A series LCR circuit containing 5.0 H inductor, 80 μF capacitor and 40 Ω resistor is connected to 230 V variab

Question

A series LCR circuit containing 5.0 H inductor, 80 μF capacitor and 40 Ω 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:

Accepted Answer

To find the angular frequencies where the power is half of the maximum power (resonant power), we use the resonant frequency ($\omega_0$) and the half-power bandwidth ($\Delta \omega$). 
1. **Resonant angular frequency ($\omega_0$):** $\omega_0 = rac{1}{\sqrt{LC}} = rac{1}{\sqrt{5.0 	imes 80 	imes 10^{-6}}} = rac{1}{\sqrt{400 	imes 10^{-6}}} = rac{1}{2 	imes 10^{-2}} = 50$ rad/s.
2. **Half-bandwidth ($\Delta \omega$):** The frequencies at which power is halved occur at $\omega = \omega_0 \pm \Delta \omega$, where $\Delta \omega = rac{R}{2L}$. Substituting the values: $\Delta \omega = rac{40}{2 	imes 5.0} = 4$ rad/s. 
3. **Calculation:** The required angular frequencies are $\omega_1 = 50 - 4 = 46$ rad/s and $\omega_2 = 50 + 4 = 54$ rad/s.

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