Schaum’s Outline of College Physics

Eugene Hecht

Chapter 27

Electrical Power - all with Video Answers

Educators

Chapter Questions

Problem 1

Compute the work and the average power required to transfer $96 \mathrm{kC}$ of charge in one hour $(1.0 \mathrm{~h})$ through a potential rise of $50 \mathrm{~V}$.
The work done equals the change in potential energy:
$$
W=q V=(96000 \mathrm{C})(50 \mathrm{~V})=4.8 \times 10^{6} \mathrm{~J}=4.8 \mathrm{MJ}
$$
Power is the rate of transferring energy:
$$
\mathrm{P}=\frac{W}{t}=\frac{4.8 \times 10^{6} \mathrm{~J}}{3600 \mathrm{~s}}=1.3 \mathrm{~kW}
$$

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Problem 2

How much current does a $60-\mathrm{W}$ light bulb draw when connected to its proper voltage of $120 \mathrm{~V}$ ?
From $\mathrm{P}=V I$,
$$
I=\frac{P}{V}=\frac{60 \mathrm{~W}}{120 \mathrm{~V}}=0.50 \mathrm{~A}
$$

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Problem 3

An electric motor takes $5.0 \mathrm{~A}$ from a $110 \mathrm{~V}$ line. Determine the power input and the energy, in $\mathrm{J}$ and $\mathrm{kW} \cdot \mathrm{h}$, supplied to the motor in $2.0 \mathrm{~h}$.
$$
\begin{aligned}
\text { Power } &=\mathrm{P}=V I=(110 \mathrm{~V})(5.0 \mathrm{~A})=0.55 \mathrm{~kW} \\
\text { Energy } &=\mathrm{P} t=(550 \mathrm{~W})(7200 \mathrm{~s})=4.0 \mathrm{MJ} \\
&=(0.55 \mathrm{~kW})(2.0 \mathrm{~h})=1.1 \mathrm{~kW} \cdot \mathrm{h}
\end{aligned}
$$

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Problem 4

An electric iron of resistance $20 \Omega$ takes a current of $5.0$ A. Calculate the thermal energy, in joules, developed in $30 \mathrm{~s}$.
$$
\begin{array}{l}
\text { Energy }=\mathrm{P} t \\
\text { Energy }=I^{2} R t=(5 \mathrm{~A})^{2}(20 \Omega)(30 \mathrm{~s})=15 \mathrm{~kJ}
\end{array}
$$

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Problem 5

An electric heater of resistance $8.0 \Omega$ draws $15 \mathrm{~A}$ from the service mains. At what rate is thermal energy developed, in W? What is the cost of operating the heater for a period of $4.0 \mathrm{~h}$ at $10 \phi / \mathrm{kW} \cdot \mathrm{h} ?$
$$
\begin{aligned}
W &=I^{2} R=(15 \mathrm{~A})^{2}(8.0 \Omega)=1800 \mathrm{~W}=1.8 \mathrm{~kW} \\
\text { Cost } &=(1.8 \mathrm{~kW})(4.0 \mathrm{~h})(10 \& / \mathrm{kW} \cdot \mathrm{h})=72 \epsilon
\end{aligned}
$$

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Problem 6

A coil develops $800 \mathrm{cal} / \mathrm{s}$ when $20 \mathrm{~V}$ is supplied across its ends. Compute its resistance.
$$
\mathrm{P}=(800 \mathrm{cal} / \mathrm{s})(4.184 \mathrm{~J} / \mathrm{cal})=3347 \mathrm{~J} / \mathrm{s}
$$
Then, because $\mathrm{P}=V^{2} / R$
$$
R=\frac{(20 \mathrm{~V})^{2}}{3347 \mathrm{~J} / \mathrm{s}}=0.12 \Omega
$$

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Problem 7

A line having a total resistance of $0.20 \Omega$ delivers $10.00 \mathrm{~kW}$ at $250 \mathrm{~V}$ to a small factory. What is the efficiency of the transmission?The line dissipates power due to its resistance. Consequently we'll need to find the current in the line. Use $\mathrm{P}=V I$ to find $I=\mathrm{P} / V$. Then
Power lost in line $=I^{2} R=\left(\frac{\mathrm{P}}{V}\right)^{2} R=\left(\frac{10000 \mathrm{~W}}{250 \mathrm{~V}}\right)^{2}(0.20 \Omega)=0.32 \mathrm{~kW}$
$$
\text { Efficiency }=\frac{\text { Power delivered by line }}{\text { Power supplied to line }}=\frac{10.00 \mathrm{~kW}}{(10.00+0.32) \mathrm{kW}}=0.970=97.0 \%
$$

