Resistors in Series. Section 28.2

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1 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

2 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

3 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

4 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

5 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

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

7 Equivalent esistance eq Parallel esistors 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

8 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

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

10 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.

11 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

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13 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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16 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 chapter summaries about circuits. BTW Current is NEVE used up. Today: eview of series & parallel use video demos and MU. How to add up resistors in series and parallel: light bulb problems. A. No new Circuit elements. B. Terms: Branch, junction Loop C. Kirchhoff s Laws. Pay attention to signs of quantities and directions of current flow.

17 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

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

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

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

21 eduction of a resistive network 2 /2 2

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

23 A freq. test question is what is the magnitude and sign of the potential difference of a relative to b.

24 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

25 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 much brighter.

26 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!

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

28 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

29 Multiloop circuits 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.

30 Multiloop circuits The circuit above has:? branches,? junctions,? loops. 3 Independent Equations. 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).

31 Kirchhoff 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 be negative flows opposite its arrow. I I 1 I2 0 I I I 0 I I I

32 Kirchoff s Voltage ule: Vn 0 Voltage rule: The voltage changes around a loop sum to zero. Arrows define positive branch-current directions. V = + 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 I 11 0

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

34 Multiloop circuit example Bottom loop: I 2 = I 1 Substitute I 2 in junction Eq: 5 3 I2(1) I (2) 0 I 3 = 2I 1 Substitute I 2 and I 3 in top loop: I 1 = 5/3 I 2 I1 I I1(1) I (1) 5 0 Solve for currents: I 1 = 5/3, I 2 = 5/3, I 3 = 10/3

35 Multiloop circuit example I1 10 4I 14 I.5I 6 0 2I 10 6I1 I I I 1 I (1.5I 1 I2 I I 0 6) ( 3I ) I1 2 I2 3 I3 1

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