The average emf during the positive half cycle of an ac supply of peak value `E_(0)` is .

Options

A. `(E_(0))/(pi)`
B. `(E_(0))/(sqrt2pi)`
C. `(E_(0))/(2pi)`
D. `(2E_(0))/(pi)`

Explanation

To find the average EMF during the positive half cycle of an AC supply with a peak value \( E_0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation of EMF**:
The instantaneous EMF \( E(t) \) of an AC supply is given by:
\[ E(t) = E_0 \sin(\omega t) \]
where \( E_0 \) is the peak value and \( \omega \) is the angular frequency.

2. **Determine the Time Period for the Positive Half Cycle**:
The positive half cycle occurs from \( t = 0 \) to \( t = \frac{T}{2} \), where \( T \) is the time period of the AC signal.

3. **Set Up the Average Calculation**:
The average EMF over the positive half cycle can be calculated using the formula:
\[ E_{\text{avg}} = \frac{1}{T/2} \int_0^{T/2} E(t) \, dt \]
Substituting \( E(t) \):
\[ E_{\text{avg}} = \frac{2}{T} \int_0^{T/2} E_0 \sin(\omega t) \, dt \]

4. **Factor Out Constants**:
Since \( E_0 \) is a constant, we can factor it out of the integral:
\[ E_{\text{avg}} = \frac{2E_0}{T} \int_0^{T/2} \sin(\omega t) \, dt \]

5. **Integrate the Sine Function**:
The integral of \( \sin(\omega t) \) is:
\[ \int \sin(\omega t) \, dt = -\frac{1}{\omega} \cos(\omega t) \]
Therefore, we evaluate the definite integral:
\[ \int_0^{T/2} \sin(\omega t) \, dt = \left[-\frac{1}{\omega} \cos(\omega t)\right]_0^{T/2} \]

6. **Evaluate the Limits**:
Substitute the limits into the integral:
\[ = -\frac{1}{\omega} \left( \cos\left(\frac{\omega T}{2}\right) - \cos(0) \right) \]
Since \( \frac{\omega T}{2} = \pi \):
\[ = -\frac{1}{\omega} \left( -1 - 1 \right) = \frac{2}{\omega} \]

7. **Substitute Back into the Average EMF Formula**:
Now substitute this result back into the average EMF formula:
\[ E_{\text{avg}} = \frac{2E_0}{T} \cdot \frac{2}{\omega} \]

8. **Relate Time Period to Angular Frequency**:
Recall that \( T = \frac{2\pi}{\omega} \):
\[ E_{\text{avg}} = \frac{2E_0}{\frac{2\pi}{\omega}} \cdot \frac{2}{\omega} = \frac{2E_0 \cdot \omega}{2\pi} = \frac{2E_0}{\pi} \]

9. **Final Result**:
Thus, the average EMF during the positive half cycle is:
\[ E_{\text{avg}} = \frac{2E_0}{\pi} \]