Solution: The two identical light bulbs in Example 26.2

In which 120-V light bulb does the filament have greater resistance: a 60-W bulb or a 120-W bulb? If the two bulbs are connected to a 120-V line in series, through which bulb will there be the greater voltage drop? What if they are connected in parallel? Explain your reasoning

Two 120-V light bulbs, one 25-W and one 200-W, were connected in series across a 240-V line. It seemed like a good idea at the time, but one bulb burned out almost immediately. Which one burned out, and why?

You connect a number of identical light bulbs to a flashlight battery. (a) What happens to the brightness of each bulb as more and more bulbs are added to the circuit if you connect them (i) in series and (ii) in parallel? (b) Will the battery last longer if the bulbs are in series or in parallel? Explain your reasoning

In the circuit shown in Fig. Q26.4, three identical light bulbs are connected to a flashlight battery. How do the brightnesses of the bulbs compare? Which light bulb has the greatest current passing through it? Which light bulb has the greatest potential difference between its terminals? What happens if bulb A is unscrewed? Bulb B? Bulb C? Explain your reasoning.

If two resistors R1 and R2 1R2 7 R12 are connected in series as shown in Fig. Q26.5, which of the following must be true? In each case justify your answer. (a) I1 = I2 = I3. (b) The current is greater in R1 than in R2. (c) The electrical power consumption is the same for both resistors. (d) The electrical power consumption is greater in R2 than in R1. (e) The potential drop is the same across both resistors. (f) The potential at point a is the same as at point c. (g) The potential at point b is lower than at point c. (h) The potential at point c is lower than at point b.

If two resistors R1 and R2 1R2 7 R12 are connected in parallel as shown in Fig. Q26.6, which of the following must be true? In each case justify your answer. (a) I1 = I2. (b) I3 = I4. (c) The current is greater in R1 than in R2. (d) The rate of electrical energy consumption is the same for both resistors. (e) The rate of electrical energy consumption is greater in R2 than in R1. (f) Vcd = Vef = Vab. (g) Point c is at higher potential than point d. (h) Point f is at higher potential than point e. (i) Point c is at higher potential than point e

A battery with no internal resistance is connected across identical light bulbs as shown in Fig. Q26.7. When you close the switch S, will the brightness of bulbs B1 and B2 change? If so, how will it change? Explain.

A resistor consists of three identical metal strips connected as shown in Fig. Q26.8. If one of the strips is cut out, does the ammeter reading increase, decrease, or stay the same? Why?

A light bulb is connected in the circuit shown in Fig. Q26.9. If we close the switch S, does the bulbs brightness increase, decrease, or remain the same? Explain why

A real battery, having nonnegligible internal resistance, is connected across a light bulb as shown in Fig. Q26.10. When the switch S is closed, what happens to the brightness of the bulb? Why?

If the battery in Discussion Question Q26.10 is ideal with no internal resistance, what will happen to the brightness of the bulb when S is closed? Why?

Consider the circuit shown in Fig. Q26.12. What happens to the brightnesses of the bulbs when the switch S is closed if the battery (a) has no internal resistance and (b) has nonnegligible internal resistance? Explain why

Is it possible to connect resistors together in a way that cannot be reduced to some combination of series and parallel combinations? If so, give examples. If not, state why not.

The battery in the circuit shown in Fig. Q26.14 has no internal resistance. After you close the switch S, will the brightness of bulb B1 increase, decrease, or stay the same?

In a two-cell flashlight, the batteries are usually connected in series. Why not connect them in parallel? What possible advantage could there be in connecting several identical batteries in parallel?

Identical light bulbs A, B, and C are connected as shown in Fig. Q26.16. When the switch S is closed, bulb C goes out. Explain why. What happens to the brightness of bulbs A and B? Explain.

The emf of a flashlight battery is roughly constant with time, but its internal resistance increases with age and use. What sort of meter should be used to test the freshness of a battery?

Will the capacitors in the circuits shown in Fig. Q26.18 charge at the same rate when the switch S is closed? If not, in which circuit will the capacitors charge more rapidly? Explain.

Verify that the time constant RC has units of time.

For very large resistances it is easy to construct R-C circuits that have time constants of several seconds or minutes. How might this fact be used to measure very large resistances, those that are too large to measure by more conventional means?

