In an `AC` circuit, the instantaneous values of e.m.f and current are `e=200sin314t` volt and `i=sin(314t+(pi)/3)` ampere. The average power consumed in watt is

Options

A. `200`
B. `100`
C. `50`
D. `25`

Explanation

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

### Step 1: Identify the given parameters
The instantaneous values of e.m.f (voltage) and current are given as:
- \( e(t) = 200 \sin(314t) \) volts
- \( i(t) = \sin(314t + \frac{\pi}{3}) \) amperes

### Step 2: Determine the peak values
From the equations, we can identify:
- Peak voltage \( E_0 = 200 \) volts
- Peak current \( I_0 = 1 \) ampere

### Step 3: Calculate the RMS values
The RMS (Root Mean Square) values are calculated using the formula:
\[ V_{RMS} = \frac{E_0}{\sqrt{2}} \quad \text{and} \quad I_{RMS} = \frac{I_0}{\sqrt{2}} \]
Substituting the peak values:
\[ V_{RMS} = \frac{200}{\sqrt{2}} \quad \text{and} \quad I_{RMS} = \frac{1}{\sqrt{2}} \]

### Step 4: Determine the phase difference
From the current equation, we can see that the phase difference \( \phi \) between the voltage and the current is:
\[ \phi = \frac{\pi}{3} \text{ radians} \quad \text{(or 60 degrees)} \]

### Step 5: Calculate the average power
The average power \( P \) consumed in an AC circuit can be calculated using the formula:
\[ P = V_{RMS} \cdot I_{RMS} \cdot \cos(\phi) \]
Substituting the values we calculated:
\[ P = \left(\frac{200}{\sqrt{2}}\right) \cdot \left(\frac{1}{\sqrt{2}}\right) \cdot \cos\left(\frac{\pi}{3}\right) \]
We know that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):
\[ P = \left(\frac{200}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}\right) \cdot \frac{1}{2} \]
\[ P = \left(\frac{200}{2}\right) = 100 \text{ watts} \]

### Final Result
The average power consumed in the circuit is:
\[ \boxed{100 \text{ watts}} \]