line. Fom E. (23.) the wok done on duing this displcement is given y W 5 3 F 5 2 F cos f dl 5 3 But the figue shows tht cos f dl 5 d. Tht is, the wok done duing smll displcement d l depends only on the chnge d in the distnce etween the chges, which is the dil component of the displcement. Thus E. (23.8) is vlid even fo this moe genel displcement; the wok done on y the electic field 0 poduced y depends only on nd, not on the detils of the pth. Also, if etuns to its stting point y diffeent pth, the totl wok done in the ound-tip displcement is zeo (the integl in E. (23.8) is fom ck to ). These e the needed chcteistics fo consevtive foce, s we defined it in ection 7.3. Thus the foce on is consevtive foce. We see tht Es. (23.2) nd (23.8) e consistent if we define / to e the potentil enegy U when is t point, distnce fom, nd we define / to e the potentil enegy when is t point, distnce fom Test chge moves fom to long n ity pth. U 2 cos f dl d f F 2 2 dl d 23.5 Test chge moves long stight line extending dilly fom chge. As it moves fom to, the distnce vies fom to. Test chge moves fom to long dil line fom. 23.6 The wok done on chge y the electic field of chge does not depend on the pth tken, ut only on the distnces nd. y y F 5 E y y
784 CHAPTER 23 Electic Potentil 23. Electic Potentil Enegy 785 23.7 Gphs of the potentil enegy U of two point chges nd vesus thei seption. () nd hve the sme sign. U U o () nd hve opposite signs. o U. 0 As 0, U `. As `, U 0. U, 0 As 0, U 2`. As `, U 0.. Thus the potentil enegy U when the test chge chge is U 5 (electic potentil enegy of two point chges nd ) is t ny distnce fom (23.9) Note tht we hve not ssumed nything out the signs of nd ; E. (23.9) is vlid fo ny comintion of signs. The potentil enegy is positive if the chges nd hve the sme sign (Fig. 23.7) nd negtive if they hve opposite signs (Fig. 23.7). CAUTIN Electic potentil enegy vs. electic foce Be ceful not to confuse E. (23.9) fo the potentil enegy of two point chges with the simil expession in E. (23.7) fo the dil component of the electic foce tht one chge exets on the othe. The potentil enegy U is popotionl to /, while the foce component F is popotionl to / 2. Potentil enegy is lwys defined eltive to some efeence point whee U 5 0. In E. (23.9), U is zeo when nd e infinitely f pt nd 5`. Theefoe U epesents the wok tht would e done on the test chge y the field of if moved fom n initil distnce to infinity. If nd hve the sme sign, the intection is epulsive, this wok is positive, nd U is positive t ny finite seption (Fig. 23.7). If the chges hve opposite signs, the intection is ttctive, the wok done is negtive, nd U is negtive (Fig. 23.7). We emphsize tht the potentil enegy U given y E. (23.9) is shed popety of the two chges nd ; it is conseuence of the intection etween these two odies. If the distnce etween the two chges is chnged fom to, the chnge in potentil enegy is the sme whethe is held fixed nd is moved o is held fixed nd is moved. Fo this eson, we neve use the phse the electic potentil enegy of point chge. (Likewise, if mss m is t height h ove the eth s sufce, the gvittionl potentil enegy is shed popety of the mss m nd the eth. We emphsized this in ections 7. nd 2.3.) Guss s lw tells us tht the electic field outside ny spheiclly symmetic chge distiution is the sme s though ll the chge wee concentted t the cente. Theefoe E. (23.9) lso holds if the test chge is outside ny spheiclly symmetic chge distiution with totl chge t distnce fom the cente. The vlues of the enegies on the ight-hnd side of this expession e K 5 2 mv 2 5 2 9. 3 023 kg 23.00 3 0 6 m/s 2 2 5 4.0 3 0 28 J U 5 5 9.0 3 0 9 N # m 2 /C 2 2 3.20 3 029 C2.60 3 0 29 C2.00 3 0 20 m 5 4.6 3 0 28 J U 5 9.0 3 0 9 N # m2 /C 2 2 3.20 3 029 C2.60 3 0 29 C2 2.00 3 0 20 m 5 2.30 3 0 28 J Hence the finl kinetic enegy is K 5 2 mv 2 5 K U 2 U 5 4.0 3 0 28 J 4.6 3 0 28 J 2 2.30 3 0 28 J 5 6.4 3 0 28 J nd the finl speed of the positon is 2K v 5 Å m 5 2 6.4 3 0 28 J 2 Å 9. 3 0 23 kg 5 3.8 3 06 m/s The foce is epulsive, so the positon speeds up s it moves wy fom the sttiony lph pticle. () When the finl positions of the positon nd lph pticle e vey f pt, the seption ppoches infinity nd the finl potentil enegy U ppoches zeo. Then the finl kinetic enegy of the positon is K 5 K U 2 U 5 4.0 3 0 28 J 4.6 3 0 28 J 2 0 5 8.7 3 0 28 J nd its finl speed is 2K v 5 Å m 5 2 8.7 3 0 28 J 2 Å 9. 3 0 23 kg 5 4.4 3 06 m/s Comping to pt (), we see tht s the positon moves fom 5 2.00 3 0 20 m to infinity, the dditionl wok done on it y the electic field of the lph pticle inceses the speed y only out 6%. This is ecuse the electic foce deceses pidly with distnce. (c) If the moving chge is negtive, the foce on it is ttctive the thn epulsive, nd we expect it to slow down the thn speed up. The only diffeence in the ove clcultions is tht oth potentil-enegy untities e negtive. Fom pt (), t distnce 5 2.00 3 0 20 m we hve K 5 K U 2 U 5 4.0 3 0 28 J 24.6 3 0 28 J2 2 22.30 3 0 28 J2 5.79 3 0 28 J v 5 Å 2K m 5 2.0 3 06 m/s Fom pt (), t 5`the kinetic enegy of the electon would seem to e K 5 K U 2 U 5 4.0 3 0 28 J 24.6 3 0 28 J 2 2 0 525. 3 0 29 J But kinetic enegies cn neve e negtive! This esult mens tht the electon cn neve ech 5`; the ttctive foce ings the electon to hlt t finite distnce fom the lph pticle. The electon will then egin to move ck towd the lph pticle. You cn solve fo the distnce t which the electon comes momentily to est y setting K eul to zeo in the eution fo consevtion of mechnicl enegy. EVALUATE: It s useful to compe ou clcultions with Fig. 23.7. In pts () nd (), the chges hve the sme sign; since., the potentil enegy U is less thn U. In pt (c), the chges hve opposite signs; since., the potentil enegy U is gete (tht is, less negtive) thn U. Exmple 23. Consevtion of enegy with electic foces A positon (the ntipticle of the electon) hs mss of 9. 3 0 23 kg nd chge e 5.60 3 0 29 C. uppose positon moves in the vicinity of n lph pticle, which hs chge 2e 5 3.20 3 0 29 C. The lph pticle is moe thn 7000 times s mssive s the positon, so we ssume tht it is t est in some inetil fme of efeence. When the positon is.00 3 0 20 m fom the lph pticle, it is moving diectly wy fom the lph pticle t speed of 3.00 3 0 6 m/s. () Wht is the positon s speed when the two pticles e 2.00 3 0 20 m pt? () Wht is the positon s speed when it is vey f wy fom the lph pticle? (c) How would the sitution chnge if the moving pticle wee n electon (sme mss s the positon ut opposite chge)? LUTIN IDENTIFY: The electic foce etween the positon nd the lph pticle is consevtive, so mechnicl enegy (kinetic plus potentil) is conseved. ET UP: The kinetic nd potentil enegies t ny two points nd e elted y E. (23.3), K U 5 K U, nd the potentil enegy t ny distnce is given y E. (23.9). We e given complete infomtion out the system t point whee the two chges e.00 3 0 20 m pt. We use Es. (23.3) nd (23.9) to find the speed t two diffeent vlues of in pts () nd (), nd fo the cse whee the chge e is eplced y 2e in pt (c). EXECUTE: () In this pt, 5 2.00 3 0 20 m nd we wnt to find the finl speed v of the positon. This ppes in the expession fo the finl kinetic enegy, K 5 2 mv 2 ; solving the enegyconsevtion eution fo K, we hve K 5 K U 2 U Electic Potentil Enegy with evel Point Chges uppose the electic field in which chge moves is cused y sevel point chges, 2, 3, c t distnces, 2, 3, c fom, s in Fig. 23.8. Fo exmple, could e positive ion moving in the pesence of othe ions (Fig. 23.9). The totl electic field t ech point is the vecto sum of the fields due to the individul chges, nd the totl wok done on duing ny displcement is the sum of the contiutions fom the individul chges. Fom E. (23.9) we conclude tht the potentil enegy ssocited with the test chge t point in Fig. 23.8 is the lgeic sum (not vecto sum): U 5 2 2 3 3 c 2 5 i (point chge nd collection (23.0) of chges ) When is t diffeent point, the potentil enegy is given y the sme expession, ut, 2, c e the distnces fom, 2, c to point. The wok i i i 23.8 The potentil enegy ssocited with chge t point depends on the othe chges, 2, nd 3 nd on thei distnces, 2, nd fom point. 3 2 2 3 3 0
