Problems. 714 Chapter 25 Electric Potential

Transcription

1 714 Chpter 25 lectric Potentil direction of the electric field? () out of the pge (b) into the pge (c) towrd the right (d) towrd the left (e) towrd the top of the pge (f) towrd the bottom of the pge (g) the field is zero 4. O A prticle with chrge 40 nc is on the is t the point with coordinte 0. A second prticle, with chrge 20 nc, is on the is t 500 mm. (i) Is there point t finite distnce where the electric field is zero? () Yes; it is to the left of 0. (b) Yes; it is between 0 nd 500 mm. (c) Yes; it is to the right of 500 mm. (d) No. (ii) Is the electric potentil zero t this point? () No; it is positive. (b) Yes. (c) No; it is negtive. (d) No such point eists. (iii) Is there point t finite distnce where the electric potentil is zero? () Yes; it is to the left of 0. (b) Yes; it is between 0 nd 500 mm. (c) Yes; it is to the right of 500 mm. (d) No. (iv) Is the electric field zero t this point? () No; it points to the right. (b) Yes. (c) No; it points to the left. (d) No such point eists. 5. The potentil energ of pir of chrged prticles with the sme sign is positive, wheres the potentil energ of pir of chrged prticles with opposite signs is negtive. Give phsicl eplntion of this sttement. 6. Describe the equipotentil surfces for () n infinite line of chrge nd (b) uniforml chrged sphere. 7. O In certin region of spce, the electric field is zero. From this fct, wht cn ou conclude bout the electric potentil in this region? () It is zero. (b) It is constnt. (c) It is positive. (d) It is negtive. (e) None of these nswers is necessril true. 8. O A filment running long the is from the origin to 80 cm crries electric chrge with uniform densit. At the point P with coordintes ( 80 cm, 80 cm), this filment cretes potentil 100 V. Now we dd nother filment long the is, running from the origin to 80 cm, crring the sme mount of chrge with the sme uniform densit. At the sme point P, does the pir of filments crete potentil () greter thn 200 V, (b) 200 V, (c) between 141 V nd 200 V, (d) 141 V, (e) between 100 V nd 141 V, (f) 100 V, (g) between 0 nd 100 V, or (h) 0? 9. O In different eperimentl trils, n electron, proton, or doubl chrged ogen tom (O ) is fired within vcuum tube. The prticle s trjector crries it through point where the electric potentil is 40 V nd then through point t different potentil. Rnk ech of the following cses ccording to the chnge in kinetic energ of the prticle over this prt of its flight, from the lrgest increse to the lrgest decrese in kinetic energ. () An electron moves from 40 V to 60 V. (b) An electron moves from 40 V to 20 V. (c) A proton moves from 40 V to 20 V. (d) A proton moves from 40 V to 10 V. (e) An O ion moves from 40 V to 50 V. (f) An O ion moves from 40 V to 60 V. For comprison, include lso in our rnking (g) zero chnge nd (h) 10 electron volts of chnge in kinetic energ. In our rnking, displ n cses of equlit. 10. Wht determines the mimum potentil to which the dome of Vn de Grff genertor cn be rised? 11. O (i) A metllic sphere A of rdius 1 cm is severl centimeters w from metllic sphericl shell B of rdius 2 cm. Chrge 450 nc is plced on A, with no chrge on B or nwhere nerb. Net, the two objects re joined b long, thin, metllic wire (s shown in Fig ), nd finll the wire is removed. How is the chrge shred between A nd B? () 0 on A, 450 nc on B (b) 50 nc on A nd 400 nc on B, with equl volume chrge densities (c) 90 nc on A nd 360 nc on B, with equl surfce chrge densities (d) 150 nc on A nd 300 nc on B (e) 225 nc on A nd 225 nc on B (f) 450 nc on A nd 0 on B (g) in some other predictble w (h) in some unpredictble w (ii) A metllic sphere A of rdius 1 cm with chrge 450 nc hngs on n insulting thred inside n unchrged thin metllic sphericl shell B of rdius 2 cm. Net, A is mde temporril to touch the inner surfce of B. How is the chrge then shred between them? Choose from the sme possibilities. Arnold Arons, the onl phsics techer et to hve his picture on the cover of Time mgzine, suggested the ide for this question. 