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Physics 212 Lecture 11

Physics 212 Lecture 11. Today's Concept: RC Circuits (Circuits with resistors & capacitors & batteries). Music. Who is the Artist? Professor Longhair John Cleary Allen Toussaint Fats Domino Tuts Washington. Salute to Mardi Gras (today!). “Piano Players Rarely Ever Play Together”.

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Physics 212 Lecture 11

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  1. Physics 212 Lecture 11 Today's Concept: RC Circuits (Circuits with resistors & capacitors & batteries)

  2. Music • Who is the Artist? • Professor Longhair • John Cleary • Allen Toussaint • Fats Domino • Tuts Washington Salute to Mardi Gras (today!) “Piano Players Rarely Ever Play Together”

  3. “thought this section would be easy, this made it hard” “This was the first time I was virtually lost on the checkpoints. I am hoping that a clearer explanation of things in class will help.” “Everything is a little jumbled. Please go over what changes and what stays the same in the long and short term of these systems” Most of our time on checkpoints and examples “Urrrg, there were a lot of crazy equations today, and I feel like they went through the derivations kinda fast... Yup, im confused!” “Differential Equation!! I dont know what is going on there!!!!”“When did differential equations become a prerequisite? ” First, remember this is just Kirchoff. We have to learn the solutions of two simple d.e.s Your Comments “This stuff really has me charged up. I can’t resist!” “Trying to conceptually join resistors and capacitors is currently demanding all my power . . .” “If this is on the test we R going to get a C” 05 A man walks into a bar with a pair of jumper cables and orders a drink. The bar tender looks at him and says, "Alright, I'll serve you, but don't start anything!" That joke had to do with electricity and circuits, right?

  4. Today’s Plan: • Examples with switches closing and opening • What changes? • What is constant? • Example problem • Exponentials Key Concepts: • Understanding the behavior of capacitors in circuits with resistors • Understanding the RC time constant 07

  5. The 212 Differential Equations • We describe the world (electrical circuits, problems in heat transfer, control systems, etc., etc.) using differential equations • You only need to know the solutions of two basic differential equations

  6. RC Circuit (Charging) a C b Vbattery R C a Vbattery R b • Capacitor uncharged, Switch is moved to position “a” • Kirchoff’s Voltage Rule • Short Term (q = q0 = 0) • Long Term (Ic =0) Intermediate 11

  7. Solving the Differential Equation a C b Vbattery R • What do we do with ?? • First consider (without Vb – a constant) • Guess the solution (in 212 there are only 2 choices!) • Is this it?A) Add another exponentialB) Add a constantC) This IS it and 11

  8. A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1a & Checkpoint 1b • V1 = V V2 = V B) V1 = 0 V2 = V Close S1, V1 = voltage across C immediately after V2 = voltage across C a long time after C) V1 = 0 V2 = 0 D) V1 = V V2 = 0 13

  9. A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. After the switch S1 has been closed for a long time Immediately after the switch S1 is closed: VR = 0 I = 0 V2 = V V = Q/C Q = 0 V1 = 0 Checkpoint 1a & Checkpoint 1b • V1 = V V2 = V B) V1 = 0 V2 = V Close S1, V1 = voltage across C immediately after V2 = voltage across C a long time after C) V1 = 0 V2 = 0 D) V1 = V V2 = 0 13

  10. R R I=0 I C V C V VC = Q/C = 0 VC = V At t = big At t = 0 Close S1 at t=0 (leave S2 open) R C V 2R S1 S2 15

  11. RC Circuit (Discharging) a C - + I b Vbattery R V a C b Vbattery R • Capacitor has q0 = CVbattery, Switch is moved to position “b” • Kirchoff’s Voltage Rule • Short Term (q=q0) • Long Term (Ic =0) Intermediate -I 19

  12. A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1c IR + - After being closed a long time, switch 1 is opened and switch 2 is closed. What is the current through the right resistor immediately after switch 2 is closed? A. IR = 0 B. IR = V/3R C. IR = V/2R D. IR = V/R A B C D “The loop is no longer closed so there will be no voltage” “I = V / total resistance which is 3R.” “The capacitor initially acts as a battery with voltage V, so Ohm’s law gives a current of V/2R” “The capacitor has potential difference V across its plates. When that charge is given a path from one plate to another, it will travel through the resistor until the plates are at equal potential” 22

  13. A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. I V C V 2R Checkpoint 1c IR + - After being closed a long time, switch 1 is opened and switch 2 is closed. What is the current through the right resistor immediately after switch 2 is closed? A. IR = 0 B. IR = V/3R C. IR = V/2R D. IR = V/R A B C D 22

  14. I V C V 2R I = V/2R Open S1 at t=big and close S2 R C V 2R S1 S2 23

  15. A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1d Now suppose both switches are closed. What is the voltage across the capacitor after a very long time? A. VC = 0 B. VC = V C. VC = 2V/3 A B C “Everything eventually goes to zero.” “Fully charged with large time.” “the resistance only allows for so much of the potential to be stored in the cap” 26

