Force field (physics)

# Force field (physics)

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In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

a force field is a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

that describes a non-contact force
Non-contact force
A non-contact force is a force applied to an object by another body that is not in direct contact with it. The most familiar example of a non-contact force is gravity. In contrast a contact force is a force applied to a body by another body that is in contact with it...

acting on a particle at various positions in space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

. Specifically, a force field is a vector field , where is the force that a particle would feel if it were at the point ..

## Examples of force fields

• In Newtonian gravity, a particle of mass M creates a gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

, where the radial unit vector points away from the particle. The gravitational force experienced by a particle of mass m is given by . This field is the force field for gravity.
• An electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

is a vector field. It exerts a force on a point charge q given by . This field is the electric force field.

## Restriction to position-dependent forces

Some forces, including friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

, air drag, and the magnetic force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

on a charged particle, depend on the particle's velocity as well as its position. Therefore these forces are not characterized by a force field.

## Work done by a force field

As a particle moves through a force field along a path C, the work done by the force is a line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

This value is independent of the velocity that the particle travels along the path. For a conservative force field
Conservative force
A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done by a conservative force is zero.It is possible to define a numerical value of...

, it is also independent of the path itself, but depends only on the starting and ending points. Therefore, if the starting and ending points are the same, the work is zero for a conservative force field.