In the Ac circuit, the current is expressed as / = 100 sin `200pit.` In this circuit the current rises from zero to peak value in time

Options

A. `(1)/(400)s`
B. `(1)/(300)s`
C. `(1)/(100)s`
D. `(1)/(200)s`

Answer

To solve the problem of how long it takes for the current in the AC circuit to rise from zero to its peak value, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Current Expression**:
The current in the AC circuit is given by the equation:
\[ I = 100 \sin(200 \pi t) \]
Here, \(100\) is the peak value of the current (denoted as \(I_0\)), and \(200 \pi\) is the angular frequency (\(\omega\)).

2. **Understand the Peak Value Condition**:
The peak value of the current occurs when the sine function equals 1. Therefore, we need to find when:
\[ \sin(200 \pi t) = 1 \]

3. **Find the Angle Corresponding to the Peak Value**:
The sine function reaches its maximum value of 1 at:
\[ 200 \pi t = \frac{\pi}{2} \]

4. **Solve for Time \(t\)**:
To find the time \(t\), we can rearrange the equation:
\[ t = \frac{\frac{\pi}{2}}{200 \pi} \]
Simplifying this gives:
\[ t = \frac{1}{400} \text{ seconds} \]

5. **Conclusion**:
The time taken for the current to rise from zero to its peak value is:
\[ t = \frac{1}{400} \text{ seconds} \]

### Final Answer:
The time taken for the current to rise from zero to its peak value is \( \frac{1}{400} \) seconds.

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