When a capacitor, battery, and resistor are connected in series, does the resistor affect the maximum charge stored on the capacitor? Why or why not? What purpose does the resistor serve?

A uniform wire of resistance R is cut into three equal lengths. One of these is formed into a circle and connected between the other two (Fig. E26.1). What is the resistance between the opposite ends a and b?

A machine part has a resistor X protruding from an opening in the side. This resistor is connected to three other resistors, as shown in Fig. E26.2. An ohmmeter connected across a and b reads 2.00 . What is the resistance of X?

A resistor with R1 = 25.0 is connected to a battery that has negligible internal resistance and electrical energy is dissipated by R1 at a rate of 36.0 W. If a second resistor with R2 = 15.0 is connected in series with R1, what is the total rate at which electrical energy is dissipated by the two resistors?

A 42@ resistor and a 20@ resistor are connected in parallel, and the combination is connected across a 240-V dc line. (a) What is the resistance of the parallel combination? (b) What is the total current through the parallel combination? (c) What is the current through each resistor?

A triangular array of resistors is shown in Fig. E26.5. What current will this array draw from a 35.0-V battery having negligible internal resistance if we connect it across (a) ab; (b) bc; (c) ac? (d) If the battery has an internal resistance of 3.00 , what current will the array draw if the battery is connected across bc?

For the circuit shown in Fig. E26.6 both meters are idealized, the battery has no appreciable internal resistance, and the ammeter reads 1.25 A. (a) What does the voltmeter read? (b) What is the emf E of the battery?

For the circuit shown in Fig. E26.7 find the reading of the idealized ammeter if the battery has an internal resistance of \(3.26 \ \Omega\).

Three resistors having resistances of 1.60 , 2.40 , and 4.80 are connected in parallel to a 28.0-V battery that has negligible internal resistance. Find (a) the equivalent resistance of the combination; (b) the current in each resistor; (c) the total current through the battery; (d) the voltage across each resistor; (e) the power dissipated in each resistor. (f) Which resistor dissipates the most power: the one with the greatest resistance or the least resistance? Explain why this should be

Now the three resistors of Exercise 26.8 are connected in series to the same battery. Answer the same questions for this situation.

Power Rating of a Resistor. The power rating of a resistor is the maximum power the resistor can safely dissipate without too great a rise in temperature and hence damage to the resistor. (a) If the power rating of a 15@k resistor is 5.0 W, what is the maximum allowable potential difference across the terminals of the resistor? (b) A 9.0@k resistor is to be connected across a 120-V potential difference. What power rating is required? (c) A 100.0@ and a 150.0@ resistor, both rated at 2.00 W, are connected in series across a variable potential difference. What is the greatest this potential difference can be without overheating either resistor, and what is the rate of heat generated in each resistor under these conditions?

In Fig. E26.11, R1 = 3.00 , R2 = 6.00 , and R3 = 5.00 . The battery has negligible internal resistance. The current I2 through R2 is 4.00 A. (a) What are the currents I1 and I3? (b) What is the emf of the battery?

In Fig. E26.11 the battery has emf 35.0 V and negligible internal resistance. R1 = 5.00 . The current through R1 is 1.50 A, and the current through R3 = 4.50A. What are the resistances R2 and R3?

Compute the equivalent resistance of the network in Fig. E26.13, and find the current in each resistor. The battery has negligible internal resistance

Compute the equivalent resistance of the network in Fig. E26.14, and find the current in each resistor. The battery has negligible internal resistance.

In the circuit of Fig. E26.15, each resistor represents a light bulb. Let R1 = R2 = R3 = R4 = 4.50 and E = 9.00 V. (a) Find the current in each bulb. (b) Find the power dissipated in each bulb. Which bulb or bulbs glow the brightest? (c) Bulb R4 is now removed from the circuit, leaving a break in the wire at its position. Now what is the current in each of the remaining bulbs R1, R2, and R3? (d) With bulb R4 removed, what is the power dissipated in each of the remaining bulbs? (e) Which light bulb(s) glow brighter as a result of removing R4? Which bulb(s) glow less brightly? Discuss why there are different effects on different bulbs.

Consider the circuit shown in Fig. E26.16. The current through the 6.00@ resistor is 4.00 A, in the direction shown. What are the currents through the 25.0@ and 20.0@ resistors?

