B Difference Between Potential and Potential Energy

[warhammer]

In generic terms and expressions without going into specificity or nature of fields/forces in order to highlight the same, how exactly could we characterise the distinction between 'Potential' & 'Potential Energy'?

 

[andrewkirk]

Potential is a property of a point in space, arising from a force field such as gravitational or electrical. Potential energy is a property of a body, or in some cases a system.
The potential energy of a body at a point in space, arising from force field F, equals the potential at the point from F, multiplied by the body's property relevant to F. For gravitational that will be mass and for electrical it will be charge.

For example, the Newtonian gravitational potential at distance R from the centre of a mass M is
$$V=-\frac{MG}{R}$$
and the the gravitational potential energy of a body of mass $m$ at that distance is
$$mV =-\frac{mMG}{R}$$

EDIT: Fixed denominator that had $R^2$ instead of the correct $R$. Thanks to @Delta2 for pointing that out.
Also changed sign, so that potential and PE increase with distance from mass M.

 

[vanhees71]

For a particle moving in an external field (electrostatic, gravitational) you have an equation of motion of the form
$$m \ddot{\vec{x}}=\vec{F}(\vec{x}).$$
If $\vec{F}$ is conservative, i.e., if there is a scalar field $V$ such that
$$\vec{F}(\vec{x})=-\vec{\nabla} V(\vec{x}),$$
you have
$$m \ddot{\vec{x}}=-\vec{\nabla} V(\vec{x}).$$
Multiplying with $\dot{\vec{x}}$ and integrating from $t_1$ to $t_2$ you get
$$\frac{m}{2} \dot{\vec{x}}^2(t_2) - \frac{m}{2} \dot{\vec{x}}^2(t_1)=-V[\vec{x}(t_2)]+V(\vec{x}(t_1),$$
or ordering expressions at $t_1$ and $t_2$
$$\frac{m}{2} \dot{\vec{x}}^2(t_2) + V[\vec{x}(t_2)] = \frac{m}{2} \dot{\vec{x}}^2(t_1) + V[\vec{x}(t_1)],$$
i.e., the total energy
$$E=T+V=\frac{m}{2} \dot{\vec{x}}^2 + V(\vec{x})=\text{const}.$$
$T=m \dot{\vec{x}}^2/2$ is called kinetic and $V(\vec{x})$ potential energy.

 

[sophiecentaur]

distinction between 'Potential' & 'Potential Energy'?

Isn't it simpler than all this? Afaiaa, Potential is the Potential Energy per unit Mass (/Charge if you're talking Electrics)

 

[vanhees71]

The potential of a force is by definition a scalar field, $V$, such that $\vec{F}(\vec{x})=-\vec{\nabla} V(\vec{x})$. That's very simple, provided such a potential exists for a given $\vec{F}$. You cannot make it simpler without confusing the subject unnecessarily.

 

[weirdoguy]

The potential of a force is by definition a scalar field, , such that .

And a lot of textbook would call it potential energy.

 

[Delta2]

@andrewkirk the denominator in your formulas should be $R$ not $R^2$.

 

[vanhees71]

And a lot of textbook would call it potential energy.

So what?

 

[Delta2]

Usually the potential is potential energy per unit charge or mass or something. For example in electrostatics it is $\mathbf{E}=-\nabla V$ and $\mathbf{F}=\mathbf{E}q=-q\nabla V$.

However if we take the case of a spring then the potential is the same as the potential energy $V=\frac{1}{2}kx^2$ and $F=-\nabla V=-kx$

 

[weirdoguy]

So what?

Well, the OP asks about the difference between potential and potential energy. None of your posts adresses the issue.

 

[vanhees71]

Usually the potential is potential energy per unit charge or mass or something. For example in electrostatics it is $\mathbf{E}=-\nabla V$ and $\mathbf{F}=\mathbf{E}q=-q\nabla V$.

However if we take the case of a spring then the potential is the same as the potential energy $V=\frac{1}{2}kx^2$ and $F=-\nabla V=-kx$

The potential is the potential of the vector field under consideration. There's an electrostatic potential for the electrostatic field, another potential for the gravitational field, and also a potential for all other kinds of "conservative forces", namely those forces which have a potential ;-).

 

[vanhees71]

Well, the OP asks about the difference between potential and potential energy. None of your posts adresses the issue.

I don't understand, what's unclear. If the force has a potential, that's the potential energy part of the total, conserved energy. That's what I said above. Making many words without math leads to such nonsensical confusions!

 

[Delta2]

The potential is the potential of the vector field under consideration. There's an electrostatic potential for the electrostatic field, another potential for the gravitational field, and also a potential for all other kinds of "conservative forces", namely those forces which have a potential ;-).

