Chapter 20

A lightning bolt delivers a charge of \(35 \mathrm{C}\) to the ground in a time of \(1.0 \times 10^{-3} \mathrm{~s}\). What is the current?

A defibrillator is used during a heart attack to restore the heart to its normal beating pattern (see Section 19.5). A defibrillator passes 18 A of current through the torso of a person in \(2.0 \mathrm{~ms}\). (a) How much charge moves during this time? (b) How many electrons pass through the wires connected to the patient?

A battery charger is connected to a dead battery and delivers a current of 6.0 A for 5.0 hours, keeping the voltage across the battery terminals at \(12 \mathrm{~V}\) in the process. How much energy is delivered to the battery?

Suppose that the resistance between the walls of a biological cell is \(5.0 \times 10^{9} \Omega\). (a) What is the current when the potential difference between the walls is \(75 \mathrm{mV} ?\) (b) If the current is composed of Na+ ions \((q=+e)\), how many such ions flow in \(0.50 \mathrm{~s}\) ?

The heating element of a clothes dryer has a resistance of \(11 \Omega\) and is connected across a \(240-\mathrm{V}\) electrical outlet. What is the current in the heating element?

A resistor is connected across the terminals of a \(9.0\) -V battery, which delivers \(1.1 \times 10^{5} \mathrm{~J}\) of energy to the resistor in six hours. What is the resistance of the resistor?

The resistance of a bagel toaster is \(14 \Omega\). To prepare a bagel, the toaster is operated for one minute from a \(120-\mathrm{V}\) outlet. How much energy is delivered to the toaster?

A car battery has a rating of 220 ampere \(\cdot\) hours \((\mathrm{A} \cdot \mathrm{h}) .\) This rating is one indication of the total charge that the battery can provide to a circuit before failing. (a) What is the total charge (in coulombs) that this battery can provide? (b) Determine the maximum current that the battery can provide for 38 minutes.

A beam of protons is moving toward a target in a particle accelerator. This beam constitutes a current whose value is \(0.50 \mu \mathrm{A}\). (a) How many protons strike the target in 15 s? (b) Each proton has a kinetic energy of \(4.9 \times 10^{-12} \mathrm{~J}\). Suppose the target is a 15 -gram block of aluminum, and all the kinetic energy of the protons goes into heating it up. What is the change in temperature of the block that results from the 15 -s bombardment of protons?

A cylindrical copper cable carries a current of \(1200 \mathrm{~A}\). There is a potential difference of \(1.6 \times 10^{-2} \mathrm{~V}\) between two points on the cable that are \(0.24 \mathrm{~m}\) apart. What is the radius of the cable?

Two wires have the same length and the same resistance. One is made from aluminum and the other from copper. Obtain the ratio of the cross-sectional area of the aluminum wire to that of the copper wire.

A coil of wire has a resistance of \(38.0 \Omega\) at \(25^{\circ} \mathrm{C}\) and \(43.7 \Omega\) at \(55^{\circ} \mathrm{C}\). What is the temperature coefficient of resistivity?

The temperature coefficient of resistivity for the metal gold is \(0.0034\left(\mathrm{C}^{\circ}\right)^{-1},\) and for tungsten it is \(0.0045\left(\mathrm{C}^{\circ}\right)^{-1}\). The resistance of a gold wire increases by \(7.0 \%\) due to an increase in temperature. For the same increase in temperature, what is the percentage increase in the resistance of a tungsten wire?

A toaster uses a Nichrome heating wire. When the toaster is turned on at \(20^{\circ} \mathrm{C}\), the initial current is \(1.50 \mathrm{~A}\). A few seconds later, the toaster warms up and the current now has a value of \(1.30 \mathrm{~A}\). The average temperature coefficient of resistivity for Nichrome wire is \(4.5 \times 10^{-4}\left(\mathrm{C}^{0}\right)^{-1} .\) What is the temperature of the heating wire?

The filament in an incandescent light bulb is made from tungsten. The light bulb is plugged into a \(120-\mathrm{V}\) outlet and draws a current of \(1.24 \mathrm{~A}\). If the radius of the tungsten wire is \(0.0030 \mathrm{~mm}\), how long must the wire be?

Two wires have the same cross-sectional area and are joined end to end to form a single wire. One is tungsten, which has a temperature coefficient of resistivity of \(\alpha=0.0045\left(\mathrm{C}^{\circ}\right)^{-1} .\) The other is carbon, for which \(\alpha=-0.0005\left(\mathrm{C}^{\circ}\right)^{-1} .\) The total resistance of the composite wire is the sum of the resistances of the pieces. The total resistance of the composite does not change with temperature. What is the ratio of the lengths of the tungsten and carbon sections? Ignore any changes in length due to thermal expansion.

A digital thermometer employs a thermistor as the temperature-sensing element. A thermistor is a kind of semiconductor and has a large negative temperature coefficient of resistivity \(\alpha\). Suppose \(\alpha=-0.060\left(\mathrm{C}^{\circ}\right)^{-1}\) for the thermistor in a digital thermometer used to measure the temperature of a patient. The resistance of the thermistor decreases to \(85 \%\) of its value at the normal body temperature of \(37.0^{\circ} \mathrm{C}\). What is the patient's temperature?

The heating element in an iron has a resistance of \(24 \Omega\). The iron is plugged into a \(120-\mathrm{V}\) outlet. What is the power delivered to the iron?

A portable CD player operates with a voltage of \(4.5 \mathrm{~V},\) and its power usage is \(0.11 \mathrm{~W}\). Find the current in the player.

A blow-dryer and a vacuum cleaner each operate with a voltage of \(120 \mathrm{~V}\). The current rating of the blow-dryer is \(11 \mathrm{~A},\) and that of the vacuum cleaner is \(4.0 \mathrm{~A} .\) Determine the power consumed by (a) the blow-dryer and (b) the vacuum cleaner. (c) Determine the ratio of the energy used by the blow-dryer in 15 minutes to the energy used by the vacuum cleaner in one-half hour.

There are approximately 110 million TVs in the United States. Each uses, on average, 75 W of power and is turned on for 6.0 hours a day. If electrical energy costs \(\$ 0.12\) per \(\mathrm{kWh}\), how much money is spent every day in keeping the TVs turned on?

In doing a load of clothes, a clothes dryer uses \(16 \mathrm{~A}\) of current at \(240 \mathrm{~V}\) for \(45 \mathrm{~min}\). A personal computer, in contrast, uses \(2.7\) A of current at \(120 \mathrm{~V}\). With the energy used by the clothes dryer, how long (in hours) could you use this computer to "surf" the Internet?

A piece of Nichrome wire has a radius of \(6.5 \times 10^{-4} \mathrm{~m}\). It is used in a laboratory to make a heater that uses \(4.00 \times 10^{2} \mathrm{~W}\) of power when connected to a voltage source of \(120 \mathrm{~V}\). Ignoring the effect of temperature on resistance, estimate the necessary length of wire.

An electric heater is used to boil small amounts of water and consists of a \(15-\Omega\) coil that is immersed directly in the water. It operates from a \(120-\mathrm{V}\) socket. How much time is required for this heater to raise the temperature of \(0.50 \mathrm{~kg}\) of water from \(13^{\circ} \mathrm{C}\) to the normal boiling point?

According to Equation \(20.7,\) an ac voltage \(V\) is given as a function of time \(t\) by \(V=V_{0} \sin 2 \pi f t,\) where \(V_{0}\) is the peak voltage and \(f\) is the frequency (in hertz). For a frequency of \(60.0 \mathrm{~Hz}\), what is the smallest value of the time at which the voltage equals one-half of the peak value?

The current in a circuit is ac and has a peak value of 2.50 A. Determine the rms current.

The rms current in a copy machine is \(6.50 \mathrm{~A},\) and the resistance of the machine is \(18.6 \Omega\). What is (a) the average power and (b) the peak power delivered to the machine?

The average power used by a stereo speaker is \(55 \mathrm{~W}\). Assuming that the speaker can be treated as a \(4.0-\Omega\) resistance, find the peak value of the ac voltage applied to the speaker.

An electric furnace runs nine hours a day to heat a house during January ( 31 days). The heating element has a resistance of \(5.3 \Omega\) and carries a current of \(25 \mathrm{~A}\). The cost of electricity is \(\$ 0.12 / \mathrm{kWh}\). Find the cost of running the furnace for the month of January.

A light bulb is connected to a \(120.0-\mathrm{V}\) wall socket. The current in the bulb depends on the time \(t\) according to the relation \(I=(0.707 \mathrm{~A}) \sin [(314 \mathrm{~Hz}) t] .\) (a) What is the frequency of the alternating current? (b) Determine the resistance of the bulb's filament. (c) What is the average power delivered to the light bulb?

The recovery time of a hot water heater is the time required to heat all the water in the unit to the desired temperature. Suppose that a 52-gal \(\left(1.00 \mathrm{gal}=3.79 \times 10^{-3} \mathrm{~m}^{3}\right)\) unit starts with cold water at \(11^{\circ} \mathrm{C}\) and delivers hot water at \(53^{\circ} \mathrm{C}\). The unit is electric and utilizes a resistance heater \((120 \mathrm{~V} \mathrm{ac}, 3.0 \Omega)\) to heat the water. Assuming that no heat is lost to the environment, determine the recovery time (in hours) of the unit.

A light bulb is connected to a \(120.0-\mathrm{V}\) wall socket. The current in the bulb depends on the time \(t\) according to the relation \(I=(0.707 \mathrm{~A}) \sin [(314 \mathrm{~Hz}) t] .\) (a) What is the frequency of the alternating current? (b) Determine the resistance of the bulb's filament. (c) What is the average power delivered to the light bulb?

To save on heating costs, the owner of a greenhouse keeps \(660 \mathrm{~kg}\) of water around in barrels. During a winter day, the water is heated by the sun to \(10.0^{\circ} \mathrm{C}\). During the night the water freezes into ice at \(0.0^{\circ} \mathrm{C}\) in nine hours. What is the minimum ampere rating of an electric heating system \((240 \mathrm{~V})\) that would provide the same heating effect as the water does?

