Solved: The resistance between points a and b in Figure

The terminals of a battery are connected across two different resistors in series. Which of the following statements are correct? (There may be more than one correct statement.) (a) The smaller resistor carries more current. (b) The larger resistor carries less current. (c) The current in each resistor is the same. (d) The voltage difference across each resistor is the same. (e) The voltage difference is greatest across the resistor closest to the positive terminal.

The terminals of a battery are connected across two different resistors in parallel. Which of the following statements are correct? (There may be more than one correct statement.) (a) The larger resistor carries more current than the smaller resistor. (b) The larger resistor carries less current than the smaller resistor. (c) The voltage difference across each resistor is the same. (d) The voltage difference across the larger resistor is greater than the voltage difference across the smaller resistor. (e) The voltage difference is greater across the resistor closer to the battery.

A 1.00- and a 2.00- resistor are in parallel. What is the equivalent single resistance? (a) 1.50 (b) 3.00 (c) 0.667 (d) 0.333 (e) 1.33

Two lightbulbs are in series, one operating at 120 W and the other operating at 60.0 W. If the voltage drop across the series combination is 120 V, what is the current in the circuit? (a) 1.0 A (b) 1.5 A (c) 2.0 A (d) 2.5 A (e) 3.0 A

When connected to a 120 V source, an electric heater is rated at 1 200 W and a toaster at 6.0 102 W. If both appliances are operating in parallel on a 120-V circuit, what total current is delivered by an external source? (a) 12 A (b) 24 A (c) 32 A (d) 8.0 A (e) 15 A

What is the current in the 10.0- resistance in the circuit shown in Figure MCQ18.6? (a) 0.59 A (b) 1.0 A (c) 11.2 A (d) 16.0 A (e) 5.3 A

Which of the following equations is not a consequence of Kirchhoffs laws, when applied to Figure MCQ18.7? (a) 9 _ I1 _ 2I2 _ 0 (b) I1 _ I2 _ I3 _ 0 (c) 4 _ 2I2 _ 3I3 _ 0 (d) 5 _ I1 _ 3I3 _ 0 (e) All these equations are correct.

If the terminals of a battery are connected across two identical resistors in series, the total power delivered by the battery is 8.0 W. If the same battery is connected across the resistors in parallel, what is the total power delivered by the battery? (a) 16 W (b) 32 W (c) 2.0 W (d) 4.0 W (e) 8.0 W

What is the time constant of the circuit shown in Figure MCQ18.9? Each of the fi ve resistors has resistance R, and each of the fi ve capacitors has capacitance C. The internal resistance of the battery is negligible. (a) RC (b) 5RC (c) 10RC (d) 25RC (e) None of these answers is correct.

The capacitor in Figure MCQ18.10 has a large capacitance and is initially uncharged. After the switch is closed, what happens to the lightbulb? (a) It never glows because the capacitor represents an open circuit. (b) It glows only after the capacitor is fully charged. (c) It glows for a very short time as the capacitor is being charged. (d) It glows continuously. (e) It glows intermittently.

Several resistors are connected in parallel. Which of the following statements are true of the corresponding equivalent resistance? (There may be more than one correct statement.) (a) It is greater than the resistance of any of the individual resistors. (b) It is less than the resistance of any of the individual resistors. (c) It is dependent on the voltage applied across the series. (d) It is equal to the sum of the resistances of all the resistors in the series. (e) It is equal to the reciprocal of the sum of the inverses of the resistances of all the resistors.

Several resistors are connected in series. Which of the following statements are true of the corresponding equivalent resistance? (There may be more than one correct statement.) (a) It is greater than the resistance of any of the individual resistors. (b) It is less than the resistance of any of the individual resistors. (c) It is dependent on the voltage applied across the group. (d) It is equal to the sum of the resistances of all the individual resistors. (e) It is equal to the reciprocal of the sum of the inverses of the resistances of all the resistors.

