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Force field (physics)

## Summary

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 ${\displaystyle {\vec {F}}}$, where ${\displaystyle {\vec {F}}({\vec {x}})}$ is the force that a particle would feel if it were at the point ${\displaystyle {\vec {x}}}$.[1]

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.

## 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 ${\displaystyle {\vec {g}}={\frac {-GM}{r^{2}}}{\hat {r}}}$ , where the radial unit vector ${\displaystyle {\hat {r}}}$  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 ${\displaystyle {\vec {F}}=m{\vec {g}}}$ , where g is the standard gravity.[3][4]
• An electric field ${\displaystyle {\vec {E}}}$  is a vector field. It exerts a force on a point charge q given by ${\displaystyle {\vec {F}}=q{\vec {E}}}$ .[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

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

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:

${\displaystyle \oint _{C}{\vec {F}}\cdot d{\vec {r}}=0}$

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:

${\displaystyle {\vec {F}}=-\nabla \phi }$

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:

${\displaystyle W=\phi (b)-\phi (a)}$