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Formalisms for both non-relativistic as well as relativistic versions offield emission of electrons in presence of strong quantizing magnetic field,relevant for strongly magnetized neutron stars or magnetars are developed. Inthe non-relativistic scenario, where electrons obey Schr{$\ddot{\rm{o}}$}dingerequation, we have noticed that when Landau levels are populated for electronsin presence of strong quantizing magnetic field the transmission probabilityexactly vanishes unless the electrons are spin polarized in the oppositedirection to the external magnetic field. On the other hand, the cold electronemission under the influence of strong electrostatic field at the poles istotally forbidden from the surface of those compact objects for which thesurface magnetic field strength is $\gg 10^{15}$G (in the eventuality that theymay exist). Whereas in the relativistic case, where the electrons obey Diracequation, the presence of strong quantizing magnetic field completely forbidsthe emission of electrons from the surface of compact objects if $B >10^{13}$G.

Fowler-Nordheim Electron Cold Emission Formalism in Presence of Strong Magnetic Field