QUESTIONS 677 Checkpoint 4 The figure shows two large, parallel, nonconucting sheets with ientical (positive) uniform surface charge ensities, an a sphere with a uniform (positive) volume charge ensit. Rank the four numbere points accoring to the magnitue of the net electric fiel there, greatest first. 1 2 3 4 Review & Summar Gauss Law Gauss law an Coulomb s law are ifferent was of escribing the relation between charge an electric fiel in static situations. Gauss law is q enc (Gauss law), (23-6) in which q enc is the net charge insie an imaginar close surface (a Gaussian surface) an is the net flu of the electric fiel through the surface: (electric flu through a E : A : (23-4) Gaussian surface). Coulomb s law can be erive from Gauss law. Applications of Gauss Law Using Gauss law an, in some cases, smmetr arguments, we can erive several important results in electrostatic situations. Among these are: 1. An ecess charge on an isolate conuctor is locate entirel on the outer surface of the conuctor. 2. The eternal electric fiel near the surface of a charge conuctor is perpenicular to the surface an has a magnitue that epens on the surface charge ensit s : E ś Within the conuctor, E. (conucting surface). (23-11) 3. The electric fiel at an point ue to an infinite line of charge with uniform linear charge ensit l is perpenicular to the line of charge an has magnitue (line of charge), (23-12) where r is the perpenicular istance from the line of charge to the point. 4. The electric fiel ue to an infinite nonconucting sheet with uniform surface charge ensit s is perpenicular to the plane of the sheet an has magnitue E s (sheet of charge). (23-13) 2 5. The electric fiel outsie a spherical shell of charge with raius R an total charge q is irecte raiall an has magnitue E 1 E 4p l 2p r q r 2 (spherical shell, for r R). (23-15) Here r is the istance from the center of the shell to the point at which E is measure. (The charge behaves, for eternal points, as if it were all locate at the center of the sphere.) The fiel insie a uniform spherical shell of charge is eactl zero: E (spherical shell, for r R). (23-16) 6. The electric fiel insie a uniform sphere of charge is irecte raiall an has magnitue E q 4p R3 r. (23-2) Questions 1 A surface has the area vector A : (2î 3ĵ) m 2. What is the flu of a uniform electric fiel through the area if the fiel is E : 4î N/C an E : 4kˆ N/C? 2 Figure 23-22 shows, in cross section, three soli cliners, each of length L an uniform charge Q. Concentric with each cliner is a clinrical Gaussian surface, with all three surfaces having the same raius. Rank the Gaussian surfaces accoring to the electric fiel at an point on the surface, greatest first. 3 Figure 23-23 shows, in cross section, a central metal ball, two spherical metal shells, an three spherical Gaussian surfaces of raii R,2R,an 3R,all with the same center.the uniform charges on the three objects are: ball, Q;smaller shell,3q;larger shell,5q.rank the Gaussian surfaces accoring to the magnitue of the electric fiel at an point on the surface, greatest first. Shell 3R Cliner Gaussian surface R (c) Figure 23-22 Question 2. Figure 23-23 Question 3. Gaussian surface 2R
678 CHAPTER 23 GAUSS LAW 4 Figure 23-24 shows, in cross section, two Gaussian spheres an two Gaussian cubes that are centere on a positivel charge particle. Rank the net flu through the four q Gaussian surfaces, greatest first. a c Rank the magnitues of the electric b fiels on the surfaces, greatest first, an inicate whether the magnitues are uniform or variable along each surface. Figure 23-24 Question 4. 5 In Fig. 23-25, an electron is release between two infinite nonconucting sheets that are horizontal an have uniform surface charge ensities s ( ) an s ( ),as inicate.the electron is subjecte to the following three situations involving surface charge ensities an sheet separations. Rank the magnitues of the electron s acceleration,greatest first. Situation s ( ) s ( ) Separation 1 4s 4s 2 7s s 4 3 3s 5s 9 σ () e σ () Figure 23-25 Question 5. 6 Three infinite nonconucting sheets, with uniform positive surface charge ensities s,2s, an 3s,are arrange to be parallel like the two sheets in Fig. 23-19a. What is their orer,from left to right, if the electric fiel E : prouce b the arrangement has magnitue E in one region an E 2s/ in another region? 