Analysis of planar field-stimulated vacuum space-charge using a dimensionless equation

In field emission contexts, any space-charge present has been field emitted. The theory of this field emitted vacuum space-charge (FEVSC) differs significantly from that of thermally emitted space-charge; it is best to regard the theories as separate. So important aspects of the theory can be put into the form of a dimensionless quadratic equation that has exact solutions. These do not constitute a complete solution of the total problem, but they provide useful insight into the complete solution. The equation has very different approximate solutions in the high and low-space-charge limits, and this brings out clearly the idea of "space-charge regimes". The equation applies both to electron and ion emission, provided appropriate interpretations are made of the symbols involved.