Linear theory of interaction of two sheet electron beams in longitudinal magnetic field

Linear theory of interaction between two infinitely thin sheet electron beams has been built in hydrodynamic approximation. It’s supposed that electron beams move in space between two conducting planes. Research takes into account the effect of space charge and finite longitudinal focusing magnetic field. The considered model allows to describe instability to high-frequency (HF) perturbations in the given system. The eight-power dispersion equation has been derived for given system. Since it’s not possible to write down its analytical solution, that dispersion equation has been solved graphically. From the point of view of the theory of coupled waves four waves can be exited in each of the beams – fast and slow space-charge waves and fast and slow cyclotron waves. The effect of finite magnetic field on development of the instability has been investigated in case of two interaction electron beams on bases of dispersion equation solution. Investigation of the dispersion characteristics allow to conclude that presence of the permanent longitudinal magnetic field leads to occurrence of new instability regions, that is caused by not only interaction of space-charge waves, but also cyclotron waves in beams.