Chapter 25

Lightning Strikes. During lightning strikes from a cloud to the ground, currents as high as \(25,000\) A can occur and last for about 40\(\mu\) s. How much charge is transferred from the cloud to the earth during such a strike?

A silver wire 2.6 \(\mathrm{mm}\) in diameter transfers a charge of 420 \(\mathrm{C}\) in 80 \(\mathrm{min}\) . Silver contains \(5.8 \times 10^{28}\) free electrons per cubic meter. (a) What is the current in the wire? (b) What is the magnitude of the drift velocity of the electrons in the wire?

A 5.00 -A current runs through a 12 -gauge copper wire (diameter 2.05 \(\mathrm{mm}\) ) and through a light bulb. Copper has \(8.5 \times\) \(10^{28}\) free electrons per cubic meter, (a) How many electrons pass through the light bulb each second? (b) What is the current density in the wire? (c) At what speed does a typical electron pass by any given point in the wire? (d) If you were to use wire of twice the diameter, which of the above answers would change? Would they increase or decrease?

An 18 -gauge copper wire (diameter 1.02 \(\mathrm{mm} )\) carries a current with a current density of \(1.50 \times 10^{6} \mathrm{A} / \mathrm{m}^{2} .\) The density of free electrons for copper is \(8.5 \times 10^{28}\) electrons per cubic meter. ,Calculate (a) the current in the wire and (b) the drift velocity of electrons in the wire.

Copper has \(8.5 \times 10^{28}\) free electrons per cubic meter. A 71.0 -cm length of 12 -gauge copper wire that is 2.05 \(\mathrm{mm}\) in diameter carries 4.85 A of current. (a) How much time does it take for an electron to travel the length of the wire? (b) Repeat part (a) for 6-gauge copper wire (diameter 4.12 \(\mathrm{mm}\) ) of the same length that carries the same current.(c) Generally speaking, how does changing the diameter of a wire that carries a given amount of current affect the drift velocity of the electrons in the wire?

The current in a wire varies with time according to the relationship \(I=55 \mathrm{A}-\left(0.65 \mathrm{A} / \mathrm{s}^{2}\right) t^{2}\) . (a) How many coulombs of charge pass a cross section of the wire in the time interval between \(t=0\) and \(t=8.0 \mathrm{s} ?\) (b) What constant current would transport the same charge in the same time interval?

Current passes through a solution of sodium chloride. In \(1.00 \mathrm{s}, 2.68 \times 10^{16} \mathrm{Na}^{+}\) ions arrive at the negative electrode and \(3.92 \times 10^{16} \mathrm{Cl}^{-}\) ions arrive at the positive electrode. (a) What is the current passing between the electrodes? (b) What is the direction of the current?

Transmission of Nerve Impulses. Nerve cells transmit electric signals through their long tubular axons. These signals propagate due to a sudden rush of \(\mathrm{Na}^{+}\) ions, each with charge \(+e,\) into the axon. Measurements have revealed that typically about \(5.6 \times 10^{11} \mathrm{Na}^{+}\) ions enter each meter of the axon during a time of 10 \(\mathrm{ms}\) . What is the current during this inflow of charge in a meter of axon?

A 1.50 -m cylindrical rod of diameter 0.500 \(\mathrm{cm}\) is connected to a power supply that maintains a constant potential difference of 15.0 \(\mathrm{V}\) across its ends, while an ammeter measures the current through it. You observe that at room temperature \(\left(20.0^{\circ} \mathrm{C}\right)\) the ammeter reads \(18.5 \mathrm{A},\) while at \(92.0^{\circ} \mathrm{C}\) it reads 17.2 \(\mathrm{A} .\) You can ignore any thermal expansion of the rod. Find (a) the resistivity at \(20.0^{\circ} \mathrm{C}\) and \((\mathrm{b})\) the temperature coefficient of resistivity at \(20^{\circ} \mathrm{C}\) for the material of the rod.

A copper wire has a square cross section 2.3 \(\mathrm{mm}\) on a side. The wire is 4.0 \(\mathrm{m}\) long and carries a current of 3.6 \(\mathrm{A}\) . The density of free electrons is \(8.5 \times 10^{28} / \mathrm{m}^{3} .\) Find the magnitudes of (a) the current density in the wire and (b) the electric field in the wire. (c) How much time is required for an electron to travel the length of the wire?

