1cc. An electron is positioned and then released on the x-axis,
where the electric potential has the value –20 V. Which of the following
statements describes the subsequent motion of the electron?a) The
electron will move to the left (negative x-direction) because it is
negatively charged.b) The electron will move to the right (positive
x-direction) because it is negatively charged.c) The electron will move
to the left (negative x-direction) because the electric potential is
negative.d) The electron will move to the right (positive x-direction)
because the electric potential is negative.e) Not enough information is
given to predict the motion of the electron. Get solution
1mcq. A positive charge is released and moves along an electric field line. This charge moves to a position ofa) lower potential and lower potential energy.b) lower potential and higher potential energy.c) higher potential and lower potential energy.d) higher potential and higher potential energy. Get solution
2cc. A cathode ray tube uses a potential difference of 5.0 kV to accelerate electrons and produce an electron beam that makes images on a phosphor screen. What is the speed of these electrons as a percentage of the speed of light?a) 0.025%b) 0.22%c) 1.3%d) 4.5%e) 14% Get solution
2mcq. A proton is placed midway between points A and B. The potential at point A is –20 V, and the potential at point B +20 V. The potential at the midpoint is 0 V. The proton willa) remain at rest.b) move toward point B with constant velocity.c) accelerate toward point A.d) accelerate toward point B.e) move toward point A with constant velocity. Get solution
3cc. In the figure, the lines represent equipotential lines. A charged object is moved from point P to point Q. How does the amount of work done on the object compare for these three cases?(1) ...(2) ...(3) ...a) All three cases involve the same work.b) The most work is done in case 1.c) The most work is done in case 2.d) The most work is done in case 3.e) Cases 1 and 3 involve the same amount of work, which is more than is involved in case 2. Get solution
3mcq. What would be the consequence of setting the potential at +100 V at infinity, rather than taking it to be zero there?a) Nothing; the field and the potential would have the same values at every finite point.b) The electric potential would become infinite at every finite point, and the electric field could not be defined.c) The electric potential everywhere would be 100 V higher, and the electric field would be the same.d) It would depend on the situation. For example, the potential due to a positive point charge would drop off more slowly with distance, so the magnitude of the electric field would be less. Get solution
4cc. What is the electric potential 45.5 cm away from a point charge of 12.5 pC?a) 0.247 Vb) 1.45 Vc) 4.22 Vd) 10.2 Ve) 25.7 V Get solution
4mcq. In which situation is the electric potential the highest?a) at a point 1 m from a point charge of 1 Cb) at a point 1 m from the center of a uniformly charged spherical shell with a radius of 0.5 m and a total charge of 1 Cc) at a point 1 m from the center of a uniformly charged rod with a length of 1 m and a total charge of 1 Cd) at a point 2 m from a point charge of 2 Ce) at a point 0.5 m from a point charge of 0.5 C Get solution
5cc. Two protons are located in space in the three ways shown in the figure. Rank the three cases from highest to lowest net electric potential, V, produced at point P.(1) ...(2) ...(3) ...a) 2 > 3 > 1b) All three potentials are the same.c) 3 > 2 > 1d) The potentials are equal for cases 1 and 3, with the potential for case 2 lower.e) 1 > 2 > 3 Get solution
5mcq. The amount of work done to move a positive point charge q on an equipotential surface of 1000 V relative to that done to move the charge on an equipotential surface of 10 V isa) the same.b) less.c) more.d) dependent on the distance the charge moves. Get solution
6cc. Three identical positive point charges are located at fixed points in space. Then charge q2 is moved from its initial location to a final location as shown in the figure. Four different paths, marked (a) through (d), are shown. Path (a) follows the shortest line; path (b) takes q2 around q3; path (c) takes q2 around q3 and q1; path (d) takes q2 out to infinity and then to the final location. Which path requires the least work?...a) path (a)b) path (b)c) path (c)d) path (d)e) The work is the same for all the paths. Get solution
6mcq. A solid conducting sphere of radius R is centered about the origin of an xyz-coordinate system. A total charge Q is distributed uniformly on the surface of the sphere. Assuming, as usual, that the electric potential is zero at an infinite distance, what is the electric potential at the center of the conducting sphere?a) zerob) Q/ε0Rc) Q/2πε0Rd) Q/4 πε0R Get solution
7cc. Suppose an electric potential is described by V(x, y, z) = –(5x2 + y + z) in volts. Which of the following expressions describes the associated electric field, in units of volts per meter?a) ...b) ...c) ...d) ...e) ... Get solution
7mcq. Which of the following angles between an electric dipole moment and an applied electric field will result in the most stable state?a) 0 radb) π/2 radc) π radd) The electric dipole moment is not stable under any condition in an applied electric field. Get solution
8cc. In the figure, the lines represent equipotential lines. How does the magnitude of the electric field, E, at point P compare for the three cases?(1) ...(2) ...(3) ...a) E1 = E2 = E3b) E1 > E2 > E3c) E1 E2 E3d) E3 > E1 > E2e) E3 E1 E2 Get solution
8mcq. A positive point charge is to be moved from point A to point B in the vicinity of an electric dipole. Which of the three paths shown in the figure will result in the most work being done by the dipole’s electric field on the point charge?...a) path 1b) path 2c) path 3d) The work is the same on all three paths. Get solution
9cc. In the figure, the lines represent equipotential lines. A positive charge is placed at point P, and then another positive charge is placed at point Q. Which set of vectors best represents the relative magnitudes and directions of the electric field forces exerted on the positive charges at P and Q?...a) ...b) ...c) ...d) ...e) ... Get solution
9mcq. Each of the following pairs of charges are separated by a distance d. Which pair has the highest potential energy?a) +5 C and +3 Cb) +5 C and –3 Cc) –5 C and +3 Cd) All pairs have the same potential energy. Get solution
10cc. In the figure, the lines represent equipotential lines. What is the direction of the electric field at point P?...a) upb) downc) leftd) righte) The electric field at P is zero. Get solution
10mcq. A negatively charged particle revolves in a clockwise direction around a positively charged sphere. The work done on the negatively charged particle by the electric field of the sphere isa) positive.b) negative.c) zero. Get solution
11cc. Three pairs of parallel plates have the same plate separation and potentials on each plate as indicated in the drawing. The electric field, E, is uniform between each pair of plates and perpendicular to them. Rank the magnitude of E between the plates, from highest to lowest.(1) ...(2) ...(3) ...a) 1 > 2 > 3b) 3 > 2 > 1c) The magnitudes for 3 and 2 are equal and greater than the magnitude for 1.d) The three magnitudes are equal.e) The magnitude for 2 is greater than that for 1 and 3, which are the same. Get solution
11mcq. A hollow conducting sphere of radius R is centered about the origin of an xyz-coordinate system. A total charge Q is distributed uniformly over the surface of the sphere. Assuming, as usual, that the electric potential is zero at an infinite distance, what is the electric potential at the center of the sphere?a) zerob) 2kQ/Rc) kQ/Rd) kQ/2Re) kQ/4R Get solution
12mcq. A solid conducting sphere of radius R has a charge Q evenly distributed over its surface, producing an electric potential V0 at the surface. How much charge must be added to the sphere to increase the potential at the surface to 2V0?a) Q/2b) Qc) 2Qd) Q2e) 2Q2 Get solution
13mcq. Which one of the following statements is not true?a) Equipotential lines are parallel to the electric field lines.b) Equipotential lines for a point charge are circular.c) Equipotential surfaces exist for any charge distribution.d) When a charge moves on an equipotential surface, the work done on the charge is zero. Get solution
