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Chapter 21: Problem 66

A portable heater is connected to a \(60-\mathrm{Hz}\) ac outlet. How many times per second is the instantancous power a maximum?

Short Answer

Expert verified
Answer: The instantaneous power in an AC circuit reaches its maximum value 120 times per second at a frequency of 60 Hz.

Step by step solution

01

Understand the relationship between power and frequency in an AC circuit

In an AC circuit, the instantaneous power is given by the formula \(P(t) = VI\cos(\phi)\cos(2\pi ft)\). Here, \(V\) is the instantaneous voltage, \(I\) is the instantaneous current, \(\phi\) is the phase angle between voltage and current, and \(f\) is the frequency. \(P(t)\) is a maximum when \(\cos(2\pi ft) = \pm 1\).
02

Calculate the number of times the power hits a maximum in a single cycle

Within one AC cycle, which occurs each \(\frac{1}{f}\) seconds, \(\cos(2\pi ft)\) will equal \(\pm 1\) twice. Since there are \(2\pi\) radians in a full cycle and \(\cos(x) = \pm 1\) where \(x=2n\pi\) (where n is an integer), we can deduce that the power would maximize twice at \(2n\pi = 2\pi\) and \(2n\pi = 4\pi\) within that cycle.
03

Multiply the number of maximum power instances in a single cycle by the frequency to find the final answer

As established in step 2, the power maximizes twice in a single cycle. Therefore, for a frequency of \(60 Hz\), the power will hit its maximum value \(60 * 2 = 120\) times per second.

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Most popular questions from this chapter

The instantaneous sinusoidal emf from an ac generator with an mos emf of $4.0 \mathrm{V}$ oscillates between what values?
(a) What mas current is drawn by a \(4200-\mathrm{W}\) electric room heater when running on \(120 \mathrm{V}\) rms? (b) What is the power dissipation by the heater if the voltage drops to \(105 \mathrm{V}\) rms during a brownout? Assume the resistance stays the same.
An x-ray machine uses \(240 \mathrm{kV}\) rms at \(60.0 \mathrm{mA}\) ms when it is operating. If the power source is a \(420-\mathrm{V}\) rms line, (a) what must be the turns ratio of the transformer? (b) What is the rms current in the primary? (c) What is the average power used by the \(x\) -ray tube?
Finola has a circuit with a \(4.00-\mathrm{k} \Omega\) resistor, a \(0.750-\mathrm{H}\) inductor, and a capacitor of unknown value connected in series to a \(440.0-\mathrm{Hz}\) ac source. With an oscilloscope, she measures the phase angle to be \(25.0^{\circ} .\) (a) What is the value of the unknown capacitor? (b) Finola has several capacitors on hand and would like to use one to tune the circuit to maximum power. Should she connect a second capacitor in parallel across the first capacitor or in series in the circuit? Explain. (c) What value capacitor does she need for maximum power?
Make a figure analogous to Fig. 21.4 for an ideal inductor in an ac circuit. Start by assuming that the voltage across an ideal inductor is \(v_{\mathrm{L}}(t)=V_{\mathrm{L}}\) sin \(\omega t .\) Make a graph showing one cycle of \(v_{\mathrm{L}}(t)\) and \(i(t)\) on the same axes. Then, at each of the times \(t=0, \frac{1}{8} T, \frac{2}{8} T, \ldots, T,\) indicate the direction of the current (or that it is zero), whether the current is increasing, decreasing, or (instantaneously) not changing, and the direction of the induced emf in the inductor (or that it is zero).
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