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Chapter 20 Problems 1, 2, 3 = straightforward, intermediate, challenging = full solution available in Student Solutions Manual/Study Guide = coached solution with hints available at www.pop4e.com = computer useful in solving problem = paired numerical and symbolic problems = biomedical application Section 20.1 Potential Difference and Electric Potential 1. (a) Calculate the speed of a proton that is accelerated from rest through a potential difference of 120 V. (b) Calculate the speed of an electron that is accelerated through the same potential difference. 2. How much work is done (by a battery, generator, or some other source of potential difference) in moving Avogadro’s number of electrons from an initial point where the electric potential is 9.00 V to a point where the potential is –5.00 V? (The potential in each case is measured relative to a common reference point.) Section 20.2 Potential Differences in a Uniform Electric Field 3. A uniform electric field of magnitude 250 V/m is directed in the positive x direction. A +12.0-μC charge moves from the origin to the point (x, y) = (20.0 cm, 50.0 cm). (a) What is the change in the potential energy of the charge–field system? (b) Through what potential difference does the charge move? 4. The difference in potential between the accelerating plates in the electron gun of a TV picture tube is about 25 000 V. If the distance between these plates is 1.50 cm, what is the magnitude of the uniform electric field in this region? 5. An electron moving parallel to the x axis has an initial speed of 3.70 × 106 m/s at the origin. Its speed is reduced to 1.40 × 105 m/s at the point x = 2.00 cm. Calculate the potential difference between the origin and that point. Which point is at the higher potential? 6. Review problem. A block having mass m and positive charge Q is connected to an insulating spring having constant k. The block lies on a frictionless, insulating horizontal track, and the system is immersed in a uniform electric field of magnitude E, directed as shown in Figure P20.6. If the block is released from rest when the spring is unstretched (at x = 0), (a) by what maximum amount does the spring expand? (b) What is the equilibrium position of the block? (c) Show that the block’s motion is simple harmonic and determine its period. (d) Repeat part (a), assuming that the coefficient of kinetic friction between block and surface is μk. what finite value(s) of x is the electric field zero? (b) For what finite value(s) of x is the electric potential zero? Figure P20.6 Section 20.3 Electric Potential and Electric Potential Energy Due to Point Charges Note: Unless stated otherwise, assume that the reference level of potential is V = 0 at r = ∞. 7. (a) Find the potential at a distance of 1.00 cm from a proton. (b) What is the potential difference between two points that are 1.00 cm and 2.00 cm from a proton? (c) Repeat parts (a) and (b) for an electron. 10. Compare this problem with Problem 19.55. Four identical point charges (q = +10.0 μC) are located on the corners of a rectangle as shown in Figure P19.55. The dimensions of the rectangle are L = 60.0 cm and W = 15.0 cm. Calculate the change in electric potential energy of the system as the charge at the lower left corner in Figure P19.55 is brought to this position from infinitely far away. Assume that the other three charges remain fixed in position. 11. The three charges in Figure P20.11 are at the vertices of an isosceles triangle. Calculate the electric potential at the midpoint of the base, taking q = 7.00 μC. 8. Given two 2.00-μC charges as shown in Figure P20.8 and a positive test charge q = 1.28 × 10–18 C at the origin, (a) what is the net force exerted by the two 2.00-μC charges on the test charge q? (b) What is the electric field at the origin due to the two 2.00-μC charges? (c) What is the electrical potential at the origin due to the two 2.00μC charges? Figure P20.8 9. A charge +q is at the origin. A charge –2q is at x = 2.00 m on the x axis. (a) For Figure P20.11 12. Compare this problem with Problem 19.14. Two point charges each of magnitude 2.00 μC are located on the x axis. One is at x = 1.00 m and the other is at x = –1.00 m. (a) Determine the electric potential on the y axis at y = 0.500 m. (b) Calculate the change in electric potential energy of the system as a third charge of –3.00 μC is brought from infinitely far away to a position on the y axis at y = 0.500 m. 13. Show that the amount of work required to assemble four identical point charges of magnitude Q at the corners of a square of side s is 5.41keQ2/s. 14. Two charged particles create influences at the origin, described by the expressions 7.00 10 9 C 8.99 10 9 N m 2 /C 2 cos 70.0î 2 0.070 0 m 9 7.00 10 9 C ˆj 8.00 10 C ˆj sin 70 . 