The electric current in a circuit is given by
i=3t
Here, t is in second and I in ampere. The rms current for the period to=0 to t=1 s is

Options

A. 3A
B. 2A
C. `sqrt3`A
D. `3sqrt3A`

Answer

To find the RMS (Root Mean Square) current for the given current function \( i(t) = 3t \) over the time interval from \( t = 0 \) to \( t = 1 \) second, we can follow these steps:

### Step 1: Understand the formula for RMS current
The RMS current is given by the formula:
\[ I_{rms} = \sqrt{\frac{1}{T} \int_0^T i^2(t) \, dt} \]
where \( T \) is the period of the current and \( i(t) \) is the instantaneous current.

### Step 2: Identify the function and limits
In this case, the function is \( i(t) = 3t \) and the limits for integration are from \( t = 0 \) to \( t = 1 \) second.

### Step 3: Calculate \( i^2(t) \)
First, we need to square the current function:
\[ i^2(t) = (3t)^2 = 9t^2 \]

### Step 4: Set up the integral for \( i^2(t) \)
Now we set up the integral:
\[ \int_0^1 i^2(t) \, dt = \int_0^1 9t^2 \, dt \]

### Step 5: Evaluate the integral
To evaluate the integral:
\[ \int 9t^2 \, dt = 9 \cdot \frac{t^3}{3} = 3t^3 \]
Now, we evaluate it from 0 to 1:
\[ \left[ 3t^3 \right]_0^1 = 3(1^3) - 3(0^3) = 3 - 0 = 3 \]

### Step 6: Calculate the RMS current
Now, substitute the value of the integral back into the RMS formula:
\[ I_{rms} = \sqrt{\frac{1}{1} \cdot 3} = \sqrt{3} \]

### Final Answer
Thus, the RMS current for the period from \( t = 0 \) to \( t = 1 \) second is:
\[ I_{rms} = \sqrt{3} \text{ A} \]