Definition:Newtonian Potential
Theorem
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 electric potential 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.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $9$. The Vector Field $\map \grad {k / r}$