Yes it is just that the electrostatic field for example gives the force per unit charge, not just the force. So it is $E=-\nabla V$ but for the electrostatic force on a point charge q it is $F=-q\nabla V$. Just saying because in your general approach in post #3 you equate $\vec{F}$ to $-\nabla V$ (and to $m\ddot x$) which seems to imply that F is just the force, and not the force per unit charge (or mass).

 

[vanhees71]

Sigh. This is really a superfluous discussion about semantics. There are many situations, where vector fields can be described as gradients of a scalar field. Since often these fields obey the Poisson equation, they are called potentials.

In mechanics there are potentials for many forces. In the energy balance they are called potential energy to distinguish them from the other part, the kinetic energy. These are just words. What's important is to understand the mathematics given in #3.

Of course, it's also important to study "potential theory", including Helmholtz's fundamental theorem of vector calculus too, but that's only rarely really needed in mechanics. There you usually find the potentials for the usually treated forces easily by direct calculation.

 

[Delta2]

Sigh. This is really a superfluous discussion about semantics.

No it is not just semantics. There is a difference of a factor of q (or m if we talk about gravitational field) between what you call potential energy and what is the potential energy.

 

[vanhees71]

Read my postings again carefully!

 

[weirdoguy]

Read my postings again carefully!

I think that you are the one who should start reading what people are asking about. At the high school and undegrad level potential and potential energy are not the same, and thus should be carefully distinguished. Throwing around Poisson and Helmholtz helps no one.

 

[PeterDonis]

how exactly could we characterise the distinction between 'Potential' & 'Potential Energy'?

Why do you think there is such a distinction?

In generic terms and expressions without going into specificity or nature of fields/forces in order to highlight the same

I don't see how this can be done. Without some specific examples of the distinction you think you see, how is anyone supposed to answer your question?

 

[PeterDonis]

I think that you are the one who should start reading what people are asking about. At the high school and undegrad level potential and potential energy are not the same

Then my question to you is the same as my question to the OP: why do you think they are different? A specific reference to a "high school" or "undergrad level" textbook that describes the distinction you are talking about would be helpful.

 

[vanhees71]

I think that you are the one who should start reading what people are asking about. At the high school and undegrad level potential and potential energy are not the same, and thus should be carefully distinguished. Throwing around Poisson and Helmholtz helps no one.

That's interesting. I learned already in high school what the potential of a force is and that this potential is called the potential energy in the expression for the total energy, admittedly restricted to 1D motion, where you don't need gradients and line integrals but usual derivatives and 1D integrals do. It cannot be made simpler than that. Otherwise it leads to useless confusion and endless discussions about imprecisely defined words.

 

[weirdoguy]

Because that is what I have been tought in high school and at the university and that is what I teach all my students beacuse that is what they have to know during their high-school exams. I believe Resnick&Halliday talk about the two. And only high-school textbooks I can reference are in polish.

 

[PeterDonis]

Because that is what I have been tought in high school

Can you give an example of a "potential" that you have been taught in high school that is not associated with a potential energy in the way @vanhees71 describes?

 

[weirdoguy]

Otherwise it leads to useless confusion and endless discussions about imprecisely defined words.

Well, mixing electric potential and electric potential energy (which is potential times charge) is definitely not confusing...

 

[weirdoguy]

that is not associated with a potential energy

No one is saying that potential is not associated with potential energy, I'm saying that it is not the same (at least in high-school). E.g. electric potential is defined as $\frac{E_p}{q}$.

 

[vanhees71]

Well, mixing electric potential and electric potential energy (which is potential times charge) is definitely not confusing...

I have not even mentioned the electric potential, which is the potential of an electrostatic field and not of a force. This confusion has been brought in by somebody else. Once more: the potential of the force (!) IS the same as potential energy.

 

[PeterDonis]

No one is saying that potential is not associated with potential energy, I'm saying that it is not the same (at least in high-school). E.g. electric potential is defined as $\frac{E_p}{q}$.

Ok, but saying the two are "not the same" with this definition seems like a quibble. Why is it important to make this distinction between the energy and the energy per unit charge?

 

[PeterDonis]

I have not even mentioned the electric potential, which is the potential of an electrostatic field and not of a force.

I don't think you have carefully considered your position.

The "electric potential" that has been defined is the electrostatic potential energy per unit charge. Its gradient is the electrostatic force per unit charge. The only difference between these quantities and what you are calling "potential of the force" and the "force" itself is the "per unit charge" part. To me that doesn't seem like enough of a difference to warrant the position you are taking here.

 

[vanhees71]

I don't think you have carefully considered your position.

The "electric potential" that has been defined is the electrostatic potential energy per unit charge. Its gradient is the electrostatic force per unit charge. The only difference between these quantities and what you are calling "potential of the force" and the "force" itself is the "per unit charge" part. To me that doesn't seem like enough of a difference to warrant the position you are taking here.

Oh come on! The OP was about mechanics, and there the potential is the potential of a force (or an interaction force between particles). I have clearly defined in #3 what I mean by potential in the context of the question. How can it be misunderstood that I mean different quantities, which where not talked about before in the thread?

