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- 1. RC Circuits Physics 102: Lecture 7
- 2. Recall …. <ul><li>First we covered circuits with batteries and capacitors </li></ul><ul><ul><li>series, parallel </li></ul></ul><ul><li>Then we covered circuits with batteries and resistors </li></ul><ul><ul><li>series, parallel </li></ul></ul><ul><ul><li>Kirchhoff’s Loop and Junction Relations </li></ul></ul><ul><li>Today: circuits with batteries, resistors, and capacitors </li></ul>
- 3. RC Circuits <ul><li>RC Circuits </li></ul><ul><li>Charging Capacitors </li></ul><ul><li>Discharging Capacitors </li></ul><ul><li>Intermediate Behavior </li></ul>
- 4. RC Circuits <ul><li>Circuits that have both resistors and capacitors : </li></ul><ul><li>With resistance in the circuits, capacitors do not charge and discharge instantaneously – it takes time (even if only fractions of a second). </li></ul>R K C S + + + R Na R Cl ε K ε Na ε Cl
- 5. Capacitors <ul><li>Charge (and therefore voltage) on Capacitors cannot change instantly: remember V C = Q/C </li></ul><ul><li>Short term behavior of Capacitor: </li></ul><ul><ul><li>If the capacitor starts with no charge, it has no potential difference across it and acts as a wire </li></ul></ul><ul><ul><li>If the capacitor starts with charge, it has a potential difference across it and acts as a battery. </li></ul></ul><ul><li>Long term behavior of Capacitor: Current through a Capacitor eventually goes to zero. </li></ul><ul><ul><li>If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit </li></ul></ul><ul><ul><li>If capacitor is discharging, potential difference goes to zero and no current flows </li></ul></ul>
- 6. Charging Capacitors <ul><li>Capacitor is initially uncharged and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? </li></ul><ul><li>What is current I in circuit a long time later? </li></ul> R C S
- 7. Charging Capacitors: t=0 <ul><li>Capacitor is initially uncharged and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? </li></ul><ul><ul><li>Capacitor initially uncharged </li></ul></ul><ul><ul><li>Therefore V C is initially 0 </li></ul></ul><ul><ul><li>Therefore C behaves as a wire (short circuit) </li></ul></ul><ul><ul><li>KLR: – I 0 R = 0 </li></ul></ul><ul><ul><ul><li>I 0 = /R </li></ul></ul></ul> R C S R
- 8. Charging Capacitors: t>0 <ul><ul><li>I 0 = /R </li></ul></ul><ul><ul><li>Positive charge flows </li></ul></ul><ul><ul><ul><li>Onto bottom plate (+Q) </li></ul></ul></ul><ul><ul><ul><li>Away from top plate (-Q) </li></ul></ul></ul><ul><ul><ul><li>As charge builds up, V C rises (V C =Q/C) </li></ul></ul></ul><ul><ul><ul><li>Loop: – V C – I R = 0 </li></ul></ul></ul><ul><ul><ul><ul><li>I = ( -V C )/R </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Therefore I falls as Q rises </li></ul></ul></ul></ul><ul><ul><li>When t is very large ( ) </li></ul></ul><ul><ul><ul><li>I = 0: no current flow into/out of capacitor for t large </li></ul></ul></ul><ul><ul><ul><li>V C = </li></ul></ul></ul> R C + - Demo R
- 9. ACT/Preflight 7.1 <ul><li>Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S 1 is closed? </li></ul>2R C R S 2 1) I b = 0 2) I b = E /(3R) 3) I b = E /(2R) 4) I b = E /R S 1 I b + - + + - -
- 10. ACT/Preflight 7.3 <ul><li>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 ? </li></ul>1) I b = 0 2) I b = E /(3R) 3) I b = E /(2R) 4) I b = E /R 2R C R S 2 S 1 I b + - + + - -
- 11. Discharging Capacitors <ul><li>Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? </li></ul><ul><li>What is current I in circuit a long time later? </li></ul>R C S
- 12. Discharging Capacitors <ul><li>Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? </li></ul><ul><ul><li>KLR: Q/C – I 0 R = 0 </li></ul></ul><ul><ul><li>So, I 0 = Q/RC </li></ul></ul><ul><li>What is current I in circuit a long time later? </li></ul><ul><ul><li>I = 0 </li></ul></ul>R C + - Demo
