 # Equipotential surface

An equipotential or isopotential surface in mathematics and physics refers to a region in space where every point in it is at the same potential. An equipotential surface is defined as a surface on which the potential of a conservative field has a constant value in all points of the field. In other words, when a material object moves within a field always staying on the same equipotential surface, its potential energy remains constant and the field does not perform work on it.

When a body moves inside the field, keeping always on the same equipotential surface, its potential energy remains constant and the field does not work on it.

In a conservative field there are an infinite number of equipotential surfaces, one for each value of the potential: they fill the space and they are disjoint, that is each point of the space always belongs to one and only one equipotential surface.

The concept of equipotential surface for the gravitational field finds application in the principle of communicating vessels: the free surface of a liquid is always an equipotential surface, because this is the configuration that minimizes the overall potential energy of the liquid. In fact, if the potential at the surface of the liquid were not constant, one could always lower the potential energy by moving liquid from a higher potential region to a lower potential region. In a limited region on the Earth’s surface, the gravitational field can be approximated by a constant vector $$\vec{g}$$ pointing downward: in this approximation the equipotential surfaces of the field are horizontal planes, and the principle of communicating vessels can be expressed in its best known form, that is, that the level of water in one or more communicating vessels reaches at all points the same height.

The geoid is, by definition, the equipotential surface of the Earth’s gravitational field at a potential level conventionally defined to correspond to the mean sea level. Since the Earth’s gravitational field is influenced by the irregular disposition of the masses of the oceans and continents, the exact shape of the geoid is very complex; in first approximation it is a rotation ellipsoid with a modest flattening (about 1/300), whose minor axis corresponds to the axis of rotation of the Earth.

Punctually each vector representing a generic force field is tangent to the lines of force themselves and is perpendicular to the equipotential surfaces. Approximating therefore the geoid with a sphere of radius R, the gravitational field can be approximated punctually with a constant vector $$\vec{g}$$ always and only towards the center of mass of the sphere (approximated therefore with the geometric center of the sphere and therefore the vector $$\vec{g}$$ is always and only radial). In this approximation a generic equipotential surface of the gravitational field corresponds to a spherical surface of radius equal to “r”. This approximation is valid for any planet or star of nearly spherical shape. For celestial bodies with irregular shapes or non-uniform clusters the same rule is valid ($$\vec{g}$$ is always and only towards the center of mass), but the definition of an equipotential surface is very complex and in most cases not representable or calculable.