Question

In a LCR circuit capacitance is changed from \(\mathrm{C}\) to \(2 \mathrm{C}\). For the resonant frequency to remain unchanged, the inductance should be change from \(L\) to (a) \(4 \mathrm{~L}\) (b) \(2 \mathrm{~L}\) (c) \(L / 2\) (d) \(\mathrm{L} / 4\)

Short answer

The inductance should change from L to L/2 for the resonant frequency to remain unchanged. The correct answer is (c) L/2.

Step-by-step solution

Formulate resonance frequency formulas for the initial and final LCR circuits

For a series LCR circuit, the resonance frequency (\(f_0\)) is given by the formula: \(f_0 = \dfrac{1}{2 \pi \sqrt{L C}}\) For the initial LCR circuit, we have capacitance C and inductance L: \(f_0 = \dfrac{1}{2 \pi \sqrt{L C}}\) For the final LCR circuit, the capacitance is doubled, and we will assume the inductance changes to L': \(f_0 = \dfrac{1}{2 \pi \sqrt{L' (2C)}}\)

Equate the two resonance frequency formulas

Since the resonance frequency remains unchanged, we can equate the two formulas: \(\dfrac{1}{2 \pi \sqrt{L C}} = \dfrac{1}{2 \pi \sqrt{L' (2C)}}\)

Solve for L'

Now we need to solve for L' in terms of L. First, we can cancel out constants on both sides of the equation: \(\sqrt{L C} = \sqrt{L' (2C)}\) Now square both sides to get rid of the square roots: \(L C = L' (2C)\) Since C ≠ 0, we can divide both sides by C: \(L = 2 L'\) Finally, we solve for L': \(L' = \dfrac{L}{2}\) So the inductance should change from L to L/2 for the resonance frequency to remain unchanged. The correct answer is (c) L/2.