In many part of physics, me talk about potential (electrical potential, gravitation potential, elastic potential…). All those definition looks very different, and I would like to know how all those quantity are related. The mathematical definition of a potential is : $F$ is a potential if $F=\nabla f$ for some scalar field $f$. But I don't understand how I can relate this to all potential that exist in physic. Also, many people relate potential and energy, what doe it mean exactly? Because if a potential $U$ is something that can be written as a gradient, every integrable function (at least in $\mathbb R$) could be a gradient. So what the thing with these potential?

# [Physics] Could someone explain what is a potential

definitionpotentialpotential energy

#### Related Solutions

In physics *voltage* is usually defined as *potential difference between two points of a circuit*:
$$V_{ab}=\phi_a-\phi_b.$$
In other words:

*potential*is a potential energy per unit charge measured in respect to an agreed reference point (e.g., an infinitely remote point), i.e. work done when bringing the charge from this point*voltage*is the work done when moving a charge between the two specified points.

One typically has the same reference point for all the potentials, but voltages can be defined between any two points.

Finally, potential is an extremely general concept, used to describe electric and electromagnetic phenomena well beyond circuit theory, whereas *voltage* is a term mainly limited to specific applications.

**Update**

The question has been further discussed in the comments, and it was agreed that much of my answer indeed reiterates the information already given in the OP. I therefore state below the main point, as it came out of the discussion:

One can say that potential is a voltage between the point of interest and the reference point. I think the real difference is indeed the usage: one will rarely speak of voltage in atomic or nuclear physics, but in case of semicodnuctor nanostructures the term is rather used. Some may say that potential is "more scientific term", as opposed to engineering.

We don’t measure $\Phi$. We measure $\mathbf{E}$.

Imagine you are doing an experiment inside of a cavity in a conductor. There’s no field inside of your conductor, so all of your electric fields in your experiment are generated locally. (A practical implementation of this is a “Faraday cage.”)

Now, somebody outside of your experiment charges up your conductor. All of the excess charge moves to the exterior surface of the conductor; it has zero effect on the fields in your cavity. Your experiment proceeds in exactly the same way, because $\mathbf{E}$ is the same, even though $\Phi$ has changed.

It’s a useful convention to define $\Phi$ so that $\Phi\to 0$ in the limit where you are very far from any nonzero charge distribution. But for describing dynamics, we don’t care about that convention. We only care about $\nabla\Phi$.

## Best Answer

I think the most general definition of a

potentialis some quantity which, when differentiated in a certain way, yields some other quantity which we're interested in.Other answers have already given examples of potentials which, when differentiated with respect to position, yield a force (i.e. $\vec F = -\nabla U$, where the minus sign is just conventional). Such objects are called

potential energies, and are special because they contribute to the total energy of a system.Other examples of potentials are the scalar and vector potentials $\phi$ and $\vec A$ from electromagnetism. In electrostatics, we define $\phi$ such that $\vec E =-\nabla \phi$, and in electrodynamics we define $\vec B = \nabla \times \vec A$. Again, we see that we obtain physical quantities (in this case $\vec E$ and $\vec B$) by differentiating the potentials ($\phi$ and $\vec A$). Note that these potentials don't automatically correspond to energies - as it turns out $\phi$ can be interpreted as the electrostatic potential energy per unit charge (in electrostatics, at least), but the same is not true for $\vec A$.

We can also consider the thermodynamic potentials, which include the internal energy $U$, the Helmholtz potential $F$, the enthalpy $H$, and the Gibbs potential $G$. Each of these can be interpreted as a kind of energy under certain conditions, but we refer to them as potentials because when we differentiate them with respect to different variables, we get other thermodynamic quantities like pressure, temperature, and volume: $$ p = -\left(\frac{\partial U}{\partial V}\right)_{S,N} = - \left(\frac{\partial F}{\partial V}\right)_{T,N}$$ $$T = \left(\frac{\partial U}{\partial S}\right)_{V,N} = \left(\frac{\partial H}{\partial S}\right)_{p,N}$$ $$V = \left(\frac{\partial H}{\partial p}\right)_{S,N} = \left(\frac{\partial G}{\partial p}\right)_{T,N}$$

so on and so forth.

As a final example that has nothing to do with energy, consider the velocity potential $\Phi$ which is used when dealing with irrotational fluid flows. The flow velocity $\vec u$ of the fluid is given by $\vec u = \nabla \Phi$; this can simplify the Navier-Stokes equations, and is analogous to the use of the magnetic scalar potential in magnetostatics.

You may be asking yourself why we ever use potentials rather than computing the quantities we're interested in directly; the answer is that often times, the math works out in such a way that the potentials are substantially easier to calculate. For instance, potential energies are scalars while forces are vectors; the vector potential $\vec A$ obeys a simpler differential equation than $\vec B$ because the equations for the various components can be decoupled from one another; the thermodynamic potentials can be obtained from the partition function and various straightforward Legendre transformations.