In an $$AC$$ circuit, the $$rms$$  value of current, $${i_{rms}}$$  is related to the peak current, $${i_0}$$ by the relation

Solution :
Root mean square value of an alternating current is defined as the square root of the average of $${i^2},$$ during a complete cycle it may be taken by
$$\eqalign{ & \overline {{i^2}} = \frac{{\int_0^{\frac{{2\pi }}{\omega }} {{i^2}} dt}}{{\frac{{2\pi }}{\omega }}} \cr & = \frac{{\int_0^{\frac{{2\pi }}{\omega }} {i_0^2} {{\sin }^2}\omega t\,dt}}{{\frac{{2\pi }}{\omega }}} \cr & = \frac{{i_0^2\omega }}{{2\pi }}\int_0^{\frac{{2\pi }}{\omega }} {\frac{1}{2}} \left( {1 - \cos 2\omega t} \right)dt \cr & = \frac{{i_0^2\omega }}{{4\pi }}\left[ {t - \frac{{\sin 2\omega t}}{{2\omega }}} \right]_0^{\frac{{2\pi }}{\omega }} \cr & = \frac{{i_0^2\omega }}{{4\pi }}\left( {\frac{{2\pi }}{\omega }} \right) = \frac{{i_0^2}}{2} \cr & \therefore {i_{rms}} = \sqrt {\overline {{i^2}} } = \frac{{{i_0}}}{{\sqrt 2 }} \cr} $$