In an AC circuit , V o , I o and cos θ are voltage amplitude, current amplitude and power factor respectively, the power consumption is

Options

A. `(1)/(2)V_(0)I_(0)costheta`
B. `(1//sqrt2)V_(0)I_(0)costheta`
C. `V_(0)I_(0)costheta`
D. `(1//sqrt20V_(0)I_(0)sintheta`

Answer

To determine the power consumption in an AC circuit given the voltage amplitude \( V_0 \), current amplitude \( I_0 \), and power factor \( \cos \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- \( V_0 \): Voltage amplitude (peak voltage).
- \( I_0 \): Current amplitude (peak current).
- \( \cos \theta \): Power factor, which is the cosine of the phase angle between the voltage and current.

2. **Convert Amplitudes to RMS Values**:
- The root mean square (RMS) value of voltage \( V_{\text{rms}} \) is given by:
\[ V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \]
- The RMS value of current \( I_{\text{rms}} \) is given by:
\[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \]

3. **Use the Power Consumption Formula**:
- The power consumption \( P \) in an AC circuit can be calculated using the formula:
\[ P = V_{\text{rms}} \times I_{\text{rms}} \times \cos \theta \]

4. **Substitute the RMS Values**:
- Substitute \( V_{\text{rms}} \) and \( I_{\text{rms}} \) into the power formula:
\[ P = \left(\frac{V_0}{\sqrt{2}}\right) \times \left(\frac{I_0}{\sqrt{2}}\right) \times \cos \theta \]

5. **Simplify the Expression**:
- Simplifying the above expression gives:
\[ P = \frac{V_0 I_0 \cos \theta}{2} \]

6. **Final Answer**:
- Thus, the power consumption in the AC circuit is:
\[ P = \frac{1}{2} V_0 I_0 \cos \theta \]

### Conclusion:
The power consumption in the AC circuit is \( \frac{1}{2} V_0 I_0 \cos \theta \).