12. Stud Figure 23.3 nd the ccompning tet discussion of chrging b induction. When the grounding wire is touched to the rightmost point on the sphere in Figure 23.3c, electrons re drined w from the sphere to leve the sphere positivel chrged. Suppose the grounding wire is touched to the leftmost point on the sphere insted. Will electrons still drin w, moving closer to the negtivel chrged rod s the do so? Wht kind of chrge, if n, remins on the sphere? Problems The Problems from this chpter m be ssigned online in WebAssign. Sign in t nd go to ThomsonNOW to ssess our understnding of this chpter s topics with dditionl quizzing nd conceptul questions. 1, 2, 3 denotes strightforwrd, intermedite, chllenging; denotes full solution vilble in Student Solutions Mnul/Stud Guide; denotes coched solution with hints vilble t denotes developing smbolic resoning; denotes sking for qulittive resoning; denotes computer useful in solving problem Section 25.1 lectric Potentil nd Potentil Difference 1. () Clculte the speed of proton tht is ccelerted from rest through potentil difference of 120 V. (b) Clculte the speed of n electron tht is ccelerted through the sme potentil difference. 2. How much work is done (b btter, genertor, or some other source of potentil difference) in moving Avogdro s number of electrons from n initil point where the electric potentil is 9.00 V to point where the poten- 2 = intermedite; 3 = chllenging; = SSM/SG; = ThomsonNOW; = smbolic resoning; = qulittive resoning

2 Problems 715 til is 5.00 V? (The potentil in ech cse is mesured reltive to common reference point.) Section 25.2 Potentil Difference in Uniform lectric Field 3. The difference in potentil between the ccelerting pltes in the electron gun of television picture tube is bout V. If the distnce between these pltes is 1.50 cm, wht is the mgnitude of the uniform electric field in this region? 4. A uniform electric field of mgnitude 325 V/m is directed in the negtive direction in Figure P25.4. The coordintes of point A re ( 0.200, 0.300) m nd those of point B re (0.400, 0.500) m. Clculte the potentil difference V B V A, using the blue pth. A Figure P An electron moving prllel to the is hs n initil speed of m/s t the origin. Its speed is reduced to m/s t the point 2.00 cm. Clculte the potentil difference between the origin nd tht point. Which point is t the higher potentil? 6. Strting with the definition of work, prove tht t ever point on n equipotentil surfce the surfce must be perpendiculr to the electric field there. 7. Review problem. A block hving mss m nd chrge Q is connected to n insulting spring hving constnt k. The block lies on frictionless, insulting horizontl trck, nd the sstem is immersed in uniform electric field of mgnitude directed s shown in Figure P25.7. If the block is relesed from rest when the spring is unstretched (t 0), () b wht mimum mount does the spring epnd? (b) Wht is the equilibrium position of the block? (c) Show tht the block s motion is simple hrmonic nd determine its period. (d) Wht If? Repet prt (), ssuming the coefficient of kinetic friction between block nd surfce is m k. k m, Q B zontl tble. The prticle is relesed from rest when the string mkes n ngle u 60.0 with uniform electric field of mgnitude 300 V/m. Determine the speed of the prticle when the string is prllel to the electric field (point in Fig. P25.8). P L m q u Top View Figure P An insulting rod hving liner chrge densit l 40.0 mc/m nd liner mss densit m kg/m is relesed from rest in uniform electric field 100 V/m directed perpendiculr to the rod (Fig. P25.9). () Determine the speed of the rod fter it hs trveled 2.00 m. (b) Wht If? How does our nswer to prt () chnge if the electric field is not perpendiculr to the rod? plin. l, m Figure P25.9 Section 25.3 lectric Potentil nd Potentil nerg Due to Point Chrges Note: Unless stted otherwise, ssume the reference level of potentil is V 0 t r. 10. Given two prticles with 2.00-mC chrges s shown in Figure P25.10 nd prticle with chrge q C t the origin, () wht is the net force eerted b the two 2.00-mC chrges on the test chrge q? (b) Wht is the electric field t the origin due to the two 2.00-mC prticles? (c) Wht is the electric potentil t the origin due to the two 2.00-mC prticles? 