Chapter 26

(I) Calculate the terminal voltage for a battery with an internal resistance of \(0.900 \Omega\) and an emf of \(6.00 \mathrm{~V}\) when the battery is connected in series with \((a)\) an \(81.0-\Omega\) resistor, and \((b)\) an \(810-\Omega\) resistor.

IThe Problems in this Section are ranked I, II, or III according to estimated difficulty, with (I) Problems being easiest. Level (III) Problems are meant mainly as a challenge for the best students, for "extra credit." The Problems are arranged by Sections, meaning that the reader should have read up to and including that Section, but this Chapter also has a group of General Problems that are not arranged by Section and not ranked.] $$ \begin{array}{l}{\text { (I) Calculate the terminal voltage for a battery with an }} \\ {\text { internal resistance of } 0.900 \Omega \text { and an emf of } 6.00 \mathrm{V} \text { when the }} \\ {\text { battery is connected in series with }(a) \text { an } 81.0-\Omega \text { resistor, and }} \\ {\text { (b) an } 810-\Omega \text { resistor. }}\end{array} $$

(I) Four \(1.50-\mathrm{V}\) cells are connected in series to a \(12-\Omega\) lightbulb. If the resulting current is \(0.45 \mathrm{~A},\) what is the internal resistance of each cell, assuming they are identical and neglecting the resistance of the wires?

(1) Four \(1.50-\mathrm{V}\) cells are connected in series to a \(12-\Omega\) light bulb. If the resulting current is 0.45 \(\mathrm{A}\) , what is the internal resistance of each cell, assuming they are identical and neglecting the resistance of the wires?

(II) A 1.5-V dry cell can be tested by connecting it to a lowresistance ammeter. It should be able to supply at least 25 A. What is the internal resistance of the cell in this case, assuming it is much greater than that of the ammeter?

(II) A \(1.5-\mathrm{V}\) dry cell can be tested by connecting it to a low- resistance ammeter. It should be able to supply at least 25 A. What is the internal resistance of the cell in this case, assuming it is much greater than that of the ammeter?

(II) What is the internal resistance of a \(12.0-\mathrm{V}\) car battery whose terminal voltage drops to 8.4 \(\mathrm{V}\) when the starter draws 95 \(\mathrm{A}\) ? What is the resistance of the starter?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (I) \(\mathrm{A} 650-\Omega\) and a \(2200-\Omega\) resistor are connected in series with a 12-V battery. What is the voltage across the \(2200-\Omega\) resistor?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (I) Three \(45-\Omega\) lightbulbs and three \(65-\Omega\) lightbulbs are connected in series. ( \(a\) ) What is the total resistance of the circuit? \((b)\) What is the total resistance if all six are wired in parallel?

(I) Three \(45-\Omega\) lightbulbs and three \(65-\Omega\) lightbulbs are connected in series \((a)\) What is the total resistance of the circuit? (b) What is the total resistance if all six are wired in parallel?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (I) Suppose that you have a \(680-\Omega,\) a \(720-\Omega,\) and a \(1.20-\mathrm{k} \Omega\) resistor. What is \((a)\) the maximum, and \((b)\) the minimum resistance you can obtain by combining these?

(I) Suppose that you have a \(680-\Omega,\) a \(720-\Omega,\) and a \(1.20-\mathrm{k} \Omega\) resistor. What is \((a)\) the maximum, and \((b)\) the minimum resistance you can obtain by combining these?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (I) How many \(10-\Omega\) resistors must be connected in series to give an equivalent resistance to five \(100-\Omega\) resistors connected in parallel?

(I) How many \(10-\Omega\) resistors must be connected in series to give an equivalent resistance to five \(100-\Omega\) resistors connected in parallel?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (II) Suppose that you have a 9.0-V battery and you wish to apply a voltage of only \(4.0 \mathrm{~V}\). Given an unlimited supply of \(1.0-\Omega\) resistors, how could you connect them so as to make a "voltage divider" that produces a \(4.0-\mathrm{V}\) output for a \(9.0-\mathrm{V}\) input?

