The time required for a 50Hz alternating current to increase from zero to `70.7%` of its peak value is-

Options

A. (a)2.5 ms
B. (b)10 ms
C. (c)20 ms
D. (d)14.14 ms

Answer

To solve the problem of finding the time required for a 50 Hz alternating current to increase from zero to 70.7% of its peak value, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the AC Current Equation**:
The equation for alternating current (AC) is given by:
\[ I(t) = I_0 \sin(\omega t) \]
where \(I_0\) is the peak current, and \(\omega\) is the angular frequency.

2. **Determine the Value of 70.7% of Peak Current**:
We need to find the time when the current reaches 70.7% of its peak value:
\[ I(t) = 0.707 I_0 \]
This is equivalent to:
\[ 0.707 I_0 = I_0 \sin(\omega t) \]
Dividing both sides by \(I_0\) (assuming \(I_0 \neq 0\)):
\[ 0.707 = \sin(\omega t) \]

3. **Find the Angle Corresponding to 0.707**:
The value \(0.707\) corresponds to an angle of \(45^\circ\) (or \(\frac{\pi}{4}\) radians):
\[ \omega t = \frac{\pi}{4} \]

4. **Calculate Angular Frequency (\(\omega\))**:
The angular frequency is related to the frequency \(f\) by the formula:
\[ \omega = 2\pi f \]
Given that the frequency \(f = 50 \, \text{Hz}\):
\[ \omega = 2\pi \times 50 = 100\pi \, \text{rad/s} \]

5. **Substitute \(\omega\) into the Equation**:
Now we can substitute \(\omega\) into the equation:
\[ 100\pi t = \frac{\pi}{4} \]

6. **Solve for Time \(t\)**:
Dividing both sides by \(100\pi\):
\[ t = \frac{\frac{\pi}{4}}{100\pi} = \frac{1}{400} \, \text{s} \]

7. **Convert Time to Milliseconds**:
Converting seconds to milliseconds:
\[ t = \frac{1}{400} \, \text{s} = 0.0025 \, \text{s} = 2.5 \, \text{ms} \]

8. **Final Answer**:
The time required for the alternating current to increase from zero to 70.7% of its peak value is:
\[ \boxed{2.5 \, \text{ms}} \]