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(D) The resonant frequency $f$ of a series $LCR$ circuit is given by the formula:$f = \frac{1}{2 \pi \sqrt{LC}}$From this expression,we can see that t

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$\left(\frac{1}{3}\right)^{1/4} f_0$

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(D) The resonant frequency $f$ of a series $LCR$ circuit is given by the formula:$f = \frac{1}{2 \pi \sqrt{LC}}$From this expression,we can see that the resonant frequency is inversely proportional to the square root of the inductance $L$ when the capacitance $C$ is kept constant:$f \propto \frac{1}{\sqrt{L}}$Let the initial resonant frequency be $f_0$ with inductance $L$. The new resonant frequency $f'$ with new inductance $L' = \sqrt{3} L$ is:$\frac{f'}{f_0} = \sqrt{\frac{L}{L'}} = \sqrt{\frac{L}{\sqrt{3} L}} = \sqrt{\frac{1}{\sqrt{3}}} = \left(\frac{1}{3^{1/2}}\right)^{1/2} = \left(\frac{1}{3}\right)^{1/4}$Therefore,the new resonant frequency is $f' = \left(\frac{1}{3}\right)^{1/4} f_0$.Note that the change in resistance $R$ does not affect the resonant frequency of an $LCR$ circuit.

$A$ series $LCR$ circuit is connected across a source of alternating $emf$ of changing frequency and resonates at frequency $f_0$. Keeping capacitance constant,if the inductance $(L)$ is increased by $\sqrt{3}$ times and resistance $(R)$ is increased by $1.4$ times,the resonant frequency now is

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