7 Figure 23-26 shows four situations in which four ver long ros eten into an out of the page (we see onl their cross sections). The value below each cross section gives that particular ro s uniform charge ensit in microcoulombs per meter. The ros are separate b either or 2 as rawn, an a central point is shown miwa between the inner ros. Rank the situations accoring to the magnitue of the net electric fiel at that central point, greatest first. 3 2 2 3 8 Figure 23-27 shows four soli spheres, each with charge Q uniforml istribute through its volume. Rank the spheres accoring to their volume charge ensit, greatest first. The figure also shows a point P for each sphere, all at the same istance from the center of the sphere. Rank the spheres accoring to the magnitue of the electric fiel the prouce at point P,greatest first. P P P P (c) () 3 2 1 A B Figure 23-27 Question 8. 9 A small charge ball lies within the hollow of a metallic spherical shell of raius R. For three situations,the net charges on the ball an shell, respectivel, are (1) 4q, ; (2) 6q, 1q; (3) 16q, 12q. Rank the situations accoring to the charge on the inner surface of the shell an the outer surface, most positive first. 1 Rank the situations of Question 9 accoring to the magnitue of the electric fiel halfwa through the shell an at a point 2R from the center of the shell, greatest first. 11 Figure 23-28 shows a section of three long charge cliners centere on the same ais. Central cliner A has a uniform charge q A 3q.What uniform charges q B an q C shoul be on cliners B an C so that (if possible) the net electric fiel is zero at point 1, point 2, an (c) point 3? C Figure 23-28 Question 11. 12 Figure 23-29 shows four Gaussian surfaces consisting of ientical clinrical misections but ifferent en caps. The surfaces are in a uniform electric fiel E : that is irecte parallel to the central ais of each clinrical misection. The en caps have these shapes: S 1,conve hemispheres; S 2,concave hemispheres;s 3,cones;S 4,flat isks. Rank the surfaces accoring to the net electric flu through them an the electric flu through the top en caps, greatest first. 2 4 4 2 (c) 8 2 2 8 E () 6 5 5 6 Figure 23-26 Question 7. S 1 S 2 S 3 S 4 Figure 23-29 Question 12.
PROBLEMS 679 Problems SSM Tutoring problem available (at instructor s iscretion) in WilePLUS an WebAssign Worke-out solution available in Stuent Solutions Manual WWW Worke-out solution is at Number of ots inicates level of problem ifficult ILW Interactive solution is at Aitional information available in The Fling Circus of Phsics an at flingcircusofphsics.com http://www.wile.com/college/hallia Moule 23-1 Electric Flu 1 SSM The square surface shown in Fig. 23-3 measures 3.2 mm on each sie. It is immerse in a uniform electric fiel with magnitue E 18 N/C an with fiel lines at an angle of u 35 with a normal to the surface, as shown. Take that normal to be irecte outwar, as though the surface were one face of a bo. Calculate the electric flu through the surface. 2 An electric fiel given b E : 4.î 3.( 2 2.)ĵ pierces a Gaussian cube of ege length 2. m an positione as shown in Fig. 23-7. (The magnitue E is in newtons per coulomb an the position is in meters.) What is the electric flu through the top face, bottom face, (c) left face, an () back face? (e) What is the net electric flu through the cube? Normal Figure 23-3 Problem 1. 3 The cube in Fig. 23-31 has ege length 1.4 m an is oriente as shown Figure 23-31 Problems 3, in a region of uniform electric fiel. Fin 6, an 9. the electric flu through the right face if the electric fiel, in newtons per coulomb, is given b 6.î, 2.ĵ, an (c) 3.î 4.kˆ. () What is the total flu through the cube for each fiel? Moule 23-2 Gauss Law 4 In Fig. 23-32, a butterfl net is a in a uniform electric fiel of magnitue E 3. mn/c. The rim, a circle of raius a 11 cm, is aligne perpenicular to the fiel. The net contains no net charge. Fin the electric flu through the netting. Figure 23-32 Problem 4. 5 In Fig. 23-33, a proton is a istance /2 irectl above the center of a square of sie.what is the magnitue of the electric flu through the square? (Hint: Think of the square as one face of a cube with ege.) Figure 23-33 Problem 5. 