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Problem 8

A hoist motor supplied by a $240-\mathrm{V}$ source requires $12.0 \mathrm{~A}$ to lift an $800-\mathrm{kg}$ load at a rate of $9.00 \mathrm{~m} / \mathrm{min}$. Determine the power input to the motor and the power output, both in horsepower, and the overall efficiency of the system.
$$
\begin{array}{l}
\text { Power input }=I V=(12.0 \mathrm{~A})(240 \mathrm{~V})=2880 \mathrm{~W}=(2.88 \mathrm{~kW})(1.34 \mathrm{hp} / \mathrm{kW})=3.86 \mathrm{hp} \\
\text { Power output }=F v=(800 \times 9.81 \mathrm{~N})\left(\frac{9.00 \mathrm{~m}}{\min }\right)\left(\frac{1.00 \mathrm{~min}}{60.0 \mathrm{~s}}\right)\left(\frac{1.00 \mathrm{hp}}{746 \mathrm{~J} / \mathrm{s}}\right)=1.58 \mathrm{hp} \\
\text { Efficiency }=\frac{1.58 \text { hpoutput }}{3.86 \mathrm{hp} \text { input }}=0.408=40.8 \%
\end{array}
$$

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Problem 9

The lights on a car are inadvertently left on. They dissipate $95.0 \mathrm{~W}$. About how long will it take for the fully charged $12.0-\mathrm{V}$ car battery to run down if the battery is rated at 150 amperehours $(\mathrm{A} \cdot \mathrm{h}) ?$
As an approximation, assume the battery maintains $12.0 \mathrm{~V}$ until it goes dead. Its $150-\mathrm{A} \cdot \mathrm{h}$ rating means it can supply the energy equivalent of a 150 - A current that flows for $1.00 \mathrm{~h}(3600 \mathrm{~s})$. Therefore, the total energy the battery can supply is
Total output energy $=($ Power $)($ Time $)=(V I) t=(12.0 \mathrm{~V} \times 150 \mathrm{~A})(3600 \mathrm{~s})=6.48 \times 10^{6} \mathrm{~J}$
The energy consumed by the lights in a time $t$ is
Energy dissipated $=(95 \mathrm{~W})(t)$
Equating these two energies and solving for $t$, we find $t=6.82 \times 10^{4} \mathrm{~s}=18.9 \mathrm{~h}$.

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Problem 10

What is the cost of electrically heating 50 liters of water from $40^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ at $8.0 \mathrm{e} / \mathrm{kW} \cdot \mathrm{h}$ ?
Heat gained by water $=($ Mass $) \times($ Specific heat $) \times($ Temperature rise)
$$
\begin{array}{c}
=(50 \mathrm{~kg}) \times\left(1000 \mathrm{cal} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\right) \times\left(60^{\circ} \mathrm{C}\right)=3.0 \times 10^{6} \mathrm{cal} \\
\operatorname{Cost}=\left(3.0 \times 10^{6} \mathrm{cal}\right)\left(\frac{4.184 \mathrm{~J}}{1 \mathrm{cal}}\right)\left(\frac{1 \mathrm{~kW} \cdot \mathrm{h}}{3.6 \times 10^{6} \mathrm{~J}}\right)\left(\frac{8.0 \not \mathrm{e}}{1 \mathrm{~kW} \cdot \mathrm{h}}\right)=28 \not \mathrm{q}
\end{array}
$$

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Problem 11

A resistive heater is labeled $1600 \mathrm{~W} / 120 \mathrm{~V}$. How much current does the heater draw from a $120-\mathrm{V}$ source?

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Problem 12

A bulb is stamped $40 \mathrm{~W} / 120 \mathrm{~V}$. What is its resistance when lighted by a $120-\mathrm{V}$ source?

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Problem 13

A spark of artificial 10.0-MV lightning had an energy output of $0.125 \mathrm{MW} \cdot \mathrm{s}$. How many coulombs of charge flowed?