786 CHAPTER 23 Electic Potentil 23.2 Electic Potentil 787 23.9 This ion engine fo spcecft uses electic foces to eject stem of positive xenon ions Xe 2 t speeds in excess of 30 km/s. The thust poduced is vey low (out 0.09 newton) ut cn e mintined continuously fo dys, in contst to chemicl ockets, which poduce lge thust fo shot time (see Fig. 8.33). uch ion engines hve een used fo mneuveing inteplnety spcecft. done on chge when it moves fom to long ny pth is eul to the diffeence U 2 U etween the potentil enegies when is t nd t. We cn epesent ny chge distiution s collection of point chges, so E. (23.0) shows tht we cn lwys find potentil-enegy function fo ny sttic electic field. It follows tht fo evey electic field due to sttic chge distiution, the foce exeted y tht field is consevtive. Eutions (23.9) nd (23.0) define U to e zeo when ll the distnces, 2, c e infinite tht is, when the test chge is vey f wy fom ll the chges tht poduce the field. As with ny potentil-enegy function, the point whee U 5 0 is ity; we cn lwys dd constnt to mke U eul zeo t ny point we choose. In electosttics polems it s usully simplest to choose this point to e t infinity. When we nlyze electic cicuits in Chptes 25 nd 26, othe choices will e moe convenient. Eution (23.0) gives the potentil enegy ssocited with the pesence of the test chge in the 0 field poduced y, 2, 3,...But thee is lso potentil enegy involved in ssemling these chges. If we stt with chges, 2, 3,...ll septed fom ech othe y infinite distnces nd then ing them togethe so tht the distnce etween i nd j is ij, the totl potentil enegy U is the sum of the potentil enegies of intection fo ech pi of chges. We cn wite this s U 5 i,j i j ij (23.) This sum extends ove ll pis of chges; we don t let i 5 j (ecuse tht would e n intection of chge with itself), nd we include only tems with i, j to mke sue tht we count ech pi only once. Thus, to ccount fo the intection etween 3 nd 4, we include tem with i 5 3 nd j 5 4 ut not tem with i 5 4 nd j 5 3. Intepeting Electic Potentil Enegy As finl comment, hee e two viewpoints on electic potentil enegy. We hve defined it in tems of the wok done y the electic field on chged pticle moving in the field, just s in Chpte 7 we defined potentil enegy in tems of the wok done y gvity o y sping. When pticle moves fom point to point, the wok done on it y the electic field is W 5 U 2 U. Thus the potentil-enegy diffeence U 2 U euls the wok tht is done y the electic foce when the pticle moves fom to. When U is gete thn U, the field does positive wok on the pticle s it flls fom point of highe potentil enegy 2 to point of lowe potentil enegy 2. An ltentive ut euivlent viewpoint is to conside how much wok we would hve to do to ise pticle fom point whee the potentil enegy is U to point whee it hs gete vlue U (pushing two positive chges close togethe, fo exmple). To move the pticle slowly (so s not to give it ny kinetic enegy), we need to exet n dditionl extenl foce F ext tht is eul nd opposite to the electic-field foce nd does positive wok. The potentilenegy diffeence U 2 U is then defined s the wok tht must e done y n extenl foce to move the pticle slowly fom to ginst the electic foce. Becuse F ext is the negtive of the electic-field foce nd the displcement is in the opposite diection, this definition of the potentil diffeence U 2 U is euivlent to tht given ove. This ltentive viewpoint lso woks if U is less thn U, coesponding to loweing the pticle; n exmple is moving two positive chges wy fom ech othe. In this cse, U 2 U is gin eul to the wok done y the extenl foce, ut now this wok is negtive. We will use oth of these viewpoints in the next section to intepet wht is ment y electic potentil, o potentil enegy pe unit chge. Exmple 23.2 A system of point chges Two point chges e locted on the x-xis, 52et x 5 0 nd EXECUTE: () The wok tht must e done on 3 y n extenl 2 5et x 5. () Find the wok tht must e done y n extenl foce to ing thid point chge 3 5e fom infinity to potentil enegy U ssocited with 3 when it is t x 5 2 nd the foce F ext is eul to the diffeence etween two untities: the x 5 2. () Find the totl potentil enegy of the system of thee potentil enegy when it is infinitely f wy. The second of these chges. is zeo, so the wok tht must e done is eul to U. The distnces etween the chges e 3 5 2 nd 23 5, so fom E. (23.0), LUTIN IDENTIFY: This polem involves the eltionship etween the wok done to move point chge nd the chnge in potentil enegy. It lso involves the expession fo the potentil enegy of collection of point chges. ET UP: Figue 23.0 shows the finl ngement of the thee chges. To find the wok euied to ing 3 in fom infinity, we use E. (23.0) to find the potentil enegy ssocited with 3 in the pesence of nd 2. We then use E. (23.) to find the totl potentil enegy of the system. 23.0 u sketch of the sitution fte the thid chge hs een ought in fom infinity. Test You Undestnding of ection 23. Conside the system of thee point chges in Exmple 2.4 (ection 2.3) nd shown in Fig. 2.4. () Wht is the sign of the totl potentil enegy of this system? (i) positive; (ii) negtive; (iii) zeo. () Wht is the sign of the totl mount of wok you would hve to do to move these chges infinitely f fom ech othe? (i) positive; (ii) negtive; (iii) zeo. 23.2 Electic Potentil In ection 23. we looked t the potentil enegy U ssocited with test chge in n electic field. Now we wnt to descie this potentil enegy on pe unit chge sis, just s electic field descies the foce pe unit chge on chged pticle in the field. This leds us to the concept of electic potentil, often clled simply potentil. This concept is vey useful in clcultions involving enegies of chged pticles. It lso fcilittes mny electic-field clcultions ecuse electic potentil is closely elted to the electic field. When we need to detemine n electic field, it is often esie to detemine the potentil fist nd then find the field fom it. Potentil is potentil enegy pe unit chge. We define the potentil V t ny point in n electic field s the potentil enegy U pe unit chge ssocited with test chge t tht point: V 5 U o U 5 V W 5 U 5 3 3 2 23 2 5 e 2e 2 e 2 5 e 2 8pP 0 If 3 is ought in fom infinity long the x-xis, it is ttcted y ut is epelled moe stongly y 2 ; hence positive wok must e done to push 3 to the position t x 5 2. () The totl potentil enegy of the ssemlge of thee chges is given y E. (23.): U 5 i j 4pP 5 0 i,j ij 2 3 2 3 2 3 23 2 5 2e 2e2 2e 2e2 e 2e2 2 2 5 2e 2 8pP 0 EVALUATE: ince ou esult in pt () is negtive, the system hs lowe potentil enegy thn it would if the thee chges wee infinitely f pt. An extenl foce would hve to do negtive wok to ing the thee chges fom infinity to ssemle this entie ngement nd would hve to do positive wok to move the thee chges ck to infinity. (23.2) Potentil enegy nd chge e oth scls, so potentil is scl untity. Fom E. (23.2) its units e found y dividing the units of enegy y those of chge. The I unit of potentil, clled one volt ( V) in hono of the Itlin NLINE.3 Electicl Potentil Enegy nd Potentil
