A real battery, having nonnegligible internal resistance,

Problem 1DQ In which 120V 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.

BIO Attenuator Chains and Axons. The infinite network of resistors shown in Fig. P26.91 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. 26.91 is \(V_{a b}\), then the potential difference between points \(c\) and \](d\) is \(V_{c d}=V_{a b} /(1+\beta)\), where \(\beta=2 R_1\left(R_{\mathrm{T}}+R_2\right) / R_{\mathrm{T}} R_2\) and \(R_{\mathrm{T}}\), the total resistance of the network, is given in Challenge Problem 26.91. (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 \(V_0\), show that the potential difference between the upper and lower wires \(n\) segments from the left end is \(V_n=V_0 /(1+\beta)^n\). If \(R_1=R_2\), how many segments are needed to decrease the potential difference \(V_n\) to less than 1.0 % of \(V_0\)? (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. P26.91 represents a short segment of the axon of length \(\Delta x\). The resistors \(R_1\) 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 \(R_2\). For an axon segment of length \(\Delta x=1.0 \mu \mathrm{m}\), \(R_1=6.4 \times 10^3 \Omega\) and \(R_2=8.0 \times 10^8 \Omega\) (the membrane wall is a good insulator). Calculate the total resistance \(R_{\mathrm{T}}\) and \(\beta\) 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 \(1 \mathrm{~m}\) but only about \(10^{-7} \mathrm{~m}\) in radius.) (d) By what fraction does the potential difference between the inside and outside of the axon decrease over a distance of \(2.0 \mathrm{~mm}\)? (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 axon's length. This reinforcement mechanism is slow, so a signal propagates along the axon at only about \(30 \mathrm{~m} / \mathrm{s}\). In situations where faster response is required, axons are covered with a segmented sheath of fatty myelin. The segments are about \(2 \mathrm{~mm}\) long, separated by gaps called the nodes of Ranvier. The myelin increases the resistance of a \(1.0-\mu \mathrm{m}\)-long segment of the membrane to \(R_2=3.3 \times 10^{12} \Omega\). 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.

Problem 1E 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 a ? nd ?b??

Problem 2DQ 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?

Problem 3DQ 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.

Problem 2E 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??

Problem 4DQ 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 ? ?? Bulb? ? Explain your reasoning.

Problem 4E A 32 ? 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?

Problem 3E A resistor with ?R?1 = 25.0 ? is connected to a battery that has negligible internal resistance and electrical energy is dissipated by ?R?1 at a rate of 36.0 W. If a second resistor with ?R?2 = 15.0 ? is connected in series with ?R?1, what is the total rate at which electrical energy is dissipated by the two resistors?

Problem 5DQ If two resistors ?R?1 and ?R?2 (?R?2 > ?R?1) are connected in series as shown in ?Fig. Q26.5?, which of the following must be true? In each case justify your answer. (a) I?1 = ?I?2 = ?I?3. (b) The current is greater in ?R?1 than in ?R?2. (c) The electrical power consumption is the same for both resistors. (d) The electrical power consumption is greater in ?R?2 than in ?R?1. (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?.

Problem 7DQ Why do the lights on a car become dimmer when the starter is operated?

Problem 6DQ If two resistors ?R?1 and ?R?2 (?R?2 > ?R?1) are connected in parallel as shown in ?Fig. Q26.6?, which of the following must be true? In each case justify your answer. (a) I?1 = ?I?2. (b) ?I?3 = ?I?4. (c) The current is greater in ?R?1 than in ?R?2. (d) The rate of electrical energy consumption is the same for both resistors. (e) The rate of electrical energy consumption is greater in ?R?2 than in ?R?1. (f) ?V?cd? ?= ?V?ef? ?= ?V?ab. (g) Point ?c ?is at higher potential than point ?d?. (h) Point ?f ?is at higher potential than point ?e?. (i) Point? ?is at higher potential than point ?e?.

Problem 5E 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??

Problem 6E 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 of the battery?

Problem 7E 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 ?.

Problem 8DQ 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?

Three resistors having resistances of 1.60 \(\Omega\), 2.40 \(\Omega\), and 4.80 \(\Omega\) 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.

Problem 10E 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?

Problem 10DQ 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?

Problem 11DQ 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 ? s closed? Why? Q26.10 ?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?

