A series `LCR` circuit with `R = 20 Omega, L = 1.5 H` 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 in `kW` transferred to the circuit in one complete cycle?

Options

A. 200 W
B. 2000 W
C. 100 W
D. 4000 W

Explanation

To find the average power transferred to the circuit in one complete cycle when the frequency of the supply equals the natural frequency of the circuit, we can follow these steps:

### Step 1: Identify the given parameters
- Resistance, \( R = 20 \, \Omega \)
- Inductance, \( L = 1.5 \, \text{H} \)
- Capacitance, \( C = 35 \, \mu\text{F} = 35 \times 10^{-6} \, \text{F} \)
- Voltage, \( V = 200 \, \text{V} \)

### Step 2: Calculate the natural frequency of the circuit
The natural frequency \( f_0 \) of an LCR circuit is given by the formula:
\[ f_0 = \frac{1}{2\pi\sqrt{LC}} \]
Substituting the values of \( L \) and \( C \):
\[ f_0 = \frac{1}{2\pi\sqrt{1.5 \times 35 \times 10^{-6}}} \]

### Step 3: Calculate the impedance at resonance
When the frequency of the supply equals the natural frequency of the circuit, the impedance \( Z \) of the circuit is equal to the resistance \( R \):
\[ Z = R = 20 \, \Omega \]

### Step 4: Calculate the average power
The average power \( P \) in an AC circuit is given by the formula:
\[ P = \frac{V_{\text{rms}}^2}{R} \]
Here, \( V_{\text{rms}} = 200 \, \text{V} \). Thus,
\[ P = \frac{(200)^2}{20} \]
Calculating this gives:
\[ P = \frac{40000}{20} = 2000 \, \text{W} \]

### Step 5: Convert power to kilowatts
To convert watts to kilowatts, we divide by 1000:
\[ P = \frac{2000}{1000} = 2 \, \text{kW} \]

### Final Answer
The average power transferred to the circuit in one complete cycle is \( 2 \, \text{kW} \).

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