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 F {\displaystyle \mathbf {F} }, where F ( r ) {\displaystyle \mathbf {F} (\mathbf {r} )} is the force that a particle would feel if it were at the position r {\displaystyle \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 g = − G M r 2 r ^ {\displaystyle \mathbf {g} ={\frac {-GM}{r^{2}}}{\hat {\mathbf {r} }}}, where the radial unit vector r ^ {\displaystyle {\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 F = m g {\displaystyle \mathbf {F} =m\mathbf {g} }, where g is Earth's gravity.
  • An electric field E {\displaystyle \mathbf {E} } exerts a force on a point charge q, given by F = q E {\displaystyle \mathbf {F} =q\mathbf {E} }.
  • In a magnetic field B {\displaystyle \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: F = q v × B {\displaystyle \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 = ∫ C F ⋅ d r {\displaystyle 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:

∮ C F ⋅ d r = 0 {\displaystyle \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:

F = − ∇ ϕ {\displaystyle \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 = ϕ ( b ) − ϕ ( a ) {\displaystyle W=\phi (b)-\phi (a)}

See also

External links

  • , , University of Texas at Austin