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

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

Physics 212 Lecture 11, Slide 3 Your Comments 05 “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. “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 “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 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? “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”

Physics 212 Lecture 11, Slide 4 Today’s Plan: 1)Examples with switches closing and opening -What changes? -What is constant? 2)Example problem 3)Exponentials Key Concepts: 1)Understanding the behavior of capacitors in circuits with resistors 2)Understanding the RC time constant 07

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 Physics 212 Lecture 11, Slide 5

Physics 212 Lecture 11, Slide 6 RC Circuit (Charging) Kirchoff’s Voltage Rule Short Term (q = q 0 = 0) Long Term (I c =0) R V battery C a b Intermediate 11 a R V battery C b Capacitor uncharged, Switch is moved to position “a”

Physics 212 Lecture 11, Slide 7 Solving the Differential Equation 11 a R V batter y C b What do we do with First consider (without V b – a constant) Guess the solution (in 212 there are only 2 choices!) Is this it? A) Add another exponential B) Add a constant C) This IS it ?? and

Physics 212 Lecture 11, Slide 8 13 Checkpoint 1a Checkpoint 1b & Close S1, V 1 = voltage across C immediately after V 2 = voltage across C a long time after A)V 1 = V V 2 = V B) V 1 = 0 V 2 = V C) V 1 = 0 V 2 = 0 D) V 1 = V V 2 = 0 A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open.

Physics 212 Lecture 11, Slide 9 13 Checkpoint 1a Checkpoint 1b & Close S1, V 1 = voltage across C immediately after V 2 = voltage across C a long time after A)V 1 = V V 2 = V B) V 1 = 0 V 2 = V C) V 1 = 0 V 2 = 0 D) V 1 = V V 2 = 0 Immediately after the switch S 1 is closed: Immediately after the switch S 1 is closed: Q = 0 V 1 = 0 V = Q/C After the switch S 1 has been closed for a long time After the switch S 1 has been closed for a long time I = 0 V 2 = V V R = 0 A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open.

Physics 212 Lecture 11, Slide 10 V 2R C R S1S2 Close S 1 at t=0 (leave S 2 open) At t = big V C R I=0 V C = V At t = 0 V C R I V C = Q/C = 0 = 0 15

R V battery C a b R C a b Physics 212 Lecture 11, Slide 11 RC Circuit (Discharging) Kirchoff’s Voltage Rule Short Term (q=q 0 ) Long Term (I c =0) Intermediate V 19 Capacitor has q 0 = CV battery, Switch is moved to position “b” I +--I

“The loop is no longer closed so there will be no voltage” I = V / total resistance which is 3R. “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 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 “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” Physics 212 Lecture 11, Slide 12 ABCDABCD 22 Checkpoint 1c + - IRIRIRIR A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. 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. I R = 0B. I R = V/3RC. I R = V/2RD. I R = V/R

Physics 212 Lecture 11, Slide 13 V 2R C V I ABCDABCD 22 Checkpoint 1c + - IRIRIRIR A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. 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. I R = 0B. I R = V/3RC. I R = V/2RD. I R = V/R

Physics 212 Lecture 11, Slide 14 2R C R S1S2 Open S1 at t=big and close S2 V I = V/2R 2R C V I 23 V

“Everything eventually goes to zero.” Fully charged with large time. “Fully charged with large time.” the resistance only allows for so much of the potential to be stored in the cap “the resistance only allows for so much of the potential to be stored in the cap” Physics 212 Lecture 11, Slide 15 ABCABC 26 Checkpoint 1d A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Now suppose both switches are closed. What is the voltage across the capacitor after a very long time? A. V C = 0B. V C = VC. V C = 2V/3

Physics 212 Lecture 11, Slide 16 ABCABC 26 Checkpoint 1d After both switches have been closed for a long time After both switches have been closed for a long time The current through the capacitor is zero The current through the capacitor is zero The current through R = current through 2R The current through R = current through 2R V capacitor = V 2R V capacitor = V 2R V 2R = 2/3 V V 2R = 2/3 V A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Now suppose both switches are closed. What is the voltage across the capacitor after a very long time? A. V C = 0B. V C = VC. V C = 2V/3

Physics 212 Lecture 11, Slide 17 2R C R S1S2 Close both S1 and S2 and wait a long time… V 2R C R V No current flows through the capacitor after a long time. This will always be the case in any static circuit!! VCVCVCVC I I = V/(3R) V C = V 2R 27 V C = I(2R) V C = (2/3)V

