Force field (physics) explained

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

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

Examples

g= (-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 F = m g , where g is Earth's gravity.^{[3] [4]} E exerts a force on a point charge q, given by F = q E .^[5] 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: F = q v x 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} F ⋅ d 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} F ⋅ d 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: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)

See also

• Classical mechanics
• Field line
• Force
• Mechanical work

External links

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

Notes and References

1. https://books.google.com/books?id=akbi_iLSMa4C&pg=PA211 Mathematical methods in chemical engineering, by V. G. Jenson and G. V. Jeffreys, p211
2. Book: General relativity from A to B. Robert. Geroch. University of Chicago Press. 1981. 0-226-28864-1. 181., Chapter 7, page 181
3. https://books.google.com/books?id=LiRLJf2m_dwC&pg=PA288 Vector calculus, by Marsden and Tromba, p288
4. https://books.google.com/books?id=bCP68dm49OkC&pg=PA104 Engineering mechanics, by Kumar, p104
5. https://books.google.com/books?id=9ue4xAjkU2oC&pg=PA1055 Calculus: Early Transcendental Functions, by Larson, Hostetler, Edwards, p1055