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We consider the limiting current from an emitting patch whose size is muchsmaller than the anode-cathode spacing. The limiting current is formulated interms of an integral equation. It is solved iteratively, first to numericallyrecover the classical one-dimensional Child-Langmuir law, including Jaffe'sextension to a constant, nonzero electron emission velocity. We extend to2-dimensions in which electron emission is restricted to an infinitely longstripe with infinitesimally narrow stripe width, so that the emitted electronsform an electron sheet. We next extend to 3-dimensions in which electronemission is restricted to a square tile (or a circular patch) with aninfinitesimally small tile size (or patch radius), so that the emittedelectrons form a needle-like line charge. Surprisingly, for the electron needleproblem, we only find the null solution for the total line charge current,regardless of the assumed initial electron velocity. For the electron sheetproblem, we also find only the null solution for the total sheet current if theelectron emission velocity is assumed to be zero, and the total maximum sheetcurrent becomes a finite, nonzero value if the electron emission velocity isassumed to be nonzero. These seemingly paradoxical results are shown to beconsistent with the earlier works of the Child-Langmuir law of higherdimensions. They are also consistent with, or perhaps even anticipated by, themore recent theories and simulations on thermionic cathodes that used realisticwork function distributions to account for patchy, nonuniform electronemission. The mathematical subtleties are discussed.

On the Child-Langmuir Law in One, Two, and Three Dimensions