Physics 102: Lecture 7 C Circuits Physics 102: Lecture 7, Slide 1
C Circuits Circuits that have both resistors and capacitors: K Na Cl C ε K ε Na ε Cl S With resistance in the circuits, capacitors do not charge and discharge instantaneously it takes time (even if only fractions of a second). Physics 102: Lecture 7, Slide 2
C Circuits Used to controllably store and release energy Today: C Circuits Charging Capacitors Discharging Capacitors Intermediate Behavior Physics 102: Lecture 7, Slide 3
Charging Capacitors Storing energy to use later 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? C S Physics 102: Lecture 7, Slide 4
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 =0 (since V=Q/C) Therefore C behaves as a wire (short circuit) Ohm s law! I 0 = / Physics 102: Lecture 7, Slide 5 C S
Charging Capacitors: t > 0 I 0 = / Positive charge flows Onto bottom plate (Q) Away from top plate (Q) C As charge builds up, V C rises (V C =Q/C) Loop: V C I = 0 Physics 102: Lecture 7, Slide 6 I = ( V C )/ Therefore I falls as Q rises When t is very large () I = 0: no current flow into/out of capacitor at long times V C =
ACT/CheckPoint 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? 1) I b = 0 2) I b = E /(3) 3) I b = E /(2) 4) I b = E / I b 2 C S 2 S 1 Physics 102: Lecture 7, Slide 7
ACT/CheckPoint 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? 2 1) I b = 0 2) I b = E/(3) 3) I b = E/(2) 4) I b = E/ I b C S 2 S 1 Physics 102: Lecture 7, Slide 8
Discharging Capacitors Time to use that stored energy! 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? S C Physics 102: Lecture 7, Slide 9
Physics 102: Lecture 7, Slide 10 Discharging Capacitors Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? KL: Q/C I 0 = 0 So, I 0 = Q/C What is current I in circuit a long time later? I = 0 I C
ACT/CheckPoint 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? 2 1) I = 0 2) I = /(3) 3) I = /(2) 4) I = / C I S 1 S 2 Physics 102: Lecture 7, Slide 11
Summary: charging & discharging 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 is eventually 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 is zero and no current flows Physics 102: Lecture 7, Slide 12
Q C Circuits: Charging The switches are originally open and the capacitor is uncharged. Then switch S 1 is closed. Loop: I(t) q(t) / C = 0 Just after : q =0 Capacitor is uncharged I 0 = 0 I 0 = / Long time after: I= 0 Capacitor is fully charged q /C =0 q = C q 1 C S 1 I C S 2 Intermediate (more complex) q(t) = q (1 e t/c ) I(t) = I 0 e t/c f( x) q 0.5 Physics 102: Lecture 7, Slide 13 0 0 1 2 3 4 t
C Circuits: Discharging Loop: q(t) / C I(t) = 0 Just after : q=q 0 Capacitor is still fully charged q 0 / C I 0 = 0 I 0 = q 0 / (C) Long time after: I=0 Capacitor is discharged q / C = 0 q = 0 Intermediate (more complex) 1 1 C S 1 I C S 2 q(t) = q 0 e t/c I(t) = I 0 e t/c q f( x) 0.5 Physics 102: Lecture 7, Slide 14 0.0183156 0 1 t 2 3 4 0 x 4
What is the time constant? The time constant = C. Given a capacitor starting with no charge, the time constant is the amount of time an C circuit takes to charge a capacitor to about 63.2% of its final value. The time constant is the amount of time an C circuit takes to discharge a capacitor by about 63.2% of its original value. Physics 102: Lecture 7, Slide 15
Time Constant Demo Each circuit has a 1 F capacitor charged to 100 Volts. When the switch is closed: Which system will be brightest? Which lights will stay on longest? Which lights consumes more energy? 1 2 Physics 102: Lecture 7, Slide 16 = 2C = C/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 is eventually 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 is zero and no current flows. Intermediate behavior: Charge and current exponentially approach their longterm values = C Physics 102: Lecture 7, Slide 17
Practice: defibrillator 1=1000 W I b C=1 mf body S 2 S 1 A 500 V battery is used to charge the 1 mf capacitor for 2 seconds. How much charge is stored on the capacitor? q(t) = q (1 e t/c ) q = CV = q 2 s = Physics 102: Lecture 7, Slide 18 V C = Q/C =
Practice: defibrillator 1=1000 W I b C=1 mf body S 2 S 1 A 500 V battery is used to charge the 1 mf capacitor for 2 seconds. How much energy is stored in the capacitor? U = Physics 102: Lecture 7, Slide 19
ACT: defibrillator 1=1000 W I b C=1 mf body S 2 S 1 What is the current through the patient right after S2 is closed if body =100 W? (A) 0 A (B) 4 A (C) 0.25 A Physics 102: Lecture 7, Slide 20
ACT: defibrillator 1=1000 W I b C=1 mf body S 2 S 1 What is the current through the patient 0.1 s after S2 is closed if body =100 W? (A) 4 A (B) 3.6 A (C) 1.5 A Physics 102: Lecture 7, Slide 21
C Summary Charging q(t) = q (1e t/c ) V(t) = V (1e t/c ) I(t) = I 0 e t/c Discharging q(t) = q 0 e t/c V(t) = V 0 e t/c I(t) = I 0 e t/c Time Constant = C Large means long time to charge/discharge Short term: Charge doesn t change (often zero or max) Long term: Current through capacitor is zero. Physics 102: Lecture 7, Slide 22