A wire 6.50 \(\mathrm{m}\) long with diameter of 2.05 \(\mathrm{mm}\) has a resistance of 0.0290\(\Omega .\) What material is the wire most likely made of?

A ductile metal wire has resistance \(R .\) What will be the resistance of this wire in terms of \(R\) if it is stretched to three times its original length, assuming that the density and resistivity of the material do not change when the wire is stretched? (Hint: The amount of metal does not change, so stretching out the wire will affect its cross-sectional area.)

A tightly coiled spring having 75 coils, each 3.50 \(\mathrm{cm}\) in diameter, is made of insulated metal wire 3.25 \(\mathrm{mm}\) in diameter. An ohmmeter connected across its opposite ends reads 1.74\(\Omega\) ? What is the resistivity of the metal?

You apply a potential difference of 4.50 \(\mathrm{V}\) between the ends of a wire that is 2.50 \(\mathrm{m}\) in length and 0.654 \(\mathrm{mm}\) in radius. The resulting current through the wire is 17.6 \(\mathrm{A}\) . What is the resistivity of the wire?

A current-carrying gold wire has diameter 0.84 \(\mathrm{mm}\) . The electric field in the wire is 0.49 \(\mathrm{V} / \mathrm{m} .\) What are (a) the current carried by the wire; (b) the potential difference between two points in the wire 6.4 \(\mathrm{m}\) apart; \((\mathrm{c})\) the resistance of a \(6.4-\mathrm{m}\) length of this wire?

A strand of wire has resistance 5.60\(\mu \Omega .\) Find the net resistance of 120 such strands if they are (a) placed side by side to form a cable of the same length as a single strand, and (b) connected end to end to form a wire 120 times as long as a single strand.

The following measurements were made on a Thyrite resistor: $$\begin{array}{llll}{I(\mathbf{A})} & {0.50} & {1.00} & {2.00} & {4.00} \\\ {V_{a b}(\mathbf{V})} & {2.55} & {3.11} & {3.77} & {4.58}\end{array}$$ (a) Graph \(V_{a b}\) as a function of \(I .(\mathrm{b})\) Does Thyrite obey Ohm's law? How can you tell? (c) Graph the resistance \(R=V_{a b} / I\) as a function of \(I\) .

The following measurements of current and potential difference were made on a resistor constructed of Nichrome wire: $$\begin{array}{llll}{\boldsymbol{I}(\mathbf{A})} & {0.50} & {1.00} & {2.00} & {4.00} \\ {\boldsymbol{V}_{a b}(\mathbf{V})} & {1.94} & {3.88} & {7.76} & {15.52}\end{array}$$ (a) Graph \(V_{a b}\) as a function of \(I\) (b) Does Nichrome obey Ohm's law? How can you tell? (c) What is the resistance of the resistor in ohms?

Light Bulbs. The power rating of a light bulb (such as a \(100-\mathrm{W}\) bulb) is the power it dissipates when connected across a \(120-\mathrm{V}\) potential difference. What is the resistance of (a) a \(100-\mathrm{W}\) bulb and (b) a \(60-\mathrm{W}\) bulb? (c) How much current does each bulb draw in normal use?

A battery-powered global positioning system (GPS) receiver operating on 9.0 \(V\) draws a current of 0.13 \(A\). How much electrical energy does it consume during \(1.5 \mathrm{~h} ?\)

Electric Eels. Electric eels generate electric pulses along their skin that can be used to stun an enemy when they come into contact with it. Tests have shown that these pulses can be up to 500 \(\mathrm{V}\) and produce currents of 80 \(\mathrm{mA}\) (or even larger). A typical pulse lasts for 10 \(\mathrm{ms}\) . What power and how much energy are delivered to the unfortunate enemy with a single pulse, assuming a steady current?

Treatment of Heart Failure. A heart defibrillator is used to enable the heart to start beating if it has stopped. This is done by passing a large current of 12 A through the body at 25 \(\mathrm{V}\) for a very short time, usually about 3.0 \(\mathrm{ms}\) . (a) What power does the defibrillator deliver to the body, and (b) how much energy is transferred?