6 At each point on the surface of the cube shown in Fig. 23-31, the electric fiel is parallel to the z ais. The length of each ege of the cube is 3. m. On the top face of the cube the fiel is /2 z θ E : 34kˆ N/C, an on the bottom face it is E : 2kˆ N/C. Determine the net charge containe within the cube. 7 A particle of charge 1.8 mc is at the center of a Gaussian cube 55 cm on ege.what is the net electric flu through the surface? 8 When a shower is turne on in a close bathroom, the splashing of the water on the bare tub can fill the room s air with negativel charge ions an prouce an electric fiel in the air as great as 1 N/C. Consier a bathroom with imensions 2.5 m 3. m 2. m. Along the ceiling, floor, an four walls, approimate the electric fiel in the air as being irecte perpenicular to the surface an as having a uniform magnitue of 6 N/C.Also, treat those surfaces as forming a close Gaussian surface aroun the room s air. What are the volume charge ensit r an the number of ecess elementar charges e per cubic meter in the room s air? 9 ILW Fig. 23-31 shows a Gaussian surface in the shape of a cube with ege length 1.4 m. What are the net flu through the surface an the net charge q enc enclose b the surface if E : (3.ĵ) N/C, with in meters? What are (c) an () q enc if E : z [ 4.î (6. 3.)ĵ] N/C? 1 Figure 23-34 shows a close Gaussian surface in the shape of a cube of ege length 2. m. It lies in a region where the nonuniform electric fiel is given b E : (3. 4.)î 6.ĵ 7.kˆ N/C, with in meters. What is the net charge containe b the cube? 11 Figure 23-35 shows a close Gaussian surface in the shape of a cube of ege length 2. m, with one corner at 1 5. m, 1 4. m.the cube lies in a region where the electric fiel vector is given b E : 3.î 4. 2 ĵ 3.kˆ N/C, with in meters.what is the net charge containe b the cube? 1 z 1 Figure 23-35 Problem 11. 12 Figure 23-36 shows two nonconucting spherical shells fie in place. Shell 1 has uniform surface charge ensit 6. mc/m 2 on its outer surface an raius 3. cm; shell 2 has uniform surface charge ensit 4. mc/m 2 on its outer surface an raius 2. cm; the shell centers are separate b L 1 cm. In unit-vector notation, what is the net electric fiel at 2. cm? Figure 23-34 Problem 1. Shell 1 Shell 2 L Figure 23-36 Problem 12.
68 CHAPTER 23 GAUSS LAW 13 SSM The electric fiel in a certain region of Earth s atmosphere is irecte verticall own. At an altitue of 3 m the fiel has magnitue 6. N/C; at an altitue of 2 m, the magnitue is 1 N/C. Fin the net amount of charge containe in a cube 1 m on ege, with horizontal faces at altitues of 2 an 3 m. 14 Flu an nonconucting shells. A charge particle is suspene at the center of two concentric spherical shells that are ver thin an mae of nonconucting material. Figure 23-37a shows a cross section. Figure 23-37b gives the net flu through a Gaussian sphere centere on the particle, as a function of the raius r of the sphere. The scale of the vertical ais is set b s 5. 1 5 N?m 2 /C. What is the charge of the central particle? What are the net charges of shell A an (c) shell B? A B Figure 23-38 Problem 16. Φ(1 5 N m 2 /C) Φ s Φ s Figure 23-37 Problem 14. 15 A particle of charge q is place at one corner of a Gaussian cube. What multiple of q/ gives the flu through each cube face forming that corner an each of the other cube faces? 16 The bo-like Gaussian surface shown in Fig. 23-38 encloses a net charge of 24. C an lies in an electric fiel given b E : [(1. 2.)î 3.ĵ bzkˆ ] N/C, with an z in meters an b a constant.the bottom face is in the z plane; the top face is in the horizontal plane passing through 2 1. m. For 1 1. m, 2 4. m, z 1 1. m, an z 2 3. m, what is b? z z 2 Moule 23-3 A Charge Isolate Conuctor 17 SSM A uniforml charge conucting sphere of 1.2 m iameter has surface charge ensit 8.1 mc/m 2.Fin the net charge on the sphere an the total electric flu leaving the surface. 18 The electric fiel just above the surface of the charge conucting rum of a photocoping machine has a magnitue E of 2.3 1 5 N/C. What is the surface charge ensit on the rum? 19 Space vehicles traveling through Earth s raiation belts can intercept a significant number of electrons. The resulting charge builup can amage electronic components an isrupt operations. Suppose a spherical metal satellite 1.3 m in iameter accumulates 2.4 mc of charge in one orbital revolution. Fin the resulting surface charge ensit. Calculate the magnitue of the electric fiel just outsie the surface of the satellite, ue to the surface charge. z 1 1 2 2 z 2 2 z 1 r 2 Flu an conucting shells. A charge particle is hel at the center of two concentric conucting spherical shells. Figure 23-39a shows a cross section. Figure 23-39b gives the net flu through a Gaussian sphere centere on the particle, as a function of the raius r of the sphere. The scale of the vertical ais is set b s 5. 