Question

A series LCR circuit containing $5.0\text{H}$ inductor, $80\mu \text{F}$ capacitor and $40\Omega$ resistor is connected to $230\text{V}$ variable frequency AC source. The angular frequencies of the source at which power transferred to the circuit is half the power at the resonant angular frequency are likely to be -

• A. $42\text{rad/s}$ and $58\text{rad/s}$
• B. $25\text{rad/s}$ and $75\text{rad/s}$
• C. $50\text{rad/s}$ and $25\text{rad/s}$
• D. $46\text{rad/s}$ and $54\text{rad/s}$

Answer 1

The correct option is D $46\text{rad/s}$ and $54\text{rad/s}$

The resonance frequency of LCR series circuit is given as,

${\omega }_0=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{5\times 80\times {10}^{-6}}}=50\text{rad/s}$

Now, half power frequencies are given as,

$\omega ={\omega }_0\pm \frac{R}{2L}$

So,

${\omega }_L=50-\frac{40}{2\times 5}=46\text{rad/s}$

${\omega }_H=50+\frac{40}{2\times 5}=54\text{rad/s}$

Hence, option $\left(D\right)$ is the correct answer.