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 find the time taken by the alternating current to reach its root mean square (RMS) value from the given equation of the current. Let's break down the solution step by step.

### Step 1: Identify the given equation
The equation of the alternating current is given as:
\[ I(t) = 10\sqrt{2} \sin(100\pi t + \frac{\pi}{6}) \]

### Step 2: Determine the peak current (\(I_0\))
From the equation, we can identify the peak current:
\[ I_0 = 10\sqrt{2} \]

### Step 3: Calculate the RMS current (\(I_{rms}\))
The RMS value of the current is given by the formula:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} \]
Substituting the value of \(I_0\):
\[ I_{rms} = \frac{10\sqrt{2}}{\sqrt{2}} = 10 \, \text{A} \]

### Step 4: Set up the equation to find the time \(t\)
We want to find the time \(t\) when the current reaches its RMS value. Thus, we set:
\[ I(t) = I_{rms} \]
This gives us:
\[ 10\sqrt{2} \sin(100\pi t + \frac{\pi}{6}) = 10 \]

### Step 5: Simplify the equation
Dividing both sides by 10:
\[ \sqrt{2} \sin(100\pi t + \frac{\pi}{6}) = 1 \]
Now, dividing both sides by \(\sqrt{2}\):
\[ \sin(100\pi t + \frac{\pi}{6}) = \frac{1}{\sqrt{2}} \]

### Step 6: Solve for the angle
The sine function equals \(\frac{1}{\sqrt{2}}\) at:
\[ \sin(\frac{\pi}{4}) \]
Thus, we can set up the equation:
\[ 100\pi t + \frac{\pi}{6} = \frac{\pi}{4} \]

### Step 7: Isolate \(t\)
Rearranging the equation:
\[ 100\pi t = \frac{\pi}{4} - \frac{\pi}{6} \]
To combine the fractions, we find a common denominator (which is 12):
\[ \frac{\pi}{4} = \frac{3\pi}{12} \]
\[ \frac{\pi}{6} = \frac{2\pi}{12} \]
Thus:
\[ 100\pi t = \frac{3\pi}{12} - \frac{2\pi}{12} = \frac{\pi}{12} \]

### Step 8: Solve for \(t\)
Now, divide both sides by \(100\pi\):
\[ t = \frac{\pi}{12} \cdot \frac{1}{100\pi} = \frac{1}{1200} \]

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

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