2.00 mc m q 2.00 mc m Figure P Figure P A prticle hving chrge q 2.00 mc nd mss m kg is connected to string tht is L 1.50 m long nd tied to the pivot point P in Figure P25.8. The prticle, string, nd pivot point ll lie on frictionless, hori- 11. () Find the potentil t distnce of 1.00 cm from proton. (b) Wht is the potentil difference between two points tht re 1.00 cm nd 2.00 cm from proton? (c) Wht If? Repet prts () nd (b) for n electron. 12. A prticle with chrge q is t the origin. A prticle with chrge 2q is t 2.00 m on the is. () For wht finite vlue(s) of is the electric field zero? (b) For wht finite vlue(s) of is the electric potentil zero? 2 = intermedite; 3 = chllenging; = SSM/SG; = ThomsonNOW; = smbolic resoning; = qulittive resoning

3 716 Chpter 25 lectric Potentil 13. At certin distnce from chrged prticle, the mgnitude of the electric field is 500 V/m nd the electric potentil is 3.00 kv. () Wht is the distnce to the prticle? (b) Wht is the mgnitude of the chrge? 14. Two chrged prticles, Q nc nd Q nc, re seprted b 35.0 cm. () Wht is the potentil energ of the pir? plin the significnce of the lgebric sign of our nswer. (b) Wht is the electric potentil t point midw between the chrged prticles? 15. The three chrged prticles in Figure P25.15 re t the vertices of n isosceles tringle. Clculte the electric potentil t the midpoint of the bse, tking q 7.00 mc. q 4.00 cm 20. Compre this problem with Problem 19 in Chpter 23. Five equl negtive chrged prticles q re plced smmetricll round circle of rdius R. Clculte the electric potentil t the center of the circle. 21. Compre this problem with Problem 35 in Chpter 23. Three prticles with equl positive chrges q re t the corners of n equilterl tringle of side s shown in Figure P () At wht point, if n, in the plne of the prticles is the electric potentil zero? (b) Wht is the electric potentil t the point P due to the two prticles t the bse of the tringle? 22. Two chrged prticles of equl mgnitude re locted long the is equl distnces bove nd below the is s shown in Figure P () Plot grph of the potentil t points long the is over the intervl 3 3. You should plot the potentil in units of k e Q /. (b) Let the chrge of the prticle locted t be negtive. Plot the potentil long the is over the intervl 4 4. q q 2.00 cm Figure P Compre this problem with Problem 16 in Chpter 23. Two chrged prticles ech of mgnitude 2.00 mc re locted on the is. One is t 1.00 m, nd the other is t 1.00 m. () Determine the electric potentil on the is t m. (b) Clculte the chnge in electric potentil energ of the sstem s third chrged prticle of 3.00 mc is brought from infinitel fr w to position on the is t m. 17. Compre this problem with Problem 47 in Chpter 23. Four identicl chrged prticles (q 10.0 mc) re locted on the corners of rectngle s shown in Figure P The dimensions of the rectngle re L 60.0 cm nd W 15.0 cm. Clculte the chnge in electric potentil energ of the sstem s the prticle t the lower left corner in Figure P23.47 is brought to this position from infinitel fr w. Assume the other three prticles in Figure P23.47 remin fied in position. 18. Two chrged prticles hve effects t the origin, described b the epressions N # m 2 >C 2 c C cos 70 î m2 nd C m2 2 sin 70 ĵ C m2 2 ĵ d N # m 2 >C 2 c C 0.07 m C 0.03 m d () Identif the loctions of the prticles nd the chrges on them. (b) Find the force on prticle with chrge 16.0 nc plced t the origin. (c) Find the work required to move this third chrged prticle to the origin from ver distnt point. 