Chapter 25

On your first day at work as an electrical technician, you are asked to determine the resistance per meter of a long piece of wire. The company you work for is poorly equipped. You find a battery, a voltmeter, and an ammeter, but no meter for directly measuring resistance (an ohmmeter). You put the leads from the voltmeter across the terminals of the battery, and the meter reads 12.6 V. You cut off a 20.0-m length of wire and connect it to the battery, with an ammeter in series with it to measure the current in the wire. The ammeter reads 7.00 A. You then cut off a 40.0-m length of wire and connect it to the battery, again with the ammeter in series to measure the current. The ammeter reads 4.20 A. Even though the equipment you have available to you is limited, your boss assures you of its high quality: The ammeter has very small resistance, and the voltmeter has very large resistance. What is the resistance of 1 meter of wire?

A 2.0-m length of wire is made by welding the end of a 120-cm-long silver wire to the end of an 80-cm-long copper wire. Each piece of wire is 0.60 mm in diameter. The wire is at room temperature, so the resistivities are as given in Table 25.1. A potential difference of 9.0 V is maintained between the ends of the 2.0-m composite wire. What is (a) the current in the copper section; (b) the current in the silver section; (c) the magnitude of $\overrightarrow{E}$ in the copper; (d) the magnitude of $\overrightarrow{E}$ in the silver; (e) the potential difference between the ends of the silver section of wire?

A 3.00-m length of copper wire at 20${}^{\circ }$C has a 1.20-mlong section with diameter 1.60 mm and a 1.80-m-long section with diameter 0.80 mm. There is a current of 2.5 mA in the 1.60- mm-diameter section. (a) What is the current in the 0.80-mmdiameter section? (b) What is the magnitude of $\overrightarrow{E}$ in the 1.60-mm-diameter section? (c) What is the magnitude of $\overrightarrow{E}$ in the 0.80-mm-diameter section? (d) What is the potential difference between the ends of the 3.00-m length of wire?

A resistor with resistance $R$ is connected to a battery that has emf 12.0 V and internal resistance $r=$ 0.40$\mathrm{\Omega }$. For what two values of $R$ will the power dissipated in the resistor be 80.0 W?

The region between two concentric conducting spheres with radii $a$ and $b$ is filled with a conducting material with resistivity $\rho$. (a) Show that the resistance between the spheres is given by $$R=\frac{\rho }{4\pi }(\frac{1}{a}-\frac{1}{b})$$(b) Derive an expression for the current density as a function of radius, in terms of the potential difference $V_{ab}$ between the spheres. (c) Show that the result in part (a) reduces to Eq. (25.10) when the separation $L=b-a$ between the spheres is small.

The potential difference across the terminals of a battery is 8.40 V when there is a current of 1.50 A in the battery from the negative to the positive terminal. When the current is 3.50 A in the reverse direction, the potential difference becomes 10.20 V. (a) What is the internal resistance of the battery? (b) What is the emf of the battery?

The average bulk resistivity of the human body (apart from surface resistance of the skin) is about 5.0$\mathrm{\Omega }$ $\cdot$ m. The conducting path between the hands can be represented approximately as a cylinder 1.6 m long and 0.10 m in diameter. The skin resistance can be made negligible by soaking the hands in salt water. (a) What is the resistance between the hands if the skin resistance is negligible? (b) What potential difference between the hands is needed for a lethal shock current of 100 mA? (Note that your result shows that small potential differences produce dangerous currents when the skin is damp.) (c) With the current in part (b), what power is dissipated in the body?

A person with body resistance between his hands of 10 k$\mathrm{\Omega }$ accidentally grasps the terminals of a 14-kV power supply. (a) If the internal resistance of the power supply is 2000 $\mathrm{\Omega }$, what is the current through the person's body? (b) What is the power dissipated in his body? (c) If the power supply is to be made safe by increasing its internal resistance, what should the internal resistance be for the maximum current in the above situation to be 1.00 mA or less?

A typical cost for electrical power is $0.120 per kilowatthour. (a) Some people leave their porch light on all the time. What is the yearly cost to keep a 75-W bulb burning day and night? (b) Suppose your refrigerator uses 400 W of power when it's running, and it runs 8 hours a day. What is the yearly cost of operating your refrigerator?

Unlike the idealized ammeter described in Section 25.4, any real ammeter has a nonzero resistance. (a) An ammeter with resistance $R_A$ is connected in series with a resistor $R$ and a battery of emf $\epsilon$ and internal resistance r. The current measured by the ammeter is $I_A$. Find the current through the circuit if the ammeter is removed so that the battery and the resistor form a complete circuit. Express your answer in terms of $I_A$, $r$, $R_A$, and $R$. The more "ideal" the ammeter, the smaller the difference between this current and the current IA. (b) If $R$ = 3.80 $\mathrm{\Omega }$, $\epsilon$ = 7.50 V, and $r$ = 0.45 $\mathrm{\Omega }$, find the maximum value of the ammeter resistance $R_A$ so that $I_A$ is within 1.0% of the current in the circuit when the ammeter is absent. (c) Explain why your answer in part (b) represents a $maximum$ value.