The capacitor in Figure MCQ18.13 is initially uncharged. When the switch is closed at t _ 0 s, what is the voltage drop across the capacitor after one time constant has elapsed? (a) 2.05 V (b) 3.25 V (c) 6.63 V (d) 3.79 V (e) 4.43 V

Is the direction of current in a battery always from the negative terminal to the positive one? Explain.

Given three lightbulbs and a battery, sketch as many different circuits as you can.

Suppose the energy transferred to a dead battery during charging is W. The recharged battery is then used until fully discharged again. Is the total energy transferred out of the battery during use also W?

A short circuit is a circuit containing a path of very low resistance in parallel with some other part of the circuit. Discuss the effect of a short circuit on the portion of the circuit it parallels. Use a lamp with a frayed line cord as an example.

If you have your headlights on while you start your car, why do they dim while the car is starting?

If electrical power is transmitted over long distances, the resistance of the wires becomes signifi cant. Why? Which mode of transmission would result in less energy loss, high current and low voltage or low current and high voltage? Discuss.

Connecting batteries in series increases the emf applied to a circuit. What advantage might there be to connecting them in parallel?

Two sets of Christmas lights are available. For set A, when one bulb is removed, the remaining bulbs remain illuminated. For set B, when one bulb is removed, the remaining bulbs do not operate. Explain the difference in wiring for the two sets.

Suppose a parachutist lands on a high-voltage wire and grabs the wire as she prepares to be rescued. Will she be electrocuted? If the wire then breaks, should she continue to hold onto the wire as she falls to the ground?

(a) Two resistors are connected in series across a battery. Is the power delivered to each resistor (i) the same or (ii) not necessarily the same? (b) Two resistors are connected in parallel across a battery. Is the power delivered to each resistor (i) the same or (ii) not necessarily the same?

Why is it possible for a bird to sit on a high-voltage wire without being electrocuted? (See Fig. CQ18.11.)

A ski resort consists of a few chairlifts and several interconnected downhill runs on the side of a mountain, with a lodge at the bottom. The lifts are analogous to batteries, and the runs are analogous to resistors. Describe how two runs can be in series. Describe how three runs can be in parallel. Sketch a junction of one lift and two runs. One of the skiers is carrying an altimeter. State Kirchhoffs junction rule and Kirchhoffs loop rule for ski resorts.

Embodied in Kirchhoffs rules are two conservation laws. What are they?

Why is it dangerous to turn on a light when you are in a bathtub?

A battery having an emf of 9.00 V delivers 117 mA when connected to a 72.0- load. Determine the internal resistance of the battery.

Three 9.0- resistors are connected in series with a 12-V battery. Find (a) the equivalent resistance of the circuit and (b) the current in each resistor. (c) Repeat for the case in which all three resistors are connected in parallel across the battery.

A lightbulb marked 75 W [at] 120 V is screwed into a socket at one end of a long extension cord in which each of the two conductors has a resistance of 0.800 . The other end of the extension cord is plugged into a 120-V outlet. Draw a circuit diagram, and fi nd the actual power of the bulb in the circuit described.

(a) Find the current in a 8.00- resistor connected to a battery that has an internal resistance of 0.15 if the voltage across the battery (the terminal voltage) is 9.00 V. (b) What is the emf of the battery?

(a) Find the equivalent resistance between points a and b in Figure P18.5. (b) Calculate the current in each resistor if a potential difference of 34.0 V is applied between points a and b. FIGURE CQ18.11 Birds on a highvoltage wire.

Consider the combination of resistors shown in Figure P18.6. (a) Find the equivalent resistance between point a and b. (b) If a voltage of 35 V is applied between points a and b, fi nd the current in each resistor.

What is the equivalent resistance of the combination of identical resistors between points a and b in Figure P18.7?

