mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, an equipotential or isopotential refers to a

region In geography, regions, otherwise referred to as areas, zones, lands or territories, are portions of the Earth's surface that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and ...

in space where every point is at the same

potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...

. This usually refers to a

scalar potential In mathematical physics, scalar potential describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one p ...

(in that case it is a

level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~. When the number of independent variables is two, a level set is call ...

of the potential), although it can also be applied to

vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field \mathbf, a ' ...

s. An equipotential of a scalar potential function in -

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

al space is typically an ()-dimensional space. The

del operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...

illustrates the relationship between a vector field and its associated scalar potential field. An equipotential region might be referred as being 'of equipotential' or simply be called 'an equipotential'. An equipotential region of a scalar potential in three-dimensional space is often an equipotential surface (or ''potential isosurface''), but it can also be a three-dimensional mathematical solid in space. The

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

of the scalar potential (and hence also its opposite, as in the case of a vector field with an associated potential field) is everywhere perpendicular to the equipotential surface, and zero inside a three-dimensional equipotential region.

Types

Electrical conductor In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. The flow of negatively c ...

s offer an intuitive example. If ''a'' and ''b'' are any two points within or at the surface of a given conductor, and given there is no flow of charge being exchanged between the two points, then the

potential difference Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge ...

is zero between the two points. Thus, an equipotential would contain both points ''a'' and ''b'' as they have the same

. Extending this definition, an isopotential is the locus of all points that are of the same potential. In

electrostatics Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...

and steady

electric current An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...

s, the

electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...

(and hence the current, if any) is perpendicular to the equipotential surfaces of the

electric potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...

(

voltage Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...

). In electrostatics, a conductor is a three-dimensional equipotential region. In the case of a hollow conductor (

Faraday cage A Faraday cage or Faraday shield is an enclosure used to block some electromagnetic fields. A Faraday shield may be formed by a continuous covering of conductive material, or in the case of a Faraday cage, by a mesh of such materials. Faraday cag ...

), the equipotential region includes the space inside.

Gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...

is perpendicular to the equipotential surfaces of the

gravity potential In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the co ...

. In gravity, a hollow sphere has a three-dimensional equipotential region inside, with no gravity from the sphere (see

shell theorem In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion a ...

). A ball will not be accelerated left or right by the force of gravity if it is resting on a flat, horizontal surface, because it is an equipotential surface. For the

gravity of Earth The gravity of Earth, denoted by , is the net force, net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a Eucl ...

(including its centrifugal potential owing to Earth's rotation), the corresponding

geopotential Geopotential (symbol ''W'') is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the of the potential energy per unit mass, so that the gravity vect ...

isosurface (the ''

equigeopotential Geopotential (symbol ''W'') is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the of the potential energy per unit mass, so that the gravity vector ...

'') that best fits

mean sea level A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

is called the ''

geoid The geoid ( ) is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is exte ...

''.

References

External links

Electric Field Applet
{{Authority control Multivariable calculus Mathematical physics Potentials de:Äquipotentialfläche

Types

See also

References

External links