Force field (physics)

For other uses, see Force field.

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 corresponding with a non-contact force acting on a particle at various positions in space. Specifically, a force field is a vector field <math>\vec{F}</math>, where <math>\vec{F}(\vec{x})</math> is the force that a particle would feel if it were at the point <math>\vec{x}</math>.<ref>Mathematical methods in chemical engineering, by V. G. Jenson and G. V. Jeffreys, p211</ref>

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.<ref>Geroch, Robert (1981). General relativity from A to B. University of Chicago Press. p. 181. ISBN 0-226-28864-1., Chapter 7, page 181 </ref> In Newtonian gravity, a particle of mass M creates a gravitational field <math>\vec{g}=\frac{-G M}{r^2}\hat{r}</math>, where the radial unit vector <math>\hat{r}</math> 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 <math>\vec{F} = m \vec{g}</math>, where g is Earth's gravity.<ref>Vector calculus, by Marsden and Tromba, p288</ref><ref>Engineering mechanics, by Kumar, p104</ref>
• An electric field <math>\vec{E}</math> exerts a force on a point charge q, given by <math>\vec{F} = q\vec{E}</math>.<ref>Calculus: Early Transcendental Functions, by Larson, Hostetler, Edwards, p1055</ref>
• In a magnetic field <math>\vec{B}</math>, a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation: <math>\vec{F} = q\vec{v}\times\vec{B}</math>.

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:

<math> W = \int_C \vec{F} \cdot d\vec{r}</math>

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:

<math> \oint_C \vec{F} \cdot d\vec{r} = 0</math>

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:

<math> \vec{F} = -\nabla \phi</math>

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:

<math> W = \phi(b) - \phi(a) </math>

See also

• Classical mechanics
• Field line
• Force
• Mechanical work

External links

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