[UPSC IFoS 2018] Electrostatics and Magnetostatics (Q91, 5+5+5=15M)

(i) The equation for an alternating current is I = 42\cdot 42 \sin (314t). Find the following : Maximum value of current, Frequency, RMS value and Average value

(ii) A condenser of capacity 1~\mu\text{F} is first charged and then discharged through a resistance of 1\text{ M}\Omega. Calculate the time in which the charge on the condenser will fall to 50\% of its initial value.

(iii) Consider the displacement vector \vec{D}, given by \vec{D} = (10xyz^2 + 4x)\hat{i} + (5x^2 z^2)\hat{j} + (10x^2 yz)\hat{k}\text{ nC/m}^2 Find the total charge enclosed in a cube of volume 10^{-9}\text{ m}^3 located at the point (1, 2, 3).