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1 Problems Page of 16 CHAPTER 23 PROBLEMS Tutoring problem available (at instructor's discretion) in WileyPLUS and WebAssign SSM Worked-out solution available in Student Solutions Manual - Number of dots indicates level of problem difficulty ILW Interactive solution Additional information available in The Flying Circus of Physics sec. 23-3 Flux of an Electric Field 1. The square surface shown in Fig. 23-26 measures 3.2 mm on each side. It is immersed in a uniform electric with a normal to the surface, field with magnitude and with field lines at an angle of as shown. Take that normal to be directed "outward," as though the surface were one face of a box. Calculate the electric flux through the surface. SSM FIGURE 23-26 Problem 1. 2. An electric field given by positioned as shown in Fig. 23-5. (The magnitude pierces a Gaussian cube of edge length and is in newtons per coulomb and the position x is in meters.) What is the electric flux through the (a) top face, (b) bottom face, (c) left face, and (d) back face? (e) What is the net electric flux through the cube? 3. The cube in Fig. 23-27 has edge length and is oriented as shown in a region of uniform electric field. Find the electric flux through the right face if the electric field, in newtons per coulomb, is given by (a) (b) , and (c) . (d) What is the total flux through the cube for each field? http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 2 of 16 FIGURE 23-27 Problems 3, 4, and 11. sec. 23-4 Gauss' Law 4. At each point on the surface of the cube shown in Fig. 23-27, the electric field is parallel to the z axis. The length of each edge of the cube is face . On the top face of the cube , and on the bottom . Determine the net charge contained within the cube. on edge. What is the net electric 5. A point charge of is at the center of a cubical Gaussian surface flux through the surface? 6. In Fig. 23-28, a butterfly net is in a uniform electric field of magnitude . The rim, a circle of radius , is aligned perpendicular to the field. The net contains no net charge. Find the electric flux through the netting. FIGURE 23-28 Problem 6. 7. In Fig. 23-29, a proton is a distance directly above the center of a square of side d. What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge d.) FIGURE 23-29 Problem 7. 8. Figure 23-30 shows two nonconducting spherical shells fixed in place. Shell 1 has uniform surface charge density on its outer surface and radius on its outer surface and radius vector notation, what is the net electric field at ; shell 2 has uniform surface charge density . In unit- ; the shell centers are separated by ? FIGURE 23-30 Problem 8. http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 3 of 16 9. It is found experimentally that the electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of the field has magnitude ; at an altitude of , the magnitude is . Find the net amount of charge contained in a cube on edge, with horizontal faces at altitudes of 200 and . SSM 10. When a shower is turned on in a closed bathroom, the splashing of the water on the bare tub can fill the room's air with negatively charged ions and produce an electric field in the air as great as . Consider a bathroom with dimensions . Along the ceiling, floor, and four walls, approximate the electric field in the air as being directed perpendicular to the surface and as having a uniform magnitude of . Also, treat those surfaces as forming a closed Gaussian surface around the room's air. What are (a) the volume charge density and (b) the number of excess elementary charges e per cubic meter in the room's air? Flying Circus of Physics - Danger of spraying water 11. Fig. 23-27 shows a Gaussian surface in the shape of a cube with edge length flux through the surface and (b) the net charge with y in meters? What are (c) and (d) if enclosed by the surface if . What are (a) the net , ? 12. Flux and nonconducting shells. A charged particle is suspended at the center of two concentric spherical shells that are very thin and made of nonconducting material. Figure 23-31a shows a cross section. Figure 23-31b gives the net flux through a Gaussian sphere centered on the particle, as a function of the radius of the sphere. The scale of the vertical axis is set by . (a) What is the charge of the central particle? What are the net charges of (b) shell A and (c) shell B? FIGURE 23-31 Problem 12. 13. A particle of charge is placed at one corner of a Gaussian cube. What multiple of through (a) each cube face forming that corner and (b) each of the other cube faces? 14. Figure 23-32 shows a closed Gaussian surface in the shape of a cube of edge length region where the electric field is given by What is the net charge contained by the cube? gives the flux . It lies in a , with x in meters. FIGURE 23-32 Problem 14. 