Interchange instability is an important dynamic mechanism in plasma physics and has been advanced as an explanation of a variety of phenomena in the magnetospheric physics. This work derives a new instability criterion for interchange motion in a plasma that connects to a finite-conductivity wall. The new criterion is for a arbitrary magnetic <beta> (ratio between thermal pressure and magnetic pressure averaged within flux tube) system, which contains background shear flow, whereas most classical criteria did not consider all of these conditions. Thus this new result is more appropriate to be applied in a real plasma system like the Earth's plasma sheet, which exhibits a wide range of <beta> values and background shear flow. Based on magnetosphere-ionosphere coupling theory and ideal MHD adiabatic theory in the inner plasma sheet, a theoretical model was constructed in the ionosphere region. A finite boundary layer was set up between two regions of uniform-content flux tubes, and a perturbation on the boundary layer was investigated. Both analytical and numerical approaches are used to study the stability of the plasma configuration. The flux tubes are interchange unstable when the angle between the gradient of flux tube volume, defined as V = dsB , and the gradient of adiabatic specific entropy PV 5/3 is larger than arccos<b> 1lnPV5/3 1ln V/ 21+5<b>/6 . Combining this new criterion with the statistical calculation of the plasma sheet characteristics by using the Tsyganenko magnetic field model (the 1996 version) and the Tsyganenko-Mukai plasma model, it is found that, in the Earth's inner plasma sheet, the angle between the two gradients is typically of the order of 15°, which indicates that the statistical-average Earth's plasma sheet is interchange stable. This result is applicable to the study of interchange instability and plasma transport in the global-MHD and other large-scale magnetosphere simulations, and provides a theoretical base for the study of analogous dynamic processes in the magnetospheres of other planets like Jupiter.