A series LCR circuit containing a resistor of 5 , an inductor of 50 mH, and an unknown capacitor is connected to an AC voltage source given by V = 20 ( t) volts. The angular frequency is tuned such that the circuit is at resonance. At this frequency, the amplitude of the voltage across the inductor is measured to be 400 V. The value of the capacitance is _____ F .

Step-by-step solution

At resonance, the impedance of the series LCR circuit is purely resistive ( Z = R ). The maximum current amplitude I_m in the circuit is: I_m = V_m R = 20 5 = 4 A The voltage amplitude across the inductor is given by: V_ Lm = I_m X_L 400 = 4 X_L X_L = 100 Since X_L = L , the resonant angular frequency is: = X_L L = 100 50 10⁻³ = 2000 rad s ⁻¹ At resonance, the capacitive reactance equals the inductive reactance: X_C = X_L = 100 Using X_C = 1 C : C = 1 X_C = 1 2000 100 = 1 200000 F C = 5 10⁻⁶ F Therefore, the capaci