Chapter 26

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

Suppose that you wish to apply a \(0.25-\mathrm{V}\) potential difference between two points on the human body. The resistance is about \(1800 \Omega,\) and you only have a \(1.5-\mathrm{V}\) battery. How can you connect up one or more resistors to produce the desired voltage?

A three-way lightbulb can produce \(50 \mathrm{~W}, 100 \mathrm{~W},\) or \(150 \mathrm{~W}\) at \(120 \mathrm{~V}\). Such a bulb contains two filaments that can be connected to the \(120 \mathrm{~V}\) individually or in parallel. (a) Describe how the connections to the two filaments are made to give each of the three wattages. (b) What must be the resistance of each filament?

Suppose you want to run some apparatus that is \(65 \mathrm{~m}\) from an electric outlet. Each of the wires connecting your apparatus to the \(120-\mathrm{V}\) source has a resistance per unit length of \(0.0065 \Omega / \mathrm{m} .\) If your apparatus draws \(3.0 \mathrm{~A},\) what will be the voltage drop across the connecting wires and what voltage will be applied to your apparatus?

A heart pacemaker is designed to operate at 72 beats/min using a \(6.5-\mu \mathrm{F}\) capacitor in a simple \(R C\) circuit. What value of resistance should be used if the pacemaker is to fire (capacitor discharge) when the voltage reaches \(75 \%\) of maximum?

Suppose that a person's body resistance is \(950 \Omega .(a)\) What current passes through the body when the person accidentally is connected to \(110 \mathrm{~V} ?(b)\) If there is an alternative path to ground whose resistance is \(35 \Omega\), what current passes through the person? \((c)\) If the voltage source can produce at most \(1.5 \mathrm{~A}\), how much current passes through the person in case \((b) ?\)

Suppose that a person's body resistance is 950\(\Omega\) (a) What current passes through the body when the person accidentally is connected to 110 \(\mathrm{V} ?(b)\) If there is an alternative path to ground whose resistance is \(35 \Omega,\) what current passes through the person? \((c)\) If the voltage source can produce at most 1.5 \(\mathrm{A}\) , how much current passes through the person in case \((b) ?\)

A Wheatstone bridge is a type of "bridge circuit" used to make measurements of resistance. The unknown resistance to be measured, \(R_{x},\) is placed in the circuit with accurately known resistances \(R_{1}, R_{2},\) and \(R_{3}\) (Fig. \(\left.26-65\right) .\) One of these, \(R_{3},\) is a variable resistor which is adjusted so that when the switch is closed momentarily, the ammeter ( \(\mathrm{A}\) shows zero current flow. (a) Determine \(R_{x}\) in terms of \(R_{1}, R_{2},\) and \(R_{3}\) (b) If a Wheatstone bridge is "balanced" when \(R_{1}=630 \Omega, \quad R_{2}=972 \Omega\) and \(R_{3}=78.6 \Omega,\) what is the value of the unknown resistance?

An unknown length of platinum wire \(1.22 \mathrm{~mm}\) in diameter is placed as the unknown resistance in a Wheatstone bridge (see Problem 71 , Fig. \(26-65\) ). Arms 1 and 2 have resistance of \(38.0 \Omega\) and \(29.2 \Omega,\) respectively. Balance is achieved when \(R_{3}\) is \(3.48 \Omega .\) How long is the platinum wire?

Terminal voltage of mercury cell \((3.99 \mathrm{~V})\) is closer to \(4.0 \mathrm{~V}\) than terminal voltage of dry cell \((3.84 \mathrm{~V})\)

The internal resistance of a \(1.35-\mathrm{V}\) mercury cell is 0.030\(\Omega\) , whereas that of a 1.5 \(\mathrm{N}\) dry cell is 0.35\(\Omega .\) Explain why three mercury cells can more effectively power a \(2.5-\mathrm{W}\) hearing aid that requires 4.0 \(\mathrm{V}\) than can three dry cells.

How many \(\frac{1}{2}\) -W resistors, each of the same resistance, must be used to produce an equivalent \(3.2-\mathrm{k} \Omega, 3.5-\mathrm{W}\) resistor? What is the resistance of each, and how must they be connected? Do not exceed \(P=\frac{1}{2} \mathrm{~W}\) in each resistor.

How many \(\frac{1}{2}-\mathrm{W}\) resistors, each of the same resistance, must be used to produce an equivalent \(3.2-\mathrm{k} \Omega, 3.5-\mathrm{W}\) resistor? What is the resistance of each, and how must they be connected? Do not exceed \(P=\frac{1}{2} \mathrm{W}\) in each resistor.

