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. 20 kW
B. 2kW
C. zero
D. 25kW

Answer

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

### Step 1: Identify the given values
We have the following equations for the instantaneous values of e.m.f (E) and current (I):
- \( e = 200 \sin(314t) \) volts
- \( i = \sin(314t + \frac{\pi}{3}) \) amperes

From these equations, we can identify:
- Maximum voltage \( V_0 = 200 \) volts
- Maximum current \( I_0 = 1 \) ampere
- Phase difference \( \phi = \frac{\pi}{3} \) radians

### Step 2: Calculate RMS values
The root mean square (RMS) values for voltage and current are calculated using the formulas:
- \( V_{rms} = \frac{V_0}{\sqrt{2}} \)
- \( I_{rms} = \frac{I_0}{\sqrt{2}} \)

Substituting the values:
- \( V_{rms} = \frac{200}{\sqrt{2}} \)
- \( I_{rms} = \frac{1}{\sqrt{2}} \)

### Step 3: Calculate the average power
The average power \( P \) consumed in an AC circuit can be calculated using the formula:
\[ P = V_{rms} \times I_{rms} \times \cos(\phi) \]

We know:
- \( \cos(\phi) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)

Substituting the RMS values and the cosine of the phase difference:
\[ P = \left(\frac{200}{\sqrt{2}}\right) \times \left(\frac{1}{\sqrt{2}}\right) \times \frac{1}{2} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ P = \frac{200 \times 1}{2} = 100 \]

### Step 5: Final calculation
Now, we divide by \( 2 \) (from the \( \sqrt{2} \) terms):
\[ P = \frac{100}{2} = 50 \text{ watts} \]

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

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