(II) Suppose that you have a \(9.0-\mathrm{V}\) battery and you wish to apply a voltage of only 4.0 \(\mathrm{V}\) . Given an unlimited supply of \(1.0-\Omega\) resistors, how could you connect them so as to make a "voltage divider" that produces a \(4.0-\mathrm{V}\) output for a \(9.0-\mathrm{V}\) input?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (II) Three \(1.70-\mathrm{k} \Omega\) resistors can be connected together in four different ways, making combinations of series and/or parallel circuits. What are these four ways, and what is the net resistance in each case?

(II) Three \(1.70-\mathrm{k} \Omega\) resistors can be connected together in four different ways, making combinations of series and/or parallel circuits. What are these four ways, and what is the net resistance in each case?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (II) A battery with an emf of \(12.0 \mathrm{~V}\) shows a terminal voltage of \(11.8 \mathrm{~V}\) when operating in a circuit with two lightbulbs, each rated at \(4.0 \mathrm{~W}\) (at \(12.0 \mathrm{~V}\) ), which are connected in parallel. What is the battery's internal resistance?

(II) A battery with an emf of 12.0 \(\mathrm{V}\) shows a terminal voltage of 11.8 \(\mathrm{V}\) when operating in a circuit with two light-bulbs, each rated at \(4.0 \mathrm{W}(\) at 12.0 \(\mathrm{V}),\) which are connected in parallel. What is the battery's internal resistance?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (II) Eight identical bulbs are connected in series across a 110-V line. ( \(a\) ) What is the voltage across each bulb? (b) If the current is \(0.42 \mathrm{~A}\), what is the resistance of each bulb, and what is the power dissipated in each?

(II) Eight identical bulbs are connected in series across a \(110-\mathrm{V}\) line. \((a)\) What is the voltage across each bulb? \((b)\) If the current is \(0.42 \mathrm{A},\) what is the resistance of each bulb, and what is the power dissipated in each?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (II) Eight bulbs are connected in parallel to a \(110-\mathrm{V}\) source by two long leads of total resistance \(1.4 \Omega\). If \(240 \mathrm{~mA}\) flows through each bulb, what is the resistance of each, and what fraction of the total power is wasted in the leads?

(II) Eight bulbs are connected in parallel to a \(110-\mathrm{V}\) source by two long leads of total resistance 1.4\(\Omega .\) If 240 \(\mathrm{mA}\) flows through each bulb, what is the resistance of each, and what fraction of the total power is wasted in the leads?

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (II) The performance of the starter circuit in an automobile can be significantly degraded by a small amount of corrosion on a battery terminal. Figure \(26-38 \mathrm{a}\) depicts a properly functioning circuit with a battery (12.5-V emf, \(0.02-\Omega\) internal resistance) attached via corrosion-free cables to a starter motor of resistance \(\quad R_{\mathrm{S}}=0.15 \Omega\). Suppose that later, corrosion between a battery terminal and a starter cable introduces an extra series resistance of just \(R_{\mathrm{C}}=0.10 \Omega\) into the circuit as suggested in Fig. \(26-38 \mathrm{~b} .\) Let \(P_{0}\) be the power delivered to the starter in the circuit free of corrosion, and let \(P\) be the power delivered to the circuit with corrosion. Determine the ratio \(P / P_{0}\).

(II) The performance of the starter circuit in an automobile can be significantly degraded by a small amount of corrosion on a battery terminal. Figure 38 depicts a properly functioning circuit with a battery \((12.5-\mathrm{V}\) emf, \(0.02-\Omega\) internal resistance \()\) attached via corrosion-free cables to a starter motor of resistance \(R_{\mathrm{S}}=0.15 \Omega\) Suppose that later, corrosion between a battery terminal and a starter cable introduces an extra series resistance of just \(R_{C}=0.10 \Omega\) into the circuit as suggested in Fig. 38 \(\mathrm{b}\) . Let \(P_{0}\) be the power delivered to the starter in the circuit free of corrosion, and let \(P\) be the power delivered to the circuit with corrosion. Determine the ratio \(P / P_{0}\) .

