31. Alternating Current

The frequency of the ac voltage source (peak voltage Vo) in an LRC circuit is tuned to the circuit’s resonant frequency f₀ = 1 / (2π√LC). (a) Show that the peak voltage across the capacitor is Vco = VoTo/ (2πτ), where To ( =1/fo) is the period of the resonant frequency and τ = RC is the time constant for charging the capacitor C through a resistor R. (b) Define β = To/ (2πτ) so that Vco = βVo. Then β is the “amplification” of the source voltage across the capacitor. If a particular LRC circuit contains a 2.0-nF capacitor and has a resonant frequency of 5.0 kHz, what value of R will yield β = 125?

For the circuit of FIGURE EX32.32, What is the resonance frequency, in both rad/s and Hz?

(II) (a) Show that oscillation of charge Q on the capacitor of an LRC circuit has amplitude

$Q_0=\frac{V_0}{\sqrt{(\omega R{)}^2+{({\omega }^{2}L-\frac{1}{C})}^2}}.$

In a series $LRC$ circuit, what is the primary effect of resonance on the circuit's response?

A series LRC circuit is formed with a power source operating at VRMS = 100 V, and is formed with a 15 Ω resistor, a 0.05 H inductor, and a 200 µF capacitor. What is the voltage across the inductor in resonance? The voltage across the capacitor?

A television channel is assigned the frequency range from 54 MHz to 60 MHz. A series RLC tuning circuit in a TV receiver resonates in the middle of this frequency range. The circuit uses a 16 pF capacitor. What is the value of the inductor?

Which of the following events is an example of resonance in a series $LRC$ circuit?

(II) A 25-mH coil whose resistance is 0.80 Ω is connected to a capacitor C and a 420-Hz source voltage. If the current and voltage are to be in phase, what value must C have?

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled?

Determine a formula for the average power $\bar{P}$ dissipated in an LRC circuit in terms of L, R, C, ω and V₀. At what frequency is the power a maximum?

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled and, at the same time, the inductor value is halved?

For the circuit of FIGURE EX32.33, What is the resonance frequency, in both rad/s and Hz?

To detect vehicles at traffic lights, wire loops with dimensions on the order of 2 m are often buried horizontally under roadways. Assume the self-inductance of such a coil is L = 5.0 mH and that it is part of an LRC circuit as shown in Fig. 30–40 with C = 0.10 μF and R = 38 Ω. The ac voltage has frequency f and rms voltage V_rms. (a) The frequency f is chosen to match the resonant frequency f₀ of the circuit. Find f₀ and determine what the rms voltage (V_R)_rms across the resistor will be when f = f₀. (b) Assume that f, C, and R never change, but that, when a car is located above the buried coil, the coil’s self-inductance decreases by 10% (due to induced eddy currents in the car’s metal parts). Determine by what factor the voltage (V_R)_rms decreases in the presence of a car in comparison to no car above the loop and thus how it detects the presence of a car. (c) Describe how the eddy currents induced in the car reduce L. [Hint: Recall Eq. 30–4, the definition of inductance.]

What is the resonant frequency of the LRC circuit of Example 30–11? At what rate is energy taken from the generator, on the average, at this frequency?

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the resistor value is doubled?