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Chapter 30: Problem 71

Show that the power dissipated in a resistor connected to an AC power source with frequency \(\omega\) oscillates with frequency \(2 \omega\).

Short Answer

Expert verified
Answer: The frequency of power dissipation in a resistor connected to an AC power source with frequency ω is 2ω.

Step by step solution

01

1. Voltage across the resistor

The voltage across the resistor connected to an AC power source can be represented as: \(v(t) = V_m \cos(\omega t)\) where \(V_m\) is the maximum voltage and \(\omega\) is the angular frequency of the source.
02

2. Current through the resistor

According to Ohm's Law, the current through the resistor can be found by dividing the voltage across the resistor by its resistance: \(i(t) = \frac{v(t)}{R} = \frac{V_m \cos(\omega t)}{R}\) where \(R\) is the resistance of the resistor.
03

3. Power dissipated in the resistor

The power dissipated in the resistor can be found using the formula: \(P(t) = v(t) \cdot i(t) = \frac{V^2_m}{R}\cos^2(\omega t)\)
04

4. Expressing power in terms of trigonometric identity

We can simplify the expression further by using the double-angle trigonometric identity: \(\cos^2(\omega t) = \frac{1 + \cos(2\omega t)}{2}\) So, substituting the trigonometric identity into the power equation, we get: \(P(t) = \frac{V^2_m}{R}\left (\frac{1 + \cos(2\omega t)}{2}\right)\)
05

5. Frequency of power dissipation

Now, we can see that the power dissipation function has a term \(\cos(2\omega t)\), which means that the power dissipated in the resistor oscillates with a frequency of \(2\omega\). So, we have shown that the power dissipated in a resistor connected to an AC power source with frequency \(\omega\) oscillates with frequency \(2\omega\).

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Most popular questions from this chapter

A 75,000 -W light bulb (yes, there are such things!) operates at \(I_{\mathrm{rms}}=200 . \mathrm{A}\) and \(V_{\mathrm{rms}}=440 . \mathrm{V}\) in a \(60.0-\mathrm{Hz} \mathrm{AC}\) circuit. Find the resistance, \(R\), and self- inductance, \(L\), of this bulb. Its capacitive reactance is negligible.

A series RLC circuit has a source of time-varying emf providing \(12.0 \mathrm{~V}\) at a frequency \(f_{0}\), with \(L=7.00 \mathrm{mH}\), \(R=100 . \Omega,\) and \(C=0.0500 \mathrm{mF}\) a) What is the resonant frequency of this circuit? b) What is the average power dissipated in the resistor at this resonant frequency?

Which statement about the phase relation between the electric and magnetic fields in an LC circuit is correct? a) When one field is at its maximum, the other is also, and the same for the minimum values. b) When one field is at maximum strength, the other is at minimum (zero) strength. c) The phase relation, in general, depends on the values of \(L\) and \(C\).

An LC circuit consists of a 1.00 -mH inductor and a fully charged capacitor. After \(2.10 \mathrm{~ms}\), the energy stored in the capacitor is half of its original value. What is the capacitance?

a) A loop of wire \(5.00 \mathrm{~cm}\) in diameter is carrying a current of \(2.00 \mathrm{~A}\). What is the energy density of the magnetic field at its center? b) What current has to flow in a straight wire to produce the same energy density at a point \(4.00 \mathrm{~cm}\) from the wire?
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