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

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

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.

## See also

- Equipotential surfaceEquipotential surfaceEquipotential surfaces are surfaces of constant scalar potential. They are used to visualize an -dimensional scalar potential function in dimensional space...
- Potential gradientPotential gradientA 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 mapIsopotential mapIsopotential 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 potentialScalar potentialA scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...