Fundamentals Physics Chapter 25 Capacitance PDF

Fundamentals Physics Eleventh Edition Halliday Chapter 25 Capacitance 25-1 Capacitance (1 of 6) Learning Objectives 25.01 Sketch a schematic diagram of a circuit with a parallel- plate capacitor, a battery, and an open or closed switch. 25.02 In a circuit with a battery,...

Fundamentals Physics Eleventh Edition Halliday Chapter 25 Capacitance 25-1 Capacitance (1 of 6) Learning Objectives 25.01 Sketch a schematic diagram of a circuit with a parallel- plate capacitor, a battery, and an open or closed switch. 25.02 In a circuit with a battery, an open switch, and an uncharged capacitor, explain what happens to the conduction electrons when the switch is closed. 25.03 For a capacitor, apply the relationship between the magnitude of charge q on either plate (“the charge on the capacitor”), the potential difference V between the plates (“the potential across the capacitor”), and the capacitance C of the capacitor. Copyright ©2018 John Wiley & Sons, Inc 2 25-1 Capacitance (2 of 6) A capacitor consists of two isolated conductors (the plates) with charges +q and −q. Its capacitance C is defined from q = CV. where V is the potential difference between the plates. Copyright ©2018 John Wiley & Sons, Inc 3 25-1 Capacitance (3 of 6) A parallel-plate capacitor, made up of two plates of area A separated by a distance d. The charges on the facing plate surfaces have the same magnitude q but opposite signs Copyright ©2018 John Wiley & Sons, Inc 4 25-1 Capacitance (4 of 6) As the field lines show, the electric field due to the charged plates is uniform in the central region between the plates. The field is not uniform at the edges of the plates, as indicated by the “fringing” of the field lines there. Copyright ©2018 John Wiley & Sons, Inc 5 25-1 Capacitance (5 of 6) Charging Capacitor When a circuit with a battery, an open switch, and an uncharged capacitor is completed by closing the switch, conduction electrons shift, leaving the capacitor plates with opposite charges. Copyright ©2018 John Wiley & Sons, Inc 6 25-1 Capacitance (6 of 6) In Figure. a, a battery B, a switch S, an uncharged capacitor C, and interconnecting wires form a circuit. The same circuit is shown in the schematic diagram of Figure. b, in which the symbols for a battery, a switch, and a capacitor represent those devices. The battery maintains potential difference V between its terminals. The terminal of higher potential is labeled + and is often called the positive terminal; the terminal of lower potential is labeled − and is often called the negative terminal. Copyright ©2018 John Wiley & Sons, Inc 7 25-2 Calculating the Capacitance (1 of 12) Learning Objectives 25.04 Explain how Gauss’ law is used to find the capacitance of a parallel-plate capacitor. 25.05 For a parallel-plate capacitor, a cylindrical capacitor, a spherical capacitor, and an isolated sphere, calculate the capacitance. Copyright ©2018 John Wiley & Sons, Inc 8 25-2 Calculating the Capacitance (2 of 12) Calculating electric field and potential difference To relate the electric field E between the plates of a capacitor to the charge q on either plate, we shall use Gauss’ law: ur ur ÑE ×d A = q. e0 ò the potential difference between the plates of a capacitor is related to the field E by f V f − Vi = −  E  ds, i Copyright ©2018 John Wiley & Sons, Inc 9 25-2 Calculating the Capacitance (3 of 12) Letting V represent the difference V f = Vi , we can then recast the above equation as: + V =  E ds − A charged parallel-plate capacitor. A Gaussian surface encloses the charge on the positive plate. The integration is taken along a path extending directly from the negative plate to the positive plate. Copyright ©2018 John Wiley & Sons, Inc 10 25-2 Calculating the Capacitance (4 of 12) Parallel-Plate Capacitor We assume, as Figure