Figure: An animation illustrating vector field and field lines

Caption: An animation illustrating how field lines are are generated for a vector field.

Features:

1. Field lines are visualization tool for vector fields. They were introduced by Michael Faraday (1791--1867), the inventor of the earliest electric motor and electric generator (see Wikipedia: Michael (1791--1867); NASA: Magnetic Field Lines -- History).

2. They are generated by drawing a curve that is tangent to the direction of a vector field at every point.

   A field line points in the direction of the vector field: i.e., NOT opposite the direction of the vector field.

3. Note field lines can NEVER cross, except when a vector field goes to zero.

4. A vector field forms a continuum, and so has value at every point.

   However, in a diagram, a vector field can only be represented by a finite discrete set of representive arrows →.

   Similarly, field lines form continuum and can only be represented in a diagram by a representative set of curves.

5. Field lines do NOT necessarily have any physical meaning (in a direct sense). If they do NOT have a physical meaning, they can be regarded just a mathematical construction acting as visualization tool (see Wikipedia: Field line: Physical significance).

   However, in certain cases they do have a physical meaning:

   1. A fluid in steady flow (i.e., a flow NOT changing in time) consists of fluid elements (AKA fluid parcels) following the field lines which in this context are called streamlines.

   2. Iron filings in magnetic field (e.g., of a bar magnet) will tend to arrange themselves along magnetic field lines. This is two-stage process: see Wikipedia: Field line: Physical significance.

   3. Charged particles acted on only by the magnetic force will helix along magnetic field lines. If they have NO component of velocity parallel to the magnetic field lines, the helix degenerates into a circular or spiraling motion. The helixing is often seen in astrophysical systems.

   4. There may other cases of physical meaning that are beyond yours truly's strength at the moment.