The electric current in an `AC` circuit is given by `I=I_(0)sin omegat`.What is the time taken by the current to change from its maximum value to the `rms` value?

Explanation

To find the time taken by the current in an AC circuit to change from its maximum value to its RMS value, we can follow these steps:

### Step 1: Understand the given current equation
The electric current in the AC circuit is given by:
\[ I(t) = I_0 \sin(\omega t) \]
where \( I_0 \) is the maximum current, and \( \omega \) is the angular frequency.

### Step 2: Determine the maximum current value
The maximum current \( I_{\text{max}} \) occurs when \( \sin(\omega t) = 1 \):
\[ I_{\text{max}} = I_0 \]

### Step 3: Determine the RMS value
The RMS (Root Mean Square) value of the current is given by:
\[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \]

### Step 4: Set up the equation for the current to equal the RMS value
We need to find the time \( t_1 \) when the current is at its maximum value and the time \( t_2 \) when the current equals the RMS value:
\[ I(t_2) = I_{\text{rms}} \]
Substituting the RMS value:
\[ I_0 \sin(\omega t_2) = \frac{I_0}{\sqrt{2}} \]

### Step 5: Simplify the equation
Dividing both sides by \( I_0 \) (assuming \( I_0 \neq 0 \)):
\[ \sin(\omega t_2) = \frac{1}{\sqrt{2}} \]

### Step 6: Solve for \( t_2 \)
The sine function equals \( \frac{1}{\sqrt{2}} \) at angles:
\[ \omega t_2 = \frac{\pi}{4} \quad \text{and} \quad \omega t_2 = \frac{3\pi}{4} \]
Thus, we can express \( t_2 \) as:
\[ t_2 = \frac{\pi}{4\omega} \quad \text{or} \quad t_2 = \frac{3\pi}{4\omega} \]

### Step 7: Find the time \( t_1 \) when the current is maximum
The maximum current occurs at:
\[ \omega t_1 = \frac{\pi}{2} \]
So,
\[ t_1 = \frac{\pi}{2\omega} \]

### Step 8: Calculate the time difference \( \Delta t \)
The time taken for the current to change from its maximum value to the RMS value is:
\[ \Delta t = t_2 - t_1 \]
Using the first \( t_2 \):
\[ \Delta t = \frac{\pi}{4\omega} - \frac{\pi}{2\omega} \]
Finding a common denominator:
\[ \Delta t = \frac{\pi}{4\omega} - \frac{2\pi}{4\omega} = -\frac{\pi}{4\omega} \]
This indicates that we should consider the time from maximum to the first occurrence of RMS value:
\[ \Delta t = \frac{\pi}{4\omega} \]

### Final Answer
The time taken by the current to change from its maximum value to its RMS value is:
\[ \Delta t = \frac{T}{8} \]
where \( T = \frac{2\pi}{\omega} \) is the time period of the AC current.