  The braiding of the solar coronal magnetic field via photospheric motions -
with subsequent relaxation and magnetic reconnection -- is one of the most
widely debated ideas of solar physics. We readdress the theory in the light of
developments in three-dimensional magnetic reconnection theory. It is known
that the integrated parallel electric field along field lines is the key
quantity determining the rate of reconnection, in contrast with the
two-dimensional case where the electric field itself is the important quantity.
We demonstrate that this difference becomes crucial for sufficiently complex
magnetic field structures.
  A numerical method is used to relax a braided magnetic field to an ideal
force-free equilibrium; that equilibrium is found to be smooth, with only
large- scale current structures. However, the equilibrium is shown to have a
highly filamentary integrated parallel current structure with extremely short
length- scales. An analytical model is developed to show that, in a coronal
situation, the length scales associated with the integrated parallel current
structures will rapidly decrease with increasing complexity, or degree of
braiding, of the magnetic field. Analysis shows the decrease in these length
scales will, for any finite resistivity, eventually become inconsistent with
the stability of a force- free field. Thus the inevitable consequence of the
magnetic braiding process is shown to be a loss of equilibrium of the coronal
field, probably via magnetic reconnection events.
