An alternating current `'I'` is given by `I = i_(0) sin 2 pi (t//T + 1//4)`. Then the average current in the first one quarter time period to

Options

A. `(2i_(0))/(pi)`
B. `(i_(0))/(pi)`
C. `(i_(0))/(2 pi)`
D. `(3 i_(0))/(pi)`

Explanation

To find the average current over the first quarter of the time period for the given alternating current \( I = i_0 \sin\left(2\pi\left(\frac{t}{T} + \frac{1}{4}\right)\right) \), we can follow these steps:

### Step 1: Define the Average Current Formula
The average current \( \overline{I} \) over a time interval \( [0, T/4] \) can be expressed as:
\[ \overline{I} = \frac{1}{T/4} \int_0^{T/4} I \, dt \]

### Step 2: Substitute the Expression for \( I \)
Substituting the expression for \( I \) into the average current formula:
\[ \overline{I} = \frac{4}{T} \int_0^{T/4} i_0 \sin\left(2\pi\left(\frac{t}{T} + \frac{1}{4}\right)\right) dt \]

### Step 3: Simplify the Sine Function
The sine function can be simplified:
\[ \sin\left(2\pi\left(\frac{t}{T} + \frac{1}{4}\right)\right) = \sin\left(\frac{2\pi t}{T} + \frac{\pi}{2}\right) = \cos\left(\frac{2\pi t}{T}\right) \]
Thus, we have:
\[ \overline{I} = \frac{4}{T} \int_0^{T/4} i_0 \cos\left(\frac{2\pi t}{T}\right) dt \]

### Step 4: Factor Out Constants
Since \( i_0 \) is a constant, we can factor it out of the integral:
\[ \overline{I} = \frac{4i_0}{T} \int_0^{T/4} \cos\left(\frac{2\pi t}{T}\right) dt \]

### Step 5: Integrate the Cosine Function
To integrate \( \cos\left(\frac{2\pi t}{T}\right) \):
\[ \int \cos\left(\frac{2\pi t}{T}\right) dt = \frac{T}{2\pi} \sin\left(\frac{2\pi t}{T}\right) \]
Now, we evaluate the definite integral from \( 0 \) to \( T/4 \):
\[ \int_0^{T/4} \cos\left(\frac{2\pi t}{T}\right) dt = \left[\frac{T}{2\pi} \sin\left(\frac{2\pi t}{T}\right)\right]_0^{T/4} \]

### Step 6: Evaluate the Limits
Calculating the limits:
\[ = \frac{T}{2\pi} \left(\sin\left(\frac{2\pi (T/4)}{T}\right) - \sin(0)\right) = \frac{T}{2\pi} \left(\sin\left(\frac{\pi}{2}\right) - 0\right) = \frac{T}{2\pi} \]

### Step 7: Substitute Back into the Average Current Formula
Now substitute this result back into the average current formula:
\[ \overline{I} = \frac{4i_0}{T} \cdot \frac{T}{2\pi} = \frac{4i_0}{2\pi} = \frac{2i_0}{\pi} \]

### Final Result
Thus, the average current over the first quarter of the time period is:
\[ \overline{I} = \frac{2i_0}{\pi} \]