Physics 2112 Unit 11 Today s oncept: ircuits Unit 11, Slide 1
Stuff you asked about.. what happens when one resistor is in parallel and one is in series with the capacitor Differential equations are tough trickier than the last few prelectures difference between equations for charging and discharging. How much are we expected to know about all these equations. The time constant and exponential decay The concepts were pretty easy to understand. I feel like adding time into the mix will cause some confusion, but at this point in time (no pun intended) I understand the majority of it (minus the equations for the time dependent graphs. more explanation and examples please. conceptually it makes sense to me, but mathematically i'm a little confused. I need to see some example problems being worked out. Unit 11, Slide 2
Kirchoff s oltage ule q battery I 0 Differential Equation ircuit (harging) apacitor uncharged, Switch is moved to position a battery q dq dt 0 a battery b bat q dq dt 1 ( q ) q( t) 1 e t / I( t) I 0 e t / Unit 11, Slide 3
Question A circuit is wired up as shown below. The capacitor is initially uncharged and switched S1 and S2 are initially open. Then S1 is closed but S2 remains open. What are - o (voltage across immediately after) - f voltage across a long time after) A) o f B) o 0 f ) o 0 f < D) o f 0 S2 Unit 11, Slide 4
Question A circuit is wired up as shown below. The capacitor is initially uncharged and switched S1 and S2 are initially open. Then S1 is closed but S2 remains open. What are - I o (current into immediately after) - I f (current into a long time after) A) I o / I f / B) I o 0 I f / ) I o =/ I f 0 D) I o / I f < / S2 Unit 11, Slide 5
lose S 1 at t = 0 (leave S 2 open) 2 S1 S2 I I 0 Q/ 0 At t 0 At t big Unit 11, Slide 6
Question A circuit is wired up as shown below. The capacitor is initially uncharged and switched S1 and S2 are initially open. Then both S1 and S2 are closed at once. What are - o (voltage across immediately after) - f voltage across a long time after) A) o f B) o 0 f ) o 0 f < D) o f 0 S2 Unit 11, Slide 7
Example 11.1 (harging apacitor) 1 = 1uF 10 1 = 1.2MW What is the charge on the capacitor 1second after the switch is closed? What is the current through the resistor 1 second after the switch is closed? onceptual Idea: Use charging equations for a capacitor. Plan: Find time constant, t Find Q f and I o Put in t=1sec
Question (harging apacitor) 10 What is the charge on the capacitor 1second after the switch is closed? 1 = 1uF 1 = 1.2MW Based on what we now so far, what do you think the charge on the capacitor will be at t = 1sec? A) a little bit less than 0.37*10*1uF B) a little bit more than 0.37*10*1uF ) a little bit less than 0.63*10*1uF D) a little bit more than 0.63*10*1uF E) 0
Question (harging apacitor) 10 What is the charge on the capacitor 1second after the switch is closed? 1 = 1uF 1 = 1.2MW Based on what we now so far, what do you think the current through the resistor will be at t = 1sec? A) a little bit less than 0.37*10/1.2MW B) a little bit more than 0.37*10/1.2MW ) a little bit less than 0.63*10/1.2MW D) a little bit more than 0.63*10/1.2MW E) 0
Example 11.2 (harging apacitor) 1 = 1uF 10 1 = 1.2MW What is the charge on the capacitor 10 seconds after the switch is closed? What is the current through the resistor 10 seconds after the switch is closed? onceptual Idea: Use charging equations for a capactor. Plan: Find time constant, t Find Q f and I o Put in t=1sec
Question (harging apacitor) 10 What is the charge on the capacitor 10 second after the switch is closed? 1 = 1uF 1 = 1.2MW Based on what we now so far, what do you think the charge on the capacitor will be at t = 10sec? A) a little bit less than 10*1uF B) a little bit more than 10*1uF ) a little bit less than 0.63*10*1uF D) a little bit more than 0.63*10*1uF E) ~0
Question (harging apacitor) 10 What is the current into the capacitor 10 second after the switch is closed? 1 = 1uF 1 = 1.2MW Based on what we now so far, what do you think the current into the capacitor will be at t = 10sec? A) a little bit less than 10/1.2MW B) a little bit more than 10/1.2MW ) a little bit less than 0.63*10/1.2MW D) a little bit more than 0.63*10/1.2MW E) ~0
10 1 1 Question In the circuit to the left, after the switch is closed, the capacitor takes a certain amount of time t 1 to reach 63% of its full charge Q 1. What if the capacitor had been twice as big ( 2 =2* 1 ). What could you say about the final full charge and the time it will take to reach 63% of the maximum level? A. The final charge will be greater and the charging time will be greater. B. The final charge will be the same and the charging time will be greater.. The final charge will be less and the charging time will be greater. D. The final charge will be the greater and the charging time will be less. E. The final charge will be less and the charging time will be less.
