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Transcript
1 DC Circuits 2 3 26-1 EMF and Terminal Voltage • To have a current in an electric circuit, we need a device such as a battery or an electric generator that transforms one type of energy (chemical, mechanical, light, etc.) into electric energy. • Such a device is called a source of electromagnetic force or emf. • The potential difference between the terminals of such a source, when no current flows to an external circuit, is called the emf (e ) the source. 4 Terminal Voltage • r: internal resistance in a battery • Terminal voltage: Vab = Va – Vb • When no current is drawn from the battery, the terminal voltage equals the emf, which is determined by the chemical reactions in the battery: Vab = E. • However, when a current I flows from the battery there is an internal drop in voltage equal to Ir. • Thus, the terminal voltage (the actual voltage delivered) is Vab = e – Ir 5 6 Example 26-1 Battery with internal resistance. A 65.0-W resistor is connected to the terminals of a battery whose emf is 12.0 V and whose internal resistance is 0.5 W. Calculate (a) the current in the circuit, (b) the terminal voltage of the battery, Vab, and (c) the power dissipated in the resistor R and the battery’s internal resistance r. 7 Terminals Battery + _ 8 26-2 Resistors in Series and in Parallel 9 Definition of Resistors in Series Connected resistances are said to be in series when a potential difference that is applied across their combination is the sum of the resulting potential differences across all resistances. 10 Resistors in Series The same current runs through each resistor. Since each resistor may be different, each can have a different potential across it. 11 If one bulb burns out (i.e., the circuit is broken) all bulbs cease to function. They get no current. 12 Resistors in Series Req = R1 + R2 + R3 13 Definition of Resistors in Parallel Connected resistors are said to be in parallel when a potential difference that is applied across their combination results in that same potential difference across each resistance. 14 Resistors in Parallel The current splits at a junction. Therefore, I = I 1 + I2 + I3 . Each resistor is connected directly to the terminals of the battery. Therefore, each has the same potential difference. 15 If one bulb burns out (i.e. the circuit is broken) the other devices keep operating. They still get current. 16 Resistors in Parallel 1 1 1 1 = + + Req R1 R2 R3 17 Conceptual Example 26-2 Series or parallel? (a) The light bulbs in the figure are identical and have a resistance R. Which configuration produces more light? (b) Which way do you think the headlights in a car are wired? 18 19 Example 26-3 Series and parallel resistors. Two 100-W resistors are connected (a) in series, and (b) in parallel, to a 24.0-V battery. What is the current through each resistor and what is the equivalent resistance of each circuit? 20 21 Example 26-4 Circuit with series and parallel. How much current flows from the battery shown? 22 23 Example 26-5 Current in one branch. How much current is flowing through the 500-W resistor shown? 24 25 Conceptual Example 26-6 Bulb brightness in a circuit. The circuit shown has three identical light bulbs, each of resistance R. How will the brightness of bulbs A and B compare with that of bulb C? 26 27 IC = IA + IB IA IC IB 28 Example 26-7 Resistor “ladder.” Estimate the equivalent resistance of the “ladder” of 100-W resistors shown in figure (a). In other words, what resistance would an ohmmeter read if it were connected between points A and B? (b) What is the current through each of the three resistors on the left if a 50.0 V battery is connected between points A and B? 29 Series (a) 30 Parallel 31 Series 32 Parallel 33 26-3 Kirchhoff’s Rules • Kirchhoff’s first or junction rule: At any junction point, the sum of all currents entering the junction must equal the sum of all currents leaving the junction. • Kirchhoff’s second or loop rule: The sum of the changes in potential around any closed path of a circuit must be zero. 34 The potential (voltage) is said to drop going across a resistor in the direction of current. The charges lose energy in the form of heat (P = I2R). + Vba = - IR _ + Vba (fi) 35 I = .017 A The algebraic sum of the changes in potential must be zero. 36 Example 26-8 Using Kirchhoff’s rules. Calculate the currents I1, I2, and I3 in each of the branches of the circuit shown. 37 Junction rule: I3 = I1 + I2 38 Start Loop 1 Loop 1: a – h – d – c – b – a Loop Rule 39 Loop 2: a – h – d – e – f – g - a Start Loop 2 40 Solving Problems with Kirchhoff’s Rules Kirchhoff’s Rules 1. Label + and – for each battery. The long side of a battery symbol is +. 2. Label the current in each branch of the circuit with a symbol and an arrow. The direction of the arrow can be chosen arbitrarily. If the current is actually in the opposite direction, it will come out minus in the solution. 3. Apply Kirchhoff’s junction rule at each junction, and the loop rule for one or more loops. 41 Problem Solving 4. In applying the loop rule, follow each loop in one direction only. Pay careful attention to signs. (a) For a resistor, the sign of the potential difference is negative if your chosen loop direction is the same as the chosen current direction through that resistor and vice versa. (b) For a battery, the sign of the potential difference is positive if your loop moves from the negative terminal toward the positive and vice versa. 5. Solve the equations algebraically for the unknowns. 