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(A) The resonant angular frequency of an $LC$ circuit is given by the formula: $\omega = \frac{1}{\sqrt{LC}}$.Let the initial inductance be $L$ and th

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$\frac{\omega}{2 \sqrt{2}}$

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(A) The resonant angular frequency of an $LC$ circuit is given by the formula: $\omega = \frac{1}{\sqrt{LC}}$.Let the initial inductance be $L$ and the initial capacitance be $C$. The initial angular frequency is $\omega = \frac{1}{\sqrt{LC}}$.According to the problem,the new inductance $L' = 2L$ and the new capacitance $C' = 4C$.The new angular frequency $\omega'$ is given by: $\omega' = \frac{1}{\sqrt{L'C'}} = \frac{1}{\sqrt{(2L)(4C)}} = \frac{1}{\sqrt{8LC}}$.We can simplify this as: $\omega' = \frac{1}{\sqrt{8} \sqrt{LC}} = \frac{1}{2\sqrt{2} \sqrt{LC}}$.Since $\omega = \frac{1}{\sqrt{LC}}$,we substitute this into the expression for $\omega'$:$\omega' = \frac{\omega}{2\sqrt{2}}$.Therefore,the correct option is $A$.

In an $LC$ circuit,the angular frequency at resonance is $\omega$. What will be the new angular frequency when the inductor's inductance is made two times and the capacitor's capacitance is made four times?

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