When resistors 1 and 2 are connected in series, the

In Fig. 27-25, the ideal batteries have emfs V and V. What are (a) the current, the dissipation rate in (b) resistor 1 (4.0 ) and (c) resistor 2 (8.0 0), and the energy transfer rate in (d) battery 1 and (e) battery 2? Is energy being supplied or absorbed by (f) battery 1 and (g) battery 2? + +12R1R2 Figure 27-25 Problem 1.

In Fig. 27-26, the ideal batteries have emfs #1 ! 150 V and #2 ! 50 V and the resistances are R1 ! 3.0 0 and R2 ! 2.0 0. If the potential at P is 100 V, what is it at Q? PR1R2 1 2 Figure 27-26 Problem 2.

A car battery with a 12 V emf and an internal resistance of 0.040 is being charged with a current of 50 A. What are (a) the potential difference V across the terminals, (b) the rate Pr of energy dissipation inside the battery, and (c) the rate Pemf of energy conversion to chemical form? When the battery is used to supply 50 A to the starter motor, what are (d) V and (e) Pr?

Figure 27-27 shows a circuit of four resistors that are connected to a larger circuit. The graph below the circuit shows the electric potential V(x) as a function of position x along the lower branch of the circuit, through resistor 4; the potential VA is 12.0 V. The graph above the circuit shows the electric potential V(x) versus position x along the upper branch of the circuit, through resistors 1, 2, and 3; the potential differences are 'VB ! 2.00 V and 'VC ! 5.00 V. Resistor 3 has a resistance of 200 0. What is the resistance of (a) resistor 1 and (b) resistor 2? x x 4 1 2 3 VB VC V V (V) Figure 27-27 Problem 4.

A 5.0 A current is set up in a circuit for 6.0 min by a rechargeable battery with a 6.0 V emf. By how much is the chemical energy of the battery reduced?

A standard flashlight battery can deliver about 2.0 W&h of energy before it runs down. (a) If a battery costs US$0.80, what is the cost of operating a 100 W lamp for 8.0 h using batteries? (b) What is the cost if energy is provided at the rate of US$0.06 per kilowatt-hour?

A wire of resistance 5.0 0 is connected to a battery whose emf # is 2.0 V and whose internal resistance is 1.0 0. In 2.0 min, how much energy is (a) transferred from chemical form in the battery, (b) dissipated as thermal energy in the wire, and (c) dissipated as thermal energy in the battery?

A certain car battery with a 12.0 V emf has an initial charge of 120 A&h. Assuming that the potential across the terminals stays constant until the battery is completely discharged, for how many hours can it deliver energy at the rate of 100 W?

(a) In electron-volts, how much work does an ideal battery with a 12.0 V emf do on an electron that passes through the battery from the positive to the negative terminal? (b) If 3.40 $ 1018 electrons pass through each second, what is the power of the battery in watts?

(a) In Fig. 27-28, what value must R have if the current in the circuit is to be 1.0 mA? Take #1 ! 2.0 V, #2 ! 3.0 V, and r1 ! r2 ! 3.0 0. (b) What is the rate at which thermal energy appears in R?+ r2 2 R Figure 27-28 Problem 10.

In Fig. 27-29, circuit section AB absorbs energy at a rate of 50 W when current i ! 1.0 A through it is in the indicated direction. Resistance R ! 2.0 0. (a) What is the potential difference between A and B? Emf device X lacks internal resistance. (b) What is its emf? (c) Is point B connected to the positive terminal of X or to the negative terminal? i A B R Figure 27-29 Problem 11

Figure 27-30 shows a resistor of resistance R ! 6.00 0 connected to an ideal battery of emf # ! 12.0 V by means of two copper wires. Each wire has length 20.0 cm and radius 1.00 mm. In dealing with such circuits in this chapter, we generally neglect the potential differences along the wires and the transfer of energy to thermal energy in them. Check the validity of this neglect for the circuit of Fig. 27- 30: What is the potential difference across (a) the resistor and (b) each of the two sections of wire? At what rate is energy lost to thermal energy in (c) the resistor and (d) each section of wire? Wire 1 Wire 2 R Figure 27-30 Problem 12

A 10-km-long underground cable extends east to west and consists of two parallel wires, each of which has resistance 13 /km. An electrical short develops at distance x from the west end when a conducting path of resistance R connects the wires (Fig. 27-31). The resistance of the wires and the short is then 100 0 when measured from the east end and 200 0 when measured from the west end. What are (a) x and (b) R? West East Conducting path x Figure 27-31 Problem 13.