Consider the circuit shown in Figure P18.8. (a) Calculate the equivalent resistance of the 10.0- and 5.00- resistors connected in parallel. (b) Using the result of part (a), calculate the combined resistance of the 10.0- , 5.00- , and 4.00- resistors. (c) Calculate the equivalent resistance of the combined resistance found in part (b) and the parallel 3.00- resistor. (d) Combine the equivalent resistance found in part (c) with the 2.00- resistor. (e) Calculate the total current in the circuit. (f) What is the voltage drop across the 2.00- resistor? (g) Subtracting the result of part (f) from the battery voltage, fi nd the voltage across the 3.00- resistor. (h) Calculate the current in the 3.00- resistor.

Consider the circuit shown in Figure P18.9. Find (a) the current in the 20.0- resistor and (b) the potential difference between points a and b.

Consider the two circuits shown in Figure P18.10 in which the lightbulbs and batteries are identical. The resistance of each lightbulb is R. (a) Find the currents in each lightbulb. (b) How does the brightness of B compare with that of C? Explain. (c) How does the brightness of A compare with that of B and C? Explain.

The resistance between terminals a and b in Figure P18.11 is 75 . If the resistors labeled R have the same value, determine R.

Three identical resistors are connected as shown in Figure P18.12. The maximum power that can safely be delivered by a battery connected between a and b is 24 W. (a) What is the equivalent resistance between points a and b? (b) Find an expression for the maximum voltage that can be applied between a and b. (c) What is the power delivered to each resistor when the maximum voltage is applied between a and b?

Find the current in the 12- resistor in Figure P18.13.

(a) Is it possible to reduce the circuit shown in Figure P18.14 (page 620) to a single equivalent resistor connected across the battery? Explain. (b) Find the current in the 2.00- resistor. (c) Calculate the power delivered by the battery to the circuit.

(a) You need a 45- resistor, but the stockroom has only 20- and 50- resistors. How can the desired resistance be achieved under these circumstances? (b) What can you do if you need a 35- resistor? Note: For some circuits, the currents are not necessarily in the direction shown.

(a) Find the current in each resistor of Figure P18.16 by using the rules for resistors in series and parallel. (b) Write three independent equations for the three currents using Kirchoffs laws: one with the node rule; a second using the loop rule through the battery, the 6.0- resistor, and the 24.0- resistor; and the third using the loop rule through the 12.0- and 24.0- resistors. Solve to check the answers found in part (a).

The ammeter shown in Figure P18.17 reads 2.00 A. Find I1, I2, and e.

Determine the potential difference _Vab for the circuit in Figure P18.18.

Figure P18.19 shows a circuit diagram. Determine (a) the current, (b) the potential of wire A relative to ground, and (c) the voltage drop across the 1 500- resistor.

In the circuit of Figure P18.20, the current I1 is 3.0 A and the values of e and R are unknown. What are the currents I2 and I3?

(a) In Figure P18.21 fi nd the current in each resistor and (b) the power delivered to each resistor.

Four resistors are connected to a battery with a terminal voltage of 12 V, as shown in Figure P18.22. (a) How would you reduce the circuit to an equivalent single resistor connected to the battery? Use this procedure to fi nd the equivalent resistance of the circuit. (b) Find the current delivered by the battery to this equivalent resistance. (c) Determine the power delivered by the battery. (d) Determine the power delivered to the 50.0- resistor.

Using Kirchhoffs rules, (a) fi nd the current in each resistor shown in Figure P18.23 and (b) fi nd the potential difference between points c and f.

Two 1.50-V batterieswith their positive terminals in the same directionare inserted in series into the barrel of a fl ashlight. One battery has an internal resistance of 0.255 , the other an internal resistance of 0.153 . When the switch is closed, a current of 0.600 A passes through the lamp. (a) What is the lamps resistance? (b) What fraction of the power dissipated is dissipated in the batteries?

(a) Can the circuit shown in Figure P18.25 be reduced to a single resistor connected to the batteries? Explain. (b) Calculate each of the unknown currents I1, I2, and I3 for the circuit.

A dead battery is charged by connecting it to the live battery of another car with jumper cables (Fig. P18.26). Determine the current in the starter and in the dead battery.

(a) Can the circuit shown in Figure P18.27 be reduced to a single resistor connected to the batteries? Explain. (b) Find the magnitude of the current and its direction in each resistor.

