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(D) The resonance frequency $f$ of a series $LCR$ circuit is given by the formula: $f = \frac{1}{2 \pi \sqrt{LC}}$.Initially,$f_1 = \frac{1}{2 \pi \sq

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$4$

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(D) The resonance frequency $f$ of a series $LCR$ circuit is given by the formula: $f = \frac{1}{2 \pi \sqrt{LC}}$.Initially,$f_1 = \frac{1}{2 \pi \sqrt{LC}}$.Finally,the resonance frequency becomes $f_2 = 2f_1$ by changing $C$ to $C^{\prime}$.Thus,$f_2 = \frac{1}{2 \pi \sqrt{LC^{\prime}}}$.Substituting $f_2 = 2f_1$ into the equation:$\frac{1}{2 \pi \sqrt{LC^{\prime}}} = 2 	imes \frac{1}{2 \pi \sqrt{LC}}$.Canceling common terms $2 \pi$ and $\sqrt{L}$ from both sides:$\frac{1}{\sqrt{C^{\prime}}} = \frac{2}{\sqrt{C}}$.Squaring both sides:$\frac{1}{C^{\prime}} = \frac{4}{C}$.Therefore,the ratio $\frac{C}{C^{\prime}} = 4$.Note: The change in resistance $R$ does not affect the resonance frequency of an $LCR$ circuit.

In an $LCR$ circuit,the resonance frequency of the circuit increases to two times its initial value by changing the capacitance from $C$ to $C^{\prime}$ and the resistance from $100 \ \Omega$ to $400 \ \Omega$,while the inductance $L$ is kept constant. The ratio $C / C^{\prime}$ is:

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