An electron is an indivisible particle and its electric field has to be a constant too. Going through a slit how the electric field goes through the wall and how it will be regenerated behind the slit? The question has a variation for endless walls and for finite walls.
There where some other questions about infinity or not of fields from particles: The maximum distance for which Coulomb's law has been verified?, What happens when a field turns on or off?, What is the largest distance for which the influence of the electric field of a single electron was measured?
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