Axisymmetric high-frequency capacitor field ($E_z(r)\propto J_0(\omega r/c)$)

If a thin capacitor interior is modeled with axially uniform axisymmetric fields, Maxwell's equations give $r^{-1}(r{E}_z^{\prime }{)}^{\prime }+(\omega /c{)}^2{E}_z=0$. The solution regular at the axis is proportional to $J_0(\omega r/c)$. Normalizing to a central sinusoidal voltage gives $E_z=-(V_0/L)J_0(\omega r/c)\sin \omega t$ in this ideal interior model. It can change sign across the radius when propagation matters. Feed, edge and radiation boundary conditions are needed for a full driven-device solution.