1. Physics 212 Lecture 19, Slide 1 Physics 212 Lecture 19 LC and RLC Circuits

2. Physics 212 Lecture 19, Slide 2 C I L Q LC Circuit+- +- Circuit Equation: where

3. Physics 212 Lecture 19, Slide 3 C L Same thing if we notice that and k x m F F = -kx a

4. Physics 212 Lecture 19, Slide 4 C L Time Dependence I + + - -

5. Physics 212 Lecture 19, Slide 5 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 Pendulum… 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

6. Physics 212 Lecture 19, Slide 6 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

7. Physics 212 Lecture 19, Slide 7 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

8. Physics 212 Lecture 19, Slide 8 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 + + - -

9. Physics 212 Lecture 19, Slide 9 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

10. Physics 212 Lecture 19, Slide 10 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

11. CRL I(t) Use I = dQ/dt and divide by L: Add resistance

12. Physics 212 Lecture 19, Slide 12 Damped oscillations

13. Physics 212 Lecture 19, Slide 13Calculation 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

14. Physics 212 Lecture 19, Slide 14Calculation 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 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: all current goes through inductor in direction shown C R L V ILILILIL The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.

15. Physics 212 Lecture 19, Slide 15Calculation 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 !!

16. Physics 212 Lecture 19, Slide 16Calculation 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.

17. Physics 212 Lecture 19, Slide 17Calculation 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.

18. Physics 212 Lecture 19, Slide 18Calculation 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

19. Physics 212 Lecture 19, Slide 19 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: