in Cartesian coordinates, where . The divergence of this field is

(255) |

If
then is a
*solenoidal* vector field. In other words, field-lines of
never begin or end. This is certainly the
case in Eq. (254) where the field-lines are a set of concentric circles centred
on the -axis. What about magnetic
fields generated by permanent magnets (the modern equivalent of loadstones)?
Do they also never begin or end? Well, we know that a conventional bar
magnet has both a North and South magnetic pole (like the Earth). If we track the
magnetic field-lines with a small compass they all emanate from the South
pole, spread out, and eventually reconverge on the North pole (see Fig. 31). It appears likely
(but we cannot prove it with a compass) that the field-lines inside the magnet
connect from the North to the South pole so as to form closed loops which never begin or end.

Can we produce an isolated North or South magnetic pole: for instance, by snapping
a bar magnet in two? A compass needle would always point towards an isolated
North pole, so this would act like a negative ``magnetic charge.''
Likewise, a compass needle would always point away from an isolated South
pole, so this would act like a positive ``magnetic charge.''
It is clear from Fig. 32
that if we take a closed surface containing an isolated
magnetic pole, which is usually termed a *magnetic monopole*, then
: the flux will be positive for an isolated
South pole, and negative for an isolated North pole.
It follows from Gauss'
theorem that if
then
. Thus, the statement that magnetic fields are solenoidal, or that
, is equivalent to the statement that *there are no magnetic monopoles*. It is not clear, *a priori*, that this is
a true statement. In fact, it is quite possible to formulate electromagnetism so as to
allow for magnetic monopoles. However, as far as we know, there are no magnetic
monopoles in the Universe.
At least, if there are any then they are all
hiding from us!
We know that if we try to make a magnetic monopole by snapping a bar magnet in two
then we just end up with two smaller bar magnets. If we snap one of these smaller
magnets in two then we end up with two even smaller bar magnets. We can continue
this process down to the atomic level without ever producing a magnetic monopole.
In fact, permanent magnetism
is generated by electric currents circulating on the atomic
scale, so this type of
magnetism is not fundamentally different to the magnetism generated
by macroscopic currents.

In conclusion, *all* steady magnetic fields in the Universe are generated by
circulating electric currents of some description. Such fields are solenoidal:
that is, they never begin or end, and satisfy the field equation

(256) |