suggests, that the plates of our parallel-plate capacitor are so large and so close together that we can neglect the fringing of the electric field at the edges of the plates, taking E to be constant throughout the region between the plates. We draw a Gaussian surface that encloses just the charge q on the positive plate q =  0 EA Copyright ©2018 John Wiley & Sons, Inc 11 25-2 Calculating the Capacitance (5 of 12) where A is the area of the plate. And therefore, + d V =  E ds = E  ds = Ed. − 0 Now if we substitute q in the above relations to q = CV, we get, 0 A C= ( parallel-plate capacitor ). d Copyright ©2018 John Wiley & Sons, Inc 12 25-2 Calculating the Capacitance (6 of 12) A charged parallel-plate capacitor. A Gaussian surface encloses the charge on the positive plate. The integration is taken along a path extending directly from the negative plate to the positive plate. Copyright ©2018 John Wiley & Sons, Inc 13 25-2 Calculating the Capacitance (7 of 12) Cylindrical Capacitor Figure shows, in cross section, a cylindrical capacitor of length L formed by two coaxial cylinders of radii a and b. We assume that L ? b so that we can neglect the fringing of the electric field that occurs at the ends of the cylinders. Each plate contains a charge of magnitude q. Here, charge and the field magnitude E is related as follows, q =  0 EA =  0 E ( 2 rL ) Copyright ©2018 John Wiley & Sons, Inc 14 25-2 Calculating the Capacitance (8 of 12) Solving for E field: + = 2q0 L ln ( ba ) a V =  E ds = − 2 0 L  q dr − b r From the relation C = Vq , we then have L C = 2 0 ( cylindrical capacitor ). ln ( b a ) Copyright ©2018 John Wiley & Sons, Inc 15 25-2 Calculating the Capacitance (9 of 12) A cross section of a long cylindrical capacitor, showing a cylindrical Gaussian surface of radius r (that encloses the positive plate) and the radial path of integration. This figure also serves to illustrate a spherical capacitor in a cross section through its center. Copyright ©2018 John Wiley & Sons, Inc 16 25-2 Calculating the Capacitance (10 of 12) Others… For spherical capacitor the capacitance is: ab C = 4 0 ( spherical capacitor ). b−a Capacitance of an isolated sphere: C = 4 0 R ( isolated sphere ). Copyright ©2018 John Wiley & Sons, Inc 17 25-2 Calculating the Capacitance (11 of 12) Checkpoint 2 For capacitors charged by the same battery, does the charge stored by the capacitor increase, decrease, or remain the same in each of the following situations? (a) The plate separation of a parallel-plate capacitor is increased. (b) The radius of the inner cylinder of a cylindrical capacitor is increased. (c) The radius of the outer spherical shell of a spherical capacitor is increased. Answer: (a) decreases (b) increases (c) increases Copyright ©2018 John Wiley & Sons, Inc 18 25-2 Calculating the Capacitance (12 of 12) A cross section of a long cylindrical capacitor, showing a cylindrical Gaussian surface of radius r (that encloses the positive plate) and the radial path of integration. This figure also serves to illustrate a spherical capacitor in a cross section through its center. Copyright ©2018 John Wiley & Sons, Inc 19 25-3 Capacitors in Parallel and in Series (1 of 9) Learning Objectives 25.06 Sketch schematic diagrams for a battery and (a) three capacitors in parallel and (b) three capacitors in series. 25.07 Identify that capacitors in parallel have the same potential difference, which is the same value that their equivalent capacitor has. 25.08 Calculate the equivalent of parallel capacitors. 25.09 Identify that the total charge stored on parallel capacitors the sum of the charges stored on the individual capacitors. Copyright ©2018 John Wiley & Sons, Inc 20 25-3 Capacitors in Parallel and in Series (2 of 9) 25.10 Identify that capacitors in series have the same charge, which is the same value that their equivalent capacitor has. 25.11 Calculate the equivalent of series capacitors. 