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(B) The resonant frequency of an $LC$ circuit is given by $f = \frac{1}{2 \pi \sqrt{LC}}$.This implies $f \propto \frac{1}{\sqrt{LC}}$.Given the initi

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$\frac{f}{2 \sqrt{3}}$

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(B) The resonant frequency of an $LC$ circuit is given by $f = \frac{1}{2 \pi \sqrt{LC}}$.This implies $f \propto \frac{1}{\sqrt{LC}}$.Given the initial values $L_1 = L$ and $C_1 = C$,the initial frequency is $f = \frac{1}{2 \pi \sqrt{LC}}$.The new values are $L_2 = 3L$ and $C_2 = C + 3C = 4C$.The new resonant frequency $f'$ is given by $f' = \frac{1}{2 \pi \sqrt{L_2 C_2}} = \frac{1}{2 \pi \sqrt{(3L)(4C)}} = \frac{1}{2 \pi \sqrt{12LC}}$.Comparing the two frequencies: $\frac{f'}{f} = \frac{\frac{1}{2 \pi \sqrt{12LC}}}{\frac{1}{2 \pi \sqrt{LC}}} = \sqrt{\frac{LC}{12LC}} = \frac{1}{\sqrt{12}} = \frac{1}{2 \sqrt{3}}$.Therefore,$f' = \frac{f}{2 \sqrt{3}}$.

An $LC$ series resonant circuit produces a resonant frequency $f$. If $L$ is tripled and $C$ is increased by $3C$ (making the new capacitance $4C$),the new resonant frequency will be:

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