In a circuit, the instantaneous values of alternating current and voltages in a circuit is given by
`I = (1)/(sqrt2) sin (100 pi t) A` and
`E = (1)/(sqrt2) sin (100 pi t + (pi)/(3)) V`.
The average power in watts consumed in the circui is

Options

A. `(sqrt3)/(4)`
B. `(1)/(2)`
C. `(1)/(8)`
D. `(1)/(4)`

Explanation

To find the average power consumed in the circuit, we will follow these steps:

### Step 1: Identify the given values
From the question, we have:
- Instantaneous current: \( I = \frac{1}{\sqrt{2}} \sin(100 \pi t) \) A
- Instantaneous voltage: \( E = \frac{1}{\sqrt{2}} \sin(100 \pi t + \frac{\pi}{3}) \) V

### Step 2: Determine the peak values of current and voltage
The peak current \( I_0 \) and peak voltage \( E_0 \) can be extracted from the equations:
- \( I_0 = \frac{1}{\sqrt{2}} \) A
- \( E_0 = \frac{1}{\sqrt{2}} \) V

### Step 3: Calculate the RMS values
The RMS (Root Mean Square) values for current and voltage are given by:
- \( I_{rms} = \frac{I_0}{\sqrt{2}} = \frac{1/\sqrt{2}}{\sqrt{2}} = \frac{1}{2} \) A
- \( E_{rms} = \frac{E_0}{\sqrt{2}} = \frac{1/\sqrt{2}}{\sqrt{2}} = \frac{1}{2} \) V

### Step 4: Determine the phase difference
The phase difference \( \phi \) between the current and voltage is given as:
- \( \phi = \frac{\pi}{3} \)

### Step 5: Calculate \( \cos \phi \)
We need to find \( \cos \phi \):
- \( \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \)

### Step 6: Use the formula for average power
The average power \( P \) consumed in the circuit can be calculated using the formula:
\[ P = I_{rms} \cdot E_{rms} \cdot \cos \phi \]

Substituting the values we calculated:
\[ P = \left( \frac{1}{2} \right) \cdot \left( \frac{1}{2} \right) \cdot \left( \frac{1}{2} \right) \]

### Step 7: Simplify the expression
\[ P = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} \text{ W} \]

### Conclusion
The average power consumed in the circuit is:
\[ \boxed{\frac{1}{8}} \text{ watts} \]

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