  The curvature of field emitter tips leads to an altered tunneling potential
that assumes significance when the radius of curvature is small. We provide
here an analytical curvature-corrected formula for the field emission current
from smooth vertically aligned emitter tips and test its applicability across a
range of apex radius, $R_a$, and local electric field, $E_a$. It is found to
give excellent results for $R_a > 10$nm with errors generally less than $10\%$.
Surprisingly, for the uncorrected potential, we find the errors to be high even
at $R_a = 100$nm ($ > 35\%$ at $E_a = 3$V/nm) and conclude that curvature
correction is essential for apex radius less than a micron.