788 CHAPTER 23 Electic Potentil 23.2 Electic Potentil 789 scientist nd electicl expeimente Alessndo Volt (745827), euls joule pe coulom: V 5 volt 5 J/C 5 joule/coulom Let s put E. (23.2), which eutes the wok done y the electic foce duing displcement fom to to the untity 2DU 52U 2 U 2, on wok pe unit chge sis. We divide this eution y, otining In this expession, i is the distnce fom the ith chge, i, to the point t which V is evluted. Just s the electic field due to collection of point chges is the vecto sum of the fields poduced y ech chge, the electic potentil due to collection of point chges is the scl sum of the potentils due to ech chge. When we hve continuous distiution of chge long line, ove sufce, o though volume, we divide the chge into elements d, nd the sum in E. (23.5) ecomes n integl: 23. The voltge of this ttey euls the diffeence in potentil V 5 V 2 V etween its positive teminl (point ) nd its negtive teminl (point ). Point Point V 5.5 volts W 52 DU (23.3) whee V 5 U / is the potentil enegy pe unit chge t point nd similly fo V. We cll V nd V the potentil t point nd potentil t point, espectively. Thus the wok done pe unit chge y the electic foce when chged ody moves fom to is eul to the potentil t minus the potentil t. The diffeence V 2 V is clled the potentil of with espect to ; we sometimes evite this diffeence s V 5 V 2 V (note the ode of the suscipts). This is often clled the potentil diffeence etween nd, ut tht s miguous unless we specify which is the efeence point. In electic cicuits, which we will nlyze in lte chptes, the potentil diffeence etween two points is often clled voltge (Fig. 23.). Eution (23.3) then sttes: V, the potentil of with espect to, euls the wok done y the electic foce when UNIT chge moves fom to. Anothe wy to intepet the potentil diffeence V in E. (23.3) is to use the ltentive viewpoint mentioned t the end of ection 23.. In tht viewpoint, U 2 U is the mount of wok tht must e done y n extenl foce to move pticle of chge slowly fom to ginst the electic foce. The wok tht must e done pe unit chge y the extenl foce is then U 2 U 2 / 5 V 2 V 5 V. In othe wods: V, the potentil of with espect to, euls the wok tht must e done to move UNIT chge slowly fom to ginst the electic foce. An instument tht mesues the diffeence of potentil etween two points is clled voltmete. In Chpte 26 we will discuss the pinciple of the common type of moving-coil voltmete. Thee e lso much moe sensitive potentilmesuing devices tht use electonic mplifiction. Instuments tht cn mesue potentil diffeence of mv e common, nd sensitivities down to 0 22 V cn e ttined. Clculting Electic Potentil To find the potentil V due to single point chge, we divide E. (23.9) y : V 5 U 5 (potentil due to point chge) (23.4) whee is the distnce fom the point chge to the point t which the potentil is evluted. If is positive, the potentil tht it poduces is positive t ll points; if is negtive, it poduces potentil tht is negtive eveywhee. In eithe cse, V is eul to zeo t 5`, n infinite distnce fom the point chge. Note tht potentil, like electic field, is independent of the test chge tht we use to define it. imilly, we divide E. (23.0) y to find the potentil due to collection of point chges: V 5 U 5 i 52 U 2 U 2 52 V 2 V 2 5 V 2 V i i (potentil due to collection of point chges) (23.5) (potentil due to continuous distiution of chge) (23.6) whee is the distnce fom the chge element d to the field point whee we e finding V. We ll wok out sevel exmples of such cses. The potentil defined y Es. (23.5) nd (23.6) is zeo t points tht e infinitely f wy fom ll the chges. Lte we ll encounte cses in which the chge distiution itself extends to infinity. We ll find tht in such cses we cnnot set V 5 0 t infinity, nd we ll need to execise ce in using nd intepeting Es. (23.5) nd (23.6). CAUTIN Wht is electic potentil? Befoe getting too involved in the detils of how to clculte electic potentil, you should stop nd emind youself wht potentil is. The electic potentil t cetin point is the potentil enegy tht would e ssocited with unit chge plced t tht point. Tht s why potentil is mesued in joules pe coulom, o volts. Keep in mind, too, tht thee doesn t hve to e chge t given point fo potentil V to exist t tht point. (In the sme wy, n electic field cn exist t given point even if thee s no chge thee to espond to it.) Finding Electic Potentil fom Electic Field When we e given collection of point chges, E. (23.5) is usully the esiest wy to clculte the potentil V. But in some polems in which the electic field is known o cn e found esily, it is esie to detemine V fom. The foce F on test chge cn e witten s F 5 0, so fom E. (23.) the wok done y the electic foce s the test chge moves fom to is given y If we divide this y V 5 d 3 nd compe the esult with E. (23.3), we find V 2 V 5 3 E # dl 5 3 E cos f dl W 5 3 F # dl 5 3 # dl (potentil diffeence s n integl of ) (23.7) The vlue of V 2 V is independent of the pth tken fom to, just s the vlue of W is independent of the pth. To intepet E. (23.7), ememe tht is the electic foce pe unit chge on test chge. If the line integl E # d l is positive, the electic field does positive wok on positive test chge s it moves fom to. In this cse the electic potentil enegy deceses s the test chge moves, so the potentil enegy pe unit chge deceses s well; hence V is less thn V nd V 2 V is positive. As n illusttion, conside positive point chge (Fig. 23.2). The electic field is diected wy fom the chge, nd V 5 / is positive t ny finite distnce fom the chge. If you move wy fom the chge, in the diection of, you move towd lowe vlues of V; if you move towd the chge, in the diection opposite, you move towd gete vlues of V. Fo the negtive point chge in Fig. 23.2, is diected towd the chge nd V 5 / is negtive t ny finite distnce fom the chge. In this cse, if you move towd the chge, you e moving in the diection of nd in the diection of decesing (moe negtive) V. Moving wy fom the chge, in the diection opposite, 23.2 If you move in the diection of, electic potentil V deceses; if you move in the diection opposite, V inceses. () A positive point chge V inceses s you move inwd. () A negtive point chge V deceses s you move inwd. V deceses s you move outwd. V inceses s you move outwd.
