# DC Circuits: Ch 19. Resistors in Series 6/1/2016

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1 DC Circuits: Ch 19 Voltage Starts out at highest point at + end of battery Voltage drops across lightbulbs and other sources of resistance. Voltage increases again at battery. I The following circuit uses a 1.5 V battery and has a 15 W lightbulb. a. Calculate the current in the circuit b. Calculate the voltage drop across the lightbulb. c. Sketch a graph of voltage vs. path (battery, top wire, resistor, bottom wire) + - Voltage highest Voltage zero Resistors in Series Same current (I) passes through all resistors (bulbs) All bulbs are equally bright (energy loss, not current loss) Voltage drop across each resistor (V 1,V 2, V 3 ) 1

2 V = V 1 + V 2 + V 3 V = IR 1 + IR 2 + IR 3 V = I(R 1 + R 2 + R 3 ) R eq = R 1 + R 2 + R 3 V = IR eq Resistors in Parallel Current splits at the junction Same Voltage across all resistors I = I 1 + I 2 + I 3 I 1 = V R 1 I = V R eq Calculate the equivalent resistance of two 200W resistors placed in: a. Series b. Parallel 1 = R eq R 1 R 2 R 3 2

3 Calculate the equivalent resistance for the circuit below. Which combination of auto headlights will produce the brightest bulbs? Assume all bulbs have a resistance of R. For the Bulbs in Series: R eq = R + R = 2R For the Bulbs in Parallel 1 = R eq R R 1 = 2 R eq R R eq = R/2 The bulbs in parallel have less resistance and will be brighter What current flows through each resistor in the following circuit? (R = 100 W) A Calculate the current through this circuit, and the voltage drop across each resistor. V ab = (0.0174A)(400W) V ab = 6.96 V V bc = (0.0174A)(290W) V bc = 5.04 V R eq = 690 W I = A 3

4 What current flows through each of the resistors in this circuit? (Both are 100 W) What current will flow through the circuit shown? (I =0.48 A, I 1 =I 2 =0.24 A) R eq = 400 W W R eq = 690 W V = IR I = V/R I = 12.0 V/690 W I = A or 17 ma What current is flowing through just the 500 W resistor? First we find the voltage drop across the first resistor: V = IR = (0.017 A)(400 W) V = 6.8 V The voltage through the resistors in parallel will be: 12.0 V 6.8 V = 5.2 V Calculate the equivalent resistance in the following circuit. To find the current across the 500 W resistor: V = IR I = V/R I = 5.2 V/500 W = A = 10 ma 4

5 Which bulb will be the brightest in this arrangement (most current)? Bulb C (current gets split running through A and B) What happens when the switch is opened? C and B will have the same brightness (I is constant in a series circuit) What resistance would be present between points A and B? (ANS: 41/15 R) EMF and Terminal Voltage Batteries - source of emf (Electromotive Force), E (battery rating) All batteries have some internal resistance r A 12-V battery has an internal resistance of 0.1 W. If 10 Amps flow from the battery, what is the terminal voltage? V ab = E Ir V ab = terminal(useful)voltage E = battery rating r = internal resistance V ab = 11 V 5

6 Calculate the current in the following circuit. R eq = 6 W W R eq = 8.7 W 1/R eq = 1/10 W + 1/8.7 W R eq = 4.8 W Everything is now in series R eq = 4.8 W W W R eq = 10.3 W V = IR I = V/R I = 9.0 V/10.3 W I = 0.87 A Now calculate the terminal(useful)voltage. V = E Ir V = 9.0 V (0.87 A)(0.50 W) V = 8.6 V Grounded Wire is run to the ground Houses have a ground wire at main circuit box Does not affect circuit behavior normally Provides path for electricity to flow in emergency 6

7 Kirchhoff s Rules 1. Junction Rule - The sum of the currents entering a junction must equal the sum of currents leaving 2. Loop Rule - The sum of the changes in potential around any closed path = 0 Kirchhoff Conventions The loop current is not a current. Just a direction that you follow around the loop. Kirchhoff Conventions Kirchhoff s Rule Ex 1 I 1 = I 2 + I 3 Junction Rule 7

8 Loop Rule Main Loop 6V (I 1 )(4W) (I 3 )(9W) = 0 Side Loop (-I 2 )(5W) + (I 3 )(9W) = 0 I 1 = I 2 + I 3 Eqn 3 6V (I 1 )(4W) (I 3 )(9W) = 0 Eqn 2 (-I 2 )(5W) + (I 3 )(9W) = 0 Eqn 3 Solve Eqn 1 (-I 2 )(5W) + (I 3 )(9W) = 0 (I 3 )(9W) = (I 2 )(5W) I 3 = 5/9 I 2 Substitution into Eqn 2 6V (I 2 + I 3 )(4W) (I 3 )(9W) = 0 6 4I 2-4I 3-9I 3 = 0 6 4I 2-13I 3 = 0 I 3 = 5/9 I 2 (from last slide) 6 4I 2-13(5/9 I 2 ) = 0 6 = 101/9 I 2 I 2 = 0.53 A I 3 = 5/9 I 2 = 0.29 A I 1 = I 2 + I 3 = 0.53 A A = 0.82 A Kirchhoff s Rule Ex 2 I 1 = I 2 + I 3 8

9 Loop Rule Main Loop 9V (I 3 )(10W) (I 1 )(5W) = 0 Side Loop (-I 2 )(5W) + (I 3 )(10W) = 0 (-I 2 )(5W) + (I 3 )(10W) = 0 (I 2 )(5W) = (I 3 )(10W) I 2 = 2I 3 I 3 = 9/25 = 0.36 A I 2 = 2I 3 = 2(0.36 A) = 0.72 A I 1 = I 2 + I 3 = 0.36A A = 1.08 A 9V (I 3 )(10W) (I 1 )(5W) = 0 9V (I 3 )(10W) (I 2 + I 3 )(5W) = I 3 5I 2 5I 3 = I 3 5I 2 = I 3 5(2I 3 ) = I 3 = 0 I 3 = 9/25 = 0.36 A Calculate all the currents in the following circuit. A = A B = 0.26 A C = A Calculate the currents in the following circuit. 2 Amps, 3 Amps, -1 Amps 9

10 Bottom Loop (clockwise) 10V (6W)I 1 (2W)I 3 = 0 Top Loop (clockwise) -14V +(6W)I 1 10 V -(4W)I 2 = 0 Work with Bottom Loop 10V (6W)I 1 (2W)I 3 = 0 I 1 + I 2 = I I 1 2(I 1 + I 2 ) = I 1 2I 1-2I 2 = I 1-2I 2 = 0 10 = 8I 1 + 2I 2 5 = 4I 1 + I 2 I 2 = 5-4I 1 Working with Top Loop -14V +(6W)I 1 10 V -(4W)I 2 = 0 24 = 6I 1-4I 2 12 = 3I 1-2I 2 12 = 3I 1-2(5-4I 1 ) 22 = 11I 1 I 1 = 22/11 = 2.0 Amps I 2 = 5-4I 1 I 2 = 5 4(2) = -3.0 Amps I 1 + I 2 = I 3 I 3 = -1.0 A Calculate all the currents in the following circuit. A = 1.56A B = A C = 1.04 A Batteries in Series If + to -, voltages add (top drawing) If + to +, voltages subtract (middle drawing = 8V, used to charge the 12V battery as in a car engine) I 1 = A I 2 = 2.6 A I 3 = 1.73 A Extra Kirchhoff Problems Batteries in Parallel Provide large current when needed (Same voltage) 10

11 Calculate I (0.5 Amps) a. Calculate the equivalent resistance. (2.26 W) b. Calculate the current in the upper and lower wires. (3.98 A) c. Calculate I 1, I 2, and I 3 (0.60 A, 2.25 A, 1.13 A) d. Sketch a graph showing the voltage through the circuit starting at the battery. Capacitors in Parallel I 1 = A I 2 = A 1 3 = 0.50 A Voltage is constant in parallel Q = CV Q = C 1 V + C 2 V Q = C eq V C eq = C 1 + C 2 + C 3 + C 1 C 2 (like using a larger plate) I 1 = A I 2 = A 1 3 = 0.50 A Capacitors in Series Charge is constant in series 1 = C eq C 1 C 2 C 3 Eq. Capacitance Ex. 1 Determine the equivalent capacitance for the circuit below if each capacitor is 5 mf. C 1 C 3 C 1 C 2 C 3 C 2 +Q -Q +Q -Q +Q -Q 11

