From the Wikipedia:

In physics, a potential may refer to the scalar potential or to the vector potential. In either case, it is a field defined in space, from which many important physical properties may be derived.

  • Leading examples are the gravitational potential and the electric potential, from which the motion of gravitating or electrically charged bodies may be obtained.
  • Specific forces have associated potentials, including the Coulomb potential, the van der Waals potential, the Lennard-Jones potential and the Yukawa potential.

It seems there should be a more specific physics definition of a potential. For example is it always attributed to a field? The voltage across a resistor is called the potential difference, and it does not seem to be a field. Does it have any relationship to potential energy? What attributes must something have to be called a potential? Do all forces have an associated potential? Are all potentials associated with a force?

Maybe the second sentence from the wiki quote is as good as it gets.

My question is: What is a basic physics general definition of a 'potential'? Is there a 'formal' definition?

  • 1
    $\begingroup$ "Potential" means "energy per {something}". $\endgroup$
    – DanielSank
    Apr 5 '18 at 18:50

Physicists use the word "potential" in different ways in different contexts, so there is no completely rigorous and general definition. There is a unifying idea, but unfortunately it's abstract enough that you probably won't understand all the jargon and concepts without some advanced physics training.

The general idea is this: consider a physical degree of freedom $x$, whose space of possible values forms some manifold $M$. Then a "potential" for $x$ is a field $V(x)$ defined on $M$ such that some kind of first derivative $V'(x_0)$ tells you how $x$ gets "pushed" if it takes on the value $x_0 \in M$.

When you first learn about potentials, it's almost always the case that the degree of freedom is the position ${\bf x}$ of a classical point particle, the manifold $M$ is physical space $\mathbb{R}^3$, the potential $V({\bf x})$ is a scalar potential energy field, and the "some kind of first derivative" is the negative gradient operator $-{\bf \nabla}$. In this case the potential $V(x)$ is simply a convenient way of encoding a position-dependent conservative force field ${\bf F}({\bf x}) = -{\bf \nabla}V({\bf x})$.

But as you get to more advanced applications, any of these can be generalized. To give a few examples:

  1. Rather than a single particle's position ${\bf x}$, the physical degree of freedom can be a field $\varphi(x)$.
  2. Rather than physical space $\mathbb{R}^3$, the manifold $M$ can be a proper subset, e.g. for a particle confined to the surface of a sphere. Even more abstractly, it doesn't need to be any kind of physical space at all; it field theory, the manifold is the set of values that the field can take on, which (regardless of the number of spatial dimensions), could be $\mathbb{R}$ for a scalar field, $\mathbb{R}^n$ for a vector field, or an even more abstract "spinor space".
  3. Rather than a scalar field, the potential $V(x)$ could be a vector field (as in the case of magnetism).
  4. Rather than the (negative) gradient operator, the "some kind of first derivative" could be the (negative) ordinary derivative (as in scalar field theory), or the vector curl (as in magnetism).
  5. Rather than a force, the "push" could be a force normalized by some suitable physical property of the degree of freedom, as in the electric field (force per unit charge) or gravitational field (force per unit mass, i.e. acceleration). More abstractly, it could be the generalized force that appears in the Euler-Lagrange equation in the Lagrangian formalism for particles, or the even more abstract one that appears in the Lagrangian formalism for fields.

Maybe the use of the word “potential” in physics could be misleading, because I think it has the following origin:

  • the “potential energy” is the part of the energy depending only on coordinates, and not on on momenta (or velocities, or derivatives); in this sense the electric scalar potential is almost a potential energy, unless to multiplicate it by the charge (for an elementary charge)
  • the electric scalar potential is also a “primitive” of the electric field, and in this sense the vector potential is a potential: a primitive of the magnetic field; it also is in some sense a potential energy, but in a more complicated fashion

I suppose that the answer you’re searching is that a potential is a primitive of a field, where the field is a physical observable quantity function of space-time point. But the examples you cite are “first type” type examples:

Specific forces have associated potentials, including the Coulomb potential, the van der Waals potential, the Lennard-Jones potential and the Yukawa potential.

they are the energy part that depends on positions, and that generates “the forces” (so they are potential energies), while scalar and vector potentials are primitives of a field (the EM one).


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