Magnetic field between the plates of a charging capacitor

The underlying principle is that a time-varying electric field induces a magnetic field. This is stated in Maxwell's equations as

$$operatorname{curl}textbf{B} = frac{1}{c^2}frac{partial textbf{E}}{partial t}.$$

Applying Stokes's theorem to a disk of radius $r$ between the plates (concentric with and parallel to the plates), we get that the line integral of the magnetic field around the edge of this disk, $2pi r B$, relates to the rate of change of the electric flux through this disk. If you neglect fringing fields and take the electric field to be uniform between the plates, then the rate of change of this electric flux is proportional to the area $pi r^2$ and also to the rate of change of the charge, $dq/dt$. Recognizing $dq/dt$ as the current $I$, and sorting out the powers of $r$, the magnetic field is proportional to $Ir$, as claimed.

As an example, suppose that the capacitor is charging up with some RC time constant. Then then $I$ will approach zero as $trightarrowinfty$, and the magnetic field will go to zero, as it should when we reach electrostatic equilibrium.