42 Example 26-9 Wheatstone bridge. A Wheatstone bridge is a type of “bridge circuit” used to make measurements of resistance. The unknown resistance to be measured, Rx, is placed in a circuit with accurately known resistances R1, R2, and R3. One of these, R3, is a variable resistor which is adjusted so that when the switch is closed momentarily, the ammeter A shows zero current flow. (a) Determine Rx in terms of R1, R2, and R3. (b) If a Wheatstone bridge is “balanced” when R1 = 630 W, R2 = 972 W, and R3 = 42.6 W, what is the value of the unknown resistance? 43 R3 is adjusted so that ammeter reads 0 amps when switch is closed. Therefore, the potential at B = potential at D. Therefore,VAB = VAD 44 EMF’s in Series and in Parallel; Charging a Battery Vca = 1.5 V + 1.5 V = 3.0V 45 EMF’s in Series and in Parallel; Charging a Battery Vca = 20 V – 12 V = 8.0 V 46 EMF’s in Series and in Parallel; Charging a Battery •Provides more energy when large currents are needed. •Each of the batteries in parallel produce one half the total current, so the loss due to internal resistance is less and the batteries will last longer. V = e - Ir 47 Example 26-10 Jump starting a car. A good car battery is being used to jump start a car with a weak battery. The good battery has an emf of 12.5 V and an internal resistance of 0.020 W. Suppose the weak battery has an emf of 10.1 V and an internal resistance of 0.10 W. Each copper jumper cable is 3.0 m long and 0.50 cm in a diameter, and can be attached as shown. Assume the starter motor can be represented as a resistor RS = 0.15 W. Determine the current through the motor (a) if only the weak battery is connected to it and (b) if the good battery is also connected. 48 (a) I1 = I3 = I Rs = 0.15 W 49 (b) Junction 50 Example 26-11 Jumper cables reversed. What would happen if the jumper cables of Example 26-10 were mistakenly connected in reverse, the positive terminal of each battery connected to the negative terminal of the other battery? Why could this be dangerous? 51 RJ = 0.0026 W 52 26-4 Circuits Containing a Resistor and a Capacitor (RC Circuits) • Until now the circuits have had a steady current. That is, one that does not change in time. • Add a capacitor. Qmax • When circuit is switched on, current flows to capacitor until it is fully charged. That is, Q = Q0 • When fully charged Vc across capacitor = e of battery. • Open switch. Capacitor discharges. That is, Q = 0. 53 Charging a capacitor Switch is closed at t = 0. e 54 Charging an RC Circuit Vc = V0 (1-e-t/RC) V0 = e Qc = Q0 (1-e-t/RC) Q0 = Ce I = I0 e-t/RC I0 e = R 55 t = RC (time constant) 56 Discharging a capacitor Switch closes at t = 0, taking battering out of circuit. e 57 Discharging an RC Circuit Vc = V0 e-t/RC V0 = e Qc = Q0 e-t/RC Q0 = Ce I = I0 e-t/RC I0 e = R 58 59 Example 26-12 RC circuit, with emf. (charging a capacitor) The capacitance in the previous circuit is C = 0.30 mF, the total resistance is 20kW, and the battery emf is 12 V. Determine (a) the time constant, (b) the maximum charge the capacitor could acquire, (c) the time it takes for the charge to reach 99 percent of this value, (d) the current I when the charge Q is half its maximum value, (e) the maximum current, and (f ) the charge Q when, the current I is 0.20 its maximum value. 60 61 62 Example 26-13 Discharging and RC circuit. In the RC circuit shown, the battery has fully charged the capacitor, so Q0 = CE. Then at t = 0 the switch is thrown from position a to b. The battery emf is 20.0 V, and the capacitance C = 1.02 mF. The current I is observed to decrease to 0.50 of its initial value in 40 ms. (a) What is the value of R? (b) What is the value of Q, the charge on the capacitor, at t = 0? (c) What is Q at t = 60 ms? 63 64 65 66 Homework Problem 1 Calculate the terminal voltage for a battery with an internal resistance of 0.900 W and an emf of 8.50 V when the battery is connected in series with (a) a 68.0-W resistor, and (b) a 680-W resistor. 67 Homework Problem 4 What is the internal resistance of a 12-V car battery whose terminal voltage drops to 9.8 V when the starter draws 60 A? What is the resistance of the starter? 68 Homework Problem 8 Suppose that you have a 500-W, a 900-W, and a 1400 W resistor. What is (a) the maximum, and (b) minimum resistance you can obtain by combining these? 69 Homework Problem 10 Three 1.20-kW resistors can be connected together in four different combinations of series and/or parallel circuits. What are the four ways and what is the net resistance in each case? 70 Homework Problem 19 Consider the network of resistors shown. Answer qualitatively: (a) What happens to the voltage across each resistor when the switch S is closed? (b) What happens to the current through each when the switch is closed? (c) What happens to the output of the battery when the switch is closed? (d) Let R1 = R2 = R3 = R4 = 100 W and V = 45.0 V. Determine the current through each resistor before and after closing the switch. Are your qualitative predictions confirmed? 71 72 73 Homework Problem 25 Determine the magnitudes and directions of the currents through R1 and R2 in the figure. 74 75 Homework Problem 29 Determine the current through each of the resistors in the figure. 76 77 Homework Problem 42 The RC switch in the figure has R = 6.7 kW and C = 6.0 mF. The capacitor has a voltage V0 at t = 0, when the switch is closed. How long does it take the capacitor to discharge to 1.0 percent of its initial voltage? 78 79 Homework Problem 43 How long does it take for the energy stored in a capacitor in a series RC circuit, shown in the figure, to reach half its maximum value? Express your answer in terms of the time constant t = RC. 80 - 81 *26-5 DC Ammeters and Voltmeters • An ammeter is used to measure current. • A voltmeter is used to measure voltage. 82 83 84 85 86 87 *26-6 Transducers and the Thermocouple • A transducer is a device that converts one type of energy into another. • A high-fidelity loudspeaker is one kind of transducer—it transforms electric energy into sound energy. 88 89 90 91