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Problem 14

A current of $1.5$ A exists in a conductor whose terminals are connected across a potential difference of 100
V. Compute the total charge transferred in one minute, the work done in transferring this charge, and the power expended in heating the conductor if all the electrical energy is converted into heat.

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Problem 15

An electric motor takes $15.0 \mathrm{~A}$ at $110 \mathrm{~V}$. Determine $(a)$ the power input and $(b)$ the cost of operating the motor for $8.00 \mathrm{~h}$ at $10.0 \mathrm{~d} / \mathrm{kW} \cdot \mathrm{h}$.

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Problem 16

A current of $10 \mathrm{~A}$ exists in a line of $0.15 \Omega$ resistance. Compute the rate of production of thermal energy in
watts.

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Problem 17

An electric broiler develops $400 \mathrm{cal} / \mathrm{s}$ when the current through it is $8.0$ A. Determine the resistance of the broiler.

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Problem 18

A 25.0-W, 120-V bulb has a cold resistance of $45.0 \Omega$. When the voltage is switched on, what is the instantaneous current? What is the current under normal operation?

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Problem 19

While carrying a current of $400 \mathrm{~A}$, a defective switch becomes overheated due to faulty surface contact. A millivoltmeter connected across the switch shows a $100-\mathrm{m} \mathrm{V}$ drop. What is the power loss due to the contact resistance?

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Problem 20

How much power does a 60-W/120-V incandescent light bulb dissipate when operated at a voltage of $115 \mathrm{~V}$ ? Neglect the bulb's decrease in resistance with lowered voltage.

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Problem 21

A house wire is to carry a current of 30 A while dissipating no more than $1.40 \mathrm{~W}$ of heat per meter of its length. What is the minimum diameter of the wire if its resistivity is $1.68 \times 10^{-8} \Omega \cdot \mathrm{m}$ ?

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Problem 22

A $10.0-\Omega$ electric heater operates on a $110-\mathrm{V}$ line. Compute the rate at which it develops thermal energy in $\mathrm{W}$ and in $\mathrm{cal} / \mathrm{s} .$

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Problem 23

An electric motor, which has 95 percent efficiency, uses $20 \mathrm{~A}$ at $110 \mathrm{~V}$. What is the horsepower output of the motor? How many watts are lost in thermal energy? How many calories of thermal energy are developed per second? If the motor operates for $3.0 \mathrm{~h}$, what energy, in MJ and in $\mathrm{kW} \cdot \mathrm{h}$, is dissipated?

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Problem 24

An electric crane uses $8.0 \mathrm{~A}$ at $150 \mathrm{~V}$ to raise a $450-\mathrm{kg}$ load at the rate of $7.0 \mathrm{~m} / \mathrm{min}$. Determine the efficiency of the system.

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Problem 25

What should be the resistance of a heating coil which will be used to raise the temperature of $500 \mathrm{~g}$ of water from $28^{\circ} \mathrm{C}$ to the boiling point in $2.0$ minutes, assuming that 25 percent of the heat is lost? The heater operates on a $110-V$ line.

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Problem 26

Compute the cost per hour at $8.0 ~ \& / \mathrm{kW} \cdot \mathrm{h}$ of electrically heating a room, if it requires $1.0 \mathrm{~kg} / \mathrm{h}$ of anthracite coal having a heat of combustion of $8000 \mathrm{kcal} / \mathrm{kg}$.

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Problem 27

Power is transmitted at $80 \mathrm{kV}$ between two stations. If the voltage can be increased to $160 \mathrm{kV}$ without a change in cable size, how much additional power can be transmitted for the same current? What effect does the power increase have on the line heating loss?

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Problem 28

A storage battery, of emf $6.4 \mathrm{~V}$ and internal resistance $0.080 \Omega$, is being charged by a current of $15 \mathrm{~A}$. Calculate $(a)$ the power loss in internal heating of the battery, $(b)$ the rate at which energy is stored in the battery, and ( $c$ ) its terminal voltage.

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Problem 29

A tank containing $200 \mathrm{~kg}$ of water was used as a constant-temperature bath. How long would it take to heat the bath from $20^{\circ} \mathrm{C}$ to $25{ }^{\circ} \mathrm{C}$ with a $250-\mathrm{W}$ immersion heater? Neglect the heat capacity of the tank frame and any heat losses to the air.

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