In the circuit shown in Fig. E26.17, the voltage across the 2.00@ resistor is 12.0 V. What are the emf of the battery and the current through the 6.00@ resistor?

In the circuit shown in Fig. E26.18, E = 36.0 V, R1 = 4.00 , R2 = 6.00 , and R3 = 3.00 . (a) What is the potential difference Vab between points a and b when the switch S is open and when S is closed? (b) For each resistor, calculate the current through the resistor with S open and with S closed. For each resistor, does the current increase or decrease when S is closed?

In the circuit in Fig. E26.19, a 20.0@ resistor is inside 100 g of pure water that is surrounded by insulating styrofoam. If the water is initially at 10.0C, how long will it take for its temperature to rise to 58.0C?

In the circuit shown in Fig. E26.20, the rate at which R1 is dissipating electrical energy is 15.0 W. (a) Find R1 and R2. (b) What is the emf of the battery? (c) Find the current through both R2 and the 10.0@ resistor. (d) Calculate the total electrical power consumption in all the resistors and the electrical power delivered by the battery. Show that your results are consistent with conservation of energy

Light Bulbs in Series and in Parallel. Two light bulbs have constant resistances of 400 and 800 . If the two light bulbs are connected in series across a 120-V line, find (a) the current through each bulb; (b) the power dissipated in each bulb; (c) the total power dissipated in both bulbs. The two light bulbs are now connected in parallel across the 120-V line. Find (d) the current through each bulb; (e) the power dissipated in each bulb; (f) the total power dissipated in both bulbs. (g) In each situation, which of the two bulbs glows the brightest? (h) In which situation is there a greater total light output from both bulbs combined?

Light Bulbs in Series. A 60-W, 120-V light bulb and a 200-W, 120-V light bulb are connected in series across a 240-V line. Assume that the resistance of each bulb does not vary with current. (Note: This description of a light bulb gives the power it dissipates when connected to the stated potential difference; that is, a 25-W, 120-V light bulb dissipates 25 W when connected to a 120-V line.) (a) Find the current through the bulbs. (b) Find the power dissipated in each bulb. (c) One bulb burns out very quickly. Which one? Why?

In the circuit shown in Fig. E26.23, ammeter A1 reads 10.0 A and the batteries have no appreciable internal resistance. (a) What is the resistance of R? (b) Find the readings in the other ammeters.

The batteries shown in the circuit in Fig. E26.24 have negligibly small internal resistances. Find the current through (a) the 30.0@ resistor; (b) the 20.0@ resistor; (c) the 10.0-V battery.

In the circuit shown in Fig. E26.25 find (a) the current in resistor R; (b) the resistance R; (c) the unknown emf E. (d) If the circuit is broken at point x, what is the current in resistor R?

Find the emfs E1 and E2 in the circuit of Fig. E26.26, and find the potential difference of point b relative to point a.

In the circuit shown in Fig. E26.27, find (a) the current in the \(3.00-\Omega\) resistor; (b) the unknown emfs \(\mathcal{E}_1\) and \(\mathcal{E}_2\); (c) the resistance R. Note that three currents are given.

In the circuit shown in Fig. E26.28, find (a) the current in each branch and (b) the potential difference Vab of point a relative to point b.

The 10.00-V battery in Fig. E26.28 is removed from the circuit and reinserted with the opposite polarity, so that its positive terminal is now next to point a. The rest of the circuit is as shown in the figure. Find (a) the current in each branch and (b) the potential difference Vab of point a relative to point b.

The 5.00-V battery in Fig. E26.28 is removed from the circuit and replaced by a 15.00-V battery, with its negative terminal next to point b. The rest of the circuit is as shown in the figure. Find (a) the current in each branch and (b) the potential difference \(V_{ab}\) of point a relative to point b.

In the circuit shown in Fig. E26.31 the batteries have negligible internal resistance and the meters are both idealized. With the switch S open, the voltmeter reads 15.0 V. (a) Find the emf E of the battery. (b) What will the ammeter read when the switch is closed?

In the circuit shown in Fig. E26.32 both batteries have insignificant internal resistance and the idealized ammeter reads 1.50 A in the direction shown. Find the emf E of the battery. Is the polarity shown correct?

In the circuit shown in Fig. E26.33 all meters are idealized and the batteries have no appreciable internal resistance. (a) Find the reading of the voltmeter with the switch S open. Which point is at a higher potential: a or b? (b) With S closed, find the reading of the voltmeter and the ammeter. Which way (up or down) does the current flow through the switch?

n the circuit shown in Fig. E26.34, the 6.0@ resistor is consuming energy at a rate of 24 J>s when the current through it flows as shown. (a) Find the current through the ammeter A. (b) What are the polarity and emf E of the unknown battery, assuming it has negligible internal resistance?

The resistance of a galvanometer coil is 25.0 , and the current required for full-scale deflection is 500 mA. (a) Show in a diagram how to convert the galvanometer to an ammeter reading 20.0 mA full scale, and compute the shunt resistance. (b) Show how to convert the galvanometer to a voltmeter reading 500 mV full scale, and compute the series resistance.

The resistance of the coil of a pivotedcoil galvanometer is 9.36 , and a current of 0.0224 A causes it to deflect full scale. We want to convert this galvanometer to an ammeter reading 20.0 A full scale. The only shunt available has a resistance of 0.0250 . What resistance R must be connected in series with the coil (Fig. E26.36)

A circuit consists of a series combination of 6.00@k and 5.00@k resistors connected across a 50.0-V battery having negligible internal resistance. You want to measure the true potential difference (that is, the potential difference without the meter present) across the 5.00@k resistor using a voltmeter having an internal resistance of 10.0 k. (a) What potential difference does the voltmeter measure across the 5.00@k resistor? (b) What is the true potential difference across this resistor when the meter is not present? (c) By what percentage is the voltmeter reading in error from the true potential difference?

A galvanometer having a resistance of 25.0 has a 1.00@ shunt resistance installed to convert it to an ammeter. It is then used to measure the current in a circuit consisting of a 15.0@ resistor connected across the terminals of a 25.0-V battery having no appreciable internal resistance. (a) What current does the ammeter measure? (b) What should be the true current in the circuit (that is, the current without the ammeter present)? (c) By what percentage is the ammeter reading in error from the true current?

A capacitor is charged to a potential of 12.0 V and is then connected to a voltmeter having an internal resistance of 3.40 M. After a time of 4.00 s the voltmeter reads 3.0 V. What are (a) the capacitance and (b) the time constant of the circuit?

You connect a battery, resistor, and capacitor as in Fig. 26.20a, where E = 36.0 V, C = 5.00 mF, and R = 120 . The switch S is closed at t = 0. (a) When the voltage across the capacitor is 8.00 V, what is the magnitude of the current in the circuit? (b) At what time t after the switch is closed is the voltage across the capacitor 8.00 V? (c) When the voltage across the capacitor is 8.00 V, at what rate is energy being stored in the capacitor?

A \(4.60-\mu F\) capacitor that is initially uncharged is connected in series with a \(7.50-\mathrm k \Omega\) resistor and an emf source with \(\mathcal{E} = 245 \ \mathrm V\) and negligible internal resistance. Just after the circuit is completed, what are (a) the voltage drop across the capacitor; (b) the voltage drop across the resistor; (c) the charge on the capacitor; (d) the current through the resistor? (e) A long time after the circuit is completed (after many time constants) what are the values of the quantities in parts (a)–(d)?

You connect a battery, resistor, and capacitor as in Fig. 26.20a, where R = 12.0 and C = 5.00 * 10-6 F. The switch S is closed at t = 0. When the current in the circuit has magnitude 3.00 A, the charge on the capacitor is 40.0 * 10-6 C. (a) What is the emf of the battery? (b) At what time t after the switch is closed is the charge on the capacitor equal to 40.0 * 10-6 C? (c) When the current has magnitude 3.00 A, at what rate is energy being (i) stored in the capacitor, (ii) supplied by the battery?

In the circuit shown in Fig. E26.43 both capacitors are initially charged to 45.0 V. (a) How long after closing the switch S will the potential across each capacitor be reduced to 10.0 V, and (b) what will be the current at that time?

A 12.4-mF capacitor is connected through a 0.895-M resistor to a constant potential difference of 60.0 V. (a) Compute the charge on the capacitor at the following times after the connections are made: 0, 5.0 s, 10.0 s, 20.0 s, and 100.0 s. (b) Compute the charging currents at the same instants. (c) Graph the results of parts (a) and (b) for t between 0 and 20 s.

