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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 vacuum cleaner motor can be viewed as an inductor with an inductance of \(100 . \mathrm{mH} .\) For a \(60.0-\mathrm{Hz} \mathrm{AC}\) voltage, \(V_{\mathrm{rms}}=115 \mathrm{~V}\), what capacitance must be in series with the motor to maximize the power output of the vacuum cleaner?

A \(200-\Omega\) resistor, a \(40.0-\mathrm{mH}\) inductor and a \(3.0-\mu \mathrm{F}\) capacitor are connected in series with a time-varying source of emf that provides \(10.0 \mathrm{~V}\) at a frequency of \(1000 \mathrm{~Hz}\). What is the impedance of the circuit? a) \(200 \Omega\) b) \(228 \Omega\) c) \(342 \Omega\) d) \(282 \Omega\)

An electromagnet consists of 200 loops and has a length of \(10.0 \mathrm{~cm}\) and a cross-sectional area of \(5.00 \mathrm{~cm}^{2}\). Find the resonant frequency of this electromagnet when it is attached to the Earth (treat the Earth as a spherical capacitor)

A \(300 .-\Omega\) resistor is connected in series with a \(4.00-\mu F\) capacitor and a source of time-varying emf providing \(V_{\mathrm{rms}}=40.0 \mathrm{~V}\) a) At what frequency will the potential drop across the capacitor equal that across the resistor? b) What is the rms current through the circuit when this occurs?

An AC power source with \(V_{\mathrm{m}}=220 \mathrm{~V}\) and \(f=60.0 \mathrm{~Hz}\) is connected in a series RLC circuit. The resistance, \(R\), inductance, \(L\), and capacitance, \(C\), of this circuit are, respectively, \(50.0 \Omega, 0.200 \mathrm{H},\) and \(0.040 \mathrm{mF}\). Find each of the following quantities: a) the inductive reactance b) the capacitive reactance c) the impedance of the circuit d) the maximum current through the circuit e) the maximum potential difference across each circuit element
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