The shape of the magnetic field produced by a horseshoe magnet is revealed by the orientation of iron filings sprinkled on a piece of paper above the magnet.
A portion of the vector field (sin y, sin x)
Right hand grip rule: a current flowing in the direction of the white arrow produces a magnetic field shown by the red arrows.
A vector field on a sphere
A Solenoid with electric current running through it behaves like a magnet.
The flow field around an airplane is a vector field in R3, here visualized by bubbles that follow the streamlines showing a wingtip vortex.
A sketch of Earth's magnetic field representing the source of the field as a magnet. The south pole of the magnetic field is near the geographic north pole of the Earth.
Vector fields are commonly used to create patterns in computer graphics. Here: abstract composition of curves following a vector field generated with OpenSimplex noise.
One of the first drawings of a magnetic field, by René Descartes, 1644, showing the Earth attracting lodestones. It illustrated his theory that magnetism was caused by the circulation of tiny helical particles, "threaded parts", through threaded pores in magnets.
A vector field that has circulation about a point cannot be written as the gradient of a function.
Hans Christian Ørsted, Der Geist in der Natur, 1854
Magnetic field lines of an iron bar (magnetic dipole)

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.

- Magnetic field

Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

- Vector field
The shape of the magnetic field produced by a horseshoe magnet is revealed by the orientation of iron filings sprinkled on a piece of paper above the magnet.

4 related topics

Alpha

Depiction of a two-dimensional vector field with a uniform curl.

Curl (mathematics)

Depiction of a two-dimensional vector field with a uniform curl.

In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space.

. This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential.

Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.

Maxwell's equations

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.

Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.
In a geomagnetic storm, a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields in Earth's atmosphere, thus causing surges in electrical power grids. (Not to scale.)
Magnetic-core memory (1954) is an application of Ampère's law. Each core stores one bit of data.
Left: A schematic view of how an assembly of microscopic dipoles produces opposite surface charges as shown at top and bottom. Right: How an assembly of microscopic current loops add together to produce a macroscopically circulating current loop. Inside the boundaries, the individual contributions tend to cancel, but at the boundaries no cancelation occurs.

The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.

, a vector field, and the magnetic field,

Vector arrow pointing from A to B

Euclidean vector

Geometric object that has magnitude (or length) and direction.

Geometric object that has magnitude (or length) and direction.

Vector arrow pointing from A to B
Illustration of tangential and normal components of a vector to a surface.
The subtraction of two vectors a and b
Scalar multiplication of a vector by a factor of 3 stretches the vector out.
The scalar multiplications −a and 2a of a vector a
The normalization of a vector a into a unit vector â

Other physical vectors, such as the electric and magnetic field, are represented as a system of vectors at each point of a physical space; that is, a vector field.

Illustration of the electric field surrounding a positive (red) and a negative (blue) charge.

Field (physics)

Physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time.

Physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time.

Illustration of the electric field surrounding a positive (red) and a negative (blue) charge.
In classical gravitation, mass is the source of an attractive gravitational field g.
The E fields and B fields due to electric charges (black/white) and magnetic poles (red/blue). Top: E field due to an electric dipole moment d. Bottom left: B field due to a mathematical magnetic dipole m formed by two magnetic monopoles. Bottom right: B field due to a pure magnetic dipole moment m found in ordinary matter (not from monopoles).
The E fields and B fields due to electric charges (black/white) and magnetic poles (red/blue). E fields due to stationary electric charges and B fields due to stationary magnetic charges (note in nature N and S monopoles do not exist). In motion (velocity v), an electric charge induces a B field while a magnetic charge (not found in nature) would induce an E field. Conventional current is used.
In general relativity, mass-energy warps space time (Einstein tensor G), and rotating asymmetric mass-energy distributions with angular momentum J generate GEM fields H
Fields due to color charges, like in quarks (G is the gluon field strength tensor). These are "colorless" combinations. Top: Color charge has "ternary neutral states" as well as binary neutrality (analogous to electric charge). Bottom: The quark/antiquark combinations.

A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field, i.e. a 1-dimensional (rank-1) tensor field.

However, it became much more natural to take the field approach and express these laws in terms of electric and magnetic fields; in 1849 Michael Faraday became the first to coin the term "field".