Physics:Force field
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 [math]\displaystyle{ \vec{F} }[/math], where [math]\displaystyle{ \vec{F}(\vec{x}) }[/math] is the force that a particle would feel if it were at the point [math]\displaystyle{ \vec{x} }[/math].[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 [math]\displaystyle{ \vec{g}=\frac{-G M}{r^2}\hat{r} }[/math], where the radial unit vector [math]\displaystyle{ \hat{r} }[/math] 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 [math]\displaystyle{ \vec{F} = m \vec{g} }[/math], where g is Earth's gravity.[3][4]
- An electric field [math]\displaystyle{ \vec{E} }[/math] exerts a force on a point charge q, given by [math]\displaystyle{ \vec{F} = q\vec{E} }[/math].[5]
- In a magnetic field [math]\displaystyle{ \vec{B} }[/math], a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation: [math]\displaystyle{ \vec{F} = q\vec{v}\times\vec{B} }[/math].
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:
- [math]\displaystyle{ W = \int_C \vec{F} \cdot d\vec{r} }[/math]
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:
- [math]\displaystyle{ \oint_C \vec{F} \cdot d\vec{r} = 0 }[/math]
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:
- [math]\displaystyle{ \vec{F} = -\nabla \phi }[/math]
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:
- [math]\displaystyle{ W = \phi(b) - \phi(a) }[/math]
See also
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. https://books.google.com/books?id=UkxPpqHs0RkC&pg=PA181., 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
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