The resonant frequency of a series LCR circui | vedclass

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(A) At resonance frequency $f$,the inductive reactance equals the capacitive reactance: $X_L = X_C \implies 2\pi f L = \frac{1}{2\pi f C}$.When the fr

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$X_{C}^{\prime} = \frac{1}{4} X_{L}^{\prime}$

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(A) At resonance frequency $f$,the inductive reactance equals the capacitive reactance: $X_L = X_C \implies 2\pi f L = \frac{1}{2\pi f C}$.When the frequency is doubled to $f' = 2f$,the new inductive reactance is $X_L' = 2\pi(2f)L = 2(2\pi f L) = 2X_L$.The new capacitive reactance is $X_C' = \frac{1}{2\pi(2f)C} = \frac{1}{2} \left( \frac{1}{2\pi f C} \right) = \frac{1}{2} X_C$.Since $X_L = X_C$,we can write $X_C = X_L$.Substituting this into the expression for $X_C'$,we get $X_C' = \frac{1}{2} X_L$.Since $X_L = \frac{1}{2} X_L'$,we have $X_C' = \frac{1}{2} (\frac{1}{2} X_L') = \frac{1}{4} X_L'$.

The resonant frequency of a series $LCR$ circuit is $f$. The circuit is now connected to a sinusoidally alternating e.m.f. of frequency $2f$. The new reactance $X_{L}^{\prime}$ and $X_{C}^{\prime}$ are related as:

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