15. Figure 23-33 shows a closed Gaussian surface in the shape of a cube of edge length , with one corner http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 4 of 16 at , . The cube lies in a region where the electric field vector is given by , with y in meters. What is the net charge contained by the cube? FIGURE 23-33 Problem 15. 16. The box-like Gaussian surface of Fig. 23-34 encloses a net charge of given by and lies in an electric field with x and z in meters and b a constant. The . For bottom face is in the xz plane; the top face is in the horizontal plane passing through , , , and , what is b? FIGURE 23-34 Problem 16. sec. 23-6 A Charged Isolated Conductor 17. Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metal satellite in diameter accumulates of charge in one orbital revolution, (a) Find the resulting surface charge density. (b) Calculate the magnitude of the electric field just outside the surface of the satellite, due to the surface charge. 18. Flux and conducting shells. A charged particle is held at the center of two concentric conducting spherical shells. Figure 23-35a shows a cross section. Figure 23-35b gives the net flux through a Gaussian sphere centered on the particle, as a function of the radius r of the sphere. The scale of the vertical axis is set by . What are (a) the charge of the central particle and the net charges of (b) shell A and (c) shell B? FIGURE 23-35 Problem 18. http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 5 of 16 19. A uniformly charged conducting sphere of diameter has a surface charge density of . (a) Find the net charge on the sphere. (b) What is the total electric flux leaving the surface of the sphere? SSM 20. The electric field just above the surface of the charged drum of a photocopying machine has a magnitude E of . What is the surface charge density on the drum, assuming the drum is a conductor? . Inside the conductor is a cavity 21. An isolated conductor of arbitrary shape has a net charge of within which is a point charge outer surface of the conductor? . What is the charge (a) on the cavity wall and (b) on the sec. 23-7 Applying Gauss' Law: Cylindrical Symmetry 22. Figure 23-36 shows a section of a long, thin-walled metal tube of radius length and (b) ? (c) Graph E versus r for the range to . , with a charge per unit . What is the magnitude E of the electric field at radial distance (a) FIGURE 23-36 Problem 22. 23. An infinite line of charge produces a field of magnitude linear charge density. SSM 24. An electron is released from rest at a perpendicular distance of nonconducting rod. That charge is uniformly distributed, with the electron's initial acceleration? at a distance of . Calculate the from a line of charge on a very long per meter. What is the magnitude of 25. (a) The drum of a photocopying machine has a length of and a diameter of . The electric field just above the drum's surface is . What is the total charge on the drum? (b) The manufacturer wishes to produce a desktop version of the machine. This requires reducing the drum length to and the diameter to . The electric field at the drum surface must not change. What must be the charge on this new drum? 26. In Fig. 23-37, short sections of two very long parallel lines of charge are shown, fixed in place, separated by . The uniform linear charge densities are for line 1 and for line 2. Where along the x axis shown is the net electric field from the two lines zero? http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 6 of 16 FIGURE 23-37 Problem 26. 27. Figure 23-38 is a section of a conducting rod of radius walled coaxial conducting cylindrical shell of radius the rod is and (d) the direction at shell? SSM ; that on the shell is and length inside a thin- and the (same) length L. The net charge on . What are the (a) magnitude E and ? What are (c) E (b) direction (radially inward or outward) of the electric field at radial distance ? What is the charge on the (e) interior and (f) exterior surface of the FIGURE 23-38 Problem 27. 28. Figure 23-39a shows a narrow charged solid cylinder that is coaxial with a larger charged cylindrical shell. Both are nonconducting and thin and have uniform surface charge densities on their outer surfaces. Figure 23-39b gives the radial component E of the electric field versus radial distance r from the common axis. The vertical axis scale is set by . What is the linear charge density of the shell? FIGURE 23-39 Problem 28. 29. Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and . The charge per unit length is on the inner shell and on the outer shell. What are the (a) magnitude E and (b) direction (radially inward or outward) of the electric field at radial distance ? What are (c) E and (d) the direction at ? 