14mcq. If a proton and an alpha particle (composed of two protons and two neutrons) are each accelerated from rest through the same potential difference, how do their resulting speeds compare?a) The proton has twice the speed of the alpha particle.b) The proton has the same speed as the alpha particle.c) The proton has half the speed of the alpha particle.d) The speed of the proton is ... times the speed of the alpha particle.e) The speed of the alpha particle is ... times the speed of the proton. Get solution
15cq. High-voltage power lines are used to transport electricity cross country. These wires are favored resting places for birds. Why don’t the birds die when they touch the wires? Get solution
16cq. You have heard that it is dangerous to stand under trees in electrical storms. Why? Get solution
17cq. Can two equipotential lines cross? Why or why not? Get solution
18cq. Why is it important, when soldering connectors onto a piece of electronic circuitry, to leave no pointy protrusions from the solder joints? Get solution
19cq. Using Gauss’s Law and the relation between electric potential and electric field, show that the potential outside a uniformly charged sphere is identical to the potential of a point charge placed at the center of the sphere and equal to the total charge of the sphere. What is the potential at the surface of the sphere? How does the potential change if the charge distribution is not uniform but has spherical (radial) symmetry? Get solution
20cq. A metal ring has a total charge q and a radius R, as shown in the figure. Without performing any calculations, predict the value of the electric potential and the electric field at the center of the circle.... Get solution
21cq. Find an integral expression for the electric potential at a point on the z-axis a distance H from a half-disk of radius R (see the figure). The half-disk has uniformly distributed charge over its surface, with charge distribution σ.... Get solution
22cq. An electron moves away from a proton. Describe how the potential it encounters changes. Describe how its potential energy is changing. Get solution
23cq. The electric potential energy of a continuous charge distribution can be found in a way similar to that used for systems of point charges in Section 23.6, by breaking the distribution up into suitable pieces. Find the electric potential energy of an arbitrary spherically symmetrical charge distribution, ρ(r). Do not assume that ρ (r) represents a point charge, that it is constant, that it is piecewise-constant, or that it does or does not end at any finite radius, r. Your expression must cover all possibilities. Your expression may include an integral or integrals that cannot be evaluated without knowing the specific form of ρ (r). (Hint: A spherical pearl is built up of thin layers of nacre added one by one.) Get solution
24. In molecules of gaseous sodium chloride, the chloride ion has one more electron than proton, and the sodium ion has one more proton than electron. These ions are separated by about 0.236 nm. How much work would be required to increase the distance between these ions to 1.00 cm? Get solution
25. A metal ball with a mass of 3.00·10–6 kg and a charge of +5.00 mC has a kinetic energy of 6.00·108 J. It is traveling directly at an infinite plane of charge with a charge distribution of +4.00 C/m2. If it is currently 1.00 m away from the plane of charge, how close will it come to the plane before stopping? Get solution
26. An electron is accelerated from rest through a potential difference of 370. V. What is its final speed? Get solution
27. How much work would be done by an electric field in moving a proton from a point at a potential of +180. V to a point at a potential of –60.0 V? Get solution
28. What potential difference is needed to give an alpha particle (composed of 2 protons and 2 neutrons) 200. keV of kinetic energy? Get solution
29. A proton, initially at rest, is accelerated through a potential difference of 500. V. What is its final velocity? Get solution
30. A 10.0-V battery is connected to two parallel metal plates placed in a vacuum. An electron is accelerated from rest from the negative plate toward the positive plate.a) What kinetic energy does the electron have just as it reaches the positive plate?b) What is the speed of the electron just as it reaches the positive plate? Get solution
31. A proton gun fires a proton from midway between two plates, A and B, which are separated by a distance of 10.0 cm; the proton initially moves at a speed of 150.0 km/s toward plate B. Plate A is kept at zero potential, and plate B at a potential of 400.0 V.a) Will the proton reach plate B?b) If not, where will it turn around?c) With what speed will it hit plate A? Get solution
32. Fully stripped (all electrons removed) sulfur (32S) ions are accelerated from rest in an accelerator that uses a total voltage of 1.00 · 109 V. 32S has 16 protons and 16 neutrons. The accelerator produces a beam consisting of 6.61·1012 ions per second. This beam of ions is completely stopped in a beam dump. What is the total power the beam dump has to absorb? Get solution
33. Two point charges are located at two corners of a rectangle, as shown in the figure....a) What is the electric potential at point A?b) What is the potential difference between points A and B? Get solution
34. Four identical point charges (+1.61 nC) are placed at the corners of a rectangle, which measures 3.00 m by 5.00 m. If the electric potential is taken to be zero at infinity, what is the potential at the geometric center of this rectangle? Get solution
35. If a Van de Graaff generator has an electric potential of 1.00·105 V and a diameter of 20.0 cm, find how many more protons than electrons are on its surface. Get solution
36. One issue encountered during the exploration of Mars has been the accumulation of static charge on land-roving vehicles, resulting in a potential of 100. V or more. Calculate how much charge must be placed on the surface of a sphere of radius 1.00 m for the electric potential just above the surface to be 100. V. Assume that the charge is uniformly distributed. Get solution
37. A charge Q = +5.60 µC is uniformly distributed on a thin cylindrical plastic shell. The radius, R, of the shell is 4.50 cm. Calculate the electric potential at the origin of the xy-coordinate system shown in the figure. Assume that the electric potential is zero at points infinitely far away from the origin.... Get solution
38. A hollow spherical conductor with a 5.00-cm radius has a surface charge of 8.00 nC.a) What is the potential 8.00 cm from the center of the sphere?b) What is the potential 3.00 cm from the center of the sphere?c) What is the potential at the center of the sphere? Get solution
39. Find the potential at the center of curvature of the (thin) wire shown in the figure. It has a (uniformly distributed) charge per unit length of λ = 3.00·10–8 C/m and a radius of curvature of R = 8.00 cm.... Get solution
40. Consider a dipole with charge q and separation d. What is the potential a distance x from the center of this dipole at an angle θ with respect to the dipole axis, as shown in the figure?... Get solution
41. A spherical water drop 50.0 µm in diameter has a uniformly distributed charge of +20.0 pC. Find (a) the potential at its surface and (b) the potential at its center. Get solution
42. Consider an electron in the ground state of the hydrogen atom, separated from the proton by a distance of 0.0529 nm.a) Viewing the electron as a satellite orbiting the proton in the electric potential, calculate the speed of the electron in its orbit.b) Calculate an effective escape speed for the electron.c) Calculate the energy of an electron having this speed, and from it determine the energy that must be given to the electron to ionize the hydrogen atom. Get solution
43. Four point charges are arranged in a square with side length 2a, where a = 2.70 cm. The charges have the same magnitude, 1.50 nC; three of them are positive and one is negative as shown in the figure. What is the value of the electric potential generated by these four point charges at point P = (0,0,c), where c = 4.10 cm?... Get solution
44. The plastic rod of length L shown in the figure has the nonuniform linear charge distribution λ = cx, where c is a positive constant. Find an expression for the electric potential at point P on the y-axis, a distance y from one end of the rod.... Get solution
