Question

Why does current in a steady state not flow in a capacitor connected across a battery but a momentary current does flow during charging or discharging of the capacitor?

Answer 1

When there is change in the electric flux there will be a displacement current and when flux is fixed the displacement as well as conduction current will be zero.
$I_D={ϵ}_0\frac{d{\varphi }_E}{dt}$

During charging and discharging, due to flux variation, some displacement current flows through the capacitor.