In these Problems neglect the internal resistance of a battery unless the Problem refers to it. (II) A close inspection of an electric circuit reveals that a \(480-\Omega\) resistor was inadvertently soldered in the place where a \(370-\Omega\) resistor is needed. How can this be fixed without removing anything from the existing circuit?

(II) A close inspection of an electric circuit reveals that a \(480-\Omega\) resistor was inadvertently soldered in the place where a \(370-\Omega\) resistor is needed. How can this be fixed without removing anything from the existing circuit?

(II) A \(75-\mathrm{W}, 110-\mathrm{V}\) bulb is connected in parallel with a \(25-\mathrm{W}, 110-\mathrm{V}\) bulb. What is the net resistance?

(II) \(\mathrm{A}\) 75-W, \(110-\mathrm{V}\) bulb is connected in parallel with a \(25-\mathrm{W}, 110-\mathrm{V}\) bulb. What is the net resistance?

(II) The two terminals of a voltage source with emf \(\mathscr{E}\) and internal resistance \(r\) are connected to the two sides of a load resistance \(R\). For what value of \(R\) will the maximum power be delivered from the source to the load?

(II) The two terminals of a voltage source with \(\operatorname{emf} \mathscr{E}\) and in ternal resistance \(r\) are connected to the two sides of a load redistance \(R .\) For what value of \(R\) will the maximum power by delivered from the source to the load?

(II) Two resistors when connected in series to a \(110-\mathrm{V}\) line use one-fourth the power that is used when they are connected in parallel. If one resistor is \(3.8 \mathrm{k} \Omega\), what is the resistance of the other?

(III) A \(2.8-\mathrm{k} \Omega\) and a 3.7 \(\mathrm{k} \Omega\) resistor are connected in parallel; this combination is connected in series with a \(1.8-\mathrm{k} \Omega\) resistor. If each resistor is rated at \(\frac{1}{2} \mathrm{W}\) (maximum without overheating), what is the maximum voltage that can be applied across the whole network?

(II) A voltage \(V\) is applied to \(n\) identical resistors connected in parallel. If the resistors are instead all connected in series with the applied voltage, show that the power transformed is decreased by a factor \(n^{2}\).

(II) Two 3.8- \(\mu\) F capacitors, two \(2.2-\mathrm{k} \Omega\) resistors, and a \(12.0-\mathrm{V}\) source are connected in series. Starting from the uncharged state, how long does it take for the current to drop from its initial value to \(1.50 \mathrm{~mA} ?\)

(II) A parallel-plate capacitor is filled with a dielectric of dielectric constant \(K\) and high resistivity \(\rho\) (it conducts very slightly). This capacitor can be modeled as a pure capacitance \(C\) in parallel with a resistance \(R\). Assume a battery places a charge \(+Q\) and \(-Q\) on the capacitor's opposing plates and is then disconnected. Show that the capacitor discharges with a time constant \(\tau=K \varepsilon_{0} \rho\) (known as the dielectric relaxation time). Evaluate \(\tau\) if the dielectric is glass with \(\rho=1.0 \times 10^{12} \Omega \cdot \mathrm{m}\) and \(K=5.0\).

(I) What is the resistance of a voltmeter on the \(250-\mathrm{V}\) scale if the meter sensitivity is \(35,000 \Omega / \mathrm{V} ?\)

(II) A galvanometer has an internal resistance of \(32 \Omega\) and deflects full scale for a \(55-\mu\) A current. Describe how to use this galvanometer to make \((a)\) an ammeter to read currents up to \(25 \mathrm{~A}\), and \((b)\) a voltmeter to give a full scale deflection $$\text { of } 250 \mathrm{~V} \text { . }$$

(II) A particular digital meter is based on an electronic module that has an internal resistance of 100 \(\mathrm{MS}\) and a full-scale sensitivity of 400 \(\mathrm{mV}\) . Two resistors connected as shown in Fig. 63 can be used to change the voltage range. Assume \(R_{1}=10 \mathrm{M} \Omega\) . Find the value of \(R_{2}\) that will result in a voltmeter with a full-scale range of 40 \(\mathrm{V} .\)

(II) A milliammeter reads 25 \(\mathrm{mA}\) full scale. It consists of a \(0.20-\Omega\) resistor in parallel with a \(33-\Omega\) galvanometer. How can you change this ammeter to a voltmeter giving a full scale reading of 25 \(\mathrm{V}\) without taking the ammeter apart? What will be the sensitivity \((\Omega / \mathrm{V})\) of your voltmeter?

(II) A 45-V battery of negligible internal resistance is connected to a \(44-\mathrm{k} \Omega\) and a \(27-\mathrm{k} \Omega\) resistor in series. What reading will a voltmeter, of internal resistance \(95 \mathrm{k} \Omega,\) give when used to measure the voltage across each resistor? What is the percent inaccuracy due to meter resistance for each case?

(II) A \(45-\mathrm{V}\) battery of negligible internal resistance is to a \(44-\mathrm{k} \Omega\) and a \(27-\mathrm{k} \Omega\) resistor in series. What reading will a voltmeter, of internal resistance 95 \(\mathrm{k} \Omega\) , give when used to measure the voltage across each resistor? What is the percent inaccuracy due to meter resistance for each case?

(II) An ammeter whose internal resistance is \(53 \Omega\) reads \(5.25 \mathrm{~mA}\) when connected in a circuit containing a battery and two resistors in series whose values are \(650 \Omega\) and \(480 \Omega\). What is the actual current when the ammeter is absent?

(II) An ammeter whose internal resistance is 53\(\Omega\) reads 5.25 \(\mathrm{mA}\) when connected in a circuit containing a battery and two resistors in series whose values are 650\(\Omega\) and 480\(\Omega .\) What is the actual current when the ammeter is absent?

(II) A battery with \(\mathscr{E}=12.0 \mathrm{~V}\) and internal resistance \(r=1.0 \Omega\) is connected to two \(7.5-\mathrm{k} \Omega\) resistors in series. An ammeter of internal resistance \(0.50 \Omega\) measures the current, and at the same time a voltmeter with internal resistance \(15 \mathrm{k} \Omega\) measures the voltage across one of the \(7.5-\mathrm{k} \Omega\) resistors in the circuit. What do the ammeter and voltmeter read?

(II) A battery with \(\operatorname{} \mathscr{E}$$=12.0 \mathrm{V} \quad\)and internal resistance \(r=1.0 \Omega\) is connected to two \(7.5-\mathrm{k} \Omega\) resistors in series. An ammeter of internal resistance 0.50\(\Omega\) measures the current, and at the same time a voltmeter with internal resistance 15 \(\mathrm{k} \Omega\) measures the voltage across one of the \(7.5-\mathrm{k} \Omega\) read? in the circuit. What do the ammeter and voltmeter read?

(II) A 12.0-V battery (assume the internal resistance \(=0\) ) is connected to two resistors in series. A voltmeter whose internal resistance is \(18.0 \mathrm{k} \Omega\) measures \(5.5 \mathrm{~V}\) and \(4.0 \mathrm{~V}\) respectively, when connected across each of the resistors. What is the resistance of each resistor?

(II) A \(12.0-\mathrm{V}\) battery (assume the internal resistance \(=0 )\) is connected to two resistors in series. A voltmeter whose internal resistance is 18.0 \(\mathrm{k} \Omega\) measures 5.5 \(\mathrm{V}\) and 4.0 \(\mathrm{V}\) respectively, when connected across each of the resistors. What is the resistance of each resistor?

(III) Two 9.4-k \(\Omega\) resistors are placed in series and connected to a battery. A voltmeter of sensitivity \(1000 \Omega / \mathrm{V}\) is on the 3.0-V scale and reads \(2.3 \mathrm{~V}\) when placed across either resistor. What is the emf of the battery? (Ignore its internal resistance.)