The problem of two‐stream instability in plasma is studied by specifying the importance of initial magnetic field associated with the motion of the charged particles and the boundary effects. In Part I the accurate initial steady state is studied when the streams of electrons and ions move with different uniform speeds in plasmas with plane and cylindrical geometry. In Part II, in order to show the effects of finiteness and inhomogeneity of the system, small transverse plasma oscillations are studied in the case of plane plasmas. The role of plasma‐sheath oscillations at the boundaries is found to be very important in driving the instabilities associated with the electromagnetic modes. The numerical estimates of the growth rates of the instability are given for the specific case of the physical data in discharge tubes.