Question 10 1 1 In the circuit to the left, after the switch is closed, the capacitor takes a certain amount of time t 1 to reach 63% of its full charge Q 1. What if the resistor had been twice as big ( 2 =2* 1 ). What could you say about the final full charge and the time it will take to reach 63% of the maximum level? A. The final charge will be greater and the charging time will be greater. B. The final charge will be the same and the charging time will be greater.. The final charge will be less and the charging time will be greater. D. The final charge will be the greater and the charging time will be less. E. The final charge will be less and the charging time will be less.
ircuit (Discharging) apacitor has q 0, Switch is moved to position b Kirchoff s oltage ule q I 0 Differential Equation q dq dt dt dq Q 0 a battery battery b I q( t) q 0 e t / I( t) I 0 e t / Unit 11, Slide 16
Example 11.3 (Discharging apacitor) 1 = 1uF 10 2 1 1 = 1.2MW The switch is held in position 1 for a long time and the capacitor becomes fully charged. It is then flipped to position 2. What is the charge on the capacitor 2 seconds after the switch is flipped? onceptual Idea: Use discharging equations for a capacitor. Plan: Find time constant, t Find Q o Put in t=2sec
Example 11.4 (A hallenge) S 1 2 3 In this circuit, assume,, and i are known. initially uncharged and then switch S is closed. What is t c, the charging time constant? What is the charge on the capacitor at any time t? There s going to be A LOT of algebra. Let s find some: Limiting ases: Unit 11, Slide 18
Question I 1 S 1 2 3 In this circuit, initially uncharged and then switch S is closed. Immediately after S is closed, what is I 1, the current through 1? ( 2 1 1 23 3 1 2 3 1 2 2 1 2 3 A B D E 1 3 ) 3 1 3 Why? Draw circuit just after S closed (knowing 0) 1 is in series with the parallel combination of 2 and 3 S 1 2 0 3 Unit 11, Slide 19
Question S 1 2 3 In this circuit, initially uncharged and then switch S is closed. After S has been closed for a long time, what is I, the current through? Why? 2 1 0 A B After a long time in a static circuit, the current through any capacitor approaches 0! This means we edraw circuit with open circuit in middle leg I 1 I 0 3 Unit 11, Slide 20
Question S 1 2 3 In this circuit, initially uncharged and then switch S is closed. After S has been closed for a long time, what is, the voltage across? 3 2 1 3 1 2 2 3 1 0 2 3 A B D E Why? 3 I 3 (/( 1 3 )) 3 1 3 I 2 I Unit 11, Slide 21
Example 11.4 (A hallenge) S 1 2 I 1 I 2 I got: t c 1 3 3 In this circuit, assume,, and i are known. initially uncharged and then switch S is closed. - What is t c, the charging time constant? - What is the charge on the capacitor at any time t? onceptual Idea: Use Kirchhoff s Laws. Get equation that relates Q and dq/dt on capacitor. Plan: Use loops to find I 1 and I 2 in terms of Q,, and s. Note dq/dt = I 1 I 2 Separate variables and integrate to get Q(t) Take derivative to find I heck the limiting cases we just determined 1 2 3 1 3 2 Q 1 (1 e 3 t / t 3 ) Unit 11, Slide 22
Prediction for Lab Bulb 2 S Bulb 1 What will happen after I close the switch? A) Both bulbs come on and stay on. B) Both bulbs come on but then bulb 2 fades out. ) Both bulbs come on but then bulb 1 fades out. D) Both bulbs come on and then both fade out. Unit 11, Slide 23
Prediction for Lab Bulb 2 S Bulb 1 Suppose the switch has been closed a long time. Now what will happen after we open the switch? A) Both bulbs come on and stay on. B) Both bulbs come on but then bulb 2 fades out. ) Both bulbs come on but then bulb 1 fades out. D) Both bulbs come on and then both fade out. Unit 11, Slide 24
How do Exponentials Work? Q( t) 0 Q e t Fraction of initial charge that remains Q( t) Q 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 How many time constants worth of time that have elapsed 0 0 1 2 3 4 5 6 7 8 9 10 t Unit 11, Slide 25
Q( t) Q 0 1 0.9 0.8 Q( t) 0 Q e t 0.7 0.6 0.5 0.4 2 Time constant: t The bigger t is, the longer it takes to get the same change 0.3 0.2 0.1 0 1 0 1 2 3 4 5 6 7 8 9 10 t Unit 11, Slide 26
Question The two circuits shown below contain identical capacitors that hold the same charge at t=0. Which of the following statements best describes the charge remaining on the two capacitors for any time after t=0? A. Q 1 < Q 2 B. Q 1 > Q 2. Q 1 = Q 2 D. Q 1 < Q 2 at first but then Q 1 > Q 2 after long time E. Q 1 < Q 2 at first but then Q 1 > Q 2 after long time Unit 11, Slide 27