30. A charge of uniform linear density is distributed along a long, thin, nonconducting rod. The rod is coaxial with a long conducting cylindrical shell (inner radius , outer radius ). The net charge on the shell is zero. (a) What is the magnitude of the electric field from the axis of the shell? What is the surface charge density on the (b) inner and (c) outer surface of the shell? 31. A long, straight wire has fixed negative charge with a linear charge density of magnitude wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of radius . The . The shell is http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 7 of 16 to have positive charge on its outside surface with a surface charge density that makes the net external electric field zero. Calculate . 32. A long, nonconducting, solid cylinder of radius function of radial distance r from the cylinder axis: the electric field at (a) and (b) ? has a nonuniform volume charge density that is a . For , what is the magnitude of sec. 23-8 Applying Gauss' Law: Planar Symmetry 33. Figure 23-40a shows three plastic sheets that are large, parallel, and uniformly charged. Figure 23-40b gives the component of the net electric field along an x axis through the sheets. The scale of the vertical axis is set by . What is the ratio of the charge density on sheet 3 to that on sheet 2? FIGURE 23-40 Problem 33. 34. Figure 23-41 shows cross sections through two large, parallel, nonconducting sheets with identical distributions of positive charge with surface charge density . In unit-vector notation, what is at points (a) above the sheets, (b) between them, and (c) below them? FIGURE 23-41 Problem 34. 35. A square metal plate of edge length and negligible thickness has a total charge of . (a) Estimate the magnitude E of the electric field just off the center of the plate (at, say, a distance of from the center) by assuming that the charge is spread uniformly over the two faces of the plate. (b) Estimate E at a distance of (large relative to the plate size) by assuming that the plate is a point charge. SSM 36. In Fig. 23-42, a small circular hole of radius nonconducting surface that has uniform charge density has been cut in the middle of an infinite, flat, . A z axis, with its origin at the hole's center, is perpendicular to the surface. In unit-vector notation, what is the electric field at point P at ? (Hint: See Eq. 22-26 and use superposition.) http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 8 of 16 FIGURE 23-42 Problem 36. 37. In Fig. 23-43, two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have excess surface charge densities of opposite signs and magnitude . In unitvector notation, what is the electric field at points (a) to the left of the plates, (b) to the right of them, and (c) between them? FIGURE 23-43 Problem 37. 38. Two large metal plates of area face each other. They are apart and have equal but opposite charges on their inner surfaces. If the magnitude E of the electric field between the plates is , what is the magnitude of the charge on each plate? Neglect edge effects. 39. An electron is shot directly toward the center of a large metal plate that has surface charge density . If the initial kinetic energy of the electron is and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be? 40. In Fig. 23-44a, an electron is shot directly away from a uniformly charged plastic sheet, at speed . The sheet is nonconducting, flat, and very large. Figure 23-44b gives the electron's vertical velocity component v versus time t until the return to the launch point. What is the sheet's surface charge density? FIGURE 23-44 Problem 40. 41. http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 9 of 16 In Fig. 23-45, a small, nonconducting ball of mass and charge (distributed uniformly through its volume) hangs from an insulating thread that makes an angle with a vertical, uniformly charged nonconducting sheet (shown in cross section). Considering the gravitational force on the ball and assuming the sheet extends far vertically and into and out of the page, calculate the surface charge density of the sheet. SSM FIGURE 23-45 Problem 41. 42. Figure 23-46 shows a very large nonconducting sheet that has a uniform surface charge density of ; it also shows a particle of charge fixed in place. If infinity) is the net electric field on the x axis is ? , at distance d from the sheet. Both are , at what coordinate , at what (a) positive and (b) negative coordinate on the x axis (other than of the sheet and particle zero? (c) If FIGURE 23-46 Problem 42. 43. 