The subject of the present theoretical and experimental investigations is the effect of the external magnetic field induction on dark current and a possibility of breakdown. The generalization of the Fowler-Nordheim equation makes it possible to take into account the influence of a magnetic field parallel to the cathode surface on the field emission current. The reduction in the breakdown voltage due to the increment in electron-impact ionization was theoretical predicted.Experimentally shown that the presence of a magnetic field about a tenth as a large as the cutoff magnetic field [18] reduces the breakdown voltage by 10% to 20% for practically all cathodes no matter what their surface treatment.

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As the title implies the article describes the possibility of taking into account the relativistic correction to the field current density of the field emission of electrons from the metal. The article provides the reader with some analytic generalization of the Fowler-Nordheim equation with the relativistic correction. The relativistic correction to the Fowler-Nordheim equation makes it possible to take into account the influence of the relativism on the field emission current. It is especially noted that the consideration of this correction is necessary in the case of sufficiently strong electric fields and relatively large interelectrode distances. It should be stressed that this correction is valid for fixed interelectrode distances that decrease with increasing electric field strength. It means that for the electric field strength of 0.1 to 1 Gv/m the interelectrode distance should not exceed values of 1 to 0.1 cm. First in the article it is spoken in detail about finding of the electron wave function. Next the field emission curent calculations are given. As a result the transmission coefficient of the potential step from the Klein-Gordon equation within the framework of the Fowler-Nordheim approximation is found. It is shown that in the case of the interelectrode distanse less than 1 cm, an analytical expression for the field electron emission current density is obtained. The conclusion that usually relativistic correction does not exceed a tenth of a percent is made. But in the case of the field electron emission from pulsars (where the work function and electric field strength are much higher) the contribution of the relativistic correction about 10 % has been established.