A parallel plate capacitor is charged by connecting it to a battery through a resistor. If I is the current in the circuit, then in the gap between the plates:

To understand the nature of the current in the gap between the plates of a charging capacitor, we need to apply Ampere-Maxwell's Law.

During the charging of a parallel plate capacitor, a conduction current $I$ flows through the connecting wires, which leads to the accumulation of charge on the capacitor plates. According to the continuity of current, the current must flow in a continuous loop.

Although there is no actual flow of charge carriers (electrons) across the vacuum or dielectric gap between the capacitor plates, a changing electric field is established in the gap as the plates charge up. A changing electric field produces a magnetic field in the same way as a conduction current does. To maintain the consistency of Ampere's Law in the region between the plates, James Clerk Maxwell introduced the concept of displacement current ($I_d$).

The displacement current in the gap is mathematically defined as:
$I_d={\epsilon }_0\frac{d{\Phi }_E}{dt}$
where ${\epsilon }_0$ is the permittivity of free space and ${\Phi }_E$ is the electric flux through the region between the plates.

The electric field $E$ between the plates of area $A$ carrying a charge $Q$ is given by:
$E=\frac{Q}{{\epsilon }_{0}A}$
The electric flux ${\Phi }_E$ is:
${\Phi }_E=E\cdot A=\frac{Q}{{\epsilon }_0}$
Substituting this flux into the expression for displacement current:
$I_d={\epsilon }_0\frac{d}{dt}\left(\frac{Q}{{\epsilon }_0}\right)=\frac{dQ}{dt}$
Since the rate of change of charge on the plates $\frac{dQ}{dt}$ is equal to the conduction current $I$ flowing in the circuit, we have:
$I_d=I$

To maintain continuity, this displacement current flows in the same direction as the conduction current $I$, completing the circuit loop across the gap.

Therefore, the displacement current of magnitude equal to $I$ flows in the same direction as $I$.