Equation of alternating current is given by `l = 10sqrt2sin(100pil+pi/6)`. The time taken by current to reach the root mean square value from t = 0 is t then value of t is

Options

A. `(1)/(1200)s`
B. `(1)/(250)s`
C. `(1)/(200)s`
D. `(1)/(800)s`

Answer

To solve the problem, we need to determine the time taken by the alternating current to reach its root mean square (RMS) value from \( t = 0 \). The given equation for the alternating current is:

\[ I(t) = 10\sqrt{2} \sin(100\pi t + \frac{\pi}{6}) \]

### Step-by-Step Solution

1. **Identify the Maximum Current (\(I_0\))**:
The maximum current \(I_0\) is given as \(10\sqrt{2}\) A.

2. **Calculate the RMS Value (\(I_{rms}\))**:
The RMS value of the current is calculated using the formula:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} = \frac{10\sqrt{2}}{\sqrt{2}} = 10 \text{ A} \]

3. **Set the Current Equal to the RMS Value**:
We need to find the time \(t\) when the current \(I(t)\) equals the RMS value:
\[ 10 = 10\sqrt{2} \sin(100\pi t + \frac{\pi}{6}) \]

4. **Simplify the Equation**:
Dividing both sides by \(10\sqrt{2}\):
\[ \frac{10}{10\sqrt{2}} = \sin(100\pi t + \frac{\pi}{6}) \]
This simplifies to:
\[ \frac{1}{\sqrt{2}} = \sin(100\pi t + \frac{\pi}{6}) \]

5. **Find the Angle Corresponding to \(\frac{1}{\sqrt{2}}\)**:
The sine function equals \(\frac{1}{\sqrt{2}}\) at:
\[ \sin\left(\frac{\pi}{4}\right) \]
Therefore, we can set up the equation:
\[ 100\pi t + \frac{\pi}{6} = \frac{\pi}{4} \]

6. **Solve for \(t\)**:
Rearranging the equation gives:
\[ 100\pi t = \frac{\pi}{4} - \frac{\pi}{6} \]
To combine the fractions, find a common denominator (which is 12):
\[ \frac{\pi}{4} = \frac{3\pi}{12}, \quad \frac{\pi}{6} = \frac{2\pi}{12} \]
Thus:
\[ 100\pi t = \frac{3\pi}{12} - \frac{2\pi}{12} = \frac{1\pi}{12} \]

7. **Isolate \(t\)**:
Dividing both sides by \(100\pi\):
\[ t = \frac{\frac{\pi}{12}}{100\pi} = \frac{1}{1200} \text{ seconds} \]

### Final Answer
The time taken by the current to reach its root mean square value from \(t = 0\) is:
\[ t = \frac{1}{1200} \text{ seconds} \]