1 5 N?m 2 /C. What are the charge of the central particle an the net charges of shell A an (c) shell B? A B Φ (1 5 N m2 /C) Φ s Φ s 2Φ s Figure 23-39 Problem 2. 21 An isolate conuctor has net charge 1 1 6 C an a cavit with a particle of charge q 3. 1 6 C.What is the charge on the cavit wall an the outer surface? Moule 23-4 Appling Gauss Law: Clinrical Smmetr 22 An electron is release 9. cm from a ver long nonconucting ro with a uniform 6. mc/m. What is the magnitue of the electron s initial acceleration? 23 The rum of a photocoping machine has a length of 42 cm an a iameter of 12 cm.the electric fiel just above the rum s surface is 2.3 1 5 N/C. What is the total charge on the rum? The manufacturer wishes to prouce a esktop version of the machine. This requires reucing the rum length to 28 cm an the iameter to 8. cm. The electric fiel at the rum surface must not change. What must be the charge on this new rum? 24 Figure 23-4 shows a section of a long, thin-walle metal tube of raius R 3. cm, with a charge per unit length of l 2. 1 8 C/m. What is the magnitue E of the electric fiel at raial istance r R/2. an r 2.R? (c) Graph E versus r for the range r to 2.R. 25 An infinite line of charge SSM prouces a fiel of magnitue 4.5 1 4 N/C at istance 2. m. Fin the linear charge ensit. 26 Figure 23-41a shows a narrow charge soli cliner that is coaial with a larger charge clinrical shell. Both are noncon- E (1 3 N/C) E s E s 1 2 3 Figure 23-41 Problem 26. R r (cm) Figure 23-4 Problem 24. r 4 5 6
PROBLEMS 681 ucting an thin an have uniform surface charge ensities on their outer surfaces. Figure 23-41b gives the raial component E of the electric fiel versus raial istance r from the common ais, an E s 3. 1 3 N/C.What is the shell s linear charge ensit? 27 A long, straight wire has fie negative charge with a linear charge ensit of magnitue 3.6 nc/m. The wire is to be enclose b a coaial, thin-walle nonconucting clinrical shell of raius 1.5 cm.the shell is to have positive charge on its outsie surface with a surface charge ensit s that makes the net eternal electric fiel zero. Calculate s. 28 A charge of uniform linear ensit 2. nc/m is istribute along a long, thin, nonconucting ro.the ro is coaial with a long conucting clinrical shell (inner raius 5. cm, outer raius 1 cm). The net charge on the shell is zero. What is the magnitue of the electric fiel 15 cm from the ais of the shell? What is the surface charge ensit on the inner an (c) outer surface of the shell? 29 SSM WWW Figure 23-42 is a section of a conucting ro of raius R 1 1.3 mm an length L 11. m insie a thin-walle coaial conucting clinrical shell of raius R 2 1.R 1 an the (same) R 1 length L.The net charge on the ro R 2 is Q 1 3.4 1 12 C; that on the shell is Q 2 2.Q 1.What Q 1 are the magnitue E an irection Q 2 (raiall inwar or out- war) of the electric fiel at raial Figure 23-42 Problem 29. istance r 2.R 2? What are (c) E an () the irection at r 5.R 1? What is the charge on the (e) interior an (f) eterior surface of the shell? 3 In Fig. 23-43, short sections of two ver long parallel lines of charge are shown, fie in place, Line 1 Line 2 separate b L 8. cm. The uniform linear charge ensities are 6. mc/m for line 1 an 2. mc/m for line 2. Where along the ais shown is the net electric fiel L/2 L/2 from the two lines zero? Figure 23-43 Problem 3. 31 ILW Two long, charge, thin-walle, concentric clinrical shells have raii of 3. an 6. cm. The charge per unit length is 5. 1 6 C/m on the inner shell an 7. 1 6 C/m on the outer shell. What are the magnitue E an irection (raiall inwar or outwar) of the electric fiel at raial istance r 4. cm? What are (c) E an () the irection at r 8. cm? 32 A long, nonconucting, soli cliner of raius 4. cm has a nonuniform volume charge ensit r that is a function of raial istance r from the cliner ais: r Ar 2.For A 2.5 mc/m 5,what is the magnitue of the electric fiel at r 3. cm an r 5. cm? Moule 23-5 Appling Gauss Law: Planar Smmetr 33 In Fig. 23-44, two large, thin metal plates are parallel an close to each other. On their inner faces, Figure 23-44 Problem 33. the plates have ecess surface charge ensities of opposite signs an magnitue 7. 