An alternating voltage given by V = 120 (2000t) is applied across a series LCR circuit containing an inductor of 0.1 H , a capacitor of 10 F , and a resistor of 40 . Let ₀ be the resonant frequency of this circuit. Which of the following options correctly gives the value of ₀ and describes the phase of the current relative to the applied voltage at the given source frequency?

Step-by-step solution

Given V = 120 (2000t) , the angular frequency of the source is = 2000 rad/s . The resonant angular frequency is given by ₀ = 1 LC . Substituting the given values L = 0.1 H and C = 10 F = 10⁻⁵ F : ₀ = 1 0.1 10⁻⁵ = 1 10⁻⁶ = 1000 rad/s The resonant frequency ₀ is: ₀ = ₀ 2 = 1000 2 = 500 Hz To find the phase relationship at the source frequency = 2000 rad/s , we calculate the inductive and capacitive reactances: X_L = L = 2000 0.1 = 200 X_C = 1 C = 1 2000 10⁻⁵ = 50 Since X_L > X_C , the circuit is predominantly inducti