Force field (physics)

For other uses, see Force field.

Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The inflection points of the cross-section are at the surface of the body.

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 $\mathbf {g} ={\frac {-GM}{r^{2}}}{\hat {\mathbf {r} }}$, 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 $\mathbf {F} =q\mathbf {E}$.
• 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

External links

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