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.]

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?

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

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?

(III) Determine a formula for the average power $\bar{P}$ dissipated in an LRC circuit in terms of L, R, C, ω and V₀. Find an approximate formula for the width of the resonance peak in average power, Δω, which is the difference in the two (angular) frequencies where $\bar{P}$ has half its maximum value. Assume a sharp peak.

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

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

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

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