In a LCR circuit capacitance is changed from C to 5C for the resonant frequency to remain unchanged, the inductance should be changed from L to value :

Question

Accepted Answer

In an LCR circuit, the resonant frequency is determined by the formula:

$$ f_0 = \frac{1}{2\pi\sqrt{LC}} $$

where $ f_0 $ is the resonant frequency, $ L $ is the inductance, and $ C $ is the capacitance.

To maintain the same resonant frequency when the capacitance is changed from $ C $ to $ 5C $, the product $ LC $ must remain constant. This means:

$$ L \times C = L' \times 5C $$

where $ L' $ is the new inductance. Solving for $ L' $, we get:

$$ L' = \frac{L \times C}{5C} = \frac{L}{5} $$

Thus, the inductance should be changed from $ L $ to $ \frac{L}{5} $ to keep the resonant frequency unchanged. This is why Option 2 is correct.

**Explanation for why other options are incorrect:**

1. **Option 1 (5L):** If the inductance were changed to $ 5L $, the product $ LC $ would become $ 5L \times 5C = 25LC $, which would increase the resonant frequency, contradicting the requirement to keep it unchanged.

2. **Option 3 ($\frac{L}{4}$):** Changing the inductance to $ \frac{L}{4} $ would result in the product $ \frac{L}{4} \times 5C = \frac{5LC}{4} $, which does not equal the original $ LC $. This would alter the resonant frequency.

3. **Option 4 ($\frac{L}{2}$):** If the inductance were changed to $ \frac{L}{2} $, the product would be $ \frac{L}{2} \times 5C = \frac{5LC}{2} $, again differing from the original $ LC $, thus changing the resonant frequency.

Therefore, only Option 2 ($\frac{L}{5}$) maintains the condition $ LC = L'C' $ necessary to keep the resonant frequency unchanged.