790 CHAPTER 23 Electic Potentil 23.2 Electic Potentil 79 moves you towd incesing (less negtive) vlues of V. The genel ule, vlid fo ny electic field, is: Moving with the diection of mens moving in the diection of decesing V, nd moving ginst the diection of mens moving in the diection of incesing V. Also, positive test chge expeiences n electic foce in the diection of 0, towd lowe vlues of V; negtive test chge expeiences foce opposite, towd highe vlues of V. Thus positive chge tends to fll fom highpotentil egion to lowe-potentil egion. The opposite is tue fo negtive chge. Notice tht E. (23.7) cn e ewitten s V 2 V 52 3 E # d l (23.8) This hs negtive sign comped to the integl in E. (23.7), nd the limits e evesed; hence Es. (23.7) nd (23.8) e euivlent. But E. (23.8) hs slightly diffeent intepettion. To move unit chge slowly ginst the electic foce, we must pply n extenl foce pe unit chge eul to 2, eul nd opposite to the electic foce pe unit chge. Eution (23.8) sys tht V 2 V 5 V, the potentil of with espect to, euls the wok done pe unit chge y this extenl foce to move unit chge fom to. This is the sme ltentive intepettion we discussed unde E. (23.3). Eutions (23.7) nd (23.8) show tht the unit of potentil diffeence V 2 is eul to the unit of electic field N/C 2 multiplied y the unit of distnce m 2. Hence the unit of electic field cn e expessed s volt pe mete V/m 2, s well s N/C: V/m 5 volt/mete 5 N/C 5 newton/coulom In pctice, the volt pe mete is the usul unit of electic-field mgnitude. Electon Volts The mgnitude e of the electon chge cn e used to define unit of enegy tht is useful in mny clcultions with tomic nd nucle systems. When pticle with chge moves fom point whee the potentil is V to point whee it is V, the chnge in the potentil enegy U is U 2 U 5 V 2 V 2 5 V If the chge euls the mgnitude e of the electon chge,.602 3 0 29 C, nd the potentil diffeence is V 5 V, the chnge in enegy is U 2 U 5.602 3 0 29 C 2 V 2 5.602 3 0 29 J This untity of enegy is defined to e electon volt ev 2 : ev 5.602 3 0 29 J The multiples mev, kev, MeV, GeV, nd TeV e often used. CAUTIN Electon volts vs. volts Rememe tht the electon volt is unit of enegy, not unit of potentil o potentil diffeence! When pticle with chge e moves though potentil diffeence of volt, the chnge in potentil enegy is ev. If the chge is some multiple of e sy Ne the chnge in potentil enegy in electon volts is N times the potentil diffeence in volts. Fo exmple, when n lph pticle, which hs chge 2e, moves etween two points with potentil diffeence of 000 V, the chnge in potentil enegy is 2 000 ev 2 5 2000 ev. To confim this, we wite U 2 U 5 V 5 2e 2000 V 2 5 2 2.602 3 0 29 C 2000 V 2 5 3.204 3 0 26 J 5 2000 ev Although we hve defined the electon volt in tems of potentil enegy, we cn use it fo ny fom of enegy, such s the kinetic enegy of moving pticle. When we spek of one-million-electon-volt poton, we men poton with kinetic enegy of one million electon volts MeV 2, eul to 0 6 2.602 3 0 29 J 2 5.602 3 0 23 J (Fig. 23.3). Exmple 23.3 Electic foce nd electic potentil A poton (chge e 5.602 3 0 29 C ) moves in stight line fom point to point inside line cceleto, totl distnce d 5 0.50 m. The electic field is unifom long this line, with mgnitude E 5.5 3 0 7 V/m 5.5 3 0 7 N/C in the diection fom to. Detemine () the foce on the poton; () the wok done on it y the field; (c) the potentil diffeence V 2 V. LUTIN IDENTIFY: This polem uses the eltionship etween electic field (which we e given) nd electic foce (which is one of ou tget viles). It lso uses the eltionship mong foce, wok, nd potentil enegy diffeence. ET UP: We e given the electic field, so it is stightfowd to find the electic foce on the poton. Clculting the wok done on the poton y this foce is lso stightfowd ecuse is unifom, which mens tht the foce is constnt. nce the wok is known, we find the potentil diffeence using E. (23.3). EXECUTE: () The foce on the poton is in the sme diection s the electic field, nd its mgnitude is F 5 E 5.602 3 0 29 C 2.5 3 0 7 N/C 2 5 2.4 3 0 22 N () The foce is constnt nd in the sme diection s the displcement, so the wok done on the poton is W 5 Fd 5 2.4 3 0 22 N 20.50 m 2 5.2 3 0 22 J 5.2 3 0 22 ev J 2.602 3 0 29 J 5 7.5 3 0 6 ev 5 7.5 MeV 23.3 This cceleto t the Femi Ntionl Acceleto Lotoy in Illinois gives potons kinetic enegy of 400 MeV 4 3 0 8 ev 2. Additionl cceletion stges incese thei kinetic enegy to 980 GeV, o 0.98 TeV 9.8 3 0 ev 2. (c) Fom E. (23.3) the potentil diffeence is the wok pe unit chge, which is V 2 V 5 W 5.2 3 022 J.602 3 0 29 C 5 7.5 3 06 J/C 5 7.5 3 0 6 V 5 7.5 MV We cn get this sme esult even moe esily y ememeing tht electon volt euls volt multiplied y the chge e. ince the wok done is 7.5 3 0 6 ev nd the chge is e, the potentil diffeence is 7.5 3 0 6 ev 2 /e 5 7.5 3 0 6 V. EVALUATE: We cn check ou esult in pt (c) y using E. (23.7) o (23.8) to clculte n integl of the electic field. The ngle f etween the constnt field nd the displcement is zeo, so E. (23.7) ecomes V 2 V 5 3 E cos f dl 5 3 E dl 5 E 3 dl The integl of dl fom to is just the distnce d, so we gin find V 2 V 5 Ed 5.5 3 0 7 V/m 20.50 m 2 5 7.5 3 0 6 V