A \(25.0-\Omega\) bulb is connected across the terminals of a \(12.0-\mathrm{V}\) battery having 3.50\(\Omega\) of internal resistance. What percentage of the power of the battery is dissipated across the internal resistance and hence is not available to the bulb?

An idealized voltmeter is connected across the terminals of a \(15.0-\mathrm{V}\) battery, and a \(75.0-\Omega\) appliance is also connected across its terminals. If the voltmeter reads \(11.3 \mathrm{V} :\) (a) how much power is being dissipated by the appliance, and (b) what is the internal resistance of the battery?

A typical small flashlight contains two batteries, each having an emf of 1.5 \(\mathrm{V}\) , connected in series with a bulb having resistance 17\(\Omega\). (a) If the internal resistance of the batteries is negligible, what power is delivered to the bulb? (b) If the batteries last for 5.0 h, what is the total energy delivered to the bulb? (c) The resistance of real batteries increases as they run down. If the initial internal resistance is negligible, what is the combined internal resistance of both batteries when the power to the bulb has decreased to half its initial value? (Assume that the resistance of the bulb is constant. Actually, it will change somewhat when the current through the filament changes, because this changes the temperature of the filament and hence the resistivity of the filament wire.)

\(\mathrm{A}^{4} 540-\mathrm{W}^{\prime \prime}\) electric heater is designed to operate from \(120-\mathrm{V}\) lines. (a) What is its resistance? (b) What current does it draw? (c) If the line line voltage drops to 110 \(\mathrm{V}\) , what power does the heater take? (Assume that the resistance is constant. Actually, it will change because of the change in temperature.) (d) The heater coils are metallic, so that the resistance of the heater decreases with decreasing temperature. If the change of resistance with temperature is taken into account, will the electrical power consumed by the heater be larger or smaller than what you calculated in part (c)? Explain.

An electrical conductor designed to carry large currents has a circular cross section 2.50 \(\mathrm{mm}\) in diameter and is 14.0 \(\mathrm{m}\) long. The resistance between its ends is 0.104\(\Omega .\) (a) What is the resistivity of the material? (b) If the electric-field magnitude in the conductor is \(1.28 \mathrm{V} / \mathrm{m},\) what is the total current? (c) If the material has \(8.5 \times 10^{28}\) free electrons per cubic meter, find the average drift speed under the conditions of part (b).

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 \(\mathrm{V} .\) You cut off a \(20.0-\mathrm{m}\) 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 \(40.0-\mathrm{m}\) 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 3.00 -m length of copper wire at \(20^{\circ} \mathrm{C}\) has a 1.20 -m-long section with diameter 1.60 \(\mathrm{mm}\) and a 1.80 -m-long section with diameter 0.80 \(\mathrm{mm}\) . There is a current of 2.5 \(\mathrm{mA}\) in the \(1.60-\) mm-diameter section. (a) What is the current in the \(0.80 \mathrm{mm}-\) diameter section? (b) What is the magnitude of \(\vec{E}\) in the 1.60 -mm-diameter section? (c) What is the magnitude of \(\vec{E}\) in the 0.80 -mm-diameter section? (d) What is the potential difference between the ends of the \(3.00-\mathrm{m}\) length of wire?

A Nichrome heating element that has resistance 28.0\(\Omega\) is connected to a battery that has emf 96.0 \(\mathrm{V}\) and internal resistance 1.2\(\Omega\) . An aluminum cup with mass 0.130 kg contains 0.200 \(\mathrm{kg}\) of water. The heating element is placed in the water and the electrical energy dissipated in the resistance of the heating element all goes into the cup and water. The element itself has very small mass. How much time does it take for the temperature of the cup and water to rise from \(21.2^{\circ} \mathrm{C}\) to \(34.5^{\circ} \mathrm{C}\) ? (The change of the resistance of the Nichrome due to its temperature change can be neglected.)