For the circuit shown in Figure P18.28, use Kirchhoffs rules to obtain equations for (a) the upper loop, (b) the lower loop, and (c) the node on the left side. In each case suppress units for clarity and simplify, combining like terms. (d) Solve the node equation for I36. (e) Using the equation found in (d), eliminate I36 from the equation found in part (b). (f) Solve the equations found in part (a) and part (e) simultaneously for the two unknowns for I18 and I12, respectively. (g) Substitute the answers found in part (f) into the node equation found in part (d), solving for I36. (h) What is the signifi cance of the negative answer for I12?

Find the potential difference across each resistor in Figure P18.29.

Show that t _ RC has units of time.

Consider the series RC-circuit shown in Active Figure 18.16 for which R _ 75.0 k , C _ 25.0 mF, and e _ 12.0 V. Find (a) the time constant of the circuit and (b) the charge on the capacitor one time constant after the switch is closed.

An uncharged capacitor and a resistor are connected in series to a source of emf. If e _ 9.00 V, C _ 20.0 mF, and R _ 100 , fi nd (a) the time constant of the circuit, (b) the maximum charge on the capacitor, and (c) the charge on the capacitor after one time constant.

Consider a series RC circuit for which R _ 1.0 M , C _ 5.0 mF, and e _ 30 V. Find the charge on the capacitor 10 s after the switch is closed.

A series combination of a 12-k resistor and an unknown capacitor is connected to a 12-V battery. One second after the circuit is completed, the voltage across the capacitor is 10 V. Determine the capacitance of the capacitor.

A charged capacitor is connected to a resistor and a switch as in Active Figure 18.17. The circuit has a time constant of 1.5 s. After the switch is closed, the charge on the capacitor is 75% of its initial charge. (a) Find the time it takes to reach this charge. (b) If R _ 250 k , what is the value of C?

A series RC circuit has a time constant of 0.960 s. The battery has an emf of 48.0 V, and the maximum current in the circuit is 0.500 mA. What are (a) the value of the capacitance and (b) the charge stored in the capacitor 1.92 s after the switch is closed?

What minimum number of 75-W lightbulbs must be connected in parallel to a single 120-V household circuit to trip a 30.0-A circuit breaker?

A lamp (R _ 150 ), an electric heater (R _ 25 ), and a fan (R _ 50 ) are connected in parallel across a 120-V line. (a) What total current is supplied to the circuit? (b) What is the voltage across the fan? (c) What is the current in the lamp? (d) What power is expended in the heater?

A heating element in a stove is designed to dissipate 3 000 W when connected to 240 V. (a) Assuming the resistance is constant, calculate the current in the heating element if it is connected to 120 V. (b) Calculate the power it dissipates at that voltage.

A coffee maker is rated at 1 200 W, a toaster at 1 100 W, and a waffl e maker at 1 400 W. The three appliances are connected in parallel to a common 120-V household circuit. (a) What is the current in each appliance when operating independently? (b) What total current is delivered to the appliances when all are operating simultaneously? (c) Is a 15-A circuit breaker suffi cient in this situation? Explain.

Assume a length of axon membrane of about 0.10 m is excited by an action potential (length excited _ nerve speed pulse duration _ 50.0 m/s 2.0 10_3 s _ 0.10 m). In the resting state, the outer surface of the axon wall is charged positively with K_ ions and the inner wall has an equal and opposite charge of negative organic ions, as shown in Figure P18.41. Model the axon as a parallelplate capacitor and take C _ kP0A/d and Q _ C _V to investigate the charge as follows. Use typical values for a cylindrical axon of cell wall thickness d _ 1.0 10_8 m, axon radius r _ 1.0 101 mm, and cell-wall dielectric constant k _ 3.0. (a) Calculate the positive charge on the outside of a 0.10-m piece of axon when it is not conducting an electric pulse. How many K_ ions are on the outside of the axon assuming an initial potential difference of 7.0 10_2 V? Is this a large charge per unit area? [Hint: Calculate the charge per unit area in terms of the number of angstroms (2) per electronic charge. An atom has a cross section of about 1 2 (1 _ 10_10 m).] (b) How much positive charge must fl ow through the cell membrane to reach the excited state of _3.0 10_2 V from the resting state of _7.0 10_2 V? How many sodium ions (Na_) is this? (c) If it takes 2.0 ms for the Na_ ions to enter the axon, what is the average current in the axon wall in this process? (d) How much energy does it take to raise the potential of the inner axon wall to _3.0 10_2 V, starting from the resting potential of _7.0 10_2 V?

