Force field

Short description: Region of space in which a force acts

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 F→, where F→(x→) is the force that a particle would feel if it were at the point x→.[1]

## Contents

* 1 Examples
* 2 Work
* 2.1 Conservative force field
* 3 See also
* 4 References
* 5 External links

## 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 energy) extends into the space around itself.[2] In Newtonian gravity, a particle of mass M creates a gravitational field g→=−GMr2r^, where the radial unit vector r^ 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 F→=mg→, where g is Earth's gravity.[3][4]
* An electric field E→ exerts a force on a point charge q, given by F→=qE→.[5]
* In a magnetic field B→, a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation: F→=qv→×B→.

## 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 done by the force is a line integral:

W=∫CF→⋅dr→

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

### Conservative force field

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:

∮CF→⋅dr→=0

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:

F→=−∇ϕ

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:

W=ϕ(b)−ϕ(a)

## See also

* Classical mechanics
* Field line
* Force
* Mechanical work

## References

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. https://books.google.com/books?id=UkxPpqHs0RkC&pg=PA181. , 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

## External links

* Conservative and non-conservative force-fields, Classical Mechanics, University of Texas at Austin

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