Question

In an LCR circuit the inductance is changed from \( L \) to \( 9L \). For the same resonant frequency, the capacitance should be changed from \( C \) to

• A. \( 9C \)
• B. \( 3C \)
• C. \( \frac{C}{9} \)
• D. \( \frac{C}{3} \)

Hint:

For the resonant frequency of an LCR circuit, if inductance increases, capacitance must decrease to maintain the same frequency.

Answer 1

The Correct Option is C

Step 1: Understanding the resonant frequency of LCR circuit.
The resonant frequency \( f \) of an LCR circuit is given by: \[ f = \frac{1}{2 \pi \sqrt{LC}} \] Step 2: Finding the new capacitance for the same frequency.
When the inductance \( L \) is changed to \( 9L \), for the frequency to remain the same, the capacitance \( C \) must change such that: \[ \frac{1}{2 \pi \sqrt{9L \cdot C'}} = \frac{1}{2 \pi \sqrt{L \cdot C}} \] Solving this, we find that \( C' = \frac{C}{9} \). Thus, the correct answer is (C) \( \frac{C}{9} \).