1 22 C/m 2.In unit-vector notation,what is the electric fiel at points to the left of the plates, to the right of them, an (c) between them? 34 In Fig. 23-45, a small circular hole of raius R 1.8 cm has been cut in the mile of an infinite, flat, nonconucting surface that has uniform charge ensit s 4.5 pc/m 2.A z ais, with its origin at the hole s center, is perpenicular to the surface. In unitvector notation, what is the electric fiel at point P at z 2.56 cm? (Hint: See Eq. 22-26 an use superposition.) 35 Figure 23-46a shows three plastic sheets that are large, parallel, an uniforml charge. Figure 23-46b gives the component of the net electric fiel along an ais through the sheets. The scale of the vertical ais is set b E s 6. 1 5 N/C.What is the ratio of the charge ensit on sheet 3 to that on sheet 2? E (1 5 N/C) z P R E s Figure 23-45 Problem 34. 1 2 3 36 Figure 23-47 shows cross sections through two large, parallel, nonconucting sheets with ientical istributions of positive charge with surface charge ensit s 1.77 1 22 C/m 2. In unit-vector notation, what is at points above the sheets, between them, an (c) below them? 37 SSM WWW A square metal plate of ege length 8. cm an negligible thickness has a total charge of 6. 1 6 C. Estimate the magnitue E of the electric fiel just off the center of the plate (at, sa, a istance of.5 mm from the center) b assuming that the charge is sprea uniforml over the two faces of the plate. Estimate E at a istance of 3 m (large relative to the plate size) b assuming that the plate is a charge particle. Figure 23-46 Problem 35. E : Figure 23-47 Problem 36.
682 CHAPTER 23 GAUSS LAW 38 In Fig. 23-48a,an electron is shot irectl awa from auni- Moule 23-6 Appling Gauss Law: Spherical Smmetr forml charge plastic sheet, at spee v s 2. 1 5 m/s.the sheet is 44 Figure 23-52 gives the magnitue E s nonconucting, flat, an ver large. Figure 23-48b gives the electron s vertical velocit component v versus time t until the return to the launch point.what is the sheet s surface charge ensit? v s of the electric fiel insie an outsie a sphere with a positive charge istribute uniforml throughout its volume.the scale of the vertical ais is 2 4 set b E s 5. 1 7 N/C. What is the r (cm) charge on the sphere? 23-52 Figure Problem 44. 9 45 Two charge concentric spherical shells have raii 1. cm an 15. cm. The charge on the inner e 3 6 12 shell is 4. 1 8 C, an that on the outer shell is 2. 1 8 C. Fin the electric fiel at r 12. cm an at r 2. cm. v (1 5 m/s) 4 Figure 23-5 shows a ver large nonconucting sheet that has a uniform surface charge ensit of s 2. mc/m 2 ;it also shows a particle of charge Q 6. mc, at istance from the sheet. Both are fie in place. If.2 m, at what positive an negative σ coorinate on the ais (other than infinit) is the net electric fiel E : net of the sheet an particle zero? (c) If.8 m, at what coorinate on the ais is E : net? v s Figure 23-48 Problem 38. t (ps) 39 SSM In Fig. 23-49, a small, nonconucting ball of mass m 1. mg an charge q 2. σ 1 8 C (istribute uniforml through its volume) hangs from an insulating threa that makes θ an angle u 3 with a vertical, uniforml charge nonconucting sheet (shown in cross section). Consiering the gravitational force on the ball an assuming the sheet etens far verticall an into an out of the page, calculate the surface charge ensit s of the sheet. m, q Figure 23-49 Problem 39. 41 An electron is shot irectl Figure 23-5 Problem 4. towar the center of a large metal plate that has surface charge ensit 2. 1 6 C/m 2.If the initial kinetic energ of the electron is 1.6 1 17 J an if the electron is to stop (ue to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be? 42 Two large metal plates of area 1. m 2 face each other, 5. cm apart, with equal charge magnitues q but opposite signs. The fiel magnitue E between them (neglect fringing) is 55 N/C. Fin q. 43 Figure 23-51 shows a cross section through a ver large nonconucting slab of thickness 9.4 mm an uniform volume charge ensit r 5.8 fc/m 3.The origin of an ais is at the slab s center. What is the magnitue of the slab s electric fiel at an coorinate of, 2. mm, (c) 4.7 mm, an () 26. mm? Q /2 Figure 23-51 Problem 43. 