# Field of Force

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## Field of Force

a bounded or unbounded region of space at every point of which a force of a certain magnitude and direction acts on a material particle placed at the point. The force depends only on the coordinates *x*, *y*, *z* of the point or on the coordinates *x, y, z* and the time *t*. In the first case the field is said to be stationary, and in the second case nonstationary. If the force has the same value at all points of the field—that is, is independent of the coordinates and time—then the field is said to be uniform.

A field of force is said to be a potential field if the work done by the forces of the field on a material particle moving in the field depends only on the initial and final positions of the particle and is independent of the particle’s trajectory. This work can be expressed in terms of the potential energy of the particle Π (*x, y, z*) by the equation *A* = Π(*x*_{1}, *y*_{1}*z*_{1}) – Π(*x*_{2}, *y*_{2}, *z*_{2}), where *x*_{1}, *y*_{1}, *z*_{1}, and *x*_{2}, *y*_{2}, *z*_{2} are the coordinates of the initial and final positions of the particle, respectively. When the particle moves in a potential field under the action of only the field forces, the law of conservation of mechanical energy holds; this law permits establishment of the relation between the particle’s velocity and position in the field of force.

An example of a potential field is the uniform field of the force of gravity, for which Π = *mgz*, where *m* is the mass of the particle and *g* is the acceleration of the force of gravity (the z-axis is directed vertically upward). A Newtonian gravitational field is also an example of a potential field. Here, Π = – *fm*/*r*, where *r* is the distance of the particle from the center of attraction and *f* is a constant for the given field.

S. M. TARG