Alternating Current: Practice Problem & Solution
31. A series LCR circuit containing $5.0 H$ inductor, $80 mu F$ capacitor and $40 Omega$ resistor is connected to 230 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: (2021)
Solution Explained:
To solve this problem, we apply the core principles of Alternating Current. Understanding the underlying formula is key to arriving at the correct answer below:
Resonant frequency $\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{5 \times 80 \times 10^{-6}}} = 50 rad/s$.
Half-power frequencies are $\omega = \omega_0 \pm \Delta\omega$, where $\Delta\omega = \frac{R}{2L}$.
$\Delta\omega = \frac{40}{2 \times 5} = 4 rad/s$.
Therefore, $\omega = 50 \pm 4$, which gives $46 rad/s$ and $54 rad/s$.
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