Field line

Visual aid to depiction of a vector field
the modern use of "field lines" as a way to depict electromagnetic and other vector fields

A field line is a graphical visual aid for visualizing vector fields. It consists of an imaginary integral curve which is tangent to the field vector at each point along its length. 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. They are used to show electric fields, magnetic fields, and gravitational fields among many other types. In fluid mechanics, field lines showing the velocity field of a fluid flow are called streamlines.

A vector field defines a direction and magnitude at each point in space. A field line is an integral curve for that vector field and may be constructed by starting at a point and tracing a line through space that follows the direction of the vector field, by making the field line tangent to the field vector at each point. 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. For example, Gauss's law states that an electric field has sources at positive charges, sinks at negative charges, and neither elsewhere, so electric field lines start at positive charges and end at negative charges. 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. However, as stated above, a special situation may occur around points where the field is zero (that cannot be intersected by field lines, because their direction would not be defined) and the simultaneous begin and end of field lines takes place. 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.

Given a vector field and a starting point a field line can be constructed iteratively by finding the field vector at that point The unit tangent vector at that point is: By moving a short distance along the field direction a new point on the line can be found

Then the field at that point is found and moving a further distance in that direction the next point of the field line is found. At each point the next point can be found by

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. But by using a small enough value for , 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 by taking each step in the opposite direction by using a negative step .

If the vector field describes a velocity field, then the field lines follow stream lines in the flow. Perhaps the most familiar example of a vector field described by field lines is the magnetic field, which is often depicted using field lines emanating from a magnet.

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 strongly, 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. It is easy to visualize as a two stage process: first, the filings are spread evenly over the magnetic field but all aligned in the direction of the field. 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. Of course the two stages described here happen concurrently until an equilibrium is achieved. Because the intrinsic magnetism of the filings modifies the field, the lines shown by the filings are only an approximation of the field lines of the original magnetic field. Magnetic fields are continuous, and do not have discrete lines.