Physics 212 Lecture 11, Slide 18 DEMO – ACT 1 V C S Bulb 1 Bulb 2 What will happen after I close the switch? A)Both bulbs come on and stay on. B)Both bulbs come on but then bulb 2 fades out. C)Both bulbs come on but then bulb 1 fades out. D)Both bulbs come on and then both fade out. R R 30 No initial charge on capacitor V(bulb 1) = V(bulb 2) = V Both bulbs light No final current through capacitor V(bulb 2) = 0

Physics 212 Lecture 11, Slide 19 DEMO – ACT 2 V C S Bulb 1 Bulb 2 Suppose the switch has been closed a long time. Now what will happen after open the switch? A)Both bulbs come on and stay on. B)Both bulbs come on but then bulb 2 fades out. C)Both bulbs come on but then bulb 1 fades out. D)Both bulbs come on and then both fade out. R R 32 Capacitor has charge (=CV) Capacitor discharges through both resistors

Physics 212 Lecture 11, Slide 20Calculation In this circuit, assume V, C, and R i 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:  V drops = 0 KCR:  I in =  I out – –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 V R1R1 R2R2 C R3R3 S 35

Physics 212 Lecture 11, Slide 21Calculation V R1R1 R2R2 C R3R3 S Immediately after S is closed: what is I 2, the current through C what is V C, the voltage across C? (A) (B) (C) (D) (A) Only I 2 = 0 (B) Only V C = 0 (C) Both I 2 and V C = 0 (D) Neither I 2 nor V C = 0 Why?? – –We are told that C is initially uncharged (V = Q/C) – –I 2 cannot be zero because charge must flow in order to charge C In this circuit, assume V, C, and R i are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? 37

Physics 212 Lecture 11, Slide 22Calculation V R1R1 R2R2 C R3R3 S Why?? – –Draw circuit just after S closed (knowing V C = 0) Immediately after S is closed, what is I 1, the current through R 1 ? V R1R1 R2R2 S R3R3 V C = 0 I1I1 – –R 1 is in series with the parallel combination of R 2 and R 3 In this circuit, assume V, C, and R i are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? 39 (A) (B) (C) (D) (E)

Physics 212 Lecture 11, Slide 23Calculation V R1R1 R2R2 C R3R3 S 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 After S has been closed “for a long time”, what is I C, the current through C ? (A) (B) (C) V R1R1 R3R3 I C = 0 VCVC I In this circuit, assume V, C, and R i are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? 41

Physics 212 Lecture 11, Slide 24Calculation V R1R1 R2R2 C R3R3 S Why?? After S has been closed “for a long time”, what is V C, the voltage across C ? (A) (B) (C) (D) (E) – –V C = V 3 = IR 3 = (V/(R 1 +R 3 ))R 3 In this circuit, assume V, C, and R i are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? V R1R1 R3R3 VCVC I I 43

Physics 212 Lecture 11, Slide 25Challenge In this circuit, assume V, C, and R i are known. C initially uncharged and then switch S is closed. V R1R1 R2R2 C R3R3 S Strategy – –Write down KVR and KCR for the circuit when S is closed 2 loop equations and 1 node equation – –Use I 2 = dQ 2 /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  = “R”  ”C” C What is  c, the charging time constant? R2R2 C R3R3 We get:

Physics 212 Lecture 11, Slide 26 “Fraction of initial charge that remains” “How many time constants worth of time that have elapsed” 45 How do exponentials work?

Physics 212 Lecture 11, Slide 27 RC = 1 RC = 2 Time constant:  = RC The bigger  is, the longer it takes to get the same change… 47 How do exponentials work?

Physics 212 Lecture 11, Slide 28 RC = 1 RC = 2 Which circuit has the largest time constant? A) A)Circuit 1 B) B)Circuit 2 C) C)Same 49  = R equiv C Checkpoint 2a 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.

Physics 212 Lecture 11, Slide Checkpoint 2b 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. Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q 1 Q 2 C. Q 1 = Q 2 D. Q 1 Q 2 after long time E. Q 1 > Q 2 at first, then Q 1 < Q 2 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”

Physics 212 Lecture 11, Slide Checkpoint 2b 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. Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q 1 Q 2 C. Q 1 = Q 2 D. Q 1 Q 2 after long time E. Q 1 > Q 2 at first, then Q 1 < Q 2 after long time

Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q 1 Q 2 C. Q 1 = Q 2 D. Q 1 Q 2 after long time E. Q 1 > Q 2 at first, then Q 1 < Q 2 after long time Physics 212 Lecture 11, Slide 31 Checkpoint 2b RC = 1 RC = 2 Q = Q 0 e -t/RC Look at plot !!! 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