  The curvature-corrected field emission current density, obtained by
linearizing at or below the Fermi energy, is investigated. Two special cases,
corresponding to the peak of the normal energy distribution and the mean normal
energy, are considered. It is found that the current density evaluated using
the mean normal energy results in errors in the net emission current below 3%
for apex radius of curvature, $R_a \geq 5$nm and for apex fields $E_a$ in the
range $3-10$ V/nm for an emitter having work-function $\phi = 4.5$eV. An
analytical expression for the net field emission current is also obtained for
locally parabolic tips using the generalized cosine law. The errors are found
to be below 6% for $R_a \geq 5$nm over an identical range of apex field
strengths. The benchmark current is obtained by numerically integrating the
current density over the emitter surface and the current density itself
computed by integrating over the energy states using the exact Gamow factor and
the Kemble form for the WKB transmission coefficient. The analytical expression
results in a remarkable speed-up in the computation of the net emission current
and is especially useful for large area field emitters having tens of thousands
of emission sites.
