Question

A series $LCR$ with $R=20\Omega$, $L=1.5\text{H}$ and $C=5\mu \text{F}$ is connected to a variable frequency $200\text{V}$ $AC$ supply. When the frequency of the supply is equal to the natural frequency of the circuit, the average power transferred to the circuit in one complete cycle is $n\times {10}^3\text{W}$. The value of $n$ is

Answer 1

When the frequency of the supply is equal to the natural frequency of the circuit, resonance occurs.

$\Rightarrow X_L=X_C$

$\therefore Z=R=20\Omega$

Now, $i_{rms}=\frac{E_{rms}}{Z}=\frac{200}{20}=10\text{A}$

Average power transferred/cycle

$P_{avg}=E_{rms}{i}_{rms}\cos 0^{\circ }$

$P_{avg}=200\times 10\times 1$

$\Rightarrow P_{avg}=2000\text{W}$

Comparing with $P_{avg}=n\times {10}^3$

$\Rightarrow n=2$