23-47 Figure shows a cross section through a very large nonconducting slab of thickness uniform volume charge density magnitude of the slab's electric field at an x coordinate of (a) 0, (b) ? , (c) , and (d) and . The origin of an x axis is at the slab's center. What is the FIGURE 23-47 Problem 43. sec. 23-9 Applying Gauss' Law: Spherical Symmetry 44. A point charge causes an electric flux of to pass through a spherical Gaussian surface of http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 10 of 16 radius centered on the charge. (a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge? 45. An unknown charge sits on a conducting solid sphere of radius center of the sphere has the magnitude charge on the sphere? SSM . If the electric field from the and is directed radially inward, what is the net 46. Figure 23-48 gives the magnitude of the electric field inside and outside a sphere with a positive charge distributed uniformly throughout its volume. The scale of the vertical axis is set by . What is the charge on the sphere? FIGURE 23-48 Problem 46. 47. Two charged concentric spherical shells have radii , and that on the outer shell is (b) at . and . The charge on the inner shell is . Find the electric field (a) at and 48. Figure 23-49 shows two nonconducting spherical shells fixed in place on an x axis. Shell 1 has uniform surface charge density surface charge density . Other than at on its outer surface and radius on its outer surface and radius , and shell 2 has uniform ; the centers are separated by , where on the x axis is the net electric field equal to zero? FIGURE 23-49 Problem 48. 49. In Fig. 23-50, a nonconducting spherical shell of inner radius and outer radius has (within its thickness) a positive volume charge density , where A is a constant and r is the distance from the center of the shell. In addition, a small ball of charge is located at that center. What value should A have if the electric field in the shell is to be uniform? SSM http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 11 of 16 FIGURE 23-50 Problem 49. 50. Figure 23-51 shows a spherical shell with uniform volume charge density ; (b) , and outer radius , (c) , (d) , inner radius . What is the magnitude of the electric field at radial distances (a) , (e) , and (f) ? FIGURE 23-51 Problem 50. 51. In Fig. 23-52, a solid sphere of radius radius and outer radius is concentric with a spherical conducting shell of inner . The sphere has a net uniform charge ; the shell has a net charge . What is the magnitude of the electric field at radial distances (a) , ? What is the net charge on the (b) , (c) , (d) , (e) , and (f) (g) inner and (h) outer surface of the shell? FIGURE 23-52 Problem 51. 52. A charged particle is held at the center of a spherical shell. Figure 23-53 gives the magnitude E of the electric field versus radial distance r. The scale of the vertical axis is set by . Approximately, what is the net charge on the shell? http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 12 of 16 FIGURE 23-53 Problem 52. 53. A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitude , directed radially outward from the center of the sphere. Here r is the radial distance from that center, and K is a constant. What is the volume density of the charge distribution? 54. Figure 23-54 shows, in cross section, two solid spheres with uniformly distributed charge throughout their volumes. Each has radius R. Point P lies on a line connecting the centers of the spheres, at radial distance from the center of sphere 1. If the net electric field at point P is zero, what is the ratio of the total charge in sphere 2 to the total charge in sphere 1? FIGURE 23-54 Problem 54. 55. A solid nonconducting sphere of radius density has a nonuniform charge distribution of volume charge , (c) , and (d) , where r is radial distance from the sphere's center. (a) What is the sphere's total charge? What is the magnitude E of the electric field at (b) ? (e) Sketch a graph of E versus r. Additional Problems 56. The chocolate crumb mystery. Explosions ignited by electrostatic discharges (sparks) constitute a serious danger in facilities handling grain or powder. Such an explosion occurred in chocolate crumb powder at a biscuit factory in the 1970s. Workers usually emptied newly delivered sacks of the powder into a loading bin, from which it was blown through electrically grounded plastic pipes to a silo for storage. Somewhere along this route, two conditions for an explosion were met: (1) The magnitude of an electric field became or greater, so that electrical breakdown and thus sparking could occur. (2) The energy of a spark was or greater so that it could ignite the powder explosively. Let us check for the first condition in the powder flow through the plastic pipes. Suppose a stream of negatively charged powder was blown through a cylindrical pipe of radius . Assume that the powder and