# Difference Between Potential and Potential Energy

• B
• warhammer
In summary, the distinction between potential and potential energy is that potential is the potential of a field, while potential energy is the property of a body or system that arises from the field.
Let me take a slightly different approach to answer the question.

Like it or not,
physics is also taught in an algebra-based context (not just calculus-based).

Along the lines of @andrewkirk 's , @sophiecentaur 's, @weirdoguy 's and @Delta2 's early comments...

In an algebra-based course [and higher-level],
"potential energy" is associated with a conservative force (e.g. gravitation, Hooke's Law, Coulomb's Law).
In introductory physics-texts ("B"-level [as opposed to "I"-level),
"potential" almost certainly refers to "electric potential [in units of Volts]"
(since gravitational potential is rarely mentioned, except maybe as an afterthought to the electric potential).
A calculus-based text may make a passing reference to the mathematical notion of a "potential" as a generalization of the "electric potential" concept
[even though "potential" is more mathematically fundamental and "electric potential" is merely a special case].

In the algebra-based context,
one describes "electric force" as an interaction on a particular charge and
"electric potential energy" as a measure of the work done by the conservative force
in reconfiguring a charge distribution in a system.
By contrast [in algebra-based electrostatics],
the "electric field" ##\vec E## and later the "electric potential" ##\Phi## [since ##V## is already taken above for potential energy] are introduced as "fields" setup by the source charge.

I think the "source [charge]" vs "target charge" distinction is important to emphasize.

When the [target] charge is placed at a particular location in space,
then we obtain the "electric force" on that target charge due to the field set up by the sources
$$\vec F_{\mbox{on q_{target}}}=q_{target} E_{\mbox{due to sources}}$$
and the "electric potential energy" of that target charge in that field [i.e. a measure of the work done if that charge were brought from infinity]
$$V_{\mbox{of q_{target}}}=q_{target} \Phi_{\mbox{due to sources}}$$
(We assume that the sources are setup once and for all... and the target charge is a test charge in the field of the sources).

So, the "electric field" and "electric potential" describe a vector field and scalar field set up by the sources.
The "electric force" and "electric potential energy" describe an interaction involving a test charge and the sources [mediated by the fields that produced the sources].
This may be good enough for the algebra-based course.

Many times we have to meet the students where they are [in their preparation].

Yes, there is calculus that relates the "electric field" and the "electric potential",
but calculus is not the explicit route taken in an algebra-based physics course to establish that relationship.

If this feature is that important, then the algebra-based class is not the appropriate class for the student.

Last edited:
nasu
What is the electric potential good for, if there is no relation to the electric field, which is describing an observable phenomenon in Nature? I've no clue, how you connect the potential with the electric field without calculus (or at least using derivatives).

Concerning "algebra-only physics", I've once given a lecture for a colleague in a non-calculus mechanics lecture. Of course, also there they used time derivatives, but it was not allowed to call it so. They just used difference quotients and then (of course without calling it so) took the limit ##\Delta t \rightarrow 0##. I found this very difficult, at least more difficult than using calculus, which is pretty intuitive on this level.

vanhees71 said:
What is the electric potential good for, if there is no relation to the electric field, which is describing an observable phenomenon in Nature? I've no clue, how you connect the potential with the electric field without calculus (or at least using derivatives).

I try to use “slope” and sketches to suggest the relation. But I don’t expect the students in that algebra-based class to evaluate a derivative operation.

Sometimes it does have to be “then a miracle occurs”. (I might suggest that those interested should study a more advanced level for details.)

The point is that the students gets glimpses of what is going on in order to solve simple problems… toward getting to what they need to know (according to those who set up the course sequence and curriculum).

We want to promote the field concept from a point charge…. But also want to evaluate voltage differences around a circuit and energies associated with charging a capacitor. Sure it’s great to fill in all the details (I know I want to… but I know I can’t expect many students to follow it all.)

“Connecting the dots” could be calculus and the operation of evaluating a derivative … but it also could be graphical sketching, or plotting with Desmos, or numerical calculation by hand or by writing a program, or other descriptive words, or analogies with another system.

While I am often the one interested in the details, many students are often more interested in how this is “useful” to their lives or livelihood. That’s just the reality on the ground.

weirdoguy and Delta2
robphy said:
While I am often the one interested in the details, many students are often more interested in how this is “useful” to their lives or livelihood. That’s just the reality on the ground.

Let's not make an organizational question (which amount of science in which classes and a distinction between mandatory and optional) a matter of content! This is the first significant failure in the current systems. Few need integration, but for those who actually need it, the dozens of examples are a waste of time.