19. Show tht the mount of work required to ssemble four identicl chrged prticles of mgnitude Q t the corners of squre of side s is 5.41k e Q 2 /s. Q 0 Q Figure P Review problem. Two insulting spheres hve rdii cm nd cm, msses kg nd kg, nd uniforml distributed chrges of 2.00 mc nd 3.00 mc. The re relesed from rest when their centers re seprted b 1.00 m. () How fst will ech be moving when the collide? Suggestion: Consider conservtion of energ nd of liner momentum. (b) Wht If? If the spheres were conductors, would the speeds be greter or less thn those clculted in prt ()? plin. 24. Review problem. Two insulting spheres hve rdii r 1 nd r 2, msses m 1 nd m 2, nd uniforml distributed chrges q 1 nd q 2. The re relesed from rest when their centers re seprted b distnce d. () How fst is ech moving when the collide? Suggestion: Consider conservtion of energ nd conservtion of liner momentum. (b) Wht If? If the spheres were conductors, would their speeds be greter or less thn those clculted in prt ()? plin. 25. Review problem. A light, unstressed spring hs length d. Two identicl prticles, ech with chrge q, re connected to the opposite ends of the spring. The prticles re held sttionr distnce d prt nd then relesed t the sme moment. The sstem then oscilltes on horizontl, frictionless tble. The spring hs bit of internl kinetic friction, so the oscilltion is dmped. The prticles eventull stop vibrting when the distnce between them is 3d. Find the increse in internl energ tht ppers in the spring during the oscilltions. Assume the sstem of the spring nd two chrged prticles is isolted. 2 = intermedite; 3 = chllenging; = SSM/SG; = ThomsonNOW; = smbolic resoning; = qulittive resoning

4 Problems In 1911, rnest Rutherford nd his ssistnts Geiger nd Mrsden conducted n eperiment in which the scttered lph prticles from thin sheets of gold. An lph prticle, hving chrge 2e nd mss kg, is product of certin rdioctive decs. The results of the eperiment led Rutherford to the ide tht most of the mss of n tom is in ver smll nucleus, with electrons in orbit round it, his plnetr model of the tom. Assume n lph prticle, initill ver fr from gold nucleus, is fired with velocit of m/s directl towrd the nucleus (chrge 79e). How close does the lph prticle get to the nucleus before turning round? Assume the gold nucleus remins sttionr. 27. Four identicl prticles ech hve chrge q nd mss m. The re relesed from rest t the vertices of squre of side L. How fst is ech prticle moving when their distnce from the center of the squre doubles? 28. How much work is required to ssemble eight identicl chrged prticles, ech of mgnitude q, t the corners of cube of side s? Section 25.4 Obtining the Vlue of the lectric Field from the lectric Potentil 29. The potentil in region between 0 nd 6.00 m is V b, where 10.0 V nd b 7.00 V/m. Determine () the potentil t 0, 3.00 m, nd 6.00 m nd (b) the mgnitude nd direction of the electric field t 0, 3.00 m, nd 6.00 m. 30. The electric potentil inside chrged sphericl conductor of rdius R is given b V k e Q /R, nd the potentil outside is given b V k e Q /r. Using r dv/dr, derive the electric field () inside nd (b) outside this chrge distribution. 31. Over certin region of spce, the electric potentil is V z 2. Find the epressions for the,, nd z components of the electric field over this region. Wht is the mgnitude of the field t the point P tht hs coordintes (1, 0, 2) m? 32. Figure P25.32 shows severl equipotentil lines, ech lbeled b its potentil in volts. The distnce between the lines of the squre grid represents 1.00 cm. () Is the mgnitude of the field lrger t A or t B? plin how ou cn tell. (b) plin wht ou cn determine bout S t B. (c) Represent wht the field looks like b drwing t lest eight field lines Figure P It is shown in mple 25.7 tht the potentil t point P distnce bove one end of uniforml chrged rod of length ling long the is is A B Q V k e / ln / 22 / 2 b Use this result to derive n epression for the component of the electric field t P. Suggestion: Replce with. Section 25.5 lectric Potentil Due to Continuous Chrge Distributions 34. Consider ring of rdius R with the totl chrge Q spred uniforml over its perimeter. Wht is the potentil difference between the point t the center of the ring nd point on its is distnce 2R from the center? 