Step 1: Concept Introduction: This question pertains to a series RLC circuit. Maximum current in such a circuit occurs at resonance. Resonance is achieved when inductive reactance (\(X_L\)) equals capacitive reactance (\(X_C\)), minimizing the circuit's total impedance to the value of resistance (\(R\)).
Step 2: Governing Equation: The condition for resonance in a series RLC circuit is: \[ X_L = X_C \] where \(X_L = 2\pi f L\) (inductive reactance) and \(X_C = \frac{1}{2\pi f C}\) (capacitive reactance). Equating these yields the resonant frequency formula or allows for calculating the capacitance (\(C\)) required for resonance at a specified frequency (\(f\)). \[ 2\pi f L = \frac{1}{2\pi f C} \implies C = \frac{1}{(2\pi f)^2 L} = \frac{1}{4\pi^2 f^2 L} \]
Step 3: Calculation Details: Provided Values: Inductance, \(L = 500 \, \text{mH} = 0.5 \, \text{H}\). Frequency, \(f = 0.4 \, \text{kHz} = 400 \, \text{Hz}\). Computation: Utilize the resonance capacitance formula with the given values: \[ C = \frac{1}{4\pi^2 f^2 L} \] \[ C = \frac{1}{4\pi^2 (400)^2 (0.5)} \] \[ C = \frac{1}{4\pi^2 (160000) (0.5)} \] \[ C = \frac{1}{320000 \pi^2} \, \text{F} \] Approximating \(\pi^2 \approx 9.87\): \[ C \approx \frac{1}{320000 \times 9.87} = \frac{1}{3158400} \, \text{F} \] \[ C \approx 3.166 \times 10^{-7} \, \text{F} \] To express in microfarads (µF), multiply by \(10^6\): \[ C \approx 3.166 \times 10^{-7} \times 10^6 \, \mu\text{F} = 0.3166 \, \mu\text{F} \] This value is approximately 0.32 µF.
Step 4: Conclusion: The capacitance necessary to achieve resonance (maximum current) is approximately 0.32 µF.