45. An electric field varies in space according to this equation: ...a) For what value of x does the electric field have its largest value, xmax?b) What is the potential difference between the points at x = 0 and x = xmax? Get solution
46. Derive an expression for electric potential along the axis (the x-axis) of a disk with a hole in the center, as shown in the figure, where R1 and R2 are the inner and outer radii of the disk. What would the potential be if R1 = 0?... Get solution
47. An electric field is established in a nonuniform rod. A voltmeter is used to measure the potential difference between the left end of the rod and a point a distance x from the left end. The process is repeated, and it is found that the data are described by the relationship ∆V = 270.x2, where ∆V has the units V/m2. What is the x-component of the electric field at a point 13.0 cm from the left end? Get solution
48. Two parallel plates are held at potentials of +200.0 V and –100.0 V. The plates are separated by 1.00 cm.a) Find the electric field between the plates.b) An electron is initially placed halfway between the plates. Find its kinetic energy when it hits the positive plate. Get solution
49. A 2.50-mg dust particle with a charge of 1.00 µC falls at a point x = 2.00 m in a region where the electric potential varies according to V(x) = (2.00 V/m2)x2 – (3.00 V/m3)x3. With what acceleration will the particle start moving after it touches down? Get solution
50. The electric potential in a volume of space is given by V(x,y,z) = x2 + xy2 + yz. Determine the electric field in this region at the coordinate (3,4,5). Get solution
51. The electric potential inside a 10.0-m-long linear particle accelerator is given by V = (3000 – 5x2/m2) V, where x is the distance from the left plate along the accelerator tube, as shown in the figure.a) Determine an expression for the electric field along the accelerator tube.b) A proton is released (from rest) at x = 4.00 m. Calculate the acceleration of the proton just after it is released.c) What is the impact speed of the proton when (and if) it collides with the plate?... Get solution
52. An infinite plane of charge has a uniform charge distribution of +4.00 nC/m2 and is located in the yz-plane at x = 0. A +11.0-nC fixed point charge is located at x = +2.00 m.a) Find the electric potential V(x) on the x-axis from 0 x b) At what position(s) on the x-axis between x = 0 and x = +2.00 m is the electric potential a minimum?c) Where on the x-axis between x = 0 m and x = +2.00 m could a positive point charge be placed and not move? Get solution
53. Use ... to derive the expression for the electric field of a point charge, q. Get solution
54. Show that an electron in a one-dimensional electrical potential, V(x) = Ax2, where the constant A is a positive real number, will execute simple harmonic motion about the origin. What is the period of that motion? Get solution
55. The electric field, ... and the electric potential, ... are calculated from the charge distribution, ... by integrating Coulomb’s Law and then the electric field. In the other direction, the field and the charge distribution are determined from the potential by suitably differentiating. Suppose the electric potential in a large region of space is given by V(r) = V0 exp (–r2/a2), where V0 and a are constants and ... is the distance from the origin.a) Find the electric field ... in this region.b) Determine the charge density, ... in this region, which gives rise to the potential and field.c) Find the total charge in this region.d) Roughly sketch the charge distribution that could give rise to such an electric field. Get solution
56. The electron beam emitted by an electron gun is controlled (steered) with two sets of parallel conducting plates: a horizontal set to control the vertical motion of the beam, and a vertical set to control the horizontal motion of the beam. The beam is emitted with an initial velocity of 2.00·107 m/s. The width of the plates is d = 5.00 cm, the separation between the plates is D = 4.00 cm, and the distance between the edge of the plates and a target screen is L = 40.0 cm. In the absence of any applied voltage, the electron beam hits the origin of the xy-coordinate system on the observation screen. What voltages need to be applied to the two sets of plates for the electron beam to hit a target placed on the observation screen at coordinates (x,y) = (0 cm,8.00 cm)? Get solution
57. Nuclear fusion reactions require that positively charged nuclei be brought into close proximity, against the electrostatic repulsion. As a simple example, suppose a proton is fired at a second, stationary proton from a large distance away. What kinetic energy must be given to the moving proton to get it to come within 1.00 ·10–15 m of the target? Assume that there is a head-on collision and that the target is fixed in place. Get solution
58. Fission of a uranium nucleus (containing 92 protons) produces a barium nucleus (56 protons) and a krypton nucleus (36 protons). The fragments fly apart as a result of electrostatic repulsion; they ultimately emerge with a total of 200. MeV of kinetic energy. Use this information to estimate the size of the uranium nucleus; that is, treat the barium and krypton nuclei as point charges and calculate the separation between them at the start of the process. Get solution
59. A deuterium ion and a tritium ion each have charge +e. What work has to be done on the deuterium ion in order to bring it within 1.00 · 10–14 m of the tritium ion? This is the distance within which the two ions can fuse, as a result of strong nuclear interactions that overcome electrostatic repulsion, to produce a helium-5 nucleus. Express the work in electron-volts. Get solution
60. Three charges, q1, q2, and q3, are located at the corners of an equilateral triangle with side length of 1.20 m. Find the work done in each of the following cases:a) to bring the first particle, q1 = 1.00 pC, to P from infinityb) to bring the second particle, q2 = 2.00 pC, to Q from infinityc) to bring the last particle, q3 = 3.00 pC, to R from infinityd) Find the total potential energy stored in the final configuration of q1, q2, and q3.... Get solution
61. Two metal balls of mass m1 = 5.00 g (diameter = 5.00 mm) and m2 = 8.00 g (diameter = 8.00 mm) have positive charges of q1 = 5.00 nC and q2 = 8.00 nC, respectively. A force holds them in place so that their centers are separated by 8.00 mm. What will their velocities be after the force has been removed and they are separated by a large distance? Get solution
62. Two protons at rest and separated by 1.00 mm are released simultaneously. What is the speed of either at the instant when the two are 10.0 mm apart? Get solution
63. A 12-V battery is connected between a hollow metal sphere with a radius of 1 m and a ground, as shown in the figure. What are the electric field and the electric potential inside the hollow metal sphere?... Get solution
64. A solid metal ball with a radius of 3.00 m has a charge of 4.00 mC. If the electric potential is zero far away from the ball, what is the electric potential at each of the following positions?a) at r = 0 m, the center of the ballb) at r = 3.00 m, on the surface of the ballc) at r = 5.00 m Get solution
65. An insulating sheet in the xz-plane is uniformly charged with a charge distribution σ = 3.50 · 10–6 C/m2. What is the change in potential when a charge of Q = 1.25 µC is moved from position A to position B in the figure?...... Get solution
66. Suppose that an electron inside a cathode ray tube starts from rest and is accelerated by the tube’s voltage of 21.9 kV. What is the speed (in km/s) with which the electron (mass = 9.11 · 10–31 kg) hits the screen of the tube? Get solution
67. A conducting solid sphere (radius of R = 18.0 cm, charge of q = 6.10 · 10–6 C) is shown in the figure. Calculate the electric potential at a point 24.0 cm from the center (point A), a point on the surface (point B), and at the center of the sphere (point C). Assume that the electric potential is zero at points infinitely far away from the origin of the coordinate system.... Get solution
68. A classroom Van de Graaff generator accumulates a charge of 1.00 · 10–6 C on its spherical conductor, which has a radius of 10.0 cm and stands on an insulating column. Neglecting the effects of the generator base or any other objects or fields, find the potential at the surface of the sphere. Assume that the potential is zero at infinity. Get solution
