An ac circuit, the current is given by `i = 4 sin (100pit + 30^(@))` ampere. The current becomes maximum first time (after t = 0) at t equal to –

Options

A. (1/200) sec
B. (1/300) sec
C. (1/50) sec
D. None of the above

Answer

To find the time at which the current becomes maximum for the given AC circuit, we start with the equation of the current:

\[ i(t) = 4 \sin(100 \pi t + 30^\circ) \]

### Step 1: Identify the condition for maximum current
The current \( i(t) \) reaches its maximum value when the sine function equals 1. Thus, we need to solve for when:

\[ 100 \pi t + 30^\circ = 90^\circ \]

### Step 2: Convert degrees to radians
To solve the equation, we can convert degrees to radians. We know that:

\[ 90^\circ = \frac{\pi}{2} \text{ radians} \]

So we rewrite the equation:

\[ 100 \pi t + 30^\circ = \frac{\pi}{2} \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( t \):

\[ 100 \pi t = \frac{\pi}{2} - 30^\circ \]

### Step 4: Convert 30 degrees to radians
Next, we convert \( 30^\circ \) to radians:

\[ 30^\circ = \frac{\pi}{6} \text{ radians} \]

Now substituting this back into the equation:

\[ 100 \pi t = \frac{\pi}{2} - \frac{\pi}{6} \]

### Step 5: Find a common denominator and simplify
To subtract the fractions, we need a common denominator. The least common multiple of 2 and 6 is 6. Thus, we rewrite:

\[ \frac{\pi}{2} = \frac{3\pi}{6} \]

Now we can perform the subtraction:

\[ 100 \pi t = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]

### Step 6: Solve for \( t \)
Now we can solve for \( t \):

\[ 100 \pi t = \frac{\pi}{3} \]

Dividing both sides by \( 100 \pi \):

\[ t = \frac{1}{300} \text{ seconds} \]

### Final Answer
Thus, the time at which the current becomes maximum for the first time after \( t = 0 \) is:

\[ t = \frac{1}{300} \text{ seconds} \]

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