The resonant frequency of a series LCR circuit is ' $\mathrm{f}$ '. The circuit is now connected to the…

The resonant frequency of a series LCR circuit is ' $\mathrm{f}$ '. The circuit is now connected to the sinusoidally alternating e.m.f. of frequency ' $2 \mathrm{f}^{\prime}$. The new reactance $\mathrm{X}_{\mathrm{L}}^{\prime}$ and $\mathrm{X}_{\mathrm{C}}^{\prime}$ are related as
  1. $\mathrm{X}_{\mathrm{C}}^{\prime}=\frac{1}{4} \mathrm{X}_{\mathrm{L}}^{\prime}$
  2. $\mathrm{X}_{\mathrm{C}}^{\prime}=2 \mathrm{X}_{\mathrm{L}}^{\prime}$
  3. $\mathrm{X}_{\mathrm{C}}^{\prime}=\mathrm{X}_{\mathrm{L}}^{\prime}$
  4. $\mathrm{X}_{\mathrm{C}}^{\prime}=\frac{1}{2} \mathrm{X}_{\mathrm{L}}^{\prime}$

Solution

In a series LCR circuit, the resonant frequency is given by: $f_0=\frac{1}{2 \pi \sqrt{L C}}$ When the circuit is connected to a sinusoidally alternating e.m.f. with a frequency $f$, the reactance of the circuit (composed of inductive reactance $X_L$ and capacitive reactance $X_C$ ) will change with frequency. At resonance, $X_L=X_C$, but off-resonance, the total reactance changes depending on the frequency of the applied e.m.f. Answer: (1) is correct, as it shows the relationship between reactance and frequency in a series LCR circuit.

Asked in: MHT CET 2020 (12 Oct Shift 2)

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