I call them disco-accidents. They are reported in the newspapers on Mondays, after the weekend. Young men often brag about their cars or nonexisting experience to drive and end their and the lives of their friends wrapped around a tree. Basic knowledge about physics could prevent a lot of those tragedies. This cannot be mentioned too often.

robphy said:
In an algebra-based course [and higher-level],
"potential energy" is associated with a conservative force (e.g. gravitation, Hooke's Law, Coulomb's Law).

robphy said:
If this feature is that important, then the algebra-based class is not the appropriate class for the student.
The only reason for bringing calculus into things is where the models are non linear. The SUVAT equations all assume uniform acceleration and so does Hooke's law calculation. I look back at my early Physics and still resent the fact that it was all based on linearity; I had to suss that out for myself; so many 'examples' unashamedly involved motor cars with no caveats. I can't have ben the only student who didn't worry about it. (Or perhaps I was regularly looking out of the window whenever it was made clear.)
Triangles are quite good enough until kids have been given calculus.

robphy
sophiecentaur said:
The only reason for bringing calculus into things is where the models are non linear. The SUVAT equations all assume uniform acceleration and so does Hooke's law calculation. I look back at my early Physics and still resent the fact that it was all based on linearity; I had to suss that out for myself; so many 'examples' unashamedly involved motor cars with no caveats. I can't have ben the only student who didn't worry about it. (Or perhaps I was regularly looking out of the window whenever it was made clear.)
Triangles are quite good enough until kids have been given calculus.
This is utter nonsense. You cannot even define velocity and acceleration without using derivatives of the position vector with respect to time. For classical mechanics you need at least linear algebra of the Euclidean affine space and derivatives as well as integrals to start. Even the simple case of constant acceleration (a good model for the motion of a mass point close to Earth, where the force is ##\vec{F}=m \vec{g}## with ##\vec{g}## the gravitational field of the Earth, which can be approximated as constant for motions close to its surface) needs very basic integration to get what you call "SUVAT equations". In many socalled "calculus-free textbooks" they somehow manage it to treat these basic integrals with some tricks without explicitly doing the integrals. It's utterly confusing, and it ends with the sad result that students rote learn these "SUVAT equations" and apply them to all kinds of problems without understanding them. I don't know, how these books treat the somewhat more complicated problem of a force that's linear to displacement as in Hooke's Law. It must be even more confusing.

Last edited:
weirdoguy
vanhees71 said:
robphy said:
The only reason...
This is utter ...
FYI: You misattributed the quote to me.

Delta2
Ugh? How could this happen? I corrected the quote by hand now:
sophiecentaur said:
The only reason for bringing calculus into things is where the models are non linear. The SUVAT equations all assume uniform acceleration and so does Hooke's law calculation. I look back at my early Physics and still resent the fact that it was all based on linearity; I had to suss that out for myself; so many 'examples' unashamedly involved motor cars with no caveats. I can't have ben the only student who didn't worry about it. (Or perhaps I was regularly looking out of the window whenever it was made clear.)

Delta2
In my primary high school years (ages 13,14,15) I did physics without calculus. It wasn't that bad and indeed the worst math you could get was a system of two linear equations with two unknowns. Oh and the only non linear formula I remember from those years was ##s=\frac{1}{2}gt^2##.

But yeah the quality of Greek state education (free education) was below mediocre back then, and I think it is at most mediocre even now. That's why everyone starting at primary high school years and intensifying at secondary high school (ages 16-18) is going to "Frontistirio" and pays in order to learn and understand something. (Google frontistiria in greece).

Last edited:
weirdoguy
vanhees71 said:
This is utter nonsense. You cannot even define velocity and acceleration without using derivatives of the position vector with respect to time.
Funnily enough, the introduction to Differential Calculus would be nearly impossible without starting with straight lines and triangles and deriving the limit of the slope of a curve as the intervals reduce to zero.

By your argument, you shouldn't try to teach any Science to a non-Mathematician. With that attitude, you'd lose some very useful potential Scientists to 'the other side' by the time they got to 16 years of age. There's some brilliant and engaging stuff available on the Arts side and thank god enough clever young people go in that direction. In any case, we're arguing about a false dichotomy (I like that term).

