 Poynting flux in a centrally discharged parallel-plate capacitor

For circular plates of radius $a$, separation $d\ll a$, and a thin axial resistor $R$, the quasistatic capacitance is $C={\epsilon }_0\pi a^2/d$. During discharge, $I=V/R=-C\dot{V}$. A circular Ampere-Maxwell law contour between the plates encloses wire current $I$ and opposing displacement current $-I{r}^2/a^2$, giving $H=I(1-r^2/a^2)/(2\pi r)$. The electric field has magnitude $V/d$, so the Poynting vector points radially inward with magnitude $V^2(1-r^2/a^2)/(2\pi rdR)$. Its flux through the cylindrical side is the displayed inward power. The field inside that cylinder releases $V^2{r}^2/(R{a}^2)$ per unit time; adding the incoming flux gives the resistor's full Joule heating rate $V^2/R$. End fringing, wire radius and magnetic-energy corrections are neglected.