Bdtyktl

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?

I think the most general definition of a potential is 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.

The most general definition of potential energy that exists is the following: potential energy is the energy associated with a particular configuration of objects. Electric potential energy is associated with a configuration of charges, gravitational potential energy is associated with a configuration of masses, and elastic potential is associated with a certain mass-spring configuration.

One extracts potential energy from a configuration by changing the positions (or other relevant attributes) of the objects in that configuration. For example, you can extract electric potential energy from a configuration of charges by moving repelling charges further away and moving attracting charges together. You can extract gravitational potential energy from a configuration of masses by moving the masses closer together. Similarly, you can extract elastic potential energy from a mass-spring configuration by moving the mass toward its equilibrium position. If you want to add potential energy to any of these configurations, then you simply do the opposite of the above.

To answer your final question, any differentiable function can, in principle, represent potential energy as a function of configuration. Whether that function describes a physical system that we've observed is a different story.

A potential in the sense of "gravitational potential" or "electric potential" is simply a scaled version of the potential energy. It's a function that describes the potential energy of a system as a function of configuration, divided by some quantity relating to one or more of the objects (for example, the gravitational potential is the gravitational potential energy divided by the mass of one of the objects, and the electric potential is the electric potential energy divided by the charge of one of the objects). It's simply a more convenient way of describing the potential energy of a system when you are constrained to vary (usually) only one of the objects' parameters. As such, it's still just a scaled form of the energy associated with a configuration, so the above reasoning still applies.

A scalar potential whose gradient is the force exists in many branches of physics, for example heat conduction, static magnetism, static electricity, etc. In all these cases the gradient (or negative gradient) of the potential field is the force that can "move" things from here to there.

For example, in static magnetism as described by Ampere's law (steady state case) $textrm{curl}textbf{H}=textbf{J}$ in regions where the current is zero we also have $textrm{curl}textbf{H}=0$ hence a scalar $psi$ such that at least in all singly connected domain we also have $textbf{H}=textrm{grad}psi$. Note the restriction to singly connected domain, it would not be true for a domain containing a toroidal magnet but would still be true locally and when calculated locally the force field is gradient and this force can move things.

Another example is steady state temperature distribution calculated from Fourier's law. We have heat (internal energy and entropy) flow as being proportional to the spatial gradient of temperature. What moves is heat (internal energy + entropy) and the driving force behind that motion is the temperature variation between neighboring points.

Potential energy $U$ can be thought of as "stored".

• Lift an object, and gravitational potential energy is stored (and released when it falls).


• Put an electric charge next to another charge, and electric potential energy is stored (and released as the charge moves away/closer due to the repulsion/attraction).


• Compress a spring, and elastic potential energy is stored (and released as it bounces back to original length).


• Similar for magnetic, chemical and other potential energies.

In all cases we have a "thing" that wants to move. Be it a mass, an electric charge, a spring. When something moves, it can do work. So this "stored" potential energy has a potential to do work. That is usually the classic definition.

Potential energy tells us "how strongly" something wants to move. But note that it is not enough with a high potential energy; you also need a point with a lower potential energy. Otherwise the "thing" wouldn't want to move anyways. A book has no use of moving sideways on the shelf since all points on the shelf give the same potential energy - but it wants to fall because at the floor there is less potential energy. The system wants to release its potential energy. So only a potential energy difference is important.

This description counts for all sorts of potential energies.

What is a potential then? The same as potential energy, just per-quantity. Meaning, per mass, per charge, etc. An electric potential is electric potential energy per coulomb, a gravitational potential is *gravitational potential energy per kilogram** etc. (It doesn't make sense to say elastic potential energy per "quantity" - what would that quantity be? Elongation? This is not easily defined and I have never seen the term elastic potential used.)

Lastly, the equation you mention is indeed true. Force $F$ an potential energy $U$ is defined as:

$$F=frac{dU}{dx}$$

And similarly, force per quantity would equal the differential potential. This formula simply says mathematically what we have described above: That the "thing" wants to move (it feels a force) towards the point of lower potential energy. If there are two point with different potential energies associated with them, then the "thing" feels a force towards the point of lower $U$.

Gradient fields have the convenient property of path invariance. Any line integral taken within that field from point A to point B will have the same value, no matter how strange of a path you might take.

Quite often our laws of physics are well described via integration. For example, in the case of studying planetary motion, we want to integrate a path of an object through a gravitational field. This can be quite mathematically taxing. Worse, you might not know exactly the path one takes through the field.

If your field happens to be a gradient field (as fields like gravitational fields are), then they are path invariant. The energy I gain or lose going from point A to point B does not depend on the path I took, just the endpoints. This means I can substitute a much easier to calculate path and arrive at the correct result.

When you have such a system, you have a potential. If you pick a reference point (such as a global minimum or a global maximum), you can assign that a value of 0. In mathematical terms, this just means adjusting the "C" that comes out of the integration to achieve the desired value. Now you can determine the value of any other point in this space with respect to this zero. This is your "potential." This is how much energy you could gain if you moved from your current position to the reference.

In many cases, the interesting questions can be answered this way. In fact, so many interesting questions can be answered this way that we tend to teach energy potentials first, and only fill in the mathematics of path invariance and gradients later.