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# Ch25 capacitance

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### Ch25 capacitance

1. 1. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Chapter 25: Capacitance p656-671 (not sec 8) Capacitance is related to the ability of two isolated conductors to hold a charge. We talk of two `plates’ but capacitors have many different shapes. To start, consider nothing between plates. We speak of the charge q on the capacitor, but the plates have equal and opposite charges +q & -q The plates are conductors => are equipotential surfaces => all points on each plate are at same potential. Ch25-1/57 Capacitance A parallel plate capacitor. Charges on facing plates are equal but of opposite sign. Charges are on inside of plates (proven on Ch23-51 for conducting plates). The field lines in the interior are constant, from +ve to -ve charges. There are fringing fields near the end. The exterior fields are much weaker (would be 0 for infinite plates - see Ch23-51). Ch25-2/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 1
2. 2. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitance The potential difference across the plates is denoted by V rather than ∆V (for historical reasons). The charge q and potential difference V on the plates are proportional to each other, i.e. Proportionality constant C is called the capacitance. Capacitance depends ONLY on the geometry of the plates, NOT on q nor V. Capacitance is a measure of charge needed on plates to attain a certain potential difference. Greater capacitance => more charge required. Alternatively, for a given voltage, more capacitance => more charge held. Ch25-3/57 The farad, F The SI unit for capacitance is the farad (F), after Englishman Michael Faraday (1791-1867) The farad is a huge unit. It is common to use the microfarad (1 µF =10-6 F) or the picofarad (1 pF = 10-12 F) (the “puff”) Ch25-4/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 2
3. 3. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Charging a capacitor Note the notation/symbols Switch S: Battery B: with the +ve side longer Capacitance C: 2 equal parallel lines even if shape very different. Battery -maintains potential difference V across terminals -sets up electric field in wires which draws e- -from plate h (becomes +ve) -to plate l (becomes negative). +ve terminal has the higher potential. Circuit incomplete until switch closed, then it is completed. Same potential V developed across h and l as across battery => no field in wires => no more current. Capacitor is fully charged. We assume (a) no leakage between plates (b) capacitor stores the full charge indefinitely. Ch25-5/57 Question Does the capacitance C of a capacitor increase, decrease or remain the same (a) when the charge q is doubled? (b) when the potential difference is tripled? Ch25-6/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 3
4. 4. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Question Does the capacitance C of a capacitor increase, decrease or remain the same (a) when the charge q is doubled? Ans: Remains the same since it is a property of the geometry, not the charge (b) when the potential difference Ans: Remains the same is tripled? since it is a property of the geometry, not the voltage Ch25-7/57 Calculating capacitance 1) assume charge q on plates 2) calculate electric field E between plates using Gauss’ Law (arranging for E and dA to be parallel) so 3) calculate the potential difference using always following a field line from -ve to +ve plate => E and ds will be in opposite directions => and 4) Calculate C from Ch25-8/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 4
5. 5. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitance of a parallel plate capacitor Assume plates large enough we can ignore fringing fields => E is constant between plates. Consider the Gaussian surface and Gauss’ Law with E=0 in plate gives which leads to substituting E = V/d and then C=q/V C increases with plate area and as plates closer together Ch25-9/57 Capacitance of a parallel plate capacitor (cont) Using k instead of would make this eqn much uglier, and this eqn is used more often. Ch25-10/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 5
6. 6. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitance of a cylindrical capacitor Cylindrical capacitor, inner radius a, outer, b and length L. Assume L>>b => ignore fringing at ends. Opposite charges q on each surface. Gaussian surface is a coaxial cylinder shown in red. E is radial => where A is area of cylinder’s curved surface (no flux thru the ends). Hence (field from a line). Take ds radially inwards (-ve to +ve) but then ds = -dr so: C = q/V => Ch25-11/57 Capacitance of a spherical capacitor Spherical capacitor, inner radius a, outer, b. Charge q on each surface. Gaussian surface is a concentric sphere shown in red. E is radial => where A is area of sphere. Hence (field from an isolated charge/sphere). Take ds radially inwards (-ve to +ve) but then ds = -dr so: C = q/V => Ch25-12/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 6
7. 7. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitance of an isolated sphere Consider a single isolated sphere, radius R, as part of a spherical capacitor with the outer radius at infinity. Rewrite the standard eqn, viz: as then let , substitute R for “a” => Ch25-13/57 Summary of capacitances parallel plates a is the co-axial cylinders inner radius, b the outer concentric spheres isolated sphere All the formulae are for εo multiplied a quantity with dimensions of length (F/m x m) Ch25-14/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 7
