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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

An ac circuit has a single resistor, capacitor, and inductor in series. The circuit uses \(100 \mathrm{W}\) of power and draws a maximum rms current of $2.0 \mathrm{A}\( when operating at \)60 \mathrm{Hz}\( and \)120 \mathrm{V}$ rms. The capacitive reactance is 0.50 times the inductive reactance. (a) Find the phase angle. (b) Find the values of the resistor, the inductor, and the capacitor.
A 22 -kV power line that is \(10.0 \mathrm{km}\) long supplies the electric energy to a small town at an average rate of \(6.0 \mathrm{MW} .\) (a) If a pair of aluminum cables of diameter \(9.2 \mathrm{cm}\) are used, what is the average power dissipated in the transmission line? (b) Why is aluminum used rather than a better conductor such as copper or silver?
Show, from \(X_{\mathrm{C}}=1 /(\omega C),\) that the units of capacitive reactance are ohms.

A capacitor is connected across the terminals of a 115 \(\mathrm{V}\) rms, \(60.0-\mathrm{Hz}\) generator. For what capacitance is the rms current \(2.3 \mathrm{mA} ?\)
A 40.0 -mH inductor, with internal resistance of \(30.0 \Omega\) is connected to an ac source $$\varepsilon(t)=(286 \mathrm{V}) \sin [(390 \mathrm{rad} / \mathrm{s}) t]$$ (a) What is the impedance of the inductor in the circuit? (b) What are the peak and rms voltages across the inductor (including the internal resistance)? (c) What is the peak current in the circuit? (d) What is the average power dissipated in the circuit? (e) Write an expression for the current through the inductor as a function of time.
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