For an ac circuit `V=15 sin omega t` and `I=20 cos omega t` the average power consumed in this circuit is

Options

A. 300 watt
B. 150 watt
C. 75 watt
D. zero

Answer

To find the average power consumed in the given AC circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Voltage and Current Equations:**
The voltage \( V \) and current \( I \) are given as:
\[ V = 15 \sin(\omega t) \]
\[ I = 20 \cos(\omega t) \]

2. **Convert the Current Equation to Sine Form:**
We can express the current in terms of sine by using the identity \( \cos(\theta) = \sin(\theta + \frac{\pi}{2}) \):
\[ I = 20 \cos(\omega t) = 20 \sin\left(\omega t + \frac{\pi}{2}\right) \]

3. **Determine the Phase Difference:**
The voltage \( V \) is \( 15 \sin(\omega t) \) and the current \( I \) is \( 20 \sin\left(\omega t + \frac{\pi}{2}\right) \). The phase difference \( \phi \) between the current and voltage is:
\[ \phi = \frac{\pi}{2} \]
This indicates that the current is leading the voltage by \( 90^\circ \).

4. **Calculate RMS Values:**
The RMS (Root Mean Square) values of voltage and current are calculated as follows:
\[ V_{\text{rms}} = \frac{V_0}{\sqrt{2}} = \frac{15}{\sqrt{2}} \quad \text{and} \quad I_{\text{rms}} = \frac{I_0}{\sqrt{2}} = \frac{20}{\sqrt{2}} \]

5. **Use the Average Power Formula:**
The average power \( P \) in an AC circuit is given by:
\[ P = V_{\text{rms}} \cdot I_{\text{rms}} \cdot \cos(\phi) \]

6. **Substitute the Values:**
We know \( \cos\left(\frac{\pi}{2}\right) = 0 \):
\[ P = \left(\frac{15}{\sqrt{2}}\right) \cdot \left(\frac{20}{\sqrt{2}}\right) \cdot \cos\left(\frac{\pi}{2}\right) \]
\[ P = \left(\frac{15 \cdot 20}{2}\right) \cdot 0 = 0 \]

7. **Conclusion:**
The average power consumed in the circuit is:
\[ P = 0 \]

### Final Answer:
The average power consumed in the circuit is \( 0 \).

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