An $$AC$$ source rated 220 V, 50 Hz is connected to a resistor. The time taken by the current to change from its maximum to the rms value is:

For an AC source with frequency $$f = 50$$ Hz, the angular frequency is $$\omega = 2\pi f = 100\pi$$ rad/s. The current varies as $$i = I_0 \sin(\omega t)$$.

The current is at its maximum value $$I_0$$ when $$\sin(\omega t_1) = 1$$, i.e., $$\omega t_1 = \frac{\pi}{2}$$. The rms value is $$I_{\text{rms}} = \frac{I_0}{\sqrt{2}}$$, which occurs when $$\sin(\omega t_2) = \frac{1}{\sqrt{2}}$$, i.e., $$\omega t_2 = \frac{3\pi}{4}$$ (taking the next instant after the maximum where the sine equals $$\frac{1}{\sqrt{2}}$$).

The time difference is $$\Delta t = t_2 - t_1 = \frac{1}{\omega}\left(\frac{3\pi}{4} - \frac{\pi}{2}\right) = \frac{1}{100\pi} \times \frac{\pi}{4} = \frac{1}{400}$$ s $$= 2.5 \times 10^{-3}$$ s $$= 2.5$$ ms.