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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 {\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]

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:

{\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.

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:

{\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} =-\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:

{\displaystyle W=\phi (b)-\phi (a)}