12 Parallel first: Given the following setup: C eq = C 1 + C 2 C eq = 5mF + 5mF = 10 mf Series: 1 = = 3/10 C eq 10 5 Ceq = 3.33 mf a. Calculate the charge on each capacitor (52.8 mc, 28.8 mc) Given the circuit below: a. Calculate the equivalent capacitance b. Calculate the charge across each capacitor c. Calculate the voltage on each capacitor (HINT: Combine them for part a, and then split them back up for b and c) Parallel capacitors C eq = 5 mf +1 mf = 6 mf Now both are series 1/ C eq = 1/6 + 1/3 C eq = 2 mf Q = CV = (2 mf)(12 V) = 24 mc For 3 mf capacitor Q = CV V = Q/C = 24 mc/3 mf = 8 V For Combined Parallel Capacitors (same voltage) V = Q/C = 24 mc/6 mf = 4 V For 5 mf Capacitor Q = CV = (5 mf)(4 V) = 20 mc For 1 mf Capacitor Q = CV = (1 mf)(4 V) = 4 mc 12

13 Given the following setup: a. Calculate the equivalent capacitance. (2.0 mf) b. Calculate the total charge that leaves the battery. (8.0 mc) c. Calculate the charge and voltage on each capacitor. (Q 1 = 8.0 mc, 2.7 V: Q 2 = Q 3 = 4.0 mc, 1.3 V) Given the following setup: a. Calculate the equivalent capacitance. (0.609 mf) b. Calculate the total charge that leaves the battery. (14.6 mc) c. Calculate the charge and voltage on each capacitor. (Q 1 = 14.6 mc, 14.6 V, Q 2 = 14.6 mc, 7.30 V, Q 3 = 6.26 mc, 2.09 V, Q 4 = 8.36 mc, 2.09 V ) a. Calculate the equivalent capacitance. (18 mf) b. Calculate the total charge that leaves the battery. (162 mc) c. Calculate the charge and voltage on each capacitor. (Q 1 = 45 mc, 9 V, Q 2 = 90 mc, 9 V, Q 3 = 27 mc, 9 V) a. Equivalent capacitance. (1.58 mf) b. Calculate the total charge that leaves the battery. (14.2 mc) c. Calculate the charge and voltage on each capacitor. (Q 1 = 14.2 mc, 2.84 V, Q 2 = 14.2 mc, 1.42 V, Q 3 = 14.2 mc, 4.73 V) a. Equivalent capacitance. (7.31 mf) b. Total charge that leaves the battery. (43.8 mc) c. Calculate the charge and voltage on each capacitor. (Q 1 = 30 mc, 6 V, Q 2 = 13.8 mc, 1.38 V, Q 3 = 13.8 mc, 4.6 V) a. Equivalent capacitance. (6.41 mf) b. Calculate the total charge that leaves the battery. (22.4 mc) c. Calculate the charge and voltage on each capacitor. (Q 1 = 22.4 mc, 2 V, Q 2 = 15 mc, 1.5 V, Q 3 = 7.5 mc, 1.5 V) 13

14 RC Circuits Capacitors store energy (flash in a camera) Resistors control how fast that energy is released Car lights that dim after you shut them off Camera flashes Difibralator DV c + DV r = 0 Q - IR = 0 (Divide by R) C Q - I = 0 RC (I = -dq/dt) Q + dq = 0 RC dt = time constant (time to reach 63% of full voltage) = RC A versatile relationship Charging the Capacitor (for capacitor) V = V o (1-e -t/rc ) Q = Q o (1-e -t/rc ) I = I o e -t/rc Discharging the Capacitor V = V o e -t/rc Q = Q o e -t/rc I = I o e -t/rc RC Circuits: Ex 1 What is the time constant for an RC circuit of resistance 200 kw and capacitance of 3.0 mf? = (200,000 W)(3.0 X 10-6 F) = 0.60 s (lower resistance will cause the capacitor to charge more quickly) 14

15 RC Circuits: Ex 2 What will happen to the bulb (resistor) in the circuit below when the switch is closed (like a car door)? Answer: Bulb will glow brightly initially, then dim as capacitor nears full charge. RC Circuits: Ex 3 An uncharged RC circuit has a 12 V battery, a 5.0 mf capacitor and a 800 kw resistor. Calculate the time constant. = RC = (5.0 X 10-6 F)(800,000 W) = 4.0 s What is the maximum charge on the capacitor? Q = CV Q = (5.0 X 10-6 F)(12 V) Q = 6 X 10-5 C or 60 mc What is the voltage and charge on the capacitor after 1 time constant (when discharging)? Calculate: a. Time constant (6 ms) b. Maximum charge on the capacitor (3.6 mc) c. Maximum current (0.60 ma) d. Time to reach 99% of maximum charge (28 ms) e. Current when charge = ½ Q max (0.300 ma) f. The charge when the current is 20% of the maximum value. (2.9 mc) V = V o e -t/rc Discharging the RC Circuit 15

