The time required for a 50 Hz sinusoidal alternating current to changed its value from zero to the r.m.s. value

Options

A. `1.5xx10^(-2)` S
B. `2.5xx10^(-3)` s
C. `10 ^(-1)` s
D. `10 ^(-6)` s

Answer

To find the time required for a 50 Hz sinusoidal alternating current to change its value from zero to the root mean square (r.m.s.) value, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between maximum current and r.m.s. value**:
The r.m.s. value (I_rms) of a sinusoidal current is given by the formula:
\[ I_{rms} = \frac{I_{max}}{\sqrt{2}} \]
where \(I_{max}\) is the maximum amplitude of the current.

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

3. **Find the time to reach the r.m.s. value**:
The current as a function of time for a sinusoidal wave can be expressed as:
\[ I(t) = I_{max} \sin(\omega t) \]
To find the time when the current reaches the r.m.s. value, we set:
\[ I_{rms} = I_{max} \sin(\omega t) \]
Substituting the expression for \(I_{rms}\):
\[ \frac{I_{max}}{\sqrt{2}} = I_{max} \sin(\omega t) \]
Dividing both sides by \(I_{max}\) (assuming \(I_{max} \neq 0\)):
\[ \frac{1}{\sqrt{2}} = \sin(\omega t) \]

4. **Solve for \(\omega t\)**:
The value of \(\sin(\theta) = \frac{1}{\sqrt{2}}\) corresponds to:
\[ \theta = \frac{\pi}{4} \, \text{radians} \]
Therefore, we have:
\[ \omega t = \frac{\pi}{4} \]

5. **Calculate time (t)**:
Now, we can solve for \(t\):
\[ t = \frac{\pi/4}{\omega} \]
Substituting \(\omega = 100\pi\):
\[ t = \frac{\pi/4}{100\pi} = \frac{1}{400} \, \text{seconds} \]
Simplifying this gives:
\[ t = 0.0025 \, \text{seconds} = 2.5 \times 10^{-3} \, \text{seconds} \]

### Final Answer:
The time required for a 50 Hz sinusoidal alternating current to change its value from zero to the r.m.s. value is:
\[ t = 2.5 \times 10^{-3} \, \text{seconds} \]