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We performed tight-binding calculation of the electronic properties of carbon nanotubes in a perpendicular electric field. Within the linear response limit, the dielectric function of a doped carbon nanotube is found to depend not only on its symmetry, but also on the Fermi level position and tube radius. Upon increasing the field, the mixing of neighboring sub bands results in metal-semiconductor transitions in both quasi-metallic and semiconducting nanotubes. The characteristic field strength of the transitions is calculated as a function of the tube radius. An optimal radius range to be used for band gap engineering is estimated for both types.