Consider the following algorithm for drawing the electric field lines
for a dipole consisting of two small but not infinitesmal spherical charge
distributions
-choose symmetrically located points at the surface of the positive
charge
- for each point:
-draw an arrow in the direction of the E field
-follow that arrow an infinitesimal distance and draw a
second arrow in
the direction of the E field at that new point
-move an infinitesimal distance in the direction of that
second arrow and
-draw a third arrows in the direction of E
-when completed draw a line tangent to the arrows
-if you do this for each of the symmetric points around the surface
of the
positive sphere, you should get the standard field line
configuration for
the dipole.
What has recently begun to perplex me is that we have only paid explicit
attention to the direction of the field and the correct proportionality
between the density of the field lines and the strength of the field seems
to have appeared by magic.
Part of the answer seems to be the rule that lines cannot start or stop
except on charges. But this still does not seem to be enough to me.