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

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

A person with body resistance between his hands of 10 \(\mathrm{k} \Omega\) accidentally grasps the terminals of a \(14-\mathrm{kV}\) power supply. (a) If the internal resistance of the power supply is \(2000 \Omega,\) 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 \(\mathrm{mA}\) or less?

The average bulk resistivity of the human body (apart from surface resistance of the skin) is about 5.0\(\Omega \cdot \mathrm{m} .\) 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 \(\mathrm{mA}\) ? (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 typical cost for electric power is \(\$ 0.120\) per kilowatt- hour. (a) Some people leave their porch light on all the time. What is the yearly cost to keep a \(75-\) W bulb buming day and night? (b) Suppose your refrigerator uses 400 \(\mathrm{W}\) of power when it's running, and it runs 8 hours a day. What is the yearly cost of operating your refrigerator?

A \(12.6-\mathrm{V}\) car battery with negligible internal resistance is connected to a series combination of a \(3.2-\Omega\) resistor that obeys Ohm's law and a thermistor that does not obey Ohm's law but instead has a current-voltage relationship \(V=\alpha I+\beta I^{2},\) with \(\alpha=\) 3.8\(\Omega\) and \(\beta=1.3 \Omega / \mathrm{A} .\) What is the current through the \(3.2-\Omega\) resistor?

A cylindrical copper cable 1.50 \(\mathrm{km}\) long is connected across a 220.0 -V potential difference. (a) What should be its diameter so that it produces heat at a rate of 75.0 \(\mathrm{W}^{\prime}\) (b) What is the electric field inside the cable under these conditions?

A Nonideal Ammeter. Unlike the idealized ammeter described in Section \(25.4,\) any real ammeter has a nonzero resistance. (a) An ammeter with resistance \(R_{\mathrm{A}}\) is connected in series with a resistor \(R\) and a battery of emf \(\mathcal{E}\) and internal resistance \(r .\) The current measured by the ammeter is \(I_{\mathrm{A}}\) . 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 \(I_{A}, r, R_{\mathrm{A}},\) and \(R .\) The more "ideal" the ammeter, the smaller the difference between this current and the current \(I_{\mathrm{A}}\) . (b) If \(R=3.80 \Omega, \mathcal{E}=7.50 \mathrm{V},\) and \(r=0.45 \Omega,\) find the maximum value of the ammeter resistance \(R_{\mathrm{A}}\) so that \(l_{\mathrm{A}}\) 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.

A \(1.50-\mathrm{m}\) cylinder of radius 1.10 \(\mathrm{cm}\) is made of a complicated mixture of materials. Its resistivity depends on the distance \(x\) from the left end and obeys the formula \(\rho(x)=\) \(a+b x^{2},\) where \(a\) and \(b\) are constants. At the left end, the resistivity is \(2.25 \times 10^{-8} \Omega \cdot \mathrm{m},\) while at the right end it is \(8.50 \times\) \(10^{-8} \Omega \cdot \mathrm{m}\) . (a) What is the resistance of this rod? (b) What is the electric field at its midpoint if it carries a \(1.75-\) A current? (c) If we cut the rod into two 75.0 -cm halves, what is the resistance of each half?

According to the U.S. National Electrical Code, copper wire used for interior wiring of houses, hotels, office buildings, and industrial plants is permitted to carry no more than a specified maximum amount of current. The table below shows the maximum current \(I_{\text { max }}\) for several common sizes of wire with varnished cambric insulation. The "wire gauge" is a standard used to describe the diameter of wires. Note that the larger the diameter of the wire, the smaller the wire gauge. $$\begin{array}{ccc}{\text { Wire gauge }} & {\text { Diameter (cm) }} & {I_{\max }(\mathbf{A})} \\ {14} & {0.163} & {18} \\\ {12} & {0.205} & {25} \\ {10} & {0.259} & {30} \\ {8} & {0.326} & {40} \\\ {6} & {0.412} & {60} \\ {5} & {0.462} & {65} \\ {4} & {0.519} & {85}\end{array}$$ (a) What considerations determine the maximum current-carrying capacity of household wiring? (b) A total of 4200 \(\mathrm{W}\) of power is to be supplied through the wires of a house to the household electrical appliances. If the potential difference across the group of appliances is 120 \(\mathrm{V}\) , determine the gauge of the thinnest permissible wire that can be used. (c) Suppose the wire used in this house is of the gauge found in part (b) and has total length 42.0 \(\mathrm{m}\) . At what rate is energy dissipated in the wires? (d) The house is built in a community where the consumer cost of electric energy is \(\$ 0.11\) per kilowatt-hour. If the house were built with wire of the next larger diameter than that found in part (b), what would be the savings in electricity costs in one year? Assume that the appliances are kept on for an average of 12 hours a day.

