An AC voltage rating 240V, 50Hz. Find the time to change current from max. value to rms value.

Options

A. `2.55 ms`
B. `2.5 ms`
C. `0.25 ms`
D. `25 ms`

Answer

To solve the problem of finding the time it takes for the current in an AC circuit to change from its maximum value to its RMS value, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- AC Voltage (V) = 240V
- Frequency (f) = 50Hz

2. **Determine the Maximum Current (I₀):**
- The maximum current (I₀) is related to the maximum voltage (V₀) and the impedance (Z) of the circuit. However, since we are not given the impedance, we can focus on the relationship between I₀ and the RMS value.
- The RMS current (I_rms) is given by the formula:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} \]

3. **Use the AC Current Equation:**
- The current in an AC circuit can be expressed as:
\[ I(t) = I_0 \sin(\omega t) \]
- Where \(\omega\) (angular frequency) is given by:
\[ \omega = 2\pi f \]

4. **Set Up the Equation for RMS Current:**
- To find the time when the current reaches the RMS value, we set:
\[ I(t) = I_{rms} = \frac{I_0}{\sqrt{2}} \]
- Substituting this into the current equation gives:
\[ I_0 \sin(\omega t) = \frac{I_0}{\sqrt{2}} \]

5. **Simplify the Equation:**
- Dividing both sides by \(I_0\) (assuming \(I_0 \neq 0\)):
\[ \sin(\omega t) = \frac{1}{\sqrt{2}} \]

6. **Find the Angle Corresponding to RMS Value:**
- The angle whose sine is \(\frac{1}{\sqrt{2}}\) is:
\[ \omega t = \frac{\pi}{4} \]

7. **Calculate the Time (t):**
- Now, substituting \(\omega = 2\pi f\):
\[ \frac{\pi}{4} = 2\pi f t \]
- Rearranging gives:
\[ t = \frac{\pi}{4 \cdot 2\pi f} = \frac{1}{8f} \]

8. **Substituting the Frequency:**
- Substituting \(f = 50 \, \text{Hz}\):
\[ t = \frac{1}{8 \cdot 50} = \frac{1}{400} \, \text{s} \]

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

### Final Answer:
The time taken for the current to change from its maximum value to its RMS value is **2.5 ms**.