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**1. Homework Statement**

A series LCR circuit with ##L=0.125/\pi## H, ##C=500/\pi## nF and ##R=23\,\Omega## is connected to a 230 V variable frequency supply. For what reactance of circuit, the power transferred to the circuit is half the power at resonance?

**2. Homework Equations**

**3. The Attempt at a Solution**

At resonance,

$$f=\frac{1}{2\pi\sqrt{LC}}=2000\,Hz$$

Hence, the power transferred at resonance is given by ##P=V^2_{rms}/R=2300\,\,W##.

When the power transferred is half, let the reactance be Z, hence,

$$P'=\frac{V^2_{rms}}{Z}\cos\phi=\frac{V^2_{rms}}{Z}\frac{R}{Z}$$

As per the question:

$$\frac{V^2_{rms}}{Z}\frac{R}{Z}=\frac{1}{2}\times 2300$$

$$\Rightarrow \frac{230\times 230 \times 23}{Z^2}=\frac{1}{2}\times 2300$$

$$\Rightarrow Z=23\sqrt{2} \,\,\Omega$$

But this is incorrect. The correct answer is ##23\,\,\Omega##.

Any help is appreciated. Thanks!