46 Assume that a ball of charge particles has a uniforml istribute negative charge ensit ecept for a narrow raial tunnel through its center, from the surface on one sie to the surface on the opposite sie. Also assume that we can position a proton anwhere along the tunnel or outsie the ball. Let F R be the magnitue of the electrostatic force on the proton when it is locate at the ball s surface, at raius R.As a multiple of R, how far from the surface is there a point where the force magnitue is.5f R if we move the proton awa from the ball an into the tunnel? 47 SSM An unknown charge sits on a conucting soli sphere of raius 1 cm. If the electric fiel 15 cm from the center of the sphere has the magnitue 3. 1 3 N/C an is irecte raiall inwar, what is the net charge on the sphere? 48 A charge particle is hel at the center of a spherical shell. Figure 23-53 gives the magnitue E of the electric fiel versus raial istance r.the scale of the vertical ais is set b E s 1. 1 7 N/C. Approimatel, what is the net charge on the shell? E (1 7 N/C) E s 1 2 3 4 5 r (cm) Figure 23-53 Problem 48. 49 In Fig. 23-54, a soli sphere of raius a 2. cm is concentric with a spherical conucting shell of inner raius b 2.a an outer raius c 2.4a. The sphere has a net uniform charge q 1 5. fc; the shell has a net charge q 2 q 1.What is the magnitue of the electric fiel at raial istances r, r a/2., (c) r a,() r 1.5a,(e)r 2.3a,an (f) r 3.5a? What is the net charge on the (g) inner an (h) outer surface of the shell? E (1 7 N/C) Figure 23-54 Problem 49. a c b
PROBLEMS 683 5 Figure 23-55 shows two nonconucting spherical shells fie in place on an ais. Shell 1 has uniform surface charge ensit 4. mc/m 2 on its outer surface an raius.5 cm, an shell 2 has uniform surface charge ensit 2. mc/m 2 on its outer surface an raius 2. cm; the centers are separate b L 6. cm. Other than at,where on the ais is the net electric fiel equal to zero? 51 SSM WWW In Fig. 23-56, a nonconucting spherical shell of inner raius a 2. cm an outer raius b 2.4 cm has (within its thickness) a positive volume charge ensit r A/r,where A is a constant an r is the istance from the center of the shell. In aition, a small ball of charge q 45. fc is locate at that center. What value shoul A have if the electric fiel in the shell (a r b) is to be uniform? 52 Figure 23-57 shows a spherical shell with uniform volume charge ensit r 1.84 nc/m 3,inner raius a 1. cm, an outer raius b 2.a. What is the magnitue ofthe electric fiel at raial istances r ; r a/2., (c) r a, () r 1.5a,(e) r b,an (f) r 3.b? Shell 1 L Shell 2 Figure 23-55 Problem 5. q Figure 23-56 Problem 51. b a a b Figure 23-57 Problem 52. 53 ILW The volume charge ensit of a soli nonconucting sphere of raius R 5.6 cm varies with raial istance r as given b r (14.1 pc/m 3 )r/r. What is the sphere s total charge? What is the fiel magnitue E at r, (c) r R/2., an () r R? (e) Graph E versus r. 54 Figure 23-58 shows, in cross section, two soli spheres with uniforml istribute charge throughout their volumes. Each has raius R.Point P lies on a line connecting the centers of the spheres, at raial R R P 1 2 Figure 23-58 Problem 54. istance R/2. from the center of sphere 1. If the net electric fiel at point P is zero, what is the ratio q 2 /q 1 of the total charges? 55 A charge istribution that is sphericall smmetric but not uniform raiall prouces an electric fiel of magnitue E Kr 4, irecte raiall outwar from the center of the sphere. Here r is the raial istance from that center, an K is a constant. What is the volume ensit r of the charge istribution? Aitional Problems 56 The electric fiel in a particular space is E : ( 2)î N/C, with in meters. Consier a clinrical Gaussian surface of raius 2 cm that is coaial with the ais. One en of the cliner is at. What is the magnitue of the electric flu through the other en of the cliner at 2. m? What net charge is enclose within the cliner? 57 A thin-walle metal spherical shell has raius 25. cm an charge 2. 1 7 C. Fin E for a point insie the shell, just outsie it, an (c) 3. m from the center. 