We consider the limiting current from an emitting patch whose size is much
smaller than the anode-cathode spacing. The limiting current is formulated in
terms of an integral equation. It is solved iteratively, first to numerically
recover the classical one-dimensional Child-Langmuir law, including Jaffe's
extension to a constant, nonzero electron emission velocity. We extend to
2-dimensions in which electron emission is restricted to an infinitely long
stripe with infinitesimally narrow stripe width, so that the emitted electrons
form an electron sheet. We next extend to 3-dimensions in which electron
emission is restricted to a square tile (or a circular patch) with an
infinitesimally small tile size (or patch radius), so that the emitted
electrons form a needle-like line charge. Surprisingly, for the electron needle
problem, we only find the null solution for the total line charge current,
regardless of the assumed initial electron velocity. For the electron sheet
problem, we also find only the null solution for the total sheet current if the
electron emission velocity is assumed to be zero, and the total maximum sheet
current becomes a finite, nonzero value if the electron emission velocity is
assumed to be nonzero. These seemingly paradoxical results are shown to be
consistent with the earlier works of the Child-Langmuir law of higher
dimensions. They are also consistent with, or perhaps even anticipated by, the
more recent theories and simulations on thermionic cathodes that used realistic
work function distributions to account for patchy, nonuniform electron
emission. The mathematical subtleties are discussed.Comment: This material has been submitted to Physics of Plasmas. After it is
published, it will be found at https://pubs.aip.org/aip/po