16 An RC circuit has a charged capacitor C = 35 mf and a resistance of 120W. How much time will elapse until the voltage falls to 10 percent of its original (maximum) value? V = V o e -t/rc 0.10V o =V o e -t/rc 0.10 =e -t/rc ln(0.10) = ln(e -t/rc ) -2.3 = -t/rc 2.3 = t/rc t = 2.3RC t = (2.3)(120W)(35 X 10-6 F) t = s or 9.7 ms If a capacitor is discharged in an RC circuit, how many time constants will it take the voltage to drop to ¼ its maximum value? A fully charged 1.02 mf capacitor is in a circuit with a 20.0 V battery and a resistor. When discharged, the current is observed to decrease to 50% of it s initial value in 40 ms. a. Calculate the charge on the capacitor at t=0 (20.4 mc) b. Calculate the resistance R (57 W) c. Calculate the charge at t = 60 ms (7.3 mc) (t = 1.39RC) The capacitor in the drawing has been fully charged. The switch is quickly moved to position b (camera flash). a. Calculate the initial charge on the capacitor. (9 mc) b. Calculate the charge on the capacitor after 5.0 ms while discharging. (5.5 mc) c. Calculate the voltage after 5.0 ms while discharging. (5.5 V) d. Calculate the current through the resistor after 5.0 ms while discharging. (0.55 A) Meters Galvanometer Can only handle a small current Full-scale Current Sensitivity (I m ) Maximum deflection Ex: Multimeter Car speedometer 16

17 Measuring I and V Measuring Current Anmeter is placed in series Current is constant in series Anmeter (Series) Voltmeter (parallel) Measuring Voltage Voltmeter placed in parallel Voltage constant in parallel circuits Measuring voltage drop across a resistor DC Anmeter Uses Shunt (parallel) resistor Shunt resistor has low resistance Most of current flows through shunt, only a little through Galvanometer I R R = I G r Meters: Ex 1 What size shunt resistor should be used if a galvanometer has a full-scale sensitivity of 50 ma and a resistance of r= 30 W? You want the meter to read a 1.0 A current. Voltage same through both (V=IR) I R R = I g r Since most of the current goes through the shunt I R ~ 1 A I R R = I g r (1 A)(R) = (50 X 10-6 A)(30 W) R = 1.5 X 10-3 W or 1.5 m W Meters: Ex 2 Design an anmeter that can test a 12 A vacuum cleaner if the galvanometer has an internal resistance of 50 W and a full scale deflection of 1 ma. I R R = I g r (12 A)(R) = (1 X 10-3 A)(50 W) R = 4.2 X 10-3 W or 4.2 m W 17

18 DC Voltmeter Resistor in series Large R for resistor (keeps current low in Galvanometer) V = I(R + r) Meters: Ex 3 What resistor should be used in a voltmeter that can read a maximum of 15 V? The galvanometer has an internal resistance of 30W and a full scale deflection of 50 ma. V = I(R + r) 15 V = (50 X 10-6 A)(R + 30W) R + 30W = 15 V 50 X 10-6 A R + 30W = 300,000 W R ~ 300,000 W Meters: Ex 4 Design a voltmeter for a 120 V appliance with and internal galvanometer resistance of 50 W and a current sensitivity of 1 ma. (ANS: R = 120,000 W) 2. Clockwise (12 V. 50 W, (10 mf and 100 W parallel)) 4. a) 0.10 A b) Graph page a) 5 V, 10 V b) Graph page W, 2.9 W X 10 6 J W % 20. R/ W W W V, 5 V, 0 V, -2 V ms ms K W 36. A> D = E > B = C m W W W and 60 V V A, 2.0 A, 15 V 56.a) 8 Amps (8 V ab ) b) 9 Amps (0 V ab ) 58. a) R =0.505 W b) R eq = W 62.Resistor(W) (V) Current (A) (bottom) (right)

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20 I 2 = A I 1 = 0.54 A I 3 = A 20

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