Question

The frequency of ac at which 16 \(\mu F\) capacitor and \( \frac{10}{\pi} \) mH inductor will have same reactance is:

• A. \( 1 \, \text{kHz} \)
• B. \( 1.25 \, \text{kHz} \)
• C. \( 1.5 \, \text{kHz} \)
• D. \( 2 \, \text{kHz} \)

Hint:

For reactance equality, use the formulas for inductive and capacitive reactance and solve for the frequency.

Answer 1

The Correct Option is B

The reactance of a capacitor is given by: \[ X_C = \frac{1}{2\pi f C} \] and the reactance of an inductor is given by: \[ X_L = 2\pi f L \] where: - \( C = 16 \, \mu F = 16 \times 10^{-6} \, F \), - \( L = \frac{10}{\pi} \, \text{mH} = \frac{10}{\pi} \times 10^{-3} \, H \). Since \( X_C = X_L \), equating both reactances: \[ \frac{1}{2\pi f C} = 2\pi f L \] Solving for \( f \): \[ f^2 = \frac{1}{4\pi^2 C L} \] Substitute the values of \( C \) and \( L \): \[ f^2 = \frac{1}{4\pi^2 (16 \times 10^{-6}) \times \left(\frac{10}{\pi} \times 10^{-3}\right)} \] \[ f^2 = \frac{1}{4 \times 3.1416^2 \times 16 \times 10^{-6} \times \frac{10}{\pi} \times 10^{-3}} \] \[ f^2 = \frac{1}{4 \times 3.1416^2 \times 16 \times 10^{-6} \times 10^{-3} \times \frac{10}{\pi}} \] \[ f = 1.25 \, \text{kHz} \] Thus, the required frequency is \( 1.25 \, \text{kHz} \).