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The force can be written as the negative gradient of a potential, : =. Proof that these three conditions are equivalent when F is a force field Main article: Conservative vector field
The simplest definition for a potential gradient F in one dimension is the following: [1] = = where ϕ(x) is some type of scalar potential and x is displacement (not distance) in the x direction, the subscripts label two different positions x 1, x 2, and potentials at those points, ϕ 1 = ϕ(x 1), ϕ 2 = ϕ(x 2).
These equations model the internucleon potential energies, or potentials. (Generally, forces within a system of particles can be more simply modelled by describing the system's potential energy; the negative gradient of a potential is equal to the vector force.) The constants for the equations are phenomenological, that is, determined by ...
The potential has units of energy per mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero. The gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential ...
In this case, the force can be defined as the negative of the vector gradient of the potential field. For example, gravity is a conservative force . The associated potential is the gravitational potential , often denoted by ϕ {\displaystyle \phi } or V {\displaystyle V} , corresponding to the energy per unit mass as a function of position.
The effective force, then, is the negative gradient of the effective potential: = = ^ where ^ denotes a unit vector in the radial direction. Important properties [ edit ]
This makes it relatively easy to break complex problems down into simple parts and add their potentials. Taking the definition of φ backwards, we see that the electric field is just the negative gradient (the del operator) of the potential. Or: = ().
This means that gravitational potential energy on a contour map is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity.