Question

Find capacitance $C$ for the given circuit.
Given: \( L = 20\text{ H}, \quad R = 100\Omega, \quad \omega = 100\text{ rad/s} \)
The applied voltage and current are: \[ V = V_0 \sin \omega t, \quad i = i_0 \sin \omega t \]

• A. $5 \times 10^{-6}$ farad
• B. $8 \times 10^{-6}$ farad
• C. $7 \times 10^{-6}$ farad
• D. $4 \times 10^{-6}$ farad

Hint:

If voltage and current in an AC circuit are given with the same sine function, it directly indicates resonance in a series RLC circuit.

Answer 1

The Correct Option is A

Concept: In a series RLC circuit: - If current and voltage are in the same phase, the circuit is in resonance. - At resonance, inductive reactance equals capacitive reactance: \[ X_L = X_C \] \[ \omega L = \frac{1}{\omega C} \]
Step 1: Use the resonance condition: \[ \omega L = \frac{1}{\omega C} \]
Step 2: Solve for capacitance $C$: \[ C = \frac{1}{\omega^2 L} \]
Step 3: Substitute the given values: \[ C = \frac{1}{(100)^2 \times 20} = \frac{1}{200000} = 5 \times 10^{-6} \text{ F} \] Step 4: Thus, the required capacitance is: \[ \boxed{5 \times 10^{-6} \text{ farad}} \]