1. RC Circuits Physics 102: Lecture 7 Exam I: Monday February 18 PRACTICE EXAM IS AVAILABLE-CLICK ON “COURSE INFO” ON WEB PAGE Exam I

2. Review, Sunday Feb. 17 3 PM, 141 Loomis I will go over a past exam On Monday, you get to vote for which exam a)Fall 2007 b)Spring 2007 c)Fall 2006 d)Spring 2006 e)Fall 2005

3. Recall …. First we covered circuits with batteries and capacitors –series, parallel Then we covered circuits with batteries and resistors –series, parallel –Kirchhoff’s Loop Law –Kirchhoff’s Junction Law Today: circuits with batteries, resistors, and capacitors

4. RC Circuits l RC Circuits l Charging Capacitors l Discharging Capacitors l Intermediate Behavior

5. RC Circuits l Circuits that have both resistors and capacitors: l With resistance in the circuits, capacitors do not charge and discharge instantaneously – it takes time (even if only fractions of a second). V R2R2 R1R1 C S S

6. Capacitors l Charge on Capacitors cannot change instantly. l Short term behavior of Capacitor: è If the capacitor starts with no charge, it has no potential difference across it and acts as a wire. è If the capacitor starts with charge, it has a potential difference across it and acts as a battery. l Current through a Capacitor eventually goes to zero. l Long term behavior of Capacitor: è If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit. è If capacitor is discharging, potential difference goes to zero and no current flows.

7. Charging Capacitors l Example: an RC circuit with a switch l When switch is first closed… (short term behavior): è No charge on the capacitor since charge can not change instantly – the capacitor acts as a wire or short circuit l After switch has been closed for a while… (long term behavior): è No current flows through the capacitor – the capacitor acts as a break in the circuit or open circuit  R C S

8. Charging Capacitors l Capacitor is initially uncharged and switch is open. Switch is then closed. What is current in circuit immediately thereafter? l What is current in circuit a long time later?  R C S

9. ACT/Preflight 7.1 2R C  R S2S2 Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S 1 is closed? 1) I b = 02) I b = E /(3R) 3) I b = E /(2R)4) I b = E /R S1S1 KVL: - E + I(2R) + q/C = 0 q = 0  I = E /(2R) IbIb + - + + - - 34% 44% 6% 17% 20

10. ACT/Preflight 7.3 Both switches are initially open, and the capacitor is uncharged. What is the current through the battery after switch 1 has been closed a long time?long time 1) I b = 02) I b = E /(3R) 3) I b = E /(2R)4) I b = E /R 2R C  R S2S2 S1S1 IbIb + - + + - - Long time  current through capacitor is zero I b =0 because the battery and capacitor are in series. KVL: - E + 0 + q  /C = 0  q  = E C 56% 28% 5% 11% 24

11. Discharging Capacitors l Example: a charged RC circuit with a switch l When switch is first closed… (short term behavior): è Full charge is on the capacitor since charge can not change instantly – the capacitor acts as a battery l After switch has been closed for a while… (long term behavior): è No current flows through the capacitor – the capacitor is fully discharged R C S

12. ACT/Preflight 7.5 After switch 1 has been closed for a long time, it is opened and switch 2 is closed. What is the current through the right resistor just after switch 2 is closed? 1) I R = 02) I R = V/(3R) 3) I R = V/(2R)4) I R = V/R KVL: -q 0 /C+IR = 0 Recall q is charge on capacitor after charging: q 0 =VC (since charged w/ switch 2 open!) -V + IR = 0  I = V/R 2R C  R S2S2 S1S1 IRIR + - + + - - + - 32% 16% 15% 37% 42

13. ACT: RC Circuits Both switches are closed. What is the final charge on the capacitor after the switches have been closed a long time? 1) Q = 02) Q = C E /3 3) Q = C E /24) Q = C E KVL (right loop): -Q/C+IR = 0 KVL (outside loop), - E + I(2R) + IR = 0 I = E /(3R) KVL: -Q/C + E /(3R) R = 0 Q = C E /3 R 2R C  S2S2 S1S1 IRIR + - + + - - + - 27

