to which has been given the name "plasma". A wire immersed in the plasma, at zero potential with respect to it, will absorb every ion and electron that strikes it. Since the electrons move about 600 times as fast as the ions, 600 times as many electrons will strike the wire as ions. If the wire is insulated it must assume such a negative potential that it receives equal numbers of electrons and ions, that is, such a potential that it repels all but 1 in 600 of the electrons headed for it." "Suppose that this wire, which we may take to be part of a grid, is made still more negative with a view to controlling the current through the tube. It will now repel all the electrons headed for it, but will receive all the positive ions that fly toward it. There will thus be a region around the wire which contains positive ions and no electrons, as shown diagrammatically in Fig. 1. The ions are accelerated as they approach the negative wire, and there will exist a potential gradient in this sheath, as we may call it, of positive ions, such that the potential is less and less negative as we recede from the wire, and at a certain distance is equal to the potential of the plasma. This distance we define as the boundary of the sheath. Beyond this distance there is no effect due to the potential of the wire." Mathematical treatment The planar sheath equation The quantitative physics of the Debye sheath is determined by four phenomena: Energy conservation of the ions: If we assume for simplicity cold ions of mass entering the sheath with a velocity , having charge opposite to the electron, conservation of energy in the sheath potential requires , where is the charge of the electron taken positively, i.e. x . Ion continuity: In the steady state, the ions do not build up anywhere, so the flux is everywhere the same: . Boltzmann relation for the electrons: Since most of the electrons are reflected, their density is given by . Poisson's equation: The curvature of the electrostatic potential is related to the net charge density as follows: . Combining these equations and writing them in terms of the dimensionless potential, position, and ion speed, we arrive at the sheath equation: . The Bohm sheath criterion The sheath equation can be