0 0.070 0 m 2 0.030 0 m 2 and 7.00 10 9 C 8.00 10 9 C 8.99 10 9 N m 2 /C 2 0.030 0 m 0.070 0 m (a) Identify the locations of the particles and the charges on them. (b) Find the force on a –16.0 nC charge placed at the origin and (c) the work required to move this third charge to the origin from a very distant point. 15. Review problem. Two insulating spheres have radii 0.300 cm and 0.500 cm, masses 0.100 kg and 0.700 kg, and uniformly distributed charges –2.00 μC and 3.00 μC. They are released from rest when their centers are separated by 1.00 m. (a) How fast will each be moving when they collide? (Suggestion: Consider conservation of energy and of linear momentum.) (b) If the spheres were conductors, would the speeds be greater or less than those calculated in part (a)? Explain. 16. Review problem. Two insulating spheres have radii r1 and r2, masses m1 and m2, and uniformly distributed charges –q1 and q2. They are released from rest when their centers are separated by a distance d. (a) How fast is each moving when they collide? (Suggestion: Consider conservation of energy and conservation of linear momentum.) (b) If the spheres were conductors, would their speeds be greater or less than those calculated in part (a)? Explain. 17. Compare this problem with Problem 19.26. Three equal positive charges q are at the corners of an equilateral triangle of side a as shown in Figure P19.26. (a) At what point, if any, in the plane of the charges is the electric potential zero? (b) What is the electric potential at the point P due to the two charges at the base of the triangle? 18. Two particles, with charges of 20.0 nC and –20.0 nC, are placed at the points with coordinates (0, 4.00 cm) and (0, –4.00 cm) as shown in Figure P20.18. A particle with charge 10.0 nC is located at the origin. (a) Find the electric potential energy of the configuration of the three fixed charges. (b) A fourth particle, with a mass of 2.00 × 10–13 kg and a charge of 40.0 nC, is released from rest at the point (3.00 cm, 0). Find its speed after it has moved freely to a very large distance away. Figure P20.18 19. Review problem. A light, unstressed spring has length d. Two identical particles, each with charge q, are connected to the opposite ends of the spring. The particles are held stationary a distance d apart and are then released at the same time. The system then oscillates on a horizontal frictionless table. The spring has a bit of internal kinetic friction, so the oscillation is damped. The particles eventually stop vibrating when the distance between them is 3d. Find the increase in internal energy that appears in the spring during the oscillations. Assume that the system of the spring and two charges is isolated. 20. In 1911, Ernest Rutherford and his assistants Hans Geiger and Ernest Marsden conducted an experiment in which they scattered alpha particles from thin sheets of gold. An alpha particle, having charge +2e and mass 6.64 × 10–27 kg, is a product of certain radioactive decays. The results of the experiment led Rutherford to the idea that most of the mass of an atom is in a very small nucleus, with electrons in orbit around it, in his planetary model of the atom. Assume that an alpha particle, initially very far from a gold nucleus, is fired with a velocity of 2.00 × 107 m/s directly toward the nucleus (charge +79e). How close does the alpha particle get to the nucleus before turning around? Assume that the gold nucleus remains stationary. Section 20.4 Obtaining Electric Field From Electric Potential 21. The potential in a region between x = 0 and x = 6.00 m is V = a + bx, where a = 10.0 V and b = –7.00 V/m. Determine (a) the potential at x = 0, 3.00 m, and 6.00 m; and (b) the magnitude and direction of the electric field at x = 0, 3.00 m, and 6.00 m. 22. The electric potential inside a charged spherical conductor of radius R is given by V = keQ/R and outside the potential is given by V = keQ/r. Using Er = –dV/dr, derive the electric field (a) inside and (b) outside this charge distribution. 23. Over a certain region of space, the electric potential is V = 5x – 3x2y + 2yz2. Find the expressions for the x, y, and z components of the electric field over this region. What is the magnitude of the field at the point P that has coordinates (1, 0, –2) m? Section 20.5 Electric Potential Due to Continuous Charge Distributions 24. Consider a ring of radius R with the total charge Q spread uniformly over its perimeter. What is the potential difference between the point at the center of the ring and a point on its axis a distance 2R from the center? 25. A rod of length L (Fig. P20.25) lies along the x axis with its left end at the origin. It has a nonuniform charge density λ = αx, where α is a positive constant. (a) What are the units of α? (b) Calculate the electric potential at A. Figure P20.25 Problems 20.25 and 20.26. 26. For the arrangement described in Problem 20.25, calculate the electric potential at point B that lies on the perpendicular bisector of the rod a distance b above the x axis. 27. Compare this problem with Problem 19.21. A uniformly charged insulating rod of length 14.0 cm is bent into the shape of a semicircle as shown in Figure P19.21. The rod has a total charge of –7.50 μC. Find the electric potential at O, the center of the semicircle. Section 20.6 Electric Potential of a Charged Conductor 28. How many electrons should be removed from an initially uncharged spherical conductor of radius 0.300 m to produce a potential of 7.50 kV at the surface? 29. A spherical conductor has a radius of 14.0 cm and charge of 26.0 μC. Calculate the electric field and the electric potential (a) r = 10.0 cm, (b) r = 20.0 cm, and (c) r = 14.0 cm from the center. 30. Electric charge can accumulate on an airplane in flight. You may have observed needle-shaped metal extensions on the wing tips and tail of an airplane. Their purpose is to allow charge to leak off before much of it accumulates. The electric field around the needle is much larger than around the body of the airplane and can become large enough to produce dielectric breakdown of the air, discharging the airplane. To model this process, assume that two charged spherical conductors are connected by a long conducting wire and that a charge of 1.20 μC is placed on the combination. One sphere, representing the body of the airplane, has a radius of 6.00 cm, and the other, representing the tip of the needle, has a radius of 2.00 cm. (a) What is the electric potential of each sphere? (b) What is the electric field at the surface of each sphere? 642 and modeled in Figure P20.34, a second identical set of plates is enmeshed with its plates halfway between those of the first set. The second set can rotate as a unit. Determine the capacitance as a function of the angle of rotation θ, where θ = 0 corresponds to the maximum capacitance. Section 20.7 Capacitance 31. (a) How much charge is on each plate of a 4.00-μF capacitor when it is connected to a 12.0-V battery? (b) If this same capacitor is connected to a 1.50-V battery, what charge is stored? 32. Two conductors having net charges of +10.0 μC and –10.0 μC have a potential difference of 10.0 V between them. (a) Determine the capacitance of the system. (b) What is the potential difference between the two conductors if the charges on each are increased to +100 μC and –100 μC? 33. An isolated charged conducting sphere of radius 12.0 cm creates an electric field of 4.90 × 104 N/C at a distance 21.0 cm from its center. (a) What is its surface charge density? (b) What is its capacitance? 34. A variable air capacitor used in a radio tuning circuit is made of N semicircular plates each of radius R and positioned a distance d from its neighbors, to which it is electrically connected. As shown in the opening photograph on page Figure P20.34 35. An air-filled capacitor consists of two parallel plates, each with an area of 7.60 cm2, separated by a distance of 1.80 mm. A 20.0-V potential difference is applied to these plates. Calculate (a) the electric field between the plates, (b) the surface charge density, (c) the capacitance, and (d) the charge on each plate. 36. A 50.0-m length of coaxial cable has an inner conductor that has a diameter of 2.58 mm and carries a charge of 8.10 μC. The surrounding conductor has an inner diameter of 7.27 mm and a charge of –8.10 μC. (a) What is the capacitance of this cable? (b) What is the potential difference between the two conductors? Assume that the region between the conductors is air. 37. A small object of mass m carries a charge q and is suspended by a thread between the vertical plates of a parallelplate capacitor. The plate separation is d. If the thread makes an angle θ with the vertical, what is the potential difference between the plates? 38. A spherical capacitor consists of a spherical conducting shell of radius b and charge –Q that is concentric with a smaller conducting sphere of radius a and charge +Q (Fig. P20.38). (a) Show that its capacitance is C ab k e b a 39. Two capacitors, C1 = 5.00 μF and C2 = 12.0 μF, are connected in parallel, and the resulting combination is connected to a 9.00-V battery. (a) What is the equivalent capacitance of the combination? What are (b) the potential difference across each capacitor and (c) the charge stored on each capacitor? 