How is that circuit wont be broken even there is an air gap in between the parallel plates?

Ans) In AC circuit capacitor offers resistance to current as aresistor would do in DC. But for DC there exists a current in between the plates called displacement current (it exists everywhere in fact)

I_d = ε₀x dΦ_e/dt

Modified amperes law = integral of B.dl = mu0 ( I + I_d)

there exists a magnetic field between the plates responsible for transfer of charges from one plate to another and hence I= 0

but I_d is not 0. At every I_d = I