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.