A resistance of `40 Omega` is connected to a source of alternating current rated 220 V, 50 Hz. Find the time taken by the current to change from its maximum value to the rms value :

Options

A. 2.5 ms
B. 1.25 ms
C. 2.5 s
D. 0.25 s

Answer

To solve the problem of finding the time taken by the current to change from its maximum value to the RMS value in a circuit with a resistance of 40 ohms connected to an AC source of 220 V and 50 Hz, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the RMS Voltage and Frequency**:
- The given voltage (V) is 220 V (RMS).
- The frequency (f) is 50 Hz.

2. **Calculate the Maximum Voltage (V0)**:
- The relationship between RMS voltage (V_RMS) and maximum voltage (V0) is given by:
\[ V_0 = V_{RMS} \times \sqrt{2} \]
- Substituting the given RMS voltage:
\[ V_0 = 220 \times \sqrt{2} \approx 220 \times 1.414 \approx 311.8 \text{ V} \]

3. **Calculate the Maximum Current (I0)**:
- Using Ohm's law, the maximum current (I0) can be calculated as:
\[ I_0 = \frac{V_0}{R} \]
- Where R is the resistance (40 ohms):
\[ I_0 = \frac{311.8}{40} \approx 7.795 \text{ A} \]

4. **Determine the RMS Current (IRMS)**:
- The RMS current (I_RMS) is given by:
\[ I_{RMS} = \frac{I_0}{\sqrt{2}} \approx \frac{7.795}{\sqrt{2}} \approx \frac{7.795}{1.414} \approx 5.5 \text{ A} \]

5. **Relate Maximum Current to Time**:
- The current in an AC circuit varies with time as:
\[ I(t) = I_0 \sin(\omega t) \]
- Where \(\omega = 2\pi f\) and \(f\) is the frequency (50 Hz):
\[ \omega = 2\pi \times 50 \approx 314.16 \text{ rad/s} \]

6. **Set Up the Equation for RMS Current**:
- We want to find the time \(t\) when the current is equal to the RMS current:
\[ I_{RMS} = I_0 \sin(\omega t) \]
- Substituting the values:
\[ 5.5 = 7.795 \sin(314.16 t) \]

7. **Solve for sin(ωt)**:
- Rearranging gives:
\[ \sin(314.16 t) = \frac{5.5}{7.795} \approx 0.705 \]

8. **Find the Angle**:
- Taking the inverse sine:
\[ \omega t = \sin^{-1}(0.705) \approx 0.785 \text{ rad} \quad (\text{which is } \frac{\pi}{4}) \]

9. **Calculate Time t**:
- Now, substituting back to find \(t\):
\[ t = \frac{\omega t}{\omega} = \frac{\frac{\pi}{4}}{314.16} \approx \frac{0.785}{314.16} \approx 0.0025 \text{ s} \]
- Converting to milliseconds:
\[ t \approx 2.5 \text{ ms} \]

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

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