Consider the model of the axon as a capacitor from Problem 41 and Figure P18.41. (a) How much energy does it take to restore the inner wall of the axon to _7.0 10_2 V, starting from _3.0 10_2 V? (b) Find the average current in the axon wall during this process.

Using Figure 18.28b and the results of Problems 18.41d and 18.42a, fi nd the power supplied by the axon per action potential. ADDITIONAL PROBLEMS

How many different resistance values can be constructed from a 2.0- , a 4.0- , and a 6.0- resistor? Show how you would get each resistance value either individually or by combining them.

Find the equivalent resistance between points a and b in Figure P18.45.

For the circuit shown in Figure P18.46, the voltmeter reads 6.0 V and the ammeter reads 3.0 mA. Find (a) the value of R, (b) the emf of the battery, and (c) the voltage across the 3.0-kV resistor. (d) What assumptions did you have to make to solve this problem?

Find (a) the equivalent resistance of the circuit in Figure P18.47, (b) each current in the circuit, (c) the potential difference across each resistor, and (d) the power dissipated by each resistor.

Three 60.0-W, 120-V lightbulbs are connected across a 120-V power source, as shown in Figure P18.48. Find (a) the total power delivered to the three bulbs and (b) the potential difference across each. Assume the resistance of each bulb is constant (even though, in reality, the resistance increases markedly with current).

An automobile battery has an emf of 12.6 V and an internal resistance of 0.080 . The headlights have a total resistance of 5.00 (assumed constant). What is the potential difference across the headlight bulbs (a) when they are the only load on the battery and (b) when the starter motor is operated, taking an additional 35.0 A from the battery?

In Figure P18.50 suppose the switch has been closed for a length of time suffi ciently long for the capacitor to become fully charged. Find (a) the steady-state current in each resistor and (b) the charge on the capacitor.

A circuit consists of three identical lamps, each of resistance R, connected to a battery as in Figure P18.51. (a) Calculate an expression for the equivalent resistance of the circuit when the switch is open. Repeat the calculation when the switch is closed. (b) Write an expression for the power supplied by the battery when the switch is open. Repeat the calculation when the switch is closed. (c) Using the results already obtained, explain what happens to the brightness of the lamps when the switch is closed.

The resistance between points a and b in Figure P18.52 drops to one-half its original value when switch S is closed. Determine the value of R.

A generator has a terminal voltage of 110 V when it delivers 10.0 A and 106 V when it delivers 30.0 A. Calculate the emf and the internal resistance of the generator.

An emf of 10 V is connected to a series RC circuit consisting of a resistor of 2.0 106 and a capacitor of 3.0 mF. Find the time required for the charge on the capacitor to reach 90% of its fi nal value.

The student engineer of a campus radio station wishes to verify the effectiveness of the lightning rod on the antenna mast (Fig. P18.55, page 624). The unknown resistance Rx is between points C and E. Point E is a true ground, but is inaccessible for direct measurement because the stratum in which it is located is several meters below Earths surface. Two identical rods are driven into the ground at A and B, introducing an unknown resistance Ry. The procedure for fi nding the unknown resistance Rx is as follows. Measure resistance R1 between points A and B. Then connect A and B with a heavy conducting wire and measure resistance R2 between points A and C. (a) Derive a formula for Rx in terms of the observable resistances R1 and R2. (b) A satisfactory ground resistance would be Rx _ 2.0 . Is the grounding of the station adequate if measurements give R1 _ 13 and R2 _ 6.0 ?