792 CHAPTER 23 Electic Potentil 23.2 Electic Potentil 793 Exmple 23.4 Potentil due to two point chges Exmple 23.6 Finding potentil y integtion An electic dipole consists of two point chges, 52 nc nd 2 522 nc, plced 0 cm pt (Fig. 23.4). Compute the potentils t points,, nd c y dding the potentils due to eithe chge, s in E. (23.5). LUTIN IDENTIFY: This is the sme ngement of chges s in Exmple 2.9 (ection 2.5). In tht exmple we clculted electic field t ech point y doing vecto sum. u tget vile in this polem is the electic potentil V t thee points. ET UP: To find V t ech point, we do the lgeic sum in E. (23.5): V 5 i 4pP 0 i i EXECUTE: At point the potentil due to the positive chge 5 9.0 3 0 9 N # m 2 /C 2 2 2 3 029 C 0.060 m 5 800 N # m/c 5 800 J/C 5 800 V nd the potentil due to the negtive chge 2 5 9.0 3 0 9 N # m 2 /C 2 2 22 3 029 C 2 2 0.040 m 522700 N # m/c 522700 J/C 522700 V The potentil V t point is the sum of these: V 5 800 V 22700 V 2 52900 V By simil clcultions you cn show tht t point the potentil due to the positive chge is 2700 V, the potentil due to the negtive chge is 2770 V, nd 2 V 5 2700 V 2770 V 2 5 930 V is is 23.4 Wht e the potentils t points,, nd c due to this electic dipole? c 4.0 cm 3.0 cm 3.0 cm 2 6.0 cm 4.0 cm At point c the potentil due to the positive chge is 5 9.0 3 0 9 N # m 2 /C 2 2 2 3 029 C 5 830 V 0.3 m The potentil due to the negtive chge is 2830 V, nd the totl potentil is zeo: V c 5 830 V 2830 V 2 5 0 The potentil is lso eul to zeo t infinity (infinitely f fom oth chges). EVALUATE: Comping this exmple with Exmple 2.9 shows tht it s much esie to clculte electic potentil ( scl) thn electic field ( vecto). We ll tke dvntge of this simplifiction wheneve possile. By integting the electic field s in E. (23.7), find the potentil t distnce fom point chge. LUTIN IDENTIFY: This polem sks us to find the electic potentil fom the electic field. ET UP: To find the potentil V t distnce fom the point chge, we let point in E. (23.7) e t distnce nd let point e t infinity (Fig. 23.5). As usul, we choose the potentil to e zeo t n infinite distnce fom the chge. EXECUTE: To cy out the integl, we cn choose ny pth we like etween points nd. The most convenient pth is stight dil line s shown in Fig. 23.5, so tht d l is in the dil diection nd hs mgnitude d. If is positive, nd d l e lwys pllel, so f50nd E. (23.7) ecomes ` ` V 2 0 5 3 E d 5 3 d 2 52 P ` V 5 This gees with E. (23.4). If is negtive, is dilly inwd while d l is still dilly outwd, so f580. ince cos 80 52, this dds minus sign to the ove esult. Howeve, the field mgnitude E is lwys positive, nd since is negtive, we must wite E 5 0 / 52/, giving nothe minus sign. The two minus signs cncel, nd the ove esult fo V is vlid fo point chges of eithe sign. Exmple 23.7 5 0 2 2 2 Moving though potentil diffeence 23.5 Clculting the potentil y integting fo single point chge. EVALUATE: We cn get the sme esult y using E. (2.7) fo the electic field, which is vlid fo eithe sign of, nd witing d l 5 ^d: ` V 2 0 5 V 5 3 E # d l ` 5 3 ^ # ^ d 5 2 3 V 5 ` 2 d Exmple 23.5 Potentil nd potentil enegy Compute the potentil enegy ssocited with point chge of 4.0 nc if it is plced t points,, nd c in Fig. 23.4. LUTIN IDENTIFY: We know the vlue of the electic potentil t ech of these points, nd we need to find the potentil enegy fo point chge plced t ech point. ET UP: Fo ny point chge, the ssocited potentil enegy is U 5 V. We use the vlues of V fom Exmple 23.4. EXECUTE: At point, U 5 V 5 4.0 3 0 29 C 22900 J/C 2 523.6 3 0 26 J At point, U 5 V 5 4.0 3 0 29 C 2930 J/C 2 5 7.7 3 0 26 J At point c, U c 5 V c 5 0 All of these vlues coespond to U nd V eing zeo t infinity. EVALUATE: Note tht no net wok is done on the 4.0-nC chge if it moves fom point c to infinity y ny pth. In pticul, let the pth e long the pependicul isecto of the line joining the othe two chges nd 2 in Fig. 23.4. As shown in Exmple 2.9 (ection 2.5), t points on the isecto the diection of is pependicul to the isecto. Hence the foce on the 4.0-nC chge is pependicul to the pth, nd no wok is done in ny displcement long it. In Fig. 23.6 dust pticle with mss m 5 5.0 3 0 29 kg 5 5.0 mg nd chge 5 2.0 nc stts fom est t point nd moves in stight line to point. Wht is its speed v t point? LUTIN IDENTIFY: This polem involves the chnge in speed nd hence kinetic enegy of the pticle, so we cn use n enegy ppoch. This polem would e difficult to solve without using enegy techniues, since the foce tht cts on the pticle vies in mgnitude s the pticle moves fom to. ET UP: nly the consevtive electic foce cts on the pticle, so mechnicl enegy is conseved: K U 5 K U EXECUTE: Fo this sitution, K nd K 5 5 0 2 mv2. We get the potentil enegies U 2 fom the potentils V 2 using E. (23.2): 23.6 The pticle moves fom point to point ; its cceletion is not constnt. 3.0 nc.0 cm Pticle.0 cm.0 cm U 5 V nd U 5 V. ustituting these into the enegy-consevtion eution nd solving fo v, we find 0 V 5 2 mv2 V 23.0 nc v 5 Å 2 V 2 V 2 m Continued
794 CHAPTER 23 Electic Potentil We clculte the potentils using E. (23.5), just s we did in Exmple 23.4: V 5 9.0 3 0 9 N # m 2 /C 2 2 3 3.0 3 029 C 0.00 m 23.0 3 029 C 2 0.020 m 2 5 350 V V 5 9.0 3 0 9 N # m 2 /C 2 2 3 3.0 3 029 C 0.020 m 23.0 3 029 C2 0.00 m 2 52350 V V 2 V 5 350 V 2 2 2350 V 2 5 2700 V Polem-olving ttegy 23. Finlly, v 5 Å 2 2.0 3 0 29 C 22700 V 2 5.0 3 0 29 kg EVALUATE: u esult mkes sense: The positive test chge gins speed s it moves wy fom the positive chge nd towd the negtive chge. To check unit consistency in the finl line of the clcultion, note tht V 5 J/C, so the numeto unde the dicl hs units of J o kg # m 2 /s 2. We cn use exctly this sme method to find the speed of n electon cceleted coss potentil diffeence of 500 V in n oscilloscope tue o 20 kv in TV pictue tue. The end-ofchpte polems include sevel exmples of such clcultions. Test You Undestnding of ection 23.2 If the electic potentil t cetin point is zeo, does the electic field t tht point hve to e zeo? (Hint: Conside point c in Exmple 23.4 nd Exmple 2.9.) 