# Definition:Newtonian Potential

## Theorem

Let $R$ be a region of space.

Let $S$ be a scalar field over $R$ such that:

$\forall \mathbf r \in R: \map S {\mathbf r} = \dfrac k r$

where:

$\mathbf r$ is the position vector of an arbitrary point in $R$ with respect to the origin
$r = \norm {\mathbf r}$ is the magnitude of $\mathbf r$
$k$ is some predetermined constant.

Then $S$ is known as a Newtonian potential.

## Examples

### Gravitational Field

The gravitational field arising from a point mass is an instance of a Newtonian potential.

### Electric Charge

The electrostatic field arising from a point charge is an instance of a Newtonian potential.

### Point Sink of Fluid Flow

A point sink in a body of incompressible fluid acts as a Newtonian potential.

## Also see

• Results about Newtonian potentials can be found here.

## Source of Name

This entry was named for Isaac Newton.

## Historical Note

The concept of a Newtonian potential arose from the work of Isaac Newton, who first formulated what is now known as Newton's Law of Universal Gravitation.

Such a scalar field is the scalar potential of a conservative vector field whose properties are exactly those of the gravitational field given rise to by a point mass.

Because such a field arises in a number of different contexts in physics, the concept of a Newtonian potential was abstracted from this, and made general.