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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

A series combination of a resistor and a capacitor are connected to a \(110-\mathrm{V}\) rms, \(60.0-\mathrm{Hz}\) ac source. If the capacitance is \(0.80 \mu \mathrm{F}\) and the rms current in the circuit is $28.4 \mathrm{mA},$ what is the resistance?
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?
A coil with an internal resistance of \(120 \Omega\) and inductance of $12.0 \mathrm{H}\( is connected to a \)60.0-\mathrm{Hz}, 110-\mathrm{V}$ ms line. (a) What is the impedance of the coil? (b) Calculate the current in the coil.
A variable inductor can be placed in series with a lightbulb to act as a dimmer. (a) What inductance would reduce the current through a \(100-\) W lightbulb to \(75 \%\) of its maximum value? Assume a \(120-\mathrm{V}\) rms, \(60-\mathrm{Hz}\) source. (b) Could a variable resistor be used in place of the variable inductor to reduce the current? Why is the inductor a much better choice for a dimmer?
In an RLC circuit, these three clements are connected in series: a resistor of \(20.0 \Omega,\) a \(35.0-\mathrm{mH}\) inductor, and a 50.0 - \(\mu\) F capacitor. The ac source of the circuit has an rms voltage of \(100.0 \mathrm{V}\) and an angular frequency of $1.0 \times 10^{3} \mathrm{rad} / \mathrm{s} .$ Find (a) the reactances of the capacitor and inductor, (b) the impedance, (c) the rms current, (d) the current amplitude, (e) the phase angle, and (f) the rims voltages across each of the circuit elements. (g) Does the current lead or lag the voltage? (h) Draw a phasor diagram.
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