  16. A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1d Now suppose both switches are closed. What is the voltage across the capacitor after a very long time? A. VC = 0 B. VC = V C. VC = 2V/3 A B C • After both switches have been closed for a long time • The current through the capacitor is zero • The current through R = current through 2R • Vcapacitor = V2R • V2R = 2/3 V 26

  17. I No current flows through the capacitor after a long time. This will always be the case in any static circuit!! R C V VC 2R VC = I(2R) VC = (2/3)V Close both S1 and S2 andwait a long time… R C V 2R S1 S2 I = V/(3R) VC = V2R 27

  18. DEMO – ACT 1 Both bulbs light V(bulb 1) = V(bulb 2) = V V(bulb 2) = 0 Bulb 2 S R R V Bulb 1 C • What will happen after I close the switch? • Both bulbs come on and stay on. • Both bulbs come on but then bulb 2 fades out. • Both bulbs come on but then bulb 1 fades out. • Both bulbs come on and then both fade out. No initial charge on capacitor No final current through capacitor 30

  19. DEMO – ACT 2 Capacitor discharges through both resistors Bulb 2 R S R V Bulb 1 C • Suppose the switch has been closed a long time. • Now what will happen after open the switch? • Both bulbs come on and stay on. • Both bulbs come on but then bulb 2 fades out. • Both bulbs come on but then bulb 1 fades out. • Both bulbs come on and then both fade out. Capacitor has charge (=CV) 32

  20. Calculation S R1 R2 R3 C V In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? • Conceptual Analysis: • Circuit behavior described by Kirchhoff’s Rules: • KVR:SVdrops = 0 • KCR:SIin = SIout • S closed and C charges to some voltage with some time constant • Strategic Analysis • Determine currents and voltages in circuit a long time after S closed 35

  21. Calculation S In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? R1 R2 R3 C V Immediately after S is closed: what is I2, the current through C what is VC, the voltage across C? (A) Only I2 = 0(B) Only VC = 0(C) Both I2 and VC = 0 (D) Neither I2 nor VC = 0 • Why?? • We are told that C is initially uncharged (V = Q/C) • I2 cannot be zero because charge must flow in order to charge C 37

  22. Calculation (A) (B) (C) (D) (E) S R1 R2 R3 V VC = 0 I1 S In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? R1 R2 R3 C V • Immediately after S is closed, what is I1, the current through R1 ? • Why?? • Draw circuit just after S closed (knowing VC = 0) • R1is in series with the parallel combination of R2and R3 39

  23. Calculation I R1 R3 VC IC = 0 V S In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? R1 R2 R3 C V After S has been closed “for a long time”, what is IC, the current through C ? (A) (B) (C) • Why?? • After a long time in a static circuit, the current through any capacitor approaches 0 ! • This means we Redraw circuit with open circuit in middle leg 41

  24. Calculation After S has been closed “for a long time”, what is VC, the voltage across C ? (A) (B) (C) (D) (E) I I R1 R3 VC V S In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? R1 R2 R3 C V • Why?? • VC = V3 = IR3 = (V/(R1+R3))R3 43

  25. Challenge R2 We get: C R3 In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. S R1 R2 R3 C V What is tc, the charging time constant? • Strategy • Write down KVR and KCR for the circuit when S is closed • 2 loop equations and 1 node equation • Use I2 = dQ2/dt to obtain one equation that looks like simple charging RC circuit ( (Q/”C”) + “R”(dQ/dt) – “V” = 0 ) • Make correspondence: “R” = ?, and “C” = ?, then t = “R” ”C” C

  26. How do exponentials work? RC = 2 Time constant: t = RC The bigger t is, the longer it takes to getthe same change… RC = 1 47

  27. The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1. RC = 2 RC = 1 Checkpoint 2a • Which circuit has the largest time constant? • Circuit 1 • Circuit 2 • Same t = RequivC 49

  28. The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1. Checkpoint 2b Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q1 < Q2 B. Q1 > Q2 C. Q1 = Q2 D. Q1 < Q2 at first, then Q1 > Q2 after long time E. Q1 > Q2 at first, then Q1 < Q2 after long time “C1 will discharge faster than C2 because of the resistance” “The smaller time constant means the charge will dissipate slower” “Charge after time doesn’t depend on resistance. Only V and C which are the same” “The charge decreases exponentially so at first 2 will be greater then eventually 1 will be greater. ” “C1 charges faster, but C2 has a higher max charge” 50

  29. The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1. Checkpoint 2b Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q1 < Q2 B. Q1 > Q2 C. Q1 = Q2 D. Q1 < Q2 at first, then Q1 > Q2 after long time E. Q1 > Q2 at first, then Q1 < Q2 after long time 50

  30. The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1. RC = 2 RC = 1 Look at plot !!! Checkpoint 2b Checkpoint 2b Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q1 < Q2 B. Q1 > Q2 C. Q1 = Q2 D. Q1 < Q2 at first, then Q1 > Q2 after long time E. Q1 > Q2 at first, then Q1 < Q2 after long time Q = Q0e-t/RC

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