Force field (physics)

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.{{cite book
• An electric field \mathbf E exerts a force on a point charge q, given by \mathbf F = q\mathbf E.
• 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

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. [https://books.google.com/books?id=LiRLJf2m_dwC&pg=PA288 Vector calculus, by Marsden and Tromba, p288]
3. [https://books.google.com/books?id=bCP68dm49OkC&pg=PA104 Engineering mechanics, by Kumar, p104]
4. [https://books.google.com/books?id=9ue4xAjkU2oC&pg=PA1055 Calculus: Early Transcendental Functions, by Larson, Hostetler, Edwards, p1055]