Force field

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

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 light mass m, close to the surface of Earth is given by \( {\vec {F}}=m{\vec {g}} \) , where g is the standard gravity.[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]

Work

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.
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, 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:

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