It's not really true that magnetic field lines cannot cross, but where they do, the magnetic field strength has to be zero.
Here's why: A magnetic field line is that path in space that
points in the direction of the magnetic field at every point along it.
Walk along a magnetic field line carrying a compass, and the needle
will always point in the direction you need to go in order to stay on
that magnetic field line (the needle has to be able to pivot up and
down as well as around in a circle like most compasses).
If two field lines crossed, then that is saying that the magnetic
field points in two different directions, at one place. There's only
one direction to the magnetic field at any place at any time, so this
Now I said that it could happen if the field is zero -- that's the
only kind of magnetic field without a direction. You get this inside of
quadrupole magnets, say. Here's how to make such a field. Start with
four bar magnets, and arrange them like so:
The magnetic field in the center of that thing will be zero, but
magnetic field lines will point in towards the center from the left and
right, and field lines will point away from the center up and down.
Now you asked about why magnetic field lines bend -- they don't
always have to. Some magnets are constructed so the magnetic field
lines are as straight as possible (at least over part of their length).
Magnetic field lines tend to follow closed loops, though. Ordinary
permanent magnets have North and South poles, and the magnetic field
lines go out of the magnetic North pole and head towards the South
pole, usually curving in the process. Magnetic fields can be produced
by electrical currents flowing in wires, and the resulting magnetic
field makes loops around the wire. You can arrange a lot of wires so
the magnetic field follows a nearly straight path for a while, but
eventually the field will bend around as no apparatus is infinitely
(published on 10/22/2007)