Equipotential
Encyclopedia
Equipotential or isopotential in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and 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...

refers to a region in space where every point in it is at the same potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

. This usually refers to a scalar potential
Scalar potential
A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...

(in that case it is a level set
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....

of the potential), although it can also be applied to vector potential
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 negative gradient is a given vector field....

s. An equipotential of a scalar potential function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

in n-dimension
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...

al space is typically an (n−1)dimensional space. The del operator
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...

illustrates the relationship between a vector field and its associated scalar potential field.

Note that 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
Equipotential surface
Equipotential surfaces are surfaces of constant scalar potential. They are used to visualize an -dimensional scalar potential function in dimensional space...

, but it can also be a three-dimensional region in space. The gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

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.

Electrical conductor
Electrical conductor
In physics and electrical engineering, a conductor is a material which contains movable electric charges. In metallic conductors such as copper or aluminum, the movable charged particles are electrons...

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 is zero between the two points. Thus, an equipotential would contain both points a and b as they have the same potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

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

Gravity is perpendicular to the equipotential surfaces of the gravity potential, and in electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....

and in the case of steady currents the 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...

(and hence the electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

, if any) is perpendicular to the equipotential surfaces of the electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

(voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

).

In gravity, a hollow sphere has a three-dimensional equipotential region inside, with no gravity (see shell theorem
Shell theorem
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body...

). 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 formed by conducting material or by a mesh of such material. Such an enclosure blocks out external static and non-static electric fields...

), the equipotential region includes the space inside.

A ball will not be accelerated by the force of gravity if it is resting on a flat, horizontal
Horizontal plane
In geometry, physics, astronomy, geography, and related sciences, a plane is said to be horizontal at a given point if it is perpendicular to the gradient of the gravity field at that point— in other words, if apparent gravity makes a plumb bob hang perpendicular to the plane at that point.In...

surface, because it is an equipotential surface.

• Equipotential surface
Equipotential surface
Equipotential surfaces are surfaces of constant scalar potential. They are used to visualize an -dimensional scalar potential function in dimensional space...

A potential gradient is the local space rate of change of the potential with respect to displacement.In electrostatics then, it is the local space rate of change of the electric potential:Units are volts per meter...

• Isopotential map
Isopotential map
Isopotential maps are a measure of electrostatic potential in space. The spatial derivatives of an electrostatic field indicate the contours of the electrostatic field, so isopotential maps show where another charged molecule might interact, using equipotential lines ....

• Scalar potential
Scalar potential
A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...