In an L-C-R circuit,the capacitance is change | vedclass

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(C) The resonant frequency of an $L-C-R$ circuit is given by the formula $\omega = \frac{1}{\sqrt{LC}}$.Let the initial resonant frequency be $\omega_

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$\frac{L}{2}$

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(C) The resonant frequency of an $L-C-R$ circuit is given by the formula $\omega = \frac{1}{\sqrt{LC}}$.Let the initial resonant frequency be $\omega_1 = \frac{1}{\sqrt{LC}}$.Let the new capacitance be $C' = 2C$ and the new inductance be $L'$. The new resonant frequency is $\omega_2 = \frac{1}{\sqrt{L'C'}} = \frac{1}{\sqrt{L'(2C)}}$.Since the resonant frequency remains unchanged,$\omega_1 = \omega_2$,which implies:$\frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{2L'C}}$Squaring both sides:$\frac{1}{LC} = \frac{1}{2L'C}$Canceling $C$ from both sides:$\frac{1}{L} = \frac{1}{2L'}$Solving for $L'$:$L' = \frac{L}{2}$.

In an $L-C-R$ circuit,the capacitance is changed from $C$ to $2C$. For the resonant frequency to remain unchanged,the inductance should be changed from $L$ to:

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