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. 100
B. 200
C. 50
D. 25

Answer

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

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

From the equation of EMF, we can see that the peak value \( E_0 \) is 200 volts. For the current, the peak value \( I_0 \) is 1 ampere.

### Step 2: Calculate the RMS values
The RMS (Root Mean Square) values are calculated using the formula:
- \( V_{rms} = \frac{E_0}{\sqrt{2}} \)
- \( I_{rms} = \frac{I_0}{\sqrt{2}} \)

Calculating these:
- \( V_{rms} = \frac{200}{\sqrt{2}} = 100\sqrt{2} \) volts
- \( I_{rms} = \frac{1}{\sqrt{2}} \) amperes

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

### Step 4: Calculate the average power
The average power \( P \) consumed in an AC circuit is given by the formula:
\[ P = V_{rms} \cdot I_{rms} \cdot \cos(\phi) \]

Substituting the values we calculated:
- \( P = \left(100\sqrt{2}\right) \cdot \left(\frac{1}{\sqrt{2}}\right) \cdot \cos\left(\frac{\pi}{3}\right) \)

### Step 5: Calculate \( \cos\left(\frac{\pi}{3}\right) \)
We know that:
- \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)

### Step 6: Substitute and simplify
Now substituting this value back into the power equation:
\[ P = 100\sqrt{2} \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{2} \]
\[ P = 100 \cdot \frac{1}{2} = 50 \text{ watts} \]

### Final Answer
The average power consumed in the circuit is \( 50 \) watts.

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