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A series LCR circuit with $\mathrm{R}=20 \Omega, \mathrm{L}=1.5 \mathrm{H}$ and $C=35 \mu \mathrm{F}$ is connected to a variable-frequency of 200 $V$ ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?
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Verified Answer
Given: $\mathrm{R}=20 \Omega, \mathrm{L}=1.5 \mathrm{H}, \mathrm{C}=35 \mu \mathrm{F}$, average emf $\mathrm{E}_{\mathrm{v}}=$ 200
To find: Average power at resonance $\mathrm{P}_{\mathrm{av}}$
Formula: $\mathrm{P}_{\mathrm{av}}=\mathrm{I}_{\mathrm{v}} \mathrm{E}_{\mathrm{v}}$
When supply frequency equals natural frequency of circuit it is resonance and $X_L=X_C$ hence
$$
\begin{aligned}
&\mathrm{I}_{\mathrm{v}}=\frac{\mathrm{E}_{\mathrm{v}}}{\mathrm{R}}=\frac{200}{20}=10 \mathrm{~A} \\
&\therefore \cos \phi=1 \text { at resonance and } \mathrm{P}_{\mathrm{av}}=\mathrm{E}_{\mathrm{v}} \mathrm{I}_{\mathrm{v}}=200 \times 10=2000 \mathrm{~W}
\end{aligned}
$$
To find: Average power at resonance $\mathrm{P}_{\mathrm{av}}$
Formula: $\mathrm{P}_{\mathrm{av}}=\mathrm{I}_{\mathrm{v}} \mathrm{E}_{\mathrm{v}}$
When supply frequency equals natural frequency of circuit it is resonance and $X_L=X_C$ hence
$$
\begin{aligned}
&\mathrm{I}_{\mathrm{v}}=\frac{\mathrm{E}_{\mathrm{v}}}{\mathrm{R}}=\frac{200}{20}=10 \mathrm{~A} \\
&\therefore \cos \phi=1 \text { at resonance and } \mathrm{P}_{\mathrm{av}}=\mathrm{E}_{\mathrm{v}} \mathrm{I}_{\mathrm{v}}=200 \times 10=2000 \mathrm{~W}
\end{aligned}
$$
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