# POWER B. Terms: EMF, terminal voltage, internal resistance, load resistance. How to add up resistors in series and parallel: light bulb problems.

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1 iclicker Quiz (1) I have completed at least 50% of the reading and study-guide assignments associated with the lecture, as indicated on the course schedule. A. True B. False Hint: this is a good time to read the pitfalls in the margins of the text. Current is NEVE used up. Today: eview of resistance use video demos and MU. Also POWE & A. New Circuit elements: Batteries. B. Terms: EMF, terminal voltage, internal resistance, load resistance. C. How to add up resistors in series and parallel: light bulb problems.

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3 B. How many of these terms are in this problem? EMF, terminal voltage, internal resistance, load resistance. C. Do we know enough to do part a? pp. A true, B. No D. If so what is the equation which helps us?

4 Circuit Analysis Simple electric circuits may contain batteries, resistors, and capacitors in various combinations. For some circuits, analysis may consist of combining resistors. Today s work In more complex complicated circuits, Kirchhoff s ules may be used for analysis. These ules are based on conservation of energy and conservation of electric charge for isolated systems. Circuits may involve direct current or alternating current. Introduction

5 Direct Current When the current in a circuit has a constant direction, the current is called direct current. Most of the circuits analyzed will be assumed to be in steady state, with constant magnitude and direction. Because the potential difference between the terminals of a battery is constant, the battery produces direct current. The battery is known as a source of emf. Section 28.1

6 EMF Electromotive Force The electromotive force (emf), ε, of a battery is the maximum possible voltage that the battery can provide between its terminals. The emf supplies energy, it does not apply a force.. Section 28.1

7 Batteries as circuit elements A battery or other constant-voltage device (e.g. power supply) is usually the energy source in a direct-current (DC) circuit. The positive terminal of the battery has higher potential than the negative terminal. The electromotive force (emf, ε) of a battery is the opencircuit voltage between its terminals when no current is flowing. Ideally, the battery has no internal resistance of its own. We often idealize the wires in a circuit to have zero resistance. +

8 eal batteries 1. Chemical energy: an electrolyte solution allows negative ions to flow toward and react with the anode ( ), while positive ions flow toward and react with the cathode (+). 2. Charge build-up prevents the reaction from proceeding unless an external circuit allows electrons accumulating at the anode to return to the cathode. 3. Dead when the reactants are used up. echargeable if the anode/cathode reactions are reversible. 4. Internal resistance tends to increase with age, use and multiple recharge cycles.

9 Typical Alkaline (Zn/MnO2) 1.5 to 1.6 V open circuit, 1.1 to 1.3 V closed circuit. 163 W-hr/kg (590 J/g) 400 kj total for AA (122.4 C) to 15 A-hr (54 kc) of charge depending on size. 85% Capacity after 4 years of non-use. Lithium Ion 3.2 V open circuit, 2.5 to 3.0 V closed circuit. 230 W-hr/kg (828 J/g) 460 kj total for AA (576 C) to 1.4 Amp-hrs (5.04 kc) of charge 95% capacity after 5 years of non-use NiCd 1.2 V open circuit 50 W-hr/kg (180 J/g) 140 kj total for AA 70% capacity after one month of non use ( cycles) Zn-Air bus battery: 200 W-hr/kg (720 J/g) 320 kw-hr (1.15 GJ) of energy

10 Internal Battery esistance, r If the internal resistance is zero, the terminal voltage equals the emf. In a real battery, there is internal resistance, r. The terminal voltage, ΔV = e Ir The emf is equivalent to the open-circuit voltage. This is the terminal voltage when no current is in the circuit. This is the voltage labeled on the battery. The actual potential difference between the terminals of the battery depends on the current in the circuit. Section 28.1

11 Load esistance The terminal voltage also equals the voltage across the external resistance. This external resistor is called the load resistance. In the previous circuit, the load resistance is just the external resistor. In general, the load resistance could be any electrical device. These resistances represent loads on the battery since it supplies the energy to operate the device containing the resistance. Section 28.1

12 Internal resistance in non-ideal batteries Terminal voltage: V V ab = ε I r < ε I = Current: ε V = < + r ε Two AA batteries yield a combined ε = 3 V. You observe a terminal voltage of 2.7 V while delivering 300 ma of current to an ultra-bright flashlight. What is the load resistance ()? (1) 1Ω (2) 3Ω (3) 9Ω (4) 10 Ω What is the internal resistance of a single AA battery? (1) 0.5Ω (2) 1Ω (3) 2Ω (4) 3Ω

13 Power: where does it go? The total power output of the battery is P = I ΔV = I ε This power is delivered to the external resistor (I 2 ) and to the internal resistor (I 2 r). P = I 2 + I 2 r The battery is a supply of constant emf. The battery does not supply a constant current since the current in the circuit depends on the resistance connected to the battery. The battery does not supply a constant terminal voltage. Section 28.1

14 Listening quiz A. A battery is a source of constant EMF B. A battery is a source of constant terminal voltage C. A battery is a source of constant current to the load.

15 Consider this problem: Do we know enough to do it? How do you approach it? Are any of the resistors in Series with others? Are any in parallel?

16 esistors in Series When two or more resistors are connected end-toend, they are said to be in series. For a series combination of resistors, the currents are the same in all the resistors because the amount of charge that passes through one resistor must also pass through the other resistors in the same time interval. The potential difference will divide among the resistors such that the sum of the potential differences across the resistors is equal to the total potential difference across the combination. Section 28.2

17 esistors in Series, cont Currents are the same I = I 1 = I 2 Potentials add ΔV = V 1 + V 2 = I 1 + I 2 = I ( ) Consequence of Conservation of Energy The equivalent resistance has the same effect on the circuit as the original combination of resistors. Section 28.2

18 Equivalent esistance Series eq = The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any individual resistance. If one device in the series circuit creates an open circuit, all devices are inoperative. Section 28.2

19 Equivalent esistance Series An Example Are all 3 representations equivalent? A. yes B. no Two resistors are replaced with their equivalent resistance. Section 28.2

20 Some Circuit Notes A local change in one part of a circuit may result in a global change throughout the circuit. For example, changing one resistor will affect the currents and voltages in all the other resistors and the terminal voltage of the battery. In a series circuit, there is one path for the current to take. In a parallel circuit, there are multiple paths for the current to take. Section 28.2

21 esistors in Parallel The potential difference across each resistor is the same because each is connected directly across the battery terminals. ΔV = ΔV 1 = ΔV 2 A junction is a point where the current can split. The current, I, that enters junction must be equal to the total current leaving that junction. I = I 1 + I 2 = (ΔV 1 / 1 ) + (ΔV 2 / 2 ) The currents are generally not the same. Consequence of conservation Section 28.2 of electric charge

22 Equivalent esistance Parallel, Examples Are all three diagrams equivalent? Equivalent resistance replaces the two original resistances. Section 28.2

23 Parallel esistors Equivalent esistance = eq The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance. The equivalent is always less than the smallest resistor in the group. Section 28.2

24 esistors in Parallel: Final Observations In parallel, each device operates independently of the others so that if one is switched off, the others remain on. In parallel, all of the devices operate on the same voltage. The current takes all the paths. The lower resistance will have higher currents. Even very high resistances will have some currents. Household circuits are wired so that electrical devices are connected in parallel. Section 28.2

25 The switch is initially open. When the switch is closed, the current measured by the ammeter will: A. increase B. decrease C. stay the same D. fall to zero.

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27 Combinations of esistors The 8.0-W and 4.0-W resistors are in series and can be replaced with their equivalent, 12.0 W The 6.0-W and 3.0-W resistors are in parallel and can be replaced with their equivalent, 2.0 W These equivalent resistances are in series and can be replaced with their equivalent resistance, 14.0 W Section 28.2

28 Consider this problem: Now do we know enough to do it? How do you approach it? Are any of the resistors in series with others? Are any in parallel?

29 Which part involves doing an integral? A. a B. b C. both a and b. D. neither. E. Squirrel

30 esistance of an object with arbitrary shape End-to-end: length dz d = d ρ = ρ area A(z) ρ = d = 2 2 π ( b a ) 0 L dz = ρ = π 2 (b a ) 2 ρ L 2 π ( b a 2 ) dz Inside-out: d length dr = d ρ = ρ area A(r) ρ b ρ = d = dr = 2π L a 2π L dr = ρ 2π rl ln b a

31 a b L dz = = = = ) ( ) ( r 2 z dz z A dz A dl d π ρ ρ ρ esistance of an object with arbitrary shape = = + = L z L u du a b L L z a b a dz ] / ) ( [ π ρ π ρ ab L b a a b L π ρ π ρ = = 1 1

32 Effective resistance: two resistors in series and 2 experience the same current but different voltages. Largest has largest V. V V + V I + I eq = = = = + I I I eq is larger than either 1 or 2.

33 1 2 Effective resistance: two resistors in parallel 1 and 2 experience the same voltage but different currents. Smallest has largest I / / + = + = + = = V V V I I V I V eq eq is smaller than smallest of 1 and 2.

34 eduction of a resistive network 2 /2 2

35

36 eduction of a resistive network a b a b a b /3 (7/3)

37 eduction of a resistive network Apply 42 V between a and c. What is I between a and c? I = 3 A What is V bc? V bc = 6 V What is I 2? I 2 = 2 A

38 Compare the brightness of the four identical bulbs in this circuit. V D is in parallel with a zero-resistance wire. The current will take the zero-resistance path and bypass D altogether. A and B are in series. So they will burn equally bright. Together, they see the full battery voltage. C experiences the full battery voltage, or twice the voltage experienced by A or B. So C is four times as bright.

39 If 1 is removed, 2 will glow (1) more brightly. (2) less brightly. (3) same brightness as before. If 1 is removed, 2 will glow (1) more brightly. (2) less brightly. (3) same brightness as before. Household devices are wired to run in parallel!

40 Strings of 50 Christmas lights in series. Assume ~100 V source and 25 W power consumption. What is the resistance of a single bulb? A. 2Ω B. 4Ω C. 8Ω D. 10 Ω

41 German physicist Worked with obert Bunsen Kirchhoff and Bunsen Invented the spectroscope and founded the science of spectroscopy Discovered the elements cesium and rubidium Invented astronomical spectroscopy Gustav Kirchhoff Section 28.3

42 Multi-loop circuits ε Branch: An independent current path experiences only one current at a given moment. It may be a simple wire or may also contain one or more circuit elements connected in series. Junction: A point where three or more circuit branches meet. Loop: A current path that begins and ends at the same circuit point, traversing one or more circuit branches, but without ever passing the same point twice.

43 Multi-loop circuits ε The circuit above has: 3 branches, 2 junctions, 3 loops. To solve for 3 unknown branch currents, we need 3 equations. To get these equations, use all but one (2 1 = 1) junction, and as many independent loops as needed ( 3 1 = 2).

44 Kirchoff s current rule: I = 0 n Current rule: The total current flowing into a junction is zero. Arrows define positive branch-current directions. A current later determined to flow opposite its arrow is negative. ε + I I I 1 2 = 0 + I I + I 0 + I + I + I = 1 2 =

45 Kirchoff s Voltage ule: Vn = 0 Voltage rule: The voltage changes around a loop sum to zero. Arrows define positive branch-current directions. V = +ε for a battery crossing from to + terminal. Use V = I when crossing a resistor in the positive direction. Use V = +I when crossing a resistor in the negative direction. Alpine loop elevation + ε I 1 1 = 0 ε

46 Single-loop circuit example + 20 I(2000) 30 I(1000) I(1500) + 25 I(500) = 15 I(5000) = 0 I I = 15/ 5000 = 3 ma V A V ground = + 20 I(2000) 30 I(1000) = 10 I(3000) = 10 (0.003)(3000) = 19 V

47 Multiloop circuit example Bottom loop: I 2 = I 1 Substitute I 2 in junction Eq: + 5 I (1) I3(2) 2 = 0 I 3 = 2I 1 + I I Substitute I 2 and I 3 in top loop: I 1 = 5/3 2 I1 3 = I1 (1) + I 2(1) 5 = 0 Solve for currents: I 1 = 5/3, I 2 = 5/3, I 3 = 10/3

48 Multiloop circuit example + 6I1 10 4I 2 14 = I =.5I I I1 I = I 5 3 = I I I (1.5I 1 + I2 3 = 1 = 5.5I 6) 1 0 ( 3I 11 = ) I1 = 2 I 2 = 3 I3 = 1

49 EMF Electromotive Force The electromotive force (emf), ε, of a battery is the maximum possible voltage that the battery can provide between its terminals. The emf supplies energy, it does not apply a force. The battery will normally be the source of energy in the circuit. The positive terminal of the battery is at a higher potential than the negative terminal. A battery is like a current pump. + charges are pumped from - terminal to +. We consider the wires to have no resistance. Section 28.1

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