Force field (physics)

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 ${\displaystyle \mathbf {F} }$, where ${\displaystyle \mathbf {F} (\mathbf {r} )}$ is the force that a particle would feel if it were at the position ${\displaystyle \mathbf {r} }$.^[1]

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 ${\displaystyle \mathbf {g} ={\frac {-GM}{r^{2}}}{\hat {\mathbf {r} }}}$, where the radial unit vector ${\displaystyle {\hat {\mathbf {r} }}}$ 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 ${\displaystyle \mathbf {F} =m\mathbf {g} }$, where g is Earth's gravity.^[3][4]
• An electric field ${\displaystyle \mathbf {E} }$ exerts a force on a point charge q, given by ${\displaystyle \mathbf {F} =q\mathbf {E} }$.^[5]
• In a magnetic field ${\displaystyle \mathbf {B} }$, a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation: ${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$.

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

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:

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

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

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:

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

See also

• Classical mechanics
• Field line
• Force
• Mechanical work

References

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

External links

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