Solve RLC Circuit Problem: Half Power at Resonance

[Saitama]

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?

Homework Equations

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!

 

[collinsmark]

The way you've defined it, Z is the impedance, not the reactance.

The reactance, X is such that Z = R + jX, and it's the X that you need to solve for.

Note that the magnitude squared of Z is, Z² = R² + X². That will come in useful.

 

[Saitama]

The way you've defined it, Z is the impedance, not the reactance.

Ah yes, thanks a lot collinsmark!