1. Given data

• Resistance: $R = 40\,\Omega$
• AC source: $220\,\text{V}, 50\,\text{Hz}$
• Frequency: $f = 50\,\text{Hz}$

We need the time taken by current to change from maximum value to rms value.

2. Expression for alternating current

For a purely resistive circuit, $i = I_0 \sin \omega t$ where:

• $I_0$ = maximum current
• $\omega = 2\pi f$

The rms current is related to maximum current by $I_{\text{rms}} = \frac{I_0}{\sqrt{2}}$

3. Condition for current changing from maximum to rms value

Maximum current occurs when $i = I_0$ This corresponds to phase $\omega t = \frac{\pi}{2}$

Now we want the next instant when current becomes $i = I_{\text{rms}} = \frac{I_0}{\sqrt{2}}$

So, $I_0 \sin \omega t = \frac{I_0}{\sqrt{2}}$ $\sin \omega t = \frac{1}{\sqrt{2}}$

After the maximum point, this occurs at $\omega t = \frac{3\pi}{4}$

Hence the phase change required is $\Delta \phi = \frac{3\pi}{4} - \frac{\pi}{2} = \frac{\pi}{4}$

4. Convert phase difference to time

We know $\omega = 2\pi f = 2\pi \times 50 = 100\pi\,\text{rad/s}$

Thus, $\Delta t = \frac{\Delta \phi}{\omega} = \frac{\pi/4}{100\pi} = \frac{1}{400}\,\text{s}$

$\Delta t = 0.0025\,\text{s} = 2.5\,\text{ms}$

5. Check options

• A: $2.5\,\text{ms}$ ✅
• B: $1.25\,\text{ms}$ ❌
• C: $2.5\,\text{s}$ ❌
• D: $0.25\,\text{s}$ ❌

6. Final answer

The time taken is $\boxed{2.5\,\text{ms}}$ So, the correct option is A.