1. Physics 212 Lecture 19, Slide 1 Physics 212 Lecture 19 LC and RLC Circuits - Oscillation frequency - Energy - Damping

2. Physics 212 Lecture 3 Music Who is the Artist? A)Stephane Grappelli B)Pearl Django C)Mark O’Connor’s Hot Swing Trio D)Miles Davis E)Cassandra Wilson

3. Physics 212 Lecture 18, Slide 3 Some Exam Stuff Exam Thu. night (Mar. 29) at 7:00 –Covers material in Lectures 9 – 18 –Bring your ID: Rooms determined by discussion section (see link) –Conflict exam at 5:15 Final Exam –“Regular”: Fri. May 4, 1:30; “Conflict”: Fri. May 11, 8:00 –Sign up for the conflict in the gradebook if you wish

4. Physics 212 Lecture 19, Slide 4 Your Comments “ ” “ I am really confused with this stuff. The PreLecture tries to tell us too much too fast. I have lost my intuition! :(” “ “The concepts today were quite intuitive. Unfortunately, there were a lot of very scary looking equations on the slides that are not on our formula sheet. Are we expected to memorize these equations and/or perform calculations with them?” WILL DO ! “ ” “ Waves (graphs) and how the signs of charge swap.” “ ” “ There are so many different combinations of circuit components now, and I’m starting to find it increasing difficult to keep straight how each circuit behaves. Please go over the check points as well; I got stuck on a few of them!!! ” “ ” “If this comment makes it onto the board today, no more puppies will be killed by horrible puns.” “Okay I get it. But one thing...does omega equal d/dt? That’s not a term though...it’s just notation...I don’t get that. Also, why don’t we just use superconducting LC circuits to supply electricity???” The LC solutions are easy to derive starting from what you know, NOT RLC

5. Physics 212 Lecture 19, Slide 5 C I L Q LC Circuit+- +- Circuit Equation: where

6. Physics 212 Lecture 19, Slide 6 What is the potential difference across the inductor at t = 0 ? A) V L = 0 B) V L = Q max /C C) V L = Q max /2C At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L C Checkpoint 1a “The current going through the inductor is going to be 0 so by ohm’s law there is no voltage difference over the inductor.” “no current, simple potential equation Q=cv ” “V = (1/2) (Q/C)”

7. Physics 212 Lecture 19, Slide 7 What is the potential difference across the inductor at t = 0 ? A) V L = 0 B) V L = Q max /C C) V L = Q max /2C At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L C The voltage across the capacitor is Q max /C Kirchhoff's Voltage Rule implies that must also be equal to the voltage across the inductor Checkpoint 1a since V L = V C

8. Physics 212 Lecture 19, Slide 8 C L Same thing if we notice that and k x m F F = -kx a Pendulum…

9. Physics 212 Lecture 19, Slide 9 C L Time Dependence I + + - -

10. Physics 212 Lecture 19, Slide 10 At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L C What is the potential difference across the inductor when the current is maximum ? A) V L = 0 B) V L = Q max /C C) V L = Q max /2C Checkpoint 1b “The change in current becomes 0 when the current is maximum.” “This is the maximum voltage across that capacitor.” “Since the voltage is dependent on the amount of charge that is left in the capacitor it is Q/2C”

11. Physics 212 Lecture 19, Slide 11 At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L C What is the potential difference across the inductor when the current is maximum ? A) V L = 0 B) V L = Q max /C C) V L = Q max /2C dI/dt is zero when current is maximum Checkpoint 1b

12. Physics 212 Lecture 19, Slide 12 At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L C How much energy is stored in the capacitor when the current is a maximum ? A) U = Q max 2 /(2C) B) U = Q max 2 /(4C) C) U = 0 Checkpoint 1c “All the energy is in the capacitor” “current is maximum when energy is evenly divided between the capacitor and inductor” “All the energy would be stored in the inductor when the current is maximum.”

13. Physics 212 Lecture 19, Slide 13 At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L C How much energy is stored in the capacitor when the current is a maximum ? A) U = Q max 2 /(2C) B) U = Q max 2 /(4C) C) U = 0 Total Energy is constant ! U Lmax = ½ LI max 2 U Cmax = Q max 2 /2C I = max when Q = 0 Checkpoint 1c

14. Physics 212 Lecture 19, Slide 14 The capacitor is charged such that the top plate has a charge +Q 0 and the bottom plate -Q 0. At time t=0, the switch is closed and the circuit oscillates with frequency  = 500 radians/s. What is the value of the capacitor C? A) C = 1 x 10 -3 F B) C = 2 x 10 -3 F C) C = 4 x 10 -3 F L C L = 4 x 10 -3 H  = 500 rad/s Checkpoint 2a + + - - “w = 1 / sqrt(LC)” “use q=vc formula” “All the charge is built up in the capacitor and so it is 4x10^-3”

15. Physics 212 Lecture 19, Slide 15 The capacitor is charged such that the top plate has a charge +Q 0 and the bottom plate -Q 0. At time t=0, the switch is closed and the circuit oscillates with frequency  = 500 radians/s. What is the value of the capacitor C? A) C = 1 x 10 -3 F B) C = 2 x 10 -3 F C) C = 4 x 10 -3 F L C L = 4 x 10 -3 H  = 500 rad/s Checkpoint 2a + + - -

16. Physics 212 Lecture 19, Slide 16 Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed? Checkpoint 2b L C closed at t=0 +Q 0 -Q 0 “Since the current is initially 0, the energy begins at zero and oscillates, allowing the energy to be negative.” “oscillates between positive and negative, does not start at zero” “It starts at 0 and cannot be negative.”