23.3 Clculting Electic Potentil 5 46 m/s When clculting the potentil due to chge distiution, we usully follow one of two outes. If we know the chge distiution, we cn use E. (23.5) o (23.6). if we know how the electic field depends on position, we cn use E. (23.7), defining the potentil to e zeo t some convenient plce. ome polems euie comintion of these ppoches. As you ed though these exmples, compe them with the elted exmples of clculting electic field in ection 2.5. You ll see how much esie it is to clculte scl electic potentils thn vecto electic fields. The mol is cle: Wheneve possile, solve polems using n enegy ppoch (using electic potentil nd electic potentil enegy) the thn dynmics ppoch (using electic fields nd electic foces). Clculting Electic Potentil IDENTIFY the elevnt concepts: Rememe tht potentil is potentil enegy pe unit chge. Undestnding this sttement cn get you long wy. ET UP the polem using the following steps:. Mke dwing tht clely shows the loctions of the chges (which my e point chges o continuous distiution of chge) nd you choice of coodinte xes. 2. Indicte on you dwing the position of the point t which you wnt to clculte the electic potentil V. ometimes this position will e n ity one (sy, point distnce fom the cente of chged sphee). EXECUTE the solution s follows:. To find the potentil due to collection of point chges, use E. (23.5). If you e given continuous chge distiution, devise wy to divide it into infinitesiml elements nd then use E. (23.6). Cy out the integtion, using ppopite limits to include the entie chge distiution. In the integl, e ceful out which geometic untities vy nd which e held constnt. 2. If you e given the electic field, o if you cn find it using ny of the methods pesented in Chpte 2 o 22, it my e esie to use E. (23.7) o (23.8) to clculte the potentil diffeence etween points nd. When ppopite, mke use of you feedom to define V to e zeo t some convenient plce, nd choose this plce to e point. (Fo point chges, this will usully e t infinity. Fo othe distiutions of chge especilly those tht themselves extend to infinity it my e convenient o necessy to define V to e zeo t some finite distnce fom the chge distiution. This is just like defining U to e zeo t gound level in gvittionl polems.) Then the potentil t ny othe point, sy, cn y found fom E. (23.7) o (23.8) with V 5 0. 3. Rememe tht potentil is scl untity, not vecto. It doesn t hve components! Howeve, you my hve to use components of the vectos nd d l when you use E. (23.7) o (23.8). EVALUATE you nswe: Check whethe you nswe gees with you intuition. If you esult gives V s function of position, mke gph of this function to see whethe it mkes sense. If you know the electic field, you cn mke ough check of you esult fo V y veifying tht V deceses if you move in the diection of. Exmple 23.8 A chged conducting sphee A solid conducting sphee of dius R hs totl chge. Find the potentil eveywhee, oth outside nd inside the sphee. LUTIN IDENTIFY: We used Guss s lw in Exmple 22.5 (ection 22.4) to find the electic field t ll points fo this chge distiution. We cn use tht esult to detemine the potentil t ll points. ET UP: We choose the oigin t the cente of the sphee. ince we know E t ll vlues of the distnce fom the cente of the sphee, we cn detemine V s function of. EXECUTE: Fom Exmple 22.5, t ll points outside the sphee the field is the sme s if the sphee wee emoved nd eplced y point chge. We tke V 5 0 t infinity, s we did fo point chge. Then the potentil t point outside the sphee t distnce fom its cente is the sme s the potentil due to point chge t the cente: V 5 The potentil t the sufce of the sphee is V sufce 5 / R. Inside the sphee, is zeo eveywhee; othewise, chge would move within the sphee. Hence if test chge moves fom ny point to ny othe point inside the sphee, no wok is done on tht chge. This mens tht the potentil is the sme t evey point inside the sphee nd is eul to its vlue / R t the sufce. EVALUATE: Figue 23.7 shows the field nd potentil s function of fo positive chge. In this cse the electic field points Ioniztion nd Coon Dischge The esults of Exmple 23.8 hve numeous pcticl conseuences. ne conseuence eltes to the mximum potentil to which conducto in i cn e ised. This potentil is limited ecuse i molecules ecome ionized, nd i ecomes conducto, t n electic-field mgnitude of out 3 3 0 6 V/m. Assume fo the moment tht is positive. When we compe the expessions in Exmple 23.8 fo the potentil V sufce nd field mgnitude E sufce t the sufce of chged conducting sphee, we note tht V sufce 5 E sufce R. Thus, if E m epesents the electic-field mgnitude t which i ecomes conductive (known s the dielectic stength of i), then the mximum potentil V m to which spheicl conducto cn e ised is V m 5 RE m Fo conducting sphee cm in dius in i, V m 5 0 22 m 23 3 0 6 V/m 2 5 30,000 V. No mount of chging could ise the potentil of conducting sphee of this size in i highe thn out 30,000 V; ttempting to ise the potentil futhe y dding ext chge would cuse the suounding i to ecome ionized nd conductive, nd the ext dded chge would lek into the i. To ttin even highe potentils, high-voltge mchines such s Vn de Gff genetos use spheicl teminls with vey lge dii (see Fig. 22.27 nd the photogph tht opens Chpte 22). Fo exmple, teminl of dius R 5 2 m hs mximum potentil V m 5 2 m 23 3 0 6 V/m 2 5 6 3 0 6 V 5 6 MV. uch mchines e sometimes plced in pessuized tnks filled with gs such s sulfu hexfluoide F 6 2 tht hs lge vlue of E m thn does i nd, theefoe, cn withstnd even lge fields without ecoming conductive. 23.3 Clculting Electic Potentil 795 dilly wy fom the sphee. As you move wy fom the sphee, in the diection of, V deceses (s it should). The electic field t the sufce hs mgnitude E sufce 5 0 / R 2. 23.7 Electic field mgnitude E nd potentil V t points inside nd outside positively chged spheicl conducto. E 5 0 R E V E 5 4pP R 2 0 V 5 R E 5 4pP 2 0 V 5
