# Q7.9  A series LCR circuit with $R=20\Omega$, $L=1.5H$  and $c=35\mu F$ is connected to a variable-frequency $200\; V$ 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?

Given,

Resistance $R=20\Omega$

Inductance $L=1.5H$

Capacitance $C=35\mu F=35*10^{-6}F$

Voltage supply $V = 200V$

At resonance, supply frequency is equal to the natural frequency, and at the natural frequency, the total impedance of the circuit is equal to the resistance of the circuit

as inductive and capacitive reactance cancels each other. in other words,

$Z = \sqrt{\left ( \omega L-\frac{1}{\omega C} \right )^2+R^2}=\sqrt{0^2+R^2}=R=20\Omega$

As

$\omega L=\frac{1}{\omega C}$

Now,

Current in the circuit

$I=\frac{V}{Z}=\frac{200}{20}=10A$

Average Power transferred in the circuit :

$P=VI=200*10=2000W$

Hence average power transferred is 2000W.

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