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RC Circuits

Exam I. Physics 102: Lecture 7. RC Circuits. Exam I: Monday February 18 PRACTICE EXAM IS AVAILABLE-CLICK ON “COURSE INFO” ON WEB PAGE. Review, Sunday Feb. 17 3 PM, 141 Loomis. I will go over a past exam On Monday, you get to vote for which exam Fall 2007 Spring 2007 Fall 2006

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RC Circuits

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

  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 • Fall 2007 • Spring 2007 • Fall 2006 • Spring 2006 • 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 • RC Circuits • Charging Capacitors • Discharging Capacitors • Intermediate Behavior

  5. RC Circuits • Circuits that have both resistors and capacitors: • With resistance in the circuits, capacitors do not charge and discharge instantaneously – it takes time (even if only fractions of a second). R1 C R2 V S S

  6. Capacitors • Charge on Capacitorscannot change instantly. • Short term behavior of Capacitor: • If the capacitor starts with no charge, it has no potential difference across it and acts as a wire. • If the capacitor starts with charge, it has a potential difference across it and acts as a battery. • Current through a Capacitor eventually goes to zero. • Long term behavior of Capacitor: • If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit. • If capacitor is discharging, potential difference goes to zero and no current flows.

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

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

  9. ACT/Preflight 7.1 Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S1 is closed? 2R + - 34% 44% 1) Ib = 0 2) Ib = E/(3R) 3) Ib = E/(2R) 4) Ib = E/R 6% 17% Ib + + C e R - - KVL: -E + I(2R) + q/C = 0 q = 0 I = E/(2R) S2 S1 20

  10. 2R + - Ib + + C e R - - S2 S1 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? 56% 28% 5% 11% 1) Ib = 0 2) Ib = E/(3R) 3) Ib = E/(2R) 4) Ib = E/R • Long time  current through capacitor is zero • Ib=0 because the battery and capacitor are in series. • KVL: - E + 0 + q/C = 0  q = EC 24

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

  12. 2R + - IR + + + e C R - - - S1 S2 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? 15% 37% 32% 16% 1) IR = 0 2) IR = V/(3R) 3) IR = V/(2R) 4) IR = V/R KVL: -q0/C+IR = 0 Recall q is charge on capacitor after charging: q0=VC (since charged w/ switch 2 open!) -V + IR = 0  I = V/R 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? 2R + - 1) Q = 0 2) Q = C E/3 3) Q = C E/2 4) Q = C E IR + + + C e - R - - 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 S1 S2 27

  14. RC 2RC q q 0 t RC Circuits: Charging The switches are originally open and the capacitor is uncharged. Then switch S1 is closed. • KVL: - e+ I(t)R + q(t) / C = 0 • Just after…: q =q0 • Capacitor is uncharged • - e + I0R = 0  I0 = e / R • Long time after: Ic= 0 • Capacitor is fully charged • - e + q/C =0  q =eC • Intermediate (more complex) q(t) = q(1-e-t/RC) I(t) = I0e-t/RC + R + e - I - C + - S2 S1 17

  15. RC 2RC q t RC Circuits: Discharging + • KVL: - q(t) / C - I(t) R = 0 • Just after…: q=q0 • Capacitor is still fully charged • -q0 / C - I0 R = 0  I0 = -q0 / (RC) • Long time after: Ic=0 • Capacitor is discharged (like a wire) • -q / C = 0  q = 0 • Intermediate (more complex) q(t) = q0 e-t/RC Ic(t) = I0 e-t/RC R + e - I - C + - S1 S2 40

  16. What is the time constant? • The time constant  = RC. • 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. • 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. 2 Example Time Constant Demo Each circuit has a 1 F capacitor charged to 100 Volts. When the switch is closed: • Which system will be brightest? • Which lights will stay on longest? • Which lights consumes more energy? 2 I=2V/R 1 Same U=1/2 CV2 1 t = 2RC t = RC/2 50

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

  19. 2R + - Ib + + C R - - S2 S1 Example Charging: Intermediate Times Calculate the charge on the capacitor 310-3 seconds after switch 1 is closed. R = 10 W V = 50 Volts C = 100mF • 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 e 38

  20. R I C E S1 R=10W C=30 mF E =20 Volts Example Practice! + - Calculate current immediately after switch is closed: -E + I0R + q0/C = 0 + + - -E + I0R + 0 = 0 - I0 = E/R Calculate current after switch has been closed for 0.5 seconds: Calculate current after switch has been closed for a long time: After a long time current through capacitor is zero! Calculate charge on capacitor after switch has been closed for a long time: -E + IR + q∞/C = 0 -E + 0 + q∞/C= 0 q∞ = EC 32

  21. ACT: RC Challenge E= 24 Volts R = 2W C = 15 mF R 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? C 2R E 1) 0.368 q0 2) 0.632 q0 3) 0.135 q0 4) 0.865 q0 S1 • q(t) = q0 e-t/RC = q0 (e-0.06/(4(1510-3))) = q0 (0.368) 45

  22. RC Summary Charging Discharging q(t) = q(1-e-t/RC) q(t) = q0e-t/RC V(t) = V(1-e-t/RC) V(t) = V0e-t/RC I(t) = I0e-t/RC I(t) = I0e-t/RC Time Constant t = RC Large t means long time to charge/discharge Short term: Charge doesn’t change (often zero or max) Long term: Current through capacitor is zero. 45

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