If `i= t^(2), 0 lt t lt T` then `r.m.s.` value of current is

Options

A. `(T^(2))/(sqrt(2))`
B. `(T^(2))/2`
C. `(T^(2))/(sqrt(5))`
D. None of these

Explanation

To find the root mean square (r.m.s.) value of the current given by the equation \( i = t^2 \) for the interval \( 0 < t < T \), we can follow these steps:

### Step 1: Understand the formula for r.m.s. value
The r.m.s. value of a current \( i(t) \) over a time period \( T \) is given by the formula:
\[ I_{r.m.s} = \sqrt{\frac{1}{T} \int_0^T i^2 \, dt} \]

### Step 2: Substitute the given current into the formula
In this case, the current is \( i(t) = t^2 \). Therefore, we need to calculate \( i^2 \):
\[ i^2 = (t^2)^2 = t^4 \]

### Step 3: Set up the integral
Now, we substitute \( i^2 \) into the r.m.s. formula:
\[ I_{r.m.s} = \sqrt{\frac{1}{T} \int_0^T t^4 \, dt} \]

### Step 4: Calculate the integral
We need to evaluate the integral \( \int_0^T t^4 \, dt \):
\[ \int_0^T t^4 \, dt = \left[ \frac{t^5}{5} \right]_0^T = \frac{T^5}{5} \]

### Step 5: Substitute the integral back into the r.m.s. formula
Now we can substitute the result of the integral back into the r.m.s. formula:
\[ I_{r.m.s} = \sqrt{\frac{1}{T} \cdot \frac{T^5}{5}} = \sqrt{\frac{T^4}{5}} = \frac{T^2}{\sqrt{5}} \]

### Final Result
Thus, the r.m.s. value of the current is:
\[ I_{r.m.s} = \frac{T^2}{\sqrt{5}} \]

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