- 13. ACT/Preflight 7.5 <ul><li>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? </li></ul>1) I R = 0 2) I R = /(3R) 3) I R = /(2R) 4) I R = /R 2R C R S 2 S 1 I R + - + + - - + -
- 14. ACT: RC Circuits <ul><li>Both switches are closed. What is the final charge on the capacitor after the switches have been closed a long time? </li></ul>1) Q = 0 2) Q = C E /3 3) Q = C E /2 4) Q = C E R 2R C S 2 S 1 I R + - + + - - + -
- 15. RC Circuits: Charging <ul><li>Loop: – I(t)R – q(t) / C = 0 </li></ul><ul><li>Just after…: q =q 0 </li></ul><ul><ul><li>Capacitor is uncharged </li></ul></ul><ul><ul><li> – I 0 R = 0 I 0 = / R </li></ul></ul><ul><li>Long time after: I c = 0 </li></ul><ul><ul><li>Capacitor is fully charged </li></ul></ul><ul><ul><li> – q /C =0 q = C </li></ul></ul><ul><li>Intermediate (more complex) </li></ul><ul><ul><li>q(t) = q (1-e -t/RC ) </li></ul></ul><ul><ul><li>I(t) = I 0 e -t/RC </li></ul></ul>C R S 1 S 2 + + + I - - - The switches are originally open and the capacitor is uncharged. Then switch S 1 is closed. t q RC 2RC 0 q
- 16. RC Circuits: Discharging <ul><li>Loop: q(t) / C + I(t) R = 0 </li></ul><ul><li>Just after…: q=q 0 </li></ul><ul><ul><li>Capacitor is still fully charged </li></ul></ul><ul><ul><li>q 0 / C + I 0 R = 0 I 0 = –q 0 / (RC) </li></ul></ul><ul><li>Long time after: I c =0 </li></ul><ul><ul><li>Capacitor is discharged (like a wire) </li></ul></ul><ul><ul><li>q / C = 0 q = 0 </li></ul></ul><ul><li>Intermediate (more complex) </li></ul><ul><ul><li>q(t) = q 0 e -t/RC </li></ul></ul><ul><ul><li>I c (t) = I 0 e -t/RC </li></ul></ul>C R S 1 - + + I - + - S 2 q RC 2RC t
- 17. What is the time constant? <ul><li>The time constant = RC . </li></ul><ul><li>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. </li></ul><ul><li>The time constant is the amount of time an RC circuit takes to discharge a capacitor by about 63.2% of its original value. </li></ul>
- 18. Time Constant Demo <ul><li>Which system will be brightest? </li></ul><ul><li>Which lights will stay on longest? </li></ul><ul><li>Which lights consumes more energy? </li></ul>Each circuit has a 1 F capacitor charged to 100 Volts. When the switch is closed: 1 Example 2
- 19. Summary of Concepts <ul><li>Charge (and therefore voltage) on Capacitors cannot change instantly: remember V C = Q/C </li></ul><ul><li>Short term behavior of Capacitor: </li></ul><ul><ul><li>If the capacitor starts with no charge, it has no potential difference across it and acts as a wire </li></ul></ul><ul><ul><li>If the capacitor starts with charge, it has a potential difference across it and acts as a battery. </li></ul></ul><ul><li>Long term behavior of Capacitor: Current through a Capacitor eventually goes to zero. </li></ul><ul><ul><li>If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit. </li></ul></ul><ul><ul><li>If capacitor is discharging, potential difference goes to zero and no current flows. </li></ul></ul><ul><li>Intermediate behavior: Charge and current exponentially approach their long-term values = RC </li></ul>
- 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: Example E – I 0 R – q 0 /C = 0 + + + - - - 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 R C ε S 1 R=10 C=30 mF ε =20 Volts I
- 21. ACT: RC Challenge <ul><li>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? </li></ul>1) 0.368 q 0 2) 0.632 q 0 3) 0.135 q 0 4) 0.865 q 0 R C ε 2R S 1 ε = 24 Volts R = 2 C = 15 mF
- 22. Charging: Intermediate Times Calculate the charge on the capacitor 3 10 -3 seconds after switch 1 is closed. <ul><ul><li>q(t) = q (1-e -t/RC ) </li></ul></ul>= 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 Example 2R C R S 2 S 1 I b + - + + - -
- 23. RC Summary Charging Discharging 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 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

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