35. A rod of length L (Fig. P25.35) lies long the is with its left end t the origin. It hs nonuniform chrge densit l, where is positive constnt. () Wht re the units of? (b) Clculte the electric potentil t A. A d Figure P25.35 Problems 35 nd For the rrngement described in Problem 35, clculte the electric potentil t point B, which lies on the perpendiculr bisector of the rod distnce b bove the is. 37. Compre this problem with Problem 27 in Chpter 23. A uniforml chrged insulting rod of length 14.0 cm is bent into the shpe of semicircle s shown in Figure P The rod hs totl chrge of 7.50 mc. Find the electric potentil t O, the center of the semicircle. 38. A wire hving uniform liner chrge densit l is bent into the shpe shown in Figure P Find the electric potentil t point O. 2R O B L R b Figure P25.38 Section 25.6 lectric Potentil Due to Chrged Conductor 39. A sphericl conductor hs rdius of 14.0 cm nd chrge of 26.0 mc. Clculte the electric field nd the electric potentil t () r 10.0 cm, (b) r 20.0 cm, nd (c) r 14.0 cm from the center. 40. How mn electrons should be removed from n initill unchrged sphericl conductor of rdius m to produce potentil of 7.50 kv t the surfce? 41. The electric field on the surfce of n irregulrl shped conductor vries from 56.0 kn/c to 28.0 kn/c. Clculte the locl surfce chrge densit t the point on the surfce where the rdius of curvture of the surfce is () gretest nd (b) smllest. 2R 2 = intermedite; 3 = chllenging; = SSM/SG; = ThomsonNOW; = smbolic resoning; = qulittive resoning

5 718 Chpter 25 lectric Potentil 42. lectric chrge cn ccumulte on n irplne in flight. You m hve observed needle-shped metl etensions on the wing tips nd til of n irplne. Their purpose is to llow chrge to lek off before much of it ccumultes. The electric field round the needle is much lrger thn the field round the bod of the irplne nd cn become lrge enough to produce dielectric brekdown of the ir, dischrging the irplne. To model this process, ssume two chrged sphericl conductors re connected b long conducting wire nd chrge of 1.20 mc is plced on the combintion. One sphere, representing the bod of the irplne, hs rdius of 6.00 cm, nd the other, representing the tip of the needle, hs rdius of 2.00 cm. () Wht is the electric potentil of ech sphere? (b) Wht is the electric field t the surfce of ech sphere? Section 25.8 Applictions of lectrosttics 43. Lightning cn be studied with Vn de Grff genertor, essentill consisting of sphericl dome on which chrge is continuousl deposited b moving belt. Chrge cn be dded until the electric field t the surfce of the dome becomes equl to the dielectric strength of ir. An more chrge leks off in sprks s shown in Figure P Assume the dome hs dimeter of 30.0 cm nd is surrounded b dr ir with dielectric strength V/m. () Wht is the mimum potentil of the dome? (b) Wht is the mimum chrge on the dome? Imge not vilble due to copright restrictions 44. A Geiger-Mueller tube is rdition detector tht consists of closed, hollow, metl clinder (the cthode) of inner rdius r nd coil clindricl wire (the node) of rdius r b (Fig. P25.44). The chrge per unit length on the node is l, nd the chrge per unit length on the cthode is l. A gs fills the spce between the electrodes. When high-energ elementr prticle psses through this spce, it cn ionize n tom of the gs. The strong electric field mkes the resulting ion nd electron ccelerte in opposite directions. The strike other molecules of the gs to ionize them, producing n vlnche of electricl dischrge. The pulse of electric current between the wire nd the clinder is counted b n eternl circuit. () Show