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44920_16_p531-567 12/27/04 7:47 AM Page 566 566 Chapter 16 Electrical Energy and Capacitance ADDITIONAL PROBLEMS 52. Three parallel-plate capacitors are constructed, each having the same plate spacing d and with C1 having plate area A1, C 2 having area A2, and C 3 having area A3. Show that the total capacitance C of the three capacitors connected in parallel is the same as that of a capacitor having plate spacing d and plate area A A1 A2 A 3. 53. Three parallel-plate capacitors are constructed, each having the same plate area A and with C 1 having plate spacing d1, C 2 having plate spacing d 2, and C 3 having plate spacing d 3. Show that the total capacitance C of the three capacitors connected in series is the same as a capacitor of plate area A and with plate spacing d d1 d 2 d 3. 54. Two capacitors give an equivalent capacitance of Cp when connected in parallel and an equivalent capacitance of C s when connected in series. What is the capacitance of each capacitor? 55. An isolated capacitor of unknown capacitance has been charged to a potential difference of 100 V. When the charged capacitor is disconnected from the battery and then connected in parallel to an uncharged 10.0-F capacitor, the voltage across the combination is measured to be 30.0 V. Calculate the unknown capacitance. 56. Two charges of 1.0 C and 2.0 C are 0.50 m apart at two vertices of an equilateral triangle as in Figure P16.56. (a) What is the electric potential due to the 1.0-C charge at the third vertex, point P ? (b) What is the electric potential due to the 2.0-C charge at P ? (c) Find the total electric potential at P. (d) What is the work required to move a 3.0-C charge from infinity to P. 59. 60. 61. P 62. 0.50 m 0.50 m 0.50 m 1.0 mC 2.0 mC Figure P16.56 63. of the outer sphere approaches infinity, the capacitance approaches the value a/ke 40a. 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 is a device that applies a strong electric shock to the chest over a time of a few milliseconds. The device contains a capacitor of a few 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 to discharge the capacitor through the patient’s chest. Assume that an energy of 300 W s is to be delivered from a 30.0-F capacitor. To what potential difference must it be charged? When a certain air-filled parallel-plate capacitor is connected across a battery, it acquires a charge of 150 C on each plate. While the battery connection is maintained, a dielectric slab is inserted into, and fills, the region between the plates. This results in the accumulation of an additional charge of 200 C on each plate. What is the dielectric constant of the slab? Capacitors C1 6.0 F and C 2 2.0 F are charged as a parallel combination across a 250-V battery. The capacitors are disconnected from the battery and from each other. They are then connected positive plate to negative plate and negative plate to positive plate. Calculate the resulting charge on each capacitor. Capacitors C1 4.0 F and C 2 2.0 F are charged as a series combination across a 100-V battery. The two capacitors are disconnected from the battery and from each other. They are then connected positive plate to positive plate and negative plate to negative plate. Calculate the resulting charge on each capacitor. The charge distribution shown in Figure P16.63 is referred to as a linear quadrupole. (a) Show that the electric potential at a point on the x-axis where x d is 57. Find the equivalent capacitance of the group of capacitors shown in Figure P16.57. 5.00 mF 3.00 mF V (b) Show that the expression obtained in (a) when x d reduces to V 2.00 mF 4.00 mF 7.00 mF 2k e Qd 2 x3 y 3.00 mF 6.00 mF 2k e Qd 2 x 3 xd 2 +Q –2Q +Q x (–d, 0) (d, 0) 48.0 V Figure P16.57 58. A spherical capacitor consists of a spherical conducting shell of radius b and charge Q concentric with a smaller conducting sphere of radius a and charge Q. (a) Find the capacitance of this device. (b) Show that as the radius b Quadrupole Figure P16.63 64. The energy stored in a 52.0-F capacitor is used to melt a 6.00-mg sample of lead. To what voltage must the capacitor be initially charged, assuming that the initial tempera- 44920_16_p531-567 12/27/04 7:47 AM Page 567 Problems ture of the lead is 20.0°C? Lead has a specific heat of 128 J/kg°C, a melting point of 327.3°C, and a latent heat of fusion of 24.5 kJ/kg. 65. Consider a parallel-plate capacitor with charge Q and area A, filled with dielectric material having dielectric constant . It can be shown that the magnitude of the attractive force exerted on each plate by the other is F Q 2/(20A). When a potential difference of 100 V exists between the plates of an air-filled 20-F parallelplate capacitor, what force does each plate exert on the other if they are separated by 2.0 mm? 66. An electron is fired at a speed v0 5.6 106 m/s and at an angle 0 45° between two parallel conducting plates that are D 2.0 mm apart, as in Figure P16.66. If the voltage difference between the plates is V 100 V, determine (a) how close, d, the electron will get to the bottom plate and (b) where the electron will strike the top plate. y Path of the electron D 0 x u0 v0 d Figure P16.66 V 567 ACTIVITIES 1. It takes an electric field of about 30 kV/cm to cause a spark in dry air. Shuffle across a rug and reach toward a doorknob. By estimating the length of the spark, determine the electric potential difference that existed between your finger and the doorknob just before you touched the knob. Try this experiment again on a very humid day, and you will find that the spark is much shorter or is imperceptible. Why? 2. Suppose you are given a battery, a capacitor, two switches, a lightbulb, and several pieces of connecting wire. On a sheet of paper, design a circuit that will do the following: (1) When switch 1 is closed and switch 2 is open, the capacitor charges, but no current moves through the lightbulb. (2) Then, when switch 1 is opened and switch 2 closed, the lightbulb is connected to the capacitor, but not to the battery. Describe the motion of charge in the circuit when switch 1 is closed and switch 2 is open. Is energy being stored in the capacitor? What measurements would you have to make to determine how much energy, if any, is stored? What happens to the lightbulb when switch 1 is opened after the capacitor has charged and switch 2 is then closed? Will the bulb light and stay lit? What happens to the charge on the capacitor when switch 2 is closed in this way?