8. 8. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Question For capacitors charged by the same battery, does the charge stored by the capacitor: increase, decrease, or stay the same when: a) plate separation is increased in a parallel plate capacitor b) radius of the inner cylinder is doubled in a cylindrical capacitor c) outer radius of a spherical capacitor is increased Ch25-15/57 Question For capacitors charged by the same battery, does the charge stored by the capacitor: increase, decrease, or stay the same when: a) plate separation is increased in a parallel plate capacitor => C decreases as d increases => q=CV decreases b) radius of the inner cylinder is doubled in a cylindrical capacitor => C & q increase as “a” increases (plates closer) c) outer radius of a spherical capacitor is increased => C & q decrease as “b” increases (plates farther apart) Ch25-16/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 8
9. 9. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Sample Problem A storage capacitor on a random access memory chip (RAM) has a capacitance of 55 fF. If charged to 5.3 V, how many excess electrons are on the -ve plate. Key idea: number of electrons on plate is n = q/e. q = CV => n = CV/e (or q= 2.9x10-13 C) Ch25-17/57 Several capacitors Replace a number of capacitors in a circuit by a single equivalent capacitor, i.e. a capacitor with same capacitance as the others. Consider combinations of capacitors in parallel or series. In parallel -directly wired together at each plate - same potential difference across all capacitors - total charge stored is the sum of charges on each component -equivalent capacitor has same total charge q and same V as actual capacitors Ch25-18/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 9
10. 10. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitors in parallel We know q = CV, hence for capacitors in parallel: so Same V across each capacitor and total charge stored is sum of individual charges Ceq is dominated by the larger Cj Ch25-19/57 Capacitors in series Capacitors wired in series, one after another. Potential difference V is applied across two ends of series. Each capacitor has same charge when in series (chain of events, charging of 1 causes charging of next because there is nowhere else for the charge to go to/come from. But potential differences may vary) The potential differences across all capacitors add up to give applied potential difference. Ch25-20/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 10
11. 11. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitors in series (cont) In series, there is only one route for charge from one capacitor to the next in a series. If there is a second route => not in series. The equivalent capacitor has the same charge q and the same total potential difference V as the actual series capacitors Ch25-21/57 Capacitors in series (cont) Since we have the same q on each capacitor in series but we know the total V is just the sum so: Ceq is dominated by the smallest Cj Ch25-22/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 11
12. 12. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Question A battery of potential V stores a charge q on a combination of two identical capacitors. What are the potential differences across each capacitor and the charge stored on each capacitor (in terms of q and V) if the capacitors are (a) in parallel or (b) in series? i) V/2, q ii) V/2, q/2 iii) V, q iv) V, q/2 Ch25-23/57 Question A battery of potential V stores a charge q on a combination of two identical capacitors. What are the potential differences across each capacitor and the charge stored on each capacitor (in terms of q and V) if the capacitors are (a) in parallel or (b) in series? i) V/2, q ii) V/2, q/2 iii) V, q iv) V, q/2 Ans: parallel => V same for all and: so (iv) is correct Ans: series => q same for all and: so (i) is correct. Ch25-24/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 12
13. 13. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Sample problem Find the equivalent capacitance of the 3 capacitors to the right. Key idea: Replace any capacitors in series or parallel by their equivalent capacitor. Which ones are in parallel or series. Are C1 and C3 in series? Are C1 and C2 in parallel? What is the equivalent capacitance C12? Ch25-25/57 Sample problem Find the equivalent capacitance of the 3 capacitors to the right. Key idea: Replace any capacitors in series or parallel by their equivalent capacitor. Which ones are in parallel or series. Are C1 and C3 in series? Ans: No. Charge from C3 can go to C1 or C2 => C1 and C3 are not in series. Are C1 and C2 in parallel? Ans: Yes since their top and bottom plates are directly wired together What is the equivalent capacitance C12? Ans: Ch25-26/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 13
14. 14. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Sample problem(cont) Are C12 and C3 in series or parallel? What is the equivalent capacitance of C123? Ch25-27/57 Sample problem(cont) Are C12 and C3 in series or parallel? Ans: Series What is the equivalent capacitance of C123? Ans: C123 = 3.57 µF if you stuff in the values. Ch25-28/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 14
15. 15. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Sample problem(cont) If potential difference is V =12.5 V, what is charge on C1? Key idea 1: work backwards to C1 from the equivalent capacitor of the set of 3, viz C123. 12.5 V is applied across C123 => Key idea 2: charge on capacitors in series is same, so q12 = q123 => Ch25-29/57 Sample problem(cont) Key idea: voltage across C12 is same as that across two capacitors in parallel => V1 = V12 = 2.58 V Hence: QED General approach: divide and conquer Ch25-30/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 15
