Start with the Maxwell equation
del dot B = 0
But what if the RHS were nonzero?
del dot B = n
Then n would be the density of monopoles.
As far as we know, n=0. But maybe not. Maybe
we've just never noticed. Theory permits nonzero n.
It is highly amusing to work out the consequences.
For starters, there would be magnetic field lines
that start on a n pole and terminate on an s pole.
(where s = -n) ... entirely in analogy to electric
field lines that start on a + charge and end on a
- charge.
You could imagine that the magnetic dipole moment
of an electron is due to an n pole and an s pole
inside the electron, separated by a small distance,
entirely in analogy to the electric dipole moment
of a water molecule, which is due to a + charge
and a - charge, separated by a small distance.
Of course not all magnetic fields would be sourced
by poles, just as not all electric fields are
sourced by charges. A changing electric field
(as in a capacitor) _or_ a current (electric charges
in motion) will produce magnetic field lines
that close on themselves, not terminating anywhere,
and are therefore not well described as terminating
on poles. This is entirely analogous to a changing
magnetic field that produces electric field lines
that close on themselves, terminating nowhere, and
are therefore not well described as terminating on
charges. And what do you think a moving monopole
would produce?
This sort of entertainment is completely unsuitable
for an introductory course.
In an intro course one makes the following assumptions:
-- del dot B is in fact zero.
-- the density of n monopoles is zero.
-- the density of s monopoles is zero.
-- you cannot cut off the north end of a bar
magnet and run away with the northness.
-- all magnetic field lines close on themselves,
without ending anywhere.
-- a magnetic dipole is not (as the name might
suggest) two monopoles, but is instead named
because its far field is analogous to an electric
dipole. The analogy is a good description of
the field, but not a good description of the
source.
One cannot prove _any_ of those assumptions. But
they are all consistent with observation, and not
likely to be overthrown anytime soon.