integrated once by multiplying by : At the sheath edge (), we can define the potential to be zero () and assume that the electric field is also zero (). With these boundary conditions, the integrations yield This is easily rewritten as an integral in closed form, although one that can only be solved numerically. Nevertheless, an important piece of information can be derived analytically. Since the left-hand-side is a square, the right-hand-side must also be non-negative for every value of , in particular for small values. Looking at the Taylor expansion around , we see that the first term that does not vanish is the quadratic one, so that we can require , or , or . This inequality is known as the Bohm sheath criterion after its discoverer, David Bohm. If the ions are entering the sheath too slowly, the sheath potential will "eat" its way into the plasma to accelerate them. Ultimately a so-called pre-sheath will develop with a potential drop on the order of and a scale determined by the physics of the ion source (often the same as the dimensions of the plasma). Normally the Bohm criterion will hold with equality, but there are some situations where the ions enter the sheath with supersonic speed. The Child–Langmuir law Although the sheath equation must generally be integrated numerically, we can find an approximate solution analytically by neglecting the term. This amounts to neglecting the electron density in the sheath, or only analyzing that part of the sheath where there are no electrons. For a "floating" surface, i.e. one that draws no net current from the plasma, this is a useful if rough approximation. For a surface biased strongly negative so that it draws the ion saturation current, the approximation is very good. It is customary, although not strictly necessary, to further simplify the equation by assuming that is much larger than unity. Then the sheath equation takes on the simple form . As before, we multiply by and integrate to obtain , or . This is easily integrated over ξ to yield , where is the (normalized) potential at the wall (relative to the sheath edge), and d is the thickness of the sheath. Changing back to the variables and and noting that the ion current into the wall is , we have . This equation is known as Child's law, after Clement D. Child (1868–1933),