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A finite-size uniform random distribution of vertically aligned fieldemitters on a planar surface is studied under the assumption that theasymptotic field is uniform and parallel to the emitter axis. A formula forfield enhancement factor is first derived for a 2-emitter system and this isthen generalized for $N$-emitters placed arbitrarily (line, array or random).It is found that geometric effects dominate the shielding of field lines. Thedistribution of field enhancement factor for a uniform random distribution ofemitter locations is found to be closely approximated by an extreme value(Gumbel-minimum) distribution when the mean separation is greater than theemitter height but is better approximated by a Gaussian for mean separationsclose to the emitter height. It is shown that these distributions can be usedto accurately predict the current emitted from a large area field emitter.

Shielding effects in random large area field emitters, the field enhancement factor distribution and current calculation