  Field emission tips with apex radius of curvature below 100nm are not
adequately described by the standard theoretical models based on the
Fowler-Nordheim and Murphy-Good formalisms. This is due to the breakdown of the
`constant electric field' assumption within the tunneling region leading to
substantial errors in current predictions. A uniformly applicable
curvature-corrected field emission theory requires that the tunneling potential
be approximately universal irrespective of the emitter shape. Using the line
charge model, it is established analytically that smooth generic emitter tips
approximately follow this universal trend when the anode is far away. This is
verified using COMSOL for various emitter shapes including the locally
non-parabolic `hemisphere on a cylindrical post'. It is also found numerically
that the curvature-corrected tunneling potential provides an adequate
approximation when the anode is in close proximity as well as in the presence
of other emitters.
