Find the average value of the square of the electromotive force `(E^(2))_(m)` over the interval from t = 0 to `t= (T)/(2)`

Answer

To find the average value of the square of the electromotive force \( (E^2)_m \) over the interval from \( t = 0 \) to \( t = \frac{T}{2} \), we can follow these steps:

### Step 1: Define the Electromotive Force
The electromotive force \( E_m \) is given by:
\[ E_m = E_0 \sin(\omega t) \]
where \( E_0 \) is the maximum electromotive force and \( \omega \) is the angular frequency.

### Step 2: Calculate \( E_m^2 \)
Now, we need to find \( E_m^2 \):
\[ E_m^2 = (E_0 \sin(\omega t))^2 = E_0^2 \sin^2(\omega t) \]

### Step 3: Set Up the Average Value Formula
The average value of a function \( f(t) \) over an interval \( [a, b] \) is given by:
\[ \text{Average value} = \frac{1}{b-a} \int_a^b f(t) \, dt \]
In our case, we need to find the average value of \( E_m^2 \) over the interval \( [0, \frac{T}{2}] \):
\[ \text{Average value} = \frac{1}{\frac{T}{2} - 0} \int_0^{\frac{T}{2}} E_0^2 \sin^2(\omega t) \, dt \]

### Step 4: Simplify the Expression
Substituting \( E_m^2 \) into the average value formula gives:
\[ \text{Average value} = \frac{2}{T} \int_0^{\frac{T}{2}} E_0^2 \sin^2(\omega t) \, dt \]

### Step 5: Use the Identity for \( \sin^2 \)
We can use the identity:
\[ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \]
to rewrite \( \sin^2(\omega t) \):
\[ \sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2} \]
Thus, we have:
\[ \int_0^{\frac{T}{2}} \sin^2(\omega t) \, dt = \int_0^{\frac{T}{2}} \frac{1 - \cos(2\omega t)}{2} \, dt \]

### Step 6: Calculate the Integral
Now, we can calculate the integral:
\[ \int_0^{\frac{T}{2}} \frac{1 - \cos(2\omega t)}{2} \, dt = \frac{1}{2} \left[ \int_0^{\frac{T}{2}} 1 \, dt - \int_0^{\frac{T}{2}} \cos(2\omega t) \, dt \right] \]
Calculating the first integral:
\[ \int_0^{\frac{T}{2}} 1 \, dt = \frac{T}{2} \]
Calculating the second integral:
\[ \int_0^{\frac{T}{2}} \cos(2\omega t) \, dt = \left[ \frac{\sin(2\omega t)}{2\omega} \right]_0^{\frac{T}{2}} = \frac{\sin(\pi)}{2\omega} - \frac{\sin(0)}{2\omega} = 0 \]
Thus, we have:
\[ \int_0^{\frac{T}{2}} \sin^2(\omega t) \, dt = \frac{1}{2} \left( \frac{T}{2} - 0 \right) = \frac{T}{4} \]

### Step 7: Substitute Back to Find the Average Value
Now substituting back into the average value formula:
\[ \text{Average value} = \frac{2}{T} \cdot E_0^2 \cdot \frac{T}{4} = \frac{E_0^2}{2} \]

### Final Answer
The average value of the square of the electromotive force over the interval from \( t = 0 \) to \( t = \frac{T}{2} \) is:
\[ \frac{E_0^2}{2} \]