Force field (physics)

• 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.^[2] In Newtonian gravity, a particle of mass M creates a gravitational field ${\displaystyle \mathbf {g} ={\frac {-GM}{r^{2}}}{\hat {\mathbf {r} }}}$ , where the radial unit vector ${\displaystyle {\hat {\mathbf {r} }}}$  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 ${\displaystyle \mathbf {F} =m\mathbf {g} }$ , where g is Earth's gravity.^[3][4]
• An electric field ${\displaystyle \mathbf {E} }$  exerts a force on a point charge q, given by ${\displaystyle \mathbf {F} =q\mathbf {E} }$ .^[5]
• In a magnetic 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 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:

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

• Classical mechanics
• Field line
• Force
• Mechanical work

 

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