16. 16. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Another sample problem Capacitor 1 with C1 =3.55 µF is charged to a potential difference of V0 = 6.3 V using a 6.3 V battery. Battery is removed and capacitor is placed in circuit shown with an uncharged capacitor C2 = 8.95 µF. When switch S is closed, charge flows between capacitors until they have same potential difference V. What is V? Are these capacitors in parallel or series or neither? Ans: neither since no applied voltage. Ch25-31/57 Another sample problem(cont) Key idea: after switch closed, the charge on C1 is shared between C1 and C2 to give equal voltages on each. Hence: and using q = CV: Key idea: the capacitors share their charge to give equal voltages. Ch25-32/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 16
17. 17. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Question In above circuit, replace C2 with two capacitors in series, C3 and C4. (a) what is the relationship between between the initial charge qo and the charges q1 and q34 ? (b) If C3 > C4, is q3 > q4, q3 = q4, q3 < q4? Ch25-33/57 Question In above circuit, replace C2 with two capacitors in series, C3 and C4. (a) what is the relationship between between the initial charge qo and the charges q1 and q34 ? Ans: (b) If C3 > C4, is q3 > q4, q3 = q4, q3 < q4? Ans: they are in series so they must be equal Ch25-34/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 17
18. 18. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Question Are C1 and C2 in parallel? What is the equivalent C12? C1 C3 C2 C4 Are C3 and C4 in parallel? What is equivalent C34? What is the equivalent C of all 4 capacitors? Ch25-35/57 Question Are C1 and C2 in parallel? What is the equivalent C12? C1 C3 Ans: yes => C12 = C1 + C2 C2 C4 Are C3 and C4 in parallel? What is equivalent C34? Ans: yes => C34 = C3 + C4 What is the equivalent C of all 4 capacitors? Ans: C12 & C34 in series => Ch25-36/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 18
19. 19. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Strategy for solving multi-capacitor problems • If capacitors, or groups of capacitors are in parallel => voltage across them is same. One can apply q=CV • if capacitors are in parallel – goal is usually to find voltage across parallel capacitors => can find qi or Ci,depending on question since qi = CiVcommon or Ci=Vcommon/qi • if capacitors are in series – goal is usually to find charge on a series of capacitors => each capacitor in the series has same charge – from this you determine C or V as required Ch25-37/57 Energy stored in an electric field Work must be done by an external agent to charge a capacitor (the battery uses its stored chemical energy). Remember: U = Wapp = q V was the general relationship. So if q’ has been transferred to a capacitor C, then V’ = q’/C and as we add a new small amount of charge q’ This work is stored as potential energy U in the capacitor U is stored in electric field between plates of capacitor. Ch25-38/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 19
20. 20. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Units of stored energy Ch25-39/57 Energy stored in an electric field(cont) +q +q d C C’ = C/2 -q 2d -q For parallel plate capacitors (Ch25-8) => field is same in both cases above since q is. for 2d case: => U’ is twice as large in larger capacitor (consistent with concept of storage in field). V’ is double to get same q in larger capacitor => U’ twice as large. Ch25-40/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 20
21. 21. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons picture p665 Sample problem hypobaric oxygen treatment Consider 2 parallel plates which form a capacitor which becomes charged to a potential difference V1 when their area of overlap was A1. What is the potential difference between them when their overlap area is A2 < A1? Key ideas: q = CV, q is constant and C varies as So how would you prevent a spark in the hospital? Ch25-41/57 Sample problem(cont) What is the ratio of the stored potential energy before vs after the movement? Key ideas: and and we just showed so easier to use 1st formula hence: Where did the extra potential energy come from? Ans: From the force moving the gurney. Ch25-42/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 21
22. 22. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Exploding wire demonstration We have a 26 µF capacitor which we charge up to 3 kV. How much energy is stored? => U = 26x10-6 F (3x103 V)2 = 234 J How much energy is that? Ch25-43/57 Energy density Define the energy density, u, of an electric field as the potential energy stored in an electric field per unit volume. This builds on concept of potential energy of capacitor being stored in electric field inside capacitor. Consider a parallel plate capacitor, ignoring fringing fields. The total energy stored is . Volume of space between plates is Ad. => We know => Also know => Ch25-44/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 22
23. 23. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Energy density (cont) We derived this for a parallel plate capacitor, but it holds in general. If an electric vector E exists at any point in space, that point is a site of stored electric potential energy. The amount stored per unit volume is given above. Ch25-45/57 Energy density (cont) Do the units make sense? Ch25-46/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 23