796 CHAPTER 23 Electic Potentil 23.3 Clculting Electic Potentil 797 23.8 The metl mst t the top of the Empie tte Building cts s lightning od. It is stuck y lightning s mny s 500 times ech ye. Exmple 23.9 ppositely chged pllel pltes Find the potentil t ny height y etween the two oppositely chged pllel pltes discussed in ection 23. (Fig. 23.9). LUTIN IDENTIFY: Fom ection 23. we know the electic potentil enegy U fo test chge s function of y. u gol hee is to find the electic potentil V due to the chges on the pltes s function of y. ET UP: Fom E. (23.5), U 5 Ey t point distnce y ove the ottom plte. We use this expession to detemine the potentil V t such point. EXECUTE: The potentil V y 2 t coodinte y is the potentil enegy pe unit chge: V y 2 5 U y 2 5 Ey We hve chosen U y 2, nd theefoe V y 2, to e zeo t point, whee y 5 0. Even if we choose the potentil to e diffeent fom zeo t, it is still tue tht V y 2 2 V 5 Ey 5 Ey The potentil deceses s we move in the diection of fom the uppe to the lowe plte. At point, whee y 5 d nd V y 2 5 V, V 2 V 5 Ed nd E 5 V 2 V d 5 V d whee V is the potentil of the positive plte with espect to the negtive plte. Tht is, the electic field euls the potentil diffeence etween the pltes divided y the distnce etween them. Fo given potentil diffeence V, the smlle the distnce d etween the two pltes, the gete the mgnitude E of the electic field. (This eltionship etween E nd V holds only fo the pln geomety we hve descied. It does not wok fo situtions such s concentic cylindes o sphees in which the electic field is not unifom.) u esult in Exmple 23.8 lso explins wht hppens with chged conducto with vey smll dius of cuvtue, such s shp point o thin wie. Becuse the mximum potentil is popotionl to the dius, even eltively smll potentils pplied to shp points in i poduce sufficiently high fields just outside the point to ionize the suounding i, mking it ecome conducto. The esulting cuent nd its ssocited glow (visile in dk oom) e clled coon. Lse pintes nd photocopying mchines use coon fom fine wies to spy chge on the imging dum (see Fig. 2.2). A lge-dius conducto is used in situtions whee it s impotnt to pevent coon. An exmple is the metl ll t the end of c dio ntenn, which pevents the sttic tht would e cused y coon. Anothe exmple is the lunt end of metl lightning od (Fig. 23.8). If thee is n excess chge in the tmosphee, s hppens duing thundestoms, sustntil chge of the opposite sign cn uild up on this lunt end. As esult, when the tmospheic chge is dischged though lightning olt, it tends to e ttcted to the chged lightning od the thn to othe ney stuctues tht could e dmged. (A conducting wie connecting the lightning od to the gound then llows the cuied chge to dissipte hmlessly.) A lightning od with shp end would llow less chge uildup nd hence would e less effective.? 23.9 The chged pllel pltes fom Fig. 23.2 y EVALUATE: u esult tells us how to mesue the chge density on the chges on the two pltes in Fig. 23.9. In Exmple 22.8 (ection 22.4), we deived the expession E 5s/P 0 fo the electic field E etween two conducting pltes hving sufce chge densities s nd 2s. etting this expession eul to E 5 V /d gives s5 P 0 V d The sufce chge density on the positive plte is diectly popotionl to the potentil diffeence etween the pltes, nd its vlue s cn e detemined y mesuing V. This techniue is useful ecuse no instuments e ville tht ed sufce chge density diectly. n the negtive plte the sufce chge density is 2s. CAUTIN Zeo potentil is ity You might think tht if conducting ody hs zeo potentil, it must necessily lso hve zeo net chge. But tht just isn t so! As n exmple, the plte t y 5 0 in Fig. 23.9 hs zeo potentil V 5 0 2 ut hs nonzeo chge pe unit e 2s. Rememe tht thee s nothing pticully specil out the plce whee potentil is zeo; we cn define this plce to e wheeve we wnt it to e. y d x Exmple 23.0 An infinite line chge o chged conducting cylinde Find the potentil t distnce fom vey long line of chge with line chge density (chge pe unit length) l. LUTIN IDENTIFY: ne ppoch to this polem is to divide the line of chge into infinitesiml elements, s we did in Exmple 2. (ection 2.5) to find the electic field poduced y such line. We could then integte s in E. (23.6) to find the net potentil V. In this cse, howeve, ou tsk is getly simplified ecuse we ledy know the electic field. ET UP: In oth Exmple 2. nd Exmple 22.6 (ection 22.4), we found tht the electic field t distnce fom long stightline chge (Fig. 23.20) hs only dil component, given y We use this expession to find the potentil y integting s in E. (23.7). EXECUTE: ince the field hs only dil component, the scl poduct # d l is eul to E d. Hence the potentil of ny point with espect to ny othe point, t dil distnces nd fom the line of chge, is V 2 V 5 3 E # d l 5 3 E d 5 l d 3 2pP 0 5 l ln 2pP 0 E 5 l 2pP 0 If we tke point t infinity nd set V 5 0, we find tht V is infinite: V 5 l ` ln 5` 2pP 0 This shows tht if we ty to define V to e zeo t infinity, then V must e infinite t ny finite distnce fom the line chge. This is not useful wy to define V fo this polem! The difficulty is tht the chge distiution itself extends to infinity. To get ound this difficulty, ememe tht we cn define V to e zeo t ny point we like. We set V 5 0 t point t n i- Exmple 23. A ing of chge Electic chge is distiuted unifomly ound thin ing of dius, with totl chge Q (Fig. 23.2). Find the potentil t point P on the ing xis t distnce x fom the cente of the ing. LUTIN IDENTIFY: We ledy know the electic field t ll points long the x-xis fom Exmple 2.0 (ection 2.5), so we could solve the polem y integting s in E. (23.7) to find V long this xis. Altentively, we could divide the ing up into infinitesiml segments nd use E. (23.6) to find V. ET UP: Figue 23.2 shows tht it s f esie to find V on the xis y using the infinitesiml-segment ppoch. Tht s ecuse 23.20 Electic field outside () long positively chged wie nd () long, positively chged cylinde. () E () ty dil distnce 0. Then the potentil V 5 V t point t dil distnce is given y V 2 0 5 l/2pp 0 2 ln 0 / 2, o V 5 l 0 ln 2pP 0 EVALUATE: Accoding to ou esult, if l is positive, then V deceses s inceses. This is s it should e: V deceses s we move in the diection of. Fom Exmple 22.6, the expession fo E with which we stted lso pplies outside long chged conducting cylinde with chge pe unit length l (Fig. 23.20). Hence ou esult lso gives the potentil fo such cylinde, ut only fo vlues of (the distnce fom the cylinde xis) eul to o gete thn the dius R of the cylinde. If we choose 0 to e the cylinde dius R, so tht V 5 0 when 5 R, then t ny point fo which. R, V 5 l R ln 2pP 0 Inside the cylinde, 5 0, nd V hs the sme vlue (zeo) s on the cylinde s sufce. R E 23.2 All the chge in ing of chge Q is the sme distnce fom point P on the ing xis. Q 5 x 2 2 x P Continued
