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  • 1
  • Initial current through battery. Final current; final voltage across C.
  • Initial current through battery. Final current; final voltage across C.
  • Initial current through battery. Final current; final voltage across C.
  • Followup…what is current a long time later? What is charge on capacitor a long time later?

Lect07 handout Lect07 handout Presentation Transcript

  • RC Circuits Physics 102: Lecture 7
  • Recall ….
    • First we covered circuits with batteries and capacitors
      • series, parallel
    • Then we covered circuits with batteries and resistors
      • series, parallel
      • Kirchhoff’s Loop and Junction Relations
    • Today: circuits with batteries, resistors, and capacitors
  • RC Circuits
    • RC Circuits
    • Charging Capacitors
    • Discharging Capacitors
    • Intermediate Behavior
  • RC Circuits
    • Circuits that have both resistors and capacitors :
    • With resistance in the circuits, capacitors do not charge and discharge instantaneously – it takes time (even if only fractions of a second).
    R K C S + + + R Na R Cl ε K ε Na ε Cl
  • Capacitors
    • Charge (and therefore voltage) on Capacitors cannot change instantly: remember V C = Q/C
    • Short term behavior of Capacitor:
      • If the capacitor starts with no charge, it has no potential difference across it and acts as a wire
      • If the capacitor starts with charge, it has a potential difference across it and acts as a battery.
    • Long term behavior of Capacitor: Current through a Capacitor eventually goes to zero.
      • If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit
      • If capacitor is discharging, potential difference goes to zero and no current flows
  • Charging Capacitors
    • Capacitor is initially uncharged and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter?
    • What is current I  in circuit a long time later?
     R C S
  • Charging Capacitors: t=0
    • Capacitor is initially uncharged and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter?
      • Capacitor initially uncharged
      • Therefore V C is initially 0
      • Therefore C behaves as a wire (short circuit)
      • KLR:  – I 0 R = 0
        • I 0 =  /R
     R C S R 
  • Charging Capacitors: t>0
      • I 0 =  /R
      • Positive charge flows
        • Onto bottom plate (+Q)
        • Away from top plate (-Q)
        • As charge builds up, V C rises (V C =Q/C)
        • Loop:  – V C – I R = 0
          • I = (  -V C )/R
          • Therefore I falls as Q rises
      • When t is very large (  )
        • I  = 0: no current flow into/out of capacitor for t large
        • V C = 
     R C + - Demo R 
  • ACT/Preflight 7.1
    • Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S 1 is closed?
    2R C  R S 2 1) I b = 0 2) I b = E /(3R) 3) I b = E /(2R) 4) I b = E /R S 1 I b + - + + - -
  • ACT/Preflight 7.3
    • Both switches are initially open, and the capacitor is uncharged. What is the current through the battery after switch 1 has been closed a long time ?
    1) I b = 0 2) I b = E /(3R) 3) I b = E /(2R) 4) I b = E /R 2R C  R S 2 S 1 I b + - + + - -
  • Discharging Capacitors
    • Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter?
    • What is current I  in circuit a long time later?
    R C S
  • Discharging Capacitors
    • Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter?
      • KLR: Q/C – I 0 R = 0
      • So, I 0 = Q/RC
    • What is current I  in circuit a long time later?
      • I  = 0
    R C + - Demo
  • ACT/Preflight 7.5
    • After switch 1 has been closed for a long time, it is opened and switch 2 is closed. What is the current through the right resistor just after switch 2 is closed?
    1) I R = 0 2) I R =  /(3R) 3) I R =  /(2R) 4) I R =  /R 2R C  R S 2 S 1 I R + - + + - - + -
  • ACT: RC Circuits
    • Both switches are closed. What is the final charge on the capacitor after the switches have been closed a long time?