40. The two capacitors of Problem 20.39 are now connected in series and to a 9.00-V battery. Find (a) the equivalent capacitance of the combination, (b) the potential difference across each capacitor, and (c) the charge on each capacitor. 41. Four capacitors are connected as shown in Figure P20.41. (a) Find the equivalent capacitance between points a and b. (b) Calculate the charge on each capacitor, taking ΔVab = 15.0 V. (b) Show that as b approaches infinity, the capacitance approaches the value a/ke = 4πε0a. Figure P20.41 Figure P20.38 Section 20.8 Combinations of Capacitors 42. Two capacitors when connected in parallel give an equivalent capacitance of Cp and an equivalent capacitance of Cs when connected in series. What is the capacitance of each capacitor? 43. Consider the circuit shown in Figure P20.43, where C1 = 6.00 μF, C2 = 3.00 μF, and ΔV = 20.0 V. Capacitor C1 is first charged by the closing of switch S1. Switch S1 is then opened, and the charged capacitor is connected to the uncharged capacitor by the closing of S2. Calculate the initial charge acquired by C1 and the final charge on each capacitor. Figure P20.44 Figure P20.43 44. Three capacitors are connected to a battery as shown in Figure P20.44. Their capacitances are C1 = 3C, C2 = C, and C3 = 5C. (a) What is the equivalent capacitance of this set of capacitors? (b) State the ranking of the capacitors according to the charge they store, from largest to smallest. (c) Rank the capacitors according to the potential differences across them, from largest to smallest. (d) If now C3 is increased, what happens to the charge stored by each of the capacitors? 45. According to its design specification, the timer circuit delaying the closing of an elevator door is to have a capacitance of 32.0 μF between two points A and B. (a) When one circuit is being constructed, the inexpensive but durable capacitor installed between these two points is found to have capacitance 34.8 μF. To meet the specification, one additional capacitor can be placed between the two points. Should it be in series or in parallel with the 34.8-μF capacitor? What should be its capacitance? (b) The next circuit comes down the assembly line with capacitance 29.8 μF between A and B. What additional capacitor should be installed in series or in parallel in that circuit to meet the specification? 46. Find the equivalent capacitance between points a and b in the combination of capacitors shown in Figure P20.46. to discharge the capacitor through the patient’s chest. Assume that an energy of 300 J is to be delivered from a 30.0-μF capacitor. To what potential difference must it be charged? Figure P20.46 Section 20.9 Energy Stored in a Charged Capacitor 47. (a) A 3.00-μF capacitor is connected to a 12.0-V battery. How much energy is stored in the capacitor? (b) If the capacitor had been connected to a 6.00-V battery, how much energy would have been stored? 48. The immediate cause of many deaths is ventricular fibrillation, an uncoordinated quivering of the heart as opposed to proper beating. An electric shock to the chest can cause momentary paralysis of the heart muscle, after which the heart will sometimes start organized beating again. A defibrillator (Fig. P20.48) is a device that applies a strong electric shock to the chest over a time interval of a few milliseconds. The device contains a capacitor of several microfarads, charged to several thousand volts. Electrodes called paddles, about 8 cm across and coated with conducting paste, are held against the chest on both sides of the heart. Their handles are insulated to prevent injury to the operator, who calls “Clear!” and pushes a button on one paddle (Adam Hart-Davis/SPL/Custom Medical Stock) Figure P20.48 A defibrillator in use. 49. Two capacitors, C1 = 25.0 μF and C2 = 5.00 μF, are connected in parallel and charged with a 100-V power supply. (a) Draw a circuit diagram and calculate the total energy stored in the two capacitors. (b) What potential difference would be required across the same two capacitors connected in series so that the combination stores the same energy as in part (a)? Draw a circuit diagram of this circuit. 50. As a person moves about in a dry environment, electric charge accumulates on the person’s body. Once it is at high voltage, either positive or negative, the body can discharge via sometimes noticeable sparks and shocks. Consider a human body well separated from ground, with the typical capacitance 150 pF. (a) What charge on the body will produce a potential of 10.0 kV? (b) Sensitive electronic devices can be destroyed by electrostatic discharge from a person. A particular device can be destroyed by a discharge releasing an energy of 250 μJ. To what voltage on the body does this energy correspond? 