In the circuit shown in the figure, C¡ = 4.0 pF, C2 = 7.0 pF, C3 = 3.0 pF, C4 = 6.0 pF, C5 = 1.0 pF, and C6 = 2.0 pF. C, C, C, Co Determine the equivalent capacitance Cequiv of the combination shown in the figure. F Cequiv
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Capacitance of the capacitors are,
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We need to calculate the equivalent capacitance of the system.
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- Two capacitors, C1 = 18.0 F and C2 = 36.0 F, are connected in series, and a 12.0-V battery is connected across them. (a) Find the equivalent capacitance, and the energy contained in this equivalent capacitor. (b) Find the energy stored in each individual capacitor. Show that the sum of these two energies is the same as the energy found in part (a). Will this equality always be true, or does it depend on the number of capacitors and their capacitances? (c) If the same capacitors were connected in parallel, what potential difference would be required across them so that the combination stores the same energy' as in part (a)? Which capacitor stores more energy in this situation, C1 or C2?Two capacitors, C1 = 18.0 F and C2 = 36.0 F, are connected in series, and a 12.0-V battery is connected across the two capacitors. Find (a) the equivalent capacitance and (b) the energy stored in this equivalent capacitance. (c) Find the energy stored in each individual capacitor. (d) Show that the sum of these two energies is the same as the energy found in part (b). (e) Will this equality always be true, or docs it depend on the number of capacitors and their capacitances? (f) If the same capacitors were connected in parallel, what potential difference would be required across them so that the combination stores the same energy as in part (a)? (g) Which capacitor stores more energy in this situation, C1 or C2?Consider the circuit shown in Figure P26.24, where C1, = 6.00 F, C2 = 3.00 F. and V = 20.0 V. Capacitor C1 is first charged by closing switch S1. Switch S1 is then opened, and the charged capacitor is connected to the uncharged capacitor by closing Calculate (a) the initial charge acquired by C, and (b) the final charge on each capacitor.
- Consider the circuit shown in Figure P20.52, where C1 = 6.00 F, C2 = 3.00 F, and V = 20.0 V. Capacitor C1 is first charged by closing switch S1. Switch S1 is then opened, and the charged capacitor is connected to the uncharged capacitor by closing S2. Calculate (a) the initial charge acquired by C1 and (b) the final charge on each capacitor. Figure P20.52If three unequal capacitors, initially uncharged, are connected in series across a battery, which of the following statements is true? (a) The equivalent capacitance is greater than any of the individual capacitances, (b) The largest voltage appeal's across the smallest capacitance, (c) The largest voltage appears across the largest capacitance. (d) The capacitor with the largest capacitance has the greatest charge, (e) The capacitor with the smallest capacitance has the smallest charge.Two capacitors, C1 = 25.0 F and C2 = 5.00 F, are connected in parallel and charged with a 100-V power supply. (a) Draw a circuit diagram and (b) calculate the total energy stored in the two capacitors. (c) What If? What potential difference would be required across the same two capacitors connected in series for the combination to store the same amount of energy as in part (b)? (d) Draw a circuit diagram of the circuit described in part (c).
- According to its design specification, the timer circuit delaying the closing of an elevator door is to have a capacitance of 32.0 F between two points A and B. When one circuit is being constructed, the inexpensive but durable capacitor installed between these two points is found to have capacitance 34.8 F. To meet the specification, one additional capacitor can be placed between the two points, (a) Should it be in series or in parallel with the 34.8-F capacitor? (b) What should be its capacitance? (c) What If? The next circuit comes down the assembly line with capacitance 29.8 F between A and B. To meet the specification, what additional capacitor should be installed in series or in parallel in that circuit?Assume a length of axon membrane of about 0.10 m is excited by an action potential (length excited = nerve speed pulse duration = 50.0 m/s 2.0 103 s = 0.10 m). In the resting state, the outer surface of the axon wall is charged positively with K+ ions and the inner wall has an equal and opposite charge of negative organic ions, as shown in Figure P18.43. Model the axon as a parallel-plate capacitor and take C = 0A/d and Q = C V to investigate the charge as follows. Use typical values for a cylindrical axon of cell wall thickness d = 1.0 108 m, axon radius r = 1.0 101 m, and cell-wall dielectric constant = 3.0. (a) Calculate the positive charge on the outside of a 0.10-m piece of axon when it is not conducting an electric pulse. How many K+ ions are on the outside of the axon assuming an initial potential difference of 7.0 102 V? Is this a large charge per unit area? Hint: Calculate the charge per unit area in terms of electronic charge e per squared (2). An atom has a cross section of about 1 2 (1 = 1010 m). (b) How much positive charge must flow through the cell membrane to reach the excited state of + 3.0 102 V from the resting state of 7.0 102 V? How many sodium ions (Na+) is this? (c) If it takes 2.0 ms for the Na+ ions to enter the axon, what is the average current in the axon wall in this process? (d) How much energy does it take to raise the potential of the inner axon wall to + 3.0 102 V, starting from the resting potential of 7.0 102 V? Figure P18.43 Problem 43 and 44.Referring to Figure CQ21.4, describe what happens to the light-bulb after the switch is closed. Assume the capacitor has a large capacitance and is initially uncharged. Also assume the light illuminates when connected directly across the battery terminals.
- A pair of capacitors with capacitances CA = 3.70 F and CB = 6.40 F are connected in a network. What is the equivalent capacitance of the pair of capacitors if they are connected a. in parallel and b. in series?A Pairs of parallel wires or coaxial cables are two conductors separated by an insulator, so they have a capacitance. For a given cable, the capacitance is independent of the length if the cable is very long. A typical circuit model of a cable is shown in Figure P27.87. It is called a lumped-parameter model and represents how a unit length of the cable behaves. Find the equivalent capacitance of a. one unit length (Fig. P27.87A), b. two unit lengths (Fig. P27.87B), and c. an infinite number of unit lengths (Fig. P27.87C). Hint: For the infinite number of units, adding one more unit at the beginning does not change the equivalent capacitance.Assume a length of axon membrane of about 0.10 m is excited by an action potential (length excited = nerve speed pulse duration = 50.0 m/s 2.0 103 s = 0.10 m). In the resting state, the outer surface of the axon wall is charged positively with K+ ions and the inner wall has an equal and opposite charge of negative organic ions, as shown in Figure P18.43. Model the axon as a parallel-plate capacitor and take C = 0A/d and Q = C V to investigate the charge as follows. Use typical values for a cylindrical axon of cell wall thickness d = 1.0 108 m, axon radius r = 1.0 101 m, and cell-wall dielectric constant = 3.0. (a) Calculate the positive charge on the outside of a 0.10-m piece of axon when it is not conducting an electric pulse. How many K+ ions are on the outside of the axon assuming an initial potential difference of 7.0 102 V? Is this a large charge per unit area? Hint: Calculate the charge per unit area in terms of electronic charge e per squared (2). An atom has a cross section of about 1 2 (1 = 1010 m). (b) How much positive charge must flow through the cell membrane to reach the excited state of + 3.0 102 V from the resting state of 7.0 102 V? How many sodium ions (Na+) is this? (c) If it takes 2.0 ms for the Na+ ions to enter the axon, what is the average current in the axon wall in this process? (d) How much energy does it take to raise the potential of the inner axon wall to + 3.0 102 V, starting from the resting potential of 7.0 102 V? Figure P18.43 Problem 43 and 44.