An emf source with E = 120 V, a resistor with R = 80.0 , and a capacitor with C = 4.00 mF are connected in series. As the capacitor charges, when the current in the resistor is 0.900 A, what is the magnitude of the charge on each plate of the capacitor?

A resistor and a capacitor are connected in series to an emf source. The time constant for the circuit is 0.780 s. (a) A second capacitor, identical to the first, is added in series. What is the time constant for this new circuit? (b) In the original circuit a second capacitor, identical to the first, is connected in parallel with the first capacitor. What is the time constant for this new circuit?

In the circuit shown in Fig. E26.47 each capacitor initially has a charge of magnitude 3.50 nC on its plates. After the switch S is closed, what will be the current in the circuit at the instant that the capacitors have lost 80.0% of their initial stored energy?

A 1.50-mF capacitor is charging through a 12.0- resistor using a 10.0-V battery. What will be the current when the capacitor has acquired 1 4 of its maximum charge? Will it be 1 4 of the maximum current?

In the circuit in Fig. E26.49 the capacitors are initially uncharged, the battery has no internal resistance, and the ammeter is idealized. Find the ammeter reading (a) just after the switch S is closed and (b) after S has been closed for a very long time.

A \(12.0-\mu \mathrm{F}\) capacitor is charged to a potential of 50.0 V and then discharged through a \(225-\Omega\) resistor. How long does it take the capacitor to lose (a) half of its charge and (b) half of its stored energy?

In the circuit shown in Fig. E26.51, C = 5.90 mF, E = 28.0 V, and the emf has negligible resistance. Initially the capacitor is uncharged and the switch S is in position 1. The switch is then moved to position 2, so that the capacitor begins to charge. (a) What will be the charge on the capacitor a long time after S is moved to position 2? (b) After S has been in position 2 for 3.00 ms, the charge on the capacitor is measured to be 110 mC. What is the value of the resistance R? (c) How long after S is moved to position 2 will the charge on the capacitor be equal to 99.0% of the final value found in part (a)?

The heating element of an electric dryer is rated at 4.1 kW when connected to a 240-V line. (a) What is the current in the heating element? Is 12-gauge wire large enough to supply this current? (b) What is the resistance of the dryer’s heating element at its operating temperature? (c) At 11 cents per kWh, how much does it cost per hour to operate the dryer?

A 1500-W electric heater is plugged into the outlet of a 120-V circuit that has a 20-A circuit breaker. You plug an electric hair dryer into the same outlet. The hair dryer has power settings of 600 W, 900 W, 1200 W, and 1500 W. You start with the hair dryer on the 600-W setting and increase the power setting until the circuit breaker trips. What power setting caused the breaker to trip?

In Fig. P26.54, the battery has negligible internal resistance and E = 48.0 V. R1 = R2 = 4.00 and R4 = 3.00 . What must the resistance R3 be for the resistor network to dissipate electrical energy at a rate of 295 W?

The two identical light bulbs in Example 26.2 (Section 26.1) are connected in parallel to a different source, one with \(\mathcal{E} = 8.0 \ \mathrm V\) and internal resistance \(0.8 \ \Omega\). Each light bulb has a resistance \(R = 2.0 \ \Omega\) (assumed independent of the current through the bulb). (a) Find the current through each bulb, the potential difference across each bulb, and the power delivered to each bulb. (b) Suppose one of the bulbs burns out, so that its filament breaks and current no longer flows through it. Find the power delivered to the remaining bulb. Does the remaining bulb glow more or less brightly after the other bulb burns out than before?

Each of the three resistors in Fig. P26.56 has a resistance of 2.4 and can dissipate a maximum of 48 W without becoming excessively heated. What is the maximum power the circuit can dissipate?

(a) Find the potential of point a with respect to point b in Fig. P26.57. (b) If points a and b are connected by a wire with negligible resistance, find the current in the 12.0-V battery.

For the circuit shown in Fig. P26.58 a 20.0@ resistor is embedded in a large block of ice at 0.00C, and the battery has negligible internal resistance. At what rate (in g>s) is this circuit melting the ice? (The latent heat of fusion for ice is 3.34 * 105 J>kg.)

