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 to change from its maximum value to its RMS value in an AC circuit with a voltage rating of 240V and a frequency of 50Hz, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Maximum Current (I0)**:
The maximum current (I0) can be derived from the maximum voltage (V0) and the impedance (Z) of the circuit. However, since the problem does not provide impedance, we will focus on the relationship between maximum current and RMS current.

2. **Understand RMS Current**:
The RMS (Root Mean Square) value of the current is given by:
\[ I_{\text{RMS}} = \frac{I_0}{\sqrt{2}} \]
This means that the RMS value is approximately 0.707 times the maximum value.

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

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

5. **Set Up the Equation for RMS Current**:
To find the time when the current reaches the RMS value, set the current equation equal to the RMS value:
\[ I_0 \sin(\omega t) = \frac{I_0}{\sqrt{2}} \]

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

7. **Find the Angle Corresponding to RMS Value**:
The sine of \(\frac{\pi}{4}\) is \(\frac{1}{\sqrt{2}}\), so:
\[ \omega t = \frac{\pi}{4} \]

8. **Solve for Time (t)**:
Rearranging gives:
\[ t = \frac{\pi}{4\omega} \]
Substituting \(\omega = 100\pi\):
\[ t = \frac{\pi}{4 \times 100\pi} = \frac{1}{400} \, \text{s} \]

9. **Convert to Milliseconds**:
\[ t = 0.0025 \, \text{s} = 2.5 \, \text{ms} \]

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

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