Problems & Exercises

A total of 600 C of charge passes through a flashlight in 0.500 h. What is the average current?

Find the current when 2.00 nC jumps between your comb and hair over a $0\text{.}\text{500 -}\mu \text{s}$ time interval.

The 200-A current through a spark plug moves 0.300 mC of charge. How long does the spark last?

During open-heart surgery, a defibrillator can be used to bring a patient out of cardiac arrest. The resistance of the path is $5\text{00 Ω}$ and a 10.0-mA current is needed. What voltage should be applied?

A clock battery wears out after moving 10,000 C of charge through the clock at a rate of 0.500 mA. (a) How long did the clock run? (b) How many electrons per second flowed?

Electron guns are used in X-ray tubes. The electrons are accelerated through a relatively large voltage and directed onto a metal target, producing X-rays. (a) How many electrons per second strike the target if the current is 0.500 mA? (b) What charge strikes the target in 0.750 s?

Repeat the above example on Example 20.3, but for a wire made of silver and given there is one free electron per silver atom.

A 14-gauge copper wire has a diameter of 1.628 mm. What magnitude current flows when the drift velocity is 1.00 mm/s? (See above example on Example 20.3 for useful information.)

Calculate the effective resistance of a pocket calculator that has a 1.35-V battery and through which 0.200 mA flows.

How many volts are supplied to operate an indicator light on a DVD player that has a resistance of $1\text{40}\Omega$, given that 25.0 mA passes through it?

A power transmission line is hung from metal towers with glass insulators having a resistance of $1\text{.}\text{00}\times {\text{10}}^9\Omega .$ What current flows through the insulator if the voltage is 200 kV? (Some high-voltage lines are DC.)

The diameter of 0-gauge copper wire is 8.252 mm. Find the resistance of a 1.00-km length of such wire used for power transmission.

Find the ratio of the diameter of aluminum to copper wire, if they have the same resistance per unit length (as they might in household wiring).

(a) To what temperature must you raise a copper wire, originally at $\text{20.0ºC}$, to double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances?

Of what material is a resistor made if its resistance is 40.0% greater at $\text{100º}\text{C}$ than at $\text{20}\text{.}\mathrm{0º}\text{C}$?

(a) Of what material is a wire made, if it is 25.0 m long with a 0.100 mm diameter and has a resistance of $\text{77}\text{.}7\Omega$ at $\text{20}\text{.}\mathrm{0º}\text{C}$? (b) What is its resistance at $\text{150º}\text{C}$?

A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase?

(a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has $\alpha =–0\text{.}\text{0600}/\text{ºC}$) when it is at the same temperature as the patient. What is a patient’s temperature if the thermistor’s resistance at that temperature is 82.0% of its value at $\text{37}\text{.}\mathrm{0º}\text{C}$ (normal body temperature)? (b) The negative value for $\alpha$ may not be maintained for very high temperatures. Discuss why and whether this is the case here. (Hint: Resistance can’t become negative.)

Unreasonable Results

(a) To what temperature must you raise a resistor made of constantan to double its resistance, assuming a constant temperature coefficient of resistivity? (b) To cut it in half? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable, or which premises are inconsistent?

What power is supplied to the starter motor of a large truck that draws 250 A of current from a 24.0-V battery hookup?

A charge of 4.00 C of charge passes through a pocket calculator’s solar cells in 4.00 h. What is the power output, given the calculator’s voltage output is 3.00 V? (See Figure 20.36.)

How many watts does a flashlight that has $6.00\times {\text{10}}^{\text{2}}\text{C}$ pass through it in 0.500 h use if its voltage is 3.00 V?

Verify that the units of a volt-ampere are watts, as implied by the equation $P=\text{IV}$.

Show that the units $1{\text{A}}^2\cdot \Omega =1\text{W}$, as implied by the equation $P=I^{2}R$.

Electrons in an X-ray tube are accelerated through $1.00\times {\text{10}}^{\text{2}}\text{kV}$ and directed toward a target to produce X-rays. Calculate the power of the electron beam in this tube if it has a current of 15.0 mA.

With a 1200-W toaster, how much electrical energy is needed to make a slice of toast (cooking time = 1 minute)? At $\text{9.0 cents/kW · h}$, how much does this cost?

Some makes of older cars have 6.00-V electrical systems. (a) What is the hot resistance of a 30.0-W headlight in such a car? (b) What current flows through it?

A cauterizer, used to stop bleeding in surgery, puts out 2.00 mA at 15.0 kV. (a) What is its power output? (b) What is the resistance of the path?

An old lightbulb draws only 50.0 W, rather than its original 60.0 W, due to evaporative thinning of its filament. By what factor is its diameter reduced, assuming uniform thinning along its length? Neglect any effects caused by temperature differences.

Integrated Concepts

Cold vaporizers pass a current through water, evaporating it with only a small increase in temperature. One such home device is rated at 3.50 A and utilizes 120 V AC with 95.0% efficiency. (a) What is the vaporization rate in grams per minute? (b) How much water must you put into the vaporizer for 8.00 h of overnight operation? (See Figure 20.38.)

Figure 20.38 This cold vaporizer passes current directly through water, vaporizing it directly with relatively little temperature increase.

Integrated Concepts

What current must be produced by a 12.0-V battery-operated bottle warmer in order to heat 75.0 g of glass, 250 g of baby formula, and $3.00\times {\text{10}}^{\text{2}}\text{g}$ of aluminum from $\text{20}\text{.}\mathrm{0º}\text{C}$ to $\text{90}\text{.}\mathrm{0º}\text{C}$ in 5.00 min?