24. 24. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitor with a Dielectric Faraday: fill space between plates of a Material Dielectric Dielectric capacitor with a dielectric, (an Constant Strength insulating material) κ kV/mm => capacitance increases by a air 1.00054 3 factor κ, the dielectric constant. polystyrene 2.6 24 κ is unity for vacuum by definition paper 3.5 16 Above a certain maximum field transformer strength the material will break oil 4.5 25 down and form a conducting path silicon 12 between the plates (sparking). water 80 => there is a maximum voltage titania which can be applied. ceramic 130 strontium The maximum field strength is titanate 310 8 called the dielectric strength, Ch25-47/57 Capacitor with a Dielectric (cont) For any capacitor we can write: where has dimensions of length. If a dielectric completely fills space between plates, then With the dielectric added For a constant charge, the charge increases on the voltage decreases the plates by a factor κ by a factor κ Ch25-48/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 24
25. 25. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitor with a Dielectric (cont) In a region completely filled by a dielectric material of dielectric constant κ, all electrostatic eqns containing the permittivity constant εo are modified, replacing εo by κεo. For example, inside a dielectric, electric field produced by a point charge is given by: and the electric field just outside any isolated conductor in a dielectric is: For a fixed distribution of charges, a dielectric weakens the electric field that would otherwise be there. Ch25-49/57 Sample problem A parallel-plate capacitor with C= 13.5 pF is charged by a battery to 12.5 V. The battery is then disconnected and a porcelain slab (κ = 6.50) is placed between the plates. What is the potential energy of the capacitor (1) before and (2) after the slab is inserted? Key idea: Ch25-50/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 25
26. 26. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Sample problem (cont) A parallel-plate capacitor with C= 13.5 pF is charged by a battery to 12.5 V. The battery is then disconnected and a porcelain slab (κ = 6.50) is placed between the plates. What is the potential energy of the capacitor (2) after the lab is inserted? Key idea 2: Since battery is disconnected, charge on capacitor cannot change. Potential changes. Capacitance becomes κC. i.e. when slab is inserted, potential energy decreases by a factor κ. The energy was used `pulling’ slab into capacitor. Ch25-51/57 Question In the previous problem, if the battery stayed connected do the following increase, decrease or stay the same: a) potential difference across the plates? b) the capacitance? c) the charge on the capacitor? d) the potential energy of the device? e) the electric field between the plates, given that charge on the capacitor is not fixed? Ch25-52/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 26
27. 27. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Question In the previous problem, if the battery stayed connected do the following increase, decrease or stay the same: a) potential difference across the plates? b) the capacitance? c) the charge on the capacitor? d) the potential energy of the device? e) the electric field between the plates, given that charge on the capacitor is not fixed? Ans: (a) constant because of battery (b) increased by factor κ. (c) increased by κ because of fixed V and increased capacitance (d) increased since V is constant and C has gone up (e) same because same potential difference and distance V=Ed Ch25-53/57 Dielectrics: an atomic view What happens when we put a dielectric in an electric field? Polar dielectrics Dipoles in an E get rotated and aligned in direction of electric field -thermal motion => not perfect alignment. Each dipole generates an E in dielectric -net effect is complex -in direction of dipole moment on axis, - in other direction & stronger closer in (goes as 1/r3). Net effect is to produce an Epolar which is in opposite direction to applied field. Ch25-54/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 27
28. 28. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Dielectrics: an atomic view (cont) Nonpolar dielectrics Dipoles are induced by external field. Net charge stays 0, but near plates there are net equal and opposite charges induced which offset charges on plates. Net effect is to produce an E’ which is in opposite direction to applied field. For both polar and nonpolar dielectrics, the net effect is to weaken any applied field, by a factor κ. Why was dielectric “pulled” into capacitor? Ans: Opposite charges on plates/dielectric are attractive Ch25-55/57 Problem solving techniques Capacitor with dielectrics as parallel capacitors +q -q A parallel-plate capacitor with two different dielectrics in the above formation can be treated as two capacitors in parallel since the upper and lower surfaces are at the same voltage and share the charge between them. Must take into account relative areas and different dielectric constants for two parts of capacitor. Ch25-56/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 28
29. 29. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics,Copyright 2005 by Wiley and Sons Capacitor with dielectrics as capacitors in series +q +q -q +q -q -q A parallel-plate capacitor with two different dielectrics in the left formation above can be treated as two capacitors in series. On the right, the `virtual pair of capacitors’ has the same charge distribution as on left since the middle pair of plates exactly cancel each other’s charge. Calculate the effective capacitance using the standard rule for a series, viz Take into account thickness and different dielectric constants for the two parts of capacitor. Ch25-57/57Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 29