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In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes $\mathrm{x}$ times its initial resonant frequency $\omega_0$. The value of $x$ is:
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$1 / 4$
The resonance frequency of LC oscillations circuit is
$\omega=\frac{1}{\sqrt{\mathrm{L}^{\prime} \mathrm{C}^{\prime}}} \Rightarrow \mathrm{L}^{\prime} \rightarrow 2 \mathrm{~L}$
$\mathrm{C}^{\prime} \rightarrow 8 \mathrm{C}$
$\omega=\frac{1}{\sqrt{2 \mathrm{~L} \times 8 \mathrm{C}}}=\frac{1}{4 \sqrt{\mathrm{LC}}} 9 \omega_0=\frac{1}{\sqrt{\mathrm{LC}}}$
$\omega=\frac{\omega_0}{4}$ So, $x=\frac{1}{4}$
$\omega=\frac{1}{\sqrt{\mathrm{L}^{\prime} \mathrm{C}^{\prime}}} \Rightarrow \mathrm{L}^{\prime} \rightarrow 2 \mathrm{~L}$
$\mathrm{C}^{\prime} \rightarrow 8 \mathrm{C}$
$\omega=\frac{1}{\sqrt{2 \mathrm{~L} \times 8 \mathrm{C}}}=\frac{1}{4 \sqrt{\mathrm{LC}}} 9 \omega_0=\frac{1}{\sqrt{\mathrm{LC}}}$
$\omega=\frac{\omega_0}{4}$ So, $x=\frac{1}{4}$
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