tht the mgnitude of the potentil difference between the wire nd the clinder is V 2k e l ln r r b b (b) Show tht the mgnitude of the electric field in the spce between cthode nd node is V ln 1r >r b 2 1 r b where r is the distnce from the is of the node to the point where the field is to be clculted. r b Cthode r Anode l l Figure P25.44 Problems 44 nd The results of Problem 44 ppl lso to n electrosttic precipittor (Figs nd P25.44). An pplied potentil difference V V V b 50.0 kv is to produce n electric field of mgnitude 5.50 MV/m t the surfce of the centrl wire. Assume the outer clindricl wll hs uniform rdius r m. () Wht should be the rdius r b of the centrl wire? You will need to solve trnscendentl eqution. (b) Wht is the mgnitude of the electric field t the outer wll? Additionl Problems 46. Review problem. From lrge distnce w, prticle of mss 2.00 g nd chrge 15.0 mc is fired t 21.0 î m/s stright towrd second prticle, originll sttionr but free to move, with mss 5.00 g nd chrge 8.50 mc. () At the instnt of closest pproch, both prticles will be moving t the sme velocit. plin wh. (b) Find this velocit. (c) Find the distnce of closest pproch. (d) Find the velocities of both prticles fter the seprte gin. 47. The liquid-drop model of the tomic nucleus suggests high-energ oscilltions of certin nuclei cn split the nucleus into two unequl frgments plus few neutrons. The fission products cquire kinetic energ from their mutul Coulomb repulsion. Clculte the electric potentil energ (in electron volts) of two sphericl frgments from urnium nucleus hving the following chrges nd rdii: 38e nd m, 54e nd m. Assume the chrge is distributed uniforml throughout the volume of ech sphericl frgment nd, immeditel before seprting, ech frgment is t rest nd their surfces re in contct. The electrons surrounding the nucleus cn be ignored. 48. In fir wether, the electric field in the ir t prticulr loction immeditel bove the rth s surfce is 120 N/C directed downwrd. () Wht is the surfce chrge densit on the ground? Is it positive or negtive? (b) Imgine the tmosphere is stripped off nd the surfce chrge densit is uniform over the plnet. Wht then is the chrge of the whole surfce of the rth? (c) Wht is the rth s electric potentil? (d) Wht is the difference in potentil between the hed nd the feet of person 1.75 m tll? (e) Imgine the Moon, with 27.3% of the rdius of the rth, hd chrge 27.3% s lrge, with the sme sign. Find the electric force tht the rth would then eert on the Moon. (f) Stte how the nswer to prt (e) compres with the grvittionl force the rth eerts on the Moon. (g) A 2 = intermedite; 3 = chllenging; = SSM/SG; = ThomsonNOW; = smbolic resoning; = qulittive resoning

6 Problems 719 dust prticle of mss 6.00 mg is in the ir ner the surfce of the sphericl rth. Wht chrge must the dust prticle crr to be suspended in equilibrium between the electric nd grvittionl forces eerted on it? Ignore buonc. (h) The rth is not perfectl sphericl. It hs n equtoril bulge due to its rottion, so the rdius of curvture of the ground is slightl lrger t the poles thn t the equtor. Would the dust prticle in prt (g) require more chrge or less chrge to be suspended t the equtor compred with being suspended t one of the poles? plin our nswer with reference to vritions in both the electric force nd the grvittionl force. 49. The Bohr model of the hdrogen tom sttes tht the single electron cn eist onl in certin llowed orbits round the proton. The rdius of ech Bohr orbit is r n 2 ( nm), where n 1, 2, 3,.... Clculte the electric potentil energ of hdrogen tom when the electron () is in the first llowed orbit, with n 1, (b) is in the second llowed orbit, with n 2, nd (c) hs escped from the tom, with r. press our nswers in electron volts. 50. On dr winter d, ou scuff our lether-soled shoes cross crpet nd get shock when ou etend the tip of one finger towrd metl doorknob. In drk room, ou see sprk perhps 5 mm long. Mke order-of-mgnitude estimtes of () our electric potentil nd (b) the chrge on our bod before ou touch the doorknob. plin our resoning. 