This relates to Gauss law of electrostatics.
We have the Gauss Law as : (we take it without proof) Proof of this requires knowledge of Direc Delta function and Divergence theorem.
where q = charge enclosed in a Closed Gaussian surface
E is perpendicular to the plates and is uniform inside the gap between the plates. This is because the gap between the plates d is << A, area of the two plates. We take a Gaussian surface with area of cross section same as A, and enclosing one of the plates of the capacitor. E and dS are both parallel and hence,
Ф = E . A = Q/ε₀
Hence, I = current = dQ/dt = d(ε₀ Φ)/ dt = ε₀ dΦ/dt
=== Additional information related to Capacitor.
Capacitance is defined as capacity to hold electric charge:
C = Q / V
V = potential difference across plates, Q = Charge on each plate
Since the distance between plates d << A, area of plates, the
electric field E between the plates is uniform and E = V / d
Hence Q = ε₀ E A
C = Q / V = ε ₀ E A / (E d) = ε₀ A / d