51. A parallel-plate capacitor has a charge Q and plates of area A. What force acts on one plate to attract it toward the other plate? Because the electric field between the plates is E = Q /Aε0, you might think that the force is F = QE = Q2/Aε0. This conclusion is wrong because the field E includes contributions from both plates, and the field created by the positive plate cannot exert any force on the positive plate. Show that the force exerted on each plate is actually F = Q2/2ε0A. (Suggestion: Let C = ε0A/x for an arbitrary plate separation x; then require that the work done in separating the two charged plates be W F dx .) The force exerted by one charged plate on another is sometimes used in a machine shop to hold a workpiece stationary. 52. A uniform electric field E = 3 000 V/m exists within a certain region. What volume of space contains an energy equal to 1.00 × 10–7 J? Express your answer in cubic meters and in liters. Section 20.10 Capacitors with Dielectrics 53. Determine (a) the capacitance and (b) the maximum potential difference that can be applied to a Teflon-filled parallel- plate capacitor having a plate area of 1.75 cm2 and plate separation of 0.040 0 mm. 54. (a) How much charge can be placed on a capacitor with air between the plates before it breaks down if the area of each of the plates is 5.00 cm2? (b) Find the maximum charge assuming polystyrene is used between the plates instead of air. 55. A commercial capacitor is to be constructed as shown in Figure 20.29a. This particular capacitor is made from two strips of aluminum foil separated by a strip of paraffin-coated paper. Each strip of foil and paper is 7.00 cm wide. The foil is 0.004 00 mm thick, and the paper is 0.025 0 mm thick and has a dielectric constant of 3.70. What length should the strips have if a capacitance of 9.50 × 10–8 F is desired before the capacitor is rolled up? (Adding a second strip of paper and rolling the capacitor effectively doubles its capacitance by allowing charge storage on both sides of each strip of foil.) 56. The supermarket sells rolls of aluminum foil, of plastic wrap, and of waxed paper. Describe a capacitor made from supermarket materials. Compute order-of-magnitude estimates for its capacitance and its breakdown voltage. 57. A parallel-plate capacitor in air has a plate separation of 1.50 cm and a plate area of 25.0 cm2. The plates are charged to a potential difference of 250 V and disconnected from the source. The capacitor is then immersed in distilled water. Determine (a) the charge on the plates before and after immersion, (b) the capacitance and potential difference after immersion, and (c) the change in energy of the capacitor. Assume that the liquid is an insulator. Section 20.11 Context Connection—The Atmosphere as a Capacitor 58. Lightning can be studied with a Van de Graaff generator, essentially consisting of a spherical dome on which charge is continuously deposited by a moving belt. Charge can be added until the electric field at the surface of the dome becomes equal to the dielectric strength of air (3 × 106 V/m). Any more charge leaks off in sparks as shown in Figure P20.58. Assume that the dome has a diameter of 30.0 cm and is surrounded by dry air. (a) What is the maximum potential of the dome? (b) What is the maximum charge on the dome? (E. R. Degginer/H. Armstrong Roberts) Figure P20.58 59. Review problem. A certain storm cloud has a potential of 1.00 × 108 V relative to a tree. If, during a lightning storm, 50.0 C of charge is transferred through this potential difference and 1.00% of the energy is absorbed by the tree, how much sap in the tree can be boiled away? Model the sap as water initially at 30.0°C. Water has a specific heat of 4 186 J/kg · °C, a boiling point of 100°C, and a latent heat of vaporization of 2.26 × 106 J/kg. Additional Problems 60. Review problem. From a large distance away, a particle of mass 2.00 g and charge 15.0 μC is fired at 21.0 î m/s straight toward a second particle, originally stationary but free to move, with mass 5.00 g and charge 8.50 μC. (a) At the instant of closest approach, both particles will be moving at the same velocity. Explain why. (b) Find this velocity. (c) Find the distance of closest approach. (d) Find the velocities of both particles after they separate again. 