Alternating current in circuit is given by `I = I_(0) sin 2 pi nt`. Then the time by the current to rise from zero to r.m.s value is equal to

Options

A. `1//2 n`
B. `1//n`
C. `1//4 n`
D. `1//8n`

Explanation

To find the time taken for the alternating current to rise from zero to its root mean square (r.m.s) value, we can follow these steps:

### Step 1: Understand the given equation
The alternating current is given by:
\[ I(t) = I_0 \sin(2 \pi n t) \]
where \( I_0 \) is the peak current, \( n \) is the frequency, and \( t \) is time.

### Step 2: Determine the r.m.s value
The r.m.s (root mean square) value of the current is given by:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} \]

### Step 3: Set the equation for r.m.s value
We want to find the time \( t \) when the current \( I(t) \) equals the r.m.s value:
\[ I(t) = I_{rms} \]
Substituting the expression for \( I(t) \) and \( I_{rms} \):
\[ I_0 \sin(2 \pi n t) = \frac{I_0}{\sqrt{2}} \]

### Step 4: Simplify the equation
We can cancel \( I_0 \) from both sides (assuming \( I_0 \neq 0 \)):
\[ \sin(2 \pi n t) = \frac{1}{\sqrt{2}} \]

### Step 5: Solve for the angle
The sine function equals \( \frac{1}{\sqrt{2}} \) at:
\[ 2 \pi n t = \frac{\pi}{4} \]
This is because \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).

### Step 6: Solve for time \( t \)
Now, we can solve for \( t \):
\[ 2 \pi n t = \frac{\pi}{4} \]
Dividing both sides by \( 2 \pi n \):
\[ t = \frac{\frac{\pi}{4}}{2 \pi n} = \frac{1}{8n} \]

### Final Answer
Thus, the time taken for the current to rise from zero to its r.m.s value is:
\[ t = \frac{1}{8n} \text{ seconds} \]

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