A resistor of 50 , inductor of self inductance ( 2 ^2 ) H and a capacitor of unknown capacity are connected in series to an a.c. source of 100 ~V , 50 Hz supply. When the voltage and current are in phase, the value of capacitance is

Step-by-step solution

Circuit resonance occurs when voltage and current are in phase in an RLC series circuit, satisfying X_L = X_C . The inductive reactance is X_L = 2 f L and capacitive reactance is X_C = 1 2 f C . Setting equal at resonance gives: 2 f L = 1 2 f C Solving for capacitance yields C = 1 4 ^2 f^2 L . Substituting given values R = 50 , L = 2 ^2 H , and f = 50 Hz : C = 1 4 ^2 (50)^2 ( 2 ^2 ) = 1 4 2500 2 = 1 20000 F Converting to microfarads: 1 20000 10^6 = 50 F . This matches option D .