69. A Van de Graaff generator has a spherical conductor with a radius of 25.0 cm. It can produce a maximum electric field of 2.00 · 106 V/m. What are the maximum voltage and charge that it can hold? Get solution
70. A proton with a speed of 1.23 · 104 m/s is moving from infinity directly toward a second proton. Assuming that the second proton is fixed in place, find the position where the moving proton stops momentarily before turning around. Get solution
71. Two metal spheres of radii r1 = 10.0 cm and r2 = 20.0 cm, respectively, have been positively charged so that each has a total charge of 100. µC.a) What is the ratio of their surface charge distributions?b) If the two spheres are connected by a copper wire, how much charge flows through the wire before the system reaches equilibrium? Get solution
72. The solid metal sphere of radius a = 0.200 m shown in the figure has a surface charge distribution of σ. The potential difference between the surface of the sphere and a point P at a distancerP = 0.500 m from the center of the sphere is ∆V = Vsurface – VP = +4π V = +12.566 V. Determine the value of σ.... Get solution
73. A particle with a charge of +5.00 µC is released from rest at a point on the x-axis, where x = 0.100 m. It begins to move as a result of the presence of a +9.00-µC charge that remains fixed at the origin. What is the kinetic energy of the particle at the instant it passes the point x = 0.200 m? Get solution
74. The sphere in the figure has a radius of 2.00 mm and carries a +2.00-µC charge uniformly distributed throughout its volume. What is the potential difference, VB – VA, if the angle between the two radii to points A and B is 60.0°? Is the potential difference dependent on the angle? Would the answer be the same if the charge distribution had an angular dependence, ρ = ρ (θ)?... Get solution
75. Two metallic spheres have radii of 10.0 cm and 5.00 cm, respectively. The magnitude of the electric field on the surface of each sphere is 3600. V/m. The two spheres are then connected by a long, thin metal wire. Determine the magnitude of the electric field on the surface of each sphere when they are connected. Get solution
76. A ring with charge Q and radius R is in the yz-plane and centered on the origin. What is the electric potential a distance x above the center of the ring? Derive the electric field from this relationship. Get solution
77. A charge of 0.681 nC is placed at x = 0. Another charge of 0.167 nC is placed at x1 = 10.9 cm on the x-axis.a) What is the combined electric potential of these two charges at x = 20.1 cm, also on the x-axis?b) At which point(s) on the x-axis does this potential have a minimum? Get solution
78. A point charge of +2.00 µC is located at (2.50 m,3.20 m). A second point charge of –3.10 µC is located at (–2.10 m,1.00 m).a) What is the electric potential at the origin?b) Along a line passing through both point charges, at what point(s) is (are) the electric potential(s) equal to zero? Get solution
79. A total charge of Q = 4.20·10–6 C is placed on a conducting sphere (sphere 1) of radius R = 0.400 m.a) What is the electric potential, V1, at the surface of sphere 1 assuming that the potential infinitely far away from it is zero? (Hint: What is the change in potential if a charge is brought from infinitely far away, where V(∞) = 0, to the surface of the sphere?)b) A second conducting sphere (sphere 2) of radius r = 0.100 m with an initial net charge of zero (q = 0) is connected to sphere 1 using a long thin metal wire. How much charge flows from sphere 1 to sphere 2 to bring them into equilibrium? What are the electric fi elds at the surfaces of the two spheres at equilibrium? Get solution
80. A thin line of charge is aligned along the positive y-axis from 0 ≤ y ≤ L, with L = 4.0 cm. The charge is not uniformly distributed but has a charge per unit length of λ = Ay, with A = 8.00·10–7 C/m2. Assuming that the electric potential is zero at infinite distance, find the electric potential at a point on the x-axis as a function of x. Give the value of the electric potential at x = 3.00 cm.... Get solution
81. Two fixed point charges are on the x-axis. A charge of –3.00 mC is located at x = +2.00 m and a charge of +5.00 mC is located at x = –4.00 m.a) Find the electric potential, V(x), for an arbitrary point on the x-axis.b) At what position(s) on the x-axis is V(x) = 0?c) Find E(x) for an arbitrary point on the x-axis. Get solution
82. One of the greatest physics experiments in history measured the charge-to-mass ratio of an electron, q/m. If a uniform potential difference is created between two plates, atomized particles—each with an integral amount of charge—can be suspended in space. The assumption is that the particles of unknown mass, M, contain a net number, n, of electrons of mass m and charge q. For a plate separation of d, what is the potential difference necessary to suspend a particle of mass M containing n net electrons? What is the acceleration of the particle if the voltage is cut in half? What is the acceleration of the particle if the voltage is doubled? Get solution
83. A uniform linear charge distribution of total positive charge Q has the shape of a half-circle of radius R, as shown in the figure.a) Without performing any calculations, predict the electric potential produced by this linear charge distribution at point O.b) Confirm, through direct calculations, your prediction of part (a).c) Make a similar prediction for the electric field.... Get solution
84. A point charge Q is placed a distance R from the center of a conducting sphere of radius a, with R > a (the point charge is outside the sphere). The sphere is grounded, that is, connected to a distant, unlimited source and/or sink of charge at zero potential. (Neither the distant ground nor the connection directly affects the electric field in the vicinity of the charge and sphere.) As a result, the sphere acquires a charge opposite in sign to Q, and the point charge experiences an attractive force toward the sphere.a) Remarkably, the electric field outside the sphere is the same as would be produced by the point charge Q plus an imaginary mirror-image point charge q, with magnitude and location that make the set of points corresponding to the surface of the sphere an equipotential of potential zero. That is, the imaginary point charge produces the same field contribution outside the sphere as the actual surface charge on the sphere. Calculate the value and location of q. (Hint: By symmetry, q must lie somewhere on the axis that passes through the center of the sphere and the location of Q.)b) Calculate the force exerted on point charge Q and directed toward the sphere, in terms of the original quantities Q, R, and a.c) Determine the actual nonuniform surface charge distribution on the conducting sphere. Get solution
85. A solid conducting sphere of radius R1 = 1.206 m has a charge of Q = 1.953 μC evenly distributed over its surface. A second solid conducting sphere of radius R2 = 0.6115 m is initially uncharged and at a distance of 10.00 m from the first sphere. The two spheres are momentarily connected with a wire, which is then removed. What is the charge on the second sphere? Get solution
86. A solid conducting sphere of radius R1 = 1.435 m has a charge of Q evenly distributed over its surface. A second solid conducting sphere of radius R2 = 0.6177 m is initially uncharged and at a distance of 10.00 m from the first sphere. The two spheres are momentarily connected with a wire, which is then removed. The resulting charge on the second sphere is 0.9356 μC. What was the original charge, Q, on the first sphere? Get solution
87. A solid conducting sphere of radius R1 has a charge of Q = 4.263 μC evenly distributed over its surface. A second solid conducting sphere of radius R2 = 0.6239 m is initially uncharged and at a distance of 10.00 m from the first sphere. The two spheres are momentarily connected with a wire, which is then removed. The resulting charge on the second sphere is 1.162 μC. What is the radius of the first sphere? Get solution
88. A solid conducting sphere of radius R = 1.895 m is charged, and the magnitude of the electric field at the surface of the sphere is 3.165·105 V/m. What is the electric potential 29.81 cm from the surface of the sphere? Get solution
89. A solid conducting sphere of radius R is charged, and the magnitude of the electric field at the surface of the sphere is 3.269·105 V/m. The electric potential 32.37 cm from the surface of the sphere is 2.843·105 V. What is the radius, R, of the sphere? Get solution
