The average value of alternating current `I=I_(0) sin omegat` in time interval `[0, pi/omega]` is

Options

A. `(2I_(0))/pi`
B. `2I_(0)`
C. `(4I_(0))/pi`
D. `I_(0)/pi`

Explanation

To find the average value of the alternating current \( I = I_0 \sin(\omega t) \) over the time interval \([0, \frac{\pi}{\omega}]\), we can follow these steps:

### Step 1: Understand the formula for average value
The average value of a function \( I(t) \) over the interval \([a, b]\) is given by:
\[ I_{\text{avg}} = \frac{1}{b-a} \int_a^b I(t) \, dt \]
In our case, \( I(t) = I_0 \sin(\omega t) \), \( a = 0 \), and \( b = \frac{\pi}{\omega} \).

### Step 2: Set up the integral
Substituting the values into the formula, we have:
\[ I_{\text{avg}} = \frac{1}{\frac{\pi}{\omega} - 0} \int_0^{\frac{\pi}{\omega}} I_0 \sin(\omega t) \, dt \]
This simplifies to:
\[ I_{\text{avg}} = \frac{\omega}{\pi} \int_0^{\frac{\pi}{\omega}} I_0 \sin(\omega t) \, dt \]

### Step 3: Factor out constants
Since \( I_0 \) is a constant, we can factor it out of the integral:
\[ I_{\text{avg}} = \frac{\omega I_0}{\pi} \int_0^{\frac{\pi}{\omega}} \sin(\omega t) \, dt \]

### Step 4: Integrate \( \sin(\omega t) \)
To integrate \( \sin(\omega t) \), we use the substitution \( u = \omega t \), which gives \( du = \omega dt \) or \( dt = \frac{du}{\omega} \). The limits change from \( t = 0 \) to \( t = \frac{\pi}{\omega} \) to \( u = 0 \) to \( u = \pi \):
\[ \int_0^{\frac{\pi}{\omega}} \sin(\omega t) \, dt = \int_0^{\pi} \sin(u) \frac{du}{\omega} = \frac{1}{\omega} \int_0^{\pi} \sin(u) \, du \]

### Step 5: Evaluate the integral
The integral \( \int_0^{\pi} \sin(u) \, du \) evaluates to:
\[ \int_0^{\pi} \sin(u) \, du = [-\cos(u)]_0^{\pi} = -\cos(\pi) - (-\cos(0)) = 1 + 1 = 2 \]
Thus, we have:
\[ \int_0^{\frac{\pi}{\omega}} \sin(\omega t) \, dt = \frac{2}{\omega} \]

### Step 6: Substitute back into the average value formula
Substituting this back into our expression for \( I_{\text{avg}} \):
\[ I_{\text{avg}} = \frac{\omega I_0}{\pi} \cdot \frac{2}{\omega} = \frac{2 I_0}{\pi} \]

### Final Answer
The average value of the alternating current \( I \) over the interval \([0, \frac{\pi}{\omega}]\) is:
\[ \boxed{\frac{2 I_0}{\pi}} \]