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Chapter 21: Problem 27

A 4.00 -mH inductor is connected to an ac voltage source of \(151.0 \mathrm{V}\) rms. If the rms current in the circuit is \(0.820 \mathrm{A},\) what is the frequency of the source?

Short Answer

Expert verified
Answer: The frequency of the AC voltage source is approximately 7.31 kHz.

Step by step solution

01

Find the inductive reactance (XL)

We can use the Ohm's law for an inductor to find the inductive reactance (\(X_L\)) by dividing the RMS voltage (\(V_{rms}\)) by the RMS current (\(I_{rms}\)): \(X_L = \frac{V_{rms}}{I_{rms}}\) Plug in the given values: \(X_L = \frac{151.0 \mathrm{V}}{0.820 \mathrm{A}} \approx 184.15 \Omega\)
02

Use the formula to find the frequency (f)

Now that we have the inductive reactance, we can use the formula that relates inductive reactance, inductance, and frequency: \(X_L = 2\pi f L\) Where \(f\) is the frequency, and \(L\) is the inductance. We can rearrange the formula to solve for the frequency: \(f = \frac{X_L}{2\pi L}\) Plug in the given values and the calculated value for \(X_L\): \(f = \frac{184.15 \Omega}{2\pi \times 4.00 \times 10^{-3} \mathrm{H}} \approx 7313.83 \ \mathrm{Hz}\)
03

Round the answer and write it in a suitable unit.

Round the frequency: \(f \approx 7.31 \times 10^3 \ \mathrm{Hz}\) Now, write the frequency in kilohertz (kHz): \(f \approx 7.31 \ \mathrm{kHz}\) So the frequency of the AC voltage source is approximately \(7.31 \ \mathrm{kHz}\).

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Most popular questions from this chapter

A portable heater is connected to a \(60-\mathrm{Hz}\) ac outlet. How many times per second is the instantancous power a maximum?
A hair dryer has a power rating of \(1200 \mathrm{W}\) at \(120 \mathrm{V}\) rms. Assume the hair dryer circuit contains only resistance. (a) What is the resistance of the heating element? (b) What is the rms current drawn by the hair dryer? (c) What is the maximum instantaneous power that the resistance must withstand?
An RLC series circuit is driven by a sinusoidal emf at the circuit's resonant frequency. (a) What is the phase difference between the voltages across the capacitor and inductor? [Hint: since they are in series, the same current \(i(t) \text { flows through them. }]\) (b) At resonance, the rms current in the circuit is \(120 \mathrm{mA}\). The resistance in the circuit is \(20 \Omega .\) What is the rms value of the applied emf? (c) If the frequency of the emf is changed without changing its rms value, what happens to the rms current? (W) tutorial: resonance)
In an RLC series circuit, these three elements are connected in scries: a resistor of \(60.0 \Omega,\) a 40.0 -mH inductor, and a \(0.0500-\mathrm{F}\) capacitor. The series elements are connected across the terminals of an ac oscillator with an rms voltage of \(10.0 \mathrm{V}\). Find the resonant frequency for the circuit.
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.
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