![Electric Field Lines](https://upload.wikimedia.org/wikipedia/commons/2/2a/Electric_Field_Lines.svg)
A field line is a graphical
visual aid for visualizing
vector fields. It consists of an imaginary directed line which is
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the field
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
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 field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
s,
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s, and
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
s among many other types. In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
field lines showing the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
field of a
fluid flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
are called
streamline
Streamline may refer to:
Business
* Streamline Air, American regional airline
* Adobe Streamline, a discontinued line tracing program made by Adobe Systems
* Streamline Cars, the company responsible for making the Burney car
Engineering
...
s.
Definition and description
![Camposcargas](https://upload.wikimedia.org/wikipedia/commons/b/bf/Camposcargas.svg)
A
vector field defines a direction and magnitude at each point in space. A field line for that vector field 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
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the field vector at each point.
A field line is usually shown as a directed line segment, with an arrow indicating the direction of the vector field. For two-dimensional fields the field lines are plane curves, and most field line diagrams are of this type. Since at each point where it is nonzero and finite the vector field has a unique direction, field lines can never intersect, so there is exactly one field line passing through each point at which the vector field is nonzero and finite.
Points where the field is zero or infinite have no field line through them, since direction cannot be defined there, but can be the ''endpoints'' of field lines.
Since there are an infinite number of points in any region, an infinite number of field lines can be drawn; but only a limited number can be shown on a field line diagram. Therefore which field lines are shown is a choice made by the person or computer program which draws the diagram, and a single vector field may be depicted by different sets of field lines. A field line diagram is necessarily an incomplete description of a vector field, since it gives no information about the field between the drawn field lines, and the choice of how many and which lines to show determines how much useful information the diagram gives.
An individual field line shows the ''direction'' of the vector field but not the ''magnitude''. 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
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
. Then the density of field lines (number of field lines per unit perpendicular area) at any location is proportional to the magnitude of the vector field at that point. Areas in which neighboring field lines are converging (getting closer together) indicates that the field is getting stronger in that direction.
In vector fields which have nonzero
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
, field lines begin on points of positive divergence (''sources'') and end on points of negative divergence (''sinks''), or extend to infinity. For example,
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
lines begin on positive charges and end on negative charges. In fields which are divergenceless (
solenoidal
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
), such as
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s, field lines have no endpoints; they are either closed loops or are endless.
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
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
states that an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
has sources at positive
charges
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aqu ...
, 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
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
has no sources or sinks (
Gauss's law for magnetism
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
), 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.
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
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
. The electric field lines in this case are straight lines that emanate from the charge uniformly in all directions in three-dimensional space. This means that their density is proportional to
, the correct result consistent with
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
for this case. However, if the electric field lines for this setup were just drawn on a two-dimensional plane, their two-dimensional density would be proportional to
, an incorrect result for this situation.
[A. Wolf, S. J. Van Hook, E. R. Weeks, ''Electric field line diagrams don't work'' Am. J. Phys., Vol. 64, No. 6. (1996), pp. 714–72]
DOI 10.1119/1.18237
/ref>
Construction
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
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
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 .
Examples
If the vector field describes a velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, 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
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, which is often depicted using field lines emanating from a magnet
A 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, such as iron, steel, nickel, ...
.
Divergence and curl
Field lines can be used to trace familiar quantities from vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
:
*Divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density 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. If the vector field is the resultant of radial inverse-square law fields with respect to one or more sources then this corresponds to the fact that the divergence of such a field is zero outside the sources. In a solenoidal vector field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
(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.
*The Kelvin–Stokes theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
shows that field lines of a vector field with zero curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
(i.e., a conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
, e.g. a gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
or an electrostatic field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field f ...
) cannot be closed loops. In other words, curl is always present when a field line forms a closed loop. It may be present in other situations too, such as a helical
Helical may refer to:
* Helix, the mathematical concept for the shape
* Helical engine, a proposed spacecraft propulsion drive
* Helical spring, a coilspring
* Helical plc, a British property company, once a maker of steel bar stock
* Helicoil
A t ...
shape of field lines.
Physical significance
While field lines are a "mere" mathematical construction, in some circumstances they take on physical significance. In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
, the velocity field lines ( streamlines) in steady flow represent the paths of particles of the fluid. In the context of , electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s or ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
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
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
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.
See also
* Force field (physics)
In physics, a force field is a vector field corresponding with a non-contact force acting on a particle at various positions in space. Specifically, a force field is a vector field \vec, where \vec(\vec) is the force that a particle would feel ...
* Field lines of Julia sets
* External ray
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.
Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray.
External rays are used in complex analysis, particularly ...
— field lines of Douady–Hubbard potential of Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
or filled-in Julia sets
* 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 t ...
* Vector field
* Line integral convolution
In scientific visualization, line integral convolution (LIC) is a method to visualize a vector field, such as fluid motion.
Features
* global method
* #Features#Integration-based method, integration-based method
* texture-based method
Convolu ...
References
Further reading
*
External links
Interactive Java applet showing the electric field lines of selected pairs of charges
{{Webarchive, url=https://web.archive.org/web/20110813101554/http://chair.pa.msu.edu/applets/E-field_W/a.htm , date=2011-08-13 by Wolfgang Bauer
course notes from a course at the Massachusetts Institute of Technology.
Vector calculus
Numerical function drawing