Electricity and Magnetism Capacitors in Series Energy Stored in a Capacitor

Transcription

1 Electricity and Magnetism Capacitors in Series Energy Stored in a Capacitor Lana Sheridan De Anza College Jan 31, 2018

2 Last time cylindrical and spherical capacitors Parallel plate capacitors Circuits and circuit diagrams Capacitors in parallel

3 Overview capacitors in series practice with capacitors in circuits Energy stored in a capacitor

4 +q + B V V 2 q C 2 Capacitors in series all store the same charge. 664 CHAPTER 25 CAPACITANCE +q Three capacitors in series: V 3 q C 3 Terminal produces charge Terminal negative charge fr +q Series c Equivalent circuit: repelled negative V (a) 1 charge q). their That equ q C 1 charge from the the same to +q + plate +q + of capacitor B V V 2 B q of capacitor 1 help C V 2 q C battery, eq leaving th +q Here are two V 3 (b) q C 3 1. When charge is Fig (a) it can Three move capac alo Terminal nected in series Series capacitors and Fig. to 25-9a.If battery the B. (a) maintains their equivalent have 2. potential The battery differenc di Capacitors in Series

5 Capacitors in Series Again, we could replace all three capacitors in the circuit with one equivalent capacitance and we can find the capacitance of this equivalent capacitor. The sum of the potential differences across capacitors in series is V, the battery s supplied potential difference. V = V 1 + V 2 + V 3 where V 1 = q/c 1, etc. Then, C eq = q V

6 Capacitors in Series Equivalent capacitance: C eq = = = = = q V q V 1 + V 2 + V 3 [ ] V1 + V 2 + V 1 3 q [ V1 q + V 2 q + V 3 q [ ] 1 C 1 C 2 C 3 ] 1

7 Capacitors in Series In general, for any number n of capacitors in series, we can always relate the effective capacitance of them all together to the individual capacitances by: 1 C eq = 1 C C C n = n 1 C i i=1 The equivalent capacitance of capacitors in series is always less than the smallest capacitance in the series.

8 Practice A 5.0 µf capacitor is connected in parallel with a 10 µf capacitor. What is the equivalent capacitance of this arrangement? A 5.0 µf capacitor is connected in series with a 10 µf capacitor. What is the equivalent capacitance of this arrangement?

9 More Practice We first reduce the circuit to a single capacitor. What is the equivalent capacitance of this arrangement? A The equival parallel cap is larger. A C 1 = 12.0 µf V C 3 = 4.50 µf (a) C 2 = 5.30 µf B V C C (b)

10 itance More Practice When solving this type of problem, take an iterative approach. itance Identify sets of capacitors that are in parallel, then series, then parallel, etc. and at each step replace with the equivalent capacitance: and b for the re 26.9a. All ly and make e connected. llel connec- a a b a b b

11 y More can be Practice connected in series or pacitance for the combination, parallel When(c) solving either this way type because of problem, take an iterative approach. Identify sets of capacitors that are in parallel, then series, then parallel, etc. and at each step replace with the equivalent capacitance: the All ake ted. ecins 6.0 a b a b a b a 6.0 b a b c d Figure 26.9 (Example 26.3) To find the equivalent capacitance

12 More Practice We first reduce the circuit to a single capacitor. What is the equivalent capacitance of this arrangement: A The equival parallel cap is larger. A C 1 = 12.0 µf V C 3 = 4.50 µf (a) C 2 = 5.30 µf B V C C (b)

13 Energy Stored in a Capacitor A charged capacitor has an electric field between the plates. This field can be thought of as storing potential energy. As you might expect, the energy stored is equal to the work done charging the capacitor. (Energy Conservation!)

14 Energy Stored in a Capacitor 26.4 Energy Stored in a Charged Capacitor 787 A charged capacitor has an electric field between the plates. This field can be thought of as storing potential energy. t occurs, there is a transformation closed, energy is stored as chemis transformed during the chemical is operating in an electric circuit. l potential energy in the battery is As you might expect, the energy stored is equal to the work done charging the capacitor. (Energy Conservation!) ted with the separation of positive r, we shall assume a charging proribed in Section 26.1 but that gives d because the energy in the final difference V across the capacitor But how much work is plates done? is given Wapproximately = q V by, yet the potential difference changes as more the area charge of the shaded is placed rectangle. on the capacitor plates. rge-transfer process. 3 Imagine the transfer the charge mechanically. You grab a small amount of posiauses this positive charge to move on the charge as it is transferred equired to transfer a small amount once this charge has been transn the plates. Therefore, work must potential difference. As more and he other, the potential difference d. The overall process is described ation 8.2 reduces to W 5 DU E ; the appears as an increase in electric The work required to move charge dq through the potential V Q q e instant during the charging pro- dq

15 dq through the potential difference V across the capacitor plates is given approximately by the area of the dw app shaded = ( V rectangle. ) dq y in the final 3 Energy Stored in a Capacitor Imagine the mechanically How much work is done? ount of posiarge to move V is transferred small amount s been transre, work must. As more and ial difference ss is described W 5 DU E ; the se in electric charging pror is DV 5 q/c. Need to integrate! dq Figure A plot of potential Q q

16 Energy Stored in a Capacitor V = q C For a fixed capacitor (plates are not changing configuration or shape), C is a constant. U E = W app = Q = q C dq Q 2 C The energy stored in a capacitor with charge Q and capacitance C: ( ) Q 2 U = 1 2 C

17 Energy Stored in a Capacitor The energy stored in a capacitor with charge Q and capacitance C: ( ) Q 2 U = 1 2 C Since Q = C ( V ) we can also write this as: U = 1 C ( V )2 2 And: U = 1 2 Q V

18 Stored Energy Example Suppose a capacitor with a capacitance 12 pf is connected to a 9.0 V battery. What is the energy stored in the capacitor s electric field once the capacitor is fully charged?

19 Energy Density It is sometimes useful to be able to compare the energy stored in different charged capacitors by their stored energy per unit volume. We can link energy density to electric field strength. This will make concrete the assertion that energy is stored in the field.

20 Energy Density It is sometimes useful to be able to compare the energy stored in different charged capacitors by their stored energy per unit volume. We can link energy density to electric field strength. This will make concrete the assertion that energy is stored in the field. For a parallel plate capacitor, energy density u is: u E = U E Ad (Ad is the volume between the capacitor plates.)

21 Energy Density and Electric Field u E = U E Ad = C( V )2 2Ad

22 Energy Density and Electric Field u E = U E Ad = C( V )2 2Ad Replace C = ɛ 0A d : u E = ɛ 0A V 2 d 2Ad = ɛ ( 0 V 2 d ) 2

23 Energy Density and Electric Field u E = U E Ad = C( V )2 2Ad Replace C = ɛ 0A d : u E = ɛ 0A V 2 d 2Ad = ɛ ( 0 V 2 d ) 2 Lastly, remember V = Ed in a parallel plate capacitor, so: u E = 1 2 ɛ 0E 2

24 Energy Density and Electric Field Energy density in a capacitor: u E = 1 2 ɛ 0E 2 The derivation of this expression assumed a parallel plate capacitor. However, it is true more generally. (General proof requires vector calculus.) It is also true for varying electric fields, in which case the energy density varies. Energy density of an electric field E 2

25 Summary capacitors in series practice with capacitors in circuits energy stored in a capacitor Homework Serway & Jewett: PREVIOUS: Ch 26, onward from page 799. Problems: 13 NEW: Ch 26. Problems: 17, 21, 25, 31, 33, 35