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

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:

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:

\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:

\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:

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

References

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

External links

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

[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

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