61. The liquid-drop model of the atomic nucleus suggests that high-energy oscillations of certain nuclei can split the nucleus into two unequal fragments plus a few neutrons. The fission products acquire kinetic energy from their mutual Coulomb repulsion. Calculate the electric potential energy (in electron volts) of two spherical fragments from a uranium nucleus having the following charges and radii: 38e and 5.50 × 10–15 m; 54e and 6.20 × 10–15 m. Assume that the charge is distributed uniformly throughout the volume of each spherical fragment and that just before separating each fragment is at rest and their surfaces are in contact. The electrons surrounding the nucleus can be ignored. 62. The Bohr model of the hydrogen atom states that the single electron can exist only in certain allowed orbits around the proton. The radius of each Bohr orbit is r = n2(0.052 9 nm) where n = 1, 2, 3, . . . . Calculate the electric potential energy of a hydrogen atom when the electron is in the (a) first allowed orbit, with n = 1; (b) second allowed orbit, with n = 2; and (c) when the electron has escaped from the atom, with r = ∞. Express your answers in electron volts. 63. Calculate the work that must be done to charge a spherical shell of radius R to a total charge Q. 64. A Geiger–Mueller tube is a radiation detector that essentially consists of a closed, hollow metal cylinder (the cathode) of inner radius ra and a coaxial cylindrical wire (the anode) of radius rb (Fig. P20.64). The charge per unit length on the anode is λ, and the charge per unit length on the cathode is –λ. A gas fills the space between the electrodes. When a high-energy elementary particle passes through this space, it can ionize an atom of the gas. The strong electric field makes the resulting ion and electron accelerate in opposite directions. They strike other molecules of the gas to ionize them, producing an avalanche of electrical discharge. The pulse of electric current between the wire and the cylinder is counted by an external circuit. (a) Show that the magnitude of the potential difference between the wire and the cylinder is r V 2 k e ln a rb (b) Show that the magnitude of the electric field in the space between cathode and anode is given by E V 1 ln ra / rb r where r is the distance from the axis of the anode to the point where the field is to be calculated. Figure P20.64 Problems 20.64, 20.65, and 20.66. 65. Assume that the internal diameter of the Geiger–Mueller tube described in Problem 20.64 is 2.50 cm and that the wire along the axis has a diameter of 0.200 mm. The dielectric strength of the gas between the central wire and the cylinder is 1.20 × 106 V/m. Use the result of Problem 20.64 to calculate the maximum potential difference that can be applied between the wire and the cylinder before breakdown occurs in the gas. 66. The results of Problem 20.64 apply also to an electrostatic precipitator (Figs. P20.64 and P20.66). This pollution-control device consists of a vertical cylindrical duct with a wire along its axis at a high negative voltage. Corona discharge ionizes the air around the wire to produce free electrons and positive and negative molecular ions. The electrons and negative ions accelerate outward. As air passes through the cylinder, the dirt particles become electrically charged by collisions and ion capture. They are then swept out of the air by the horizontal electric field between the wire and the cylinder. In a particular case, an applied voltage ΔV = Va – Vb = 50.0 kV is to produce an electric field of magnitude 5.50 MV/m at the surface of the central wire. Assume that the outer cylindrical wall has uniform radius ra = 0.850 m. (a) What should be the radius rb of the central wire? You will need to solve a transcendental equation. (b) What is the magnitude of the electric field at the outer wall? Figure P20.66 67. A model of a red blood cell portrays the cell as a capacitor with two spherical plates. It is a positively charged conducting liquid sphere of area A, separated by an insulating membrane of thickness t from the surrounding negatively charged conducting fluid. Tiny electrodes introduced into the cell show a potential difference of 100 mV across the membrane. Take the membrane’s thickness as 100 nm and its dielectric constant as 5.00. (a) Assume that a typical red blood cell has a mass of 1.00 × 10–12 kg and density 1 100 kg/m3. Calculate its volume and its surface area. (b) Find the capacitance of the cell. (c) Calculate the charge on the surfaces of the membrane. How many electronic charges does this charge represent? (Suggestion: The chapter text models the Earth’s atmosphere as a capacitor with two spherical plates.) 