3.5 Alternating current versus direct current  (Page 4/10)

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Section summary

• Direct current (DC) is the flow of electric current in only one direction. It refers to systems where the source voltage is constant.
• The voltage source of an alternating current (AC) system puts out $V={V}_{0}\phantom{\rule{0.25em}{0ex}}\text{sin 2}\pi \text{ft}$ , where $V$ is the voltage at time $t$ , ${V}_{0}$ is the peak voltage, and $f$ is the frequency in hertz.
• In a simple circuit, $I=\text{V/R}$ and AC current is $I={I}_{0}\phantom{\rule{0.25em}{0ex}}\text{sin 2}\pi \text{ft}$ , where $I$ is the current at time $t$ , and ${I}_{0}={V}_{0}\text{/R}$ is the peak current.
• The average AC power is ${P}_{\text{ave}}=\frac{1}{2}{I}_{0}{V}_{0}$ .
• Average (rms) current ${I}_{\text{rms}}$ and average (rms) voltage ${V}_{\text{rms}}$ are ${I}_{\text{rms}}=\frac{{I}_{0}}{\sqrt{2}}$ and ${V}_{\text{rms}}=\frac{{V}_{0}}{\sqrt{2}}$ , where rms stands for root mean square.
• Thus, ${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}$ .
• Ohm’s law for AC is ${I}_{\text{rms}}=\frac{{V}_{\text{rms}}}{R}$ .
• Expressions for the average power of an AC circuit are ${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}$ , ${P}_{\text{ave}}=\frac{{V}_{\text{rms}}^{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}2}}{R}$ , and ${P}_{\text{ave}}={I}_{\text{rms}}^{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}2}R$ , analogous to the expressions for DC circuits.

Conceptual questions

Give an example of a use of AC power other than in the household. Similarly, give an example of a use of DC power other than that supplied by batteries.

Why do voltage, current, and power go through zero 120 times per second for 60-Hz AC electricity?

You are riding in a train, gazing into the distance through its window. As close objects streak by, you notice that the nearby fluorescent lights make dashed streaks. Explain.

Problem exercises

(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 $\text{2600º}\text{C}$ ?

Certain heavy industrial equipment uses AC power that has a peak voltage of 679 V. What is the rms voltage?

480 V

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

Military aircraft use 400-Hz AC power, because it is possible to design lighter-weight equipment at this higher frequency. What is the time for one complete cycle of this power?

2.50 ms

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?

In this problem, you will verify statements made at the end of the power losses for [link] . (a) What current is needed to transmit 100 MW of power at a voltage of 25.0 kV? (b) Find the power loss in a $1\text{.}\text{00 -}\phantom{\rule{0.25em}{0ex}}\Omega$ transmission line. (c) What percent loss does this represent?

(a) 4.00 kA

(b) 16.0 MW

(c) 16.0%

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 $9.00 cents\text{/kW}\cdot \text{h}$ ?

What is the peak power consumption of a 120-V AC microwave oven that draws 10.0 A?

2.40 kW

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

Two different electrical devices have the same power consumption, but one is meant to be operated on 120-V AC and the other on 240-V AC. (a) What is the ratio of their resistances? (b) What is the ratio of their currents? (c) Assuming its resistance is unaffected, by what factor will the power increase if a 120-V AC device is connected to 240-V AC?

(a) 4.0

(b) 0.50

(c) 4.0

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?

Find the time after $t=0$ when the instantaneous voltage of 60-Hz AC first reaches the following values: (a) ${V}_{0}/2$ (b) ${V}_{0}$ (c) 0.

(a) 1.39 ms

(b) 4.17 ms

(c) 8.33 ms

(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) $-{V}_{\text{rms}}$ ?

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