Q.Find the time required for a 50 Hz alternating current to change its value from zero to the rms value.

[!TLDR] $i=i_0\sin\omega t$ reaches $i_{rms}=i_0/\sqrt2$ when $\sin\omega t=1/\sqrt2$, i.e. $\omega t=\pi/4$; $t=\pi/(4\times2\pi\times50 …

Let the current be $i=i_0\sin\omega t$, starting from zero. It reaches the rms value $i_{rms}=i_0/\sqrt2$ when $i_0\sin\omega t=\dfrac{i_0}{\sqrt2}$, i.e. $\sin\omega t=\dfrac{1}{\sqrt2}$, which first occurs at $\omega t=\dfrac{\pi}{4}$ rad ($45^\circ$).\n\nWith $f=50$ Hz, $\omega=2\pi f=2\pi(50)=314.16$ rad/s (or $100\pi$), so $t=\dfr …

Set $i_0\sin\omega t$ equal to $i_0/\sqrt2$, solve for the phase $\omega t=\pi/4$, and …

Assuming the rms value is reached at a quarter of the period (a common but wrong intuition carried over from average-value reasoning) instead of correctly solving $\sin\omega …