17. Physics 212 Lecture 19, Slide 17 Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed? Energy proportional to I 2  U cannot be negative Current is changing  U L is not constant Initial current is zero Checkpoint 2b L C closed at t=0 +Q 0 -Q 0

18. Physics 212 Lecture 19, Slide 18 When the energy stored in the capacitor reaches its maximum again for the first time after t=0, how much charge is stored on the top plate of the capacitor? Checkpoint 2c L C closed at t=0 +Q 0 -Q 0 A) +Q 0 B) +Q 0 /2 C) 0 D)-Q 0 /2 E)-Q 0 “V is the same as before, so Q is the same” “only 1/2 the charge is needed” “Current direction swaps around so the signs on the capacitor plates changes as well.”

19. Physics 212 Lecture 19, Slide 19 When the energy stored in the capacitor reaches its maximum again for the first time after t=0, how much charge is stored on the top plate of the capacitor? Checkpoint 2c L C closed at t=0 +Q 0 -Q 0 A) +Q 0 B) +Q 0 /2 C) 0 D)-Q 0 /2 E)-Q 0 Q is maximum when current goes to zero Current goes to zero twice during one cycle

20. Physics 212 Lecture 19, Slide 20 Physics Truth #N: Even though the answer sometimes looks complicated… …the physics under the hood is still very simple !!

21. Physics 212 Lecture 19, Slide 21 …but answer looks kind of complicated Concept makes sense… Just like LC circuit but energy and the oscillations get smaller because of R Add R: Damping

22. Physics 212 Lecture 19, Slide 22 The elements of a circuit are very simple: This is all we need to know to solve for anything !

23. Physics 212 Lecture 19, Slide 23 Start with some initial V, I, Q, V L Now take a tiny time step dt (1 ms) Repeat… A Different Approach

24. Physics 212 Lecture 19, Slide 24Calculation The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. What is Q MAX, the maximum charge on the capacitor? Conceptual Analysis –Once switch is opened, we have an LC circuit –Current will oscillate with natural frequency  0 Strategic Analysis –Determine initial current –Determine oscillation frequency   –Find maximum charge on capacitor C R L V

25. Physics 212 Lecture 19, Slide 25Calculation What is I L, the current in the inductor, immediately AFTER the switch is opened? Take positive direction as shown. (B) (C) (A) I L 0 C R L V Current through inductor immediately AFTER switch is opened IS THE SAME AS the current through inductor immediately BEFORE switch is opened ILILILIL BEFORE switch is opened: all current goes through inductor in direction shown The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.

26. Physics 212 Lecture 19, Slide 26Calculation The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. The energy stored in the capacitor immediately after the switch is opened is zero. (B) (A) TRUE (B) FALSE C R L V V C =0 BUT: V L = V C since they are in parallel V C = 0 I L (t=0+) > 0 ILILILIL dI L /dt ~ 0  V L = 0 BEFORE switch is opened: V C = 0 V C cannot change abruptly AFTER switch is opened: IMPORTANT: NOTE DIFFERENT CONSTRAINTS AFTER SWITCH OPENED CURRENT through INDUCTOR cannot change abruptly VOLTAGE across CAPACITOR cannot change abruptly U C = ½ CV C 2 = 0 !!

27. Physics 212 Lecture 19, Slide 27Calculation What is the direction of the current immediately after the switch is opened? (B) (A) clockwise (B) counterclockwise C R L V Current through inductor immediately AFTER switch is opened IS THE SAME AS the current through inductor immediately BEFORE switch is opened BEFORE switch is opened: Current moves down through L ILILILIL AFTER switch is opened: Current continues to move down through L I L (t=0+) > 0 V C (t=0+) = 0 The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.

28. Physics 212 Lecture 19, Slide 28Calculation What is the magnitude of the current right after the switch is opened? (B) (C) (D) (A) (B) (C) (D) Current through inductor immediately AFTER switch is opened IS THE SAME AS the current through inductor immediately BEFORE switch is opened C R L V V L = 0 BEFORE switch is opened: C R L V V L =0 ILILILIL ILILILIL ILILILIL V = I L R I L (t=0+) > 0 V C (t=0+) = 0 The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.

29. Physics 212 Lecture 19, Slide 29Calculation What is Q max, the maximum charge on the capacitor during the oscillations? (B) (C) (D) (A) (B) (C) (D) C R L V I L (t=0+) =V/R V C (t=0+) = 0 ILILILIL Hint: Energy is conserved The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. C L I max When I is max (and Q is 0) C When Q is max (and I is 0) L Q max

30. Physics 212 Lecture 19, Slide 30 Follow-Up 1 The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. Is it possible for the maximum voltage on the capacitor to be greater than V? (B) NO (A) YES (B) NO C R L V I max =V/R ILILILIL We can rewrite this condition in terms of the resonant frequency: OR We will see these forms again when we study AC circuits!! V max can be greater than V IF: