Potential

in the broad sense, the means, reserves, and sources that are available and capable of being mobilized, set in action, or used to attain a certain goal, carry out a plan, or solve some problem; the possibilities open to an individual, society, or state in a field, for example, economic potential or production potential.

(also potential function), a concept characteristic of a broad class of physical force fields, such as electric and gravitational fields, and, in general, fields of any physical quantities representable by vectors, such as the velocity field in a fluid. For an electrostatic field, the potential is introduced as an auxiliary function whose spatial derivatives are the components of the electric field intensity at a given point. In hydrodynamics, they are the components of the velocity at a given point. In many cases, the potential also has another important physical meaning. Thus, in an electrostatic field it is numerically equal to the negative of the energy necessary to remove a unit positive charge from a given point to infinity.

In the general case of the potential of a vector field a(x, y, z), there is a scalar function u(x, y, z) such that a = grad u; that is, a_x = ∂u/∂x, a_y = ∂u/∂y, and a_z = ∂u/∂z, where a_x, a_y, and a_z are the components of field a in the Cartesian coordinate system Oxyz. If such a function can be introduced, then vector field a is called a potential field. In electrostatics, the potential can also be designated by the function U = — u. The potential of a vector field a is not determined unambiguously but only to the accuracy offactor. Therefore, only differences in potential between various points in the field are of interest in the study of a potential field. Geometrically, the equation u(x, y, z) = c represents a surface at all points of which the potential has the same value. Such surfaces are called equipotential surfaces.

For a gravitational field formed by a point mass m placed at point A(ξ, η, ξ), the potential, here called the Newtonian potential, at point P(x, y, z) has the form

(1) u(x, y, z) = Gm/r

where and G is the gravitational constant. When fields are superposed on one another, their potentials are algebraically added. If a gravitational field is set up by some mass of density p(ξ, η, ξ) occupying a volume T, then it may be considered as a result of the superposition of the elementary fields formed by infinitesimally small bodies of mass ρdξdηdξ. The Newtonian potential of such a field is given by the integral

The potential u(x, y, z), together with its partial derivatives of the first order, is a continuous function throughout all space. Outside the body of volume T, the function u(x, y, z) satisfies the Laplace equation and, within the body, the Poisson equation.

If the attracting masses are distributed with a density p_sur over a surface S as a single layer, then the potential of the field formed by them is given by the integral

The potential of the single layer v(x, y, z) is a continuous function throughout all space outside of S. When the surface S is crossed, however, the normal derivative of the function v(x, y, z) experiences a discontinuity equal to 4πG/p_sur. By bringing together without limit two surfaces on which single layers with densities ρ_sur and —ρ_sur are located and by simultaneously increasing p_sur to infinity in such a way that the limit is lim np_sur = μ, where η is the perpendicular distance between the surfaces, we reach the concept of the potential of a double layer:

The potential of the double layer w(x, y, z) is a continuous function throughout all space outside of S. When the surface S is crossed, the function w(x, y, z) experiences a discontinuity, which is equal to 4πμ. The functions v(x, y, z) and w(x, y, z) both satisfy the Laplace equation.

If a body of volume T is an infinite cylinder of cross section D and if the density ρ of the matter in the cylinder is constant along every line parallel to the generatrices of the cylinder, then equation (2) leads to the concept of the logarithmic potential:

(5) u(x,y) = G ∫∫_D ρ In (1/r) ds

Any harmonic function can be represented as the sum of the potentials of a single and double layer; this helps to explain the importance attached to potential theory.