Integrated Concepts

Hydroelectric generators (see Figure 20.39) at Hoover Dam produce a maximum current of $8.00\times {\text{10}}^{\text{3}}\text{A}$ at 250 kV. (a) What is the power output? (b) The water that powers the generators enters and leaves the system at low speed (thus its kinetic energy does not change) but loses 160 m in altitude. How many cubic meters per second are needed, assuming 85.0% efficiency?

Figure 20.39 Hydroelectric generators at the Hoover dam. (credit: Jon Sullivan)

Integrated Concepts

A light-rail commuter train draws 630 A of 650-V DC electricity when accelerating. (a) What is its power consumption rate in kilowatts? (b) How long does it take to reach 20.0 m/s starting from rest if its loaded mass is $5\text{.}\text{30}\times {\text{10}}^4\text{kg}$, assuming 95.0% efficiency and constant power? (c) Find its average acceleration. (d) Discuss how the acceleration you found for the light-rail train compares to what might be typical for an automobile.

Integrated Concepts

(a) An immersion heater utilizing 120 V can raise the temperature of a $1.00\times {\text{10}}^{\text{2}}\text{-g}$ aluminum cup containing 350 g of water from $\text{20}\text{.}\mathrm{0º}\text{C}$ to $\text{95}\text{.}\mathrm{0º}\text{C}$ in 2.00 min. Find its resistance, assuming it is constant during the process. (b) A lower resistance would shorten the heating time. Discuss the practical limits to speeding the heating by lowering the resistance.

Integrated Concepts

(a) What is the cost of heating a hot tub containing 1500 kg of water from $\text{10}\text{.}\mathrm{0º}\text{C}$ to $\text{40}\text{.}\mathrm{0º}\text{C}$, assuming 75.0% efficiency to account for heat transfer to the surroundings? The cost of electricity is $\text{9}\text{cents/kW}\cdot \text{h}$. (b) What current was used by the 220-V AC electric heater, if this took 4.00 h?

Unreasonable Results

(a) What current is needed to transmit $1.00\times {\text{10}}^{\text{2}}\text{MW}$ of power at 10.0 kV? (b) Find the resistance of 1.00 km of wire that would cause a 0.0100% power loss. (c) What is the diameter of a 1.00-km-long copper wire having this resistance? (d) What is unreasonable about these results? (e) Which assumptions are unreasonable, or which premises are inconsistent?

Construct Your Own Problem

Consider an electric immersion heater used to heat a cup of water to make tea. Construct a problem in which you calculate the needed resistance of the heater so that it increases the temperature of the water and cup in a reasonable amount of time. Also calculate the cost of the electrical energy used in your process. Among the things to be considered are the voltage used, the masses and heat capacities involved, heat losses, and the time over which the heating takes place. Your instructor may wish for you to consider a thermal safety switch (perhaps bimetallic) that will halt the process before damaging temperatures are reached in the immersion unit.

(a) What is the hot resistance of a 25-W light bulb that runs on 120-V AC? (b) If the bulb’s operating temperature is $\text{2700º}\text{C}$, what is its resistance at $\text{2600º}\text{C}$?

A certain circuit breaker trips when the rms current is 15.0 A. What is the corresponding peak current?

A North American tourist takes his 25.0-W, 120-V AC razor to Europe, finds a special adapter, and plugs it into 240 V AC. Assuming constant resistance, what power does the razor consume as it is ruined?

A small office-building air conditioner operates on 408-V AC and consumes 50.0 kW. (a) What is its effective resistance? (b) What is the cost of running the air conditioner during a hot summer month when it is on 8.00 h per day for 30 days and electricity costs $\mathrm{9.00\; cents}\text{/kW}\cdot \text{h}$?

What is the peak current through a 500-W room heater that operates on 120-V AC power?

Nichrome wire is used in some radiative heaters. (a) Find the resistance needed if the average power output is to be 1.00 kW utilizing 120-V AC. (b) What length of Nichrome wire, having a cross-sectional area of $5.00{\text{mm}}^2$, is needed if the operating temperature is $\text{500º C}$? (c) What power will it draw when first switched on?

(a) At what two times in the first period following $t=0$ does the instantaneous voltage in 60-Hz AC equal $V_{\text{rms}}$? (b) $-V_{\text{rms}}$?

What voltage is involved in a 1.44-kW short circuit through a $0\text{.}\text{100}\text{-}\Omega$ resistance?

While taking a bath, a person touches the metal case of a radio. The path through the person to the drainpipe and ground has a resistance of $\text{4000}\Omega$. What is the smallest voltage on the case of the radio that could cause ventricular fibrillation?

(a) During surgery, a current as small as $\mathrm{\text{20.0 μA}}$ applied directly to the heart may cause ventricular fibrillation. If the resistance of the exposed heart is $\text{300}\Omega$, what is the smallest voltage that poses this danger? (b) Does your answer imply that special electrical safety precautions are needed?

A heart defibrillator passes 10.0 A through a patient’s torso for 5.00 ms in an attempt to restore normal beating. (a) How much charge passed? (b) What voltage was applied if 500 J of energy was dissipated? (c) What was the path’s resistance? (d) Find the temperature increase caused in the 8.00 kg of affected tissue.

Construct Your Own Problem

Consider a person working in an environment where electric currents might pass through their body. Construct a problem in which you calculate the resistance of insulation needed to protect the person from harm. Among the things to be considered are the voltage to which the person might be exposed, likely body resistance (dry, wet, …), and acceptable currents (safe but sensed, safe and unfelt, …).

Integrated Concepts

(a) Referring to Figure 20.31, find the time systolic pressure lags behind the middle of the QRS complex. (b) Discuss the reasons for the time lag.