Calculate the three currents I1, I2, and I3 indicated in the circuit diagram shown in Fig. P26.59.

What must the emf E in Fig. P26.60 be in order for the current through the 7.00@ resistor to be 1.80 A? Each emf source has negligible internal resistance.

Find the current through each of the three resistors of the circuit shown in Fig. P26.61. The emf sources have negligible internal resistance.

(a) Find the current through the battery and each resistor in the circuit shown in Fig. P26.62. (b) What is the equivalent resistance of the resistor network?

Consider the circuit shown in Fig. P26.63. (a) What must the emf E of the battery be in order for a current of 2.00 A to flow through the 5.00-V battery as shown? Is the polarity of the battery correct as shown? (b) How long does it take for 60.0 J of thermal energy to be produced in the 10.0@ resistor?

In the circuit shown in Fig. P26.64, \(\mathcal{E} = 24.0 V, R_1 = 6.00 \ \Omega, R_3 = 12.0 \ \Omega\), and \(R_2\) can vary between \(3.00 \ \Omega\) and \(24.0 \ \Omega\). For what value of \(R_2\) is the power dissipated by heating element \(R_1\) the greatest? Calculate the magnitude of the greatest power.

In the circuit shown in Fig. P26.65, the current in the 20.0-V battery is 5.00 A in the direction shown and the voltage across the 8.00- resistor is 16.0 V, with the lower end of the resistor at higher potential. Find (a) the emf (including its polarity) of the battery X; (b) the current I through the 200.0-V battery (including its direction); (c) the resistance R.

In the circuit shown in Fig. P26.66 all the resistors are rated at a maximum power of 2.00 W. What is the maximum emf E that the battery can have without burning up any of the resistors?

Figure P26.67 employs a convention often used in circuit diagrams. The battery (or other power supply) is not shown explicitly. It is understood that the point at the top, labeled “36.0 V,” is connected to the positive terminal of a 36.0-V battery having negligible internal resistance, and that the ground symbol at the bottom is connected to the negative terminal of the battery. The circuit is completed through the battery, even though it is not shown. (a) What is the potential difference \(V_{a b}\), the potential of point a relative to point b, when the switch S is open? (b) What is the current through S when it is closed? (c) What is the equivalent resistance when S is closed?

. Figure P26.67 employs a convention often used in circuit diagrams. The battery (or other power supply) is not shown explicitly. It is understood that the point at the top, labeled 36.0 V, is connected to the positive terminal of a 36.0-V battery having negligible internal resistance, and that the ground symbol at the bottom is connected to the negative terminal of the battery. The circuit is completed through the battery, even though it is not shown. (a) What is the potential difference Vab, the potential of point a relative to point b, when the switch S is open? (b) What is the current through S when it is closed? (c) What is the equivalent resistance when S is closed?

A resistor R1 consumes electrical power P1 when connected to an emf E. When resistor R2 is connected to the same emf, it consumes electrical power P2. In terms of P1 and P2, what is the total electrical power consumed when they are both connected to this emf source (a) in parallel and (b) in series?

The capacitor in Fig. P26.70 is initially uncharged. The switch S is closed at t = 0. (a) Immediately after the switch is closed, what is the current through each resistor? (b) What is the final charge on the capacitor?

A 2.00-mF capacitor that is initially uncharged is connected in series with a 6.00-k resistor and an emf source with E = 90.0 V and negligible internal resistance. The circuit is completed at t = 0. (a) Just after the circuit is completed, what is the rate at which electrical energy is being dissipated in the resistor? (b) At what value of t is the rate at which electrical energy is being dissipated in the resistor equal to the rate at which electrical energy is being stored in the capacitor? (c) At the time calculated in part (b), what is the rate at which electrical energy is being dissipated in the resistor?

A 6.00-mF capacitor that is initially uncharged is connected in series with a 5.00- resistor and an emf source with E = 50.0 V and negligible internal resistance. At the instant when the resistor is dissipating electrical energy at a rate of 300 W, how much energy has been stored in the capacitor?

Point a in Fig. P26.73 is maintained at a constant potential of 400 V above ground. (See Problem 26.67.) (a) What is the reading of a voltmeter with the proper range and with resistance 5.00 * 104 when connected between point b and ground? (b) What is the reading of a voltmeter with resistance 5.00 * 106 ? (c) What is the reading of a voltmeter with infinite resistance?

