In a circuit, the instantaneous values of alternating current and voltages in a circuit is given by
`I = (12)/(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. `(1)/(4)`
B. `(sqrt3)/(4)`
C. `(1)/(2)`
D. `(1)/(8)`

Explanation

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

### Step 1: Identify the peak values of current and voltage
Given:
- The instantaneous current is \( I = \frac{12}{\sqrt{2}} \sin(100 \pi t) \) A.
- The instantaneous voltage is \( E = \frac{1}{\sqrt{2}} \sin(100 \pi t + \frac{\pi}{3}) \) V.

From the current equation, the peak current \( I_0 \) is:
\[ I_0 = \frac{12}{\sqrt{2}} \text{ A} \]

From the voltage equation, the peak voltage \( E_0 \) is:
\[ E_0 = \frac{1}{\sqrt{2}} \text{ V} \]

### Step 2: Calculate the RMS values of current and voltage
The RMS (Root Mean Square) values are calculated as follows:
\[ I_{RMS} = \frac{I_0}{\sqrt{2}} = \frac{12/\sqrt{2}}{\sqrt{2}} = \frac{12}{2} = 6 \text{ A} \]
\[ E_{RMS} = \frac{E_0}{\sqrt{2}} = \frac{1/\sqrt{2}}{\sqrt{2}} = \frac{1}{2} \text{ V} \]

### Step 3: Determine the phase difference
The phase difference \( \phi \) between the current and voltage is given by the term in the voltage equation:
\[ \phi = \frac{\pi}{3} \]

### Step 4: Calculate the cosine of the phase difference
We need to calculate \( \cos \phi \):
\[ \cos \phi = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 5: Use the formula for average power
The average power \( P_{average} \) consumed in the circuit is given by the formula:
\[ P_{average} = I_{RMS} \cdot E_{RMS} \cdot \cos \phi \]
Substituting the values we calculated:
\[ P_{average} = 6 \cdot \frac{1}{2} \cdot \frac{1}{2} = 6 \cdot \frac{1}{4} = \frac{6}{4} = \frac{3}{2} \text{ W} \]

### Final Answer
The average power consumed in the circuit is:
\[ P_{average} = 1.5 \text{ W} \]

---