The original P vs PE question has no particular field of application - it's a general thing. P implies Energy and the units will always agree, somewhere in there.
vanhees71 said:
Ugh? How could this happen? I corrected the quote by hand now:
Gremlins in the works! I didn't; bother to fess up - I knew someone would re-point your finger.

nasu
sophiecentaur said:
Funnily enough, the introduction to Differential Calculus would be nearly impossible without starting with straight lines and triangles and deriving the limit of the slope of a curve as the intervals reduce to zero.
Exactly, and what's the merit not to call this a "derivative" and to work out the basic rules, how to work with them? To the contrary to avoid this step makes the physics more complicated rather than simpler as advocated by the anti-math lobby.
sophiecentaur said:
By your argument, you shouldn't try to teach any Science to a non-Mathematician. With that attitude, you'd lose some very useful potential Scientists to 'the other side' by the time they got to 16 years of age. There's some brilliant and engaging stuff available on the Arts side and thank god enough clever young people go in that direction. In any case, we're arguing about a false dichotomy (I like that term).
That's utter nonsense again. You don't need to be a mathematician to understand basic calculus on the level used in physics. My impression is that we loose a lot of very clever people exactly because one tries to make "things simpler than possible" in the STEM area, i.e., particularly the very clever pupils get the impression that these subjects are something ununderstandable and thus build up an interest for other subjects.
sophiecentaur said:
The original P vs PE question has no particular field of application - it's a general thing. P implies Energy and the units will always agree, somewhere in there.
I've still no idea, what the difference should be. Any vector field may have a potential, also forces, and that's the potential going into the total energy of a mechanical system and is thus named "potential energy".
sophiecentaur said:
Gremlins in the works! I didn't; bother to fess up - I knew someone would re-point your finger.
?

weirdoguy
vanhees71 said:
That's utter nonsense again. You don't need to be a mathematician to understand basic calculus on the level used in physics.
Only nonsense if your personal view is that way. Who is a mathematician? Someone who can count his sheep reliably or someone who can do tensor calculus?
I had no calculus until the first year of A level Maths but I knew Suvat for O level. Are you saying I should not have ben taught Suvat at all?
vanhees71 said:
Any vector field may have a potential
What are the units of that potential?

weirdoguy
It has of course the dimension of the vector field times length since ##\vec{V}=-\vec{\nabla} \Phi##.

vanhees71 said:
It has of course the dimension of the vector field times length since ##\vec{V}=-\vec{\nabla} \Phi##.
. . . . . and the Potential Energy? We are trying to establish the relationship between two quantities.
I think the answer must be that Potential Energy will only apply for fields where Work is involved i.e. a subset of Vector Fields.

An energy has the dimension of an energy, what else? Once more: The potential energy in point-particle mechanics is the potential of a force. That's what I said already in my first posting in this thread...

vanhees71 said:
An energy has the dimension of an energy, what else? Once more: The potential energy in point-particle mechanics is the potential of a force. That's what I said already in my first posting in this thread...
I still don't get what you are saying about any distinction between P and PE.
Are you prepared to accept that the (potential) Energy in a charged battery is not the Voltage (i.e. potential), but the Watt hours or that the (potential) energy in a suspended mass in a clock is mgh and not gh? How can those pairs be the same?

Sigh. Is it so difficult to understand what I write? So once more: The potential of the force is the same as potential energy. What else should potential energy be?

A voltage is the difference of a potential of the electromagnetic field. That's not the same as potential energy, which is clear already from the dimensions. In a battery you have no voltage though but an electromotive force, but that's another issue. The energy of a charged battery is in form of internal/chemical energy.

The potential energy in a suspended mass in a clock is given by the potential of the gravitational force on this mass, which is ##V=m g z## (with the ##z## axis pointing "upwards", i.e., against ##\vec{g}=-g \vec{e}_z##), while ##\Phi=g z## is the potential of the gravitational field. The force is ##\vec{F}=-\vec{\nabla} V=-m g \vec{e}_z=m \vec{g}## and the gravitational field is ##\vec{g}=-\vec{\nabla} \Phi=-g \vec{e}_z=\vec{g}##. Of course, ##V=m \Phi##.

I still don't understand what your point really is!

vanhees71 said:
The potential energy in a suspended mass in a clock is given by the potential of the gravitational force on this mass, which is V=mgz (with the z axis pointing "upwards", i.e., against G→), while Φ=gz is the potential of the gravitational field.
"Sigh" are you really saying anything different from me?You say that the Potential Energy depends on the mass and that the Potential is independent of mass. That's what I am saying. What are you arguing about? How did I say it wrong?

weirdoguy
I'm also puzzled, what we discuss about. The only point is that you keep saying "the potential". You have to state which potential you mean, that of the force or the gravitational field. They differ by a factor of ##m## in this example.

The only time I use the term "Potential" is when referring to a unit mass / charge. I think I've been consistent over this thread at least. When using the term "Field", I assume that it's force per unit Mass / Charge.
In " the Sun's Gravitational Potential Well" diagrams, I wouldn't expect the y-axis to be different, depending on what mass of planet we're discussing.

Well, it would have helped to clearly indicate the potential of what vector field you are talking. That's all I'm saying.

weirdoguy and sophiecentaur

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