The resistor R in Figure P18.56 dissipates 20 W of power. Determine the value of R.

A voltage _V is applied to a series confi guration of n resistors, each of resistance R. The circuit components are reconnected in a parallel confi guration, and voltage _V is again applied. Show that the power consumed by the series confi guration is 1/n2 times the power consumed by the parallel confi guration.

For the network in Figure P18.58, show that the resistance between points a and b is Rab 5 27 17 V. (Hint: Connect a battery with emf e across points a and b and determine e/I, where I is the current in the battery.) FIGURE P18.58 a 1.0 1.0 b 1.0 3.0 5.0

A battery with an internal resistance of 10.0 produces an open-circuit voltage of 12.0 V. A variable load resistance with a range from 0 to 30.0 is connected across the battery. (Note: A battery has a resistance that depends on the condition of its chemicals and that increases as the battery ages. This internal resistance can be represented in a simple circuit diagram as a resistor in series with the battery.) (a) Graph the power dissipated in the load resistor as a function of the load resistance. (b) With your graph, demonstrate the following important theorem: The power delivered to a load is a maximum if the load resistance equals the internal resistance of the source.

The circuit in Figure P18.60 contains two resistors, R1 _ 2.0 k and R2 _ 3.0 k , and two capacitors, C1 _ 2.0 mF and C2 _ 3.0 mF, connected to a battery with emf e _ 120 V. If there are no charges on the capacitors before switch S is closed, determine the charges q1 and q2 on capacitors C1 and C2, respectively, as functions of time, after the switch is closed. (Hint: First reconstruct the circuit so that it becomes a simple RC circuit containing a single resistor and single capacitor in series, connected to the battery, and then determine the total charge q stored in the circuit.)

Consider the circuit shown in Figure P18.61. Find (a) the potential difference between points a and b and (b) the current in the 20.0- resistor.

In Figure P18.62, R1 _ 0.100 , R2 _ 1.00 , and R3 _ 10.0 . Find the equivalent resistance of the circuit and the current in each resistor when a 5.00-V power supply is connected between (a) points A and B, (b) points A and C, and (c) points A and D.

What are the expected readings of the ammeter and voltmeter for the circuit in Figure P18.63?

Consider the two arrangements of batteries and bulbs shown in Figure P18.64. The two bulbs are identical and have resistance R, and the two batteries are identical with output voltage _V. (a) In case 1, with the two bulbs in series, compare the brightness of each bulb, the current in each bulb, and the power delivered to each bulb. (b) In case 2, with the two bulbs in parallel, compare the brightness of each bulb, the current in each bulb, and the power supplied to each bulb. (c) Which bulbs are brighter, those in case 1 or those in case 2? (d) In each case, if one bulb fails, will the other go out as well? If the other bulb doesnt fail, will it get brighter or stay the same? (Problem 64 is courtesy of E. F. Redish. For other problems of this type, visit http://www.physics.umd.edu/perg/.)

The given pair of capacitors in Figure P18.65 are fully charged by a 12.0-V battery. The battery is disconnected and the circuit closed. After 1.00 ms, how much charge remains on (a) the 3.00-mF capacitor? (b) The 2.00-mF capacitor? (c) What is the current in the resistor?

What is the equivalent resistance of the collection of resistors shown in Figure P18.66?

An electric eel generates electric currents through its highly specialized Hunters organ, in which thousands of disk-shaped cells called electrocytes are lined up in series, very much in the same way batteries are lined up inside a fl ashlight. When activated, each electrocyte can maintain a potential difference of about 150 mV at a current of 1 A for about 2.0 ms. Suppose a grown electric eel has 4.0 103 electrocytes and can deliver up to 300 shocks in rapid series over about 1 s. (a) What maximum electrical power can an electric eel generate? (b) Approximately how much energy does it release in one shock? (c) How high would a mass of 1 kg have to be lifted so that its gravitational potential energy equals the energy released in 300 such shocks?