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(B) The resonant frequency of an $L-C$ circuit is given by the formula: $f = \frac{1}{2 \pi \sqrt{LC}}$.From this relation,we can see that $f \propto

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$\frac{f}{2}$

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(B) The resonant frequency of an $L-C$ circuit is given by the formula: $f = \frac{1}{2 \pi \sqrt{LC}}$.From this relation,we can see that $f \propto \frac{1}{\sqrt{C}}$.Let the initial frequency be $f_1 = f$ and the initial capacitance be $C_1 = C$.Let the new frequency be $f_2$ and the new capacitance be $C_2 = 4C$.Using the proportionality $f \propto \frac{1}{\sqrt{C}}$,we have the ratio: $\frac{f_2}{f_1} = \sqrt{\frac{C_1}{C_2}}$.Substituting the values: $\frac{f_2}{f} = \sqrt{\frac{C}{4C}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$.Therefore,the new resonant frequency is $f_2 = \frac{f}{2}$.

$A$ resonant frequency of a circuit is $f$. If the capacitance is made four times the initial value,then the resonant frequency will become

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