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 doesn’t happen.
A field of zero is 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:
N S SN NS S N
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. So even here they don't actually cross.
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 big.
(republished on 07/13/06)
(published on 03/04/13)