= Poynting flux in a centrally discharged parallel-plate capacitor
{c}
{title2=$P_{\rm in}(r)=V^2(1-r^2/a^2)/R$}
For circular plates of radius $a$, separation $d\ll a$, and a thin axial resistor $R$, the quasistatic <capacitance> is $C=\varepsilon_0\pi a^2/d$. 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> $-Ir^2/a^2$, 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 $V^2r^2/(Ra^2)$ 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.
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