force field (physics)

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 $\mathbf r$.

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. In Newtonian gravity, a particle of mass ''M'' creates a gravitational field $\mathbf g=\frac\hat\mathbf r$, 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.
*An electric field $\mathbf E$ exerts a force on a point charge ''q'', given by $\mathbf F = q\mathbf E$.Calculus: Early Transcendental Functions, by Larson, Hostetler, Edwards, p1055
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*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:

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

See also

* Classical mechanics
* Field line
* Force
* Mechanical work

References

External links

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

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