51. The electric potentil immeditel outside chrged conducting sphere is 200 V, nd 10.0 cm frther from the center of the sphere the potentil is 150 V. () Is this informtion sufficient to determine the chrge on the sphere nd its rdius? plin. (b) The electric potentil immeditel outside nother chrged conducting sphere is 210 V, nd 10.0 cm frther from the center the mgnitude of the electric field is 400 V/m. Is this informtion sufficient to determine the chrge on the sphere nd its rdius? plin. 52. As shown in Figure P25.52, two lrge, prllel, verticl conducting pltes seprted b distnce d re chrged so tht their potentils re V 0 nd V 0. A smll conducting bll of mss m nd rdius R (where R V d) is hung midw between the pltes. The thred of length L supporting the bll is conducting wire connected to ground, so the potentil of the bll is fied t V 0. The bll hngs stright down in stble equilibrium when V 0 is sufficientl smll. Show tht the equilibrium of the bll is unstble if V 0 eceeds the criticl vlue k e d 2 mg/(4rl). Suggestion: Consider the forces on the bll when it is displced distnce V L. 53. The electric potentil everwhere on the plne is given b V where V is in volts nd nd re in meters. Determine the position nd chrge on ech of the prticles tht crete this potentil. 54. Compre this problem with Problem 28 in Chpter 23. () A uniforml chrged clindricl shell hs totl chrge Q, rdius R, nd height h. Determine the electric potentil t point distnce d from the right end of the clinder s shown in Figure P Suggestion: Use the result of mple 25.5 b treting the clinder s collection of ring chrges. (b) Wht If? Use the result of mple 25.6 to solve the sme problem for solid clinder. h Figure P Clculte the work tht must be done to chrge sphericl shell of rdius R to totl chrge Q. 56. () Use the ect result from mple 25.4 to find the electric potentil creted b the dipole described t the point (3, 0). (b) plin how this nswer compres with the result of the pproimte epression tht is vlid when is much greter thn. 57. From Guss s lw, the electric field set up b uniform line of chrge is S l 2pP 0 r b rˆ where rˆ is unit vector pointing rdill w from the line nd l is the liner chrge densit long the line. Derive n epression for the potentil difference between r r 1 nd r r Four blls, ech with mss m, re connected b four nonconducting strings to form squre with side s shown in Figure P The ssembl is plced on horizontl, nonconducting, frictionless surfce. Blls 1 nd 2 ech hve chrge q, nd blls 3 nd 4 re unchrged. Find the mimum speed of blls 3 nd 4 fter the string connecting blls 1 nd 2 is cut. R d 1 2 L 3 4 Figure P25.58 V 0 V 0 d Figure P The is is the smmetr is of sttionr, uniforml chrged ring of rdius R nd chrge Q (Fig. P25.59). A prticle with chrge Q nd mss M is locted initill t 2 = intermedite; 3 = chllenging; = SSM/SG; = ThomsonNOW; = smbolic resoning; = qulittive resoning

7 720 Chpter 25 lectric Potentil the center of the ring. When it is displced slightl, the prticle ccelertes long the is to infinit. Show tht the ultimte speed of the prticle is v 2k eq 2 1>2 MR b Q R Uniforml chrged ring Figure P The thin, uniforml chrged rod shown in Figure P25.60 hs liner chrge densit l. Find n epression for the electric potentil t P. Q v rrngement shown, epress V in terms of Crtesin coordintes using r ( 2 2 ) 1/2 nd cos u Using these results nd gin tking r W, clculte the field components nd. 62. A solid sphere of rdius R hs uniform chrge densit r nd totl chrge Q. Derive n epression for its totl electric potentil energ. Suggestion: Imgine the sphere is constructed b dding successive lers of concentric shells of chrge dq (4pr 2 dr)r nd use du V dq. 63. A disk of rdius R (Fig. P25.63) hs nonuniform surfce chrge densit s Cr, where C is constnt nd r is mesured from the center of the disk to point on the surfce