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

Options

A. `(T^(2))/(sqrt2)`
B. `(T^(2))/(2)`
C. `(T^(2))/(sqrt5)`
D. none of these

Answer

To find the root mean square (RMS) value of the current given by the function \( i(t) = t^2 \) for \( 0 < t < T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the RMS Formula**:
The RMS value of a function \( i(t) \) over a period \( T \) is given by the formula:
\[ I_{\text{rms}} = \sqrt{\frac{1}{T} \int_0^T i^2(t) \, dt} \]

2. **Substituting the Given Function**:
Given \( i(t) = t^2 \), we need to calculate \( i^2(t) \):
\[ i^2(t) = (t^2)^2 = t^4 \]

3. **Setting Up the Integral**:
Now, substitute \( i^2(t) \) into the RMS formula:
\[ I_{\text{rms}} = \sqrt{\frac{1}{T} \int_0^T t^4 \, dt} \]

4. **Calculating the Integral**:
We need to evaluate the integral \( \int_0^T t^4 \, dt \):
\[ \int t^4 \, dt = \frac{t^5}{5} \]
Now, apply the limits from 0 to \( T \):
\[ \int_0^T t^4 \, dt = \left[ \frac{T^5}{5} \right]_0^T = \frac{T^5}{5} - 0 = \frac{T^5}{5} \]

5. **Substituting Back into the RMS Formula**:
Now substitute the value of the integral back into the RMS formula:
\[ I_{\text{rms}} = \sqrt{\frac{1}{T} \cdot \frac{T^5}{5}} = \sqrt{\frac{T^5}{5T}} = \sqrt{\frac{T^4}{5}} = \frac{T^2}{\sqrt{5}} \]

6. **Final Result**:
Thus, the RMS value of the current is:
\[ I_{\text{rms}} = \frac{T^2}{\sqrt{5}} \]

### Summary:
The RMS value of the current \( i(t) = t^2 \) over the interval \( 0 < t < T \) is \( \frac{T^2}{\sqrt{5}} \).

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