Compact fluorescent bulbs are much more efficient at producing light than are ordinary incandescent bulbs. They initially cost much more, but they last far longer and use much less electricity. According to one study of these bulbs, a compact bulb that produces as much light as a 100-W incandescent bulb uses only \(23 \mathrm{~W}\) of power. The compact bulb lasts 10,000 hours, on the average, and costs \(\$ 11.00,\) whereas the incandescent bulb costs only \(\$ 0.75,\) but lasts just 750 hours. The study assumed that electricity costs \(\$ 0.080\) per kilowatt-hour and that the bulbs are on for \(4.0 \mathrm{~h}\) per day. (a) What is the total cost (including the price of the bulbs) to run each bulb for 3.0 years? (b) How much do you save over 3.0 years if you use a compact fluorescent bulb instead of an incandescent bulb? (c) What is the resistance of \(\mathrm{a}^{*} 100-\mathrm{W}^{* *}\) fluorescent bulb? (Remember, it actually uses only \(23 \mathrm{~W}\) of power and operates across \(120 \mathrm{~V}\).)

A lightning bolt strikes one end of a steel lightning rod, producing a \(15,000-\) A current burst that lasts for 65 \mus. The rod is 2.0 \(\mathrm{m}\) long and 1.8 \(\mathrm{cm}\) in diameter, and its other end is connected to the ground by 35 \(\mathrm{m}\) of \(8.0-\mathrm{mm}\) -diameter copper wire. (a) Find the potential difference between the top of the steel rod and the lower end of the copper wire during the current burst. (b) Find the total energy deposited in the rod and wire by the current burst.

A \(12.0-\mathrm{V}\) battery has an internal resistance of 0.24 \(\mathrm{s}\) and a capacity of 50.0 \(\mathrm{A} \cdot \mathrm{h}\) (see Exercise 25.47\() .\) The battery is charged by passing a 10 -A current through it for 5.0 \(\mathrm{h}\) . (a) What is the terminal voltage during charging? (b) What total electricalenergy is supplied to the battery during charging? (c) What electrical energy is dissipated in the internal resistance during charging? (d) The battery is now completely discharged through a resistor, again with a constant current of 10 \(\mathrm{A}\) . What is the external circuit resistance? (e) What total electrical energy is supplied to the external resistor? (f) What total electrical energy is dissipated in the internal resistance? (g) Why are the answers to parts (b) and (e) not the same?

A source with emf \(\mathcal{E}\) and internal resistance \(r\) is connected to an external circuit. (a) Show that the power output of the source is maximum when the current in the circuit is one-half the short-circuit current of the source. (b) If the external circuit consists of a resistance \(R,\) show that the power output is maximum when \(R=r\) and that the maximum power is \(\mathcal{E}^{2} / 4 r_{1}\)

The resistivity of a semiconductor can be modified by adding different amounts of impurities. A rod of semiconducting material of length \(L\) and cross- sectional area \(A\) lies along the \(x\) -axis between \(x=0\) and \(x=L .\) The material obeys Ohm's law, and its resistivity varies along the rod according to \(\rho(x)=\) \(\rho_{0} \exp (-x / L) .\) The end of the rod at \(x=0\) is at a potential \(V_{0}\) greater than the end at \(x=L .\) (a) Find the total resistance of the rod and the current in the rod. (b) Find the electric-field magnitude \(E(x)\) in the rod as a function of \(x .\) (c) Find the electric potential \(V(x)\) in the rod as a function of \(x .\) (d) Graph the functions \(\rho(x), E(x),\) and \(V(x)\) for values of \(x\) between \(x=0\) and \(x=L\) .