    1) Q = 0 2) Q = C E /3 3) Q = C E /2 4) Q = C E R 2R C  S 2 S 1 I R + - + + - - + -
  • RC Circuits: Charging
    • Loop:  – I(t)R – q(t) / C = 0
    • Just after…: q =q 0
      • Capacitor is uncharged
      •  – I 0 R = 0  I 0 =  / R
    • Long time after: I c = 0
      • Capacitor is fully charged
      •  – q  /C =0  q  =  C
    • Intermediate (more complex)
      • q(t) = q  (1-e -t/RC )
      • I(t) = I 0 e -t/RC
    C  R S 1 S 2 + + + I - - - The switches are originally open and the capacitor is uncharged. Then switch S 1 is closed. t q RC 2RC 0 q 
  • RC Circuits: Discharging
    • Loop: q(t) / C + I(t) R = 0
    • Just after…: q=q 0
      • Capacitor is still fully charged
      • q 0 / C + I 0 R = 0  I 0 = –q 0 / (RC)
    • Long time after: I c =0
      • Capacitor is discharged (like a wire)
      • q  / C = 0  q  = 0
    • Intermediate (more complex)
      • q(t) = q 0 e -t/RC
      • I c (t) = I 0 e -t/RC
    C  R S 1 - + + I - + - S 2 q RC 2RC t
  • What is the time constant?
    • The time constant  = RC .
    • Given a capacitor starting with no charge, the time constant is the amount of time an RC circuit takes to charge a capacitor to about 63.2% of its final value.
    • The time constant is the amount of time an RC circuit takes to discharge a capacitor by about 63.2% of its original value.
  • Time Constant Demo
    • Which system will be brightest?
    • Which lights will stay on longest?
    • Which lights consumes more energy?
    Each circuit has a 1 F capacitor charged to 100 Volts. When the switch is closed: 1 Example 2
  • Summary of Concepts
    • Charge (and therefore voltage) on Capacitors cannot change instantly: remember V C = Q/C
    • Short term behavior of Capacitor:
      • If the capacitor starts with no charge, it has no potential difference across it and acts as a wire
      • If the capacitor starts with charge, it has a potential difference across it and acts as a battery.
    • Long term behavior of Capacitor: Current through a Capacitor eventually goes to zero.
      • If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit.
      • If capacitor is discharging, potential difference goes to zero and no current flows.
    • Intermediate behavior: Charge and current exponentially approach their long-term values  = RC
  • Practice! Calculate current immediately after switch is closed: Calculate current after switch has been closed for 0.5 seconds: Calculate current after switch has been closed for a long time: Calculate charge on capacitor after switch has been closed for a long time: Example E – I 0 R – q 0 /C = 0 + + + - - - E – I 0 R – 0 = 0 I 0 = E /R After a long time current through capacitor is zero! E – IR – q ∞ /C = 0 E – 0 – q ∞ /C = 0 q ∞ = E C R C ε S 1 R=10  C=30 mF ε =20 Volts I
  • ACT: RC Challenge
    • After being closed for a long time, the switch is opened. What is the charge Q on the capacitor 0.06 seconds after the switch is opened?
    1) 0.368 q 0 2) 0.632 q 0 3) 0.135 q 0 4) 0.865 q 0 R C ε 2R S 1 ε = 24 Volts R = 2  C = 15 mF
  • Charging: Intermediate Times  Calculate the charge on the capacitor 3  10 -3 seconds after switch 1 is closed.
      • q(t) = q  (1-e -t/RC )
    = q  (1-e - 3  10 -3 /(20  100  10 -6) ) ) = q  (0.78) Recall q  =  C = (50)(100x10 -6 ) (0.78) = 3.9 x10 -3 Coulombs R = 10  V = 50 Volts C = 100  F Example 2R C R S 2 S 1 I b + - + + - -
  • RC Summary Charging Discharging q(t) = q  (1-e -t/RC ) q(t) = q 0 e -t/RC V(t) = V  (1-e -t/RC ) V(t) = V 0 e -t/RC I(t) = I 0 e -t/RC I(t) = I 0 e -t/RC Short term: Charge doesn’t change (often zero or max) Long term: Current through capacitor is zero. Time Constant  = RC Large  means long time to charge/discharge