A solar cell, \(3.0 \mathrm{~cm}\) square, has an output of \(350 \mathrm{~mA}\) at \(0.80 \mathrm{~V}\) when exposed to full sunlight. A solar panel that delivers close to \(1.3 \mathrm{~A}\) of current at an emf of \(120 \mathrm{~V}\) to an external load is needed. How many cells will you need to create the panel? How big a panel will you need, and how should you connect the cells to one another? How can you optimize the output of your solar panel?

A battery produces \(40.8 \mathrm{~V}\) when \(7.40 \mathrm{~A}\) is drawn from it, and \(47.3 \mathrm{~V}\) when \(2.80 \mathrm{~A}\) is drawn. What are the \(\mathrm{emf}\) and internal resistance of the battery?

A flashlight bulb rated at \(2.0 \mathrm{~W}\) and \(3.0 \mathrm{~V}\) is operated by a 9.0-V battery. To light the bulb at its rated voltage and power, a resistor \(R\) is connected in series as shown in Fig. 26-72. What value should the resistor have?

A potentiometer is a device to precisely measure potential differences or emf, using a "null" technique. In the simple potentiometer circuit shown in Fig. \(26-74, R^{\prime}\) represents the total resistance of the resistor from \(\mathrm{A}\) to \(\mathrm{B}\) (which could be a long uniform "slide" wire), whereas \(R\) represents the resistance of only the part from A to the movable contact at C. When the unknown emf to be measured, \(\mathscr{E}_{x},\) is placed into the circuit as shown, the movable contact \(\mathrm{C}\) is moved until the galvanometer G gives a null reading (i.e., zero) when the switch \(S\) is closed. The resistance between \(A\) and \(C\) for this situation we call \(R_{x} .\) Next, a standard \(\mathrm{emf}, \mathscr{E}_{\mathrm{s}},\) which is known precisely, is inserted into the circuit in place of \(\mathscr{E}_{x}\) and again the contact \(\mathrm{C}\) is moved until zero current flows through the galvanometer when the switch \(\mathrm{S}\) is closed. The resistance between \(A\) and \(C\) now is called \(R_{s} \cdot(a)\) Show that the unknown emf is given by $$\mathscr{E}_{x}=\left(\frac{R_{x}}{R_{\mathrm{s}}}\right) \mathscr{E}_{\mathrm{s}}$$ where \(R_{x}, R_{\mathrm{s}},\) and \(\mathscr{E}_{\mathrm{s}}\) are all precisely known. The working battery is assumed to be fresh and to give a constant voltage. (b) A slide-wire potentiometer is balanced against a 1.0182-V standard cell when the slide wire is set at \(33.6 \mathrm{~cm}\) out of a total length of \(100.0 \mathrm{~cm} .\) For an unknown source, the setting is \(45.8 \mathrm{~cm}\). What is the emf of the unknown? \((c)\) The galvanometer of a potentiometer has an internal resistance of \(35 \Omega\) and can detect a current as small as \(0.012 \mathrm{~mA}\). What is the minimum uncertainty possible in measuring an unknown voltage? (d) Explain the advantage of using this "null" method of measuring emf.

Measurements made on circuits that contain large resistances can be confusing. Consider a circuit powered by a battery \(\mathscr{E}=15.000 \mathrm{~V}\) with a \(10.00-\mathrm{M} \Omega\) resistor in series with an unknown resistor \(R\). As shown in Fig. \(26-80\), a particular voltmeter reads \(V_{1}=366 \mathrm{mV}\) when connected across the \(10.00-\mathrm{M} \Omega\) resistor, and this meter reads \(V_{2}=7.317 \mathrm{~V}\) when connected across \(R\). Determine the value of \(R\).

(II) An \(R C\) series circuit contains a resistor \(R=15 \mathrm{k} \Omega\), a capacitor \(C=0.30 \mu \mathrm{F},\) and a battery of emf \(\mathscr{E}=9.0 \mathrm{~V}\) Starting at \(t=0\), when the battery is connected, determine the charge \(Q\) on the capacitor and the current \(I\) in the circuit from \(t=0\) to \(t=10.0 \mathrm{~ms}\) (at 0.1-ms intervals). Make graphs showing how the charge \(Q\) and the current \(I\) change with time within this time interval. From the graphs find the time at which the charge attains \(63 \%\) of its final value, \(C^{\mathscr{C}},\) and the current drops to \(37 \%\) of its initial value, \(\mathscr{E} / R\).