Field line

It consists of an imaginary integral curve which is tangent to the field vector at each point along its length.

[1][2] A diagram showing a representative set of neighboring field lines is a common way of depicting a vector field in scientific and mathematical literature; this is called a field line diagram.

A vector field defines a direction and magnitude at each point in space.

[3][2][1] A field line is usually shown as a directed line segment, with an arrowhead indicating the direction of the vector field.

In order to also depict the magnitude of the field, field line diagrams are often drawn so that each line represents the same quantity of flux.

In vector fields which have nonzero divergence, field lines begin on points of positive divergence (sources) and end on points of negative divergence (sinks), or extend to infinity.

[4][5] In physics, drawings of field lines are mainly useful in cases where the sources and sinks, if any, have a physical meaning, as opposed to e.g. the case of a force field of a radial harmonic.

A gravitational field has no sources, it has sinks at masses, and it has neither elsewhere, gravitational field lines come from infinity and end at masses.

A magnetic field has no sources or sinks (Gauss's law for magnetism), so its field lines have no start or end: they can only form closed loops, extend to infinity in both directions, or continue indefinitely without ever crossing itself.

This situation happens, for instance, in the middle between two identical positive electric point charges.

There, the field vanishes and the lines coming axially from the charges end.

At the same time, in the transverse plane passing through the middle point, an infinite number of field lines diverge radially.

The concomitant presence of the lines that end and begin preserves the divergence-free character of the field in the point.

[5] Note that for this kind of drawing, where the field-line density is intended to be proportional to the field magnitude, it is important to represent all three dimensions.

For example, consider the electric field arising from a single, isolated point charge.

The electric field lines in this case are straight lines that emanate from the charge uniformly in all directions in three-dimensional space.

, the correct result consistent with Coulomb's law for this case.

along the field direction a new point on the line can be found

By repeating this and connecting the points, the field line can be extended as far as desired.

This is only an approximation to the actual field line, since each straight segment isn't actually tangent to the field along its length, just at its starting point.

, taking a greater number of shorter steps, the field line can be approximated as closely as desired.

The field line can be extended in the opposite direction from

Field lines can be used to trace familiar quantities from vector calculus: While field lines are a "mere" mathematical construction, in some circumstances they take on physical significance.

In fluid mechanics, the velocity field lines (streamlines) in steady flow represent the paths of particles of the fluid.

In the context of plasma physics, electrons or ions that happen to be on the same field line interact readily, while particles on different field lines in general do not interact.

This is the same behavior that the particles of iron filings exhibit in a magnetic field.

The iron filings in the photo appear to be aligning themselves with discrete field lines, but the situation is more complex.

Then, based on the scale and ferromagnetic properties of the filings they damp the field to either side, creating the apparent spaces between the lines that we see.

[citation needed] Of course the two stages described here happen concurrently until an equilibrium is achieved.

Magnetic fields are continuous, and do not have discrete lines.