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Chapter 30: Problem 59

When you turn the dial on a radio to tune it, you are adjusting a variable capacitor in an LC circuit. Suppose you tune to an AM station broadcasting at a frequency of \(1000 . \mathrm{kHz},\) and there is a \(10.0-\mathrm{mH}\) inductor in the tuning circuit. When you have tuned in the station, what is the capacitance of the capacitor?

Short Answer

Expert verified
Answer: The capacitance needed for tuning the radio to the given frequency is approximately \(2.53 \times 10^{-12}\,\text{F}\).

Step by step solution

01

Write down the given information

We are given the following information: - Frequency of the radio station: \(f = 1000 \,\text{kHz}\) - Inductance of the inductor: \(L = 10.0\, \text{mH}\)
02

Convert the frequency and inductance to SI units

Before we proceed with the calculations, let's convert the frequency and inductance to the appropriate SI units: - Frequency: \(f = 1000 \,\text{kHz} = 1000 \times 10^{3} \,\text{Hz} = 10^6\, \text{Hz}\) - Inductance: \(L = 10.0 \,\text{mH} = 10.0 \times 10^{-3}\, \text{H}\)
03

Apply the resonant frequency formula

We can use the formula for the resonance frequency of an LC circuit: $$ f = \dfrac{1}{2\pi\sqrt{LC}} $$ We need to rearrange this formula to get the capacitance, \(C\): $$ C = \dfrac{1}{(2\pi f)^2 L} $$
04

Substitute the values and calculate the capacitance

Now, let's substitute the values of \(f\) and \(L\) that we found earlier into the formula: $$ C =\dfrac{1}{(2\pi (10^6\,\text{Hz}))^2 (10.0 \times 10^{-3}\,\text{H})} $$ Calculate the value of the capacitance: $$ C \approx 2.53 \times 10^{-12}\,\text{F} $$
05

Write down the final answer

The capacitance of the capacitor needed to tune the radio to the given frequency is approximately \(2.53 \times 10^{-12}\,\text{F}\).

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Most popular questions from this chapter

A 2.00 - \(\mu\) F capacitor was fully charged by being connected to a 12.0 - \(V\) battery. The fully charged capacitor is then connected in series with a resistor and an inductor: \(R=50.0 \Omega\) and \(L=0.200 \mathrm{H}\). Calculate the damped frequency of the resulting circuit.

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

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 radio tuner has a resistance of \(1.00 \mu \Omega\), a capacitance of \(25.0 \mathrm{nF}\), and an inductance of \(3.00 \mathrm{mH}\) a) Find the resonant frequency of this tuner. b) Calculate the power in the circuit if a signal at the resonant frequency produces an emf across the antenna of \(V_{\mathrm{rms}}=1.50 \mathrm{mV}\)

In Solved Problem 30.1 , the voltage supplied by the source of time-varying emf is \(33.0 \mathrm{~V}\), the voltage across the resistor is \(V_{R}=I R=13.1 \mathrm{~V}\), and the voltage across the inductor is \(V_{L}=I X_{L}=30.3 \mathrm{~V}\). Does this circuit obey Kirchhoff's rules?
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