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Choose the plate polarity so that the gap electric field points along $+z$ when $V>0$. Away from the plate edges, the quasistatic electromagnetic approximation gives $E_z=V_0\sin \omega t/d$. A circular Ampere-Maxwell law contour of radius $r<a$ then givesIt is displacement current, rather than a conduction current across the air, that supplies the enclosed current term. The direction follows the right-hand rule for increasing positive electric flux; it reverses every half cycle. In the ideal extension that sets the gap electric field to zero outside the plate footprint, $B_{\varphi }={\mu }_0{ϵ}_0\omega V_0{a}^2\cos \omega t/(2dr)$ for $r>a$. That extension is an edge approximation, not an exact exterior solution for finite plates. Neglect fringing field of a capacitor and retardation, requiring $d\ll a$ and $\omega a/c\ll 1$.

A quasistatic electromagnetic approximation neglects propagation delay over a vacuum device of characteristic size $L$ when $\omega L/c\ll 1$. It does not automatically discard every time derivative: an Ampere-Maxwell law calculation in a capacitor gap must retain displacement current. In a medium use the relevant propagation and response scales. The dominant approximately static electric field can be used to calculate a smaller induced magnetic field, provided the corrections to the dominant field remain small. Near a resonance or for a large distributed circuit, the size criterion alone may not justify an instantaneous lumped model.