I try to use the 1/4pi as an opportunity. There are so many 1/r^2 laws
in physics... what do they have in common that makes them plentiful? I
think students should leave their physics class with an answer to this
question.
In general, there's some constant to make the units work, something
representing an amount, and a 1/r^2 dependence wrt distance. So why in
this case do we have a constant and a 4*pi? Its because the electric
field is so prevalent as the quantity to study in electrodynamics, and
electric fields find their source in charges via Gauss' Law. Choosing
appropriately symmetric enclosing surfaces, the proportionality constant
between the density of charge and the field generated over an enclosing
area is epsilon_zero. Or, the total charge enclosed is the field times
the area, or the field is the charge divided by the area - all carrying
the e_o factor in the appropriate fashion to make the units work. The
electric field at a radius 'r' from a point charge is thus:
E = (1/e_o) * (q/area) = (1/e_o) * (q/4pi*r^2)
since 4*pi*r^2 is the area of a spherical surface at radius 'r' from the
charge.
From here, one can revisit the law of gravitation, light and sound
intensity from a source... and give the students a sense of how
different laws of physics are based on common principles.
The real pity I see is that the 4pi is usually associated with e_o
rather than with r^2 so that the conceptual tie-in is lost and the
factor becomes a burden rather than an opportunity to look at the big
picture of 1/r^2 laws.