90. A solid conducting sphere of radius R = 1.351 m is charged, and the magnitude of the electric potential 34.95 cm from the surface of the sphere is 3.618·105 V. What is the magnitude of the electric field at the surface of the sphere? Get solution
1mcq. A positive charge is released and moves along an electric field line. This charge moves to a position ofa) lower potential and lower potential energy.b) lower potential and higher potential energy.c) higher potential and lower potential energy.d) higher potential and higher potential energy. Get solution
2cc. A cathode ray tube uses a potential difference of 5.0 kV to accelerate electrons and produce an electron beam that makes images on a phosphor screen. What is the speed of these electrons as a percentage of the speed of light?a) 0.025%b) 0.22%c) 1.3%d) 4.5%e) 14% Get solution
2mcq. A proton is placed midway between points A and B. The potential at point A is –20 V, and the potential at point B +20 V. The potential at the midpoint is 0 V. The proton willa) remain at rest.b) move toward point B with constant velocity.c) accelerate toward point A.d) accelerate toward point B.e) move toward point A with constant velocity. Get solution
3cc. In the figure, the lines represent equipotential lines. A charged object is moved from point P to point Q. How does the amount of work done on the object compare for these three cases?(1) ...(2) ...(3) ...a) All three cases involve the same work.b) The most work is done in case 1.c) The most work is done in case 2.d) The most work is done in case 3.e) Cases 1 and 3 involve the same amount of work, which is more than is involved in case 2. Get solution
3mcq. What would be the consequence of setting the potential at +100 V at infinity, rather than taking it to be zero there?a) Nothing; the field and the potential would have the same values at every finite point.b) The electric potential would become infinite at every finite point, and the electric field could not be defined.c) The electric potential everywhere would be 100 V higher, and the electric field would be the same.d) It would depend on the situation. For example, the potential due to a positive point charge would drop off more slowly with distance, so the magnitude of the electric field would be less. Get solution
4cc. What is the electric potential 45.5 cm away from a point charge of 12.5 pC?a) 0.247 Vb) 1.45 Vc) 4.22 Vd) 10.2 Ve) 25.7 V Get solution
4mcq. In which situation is the electric potential the highest?a) at a point 1 m from a point charge of 1 Cb) at a point 1 m from the center of a uniformly charged spherical shell with a radius of 0.5 m and a total charge of 1 Cc) at a point 1 m from the center of a uniformly charged rod with a length of 1 m and a total charge of 1 Cd) at a point 2 m from a point charge of 2 Ce) at a point 0.5 m from a point charge of 0.5 C Get solution
5cc. Two protons are located in space in the three ways shown in the figure. Rank the three cases from highest to lowest net electric potential, V, produced at point P.(1) ...(2) ...(3) ...a) 2 > 3 > 1b) All three potentials are the same.c) 3 > 2 > 1d) The potentials are equal for cases 1 and 3, with the potential for case 2 lower.e) 1 > 2 > 3 Get solution
5mcq. The amount of work done to move a positive point charge q on an equipotential surface of 1000 V relative to that done to move the charge on an equipotential surface of 10 V isa) the same.b) less.c) more.d) dependent on the distance the charge moves. Get solution
6cc. Three identical positive point charges are located at fixed points in space. Then charge q2 is moved from its initial location to a final location as shown in the figure. Four different paths, marked (a) through (d), are shown. Path (a) follows the shortest line; path (b) takes q2 around q3; path (c) takes q2 around q3 and q1; path (d) takes q2 out to infinity and then to the final location. Which path requires the least work?...a) path (a)b) path (b)c) path (c)d) path (d)e) The work is the same for all the paths. Get solution
6mcq. A solid conducting sphere of radius R is centered about the origin of an xyz-coordinate system. A total charge Q is distributed uniformly on the surface of the sphere. Assuming, as usual, that the electric potential is zero at an infinite distance, what is the electric potential at the center of the conducting sphere?a) zerob) Q/ε0Rc) Q/2πε0Rd) Q/4 πε0R Get solution
7cc. Suppose an electric potential is described by V(x, y, z) = –(5x2 + y + z) in volts. Which of the following expressions describes the associated electric field, in units of volts per meter?a) ...b) ...c) ...d) ...e) ... Get solution
7mcq. Which of the following angles between an electric dipole moment and an applied electric field will result in the most stable state?a) 0 radb) π/2 radc) π radd) The electric dipole moment is not stable under any condition in an applied electric field. Get solution
8cc. In the figure, the lines represent equipotential lines. How does the magnitude of the electric field, E, at point P compare for the three cases?(1) ...(2) ...(3) ...a) E1 = E2 = E3b) E1 > E2 > E3c) E1 E2 E3d) E3 > E1 > E2e) E3 E1 E2 Get solution
8mcq. A positive point charge is to be moved from point A to point B in the vicinity of an electric dipole. Which of the three paths shown in the figure will result in the most work being done by the dipole’s electric field on the point charge?...a) path 1b) path 2c) path 3d) The work is the same on all three paths. Get solution
9cc. In the figure, the lines represent equipotential lines. A positive charge is placed at point P, and then another positive charge is placed at point Q. Which set of vectors best represents the relative magnitudes and directions of the electric field forces exerted on the positive charges at P and Q?...a) ...b) ...c) ...d) ...e) ... Get solution
9mcq. Each of the following pairs of charges are separated by a distance d. Which pair has the highest potential energy?a) +5 C and +3 Cb) +5 C and –3 Cc) –5 C and +3 Cd) All pairs have the same potential energy. Get solution
10cc. In the figure, the lines represent equipotential lines. What is the direction of the electric field at point P?...a) upb) downc) leftd) righte) The electric field at P is zero. Get solution
10mcq. A negatively charged particle revolves in a clockwise direction around a positively charged sphere. The work done on the negatively charged particle by the electric field of the sphere isa) positive.b) negative.c) zero. Get solution
11cc. Three pairs of parallel plates have the same plate separation and potentials on each plate as indicated in the drawing. The electric field, E, is uniform between each pair of plates and perpendicular to them. Rank the magnitude of E between the plates, from highest to lowest.(1) ...(2) ...(3) ...a) 1 > 2 > 3b) 3 > 2 > 1c) The magnitudes for 3 and 2 are equal and greater than the magnitude for 1.d) The three magnitudes are equal.e) The magnitude for 2 is greater than that for 1 and 3, which are the same. Get solution
11mcq. A hollow conducting sphere of radius R is centered about the origin of an xyz-coordinate system. A total charge Q is distributed uniformly over the surface of the sphere. Assuming, as usual, that the electric potential is zero at an infinite distance, what is the electric potential at the center of the sphere?a) zerob) 2kQ/Rc) kQ/Rd) kQ/2Re) kQ/4R Get solution
12mcq. A solid conducting sphere of radius R has a charge Q evenly distributed over its surface, producing an electric potential V0 at the surface. How much charge must be added to the sphere to increase the potential at the surface to 2V0?a) Q/2b) Qc) 2Qd) Q2e) 2Q2 Get solution
13mcq. Which one of the following statements is not true?a) Equipotential lines are parallel to the electric field lines.b) Equipotential lines for a point charge are circular.c) Equipotential surfaces exist for any charge distribution.d) When a charge moves on an equipotential surface, the work done on the charge is zero. Get solution
14mcq. If a proton and an alpha particle (composed of two protons and two neutrons) are each accelerated from rest through the same potential difference, how do their resulting speeds compare?a) The proton has twice the speed of the alpha particle.b) The proton has the same speed as the alpha particle.c) The proton has half the speed of the alpha particle.d) The speed of the proton is ... times the speed of the alpha particle.e) The speed of the alpha particle is ... times the speed of the proton. Get solution
