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In SI, the Poynting vector is defined to be the vector product E
cross H (where H equals B/musubzero in the nonmagnetic dielectric of
the capacitor). The Poynting vector has unit W/m^2 and so some
incorrectly believe that it always gives the em power transferred per
perpendicular area.
To quote Reitz and Milford (2nd ed), "It is tempting to interpret E
cross H itself as the energy flow per unit time per unit area. The
latter interpretation, however, leads to certain inconsistencies; the
only interpretation which survives careful scrutiny is that the
integral of E cross H over a CLOSED surface represents the rate at
which electromagnetic energy crosses the CLOSED surface." (my
emphasis).
I use the example of crossed static E and B fields to show the
fallacy of assuming the Poynting vector always gives the em power
transferred per perpendicular area.
Tom Sandin