Carbon nanotubes are excellent charge transports due to the relationship between the length-diameter in the order of microns to nanometers, respectively, thus being characterized as ballistic conductors and excellent materials for field emission applications in electronic devices. We study the Fowler-Nordheim formalism and first-principles calculations of metallic (2,2) singlewall carbon nanotube with a diameter of 0.24 nm. Our results are consistent for field emission devices nanostructured with these specifications. Summarize the following results: (i) for the both methods used the same trend I-V; (ii) exponential behavior and quasi-ballistic, indicating the proportionality between I and V characteristic of (2,2) carbon nanotube armchair, (iii) tendency of inelastic tunneling under intense field electric, which is consistent with studies on inelastic electron tunneling spectroscopy (JETS); (iv) matching number Mulliken charge for using the formalism of the Fowler-Nordheim equation to reduce the computational first-principles calculations for this (2,2) carbon nanotube.