25.12 Identify that the potential applied to capacitors in series is equal to the sum of the potentials across the individual capacitors. 25.13 For a circuit with a battery and some capacitors in parallel and some in series, simplify the circuit in steps by finding equivalent capacitors, until the charge and potential on the final equivalent capacitor can be determined, and then reverse the steps to find the charge and potential on the individual capacitors. Copyright ©2018 John Wiley & Sons, Inc 21 25-3 Capacitors in Parallel and in Series (3 of 9) 25.14 For a circuit with a battery, an open switch, and one or more uncharged capacitors, determine the amount of charge that moves through a point in the circuit when the switch is closed. 25.15 When a charged capacitor is connected in parallel to one or more uncharged capacitors, determine the charge and potential difference on each capacitor when equilibrium is reached. Copyright ©2018 John Wiley & Sons, Inc 22 25-3 Capacitors in Parallel and in Series (4 of 9) Capacitors in Parallel When a potential difference V is applied across several capacitors connected in parallel, that potential difference V is applied across each capacitor. The total charge q stored on the capacitors is the sum of the charges stored on all the capacitors. q1 = C1V , q2 = C2V , and q3 = C3V. The total charge on the parallel combination of Figure. 25-8a is then q = q1 + q2 + q3 = ( C1 + C2 + C3 )V. Copyright ©2018 John Wiley & Sons, Inc 23 25-3 Capacitors in Parallel and in Series (5 of 9) The equivalent capacitance, with the same total charge q and applied potential difference V as the combination, is then Ceq = Vq = C1 + C2 + C3 , a result that we can easily extend to any number n of capacitors, as n Ceq =  C j ( n capacitors in parallel ). j =1 Copyright ©2018 John Wiley & Sons, Inc 24 25-3 Capacitors in Parallel and in Series (6 of 9) Capacitors connected in parallel can be replaced with an equivalent capacitor that has the same total charge q and the same potential difference V as the actual capacitors. Copyright ©2018 John Wiley & Sons, Inc 25 25-3 Capacitors in Parallel and in Series (7 of 9) Capacitors in Series When a potential difference V is applied across several capacitors connected in series, the capacitors have identical charge q. The sum of the potential differences across all the capacitors is equal to the applied potential difference V. q q q V1 = , V2 = , and V3 =. C1 C2 C3 The total potential difference V due to the battery is the sum 1 1 1  V = V1 + V2 + V3 = q  + + .  C1 C2 C3  Copyright ©2018 John Wiley & Sons, Inc 26 25-3 Capacitors in Parallel and in Series (8 of 9) The equivalent capacitance is then q 1 Ceq = = , V 1 C1 + 1 C2 + 1 C3 or 1 1 1 1 = + +. Ceq C1 C2 C3 n 1 1 = ( n capacitors in series ). Ceq j =1 C j Copyright ©2018 John Wiley & Sons, Inc 27 25-3 Capacitors in Parallel and in Series (9 of 9) Capacitors that are connected in series can be replaced with an equivalent capacitor that has the same charge q and the same total potential difference V as the actual series capacitors. Copyright ©2018 John Wiley & Sons, Inc 28 25 Summary (1 of 4) Capacitor and Capacitance The capacitance of a capacitor is defined as: q = CV Equation (25-1) Determining Capacitance Parallel-plate capacitor: 0A C=. Equation (25-9) d Cylindrical Capacitor: L C = 2 0. Equation (25-14) ln ( b a ) Copyright ©2018 John Wiley & Sons, Inc 29 25 Summary (2 of 4) Spherical Capacitor: ab C = 4 0. Equation (25-17) b−a Isolated sphere: C = 4 0 R. Equation (25-18) Capacitor in parallel and series In parallel: n Ceq =  C j Equation (25-19) j =1 Copyright ©2018 John Wiley & Sons, Inc 30 Copyright Copyright © 2018 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein. Copyright ©2018 John Wiley & Sons, Inc 31