Force field (physics)

For other uses, see Force field (disambiguation).

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 that describes a non-contact force acting on a particle at various positions in space. Specifically, a force field is a vector field ${\vec {F}}$ , where ${\vec {F}}({\vec {x}})$ is the force that a particle would feel if it were at the point ${\vec {x}}$ .^[1]

Examples of force fields

In Newtonian gravity, a particle of mass M creates a gravitational field ${\vec {g}}={\frac {-GM}{r^{2}}}{\hat {r}}$ , where the radial unit vector ${\hat {r}}$ points away from the particle. The gravitational force experienced by a particle of mass m is given by ${\vec {F}}=m{\vec {g}}$ . ^[2]^[3]
An electric field ${\vec {E}}$ is a vector field. It exerts a force on a point charge q given by ${\vec {F}}=q{\vec {E}}$ . ^[4]
a gravitational force field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body.,^[5]

Restriction to position-dependent forces

Some forces, including friction, air drag, and the magnetic force on a charged particle, depend on the particle's velocity as well as its position. Therefore these forces are not characterized by a force field.

Work done by a force field

As a particle moves through a force field along a path C, the work done by the force is a line integral

W=\int _{C}{\vec {F}}\cdot d{\vec {r}}

This value is independent of the velocity/momentum that the particle travels along the path. For a conservative force field, it is also independent of the path itself, but depends only on the starting and ending points. Therefore, if the starting and ending points are the same, the work is zero for a conservative field:

\oint_C \vec{F} \cdot d\vec{r} = 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:

\vec{F} = \nabla \phi

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 = \phi(b) - \phi(a)

References

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

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Force field (physics)

Examples of force fields

Restriction to position-dependent forces

Work done by a force field

See also

References