Inphysics, aforce field is avector field corresponding with anon-contact force acting on a particle at various positions inspace. Specifically, a force field is a vector field${\displaystyle \mathbf{F}}$, where${\displaystyle \mathbf{F}(\mathbf{r})}$ is the force that a particle would feel if it were at the position${\displaystyle \mathbf{r}}$.^[1]

• 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 ofenergy) extends into the space around itself.^[2] InNewtonian gravity, a particle of massM creates agravitational field${\displaystyle \mathbf{g}=\frac{-GM}{r^2}\hat{\mathbf{r}}}$, where the radialunit vector${\displaystyle \hat{\mathbf{r}}}$ points away from the particle. The gravitational force experienced by a particle of light massm, close to the surface ofEarth is given by${\displaystyle \mathbf{F}=m\mathbf{g}}$, whereg isEarth's gravity.^[3][4]
• Anelectric field${\displaystyle \mathbf{E}}$ exerts a force on apoint chargeq, given by${\displaystyle \mathbf{F}=q\mathbf{E}}$.^[5]
• In amagnetic field${\displaystyle \mathbf{B}}$, a point charge moving through it experiences a force perpendicular to its own velocity and to the direction of the field, following the relation:${\displaystyle \mathbf{F}=q\mathbf{v}\times \mathbf{B}}$.

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 pathC, thework done by the force is aline integral:$${\displaystyle W={\int }_C\mathbf{F}\cdot d\mathbf{r}}$$

This value is independent of thevelocity/momentum that the particle travels along the path.

For aconservative 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:

$${\displaystyle {\oint }_C\mathbf{F}\cdot d\mathbf{r}=0}$$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:

$${\displaystyle \mathbf{F}=-\mathrm{\nabla }\varphi }$$

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 byx =a andx =b, respectively:

$${\displaystyle W=\varphi (b)-\varphi (a)}$$

• Classical mechanics
• Field line
• Force
• Mechanical work

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

• Force

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