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

Examples of force fields

• In Newtonian gravity, a particle of mass M creates a gravitational field $\vec{g}=\frac{-G M}{r^2}\hat{r}$, 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 $\vec{F} = m \vec{g}$. This field $\vec{F}$ is the force field for gravity.
• An electric field $\vec{E}$ is a vector field. It exerts a force on a point charge q given by $\vec{F} = q\vec{E}$. This field $\vec{F}$ is the electric force field.

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 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 force field.

See also

• Field line
• Force