 

[PeterDonis]

The OP was about mechanics

Perhaps, if we include electrostatics in "mechanics".

I have clearly defined in #3 what I mean by potential in the context of the question.

So what? The question isn't about what you mean by "potential", it's about what the OP means by "potential". If you're going to just adopt a different meaning, you at least need to explain why the OP's meaning isn't a good choice. And doing that means doing what I did--suggesting that the difference between "energy" and "energy per unit charge" is not very important. And if that's the case, then neither is the difference between "potential of a force" and "potential of a force per unit charge". You have not responded to this point at all.

 

[sophiecentaur]

Once more: the potential of the force (!) IS the same as potential energy.

Ok, but saying the two are "not the same" with this definition seems like a quibble. Why is it important to make this distinction between the energy and the energy per unit charge?

I have already made this point but it got no reaction. I was assuming that the OP was basically about High school level physics. I am still convinced that the distinction is between Potential, an Intrinsic quantity (Energy per unit charge / mass) and Potential Energy, an extrinsic quantity (Energy for a particular mass or charge).

This is the equivalent sort of distinction to comparing Density with Mass - and many other pairs of quantities.

Some of the above posts imply that the Maths is the whole story. This is true except that dimensional analysis can be hard when complicated maths is involved.

 

[Doc Al]

At the high school and undegrad level potential and potential energy are not the same, and thus should be carefully distinguished.

Absolutely. A student who cannot distinguish potential from potential energy will have trouble with simple problems. Such as: Calculate the KE a given charge gains when it falls through a given potential difference.

The two concepts are intimately related but are not the same. Especially at the lower levels.

 

[weirdoguy]

Why is it important to make this distinction between the energy and the energy per unit charge?

Because for high-school students physics is confusing enough

 

[weirdoguy]

I have not even mentioned the electric potential

You tend to forget that there are different levels of physics. For a high school teacher that is obvious - there are two potentials considered in high-school: gravitational and electric. No gradients and other fancy stuff. And yes, you can teach electrostatics without using gradients and vector calculus explicitly, just look at some of the physics olimpiad problems.

 

[vanhees71]

Maybe it's another convention you use. A potential of a vector field is a scalar field, whose gradient is this field. In the context of the question posed in #1 it can only be the potential of a force. In #3 I've chosen the most simple case of the motion of a particle in an external force field, i.e., you have
$$\vec{F}(\vec{x})=-\vec{\nabla} V(\vec{x}).$$
In high school, of course, you don't have vector calculus at your disposal and that's why it's usually simplified to the motion along a straight line, i.e., you have
$$F(x)=-V'(x),$$
and also here one calls $V$ the potential of the force.

There is no way to understand electrostatics without the use of vector calculus. In high school we usually used the integral form and only very symmetric situations to calculate electrostatic fields. Of course also the potential of the electric field was introduced using the application of the motion of a charge in this field. Then of course $V(\vec{x})=q \Phi(\vec{x})$, where $q$ is the charge of the particle under consideration.

As I said, you make things more complicated in avoiding a minimum of math!

 

[fresh_42]

My 2 cents and personal opinion.

No gradients and other fancy stuff.

This is certainly the main opinion and the given fact at our schools.

However, it does not mean that it has to be the only valid point of view. Au contraire, it is the reason why STEM fields have the reputation they have and why students are forced to relearn concepts over and over again instead of treating them scientifically in the first place in my opinion. I do not like the underlying assumptions that students at school are not capable of learning things right. Such an assumption does not match my experiences.

Teach them right instead of teaching them twice!

We as a website are committed to the way subjects are learned at universities. There is a reasonable exception in the homework forums. I can not see that the homework standard should be applied everywhere else as some members try to achieve. (Evidence: this thread and https://www.physicsforums.com/threads/epsilon-delta-proof-and-algebraic-proof-of-limits.1016299/.) We should not lower our standards needlessly.

'Keep them stupid' is the reason students hate STEM. And those who don't are damned to relearn things over and over again, excused by "complicated math".

 

[robphy]

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.

 

[vanhees71]

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.

 

[robphy]

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.

 

[fresh_42]

This thread is getting off-topic.

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.

Two answers:

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.

 

[sophiecentaur]

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).

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.

 

[vanhees71]

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.

 

[robphy]

The only reason...

This is utter ...

FYI: You misattributed the quote to me.

 

[vanhees71]

Ugh? How could this happen? I corrected the quote by hand now:

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).

 

[sophiecentaur]

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.

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.

 

[vanhees71]

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.

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.

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".

Gremlins in the works! I didn't; bother to fess up - I knew someone would re-point your finger.

?

 

[sophiecentaur]

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?

Any vector field may have a potential

What are the units of that potential?

 

[vanhees71]

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

 

[sophiecentaur]

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.

 

[vanhees71]

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...