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 $$\vec{F}$$, where $$\vec{F}(\vec{x})$$ is the force that a particle would feel if it were at the point $$\vec{x}$$.

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 $$\vec{g}=\frac{-G M}{r^2}\hat{r}$$, where the radial unit vector $$\hat{r}$$ 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 $$\vec{F} = m \vec{g}$$, where g is Earth's gravity.
 * An electric field $$\vec{E}$$ exerts a force on a point charge q, given by $$\vec{F} = q\vec{E}$$.
 * In a magnetic field $$\vec{B}$$, a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation: $$\vec{F} = q\vec{v}\times\vec{B}$$.

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:
 * $$ W = \int_C \vec{F} \cdot d\vec{r}$$

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:


 * $$ \oint_C \vec{F} \cdot d\vec{r} = 0$$

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:


 * $$ \vec{F} = -\nabla \phi$$

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:


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