$6$a: A circular parallel plate capacitor $($see diagram on next page$)$ has a plate radius $\backslash ($ R $\backslash )$ and a plate separation $\backslash ($ d $\backslash ).$ Show that the Poynting vector points radially into the capacitor. Note that $\backslash (\backslash$vec$\{$B$\}\backslash )$ must be the induced magnetic field due to the displacement current.

b: The equation below describes an alternative description of energy conservation for this charging capacitor. The left hand side is the power entering the capacitor so the integral there is over the area describing the edge of the capacitor. The right hand side is the rate at which energy is stored in the capacitor. Note that $\backslash ($ u $\backslash )$ is the energy density and $\backslash ($ V $\backslash )$ is the volume of the capacitor.

$\backslash [$

$\backslash$iint $\backslash$vec$\{$S$\}\backslash$cdot $\backslash$overrightarrow$\{$d A$\}=\backslash$frac$\{$d$\}\{$d t$\}($u V$)$

$\backslash ]$

Calculate the left and right hand sides of this equation separately and verify that they match.

Ignore the fringing of the electric field lines. From this viewpoint, the energy "enters" the capacitor from the space around the wires and the plates rather than through the wires.

The Poynting vector often gives a strange interpretation of the transport of energy but it works and it is a very useful description.