14. RC Circuits: Charging KVL: -  + I(t)R + q(t) / C = 0 Just after…: q =q 0 –Capacitor is uncharged –-  + I 0 R = 0  I 0 =  / R Long time after: I c = 0 –Capacitor is fully charged –-  + q  /C =0  q  =  C Intermediate (more complex) q(t) = q  (1-e -t/RC ) I(t) = I 0 e -t/RC C  R S1S1 S2S2 + + + I - - - The switches are originally open and the capacitor is uncharged. Then switch S 1 is closed. t q RC 2RC 0 qq 1717

15. RC Circuits: Discharging KVL: - q(t) / C - I(t) R = 0 Just after…: q=q 0 –Capacitor is still fully charged –-q 0 / C - I 0 R = 0  I 0 = -q 0 / (RC) Long time after: I c =0 –Capacitor is discharged (like a wire) –-q  / C = 0  q  = 0 Intermediate (more complex) q(t) = q 0 e -t/RC I c (t) = I 0 e -t/RC C  R S1S1 - + + I - + - S2S2 q RC2RC t 40

16. What is the time constant? l The time constant  = RC. l Given a capacitor starting with no charge, the time constant is the amount of time an RC circuit takes to charge a capacitor to about 63.2% of its final value. l The time constant is the amount of time an RC circuit takes to discharge a capacitor by about 63.2% of its original value.

17. Time Constant Demo Which system will be brightest? Which lights will stay on longest? Which lights consumes more energy? 2 Each circuit has a 1 F capacitor charged to 100 Volts. When the switch is closed: 1  = 2RC  = RC/2 2 I=2V/R 1 Same U=1/2 CV 2 50

18. Summary of Concepts l Charging Capacitors è Short Term: capacitor has no charge, no potential difference, acts as a wire è Long Term: capacitor fully charged, no current flows through capacitor, acts as an open circuit l Discharging Capacitors è Short Term: capacitor is fully charged, potential difference is a maximum, acts as a battery è Long Term: capacitor is fully discharged, potential difference is zero, no current flows l Intermediate Behavior è Charge & Current exponentially approach their long-term values è  = RC

19. Charging: Intermediate Times  2R C R S2S2 S1S1 IbIb + - + + - - Calculate the charge on the capacitor 3  10 -3 seconds after switch 1 is closed. q(t) = q  (1-e -t/RC ) = q  (1-e - 3  10 -3 /(20  100  10 -6) ) ) = q  (0.78) Recall q  =  C = (50)(100x10 -6 ) (0.78) = 3.9 x10 -3 Coulombs R = 10  V = 50 Volts C = 100  F 38

20. Practice! Calculate current immediately after switch is closed: Calculate current after switch has been closed for 0.5 seconds: Calculate current after switch has been closed for a long time: Calculate charge on capacitor after switch has been closed for a long time: R C E S1S1 R=10  C=30 mF E =20 Volts - E + I 0 R + q 0 /C = 0 I + + + - - - - E + I 0 R + 0 = 0 I 0 = E /R After a long time current through capacitor is zero! - E + IR + q ∞ /C = 0 - E + 0 + q ∞ /C = 0 q ∞ = E C 32

21. ACT: RC Challenge 1) 0.368 q 0 2) 0.632 q 0 3) 0.135 q 0 4) 0.865 q 0 R C E 2R S1S1 E = 24 Volts R = 2  C = 15 mF After being closed for a long time, the switch is opened. What is the charge Q on the capacitor 0.06 seconds after the switch is opened? q(t) = q 0 e -t/RC = q 0 (e - 0.06 /(4  (15  10 -3 )) ) = q 0 (0.368) 45

22. RC Summary ChargingDischarging q(t) = q  (1-e -t/RC )q(t) = q 0 e -t/RC V(t) = V  (1-e -t/RC )V(t) = V 0 e -t/RC I(t) = I 0 e -t/RC Short term: Charge doesn’t change (often zero or max) Long term: Current through capacitor is zero. Time Constant  = RC Large  means long time to charge/discharge 45