of the disk. Find (b direct integrtion) the potentil t P. R >2 P P b L Figure P An electric dipole is locted long the is s shown in Figure P The mgnitude of its electric dipole moment is defined s p 2q. () At point P, which is fr from the dipole (r W ), show tht the electric potentil is V k ep cos u r 2 (b) Clculte the rdil component r nd the perpendiculr component u of the ssocited electric field. Note tht u (1/r)( V/ u). Do these results seem resonble for u 90 nd 0? For r 0? (c) For the dipole q q u r 1 r r 2 P Figure P25.61 r u Figure P A uniforml chrged filment lies long the is between 1.00 m nd 3.00 m s shown in Figure The totl chrge on the filment is 1.60 nc. Clculte successive pproimtions for the electric potentil t the origin b modeling the filment s () single chrged prticle t 2.00 m, (b) two nc chrged prticles t 1.5 m nd 2.5 m, nd (c) four nC chrged prticles t 1.25 m, 1.75 m, 2.25 m, nd 2.75 m. Net, write nd eecute computer progrm tht will reproduce the results of prts (), (b), nd (c) nd etend our clcultion to (d) 32 nd (e) 64 equll spced chrged prticles. (f) plin how the results compre with the potentil given b the ect epression V k eq / / ln b 65. Two prllel pltes hving chrges of equl mgnitude but opposite sign re seprted b 12.0 cm. ch plte hs surfce chrge densit of 36.0 nc/m 2. A proton is relesed from rest t the positive plte. Determine () the potentil difference between the pltes, (b) the kinetic energ of the proton when it reches the negtive plte, (c) the speed of the proton just before it strikes the negtive plte, (d) the ccelertion of the proton, nd (e) the force on the proton. (f) From the force, find the mgnitude of the electric field nd show tht it is equl to the electric field found from the chrge densities on the pltes. 66. A prticle with chrge q is locted t R, nd prticle with chrge 2q is locted t the origin. Prove tht the equipotentil surfce tht hs zero potentil is sphere centered t ( 4R/3, 0, 0) nd hving rdius r 2R/3. 2 = intermedite; 3 = chllenging; = SSM/SG; = ThomsonNOW; = smbolic resoning; = qulittive resoning

8 Answers to Quick Quizzes When n unchrged conducting sphere of rdius is plced t the origin of n z coordinte sstem tht lies in n initill uniform electric field S 0 kˆ, the resulting electric potentil is V(,, z) V 0 for points inside the sphere nd 0 3 z V 1,, z2 V 0 0 z z 2 2 3>2 for points outside the sphere, where V 0 is the (constnt) electric potentil on the conductor. Use this eqution to determine the,, nd z components of the resulting electric field. Answers to Quick Quizzes 25.1 (i), (b). When moving stright from to, nd d S s both point towrd the right. Therefore, the dot product S # d S s in qution 25.3 is positive nd V is negtive. (ii), (). From qution 25.3, U q 0 V, so if negtive test chrge is moved through negtive potentil difference, the chnge in potentil energ is positive. Work must be done to move the chrge in the direction opposite to the electric force on it to, to, to, to. Moving from to decreses the electric potentil b 2 V, so the electric field performs 2 J of work on ech coulomb of positive chrge tht moves. Moving from to decreses the electric potentil b 1 V, so 1 J of work is done b the field. It tkes no work to move the chrge from to S becuse the electric potentil does not chnge. Moving from to increses the electric potentil b 1 V; therefore, the field does 1 J of work per unit of positive chrge tht moves (i), (c). The potentil is estblished onl b the source chrge nd is independent of the test chrge. (ii), (). The potentil energ of the two-chrge sstem is initill negtive due to the product of chrges of opposite sign in qution When the sign of q 2 is chnged, both chrges re negtive nd the potentil energ of the sstem is positive (). If the potentil is constnt (zero in this cse), its derivtive long this direction is zero. 2 = intermedite; 3 = chllenging; = SSM/SG; = ThomsonNOW; = smbolic resoning; = qulittive resoning