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Physics 212 Lecture 19

Physics 212 Lecture 19. LC and RLC Circuits. -. +. +. -. Circuit Equation:. where. LC Circuit. I. C. Q. L. L. C. m. F = -kx. k. a. x. Same thing if we notice that and. L. C. Time Dependence. I. +. +. -. -. C. L. since V L = V C.

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Physics 212 Lecture 19

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  1. Physics 212 Lecture 19 LC and RLC Circuits

  2. - + + - Circuit Equation: where LC Circuit I C Q L

  3. L C m F = -kx k a x Same thing if we notice that and

  4. L C Time Dependence I + + - -

  5. C L since VL = VC Checkpoint 1a At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. What is the potential difference across the inductor at t = 0 ?A)  VL = 0 B)  VL = Qmax/C C)  VL = Qmax/2C The voltage across the capacitor is Qmax/C Kirchhoff's Voltage Rule implies that must also be equal to the voltage across the inductor Pendulum…

  6. Checkpoint 1b C L At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. What is the potential difference across the inductor when the current is maximum ?A)  VL = 0 B)  VL = Qmax/C C)  VL = Qmax/2C dI/dt is zero when current is maximum

  7. Checkpoint 1c C L At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. How much energy is stored in the capacitor when the current is a maximum ?A)  U = Qmax2/(2C) B)  U = Qmax2/(4C) C)  U = 0 Total Energy is constant ! ULmax = ½ LImax2 UCmax = Qmax2/2C I = max when Q = 0

  8. C L Checkpoint 2a The capacitor is charged such that the top plate has a charge +Q0 and the bottom plate -Q0. At time t=0, the switch is closed and the circuit oscillates with frequencyw= 500 radians/s. L = 4 x 10-3 H w = 500 rad/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

  9. C L +Q0 -Q0 Energy proportional to I 2 U cannot be negative Current is changing  UL is not constant Initialcurrent is zero Checkpoint 2b closed at t=0 Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed?

  10. closed at t=0 +Q0 -Q0 C L Q is maximum when current goes to zero Checkpoint 2c 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? • A)  +Q0 • B)  +Q0 /2 • C)  0 • -Q0/2 • -Q0 Current goes to zero twice during one cycle

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

  12. Damped oscillations

  13. Calculation V C L R The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. What is QMAX, the maximum charge on the capacitor? • Conceptual Analysis • Once switch is opened, we have an LC circuit • Current will oscillate with natural frequency w0 • Strategic Analysis • Determine initial current • Determine oscillation frequency w0 • Find maximum charge on capacitor

  14. Calculation IL V C L R 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. What is IL, the current in the inductor, immediately AFTER the switch is opened? Take positive direction as shown. • IL < 0 (B) IL = 0 (C) IL > 0 Current through inductor immediately AFTER switch is opened IS THE SAME AS the current through inductor immediately BEFORE switch is opened

  15. Calculation VC=0 AFTER switch is opened: BEFORE switch is opened: dIL/dt ~ 0  VL = 0 VC cannot change abruptly UC = ½ CVC2 = 0 !! VC = 0 VC = 0 IL The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. V C L R IL(t=0+) > 0 The energy stored in the capacitor immediately after the switch is opened is zero. • TRUE (B) FALSE BUT: VL = VC since they are in parallel IMPORTANT: NOTE DIFFERENT CONSTRAINTS AFTER SWITCH OPENED CURRENT through INDUCTOR cannot change abruptly VOLTAGE across CAPACITOR cannot change abruptly

  16. Calculation IL The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. V C L R VC(t=0+) = 0 IL(t=0+) > 0 What is the direction of the current immediately after the switch is opened? • clockwise (B) counterclockwise 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 AFTER switch is opened: Current continues to move down through L

  17. Calculation IL IL V BEFORE switch is opened: VL=0 C L V = ILR IL R The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. V C L R VC(t=0+) = 0 IL(t=0+) > 0 What is the magnitude of the current right after the switch is opened? • (B) (C) (D) Current through inductor immediately AFTER switch is opened IS THE SAME AS the current through inductor immediately BEFORE switch is opened VL = 0

  18. Calculation V C L R Imax Qmax C L C L When Q is max (and I is 0) When I is max (and Q is 0) The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. IL Hint: Energy is conserved VC(t=0+) = 0 IL(t=0+) =V/R What is Qmax, the maximum charge on the capacitor during the oscillations? • (B) (C) (D)

  19. Follow-Up 1 V C L R Vmax can be greater than V IF: OR 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? IL Imax =V/R • YES (B) NO We can rewrite this condition in terms of the resonant frequency: We will see these forms again when we study AC circuits!!

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