15cq. High-voltage power lines are used to transport electricity cross country. These wires are favored resting places for birds. Why don’t the birds die when they touch the wires? Get solution
16cq. You have heard that it is dangerous to stand under trees in electrical storms. Why? Get solution
17cq. Can two equipotential lines cross? Why or why not? Get solution
18cq. Why is it important, when soldering connectors onto a piece of electronic circuitry, to leave no pointy protrusions from the solder joints? Get solution
19cq. Using Gauss’s Law and the relation between electric potential and electric field, show that the potential outside a uniformly charged sphere is identical to the potential of a point charge placed at the center of the sphere and equal to the total charge of the sphere. What is the potential at the surface of the sphere? How does the potential change if the charge distribution is not uniform but has spherical (radial) symmetry? Get solution
20cq. A metal ring has a total charge q and a radius R, as shown in the figure. Without performing any calculations, predict the value of the electric potential and the electric field at the center of the circle.... Get solution
21cq. Find an integral expression for the electric potential at a point on the z-axis a distance H from a half-disk of radius R (see the figure). The half-disk has uniformly distributed charge over its surface, with charge distribution σ.... Get solution
22cq. An electron moves away from a proton. Describe how the potential it encounters changes. Describe how its potential energy is changing. Get solution
23cq. The electric potential energy of a continuous charge distribution can be found in a way similar to that used for systems of point charges in Section 23.6, by breaking the distribution up into suitable pieces. Find the electric potential energy of an arbitrary spherically symmetrical charge distribution, ρ(r). Do not assume that ρ (r) represents a point charge, that it is constant, that it is piecewise-constant, or that it does or does not end at any finite radius, r. Your expression must cover all possibilities. Your expression may include an integral or integrals that cannot be evaluated without knowing the specific form of ρ (r). (Hint: A spherical pearl is built up of thin layers of nacre added one by one.) Get solution
24. In molecules of gaseous sodium chloride, the chloride ion has one more electron than proton, and the sodium ion has one more proton than electron. These ions are separated by about 0.236 nm. How much work would be required to increase the distance between these ions to 1.00 cm? Get solution
25. A metal ball with a mass of 3.00·10–6 kg and a charge of +5.00 mC has a kinetic energy of 6.00·108 J. It is traveling directly at an infinite plane of charge with a charge distribution of +4.00 C/m2. If it is currently 1.00 m away from the plane of charge, how close will it come to the plane before stopping? Get solution
26. An electron is accelerated from rest through a potential difference of 370. V. What is its final speed? Get solution
27. How much work would be done by an electric field in moving a proton from a point at a potential of +180. V to a point at a potential of –60.0 V? Get solution
28. What potential difference is needed to give an alpha particle (composed of 2 protons and 2 neutrons) 200. keV of kinetic energy? Get solution
29. A proton, initially at rest, is accelerated through a potential difference of 500. V. What is its final velocity? Get solution
30. A 10.0-V battery is connected to two parallel metal plates placed in a vacuum. An electron is accelerated from rest from the negative plate toward the positive plate.a) What kinetic energy does the electron have just as it reaches the positive plate?b) What is the speed of the electron just as it reaches the positive plate? Get solution
31. A proton gun fires a proton from midway between two plates, A and B, which are separated by a distance of 10.0 cm; the proton initially moves at a speed of 150.0 km/s toward plate B. Plate A is kept at zero potential, and plate B at a potential of 400.0 V.a) Will the proton reach plate B?b) If not, where will it turn around?c) With what speed will it hit plate A? Get solution
32. Fully stripped (all electrons removed) sulfur (32S) ions are accelerated from rest in an accelerator that uses a total voltage of 1.00 · 109 V. 32S has 16 protons and 16 neutrons. The accelerator produces a beam consisting of 6.61·1012 ions per second. This beam of ions is completely stopped in a beam dump. What is the total power the beam dump has to absorb? Get solution
33. Two point charges are located at two corners of a rectangle, as shown in the figure....a) What is the electric potential at point A?b) What is the potential difference between points A and B? Get solution
34. Four identical point charges (+1.61 nC) are placed at the corners of a rectangle, which measures 3.00 m by 5.00 m. If the electric potential is taken to be zero at infinity, what is the potential at the geometric center of this rectangle? Get solution
35. If a Van de Graaff generator has an electric potential of 1.00·105 V and a diameter of 20.0 cm, find how many more protons than electrons are on its surface. Get solution
36. One issue encountered during the exploration of Mars has been the accumulation of static charge on land-roving vehicles, resulting in a potential of 100. V or more. Calculate how much charge must be placed on the surface of a sphere of radius 1.00 m for the electric potential just above the surface to be 100. V. Assume that the charge is uniformly distributed. Get solution
37. A charge Q = +5.60 µC is uniformly distributed on a thin cylindrical plastic shell. The radius, R, of the shell is 4.50 cm. Calculate the electric potential at the origin of the xy-coordinate system shown in the figure. Assume that the electric potential is zero at points infinitely far away from the origin.... Get solution
38. A hollow spherical conductor with a 5.00-cm radius has a surface charge of 8.00 nC.a) What is the potential 8.00 cm from the center of the sphere?b) What is the potential 3.00 cm from the center of the sphere?c) What is the potential at the center of the sphere? Get solution
39. Find the potential at the center of curvature of the (thin) wire shown in the figure. It has a (uniformly distributed) charge per unit length of λ = 3.00·10–8 C/m and a radius of curvature of R = 8.00 cm.... Get solution
40. Consider a dipole with charge q and separation d. What is the potential a distance x from the center of this dipole at an angle θ with respect to the dipole axis, as shown in the figure?... Get solution
41. A spherical water drop 50.0 µm in diameter has a uniformly distributed charge of +20.0 pC. Find (a) the potential at its surface and (b) the potential at its center. Get solution
42. Consider an electron in the ground state of the hydrogen atom, separated from the proton by a distance of 0.0529 nm.a) Viewing the electron as a satellite orbiting the proton in the electric potential, calculate the speed of the electron in its orbit.b) Calculate an effective escape speed for the electron.c) Calculate the energy of an electron having this speed, and from it determine the energy that must be given to the electron to ionize the hydrogen atom. Get solution
43. Four point charges are arranged in a square with side length 2a, where a = 2.70 cm. The charges have the same magnitude, 1.50 nC; three of them are positive and one is negative as shown in the figure. What is the value of the electric potential generated by these four point charges at point P = (0,0,c), where c = 4.10 cm?... Get solution
44. The plastic rod of length L shown in the figure has the nonuniform linear charge distribution λ = cx, where c is a positive constant. Find an expression for the electric potential at point P on the y-axis, a distance y from one end of the rod.... Get solution
45. An electric field varies in space according to this equation: ...a) For what value of x does the electric field have its largest value, xmax?b) What is the potential difference between the points at x = 0 and x = xmax? Get solution
46. Derive an expression for electric potential along the axis (the x-axis) of a disk with a hole in the center, as shown in the figure, where R1 and R2 are the inner and outer radii of the disk. What would the potential be if R1 = 0?... Get solution
47. An electric field is established in a nonuniform rod. A voltmeter is used to measure the potential difference between the left end of the rod and a point a distance x from the left end. The process is repeated, and it is found that the data are described by the relationship ∆V = 270.x2, where ∆V has the units V/m2. What is the x-component of the electric field at a point 13.0 cm from the left end? Get solution
48. Two parallel plates are held at potentials of +200.0 V and –100.0 V. The plates are separated by 1.00 cm.a) Find the electric field between the plates.b) An electron is initially placed halfway between the plates. Find its kinetic energy when it hits the positive plate. Get solution
49. A 2.50-mg dust particle with a charge of 1.00 µC falls at a point x = 2.00 m in a region where the electric potential varies according to V(x) = (2.00 V/m2)x2 – (3.00 V/m3)x3. With what acceleration will the particle start moving after it touches down? Get solution
50. The electric potential in a volume of space is given by V(x,y,z) = x2 + xy2 + yz. Determine the electric field in this region at the coordinate (3,4,5). Get solution
51. The electric potential inside a 10.0-m-long linear particle accelerator is given by V = (3000 – 5x2/m2) V, where x is the distance from the left plate along the accelerator tube, as shown in the figure.a) Determine an expression for the electric field along the accelerator tube.b) A proton is released (from rest) at x = 4.00 m. Calculate the acceleration of the proton just after it is released.c) What is the impact speed of the proton when (and if) it collides with the plate?... Get solution
52. An infinite plane of charge has a uniform charge distribution of +4.00 nC/m2 and is located in the yz-plane at x = 0. A +11.0-nC fixed point charge is located at x = +2.00 m.a) Find the electric potential V(x) on the x-axis from 0 x b) At what position(s) on the x-axis between x = 0 and x = +2.00 m is the electric potential a minimum?c) Where on the x-axis between x = 0 m and x = +2.00 m could a positive point charge be placed and not move? Get solution
53. Use ... to derive the expression for the electric field of a point charge, q. Get solution
54. Show that an electron in a one-dimensional electrical potential, V(x) = Ax2, where the constant A is a positive real number, will execute simple harmonic motion about the origin. What is the period of that motion? Get solution
55. The electric field, ... and the electric potential, ... are calculated from the charge distribution, ... by integrating Coulomb’s Law and then the electric field. In the other direction, the field and the charge distribution are determined from the potential by suitably differentiating. Suppose the electric potential in a large region of space is given by V(r) = V0 exp (–r2/a2), where V0 and a are constants and ... is the distance from the origin.a) Find the electric field ... in this region.b) Determine the charge density, ... in this region, which gives rise to the potential and field.c) Find the total charge in this region.d) Roughly sketch the charge distribution that could give rise to such an electric field. Get solution
56. The electron beam emitted by an electron gun is controlled (steered) with two sets of parallel conducting plates: a horizontal set to control the vertical motion of the beam, and a vertical set to control the horizontal motion of the beam. The beam is emitted with an initial velocity of 2.00·107 m/s. The width of the plates is d = 5.00 cm, the separation between the plates is D = 4.00 cm, and the distance between the edge of the plates and a target screen is L = 40.0 cm. In the absence of any applied voltage, the electron beam hits the origin of the xy-coordinate system on the observation screen. What voltages need to be applied to the two sets of plates for the electron beam to hit a target placed on the observation screen at coordinates (x,y) = (0 cm,8.00 cm)? Get solution
57. Nuclear fusion reactions require that positively charged nuclei be brought into close proximity, against the electrostatic repulsion. As a simple example, suppose a proton is fired at a second, stationary proton from a large distance away. What kinetic energy must be given to the moving proton to get it to come within 1.00 ·10–15 m of the target? Assume that there is a head-on collision and that the target is fixed in place. Get solution
58. Fission of a uranium nucleus (containing 92 protons) produces a barium nucleus (56 protons) and a krypton nucleus (36 protons). The fragments fly apart as a result of electrostatic repulsion; they ultimately emerge with a total of 200. MeV of kinetic energy. Use this information to estimate the size of the uranium nucleus; that is, treat the barium and krypton nuclei as point charges and calculate the separation between them at the start of the process. Get solution
59. A deuterium ion and a tritium ion each have charge +e. What work has to be done on the deuterium ion in order to bring it within 1.00 · 10–14 m of the tritium ion? This is the distance within which the two ions can fuse, as a result of strong nuclear interactions that overcome electrostatic repulsion, to produce a helium-5 nucleus. Express the work in electron-volts. Get solution
60. Three charges, q1, q2, and q3, are located at the corners of an equilateral triangle with side length of 1.20 m. Find the work done in each of the following cases:a) to bring the first particle, q1 = 1.00 pC, to P from infinityb) to bring the second particle, q2 = 2.00 pC, to Q from infinityc) to bring the last particle, q3 = 3.00 pC, to R from infinityd) Find the total potential energy stored in the final configuration of q1, q2, and q3.... Get solution
61. Two metal balls of mass m1 = 5.00 g (diameter = 5.00 mm) and m2 = 8.00 g (diameter = 8.00 mm) have positive charges of q1 = 5.00 nC and q2 = 8.00 nC, respectively. A force holds them in place so that their centers are separated by 8.00 mm. What will their velocities be after the force has been removed and they are separated by a large distance? Get solution
62. Two protons at rest and separated by 1.00 mm are released simultaneously. What is the speed of either at the instant when the two are 10.0 mm apart? Get solution
63. A 12-V battery is connected between a hollow metal sphere with a radius of 1 m and a ground, as shown in the figure. What are the electric field and the electric potential inside the hollow metal sphere?... Get solution
64. A solid metal ball with a radius of 3.00 m has a charge of 4.00 mC. If the electric potential is zero far away from the ball, what is the electric potential at each of the following positions?a) at r = 0 m, the center of the ballb) at r = 3.00 m, on the surface of the ballc) at r = 5.00 m Get solution
65. An insulating sheet in the xz-plane is uniformly charged with a charge distribution σ = 3.50 · 10–6 C/m2. What is the change in potential when a charge of Q = 1.25 µC is moved from position A to position B in the figure?...... Get solution
66. Suppose that an electron inside a cathode ray tube starts from rest and is accelerated by the tube’s voltage of 21.9 kV. What is the speed (in km/s) with which the electron (mass = 9.11 · 10–31 kg) hits the screen of the tube? Get solution
67. A conducting solid sphere (radius of R = 18.0 cm, charge of q = 6.10 · 10–6 C) is shown in the figure. Calculate the electric potential at a point 24.0 cm from the center (point A), a point on the surface (point B), and at the center of the sphere (point C). Assume that the electric potential is zero at points infinitely far away from the origin of the coordinate system.... Get solution
68. A classroom Van de Graaff generator accumulates a charge of 1.00 · 10–6 C on its spherical conductor, which has a radius of 10.0 cm and stands on an insulating column. Neglecting the effects of the generator base or any other objects or fields, find the potential at the surface of the sphere. Assume that the potential is zero at infinity. Get solution
69. A Van de Graaff generator has a spherical conductor with a radius of 25.0 cm. It can produce a maximum electric field of 2.00 · 106 V/m. What are the maximum voltage and charge that it can hold? Get solution
70. A proton with a speed of 1.23 · 104 m/s is moving from infinity directly toward a second proton. Assuming that the second proton is fixed in place, find the position where the moving proton stops momentarily before turning around. Get solution
71. Two metal spheres of radii r1 = 10.0 cm and r2 = 20.0 cm, respectively, have been positively charged so that each has a total charge of 100. µC.a) What is the ratio of their surface charge distributions?b) If the two spheres are connected by a copper wire, how much charge flows through the wire before the system reaches equilibrium? Get solution
72. The solid metal sphere of radius a = 0.200 m shown in the figure has a surface charge distribution of σ. The potential difference between the surface of the sphere and a point P at a distancerP = 0.500 m from the center of the sphere is ∆V = Vsurface – VP = +4π V = +12.566 V. Determine the value of σ.... Get solution
73. A particle with a charge of +5.00 µC is released from rest at a point on the x-axis, where x = 0.100 m. It begins to move as a result of the presence of a +9.00-µC charge that remains fixed at the origin. What is the kinetic energy of the particle at the instant it passes the point x = 0.200 m? Get solution
74. The sphere in the figure has a radius of 2.00 mm and carries a +2.00-µC charge uniformly distributed throughout its volume. What is the potential difference, VB – VA, if the angle between the two radii to points A and B is 60.0°? Is the potential difference dependent on the angle? Would the answer be the same if the charge distribution had an angular dependence, ρ = ρ (θ)?... Get solution
75. Two metallic spheres have radii of 10.0 cm and 5.00 cm, respectively. The magnitude of the electric field on the surface of each sphere is 3600. V/m. The two spheres are then connected by a long, thin metal wire. Determine the magnitude of the electric field on the surface of each sphere when they are connected. Get solution
76. A ring with charge Q and radius R is in the yz-plane and centered on the origin. What is the electric potential a distance x above the center of the ring? Derive the electric field from this relationship. Get solution
77. A charge of 0.681 nC is placed at x = 0. Another charge of 0.167 nC is placed at x1 = 10.9 cm on the x-axis.a) What is the combined electric potential of these two charges at x = 20.1 cm, also on the x-axis?b) At which point(s) on the x-axis does this potential have a minimum? Get solution
78. A point charge of +2.00 µC is located at (2.50 m,3.20 m). A second point charge of –3.10 µC is located at (–2.10 m,1.00 m).a) What is the electric potential at the origin?b) Along a line passing through both point charges, at what point(s) is (are) the electric potential(s) equal to zero? Get solution
79. A total charge of Q = 4.20·10–6 C is placed on a conducting sphere (sphere 1) of radius R = 0.400 m.a) What is the electric potential, V1, at the surface of sphere 1 assuming that the potential infinitely far away from it is zero? (Hint: What is the change in potential if a charge is brought from infinitely far away, where V(∞) = 0, to the surface of the sphere?)b) A second conducting sphere (sphere 2) of radius r = 0.100 m with an initial net charge of zero (q = 0) is connected to sphere 1 using a long thin metal wire. How much charge flows from sphere 1 to sphere 2 to bring them into equilibrium? What are the electric fi elds at the surfaces of the two spheres at equilibrium? Get solution
80. A thin line of charge is aligned along the positive y-axis from 0 ≤ y ≤ L, with L = 4.0 cm. The charge is not uniformly distributed but has a charge per unit length of λ = Ay, with A = 8.00·10–7 C/m2. Assuming that the electric potential is zero at infinite distance, find the electric potential at a point on the x-axis as a function of x. Give the value of the electric potential at x = 3.00 cm.... Get solution
81. Two fixed point charges are on the x-axis. A charge of –3.00 mC is located at x = +2.00 m and a charge of +5.00 mC is located at x = –4.00 m.a) Find the electric potential, V(x), for an arbitrary point on the x-axis.b) At what position(s) on the x-axis is V(x) = 0?c) Find E(x) for an arbitrary point on the x-axis. Get solution
82. One of the greatest physics experiments in history measured the charge-to-mass ratio of an electron, q/m. If a uniform potential difference is created between two plates, atomized particles—each with an integral amount of charge—can be suspended in space. The assumption is that the particles of unknown mass, M, contain a net number, n, of electrons of mass m and charge q. For a plate separation of d, what is the potential difference necessary to suspend a particle of mass M containing n net electrons? What is the acceleration of the particle if the voltage is cut in half? What is the acceleration of the particle if the voltage is doubled? Get solution
83. A uniform linear charge distribution of total positive charge Q has the shape of a half-circle of radius R, as shown in the figure.a) Without performing any calculations, predict the electric potential produced by this linear charge distribution at point O.b) Confirm, through direct calculations, your prediction of part (a).c) Make a similar prediction for the electric field.... Get solution
84. A point charge Q is placed a distance R from the center of a conducting sphere of radius a, with R > a (the point charge is outside the sphere). The sphere is grounded, that is, connected to a distant, unlimited source and/or sink of charge at zero potential. (Neither the distant ground nor the connection directly affects the electric field in the vicinity of the charge and sphere.) As a result, the sphere acquires a charge opposite in sign to Q, and the point charge experiences an attractive force toward the sphere.a) Remarkably, the electric field outside the sphere is the same as would be produced by the point charge Q plus an imaginary mirror-image point charge q, with magnitude and location that make the set of points corresponding to the surface of the sphere an equipotential of potential zero. That is, the imaginary point charge produces the same field contribution outside the sphere as the actual surface charge on the sphere. Calculate the value and location of q. (Hint: By symmetry, q must lie somewhere on the axis that passes through the center of the sphere and the location of Q.)b) Calculate the force exerted on point charge Q and directed toward the sphere, in terms of the original quantities Q, R, and a.c) Determine the actual nonuniform surface charge distribution on the conducting sphere. Get solution
85. A solid conducting sphere of radius R1 = 1.206 m has a charge of Q = 1.953 μC evenly distributed over its surface. A second solid conducting sphere of radius R2 = 0.6115 m is initially uncharged and at a distance of 10.00 m from the first sphere. The two spheres are momentarily connected with a wire, which is then removed. What is the charge on the second sphere? Get solution
86. A solid conducting sphere of radius R1 = 1.435 m has a charge of Q evenly distributed over its surface. A second solid conducting sphere of radius R2 = 0.6177 m is initially uncharged and at a distance of 10.00 m from the first sphere. The two spheres are momentarily connected with a wire, which is then removed. The resulting charge on the second sphere is 0.9356 μC. What was the original charge, Q, on the first sphere? Get solution
87. A solid conducting sphere of radius R1 has a charge of Q = 4.263 μC evenly distributed over its surface. A second solid conducting sphere of radius R2 = 0.6239 m is initially uncharged and at a distance of 10.00 m from the first sphere. The two spheres are momentarily connected with a wire, which is then removed. The resulting charge on the second sphere is 1.162 μC. What is the radius of the first sphere? Get solution
88. A solid conducting sphere of radius R = 1.895 m is charged, and the magnitude of the electric field at the surface of the sphere is 3.165·105 V/m. What is the electric potential 29.81 cm from the surface of the sphere? Get solution
89. A solid conducting sphere of radius R is charged, and the magnitude of the electric field at the surface of the sphere is 3.269·105 V/m. The electric potential 32.37 cm from the surface of the sphere is 2.843·105 V. What is the radius, R, of the sphere? Get solution
90. A solid conducting sphere of radius R = 1.351 m is charged, and the magnitude of the electric potential 34.95 cm from the surface of the sphere is 3.618·105 V. What is the magnitude of the electric field at the surface of the sphere? Get solution
