In physics, a **force field** is a vector field that describes 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 point .^{[1]}

## 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 , 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 the standard gravity.^{[3]}^{[4]} - An electric field is a vector field. It exerts a force on a point charge
*q*given by .^{[5]}

## 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

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:

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

## References

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

## External links

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