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{{short description|Region of space in which a force acts}}
{{Otheruses|Force field}}
{{Other uses|Force field (disambiguation){{!}}Force field}}
buttnut a term coined by [[Michael Faraday]] to provide an intuitive paradigm, but theoretical construct (in the [[Thomas Samuel Kuhn|Kuhnian]] sense), for the behavior of electromagnetic fields, the term '''force field''' refers to the '''[[Line of force|lines of force]]''' one object (the "source object") exerts on another object or a collection of other objects. An object might be a mass particle or an electric or magnetic charge, for example. The lines do not have to be straight, in the [[Euclidean geometry]] case, but may be curved. Faraday called these theoretical connections between objects '''lines of force''' because the objects are most directly connected to the source object along this '''line.'''


[[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.]]
A [[Conservative vector field|conservative]] '''force field''' is a special kind of [[vector field]] that can be represented as the [[gradient]] of a [[potential]].


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>\mathbf F</math>, where <math>\mathbf F(\mathbf r)</math> is the force that a particle would feel if it were at the position <math>\mathbf r</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>
Note that a force field does not exist in reality, per se, but it is really a [[Kuhnian construct]] that allows scientists to visualize the effects of objects on other objects; in other words, it makes the math easy.


==Examples of force fields==
==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
*A local [[Newtonian gravitational field]] near Earth ground typically consists of a uniform array of vectors pointing in one direction---downwards, towards the ground; its force field is represented by the Cartesian [[Vector (geometric)|vector]] <math>\vec{F}=m\cdot g (-\hat{z})</math>, where <math>\hat{z}</math> points in a [[Direction (geometry, geography)|direction]] away from the ground, and m refers to the mass, and g refers to the [[acceleration]] due to [[gravity]].
|title=General relativity from A to B
*A global [[Gravitational field]] consists of a spherical array of vectors pointing towards the center of gravity. Its classical force field, in spherical coordinates, is represented by the vector, <math>\vec{F}=\frac{G m M}{r^2}(-\hat{r})</math>, which is just [[Newton's Law of Gravity]], with the radial unit <math>\hat{r}</math> vector pointing towards the origin of the sphere (center of the Earth).
|first1=Robert
*A conservative [[Electric field]] has an [[electric charge]] (or a smeared plum pudding of electric charges) as its source object. In the case of the point charges, the force field is represented by <math>\vec{F}=\frac{kqQ}{r^2}(-\vec{R})</math>, where <math>\vec{R}</math> is the position vector that represents the straightest line between the source charge and the other charge.
|last1=Geroch
* A static [[Magnetic field]] has a magnetic charge (a [[magnetic monopole]] or a charge distribution).
|publisher=University of Chicago Press
* The electromagnetic force is given by the Lorentz force formula, which in SI units is, <math>\vec{F}=q(\vec{E} + \vec{v} \times \vec{B})</math>.
|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>\mathbf g=\frac{-G M}{r^2}\hat\mathbf r</math>, where the radial [[unit vector]] <math>\hat\mathbf 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>\mathbf F = m \mathbf 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>\mathbf E</math> exerts a force on a [[point charge]] ''q'', given by <math>\mathbf F = q\mathbf 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>\mathbf 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>\mathbf F = q\mathbf v\times\mathbf 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 \mathbf F \cdot d\mathbf 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 \mathbf F \cdot d\mathbf 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> \mathbf 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==
==See also==
*[[Field line]]
* [[Classical mechanics]]
* [[Field line]]
* [[Force]]
* [[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


{{Authority control}}
{{classicalmechanics-stub}}
{{electromagnetism-stub}}


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Latest revision as of 08:19, 14 August 2024

Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The inflection points 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 , where is the force that a particle would feel if it were at the position .[1]

Examples

[edit]
  • 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 energy) extends into the space around itself.[2] In Newtonian gravity, a particle of mass M creates a gravitational field , where the radial unit vector 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 , where g is Earth's gravity.[3][4]
  • An electric field exerts a force on a point charge q, given by .[5]
  • In a magnetic field , a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation: .

Work

[edit]

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 done by the force is a line integral:

This value is independent of the velocity/momentum that the particle travels along the path.

Conservative force field

[edit]

For a 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:

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:

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:

See also

[edit]

References

[edit]
  1. ^ Mathematical methods in chemical engineering, by V. G. Jenson and G. V. Jeffreys, p211
  2. ^ Geroch, Robert (1981). General relativity from A to B. University of Chicago Press. p. 181. ISBN 0-226-28864-1., Chapter 7, page 181
  3. ^ Vector calculus, by Marsden and Tromba, p288
  4. ^ Engineering mechanics, by Kumar, p104
  5. ^ Calculus: Early Transcendental Functions, by Larson, Hostetler, Edwards, p1055
[edit]