68. Four balls, each with mass m, are connected by four nonconducting strings to form a square with side a as shown in Figure P20.68. The assembly is placed on a horizontal, nonconducting, frictionless surface. Balls 1 and 2 each have charge q, and balls 3 and 4 are uncharged. Find the maximum speed of balls 3 and 4 after the string connecting balls 1 and 2 is cut. Figure P20.69 70. An electric dipole is located along the y axis as shown in Figure P20.70. The magnitude of its electric dipole moment is defined as p = 2qa. (a) At a point P, which is far from the dipole (r >> a), show that the electric potential is V Figure P20.68 69. The x axis is the symmetry axis of a stationary, uniformly charged ring of radius R and charge Q (Fig. P20.69). A particle with charge Q and mass M is located at the center of the ring. When it is displaced slightly, the point charge accelerates along the x axis to infinity. Show that the ultimate speed of the point charge is 2k Q 2 v e MR 1/ 2 k e p cos r2 (b) Calculate the radial component Er and the perpendicular component Eθ of the associated electric field. Note that Eθ = −(1/r)(∂V/∂θ). Do these results seem reasonable for θ = 90° and 0°? for r = 0? (c) For the dipole arrangement shown, express V in terms of Cartesian coordinates using r = (x2 + y2)1/2 and cos x y 2 y2 1/ 2 Using these results and again taking r >> a, calculate the field components Ex and Ey. Figure P20.71 72. A 2.00-nF parallel-plate capacitor is charged to an initial potential difference ΔVi = 100 V and then isolated. The dielectric material between the plates is mica, with a dielectric constant of 5.00. (a) How much work is required to withdraw the mica sheet? (b) What is the potential difference of the capacitor after the mica is withdrawn? Figure P20.70 71. Two large, parallel metal plates are oriented horizontally and separated by a distance 3d. A grounded conducting wire joins them, and initially each plate carries no charge. Now a third identical plate carrying charge Q is inserted between the two plates, parallel to them and located a distance d from the upper plate, as shown in Figure P20.71. (a) What induced charge appears on each of the two original plates? (b) What potential difference appears between the middle plate and each of the other plates? Let A represent the area of each plate. 73. A parallel-plate capacitor is constructed using a dielectric material whose dielectric constant is 3.00 and whose dielectric strength is 2.00 × 108 V/m. The desired capacitance is 0.250 μF, and the capacitor must withstand a maximum potential difference of 4 000 V. Find the minimum area of the capacitor plates. 74. A 10.0-μF capacitor is charged to 15.0 V. It is next connected in series with an uncharged 5.00-μF capacitor. The series combination is finally connected across a 50.0-V battery as diagrammed in Figure P20.74. Find the new potential differences across the 5.00-μF and 10.0-μF capacitors. Figure P20.74 75. A capacitor is constructed from two square plates of sides ℓ and separation d. A material of dielectric constant κ is inserted a distance x into the capacitor as shown in Figure P20.75. Assume that d is much smaller than x. (a) Find the equivalent capacitance of the device. (b) Calculate the energy stored in the capacitor, letting ΔV represent the potential difference. (c) Find the direction and magnitude of the force exerted on the dielectric, assuming a constant potential difference ΔV. Ignore friction. (d) Obtain a numerical value for the force assuming that ℓ = 5.00 cm, ΔV = 2 000 V, d = 2.00 mm, and the dielectric is glass (κ = 4.50). (Suggestion: The system can be considered as two capacitors connected in parallel.) Figure P20.75 Problems 20.75 and 20.76. 76. A capacitor is constructed from two square plates of sides ℓ and separation d as suggested in Figure P20.75. You may assume that d is much less than ℓ. The plates carry charges +Q0 and –Q0. A block of metal has width ℓ, length ℓ, and thickness slightly less than d. It is inserted a distance x into the capacitor. The charges on the plates are not disturbed as the block slides in. In a static situation, a metal prevents an electric field from penetrating inside it. The metal can be thought of as a perfect dielectric, with κ → ∞. (a) Calculate the stored energy as a function of x. (b) Find the direction and magnitude of the force that acts on the metallic block. (c) The area of the advancing front face of the block is essentially equal to ℓd. Considering the force on the block as acting on this face, find the stress (force per area) on it. (d) For comparison, express the energy density in the electric field between the capacitor plates in terms of Q0, ℓ, d, and ε0. 77. Determine the equivalent capacitance of the combination shown in Figure P20.77. (Suggestion: Consider the symmetry involved.) Figure P20.77 © Copyright 2004 Thomson. All rights reserved.