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 , where is the force that a particle would feel if it were at the position .Mathematical methods in chemical engineering, by V. G. Jenson and G. V. Jeffreys, p211
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., Chapter 7, page 181 In Newtonian gravity, a particle of mass M creates a gravitational field , where the radial unit vector 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 , where g is Earth's gravity.Vector calculus, by Marsden and Tromba, p288Engineering mechanics, by Kumar, p104 An electric field exerts a force on a point charge q, given by .Calculus: Early Transcendental Functions, by Larson, Hostetler, Edwards, p1055 In a magnetic field , a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation: .
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:
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:
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:
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:
See also Classical mechanics Field line Force Mechanical work
References
External links
Conservative and non-conservative force-fields, Classical Mechanics, University of Texas at Austin