In Fig. 27-32a, both batteries have emf # ! 1.20 V and the external resistance R is a variable resistor. Figure 27-32b gives the electric potentials V between the terminals of each battery as functions of R: Curve 1 corresponds to battery 1, and curve 2 corresponds to battery 2.The horizontal scale is set by Rs ! 0.20 0.What is the internal resistance of (a) battery 1 and (b) battery 2? 796 CHAPTER 27 CIRCUITS R 1 2 1 2 Rs R () 0.5 0 0.3 V (V) (a) (b) + + Figure 27-32 Problem 14.

The current in a single-loop circuit with one resistance R is 5.0 A. When an additional resistance of 2.0 is inserted in series with R, the current drops to 4.0 A.What is R?

A solar cell generates a potential difference of 0.10 V when a 500 0 resistor is connected across it, and a potential difference of 0.15 V when a 1000 0 resistor is substituted. What are the (a) internal resistance and (b) emf of the solar cell? (c) The area of the cell is 5.0 cm2 , and the rate per unit area at which it receives energy from light is 2.0 mW/cm2 . What is the efficiency of the cell for converting light energy to thermal energy in the 1000 0 external resistor?

In Fig. 27-33, battery 1 has emf V and internal resistance r1 0.016 and battery 2 has emf V and internal resistance r2 0.012 . The batteries are connected in series with an external resistance R. (a) What R value makes the terminal-to-terminal potential difference of one of the batteries zero? (b) Which battery is that?Rr1r2++1 2 Figure 27-33 Problem 17.

In Fig. 27-9, what is the potential difference Vd # Vc between points d and c if #1 ! 4.0 V, #2 ! 1.0 V, R1 ! R2 ! 10 0, and R3 ! 5.0 0, and the battery is ideal?

A total resistance of 3.00 0 is to be produced by connecting an unknown resistance to a 12.0 0 resistance. (a) What must be the value of the unknown resistance, and (b) should it be connected in series or in parallel?

When resistors 1 and 2 are connected in series, the equivalent resistance is 16.0 0. When they are connected in parallel, the equivalent resistance is 3.0 0. What are (a) the smaller resistance and (b) the larger resistance of these two resistors?

Four 18.0 0 resistors are connected in parallel across a 25.0 V ideal battery. What is the current through the battery

Figure 27-34 shows five 5.00 0 resistors. Find the equivalent resistance between points (a) F and H and (b) F and G. (Hint: For each pair of points, imagine that a battery is connected across the pair.) F HR R R R R Figure 27-34 Problem 22

In Fig. 27-35, R1 ! 100 0, R2 ! 50 0, and the ideal batteries have emfs #1 ! 6.0 V, #2 ! 5.0 V, and #3 ! 4.0 V. Find (a) the current in resistor 1, (b) the current in resistor 2, and (c) the potential difference between points a and b.R2+ 1R1+ + 2 3a bFigure 27-35 Problem 23.

In Fig. 27-36, R1 ! R2 ! 4.00 0 and R3 ! 2.50 0. Find the equivalent resistance between points D and E. (Hint: Imagine that a battery is connected across those points.) R2 R3DER1 Figure 27-36 Problem 24

Nine copper wires of length l and diameter d are connected in parallel to form a single composite conductor of resistance R. What must be the diameter D of a single copper wire of length l if it is to have the same resistance?

Figure 27-37 shows a battery connected across a uniform resistor R0. A sliding contact can move across the resistor from x ! 0 at the left to x ! 10 cm at the right. Moving the contact changes how much resistance is to the left of the contact and how much is to the right. Find the rate at which energy is dissipated in resistor R as a function of x. Plot the function for # ! 50 V, R ! 2000 0, and R0 ! 100 0. x+ RR0 Sliding contactFigure 27-37 Problem 26.

Side flash. Figure 27-38 indicates one reason no one should stand under a tree during a lightning storm. If lightning comes down the side of the tree, a portion can jump over to the person, especially if the current on the tree reaches a dry region on the bark and thereafter must travel through air to reach the ground. In the figure, part of the lightning jumps through distance d in air and then travels through the person (who has negligible resistance relative to that of air because of the highly conducting salty fluids within the body).The rest of the current travels through air alongside the tree, for a distance h. If d/h ! 0.400 and the total current is I ! 5000 A, what is the current through the person? h d I Lightning current Figure 27-38 Problem 27

The ideal battery in Fig. 27-39a has emf # ! 6.0 V. Plot 1 in Fig. 27-39b gives the electric potential difference V that can appear across resistor 1 versus the current i in that resistor when the resistor is individually tested by putting a variable potential across it. Thescale of the V axis is set by Vs ! 18.0 V, and the scale of the i axis is set by is ! 3.00 m1. Plots 2 and 3 are similar plots for resistors 2and 3, respectively, when they are individually tested by putting avariable potential across them. What is the current in resistor 2 inthe circuit of Fig. 2739aV(V)Vs01i (mA)2is3(b)R1 R2R3(a)+Figure 27-39 Problem 28.

In Fig. 27-40, R1 ! 6.00 0, R2 ! 18.0 0, and the ideal battery has emf # ! 12.0 V. What are the (a) size and (b) direction (left or right) of current i1? (c) How much energy is dissipated by all four resistors in 1.00 min? i1R1R2R2R2+ Figure 27-40 Problem 29

In Fig. 27-41, the ideal batteries have emfs V and 2 0.500 1, and the resistances are each 4.00 . What is the current in (a) resistance 2 and (b) resistance 3? +2 R3+ 1R1 R2Figure 27-41 Problems 30, 41,and 88.

In Fig. 27-42, the ideal batteries have emfs 1 5.0 V and V, the resistances are each 2.0 , and the potential is defined to be zero at the grounded point of the circuit. What are potentials (a) V1 and (b) V2 at the indicated points?+2+V2 V1Figure 27-42 Problem 31.

Both batteries in Fig. 27-43a are ideal. Emf #1 of battery 1 has a fixed value, but emf #2 of battery 2 can be varied between 1.0 V and 10 V. The plots in Fig. 27-43b give the currents through the two batteries as a function of #2. The vertical scale is set by is ! 0.20 A. You must decide which plot corresponds to which battery, but for both plots, a negativecurrent occurs when the direction of the current through the 21R2R1(a) 2 (V)Current (A)is0is5 10(b)21R2R1(a)+ +Figure 27-43 Problem 32.battery is opposite the direction of that batterys emf. What are (a)emf #1, (b) resistance R1, and (c) resistance R2?

In Fig. 27-44, the current in resistance 6 is i6 ! 1.40 A and the resistances are R1 R2 R3 2.00 , R4 16.0 , R5 8.00 , and 0 R6 ! 4.00 0.What is the emf of the ideal battery?R1 R2 R5R3 R4 R6 i6 +Figure 27-44 Problem 33.

The resistances in Figs. 27-45a and b are all 6.0 0, and the batteries are ideal 12 V batteries. (a) When switch S in Fig. 27-45a is closed, what is the change in the electric potential V1 across resistor 1, or does V1 remain the same? (b) When switch S in Fig. 27-45b is closed, what is the change in V1 across resistor 1,or does V1 remain the same?(a) (b)R1 R2SR1 R2R3S+Figure 27-45 Problem 34.

In Fig. 27-46, # ! 12.0 V, R1 2000 , R2 3000 , and R3 4000 . What are the potential differences (a) VA VB, (b) VB # VC, (c) VC # VD, and (d) VA # VC?R2ABDC+1 2 R1R2 R3A++Figure 27-46 Problem 35

In Fig. 27-47, V, V, R1 100 , and R3 300 . One point of the circuit is grounded (V ! 0). What are the (a) size and (b) direction (up or down) of the current through resistance 1, the (c) size and (d) direction (left or right) of the current through resistance 2, and the (e) size and (f) direction of the current through resistance 3? (g) What is the electric potential at point A? 2 R1R2 R3A++Figure 27-47 Problem 36.

In Fig. 27-48, the resistances are R1 ! 2.00 0, R2 ! 5.00 0, and the battery is ideal. What value of R3 maximizes the dissipation rate in resistance 3?+R2 R3R1Figure 27-48 Problems 37and 98.

Figure 27-49 shows a section of a circuit. The resistances are R1 ! 2.0 0, R2 ! 4.0 0, and R3 ! 6.0 0, and the indicated current is i ! 6.0 A. The electric potential difference between points A and B that connect the section to the rest of the circuit is VA # VB ! 78 V. (a) Is the device represented by Box absorbing or providing energy to the circuit, and (b) at what rate?A BiR1R2R3Figure 27-49 Problem 38.

In Fig. 27-50, two batteries with an emf V and an internal resistance r ! 0.300 0 are connected in parallel across a resistance R. (a) For what value of R is the dissipation rate in the resistor a maximum? (b) What is thatmaximum?+ rrRFigure 27-50Problems 39and 40

Two identical batteries of emf # ! 12.0 V and internal resistance r 0.200 are to be connected to an external resistance R, either in parallel (Fig. 27-50) or in series (Fig. 27-51). If R ! 2.00r, what is the current i in the external resistance in the (a) parallel and (b) series arrangements? (c) For which arrangement is i greater? If R ! r/2.00, what is i in the external resistance in the (d) parallel arrangement and (e) series arrangement? (f) For whicharrangement is i greater now?+ + rRFigure 27-51 Problem 40

In Fig. 27-41, #1 ! 3.00 V, #2 ! 1.00 V, R1 ! 4.00 0, R2 ! 2.00 0, R3 ! 5.00 0, and both batteries are ideal. What is the rate at which energy is dissipated in (a) R1, (b) R2, and (c) R3? What is the power of (d) battery 1 and (e) battery 2?

In Fig. 27-52, an array of n parallel resistors is connected in series to a resistor and an ideal battery. All the resistors have the same resistance. If an identical resistor were added in parallel to the parallel array, the current through the battery would change by 1.25%. What is the value of n?R RRn resistorsin paralleFigure 27-52 Problem 42.

You are given a number of 10 0 resistors, each capable of dissipating only 1.0 W without being destroyed. What is the minimum number of such resistors that you need to combine in series or in parallel to make a 10 0 resistance that is capable of dissipating at least 5.0 W?

In Fig. 27-53, R1 100 , R2 R3 50.0 , R4 75.0 , and the ideal battery has emf # ! 6.00 V. (a) What is the equivalent resistance? What is i in (b) resistance 1, (c) resistance 2, (d) resistance 3, and (e) resistance 4?R2+R1R3R4Figure 27-53Problems 44 and 48.

In Fig. 27-54, the resistances are R1 1.0 and R2 2.0 , and the ideal batteries have emfs #1 ! 2.0 V and #2 ! #3 ! 4.0 V. What are the (a) size and (b) direction (up or down) of the current in battery 1, the (c) size and (d) direction of the current in battery 2, and the (e) size and (f) direction of the current in battery 3? (g) What is the potential difference Va # Vb?R1abR1 R1Figure 27-54 Problem 45.

In Fig. 27-55a, resistor 3 is a variable resistor and the ideal battery has emf # ! 12 V. Figure 27-55b gives the current ithrough the battery as a function of R3. The horizontal scale isset by R3s ! 20 0. The curve has an asymptote of 2.0 mA as R3 :,. What are (a) resistance R1 and (b) resistance R2?R3R1i (mA)6420R3 ()R3s(a) (b)+Figure 27-55 Problem 46.

A copper wire of radius a ! 0.250 mm has an aluminum jacket of outer radius b 0.380 mm. There is a current i !2.00 A in the composite wire. Using Table 26-1, calculate the currentin (a) the copper and (b) the aluminum. (c) If a potential differenceV ! 12.0 V between the ends maintains the current, whatis the length of the composite wire?

In Fig. 27-53, the resistors have the values R1 ! 7.00 0, ! R2 12.0 , and R3 4.00 , and the ideal batterys emf is # ! 24.0 V. For what value of R4 will the rate at which the battery transfers energy to the resistors equal (a) 60.0 W, (b) the maximum possible rate Pmax, and (c) the minimum possible rate Pmin? What are (d) Pmax and (e) Pmin?

(a) In Fig. 27-56, what current does the ammeter read if 5.0 V (ideal battery), R1 ! 2.0 0, R2 ! 4.0 0, and R3 ! 6.0 0? (b) The ammeter and battery are now interchanged. Show that the ammeter reading is unchanged.+ R3AR2R1Figure 27-56 Problem 49

In Fig. 27-57, R1 2.00R, the ammeter resistance is zero, and the battery is ideal. What multiple of #/R gives the current in the ammeter?+RARRR1Figure 27-57 Problem 50

In Fig. 27-58, a voltmeter of resistance RV ! 300 0 and an ammeter of resistance RA ! 3.00 0 are being used to measure a resistance R in a circuit that also contains a resistance R0 ! 100 0 and an ideal battery with an emf of # ! 12.0 V. Resistance R is given by R ! V/i, where V is the potential across R and i is the ammeter reading. The voltmeter reading is V-, which is V plus the potential difference across the ammeter. Thus, the ratio of the two meter readings is not R but only an apparent resistance R- ! V-/i. If R ! 85.0 0, what are (a) the ammeter reading, (b) the voltmeter reading, and (c) R-? (d) If RA is decreased, does the difference between R- and R increase, decrease, or remain the same?RR0VAFigure 27-58 Problem 51

A simple ohmmeter is made by connecting a 1.50 V flashlight battery in series with a resistance R and an ammeter that reads from 0 to 1.00 mA, as shown in Fig. 27-59. Resistance R is adjusted so that when the clip leads are shorted together, the meter deflects to its full-scale value of 1.00 mA.What external resistance across theleads results in a deflection of (a)10.0%, (b) 50.0%, and (c) 90.0% of full scale? (d) If the ammeterhas a resistance of 20.0 0 and the internal resistance of the batteryis negligible, what is the value of R?01.00mARFigure 2759 Problem 52.

In Fig. 27-14, assume that # ! 3.0 V, r ! 100 0, R1 ! 250 0, and R2 ! 300 0. If the voltmeter resistance RV is 5.0 k0, what percent error does it introduce into the measurement of the potential difference across R1? Ignore the presence of the ammeter.

When the lights of a car are switched on, an ammeter in series with them reads 10.0 A and a voltmeter connected across them reads 12.0 V (Fig. 27-60). When the electric starting motor is turned on, the ammeter reading drops to 8.00 A and the lights dim somewhat. If the internal resistance of the battery is 0.0500 0 and that of the ammeter is negligible, what are (a) the emf of the battery and (b) the current through the starting motor when the lights are on?+SS Starting motorLightsArFigure 27- 60Problem 54

In Fig. 27-61, Rs is to be adjusted in value by moving the sliding contact across it until points a and b are brought to the same potential. (One tests for this condition by momentarily connecting a sensitive ammeter between a and b; if these points are at the same potential, the ammeter will not deflect.) Show that when this adjustment is made, the following relation holds: Rx ! RsR2/R1. An unknown resistance (Rx) can be measured in terms of a standard (Rs) using this device, which is called a Wheatstone bridge.+ R0baRs RxR1 R2Sliding contact Figure 27-61 Problem 55.

In Fig. 27-62, a voltmeter of resistance RV ! 300 0 and an ammeter of resistance RA ! 3.00 0 are being used to measure a resistance R in a circuit that also contains a resistance R0 ! 100 0 and an ideal battery of emf # ! 12.0 V. Resistance R is given by R ! V/i, where V is the voltmeter reading and i is the current in resistance R. However, the ammeter reading is not i but rather i-, which is i plus the current through the voltmeter. Thus, the ratio of the two meter readings is not R but only an apparent resistance R- ! V/i-. If R ! 85.0 0, what are (a) the ammeter reading, (b) the voltmeter reading, and (c) R-? (d) If RV is increased, does the difference between R- and R increase,decrease, or remain the same?RR0VAFigure 27-62 Problem 56

Switch S in Fig. 27-63 is closed at time t ! 0, to begin charging an initially uncharged capacitor of capacitance C ! 15.0 mF through a resistor of resistance R ! 20.0 0. At what time is the potential across the capacitor equal tothat across the resistor?CRS+Figure 27-63 Problems 57 and 96.

In an RC series circuit, emf # ! 12.0 V, resistance R ! 1.40 M0, and capacitance C ! 1.80 mF. (a) Calculate the time constant. (b) Find the maximum charge that will appear on the capacitor during charging. (c) How long does it take for the charge to build up to 16.0 mC?

What multiple of the time constant t gives the time taken by an initially uncharged capacitor in an RC series circuit to be charged to 99.0% of its final charge?

A capacitor with initial charge q0 is discharged through a resistor. What multiple of the time constant t gives the time the capacitor takes to lose (a) the first one-third of its charge and (b) two-thirds of its charge?

A 15.0 k0 resistor and a capacitor are connected in series, and then a 12.0 V potential difference is suddenly applied across them. The potential difference across the capacitor rises to 5.00 V in 1.30 ms. (a) Calculate the time constant of the circuit. (b) Find the capacitance of the capacitor.

Figure 27-64 shows the circuit of a flashing lamp, like those attached to barrels at highway construction sites. The fluorescent lamp L (of negligible capacitance) is connected in parallel across the capacitor C of an RC circuit. There is a current through the lamp only when the potential difference across it reaches the breakdown voltage VL; then the capacitor discharges completely through the lamp and the lamp flashes briefly. For a lamp with breakdown voltage VL ! 72.0 V, wired to a 95.0 V ideal battery and a 0.150 mF capacitor, what resistance R is needed for two flashes per second?RC L Figure 27-64 Problem 62.

In the circuit of Fig 27-65, kV, C 6.5 mF, R1 !R2 ! R3 ! 0.73 M0. With C completely uncharged, switch S is suddenly closed (at t ! 0). At t ! 0, what are (a) current i1 in resistor 1, (b) current i2 in resistor 2,and (c) current i3 in resistor 3? At t ! , (that is, after many time constants), # ! 1.2 ! what are (d) i1, (e) i2, and (f) i3? What is the potential difference V2across resistor 2 at (g) t ! 0 and (h) t ! ,? (i) Sketch V2 versus t between these two extreme times.C +SR3R2R1 Figure 27-65 Problem 63

A capacitor with an initial potential difference of 100 V is discharged through a resistor when a switch between them is closed at t ! 0.At t ! 10.0 s, the potential difference across the capacitor is 1.00 V. (a) What is the time constant of the circuit? (b) What is the potential difference across the capacitor at t ! 17.0 s?

In Fig. 27-66, R1 ! 10.0 k0, R2 15.0 k , C !0! 0.400 mF, and the + R2 R1 C Figure 27-66 Problems 65 and 99 ideal battery has emf V. First, the switch is closed a long SSM time so that the steady state is reached. Then the switch is opened at time t ! 0.What is the current in resistor 2 at t ! 4.00 ms?

Figure 27-67 displays two circuits with a charged capacitor that is to be discharged through a resistor when a switch is closed. In Fig. 27-67a, R1 ! 20.0 0 and C1 ! 5.00 mF. In Fig. 27-67b, R2 ! 10.0 0 and C2 ! 8.00 mF. The ratio of the initial charges on the two capacitors is q02/q01 ! 1.50. At time t ! 0, both switches are closed. At what time t do the two capacitors have the same charge? C1 R1 C2 R2 (a) (b)Figure 27-67 Problem 66.

The potential difference between the plates of a leaky (meaning that charge leaks from one plate to the other) 2.0 mF capacitor drops to one-fourth its initial value in 2.0 s. What is the equivalent resistance between the capacitor plates?

A 1.0 mF capacitor with an initial stored energy of 0.50 J is discharged through a 1.0 M0 resistor. (a) What is the initial charge on the capacitor? (b) What is the current through the resistor when the discharge starts? Find an expression that gives, as a function of time t, (c) the potential difference VC across the capacitor, (d) the potential difference VR across the resistor, and (e) the rate at which thermal energy is produced in the resistor.

A 3.00 M0 resistor and a 1.00 mF capacitor are connected in series with an ideal battery of emf V.At 1.00 s after the connection is made, what is the rate at which (a) the charge of the capacitor is increasing, (b) energy is being stored in the capacitor, (c) thermal energy is appearing in the resistor, and (d) energy is being delivered by the battery?

Each of the six real batteries in Fig. 27-68 has an emf of 20 V and a resistance of 4.0 . (a) What is the current through the (external) resistance R 4.0 ? (b) What is the potential difference across each battery? (c) What is the power of each battery? (d) At what rate does each battery transfer energy to internal thermal energy?R Figure 27-68 Problem 70.

In Fig. 27-69, R1 20.0 , R2 10.0 , and the ideal battery has emf # 120 V. What is the current at point a if we close (a) only switch S1, (b) only switches S1 and S2, and (c) all three switches?a S1 S2 S3R1 R1 R1R1 R2 R2+Figure 27-69 Problem 71.

In Fig. 27-70, the ideal battery has emf # 30.0 V, and the resistances are R1 R2 14 , R3 R4 R5 6.0 , R6 2.0 , and R7 1.5 .What are currents (a) i2, (b) i4, (c) i1, (d)i3, and (e) i 0 5? 2R1 R2 R7R3 R4 R5 R6i4i + 5Figure 27-70 Problem 72.

Wires A and B, having equal lengths of 40.0 m and equal diameters of 2.60 mm, are connected in series. A potential difference of 60.0 V is applied between the ends of the composite wire. The resistances are RA ! 0.127 0 and RB ! 0.729 0. For wire A, what are (a) magnitude J of the current density and (b) potential difference V? (c) Of what type material is wire A made (see Table 26-1)? For wire B, what are (d) J and (e) V? (f) Of what type material is B made?

What are the (a) size and (b) direction (up or down) of current i in Fig. 27-71, where all resistances are 4.0 0 and all batteries are ideal and have an emf of 10 V? (Hint: This can be answered using only mental calculation.)Figure 27-71 Problem 74.

Suppose that, while you are sitting in a chair, charge separation between your clothing and the chair puts you at a potential of 200 V, with the capacitance between you and the chair at 150 pF. When you stand up, the increased separation between your body and the chair decreases the capacitance to 10 pF. (a) What then is the potential of your body? That potential is reduced over time, as the charge on you drains through your body and shoes (you are a capacitor discharging through a resistance). Assume that the resistance along that route is 300 G0. If you touch an electrical component while your potential is greater than 100 V, you could ruin the component. (b) How long must you wait until your potential reaches the safe level of 100 V? If you wear a conducting wrist strap that is connected to ground, your potential does not increase as much when you stand up; you also discharge more rapidly because the resistance through the grounding connection is much less than through your body and shoes. (c) Suppose that when you stand up, your potential is 1400 V and the chair-to-you capacitance is 10 pF. What resistance in that wrist-strap grounding connection will allow you to discharge to 100 V in 0.30 s, which is less time than you would need to reach for, say, your computer?

In Fig. 27-72, the ideal batteries have emfs #1 ! 20.0 V, V, and V, and the resistances are each 2.00 . What are the (a) size and (b) direction (left or right) of current i1? (c) Does battery 1 supply or absorb energy, and (d) what is its power? (e) Does battery 2 supply or absorb energy, and (f) what is its power? (g) Does battery 3 supply or absorb energy, and (h) what is its power?i13++12+Figure 27-72 Problem 76

A temperature-stable resistor is made by connecting a resistor made of silicon in series with one made of iron. If the required total resistance is 1000 0 in a wide temperature range around 20/C, what should be the resistance of the (a) silicon resistor and (b) iron resistor? (See Table 26-1.)

In Fig. 27-14, assume that # ! 5.0 V,r ! 2.0 0, R1 ! 5.0 0, and R2 ! 4.0 0. If the ammeter resistance RA is 0.10 0, what percent error does it introduce into the measurement of the current? Assume that the voltmeter is not present.

An initially uncharged capacitor C is fully charged by a device of constant emf connected in series with a resistor R. (a) Show that the final energy stored in the capacitor is half the energy supplied by the emf device. (b) By direct integration of i 2 R over the charging time, show that the thermal energy dissipated by the resistor is also half the energy supplied by the emf device.

In Fig. 27-73, R1 ! 5.00 0, R2 ! 10.0 0, R3 ! 15.0 0, C1 ! 5.00 mF, C2 ! 10.0 mF, and the ideal battery has emf # ! 20.0 V.Assuming that the circuit is in the steady state,what is the total energy stored in the two capacitors? C1C2R2R1R3+Figure 27-73 Problem 80.

In Fig. 27-5a, find the potential difference across R2 if # ! 12 V, R1 ! 3.0 0, R2 ! 4.0 0, and R3 ! 5.0 0

In Fig. 27-8a, calculate the potential difference between a and c by considering a path that contains R, r1, and #1.

A controller on an electronic arcade game consists of a variable resistor connected across the plates of a 0.220 mF capacitor.The capacitor is charged to 5.00 V, then discharged through the resistor. The time for the potential difference across the plates to decrease to 0.800 V is measured by a clock inside the game. If the range of discharge times that can be handled effectively is from 10.0 ms to 6.00 ms, what should be the (a) lower value and (b) higher value of the resistance range of the resistor?

An automobile gasoline gauge is shown schematically in Fig. 27-74. The indicator (on the dashboard) has a resistance of 10 0. The tank unit is a float connected to a variable resistor whose resistance varies linearly with the volume of gasoline. The resistance is 140 0 when the tank is empty and 20 0 when the tank is full. Find the current in the circuit when the tank is (a) empty, (b) half-full, and (c) full.Treat the battery as idea + 12 V Rindicator Rtank Connected through chassis Indicator Tank unit Figure 27-74 Problem 84.

The starting motor of a car is turning too slowly, and the mechanic has to decide whether to replace the motor, the cable, or the battery. The cars manual says that the 12 V battery should have no more than 0.020 0 internal resistance, the motor no more than 0.200 0 resistance, and the cable no more than 0.040 0 resistance. The mechanic turns on the motor and measures 11.4 V across the battery, 3.0 V across the cable, and a current of 50 A. Which part is defective?

Two resistors R1 and R2 may be connected either in series or in parallel across an ideal battery with emf #. We desire the rate of energy dissipation of the parallel combination to be five times that of the series combination. If R1 ! 100 0, what are the (a) smaller and (b) larger of the two values of R2 that result in that dissipation rate?

The circuit of Fig. 27-75 shows a capacitor, two ideal batteries, two resistors, and a switch S. Initially S has been open for a long time. If it is then closed for a long time, what is the change in the charge on the capacitor?Assume C ! 10 mF,#1 ! 1.0 V,#2 ! 3.0 V, R1 ! 0.20 0, and R2 ! 0.40 0 2R2C+ 1R1+S Figure 27-75 Problem 87.

In Fig. 27-41, R1 ! 10.0 0, R2 ! 20.0 0, and the ideal batteries have emfs #1 ! 20.0 V and #2 ! 50.0 V. What value of R3 results in no current through battery 1?

In Fig. 27-76, R ! 10 0. What is the equivalent resistance between points A and B? (Hint: This circuit section might look simpler if you first assume that points A and B are connected to a battery.) 4.0R 2.0R6.0R3.0RRBA Figure 27-76 Problem 89.

(a) In Fig. 27-4a, show that the rate at which energy is dissipated in R as thermal energy is a maximum when R ! r. (b) Show that this maximum power is P ! #2 /4r.

In Fig. 27-77, the ideal batteries have emfs #1 ! 12.0 V and #2 ! 4.00 V, and the resistances are each 4.00 0. What are the (a) size and (b) direction (up or down) of i1 and the (c) size and (d) direction of i2? (e) Does battery 1 supply or absorb energy, and (f) what is its energy transfer rate? (g) Does battery 2 supply or absorb energy, and (h) what is its energy transfer rate?2 i1 1+ + 2Figure 27-77 Problem 91

Figure 27-78 shows a portion of a circuit through which there is a current I ! 6.00 A. The resistances are R1 ! R2 ! 2.00R3 ! 2.00R4 ! 4.00 0. What is the current i1 through resistor 1? 2.i1 R1 R2R3R4II

Thermal energy is to be generated in a 0.10 0 resistor at the rate of 10 W by connecting the resistor to a battery whose emf is 1.5 V. (a) What potential difference must exist across the resistor? (b) What must be the internal resistance of the battery

Figure 27-79 shows three 20.0 0 resistors. Find the equivalent resistance between points (a) A and B, (b) A and C, and (c) B and C. (Hint: Imagine that a battery is connected between a given pair of points.) ABCFigure 27-79 Problem 94.

A 120 V power line is protected by a 15 A fuse. What is the maximum number of 500 W lamps that can be simultaneously operated in parallel on this line without blowing the fuse because of an excess of current?

Figure 27-63 shows an ideal battery of emf # ! 12 V, a resistor of resistance R ! 4.0 0, and an uncharged capacitor of capacitance C ! 4.0 mF.After switch S is closed, what is the current through the resistor when the charge on the capacitor is 8.0 mC?

A group of N identical batteries of emf # and internal resistance r may be connected all in series (Fig. 27-80a) or all in parallel (Fig. 27-80b) and then across a resistor R. Show that both arrangements give the same current in R if R ! r. SSM + + r r R (a) + r N batteries in series + + + R (b) N batteries in parallel r r r Figure 27-80 Problem 97.

In Fig. 27-48, R1 ! R2 10.0 , and the ideal battery has emf V. (a) What value of R3 maximizes the rate at which the battery supplies energy and (b) what is that maximum rate?

In Fig. 27-66, the ideal bat- # ! 12.0 0 SSM ! tery has emf # ! 30 V, the resistances are R1 20 k and R2 10 k , and the capacitor is uncharged. When the switch is closed at time t ! 0, what is the current in (a) resistance 1 and (b) resistance 2? (c) A long time later, what is the current in resistance 2?

In Fig. 27-81, the ideal batteries have emfs #1 ! 20.0 V, #2 ! 10.0 V, 1 + +Figure 27-81 Problem 100.#3 ! 5.00 V, and #4 ! 5.00 V, and the resistances are each 2.00 0. What are the (a) size and (b) direction (left or right) ofcurrent i1 and the (c) size and (d) direction of current i2? (This can be answered with only mental calculation.) (e) At what rate is energy being transferred in battery 4, and (f) is the energy being supplied or absorbedby the battery?

In Fig. 27-82, an ideal battery of emf # ! 12.0 V is connected to a network of resistances R1 ! 6.00 0, R2 ! 12.0 0,R3 ! 4.00 0,R4 ! 3.00 0, and R5 ! 5.00 0. What is the potential difference across resistance 5? .+ R1R2R4 R5R Figure 27-82 Problem 101.

The following table gives the electric potential difference VT across the terminals of a battery as a function of current i being drawn from the battery. (a) Write an equation that represents the relationship between VT and i. Enter the data into your graphing calculator and perform a linear regression fit of VT versus i. From the parameters of the fit, find (b) the batterys emf and (c) its internal resistance. i(A): 50.0 75.0 100 125 150 175 200 VT(V): 10.7 9.00 7.70 6.00 4.80 3.00 1.70

In Fig. 27-83, #1 ! 6.00 V, #2 ! 12.0 V, R1 ! 200 0, and R2 ! 100 0. What are the (a) size and (b) direction (up or down) of the current through resistance 1, the (c) size and (d) direction of the current through resistance 2, and the (e) size and (f) direction of the current through battery 2? .++ R1R21 2Figure 27-83 Problem 103.

A three-way 120 V lamp bulb that contains two filaments is rated for 100-200-300 W. One filament burns out. Afterward, the bulb operates at the same intensity (dissipates energy at the same rate) on its lowest as on its highest switch positions but does not operate at all on the middle position. (a) How are the two filaments wired to the three switch positions? What are the (b) smaller and (c) larger values of the filament resistances?

In Fig. 27-84, R1 ! R2 ! 2.0 0, R3 ! 4.0 0, R4 ! 3.0 0, R5 ! 1.0 0, and R6 ! R7 ! R8 ! 8.0 0, and the ideal batteries have emfs #1 ! 16 V and #2 ! 8.0 V. What are the (a) size and (b) direction (up or down) of current i1 and the (c) size and (d) direction of current i2? What is the energy transfer rate in (e) battery 1 and (f) battery 2? Is energy being supplied or absorbed in (g) battery 1 and (h) battery 2? .i2i1 R1R2 R3 R4R6 R5R8 R71+ + Figure 27-84 Problem 105