In a certain circuit current changes with time according to `i=2sqrt(t)` RMS value of current between t=2s to t=4s will be

Options

A. 3A
B. `3sqrt(3)`A
C. `2sqrt(3)`
D. `sqrt(3)`A

Explanation

To find the RMS (Root Mean Square) value of the current \( i = 2\sqrt{t} \) between \( t = 2 \) seconds and \( t = 4 \) seconds, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding RMS Value**:
The RMS value of a current \( i(t) \) over a time interval \( [a, b] \) is given by:
\[ I_{\text{RMS}} = \sqrt{\frac{1}{b-a} \int_a^b i(t)^2 \, dt} \]

2. **Substituting the Current Function**:
Here, \( i(t) = 2\sqrt{t} \). Therefore, we need to calculate \( i(t)^2 \):
\[ i(t)^2 = (2\sqrt{t})^2 = 4t \]

3. **Setting Up the Integral**:
We will now set up the integral for the interval from \( t = 2 \) to \( t = 4 \):
\[ I_{\text{RMS}} = \sqrt{\frac{1}{4-2} \int_2^4 4t \, dt} \]

4. **Calculating the Integral**:
We calculate the integral \( \int_2^4 4t \, dt \):
\[ \int 4t \, dt = 2t^2 + C \]
Evaluating from 2 to 4:
\[ \left[ 2t^2 \right]_2^4 = 2(4^2) - 2(2^2) = 2(16) - 2(4) = 32 - 8 = 24 \]

5. **Substituting Back into the RMS Formula**:
Now substitute the result of the integral back into the RMS formula:
\[ I_{\text{RMS}} = \sqrt{\frac{1}{2} \cdot 24} = \sqrt{12} = 2\sqrt{3} \]

6. **Final Answer**:
Therefore, the RMS value of the current between \( t = 2 \) seconds and \( t = 4 \) seconds is:
\[ I_{\text{RMS}} = 2\sqrt{3} \, \text{A} \]