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

Problem 9E Now the three resistors of Exercise 26.8 are connected in series to the same battery. Answer the same questions for this situation. 26.8 . ?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.

Problem 11E In ?Fig. E26.11?, ?R?1 = 3.00 ?, ?R?2 = 6.00 ?, and ?R?3 = 5.00 ?. The battery has negligible internal resistance. The current ?I?2 through ?R?2 is 4.00 A. (a) What are the currents ?I?1 and ?I?3? (b) What is the emf of the battery?

Problem 12DQ 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.

Problem 13DQ 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.

Problem 12E In below Fig. the battery has emf 25.0 V and negligible internal resistance, ?R?1 = 5.00 ?. The current through ?R?1 is 1.50 A and the current through ?R?3 = 4.50 A. What are the resistances ? ?2 and ?R?3? Figure:

Problem 14DQ The direction of current in a battery can be reversed by connecting it to a second battery of greater emf with the positive terminals of the two batteries together. When the direction of current is reversed in a battery, does its emf also reverse? Why or why not?

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

Problem 14E 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 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?

Problem 15E In the circuit of ?Fig. E26.15?, each resistor represents a light bulb. Let ?R?1 = ?R?2 = R?3 = ?R?4 = 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 R?4 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 ?R?1, ?R?2, and ?R?3? (d) With bulb ?R?4 removed, what is the power dissipated in each of the remaining bulbs? (e) Which light bulb(s) glow brighter as a result of removing ?R?4? Which bulb(s) glow less brightly? Discuss why there are different effects on different bulbs.

Problem 16E 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?

Problem 16DQ The greater the diameter of the wire used in household wiring, the greater the maximum current that can safely be carried by the wire. Why is this? Does the maximum permissible current depend on the length of the wire? Does it depend on what the wire is made of? Explain your reasoning.

Problem 17DQ 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?

Problem 18DQ Is it possible to have a circuit in which the potential difference across the terminals of a battery in the circuit is zero? If so, give an example. If not, explain why not.

Problem 18E A Three-Way Light Bulb.? A three-way light bulb has three brightness settings (low, medium, and high) but only two filaments. (a) A particular three-way light bulb connected across a 120-V line can dissipate 60 W, 120 W, or 180 W. Describe how the two filaments are arranged in the bulb, and calculate the resistance of each filament. (b) Suppose the filament with the higher resistance burns out. How much power will the bulb dissipate on each of the three brightness settings? What will be the brightness (low, medium, or high) one ach setting? (c) Repeat part (b) for the situation in which the filament with the lower resistance burns out.

Problem 17E 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?

Problem 19E Working Late!? You are working late in your electronics shop and find that you need various resistors for a project. But alas, all you have is a big box of 10.0? resistors. Show how you can make each of the following equivalent resistances by a combination of your 10.0? resistors: (a) 35 ?, (b) 1.0 ?, (c) 3.33 ?, (d) 7.5 ?.

Problem 19DQ Verify that the time constant ?RC? as units of time.

Problem 20E In the circuit shown in ?Fig. E26.20?, the rate at which ?R?1 is dissipating electrical energy is 15.0 W. (a) Find ?R?1 and ?R?2. (b) What is the emf of the battery? (c) Find the current through both ?R?2 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.

Problem 21DQ 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?

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?

Problem 22E 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?

Problem 21E 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?

Problem 23E CP ?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 10.0°C, how long will it take for its temperature to rise to 58.0°C?

Problem 24E 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.

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

Problem 27E In the circuit shown in ?Fig. E26.27?, find (a) the current in the 3.00-? resistor; (b) the unknown emfs ; (c) the resistance ?R?. Note that three currents are given.

Problem 26E Find the emfs in the circuit of ? ig. E26.26,? and find the potential difference of point ?b r? elative to point ? ?.

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

Problem 29E 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 ?V? b? ?of point ?a ?relative to point ?b?.

Problem 32E 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 of the battery. Is the polarity shown correct?

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 /(\mathcal{E}\) of the battery. (b) What will the ammeter read when the switch is closed?

Problem 30E The 5.00-V battery in below Fig. is removed from the circuit and replaced by a 20.00-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 ? ab? point? ? relative to point ?b. Figure:

Problem 34E In 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 of the unknown battery, assuming it has negligible internal resistance?

Problem 35E The resistance of a galvanometer coil is 25.0 ?, and the current required for full-scale deflection is 500 µA. (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.

Problem 33E 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?

Problem 36E The resistance of the coil of a pivoted-coil 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 galvanometer having a resistance of \(25.0 \Omega\) has a \(1.00-\Omega\) shunt resistance installed to convert it to an ammeter. It is then used to measure the current in a circuit consisting of a \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?

Problem 39E In the ohmmeter in below Fig.M is a 2.50-mA meter of resistance 65.0 ?. (A 2.50-mA meter deflects full scale when the current through it is 2.50 m A.) The battery ?B? has an emf of 1.52 V and negligible internal resistance. ?R? is chosen so that when the terminals ?a? and ?b? are shorted (?Rx? = 0), the meter reads full scale. When ?a? and ?b? are open [?Rx? = ?], the meter reads zero. (a) What is the resistance of the resistor ?R?? (b) What current indicates a resistance ?Rx? of 200?? (c) What values of ?Rx? correspond to meter defections of ¼, ½ and ¾ of full scale if the deflection is proportional to the current through the galvanometer? Figure:

Problem 37E 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?

Problem 40E A 4.60-µ?F ?capacitor that is initially uncharged is connected in series with a 7.50-k? resistor and an emf source with = 245 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)?

Problem 41E 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?

Problem 42E A 12.4-µF 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? ?between 0 and 20 s.

Problem 43E CP ?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?

Problem 44E A resistor and a capacitor are connected in series to an emf source. The time constant for the circuit is 0.870 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?

Problem 46E A 1.50-µF capacitor is charging through a 12.0-? resistor using a 10.0-V battery. What will be the current when the capacitor has acquired 14 of its maximum charge? Will it be of the maximum current?

Problem 45E An emf source with = 120 V, a resistor with ? ?= 80.0 ?, and a capacitor with C ?= 4.00 µF 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?

Problem 48E A 12.0- ???F capacitor is charged to a potential of 50.0 V and then discharged through a 175-? 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 in Fig. E26.49 the capacitors are all initially uncharged, the battery has no internal resistance, and the ammeter is idealized. Find the reading of the ammeter (a) just after the switch S is closed and (b) after the switch has been closed for a very long time.

Problem 47E CP ?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?

Problem 52E 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?

Problem 50E In the circuit shown in ? ig. E26.51?, ?C ?= 5.90 mF, = 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 µC. 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)?

Problem 54E The heating element of an electric stove consists of a heater wire embedded within an electrically insulating material, which in turn is inside a metal casing. The heater wire has a resistance of 20 ? at room temperature (23.0°C) and a temperature coefficient of resistivity ??? = 2.8 × 10?1(C°)?1. The heating element operates from a 120-V line. (a) When the heating element is first turned on, what current does it draw and what electrical power does it dissipate? (b) When the heating element has reached an operating temperature of 280°C (536°F), what current does it draw and what electrical power does it dissipate?

A capacitor with \(C=1.50 \times 10^{-5} \mathrm{F}\) is connected as shown in Fig. E26.50 with a resistor with \(R=980 \Omega\) and an emf source with \(\mathcal{E}=18.0 \mathrm{V}\) and negligible internal 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. After the switch has been in position 2 for 10.0 ms, the switch is moved back to position 1 so that the capacitor begins to discharge. (a) Compute the charge on the capacitor just before the switch is thrown from position 2 back to position 1. (b) Compute the voltage drops across the resistor and across the capacitor at the instant described in part (a). (c) Compute the voltage drops across the resistor and across the capacitor just after the switch is thrown from position 2 back to position 1. (d) Compute the charge on the capacitor 10.0 ms after the switch is thrown from position 2 back to position 1.

Problem 56P A 400?, 2.4-W resistor is needed, but only several 400?, 1.2 W resistors are available (see below Exercise). (a) What two different combinations of the available units give the required resistance and power rating? (b) For each of the resistor networks from part (a), what power is dissipated in each resistor when 2.4 W is dissipated by the combination? EXERCISE: 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. P26.55, the battery has negligible internal resistance and \(\mathcal{E}=48.0 \mathrm{~V} . R_1=\) \(R_2=4.00 \Omega\) and \(R_4=3.00 \Omega\). What must the resistance \(R_3\) be for the resistor network to dissipate electrical energy at a rate of \(295 \mathrm{~W}\)?

CP A 20.0-m-long cable consists of a solid-inner, cylindrical, nickel core 10.0 cm in diameter surrounded by a solid-outer cylindrical shell of copper 10.0 cm in inside diameter and 20.0 cm in outside diameter. The resistivity of nickel is \(7.8 \times 10^{-8} \Omega \cdot \mathrm{m}\). (a) What is the resistance of this cable? (b) If we think of this cable as a single material, what is its equivalent resistivity?

Problem 58P Two identical 3.00-? wires are laid side by side and soldered together so they touch each other for half of their lengths. What is the equivalent resistance of this combination?

Problem 59P The two identical light bulbs in Example 26.2 (Section 26.1) are connected in parallel to a different source, one with 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. P26.60 has a resistance of 2.4 \(\Omega\) and can dissipate a maximum of 48 W without becoming excessively heated. What is the maximum power the circuit can dissipate?

Problem 61P If an ohmmeter is connected between points ?a? and ?b? in each of the circuits shown in below Fig, what will it read? Figure:

Problem 63P Calculate the three currents ?I?1, ?I?2, and ?I?3 indicated in the circuit diagram shown in ?Fig. P26.59?.

Problem 64P What must the emf in ?Fig. P26.60? e in order for the current through the 7.00-? resistor to be 1.80 A? Each emf source has negligible internal resistance.

Problem 62P CP ?For the circuit shown in ?Fig. P26.58 ?a 20.0-? resistor is embedded in a large block of ice at 0.00°C, 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.)

Problem 65P Find the current through each of the three resistors of the circuit shown in ?Fig. P26.61.? The emf sources have negligible internal resistance.

Problem 66P (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?

(a) Find the potential of point a with respect to point b in Fig. P26.67. (b) If points a and b are connected by a wire with negligible resistance, find the current in the 12.0-V battery.

Problem 69P A 1.00 km cable having a cross-sectional area 0.500 cm2 is to be constructed out of equal lengths of copper and aluminum. This could be accomplished either by making a 0.50-km cable of each one and welding them together end to end or by making two parallel 1.00-km cables, one of each metal (below Fig.) Calculate the resistance of the 1.00-km cable for both designs to see which one provides the least resistance. Figure:

Problem 68P Consider the circuit shown in ?Fig. P26.63?. (a) What must the emf 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. P26.71, the current in the 20.0-V battery is 5.00 A in the direction shown and the voltage across the \(8.00-\Omega\) 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.

Problem 72P Three identical resistors are connected in series. When a certain potential difference is applied across the combination, the total power dissipated is 36 W. What power would be dissipated if the three resistors were connected in parallel across the same potential difference?

In the circuit shown in Fig. P26.70 all the resistors are rated at a maximum power of 2.00 W. What is the maximum emf \(\mathcal{E}\) that the battery can have without burning up any of the resistors?

Problem 73P A resistor ?R?1 consumes electrical power ?P?1 when connected to an emf . When resistor ?R?2 is connected to the same emf, it consumes electrical power ?P?2. In terms of ?P?1 and ?P?2, what is the total electrical power consumed when they are both connected to this emf source (a) in parallel and (b) in series?

Problem 74P 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?

Problem 75P A 2.00-mF capacitor that is initially uncharged is connected in series with a 6.00-k? resistor and an emf source with = 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?

Problem 76P A 6.00-???F capacitor that is initially uncharged is connected in series with a 5.00-? resistor and an emf source with ??? = 50.0 V and negligible internal resistance. At the instant when the resistor is dissipating electrical energy at a rate of 250 W. How much energy hits been stored in the capacitor?

Problem 77P 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?ab, 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?

Problem 80P A 150-V voltmeter has a resistance of 30,000 ?. When connected in series with a large resistance ?R? across a 110-V line, the meter reads 74 V. Find the resistance ?R?.

Problem 82P A 2.36-µF capacitor that is initially uncharged is connected in series with a 5.86-? resistor and an emf source with = 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.

Problem 79P 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?

Problem 83P 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.

Problem 84P A resistor with ?R ?= 850 ? is connected to the plates of a charged capacitor with capacitance ?C ?= 4.62 µF. 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)?

Problem 81P The Wheatstone Bridge.? The circuit shown in below Fig, 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 K1 and K2 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 the galvanometer G shows zero deflection when ?M? = 850.0 ?, ?N? = 15.00 ? and ?P? = 33.48 ?. What is the unknown resistance ?X?? Figure:

Problem 85P A capacitor that is initially uncharged is connected in series with a resistor and an emf source with = 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?

Problem 78P (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 i ? s closed?

Problem 86P An ?RC? circuit has a time constant ?RC?. (a) If the circuit is discharging, how long will it take for its stored energy to be reduced to 1/?e? of its initial value? (b) If it is charging, how long will it take for the stored energy to reach 1/?e? of its maximum value?

Problem 87P Strictly speaking, Eq. (26.16) implies that an ?infinite ?amount of time is required to discharge a capacitor completely. Yet for practical purposes, a capacitor may be considered to be fully discharged after a finite length of time. To be specific, consider a capacitor with capacitance ?C ?connected to a resistor ?R ?to be fully discharged if its charge ?q ?differs from zero by no more than the charge of one electron. (a) Calculate the time required to reach this required to reach this state always the same number of time constants, independent of the values of ?C a? nd ?? Why or why not?

CALC The current in a charging capacitor is given by Eq. (26.13). (a) The instantaneous power supplied by the battery is \(\mathcal{E} i\). Integrate this to find the total energy supplied by the battery. (b) The instantaneous power dissipated in the resistor is \(i^{2} R\). Integrate this to find the total energy dissipated in the resistor. (c) Find the final energy stored in the capacitor, and show that this equals the total energy supplied by the battery less the energy dissipated in the resistor, as obtained in parts (a) and (b). (d) What fraction of the energy supplied by the battery is stored in the capacitor? How does this fraction depend on R?

Problem 90CP A Capacitor Burglar Alarm. The capacitance of a capacitor can be affected by dielectric material that, although not inside the capacitor, is near enough to the capacitor to be polarized by the fringing electric field that exists near a charged capacitor. This effect is usually of the order of picofarads (pF), but it can be used with appropriate electronic circuitry to detect a change in the dielectric material surrounding the capacitor. Such a dielectric material might be the human body, and the effect described above might be used in the design of a burglar alarm. Consider the simplified circuit shown in Fig. The voltage source has emf ??? = 1000 V. and the capacitor has capacitance ?C? = 10.0 pF. The electronic circuitry for detecting the current, represented as an ammeter in the diagram, has negligible resistance and is capable of detecting a current that persists at a level of at least 1.00 ???A for at least 200 ???s after the capacitance has changed abruptly from ?C? to C ? ?’. The burglar alarm is designed to be activated if the capacitance changes by 10%. (a) Determine the charge on the 10.0-pF capacitor when it is fully charged. (b) If the capacitor is fully charged before the intruder is detected, assuming that the time taken for the capacitance to change by 10% is short enough to be ignored, derive an equation that expresses the current through the resistor ?R as a function of the time ?t? since the capacitance has changed. (c) Determine the range of values of the resistance ?R? that will meet the design specifications of the burglar alarm. What happens if ?R? is too small? Too large? (?Hint?: You will not be able to solve this part analytically but must use numerical methods. Express ?R? as a logarithmic function of ?R? plus known quantities. Use a trial value of ?R? and calculate from the expression a new value. Continue to do this until the input and output values of ?R? agree to within three significant figures.) Figure:

Problem 89P (a) Using Eq. (26.17) for the current in a discharging capacitor, derive an expression for the instantaneous power dissipated in the resistor. (b) Integrate the expression for ?P ?to find the total energy dissipated in the resistor, and show that this is equal to the total energy initially stored in the capacitor.

Problem 91CP An Infinite Network. ?As shown in ?Fig. P26.83,? a network of resistors of resistances ?R?1 and ?R?2 extends to infinity toward the right. Prove that the total resistance R? ?T of the infinite network is equal to (?Hint: ?Since the network is infinite, the resistance of the network to the right of points ?c ? nd ?d ?is also equal to? .)

Problem 92CP 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 F? ig. P26.84)? .