798 CHAPTER 23 Electic Potentil 23.4 Euipotentil ufces 799 ll pts of the ing (tht is, ll elements of the chge distiution) e the sme distnce fom point P. EXECUTE: Figue 23.2 shows tht the distnce fom ech chge element d on the ing to the point P is 5 "x 2 2. Hence we cn tke the fcto / outside the integl in E. (23.6), nd V 5 d 3 5 "x 2 2 3 d 5 Potentil is scl untity; thee is no need to conside components of vectos in this clcultion, s we hd to do when we found Exmple 23.2 A line of chge Electic chge Q is distiuted unifomly long line o thin od of length 2. Find the potentil t point P long the pependicul isecto of the od t distnce x fom its cente. Q "x 2 2 the electic field t P. o the potentil clcultion is lot simple thn the field clcultion. EVALUATE: When x is much lge thn, the ove expession fo V ecomes ppoximtely eul to V 5 Q/ x. This coesponds to the potentil of point chge Q t distnce x. o when we e vey f wy fom chged ing, it looks like point chge. (We dew simil conclusion out the electic field of ing in Exmple 2.0.) These esults fo V cn lso e found y integting the expession fo found in Exmple 2.0 (see Polem 23.69). E x 23.22 u sketch fo this polem. in long such contou line, the gvittionl potentil enegy mgy does not chnge ecuse the elevtion y is constnt. Thus contou lines on topogphic mp e elly cuves of constnt gvittionl potentil enegy. Contou lines e close togethe in egions whee the tein is steep nd thee e lge chnges in elevtion ove smll hoizontl distnce; the contou lines e fthe pt whee the tein is gently sloping. A ll llowed to oll downhill will expeience the getest downhill gvittionl foce whee contou lines e closest togethe. By nlogy to contou lines on topogphic mp, n euipotentil sufce is thee-dimensionl sufce on which the electic potentil V is the sme t evey point. If test chge is moved fom point to point on such sufce, the electic potentil enegy V emins constnt. In egion whee n electic field is pesent, we cn constuct n euipotentil sufce though ny point. In digms we usully show only few epesenttive euipotentils, often with eul potentil diffeences etween djcent sufces. No point cn e t two diffeent potentils, so euipotentil sufces fo diffeent potentils cn neve touch o intesect. 23.23 Contou lines on topogphic mp e cuves of constnt elevtion nd hence of constnt gvittionl potentil enegy. LUTIN IDENTIFY: This is the sme sitution s in Exmple 2. (ection 2.5), whee we found n expession fo the electic field t n ity point on the x-xis. We could integte using E. (23.7) to find V. Insted, we ll integte ove the chge distiution using E. (23.6) to get it moe expeience with this ppoch. ET UP: Figue 23.22 shows the sitution. Unlike the sitution in Exmple 23., ech chge element dq is diffeent distnce fom point P. EXECUTE: As in Exmple 2., the element of chge dq coesponding to n element of length dy on the od is given y dq 5 Q/2 2 dy. The distnce fom dq to P is "x 2 y 2, nd the contiution dv tht it mkes to the potentil t P is dv 5 Q 2 To get the potentil t P due to the entie od, we integte dv ove the length of the od fom y 52to y 5 : V 5 Q 2 3 2 dy "x 2 y 2 dy "x 2 y 2 You cn look up the integl in tle. The finl esult is V 5 Q 2 ln "2 x 2 2 " 2 x 2 2 EVALUATE: We cn check ou esult y letting x ppoch infinity. In this limit the point P is infinitely f fom ll of the chge, so we expect V to ppoch zeo; we invite you to veify tht it does so. As in Exmple 23., this polem is simple thn finding t point P ecuse potentil is scl untity nd no vecto clcultions e involved. Euipotentil ufces nd Field Lines Becuse potentil enegy does not chnge s test chge moves ove n euipotentil sufce, the electic field cn do no wok on such chge. It follows tht must e pependicul to the sufce t evey point so tht the electic foce is lwys pependicul to the displcement of chge moving on the sufce. Field lines nd euipotentil sufces e lwys mutully pependicul. In genel, field lines e cuves, nd euipotentils e cuved sufces. Fo the specil cse of unifom field, in which the field lines e stight, pllel, nd eully spced, the euipotentils e pllel plnes pependicul to the field lines. Figue 23.24 shows thee ngements of chges. The field lines in the plne of the chges e epesented y ed lines, nd the intesections of the euipotentil sufces with this plne (tht is, coss sections of these sufces) e shown s lue lines. The ctul euipotentil sufces e thee-dimensionl. At ech cossing of n euipotentil nd field line, the two e pependicul. In Fig. 23.24 we hve dwn euipotentils so tht thee e eul potentil diffeences etween djcent sufces. In egions whee the mgnitude of is lge, the euipotentil sufces e close togethe ecuse the field does el- 23.24 Coss sections of euipotentil sufces (lue lines) nd electic field lines (ed lines) fo ssemlies of point chges. Thee e eul potentil diffeences etween djcent sufces. Compe these digms to those in Fig. 2.29, which showed only the electic field lines. Test You Undestnding of ection 23.3 If the electic field t cetin point is zeo, does the electic potentil t tht point hve to e zeo? (Hint: Conside the cente of the ing in Exmple 23. nd Exmple 2.0.) () A single positive chge () An electic dipole (c) Two eul positive chges 23.4 Euipotentil ufces Field lines (see ection 2.6) help us visulize electic fields. In simil wy, the potentil t vious points in n electic field cn e epesented gphiclly y euipotentil sufces. These use the sme fundmentl ide s topogphic mps like those used y hikes nd mountin climes (Fig. 23.23). n topogphic mp, contou lines e dwn though points tht e ll t the sme elevtion. Any nume of these could e dwn, ut typiclly only few contou lines e shown t eul spcings of elevtion. If mss m is moved ove the te- V 5 30 V V 5 50 V V 5 70 V V 5 230 V V 5 30 V V 5 250 V V 5 0 V V 5 50 V V 5 270 V V 5 70 V Electic field lines V 5 30 V Coss sections of euipotentil sufces V 5 50 V V 5 70 V