The current in a series circuit is \(15.0 \mathrm{~A}\). When an additional \(8.00-\Omega\) resistor is inserted in series, the current drops to 12.0 A. What is the resistance in the original circuit?

A 60.0 -W lamp is placed in series with a resistor and a 120.0 -V source. If the voltage across the lamp is \(25 \mathrm{~V},\) what is the resistance \(R\) of the resistor?

A \(36.0-\Omega\) resistor and an \(18.0-\Omega\) resistor are connected in series across a \(15.0-\) battery. What is the voltage across (a) the \(36.0-\Omega\) resistor and (b) the \(18.0-\Omega\) resistor?

The current in a \(47-\Omega\) resistor is \(0.12 \mathrm{~A}\). This resistor is in series with a \(28-\Omega\) resistor, and the series combination is connected across a battery. What is the battery voltage?

The current in a \(47-\Omega\) resistor is 0.12 A. This resistor is in series with a \(28-\Omega\) resistor, and the series combination is connected across a battery. What is the battery voltage?

Three resistors, \(25,45,\) and \(75 \Omega,\) are connected in series, and a \(0.51-\mathrm{A}\) current passes through them. What is (a) the equivalent resistance and (b) the potential difference across the three resistors?

An extension cord is used with an electric weed trimmer that has a resistance of \(15.0 \Omega .\) The extension cord is made of copper wire that has a cross- sectional area of \(1.3 \times 10^{-6} \mathrm{~m}^{2}\). The combined length of the two wires in the extension cord is \(92 \mathrm{~m}\). (a) Determine the resistance of the extension cord. (b) The extension cord is plugged into a \(120-\mathrm{V}\) socket. What voltage is applied to the trimmer itself?

Two resistances, \(R_{1}\) and \(R_{2}\), are connected in series across a \(12-\mathrm{V}\) battery. The current increases by \(0.20\) A when \(R_{2}\) is removed, leaving \(R_{1}\) connected across the battery. However, the current increases by just \(0.10 \mathrm{~A}\) when \(R_{1}\) is removed, leaving \(R_{2}\) connected across the battery. Find (a) \(R_{1}\) and (b) \(R_{2}\).

What resistance must be placed in parallel with a \(155-\Omega\) resistor to make the equivalent resistance \(115 \Omega ?\)

A \(16-\Omega\) loudspeaker and an \(8.00-\Omega\) loudspeaker are connected in parallel across the terminals of an amplifier. Assuming the speakers behave as resistors, determine the equivalent resistance of the two speakers.

Help with problems of this kind is available in Interactive Solution 20.5120 .51 at . A coffee cup heater and a lamp are connected in parallel to the same \(120-\mathrm{V}\) outlet. Together, they use a total of \(111 \mathrm{~W}\) of power. The resistance of the heater is \(4.0 \times 10^{2} \Omega\). Find the resistance of the lamp.

Two resistors, 42.0 and \(64.0 \Omega\), are connected in parallel. The current through the \(64.0-\Omega\) resistor is 3.00 A. (a) Determine the current in the other resistor. (b) What is the total power supplied to the two resistors?

A wire whose resistance is \(R\) is cut into three equally long pieces, which are then connected in parallel. In terms of \(R\), what is the resistance of the parallel combination?

The total current delivered to a number of devices connected in parallel is the sum of the individual currents in each device. Circuit breakers are resettable automatic switches that protect against a dangerously large total current by "opening" to stop the current at a specified safe value. A 1650 -W toaster, a 1090 -W iron, and a 1250 -W microwave oven are turned on in a kitchen. As the drawing shows, they are all connected through a 20 -A circuit breaker to an ac voltage of \(120 \mathrm{~V}\). (a) Find the equivalent resistance of the three devices. (b) Obtain the total current delivered by the source and determine whether the breaker will "open" to prevent an accident.

A cylindrical aluminum pipe of length \(1.50 \mathrm{~m}\) has an inner radius of \(2.00 \times 10^{-3} \mathrm{~m}\) and an outer radius of \(3.00 \times 10^{-3} \mathrm{~m}\). The interior of the pipe is completely filled with copper. What is the resistance of this unit?

The rear window defogger of a car consists of thirteen thin wires (resistivity \(\left.=88.0 \times 10^{-8} \Omega \cdot \mathrm{m}\right\\}\) embedded in the glass. The wires are connected in parallel to the \(12.0-\mathrm{V}\) battery, and each has a length of \(1.30 \mathrm{~m}\). The defogger can melt \(2.10 \times 10^{-2} \mathrm{~kg}\) of ice at \(0{ }^{\circ} \mathrm{C}\) into water at \(0{ }^{\circ} \mathrm{C}\) in two minutes. Assume that all the power delivered to the wires is used immediately to melt the ice. Find the cross-sectional area of each wire.

A \(14-\Omega\) coffee maker and a \(16-\Omega\) frying pan are connected in series across a \(120-\mathrm{V}\) source of voltage. A \(23-\Omega\) bread maker is also connected across the \(120-\mathrm{V}\) source and is in parallel with the series combination. Find the total current supplied by the source of voltage.