View source for Force field (physics)

{{short description|Region of space in which a force acts}}
{{Other uses|Force field (disambiguation){{!}}Force field}}

[[Image:GravityPotential.jpg|thumb|300px|Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The [[inflection point]]s of the cross-section are at the surface of the body.]]

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 <math>\vec{F}</math>, where <math>\vec{F}(\vec{x})</math> is the force that a particle would feel if it were at the point <math>\vec{x}</math>.<ref>[https://books.google.com/books?id=akbi_iLSMa4C&pg=PA211 Mathematical methods in chemical engineering, by V. G. Jenson and G. V. Jeffreys, p211]</ref>

==Examples==
*[[Gravity]] is the force of attraction between two objects. A gravitational force field models this influence that a massive body (or more generally, any quantity of [[Mass–energy equivalence|energy]]) extends into the space around itself.<ref>{{cite book
|title=General relativity from A to B
|first1=Robert
|last1=Geroch
|publisher=University of Chicago Press
|year=1981
|isbn=0-226-28864-1
|page=181
|url=https://books.google.com/books?id=UkxPpqHs0RkC&pg=PA181}}, [https://books.google.com/books?id=UkxPpqHs0RkC&pg=PA181 Chapter 7, page 181] </ref> In [[Newtonian gravity]], a particle of mass ''M'' creates a [[gravitational field]] <math>\vec{g}=\frac{-G M}{r^2}\hat{r}</math>, where the radial unit vector <math>\hat{r}</math> points away from the particle. The gravitational force experienced by a particle of light mass ''m'', close to the surface of [[Earth]] is given by <math>\vec{F} = m \vec{g}</math>, where ''g'' is [[Earth's gravity]].<ref>[https://books.google.com/books?id=LiRLJf2m_dwC&pg=PA288 Vector calculus, by Marsden and Tromba, p288]</ref><ref>[https://books.google.com/books?id=bCP68dm49OkC&pg=PA104 Engineering mechanics, by Kumar, p104]</ref>
*An [[electric field]] <math>\vec{E}</math> exerts a force on a [[point charge]] ''q'', given by <math>\vec{F} = q\vec{E}</math>.<ref>[https://books.google.com/books?id=9ue4xAjkU2oC&pg=PA1055 Calculus: Early Transcendental Functions, by Larson, Hostetler, Edwards, p1055]</ref>
*In a [[magnetic field]] <math>\vec{B}</math>, a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation: <math>\vec{F} = q\vec{v}\times\vec{B}</math>.

== Work ==
Work is dependent on the displacement as well as the force acting on an object. As a particle moves through a force field along a path ''C'', the [[Work (physics)|work]] done by the force is a [[line integral]]:
:<math> W = \int_C \vec{F} \cdot d\vec{r}</math>

This value is independent of the [[velocity]][[Momentum|/momentum]] that the particle travels along the path. 

=== Conservative force field ===
For a [[conservative force|conservative force field]], it is also independent of the path itself, depending only on the starting and ending points. Therefore, the work for an object travelling in a closed path is zero, since its starting and ending points are the same:

:<math> \oint_C \vec{F} \cdot d\vec{r} = 0</math>
If the field is conservative, the work done can be more easily evaluated by realizing that a conservative vector field can be written as the gradient of some scalar potential function:

:<math> \vec{F} = -\nabla \phi</math>

The work done is then simply the difference in the value of this potential in the starting and end points of the path. If these points are given by ''x'' = ''a'' and ''x'' = ''b'', respectively:

:<math> W = \phi(b) - \phi(a) </math>

==See also==
* [[Classical mechanics]]
* [[Field line]]
* [[Force]]
* [[Work (physics)|Mechanical work]]

==References==
{{Reflist}}

== External links ==
{{wikiquote}}
* [http://farside.ph.utexas.edu/teaching/301/lectures/node59.html Conservative and non-conservative force-fields], [http://farside.ph.utexas.edu/teaching/301/lectures/lectures.html Classical Mechanics], University of Texas at Austin

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[[Category:Force]]