# Circuits. The light bulbs in the circuits below are identical. Which configuration produces more light? (a) circuit I (b) circuit II (c) both the same

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1 Circuits The light bulbs in the circuits below are identical. Which configuration produces more light? (a) circuit I (b) circuit II (c) both the same Circuit II has ½ current of each branch of circuit I, so each bulb is ¼ as bright. The total power in circuit I is thus 4x that of circuit II. PHY2049: Chapter 27 31

2 Circuits The three light bulbs in the circuit are identical. What is the brightness of bulb B compared to bulb A? a) 4 times as much b) twice as much c) the same d) half as much e) 1/4 as much Use P = I 2 R. Thus 2x current in A means it is 4x brighter. PHY2049: Chapter 27 32

3 Circuit Problem (1) The light bulbs in the circuit are identical. What happens when the switch is closed? a) both bulbs go out b) the intensity of both bulbs increases c) the intensity of both bulbs decreases d) nothing changes Before switch closed: V a = 12V because of battery. V b =12 because equal resistance divides 24V in half. (a) (b) After switch closed: Nothing changes since (a) and (b) are still at same potential. PHY2049: Chapter 27 33

4 Circuit Problem (2) The light bulbs in the circuit shown below are identical. When the switch is closed, what happens to the intensity of the light bulbs? a) bulb A increases b) bulb A decreases c) bulb B increases d) bulb B decreases e) nothing changes (a) (b) Before switch closed: V a = 12V because of battery. V b =12 because equal resistance divides 24V in half. After switch closed: Nothing changes since (a) and (b) are still at same potential. PHY2049: Chapter 27 34

5 Circuit Problem (3) The bulbs A and B have the same R. What happens when the switch is closed? a) nothing happens b) A gets brighter, B dimmer c) B gets brighter, A dimmer d) both go out (a) (b) Before: V a = 24, V b = 18. Bulb A and bulb B both have 18V across them. 24V After: V a = 24, V b = 24 (forced by the batteries). Bulb A has 12V across it and bulb B has 24V across it. PHY2049: Chapter 27 35

6 Kirchhoff s Rules Junction rule (conservation of charge) Current into junction = sum of currents out of it I I 1 I 2 I = I1+ I2 + I3 Loop rule (conservation of energy) Algebraic sum of voltages around a closed loop is 0 I I3R1 I2R2 E2 I3R1+ E2 = 0 I1R1 E1 I1R1+ E2 + I2R2 = 0 PHY2049: Chapter 27 36

7 Problem Solving Using Kirchhoff s Rules Label the current in each branch of the circuit Choice of direction is arbitrary Signs will work out in the end (if you are careful!!) Apply the junction rule at each junction Keep track of sign of currents entering and leaving Apply loop rule to each loop (follow in one direction only) Resistors: if loop direction matches current direction, voltage drop Batteries: if loop direction goes through battery in normal direction, voltage gain Solve equations simultaneously You need as many equations as you have unknowns PHY2049: Chapter 27 37

8 Kirchhoff s rules Determine the magnitudes and directions of the currents through the two resistors in the figure below. Take two loops, 1 and 2, as shown Use I 1 = I 2 + I I = 0 22I I = 0 I I = 6/15 = 0.40 = 15/ 22 = 0.68 I = I + I = PHY2049: Chapter 27 38

9 Circuits Which of the equations is valid for the circuit shown below? a) 2 I 1 2I 2 = 0 b) 2 2I 1 2I 2 4I 3 = 0 c) 2 I 1 4 2I 2 = 0 d) I 3 2I 2 4I 3 = 0 e) 2 2I 1 2I 2 4I 3 = 0 1 2V 4V 6V I 1 I2 I 3 PHY2049: Chapter 27 39

10 Wheatstone Bridge An ammeter A is connected between points a and b in the circuit below. What is the current through the ammeter? a) I / 2 b) I / 4 c) zero d) need more information Before ammeter is added: The top branch divides the voltage evenly, so V a = V/2. The bottom branch also divides the voltage evenly, so V b = V/2. Thus V a = V b and current is 0 across ammeter. V PHY2049: Chapter 27 40

