  The electric field at the surface of a curved emitter is necessary to
calculate the field emission current. For smooth parabolic emitting tips where
space charge is negligible, variation of the electric field at the surface is
known to follow the generalized cosine law. Here we investigate the validity of
the cosine law in the regime where space charge due to emitted electrons is
important. Particle-in-Cell (PIC) simulations with an emission algorithm based
on the cosine law is employed for this study. It is shown that if $E_P$ and
$E_L$ be the field at the apex of tip with and without space charge
respectively, then for $\vartheta=E_P/E_L \geq 0.9$, the average relative
deviation of the electric field from the cosine law is less than $3\%$ over the
endcap. Thus, an emission scheme based on cosine law may be used in PIC
simulations of field emission of electrons from curved emitter tips in the weak
space charge regime. The relation between $\vartheta$ and normalized current
$\zeta$ for curved emitters in this regime is also investigated. A linear
relation, $\vartheta=1 - \delta \zeta$ (where $\delta$ is a constant), similar
to that obtained theoretically for flat emitting surfaces is observed but the
value of $\delta$ indicates that the extension of the theory for curved
emitters may require incorporation of the field enhancement factor.