its charge were spread uniformly through the pipe with a volume charge density . (a) Using Gauss' law, find an expression for the magnitude of the electric field in the pipe as a function directed of radial distance r from the pipe center. (b) Does E increase or decrease with increasing r? (c) Is radially inward or outward? (d) For (a typical value at the factory), find the maximum E and determine where that maximum field occurs, (e) Could sparking occur, and if so, where? (The story continues with Problem 68 in Chapter 24 .) http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 13 of 16 Flying Circus of Physics - Danger of powder floating in the air 57. Charge Q is uniformly distributed in a sphere of radius R. (a) What fraction of the charge is contained within radius ? (b) What is the ratio of the electric field magnitude at to that on the surface of the sphere? 58. Charge of uniform volume density the magnitude of the electric field (a) fills a nonconducting solid sphere of radius and (b) from the sphere's center? . What is 59. The electric field at point P just outside the outer surface of a hollow spherical conductor of inner radius and outer radius has magnitude and is directed outward. When an unknown point charge Q is introduced into the center of the sphere, the electric field at P is still directed outward but is now (a) What was the net charge enclosed by the outer surface before Q was introduced? (b) What is charge Q? After Q is introduced, what is the charge on the (c) inner and (d) outer surface of the conductor? SSM 60. Assume that a ball of charged particles has a uniformly distributed negative charge density except for a narrow radial tunnel through its center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton anywhere along the tunnel or outside the ball. Let be the magnitude of the electrostatic force on the proton when it is located at the ball's surface, at radius R. As a multiple of R, how far from the surface is there a point where the force magnitude is if we move the proton (a) away from the ball and (b) into the tunnel? 61. Charge of uniform volume density (b) fills an infinite slab between and and . What is the magnitude of the electric field at any point with the coordinate (a) ? 62. A uniform surface charge of density is distributed over the entire plane. What is the electric flux through a spherical Gaussian surface centered on the origin and having a radius of ? 63. A thin-walled metal spherical shell has radius and charge the shell, (b) just outside it, and (c) from the center. 64. The electric field in a particular space is . Find E for a point (a) inside , with x in meters. Consider a cylindrical Gaussian surface of radius that is coaxial with the x axis. One end of the cylinder is at . (a) What is the magnitude of the electric flux through the other end of the cylinder at ? (b) What net charge is enclosed within the cylinder? 65. Figure 23-55 shows, in cross section, three infinitely large nonconducting sheets on which charge is uniformly spread. The surface charge densities are , , and , and distance point P? . In unit-vector notation, what is the net electric field at http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 14 of 16 FIGURE 23-55 Problem 65. 66. The net electric flux through each face of a die (singular of dice) has a magnitude in units of that is exactly equal to the number of spots N on the face (1 through 6). The flux is inward for N odd and outward for N even. What is the net charge inside the die? 67. A Gaussian surface in the form of a hemisphere of radius lies in a uniform electric field of magnitude . The surface encloses no net charge. At the (flat) base of the surface, the field is perpendicular to the surface and directed into the surface. What is the flux through (a) the base and (b) the curved portion of the surface? 68. A point charge is at the center of a spherical cavity of radius in a chunk of metal. Use Gauss' law to find the electric field (a) from the cavity center and (b) anyplace in the metal. 69. A thin-walled metal spherical shell of radius a has a charge . Concentric with it is a thin-walled metal spherical shell of radius and charge . Find the electric field at points a distance r from the common center, where (a) , (b) , and (c) . (d) Discuss the criterion you would use to determine how the charges are distributed on the inner and outer surfaces of the shells. SSM 70. What net charge is enclosed by the Gaussian cube of Problem 2? orbits just outside a charged sphere of radius . What is 71. A proton with speed the charge on the sphere? 72. Equation 23-11 gives the electric field at points near a charged conducting surface. Apply this equation to a conducting sphere of radius r and charge q, and show that the electric field outside the sphere is the same as the field of a point charge located at the center of the sphere. 