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 find the time it takes for the current in the AC circuit to rise from zero to its peak value, we can follow these steps:

### Step 1: Understand the given equation
The current in the AC circuit is given by:
\[ I(t) = 100 \sin(200 \pi t) \]
Here, \( I_0 = 100 \) is the peak value of the current, and \( \omega = 200 \pi \) is the angular frequency.

### Step 2: Determine when the current reaches its peak value
The current reaches its peak value when the sine function equals 1. This occurs at:
\[ \sin(\omega t) = 1 \]
The sine function equals 1 at:
\[ \omega t = \frac{\pi}{2} \]

### Step 3: Substitute the value of \( \omega \)
We know that \( \omega = 200 \pi \). Substituting this into the equation gives:
\[ 200 \pi t = \frac{\pi}{2} \]

### Step 4: Solve for \( t \)
To find \( t \), we can divide both sides of the equation by \( 200 \pi \):
\[ t = \frac{\frac{\pi}{2}}{200 \pi} \]
This simplifies to:
\[ t = \frac{1}{400} \text{ seconds} \]

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

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