The Wheatstone Bridge. The circuit shown in Fig. P26.74, called a Wheatstone bridge, is used to determine the value of an unknown resistor X by comparison with three resistors M, N, and P whose resistances can be varied. For each setting, the resistance of each resistor is precisely known. With switches S1 and S2 closed, these resistors are varied until the current in the galvanometer G is zero; the bridge is then said to be balanced. (a) Show that under this condition the unknown resistance is given by X = MP>N. (This method permits very high precision in comparing resistors.) (b) If galvanometer G shows zero deflection when M = 850.0 , N = 15.00 , and P = 33.48 , what is the unknown resistance X?

(See Problem 26.67.) (a) What is the potential of point a with respect to point b in Fig. P26.75 when the switch S is open? (b) Which point, a or b, is at the higher potential? (c) What is the final potential of point b with respect to ground when S is closed? (d) How much does the charge on each capacitor change when S is closed?

A 2.36-mF capacitor that is initially uncharged is connected in series with a 5.86- resistor and an emf source with E = 120 V and negligible internal resistance. (a) Just after the connection is made, what are (i) the rate at which electrical energy is being dissipated in the resistor; (ii) the rate at which the electrical energy stored in the capacitor is increasing; (iii) the electrical power output of the source? How do the answers to parts (i), (ii), and (iii) compare? (b) Answer the same questions as in part (a) at a long time after the connection is made. (c) Answer the same questions as in part (a) at the instant when the charge on the capacitor is one-half its final value.

A 224- resistor and a 589@ resistor are connected in series across a 90.0-V line. (a) What is the voltage across each resistor? (b) A voltmeter connected across the 224- resistor reads 23.8 V. Find the voltmeter resistance. (c) Find the reading of the same voltmeter if it is connected across the 589- resistor. (d) The readings on this voltmeter are lower than the true voltages (that is, without the voltmeter present). Would it be possible to design a voltmeter that gave readings higher than the true voltages? Explain

A resistor with R = 850 is connected to the plates of a charged capacitor with capacitance C = 4.62 mF. Just before the connection is made, the charge on the capacitor is 6.90 mC. (a) What is the energy initially stored in the capacitor? (b) What is the electrical power dissipated in the resistor just after the connection is made? (c) What is the electrical power dissipated in the resistor at the instant when the energy stored in the capacitor has decreased to half the value calculated in part (a)?

A capacitor that is initially uncharged is connected in series with a resistor and an emf source with E = 110 V and negligible internal resistance. Just after the circuit is completed, the current through the resistor is 6.5 * 10-5 A. The time constant for the circuit is 5.2 s. What are the resistance of the resistor and the capacitance of the capacitor?

You set up the circuit shown in Fig. 26.22a, where R = 196 . You close the switch at time t = 0 and measure the magnitude i of the current in the resistor R as a function of time t since the switch was closed. Your results are shown in Fig. P26.80, where you have chosen to plot ln i as a function of t. (a) Explain why your data points lie close to a straight line. (b) Use the graph in Fig. P26.80 to calculate the capacitance C and the initial charge Q0 on the capacitor. (c) When i = 0.0500 A, what is the charge on the capacitor? (d) When q = 0.500 * 10-4 C, what is the current in the resistor?

You set up the circuit shown in Fig. 26.20, where C = 5.00 * 10-6 F. At time t = 0, you close the switch and then measure the charge q on the capacitor as a function of the current i in the resistor. Your results are given in the table: i 1mA2 56.0 48.0 40.0 32.0 24.0 q 1mC2 10.1 19.8 30.2 40.0 49.9 (a) Graph q as a function of i. Explain why the data points, when plotted this way, fall close to a straight line. Find the slope and y-intercept of the straight line that gives the best fit to the data. (b) Use your results from part (a) to calculate the resistance R of the resistor and the emf E of the battery. (c) At what time t after the switch is closed is the voltage across the capacitor equal to 10.0 V? (d) When the voltage across the capacitor is 4.00 V, what is the voltage across the resistor?

The electronics supply company where you work has two different resistors, R1and R2, in its inventory, and you must measure the values of their resistances. Unfortunately, stock is low, and all you have are R1 and R2 in parallel and in seriesand you cant separate these two resistor combinations. You separately connect each resistor network to a battery with emf 48.0 V and negligible internal resistance and measure the power P supplied by the battery in both cases. For the series combination, P = 48.0 W; for the parallel combination, P = 256 W. You are told that R1 7 R2. (a) Calculate R1 and R2. (b) For the series combination, which resistor consumes more power, or do they consume the same power? Explain. (c) For the parallel combination, which resistor consumes more power, or do they consume the same power?

An Infinite Network. As shown in Fig. P26.83, a network of resistors of resistances R1 and R2 extends to infinity toward the right. Prove that the total resistance RT of the infinite network is equal to RT = R1 + 2R1 2 + 2R1R2 (Hint: Since the network is infinite, the resistance of the network to the right of points c and d is also equal to RT.)

Suppose a resistor R lies along each edge of a cube (12 resistors in all) with connections at the corners. Find the equivalent resistance between two diagonally opposite corners of the cube (points a and b in Fig. P26.84).

Attenuator Chains and Axons. The infinite network of resistors shown in Fig. P26.83 is known as an attenuator chain, since this chain of resistors causes the potential difference between the upper and lower wires to decrease, or attenuate, along the length of the chain. (a) Show that if the potential difference between the points a and b in Fig. 26.83 is Vab, then the potential difference between points c and d is Vcd = Vab>11 + b2, where b = 2R11RT + R22>RTR2 and RT, the total resistance of the network, is given in Challenge Problem 26.83. (See the hint given in that problem.) (b) If the potential difference between terminals a and b at the left end of the infinite network is V0, show that the potential difference between the upper and lower wires n segments from the left end is Vn = V0>11 + b2n . If R1 = R2, how many segments are needed to decrease the potential difference Vn to less than 1.0% of V0? (c) An infinite attenuator chain provides a model of the propagation of a voltage pulse along a nerve fiber, or axon. Each segment of the network in Fig. P26.83 represents a short segment of the axon of length x. The resistors R1 represent the resistance of the fluid inside and outside the membrane wall of the axon. The resistance of the membrane to current flowing through the wall is represented by R2. For an axon segment of length x = 1.0 mm, R1 = 6.4 * 103 and R2 = 8.0 * 108 (the membrane wall is a good insulator). Calculate the total resistance RT and b for an infinitely long axon. (This is a good approximation, since the length of an axon is much greater than its width; the largest axons in the human nervous system are longer than 1 m but only about 10-7 m in radius.) (d) By what fraction does the potential difference between the inside and outside of the axon decrease over a distance of 2.0 mm? (e) The attenuation of the potential difference calculated in part (d) shows that the axon cannot simply be a passive, current-carrying electrical cable; the potential difference must periodically be reinforced along the axons length. This reinforcement mechanism is slow, so a signal propagates along the axon at only about 30 m>s. In situations where faster response is required, axons are covered with a segmented sheath of fatty myelin. The segments are about 2 mm long, separated by gaps called the nodes of Ranvier. The myelin increases the resistance of a 1.0@mm@long segment of the membrane to R2 = 3.3 * 1012 . For such a myelinated axon, by what fraction does the potential difference between the inside and outside of the axon decrease over the distance from one node of Ranvier to the next? This smaller attenuation means the propagation speed is increased.

Assume that a typical open ion channel spanning an axons membrane has a resistance of 1 * 1011 . We can model this ion channel, with its pore, as a 12-nm-long cylinder of radius 0.3 nm. What is the resistivity of the fluid in the pore? (a) 10 # m; (b) 6 # m; (c) 2 # m; (d) 1 # m.

In a simple model of an axon conducting a nerve signal, ions move across the cell membrane through open ion channels, which act as purely resistive elements. If a typical current density (current per unit cross-sectional area) in the cell membrane is 5 mA>cm2 when the voltage across the membrane (the action potential) is 50 mV, what is the number density of open ion channels in the membrane? (a) 1>cm2 ; (b) 10>cm2 ; (c) 10>mm2 ; (d) 100>mm2

Cell membranes across a wide variety of organisms have a capacitance per unit area of 1 mF>cm2 . For the electrical signal in a nerve to propagate down the axon, the charge on the membrane capacitor must change. What time constant is required when the ion channels are open? (a) 1 ms; (b) 10 ms; (c) 100 ms; (d) 1 ms.