Figure: Vector field and field lines

Caption: A cartoon image showing electric field field lines. The field lines are generated by following the electric field vector direction at every point. The field lines never cross since the electric field cannot have two directions at one point. There can be points and regions where the electric field goes to zero where electric field lines diverge from or converge to (see below). The Coulomb's law force (AKA electric force) is given by the formula (vector F) = q times (vector E), where q is a point charge and (vector E) is the electric field vector.

Further explication:

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. 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.

4. 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 (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 helixing degenerates into a circular 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.

5. As noted in the preamble, field lines can NEVER cross since that requires a vector to point two ways at one point.

   However, there can be points or regions where the vector field is zero and where field lines can diverge from or converge to. Such points or regions are for idealized cases where you do not consider the complications of quantum mechanics mechanics and the microscopic scale.

   Some examples:

   1. The point equidistant between two equal positive point charges. The electric field here is zero by symmetry. The field lines directly from the charges converge here. But in the perpendicular direction, two field lines diverge. All other field lines never reach this zero point though there is a continuum of them approaching the zero point and always missing.
   2. Consider two infinite parallel sheets of positive change with equal surface charge density. Symmetry dictates that the plane equidistant between the sheets has zero electric field and all field lines are perpendicular to the sheets and converge on the plane of zero field.
   3. An ideal conductor in an electrostatic case has zero electric field inside. So all field lines outside that reach the conductor just end at the ideal infinitely thin surface. They can either start or end there.
   4. Many other cases can be imagined.

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