73. Figure 23-56 shows a Geiger counter, a device used to detect ionizing radiation, which causes ionization of atoms. A thin, positively charged central wire is surrounded by a concentric, circular, conducting cylindrical shell with an equal negative charge, creating a strong radial electric field. The shell contains a low-pressure inert gas. A particle of radiation entering the device through the shell wall ionizes a few of the gas atoms. The resulting free electrons (e) are drawn to the positive wire. However, the electric field is so intense that, between collisions with gas atoms, the free electrons gain energy sufficient to ionize these atoms also. More free electrons are thereby created, and the process is repeated until the electrons reach the wire. The resulting "avalanche" of electrons is collected by the wire, generating a signal that is used to record the passage of the original particle of radiation. Suppose that the radius of the central wire is , the inner radius of the shell , and the length of the shell . If the electric field at the shell's inner wall is , what is the total positive charge on the central wire? http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 15 of 16 FIGURE 23-56 Problem 73. 74. Charge is distributed uniformly throughout the volume of an infinitely long solid cylinder of radius R. (a) Show that, at a distance from the cylinder axis, where is the volume charge density. (b) Write an expression for E when . . The 75. Water in an irrigation ditch of width and depth flows with a speed of mass flux of the flowing water through an imaginary surface is the product of the water's density and its volume flux through that surface. Find the mass flux through the following imaginary surfaces: (a) a surface of area wd, entirely in the water, perpendicular to the flow; (b) a surface with area , of which wd is in the water, perpendicular to the flow; (c) a surface of area , entirely in the water, perpendicular to the flow; (d) a surface of area wd, half in the water and half out, perpendicular to the flow; (e) a surface of area wd, entirely in the water, with its normal from the direction of flow. 76. A free electron is placed between two large, parallel, nonconducting plates that are horizontal and apart. One plate has a uniform positive charge; the other has an equal amount of uniform negative charge. The force on the electron due to the electric field or down) of ? SSM between the plates balances the gravitational force on the electron. What are (a) the magnitude of the surface charge density on the plates and (b) the direction (up 77. A nonconducting solid sphere has a uniform volume charge density . Let be the vector from the center of the sphere to a general point P within the sphere. (a) Show that the electric field at P is given by . (Note that the result is independent of the radius of the sphere.) (b) A spherical cavity is hollowed out of the sphere, as shown in Fig. 23-57. Using superposition concepts, show that the electric field at all points within the cavity is uniform and equal to , where is the position vector from the center of the sphere to the center of the cavity. (Note that this result is independent of the radius of the sphere and the radius of the cavity.) http://edugen.wiley.com/edugen/courses/crs1650/pc/halliday8019c23/halliday8019c23_13.... 8/27/2007 Problems Page 16 of 16 FIGURE 23-57 Problem 77. 78. A uniform charge density of is distributed throughout a spherical volume of radius . Consider a cubical Gaussian surface with its center at the center of the sphere. What is the electric flux through this cubical surface if its edge length is (a) and (b) ? 79. A spherical conducting shell has a charge of on its outer surface and a charged particle in its hollow. If the net charge on the shell is , what is the charge (a) on the inner surface of the shell and (b) of the particle? SSM 80. A charge of is spread uniformly throughout the volume of a sphere of radius magnitude of the electric field at a radial distance of (a) and (b) ? . What is the 81. A spherical ball of charged particles has a uniform charge density. In terms of the ball's radius R, at what radial distances (a) inside and (b) outside the ball is the magnitude of the ball's electric field equal to maximum magnitude of that field? 82. Charge of uniform surface density is distributed over an entire plane; charge of uniform and (b) . Determine the . of the surface density is distributed over the parallel plane defined by magnitude of the electric field at any point having a z coordinate of (a) Copyright 2008 John Wiley & Sons, Inc. 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