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# Equipotentials - Constant Potential in an Electric Field

An animation showing the lines of constant potential in an electric field between two charged metal plates.

Equipotential or isopotential in mathematics and physics refers to a region in space where every point in it is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can also be applied to vector potentials. An equipotential of a scalar potential function in n-dimensional space is typically an (n−1)dimensional space. The del operator 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, but it can also be a three-dimensional region in space. The gradient 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 conductors 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. 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 and in the case of steady currents the electric field (and hence the electric current, if any) is perpendicular to the equipotential surfaces of the electric potential (voltage).

In gravity, a hollow sphere has a three-dimensional equipotential region inside, with no gravity (see shell theorem). In electrostatics a conductor is a three-dimensional equipotential region. In the case of a hollow conductor, 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 surface, because it is an equipotential surface.

All points on an equipotential surface have the same electric potential (i.e. the same voltage).

The electric force neither helps nor hinders motion of an electric charge along an equipotential surface.

Electric field lines are always perpendicular to an equipotential surface.

Electric potential is analogous to altitude; one can make maps of each in very similar ways.

The electric field strength can be determined by looking at the local spatial gradient of the electric potential:

we define an equipotential surface as one that joins points of equal potential in the field; in other words, no work is done in moving a charge on an equipotential surface. Although our discussion and definition of potential will be almost exactly the same as in the gravitational case, we will leave that to a later episode. For now, we just need to know that:

Equipotential surfaces are perpendicular to field lines.

Any electrical conductor is an equipotential surface.