11 Wheatstone Bridge An ammeter A is connected between points a and b in the circuit below. What is the current through the ammeter? a) I / 2 b) I / 4 c) zero d) need more information Same analysis. Before ammeter is added: The top branch divides the voltage evenly, so V a = V/2. The bottom branch also divides the voltage evenly, so V b = V/2. Thus V a = V b and current is 0 across ammeter. 2R 2R PHY2049: Chapter 27 41

12 Res-Monster Maze (p. 725) All batteries are 4V All resistors are 4Ω Find current in R Hint: follow the batteries PHY2049: Chapter 27 42

13 12V Res-Monster Maze (p. 725) 8V 8V 4V I R = 2 0V 0V All batteries are 4V All resistors are 4Ω Find current in R Hint: follow the batteries PHY2049: Chapter 27 43

14 Problem solving Find the value of R that maximizes power emitted by R. I T 6Ω I 1 I 2 18V 12Ω R T 1 R I R I2 = = = I R R P R I 2 2 = 2 T I R 12R = 12 + R R = = 12R R 12 + R P 144R 2 2 = I2R= 2 ( 4 + R) Maximize R = 4Ω P = 9W 2 PHY2049: Chapter 27 44

15 Light Bulbs A three-way light bulb contains two filaments that can be connected to the 120 V either individually or in parallel. A three-way light bulb can produce 50 W, 100 W or 150 W, at the usual household voltage of 120 V. What are the resistances of the filaments that can give the three wattages quoted above? Use P = V 2 /R R 1 = /50 = 288Ω (50W) R 2 = /100 = 144Ω (100W) PHY2049: Chapter 27 45

16 Problem What is the maximum number of 100 W light bulbs you can connect in parallel in a 100 V circuit without tripping a 20 A circuit breaker? (a) 1 (b) 5 (c) 10 (d) 20 (e) 100 Each bulb draws a current of 1A. Thus only 20 bulbs are allowed before the circuit breaker is tripped. PHY2049: Chapter 27 46

17 RC Circuits Charging a capacitor takes time in a real circuit Resistance allows only a certain amount of current to flow Current takes time to charge a capacitor Assume uncharged capacitor initially Close switch at t = 0 Initial current is (no charge on capacitor) Current flows, charging capacitor Generates capacitor potential of q/c E q/ C i = R Current decreases continuously as capacitor charges! Goes to 0 when fully charged i = E/ R E PHY2049: Chapter 27 47

18 Analysis of RC Circuits Current and charge are related i = dq/ dt So can recast previous equation as differential equation i = E q/ C R General solution is (Check and see!) q = EC+ Ke dq q E + = dt RC R K = EC (necessary to make q = 0 at t = 0) Solve for charge q and current i t/ RC ( t RC 1 ) dq E q= EC e i= = e dt R / t/ RC PHY2049: Chapter 27 48

19 Charge and Current vs Time (For Initially Uncharged Capacitor) ( ) ( / ) 0 1 t RC = qt q e it ie ( ) / = 0 t RC PHY2049: Chapter 27 49

20 Exponential Behavior t = RC is the characteristic time of any RC circuit Only t / RC is meaningful t = RC Current falls to 37% of maximum value Charge rises to 63% of maximum value t =2RC Current falls to 13.5% of maximum value Charge rises to 86.5% of maximum value t =3RC Current falls to 5% of maximum value Charge rises to 95% of maximum value t =5RC Current falls to 0.7% of maximum value Charge rises to 99.3% of maximum value PHY2049: Chapter 27 50

21 Discharging a Capacitor Connect fully charged capacitor to a resistor at t = 0 S C General solution is R dq dt q = Ke t/ RC K = VC (necessary to make have full charge at t = 0) Solve for charge q and current i dq V q= VCe i = = e dt R t/ RC t/ RC q ir = 0 C q + = 0 RC PHY2049: Chapter 27 51

22 Charge and Current vs Time (For Initially Charged Capacitor) qt qe ( ) / = 0 it ie t RC ( ) / = 0 t RC PHY2049: Chapter 27 52

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