58 A uniform surface charge of ensit 8. nc/m 2 is istribute over the entire plane. What is the electric flu through a spherical Gaussian surface centere on the origin an having a raius of 5. cm? 59 Charge of uniform volume ensit r 1.2 nc/m 3 fills an infinite slab between 5. cm an 5. cm. What is the magnitue of the electric fiel at an point with the coorinate 4. cm an 6. cm? 6 The chocolate crumb mster. Eplosions ignite b electrostatic ischarges (sparks) constitute a serious anger in facilities hanling grain or power. Such an eplosion occurre in chocolate crumb power at a biscuit factor in the 197s. Workers usuall emptie newl elivere sacks of the power into a loaing bin, from which it was blown through electricall groune plastic pipes to a silo for storage. Somewhere along this route, two conitions for an eplosion were met: (1) The magnitue of an electric fiel became 3. 1 6 N/C or greater, so that electrical breakown an thus sparking coul occur. (2) The energ of a spark was 15 mj or greater so that it coul ignite the power eplosivel. Let us check for the first conition in the power flow through the plastic pipes. Suppose a stream of negativel charge power was blown through a clinrical pipe of raius R 5. cm. Assume that the power an its charge were sprea uniforml through the pipe with a volume charge ensit r. Using Gauss law,fin an epression for the magnitue of the electric fiel E : in the pipe as a function of raial istance r from the pipe center. Does E increase or ecrease with increasing r? (c) Is E : irecte raiall inwar or outwar? () For r 1.1 1 3 C/m 3 (a tpical value at the factor), fin the maimum E an etermine where that maimum fiel occurs. (e) Coul sparking occur, an if so, where? (The stor continues with Problem 7 in Chapter 24.) 61 SSM A thin-walle metal spherical shell of raius a has a charge q a.concentric with it is a thin-walle metal spherical shell of raius b a an charge q b.fin the electric fiel at points a istance r from the common center, where r a, a r b, an (c) r b. () Discuss the criterion ou woul use to etermine how the charges are istribute on the inner an outer surfaces of the shells. 62 A particle of charge q 1. 1 7 C is at the center of a spherical cavit of raius 3. cm in a chunk of metal. Fin the electric fiel 1.5 cm from the cavit center an anplace in the metal. 63 A proton at spee v 3. 1 5 m/s orbits at raius r 1. cm outsie a charge sphere. Fin the sphere s charge. 64 Equation 23-11 (E s/ ) gives the electric fiel at points near a charge conucting surface. Appl this equation to a conucting sphere of raius r an charge q,an show that the electric fiel outsie the sphere is the same as the fiel of a charge particle locate at the center of the sphere. 65 Charge Q is uniforml istribute in a sphere of raius R. What fraction of the charge is containe within the raius r R/2.? What is the ratio of the electric fiel magnitue at r R/2. to that on the surface of the sphere? 66 A charge particle causes an electric flu of 75 N?m 2 /C to pass through a spherical Gaussian surface of 1. cm raius centere on the charge. If the raius of the Gaussian surface were
684 CHAPTER 23 GAUSS LAW ouble, how much flu woul pass through the surface? What is the charge of the particle? 67 SSM The electric fiel at point P just outsie the outer surface of a hollow spherical conuctor of inner raius 1 cm an outer raius 2 cm has magnitue 45 N/C an is irecte outwar. When a particle of unknown charge Q is introuce into the center of the sphere, the electric fiel at P is still irecte outwar but is now 18 N/C. What was the net charge enclose b the outer surface before Q was introuce? What is charge Q? After Q is introuce, what is the charge on the (c) inner an () outer surface of the conuctor? 68 The net electric flu through each face of a ie (singular of ice) has a magnitue in units of 1 3 N?m 2 /C that is eactl equal to the number of spots N on the face (1 through 6). The flu is inwar for N o an outwar for N even. What is the net charge insie the ie? 69 Figure 23-59 shows, in P cross section, three infinitel σ 3 L/2 large nonconucting sheets on which charge is uniforml sprea. The surface charge 2L ensities are s 1 2. mc/m 2, s 2 4. mc/m 2, σ 2 an s 3 5. mc/m 2, an istance L 1.5 cm. In unitvector L notation, what is the net σ 1 electric fiel at point P? Figure 23-59 Problem 69. 7 Charge of uniform volume ensit r 3.2 mc/m 3 fills a nonconucting soli sphere of raius 5. cm. What is the magnitue of the electric fiel 3.5 cm an 8. cm from the sphere s center? 