A
field line is a
locusIn mathematics, a locus is a collection of points which share a property. The term locus is typically used of a condition which defines a continuous figure or figures, that is, a curve...
that is defined by a
vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
and a starting location within the field. Field lines are useful for
visualizingThe term visualization or visualisation may refer to:* Creative visualization* Flow visualization* Geovisualization* Illustration* Information graphics, visual representations of information, data, or knowledge* Information visualization...
vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
s, which are otherwise hard to depict. Note that, like longitude and latitude lines on a globe, or topographic lines on a
topographic mapA topographic map is a type of map characterized by large-scale detail and quantitative representation of relief, usually using contour lines in modern mapping, but historically using a variety of methods...
, these lines are not physical lines that are actually present at certain locations; they are merely visualization tools.
A vector field defines a direction at all points in space; a field line for that vector field may be constructed by tracing a
topographic pathIn mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
in the direction of the vector field.
A
field line is a
locusIn mathematics, a locus is a collection of points which share a property. The term locus is typically used of a condition which defines a continuous figure or figures, that is, a curve...
that is defined by a
vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
and a starting location within the field. Field lines are useful for
visualizingThe term visualization or visualisation may refer to:* Creative visualization* Flow visualization* Geovisualization* Illustration* Information graphics, visual representations of information, data, or knowledge* Information visualization...
vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
s, which are otherwise hard to depict. Note that, like longitude and latitude lines on a globe, or topographic lines on a
topographic mapA topographic map is a type of map characterized by large-scale detail and quantitative representation of relief, usually using contour lines in modern mapping, but historically using a variety of methods...
, these lines are not physical lines that are actually present at certain locations; they are merely visualization tools.
Precise definition
A vector field defines a direction at all points in space; a field line for that vector field may be constructed by tracing a
topographic pathIn mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
in the direction of the vector field. More precisely, the tangent line to the path at each point is required to be parallel to the vector field at that point.
A complete description of the
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
of all the field lines of a vector field is sufficient to completely specify the
direction of the vector field everywhere. In order to also depict the
magnitude, a selection of field lines is drawn such that the density of field lines (number of field lines per unit area) at any location is proportional to the magnitude of the vector field at that point. This is almost always the case, for example, when field lines are used to depict electric and magnetic fields.
Examples
If the vector field describes a
velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
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 fieldMagnetic fields surround magnetic materials and electric currents and are detected by the force they exert on other magnetic materials and moving electric charges...
, which is often depicted using field lines emanating from a
magnetA magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials and attracts or repels other magnets.A permanent magnet is an object made from a...
.
Divergence and curl
Field lines can be used to trace familiar quantities from
vector calculusVector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
:
- Divergence
In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
may be easily seen through field lines, assuming the lines are drawn such that the density of field lines is proportional to the magnitude of the field (see above). In this case, the divergence may be seen as the beginning and ending of field lines. In a solenoidal vector fieldIn vector calculus a solenoidal vector field is a vector field v with divergence zero:...
(i.e. a vector field where the divergence is zero everywhere), the field lines neither begin nor end; they either form closed loops, or go off to infinity in both directions. If a vector field has positive divergence in some area, there will be field lines starting from points in that area. If a vector field has negative divergence in some area, there will be field lines ending at points in that area.
- Curl may be seen as a helical
A helix is a special kind of space curve, i.e. a smooth curve in three-space. As a mental image of a helix one may take the spring...
shape of field lines.
Physical significance
While field lines are a "mere" mathematical construction, in some circumstance they take on physical significance. In the context of plasma physics,
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s or
ionAn ion is an atom or molecule where the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge...
s 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 a homeostasis is achieved. Because the intrinsic magnetism of the filings modifies the field, the lines shown by the filings are only an approximation of the equipotential lines of the original magnetic field. Magnetic fields are continuous, and do not have discrete lines.
See also
- Force field (physics)
Originally a term coined by Michael Faraday to provide an intuitive paradigm, but theoretical construct , for the behavior of electromagnetic fields, the term force field refers to the lines of force one object exerts on another object or a collection of other objects...
- External ray
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.External rays are used in complex analysis, particularly in complex dynamics and geometric function theory,-History:...
— field lines of Douady-Hubbard potential of Mandelbrot setIn mathematics the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal...
or filled-in Julia setsThe filled-in Julia set of a polynomial is defined as the set of all points of dynamical plane that have bounded orbit with respect to
where :
is set of complex numbers
...
- Line of force
A line of force in Faraday's extended sense is synonymous with Maxwell's line of induction. According to J.J. Thomson, Faraday usually discusses lines of force as chains of polarized particles in a dielectric, yet sometimes Faraday discusses them as having an existence all their own as in...