71 A Gaussian surface in the form of a hemisphere of raius R 5.68 cm lies in a uniform electric fiel of magnitue E 2.5 N/C. The surface encloses no net charge. At the (flat) base of the surface, the fiel is perpenicular to the surface an irecte into the surface. What is the flu through the base an the curve portion of the surface? 72 What net charge is enclose b the Gaussian cube of Problem 2? 75 Figure 23-61 shows a Geiger counter, a evice use to etect ionizing raiation, which causes ionization of atoms. A thin, posi- 73 A nonconucting soli sphere has a uniform volume charge ensit r. Let r be the : vector from the center of the sphere to a general point P within the sphere. Show that the electric fiel at P is given b E : a rr : /3. (Note that the result is inepenent of the raius of the sphere.) A spherical cavit is hollowe out of the sphere, as shown in Fig. 23- Figure 23-6 6. Using superposition concepts, show that Problem 73. the electric fiel at all points within the cavit is uniform an equal to E : ra : /3, where a : is the position vector from the center of the sphere to the center of the cavit. 74 A uniform charge ensit of 5 nc/m 3 is istribute throughout a spherical volume of raius 6. cm. Consier a cubical Gaussian surface with its center at the center of the sphere. What is the electric flu through this cubical surface if its ege length is 4. cm an 14. cm? tivel charge central wire is surroune b a concentric, circular, conucting clinrical shell with an equal negative charge, creating a strong raial electric fiel. The shell contains a low-pressure inert gas. A particle of raiation entering the evice through the shell wall ionizes a few of the gas atoms. The resulting free electrons (e) are rawn to the positive wire. However, the electric fiel is so intense that, between collisions with gas atoms, the free electrons gain energ sufficient to ionize these atoms also. More free electrons are thereb create, an the process is repeate until the electrons reach the wire. The resulting avalanche of electrons is collecte b the wire, generating a signal that is use to recor the passage of the original particle of raiation. Suppose that the raius of the central wire is 25 mm, the inner raius of the shell 1.4 cm, an the length of the shell 16 cm. If the electric fiel at the shell s inner wall is 2.9 1 4 N/C, what is the total positive charge on the central wire? 76 Charge is istribute uniforml throughout the volume of an infinitel long soli cliner of raius R. Show that,at a istance r R from the cliner ais, E rr, 2 where r is the volume charge ensit. Write an epression for E when r R. 77 SSM A spherical conucting shell has a charge of 14 mc on its outer surface an a charge particle in its hollow. If the net charge on the shell is 1 mc, what is the charge on the inner surface of the shell an of the particle? 78 A charge of 6. pc is sprea uniforml throughout the volume of a sphere of raius r 4. cm.what is the magnitue of the electric fiel at a raial istance of 6. cm an 3. cm? 79 Water in an irrigation itch of with w 3.22 m an epth 1.4 m flows with a spee of.27 m/s. The mass flu of the flowing water through an imaginar surface is the prouct of the water s ensit (1 kg/m 3 ) an its volume flu through that surface. Fin the mass flu through the following imaginar surfaces: a surface of area w, entirel in the water,perpenicular to the flow; a surface with area 3w/2, of which w is in the water, perpenicular to the flow; (c) a surface of area w/2, entirel in the water, perpenicular to the flow; () a surface of area w,half in the water an half out, perpenicular to the flow; (e) a surface of area w,entirel in the water, with its normal 34. from the irection of flow. 8 Charge of uniform surface ensit 8. nc/m 2 is istribute over an entire plane; charge of uniform surface ensit 3. nc/m 2 is istribute over the parallel plane efine b z 2. m. Determine the magnitue of the electric fiel at an point having a z coorinate of 1. m an 3. m. 81 A spherical ball of charge particles has a uniform charge ensit. In terms of the ball s raius R, at what raial istances insie an outsie the ball is the magnitue of the ball s 1 electric fiel equal to of the maimum magnitue of that fiel? 4 Particle e Charge wire Charge clinrical shell Signal Figure 23-61 Problem 75.