Seeming Contradiction of Potential

[Zack K]

If we have a conservative vector field, then we can describe it as $\textbf{F}=\nabla\phi$ where $\phi$ is some potential.

This here is the derivation of Newtons law of gravity:

Where $\nabla u$ is the gravitational potential. If we were to ignore it as a gravitational field, why is it force per unit mass? Should it not be the same expression as an arbitrary conservative field?

 

[jbriggs444]

Summary:: I have gotten confused over two definitions of potential I have found. These two definitions of potential seem to argue different expressions.

If we have a conservative vector field, then we can describe it as $\textbf{F}=\nabla\phi$ where $\phi$ is some potential.

This here is the derivation of Newtons law of gravity:View attachment 261776

Where $\nabla u$ is the gravitational potential. If we were to ignore it as a gravitational field, why is it force per unit mass? Should it not be the same expression as an arbitrary conservative field?

The potential energy field is is in units of energy per unit mass. So the associated force field is in units of force per unit mass.

If the potential field were in units of energy per cherry pie then the associated force field would be units of force per cherry pie.

If the potential field were in units of energy per coulomb then the associated force field would be in units of force per coulomb.

If the potential field were in units of energy then the associated force field would be in units of force.

Nothing more exciting than that going on here.

 

[etotheipi]

$\mathbf{F} = - \nabla U$, if $U$ is potential energy
$\mathbf{g} = - \nabla \phi$, if $\phi$ is potential

@jbriggs444 basically said all there is to say. The gravitational field is the limit as the test mass mass $m$ goes to zero of $\frac{1}{m}$ times the gravitational force. The gravitational potential is the limit as the test mass $m$ goes to zero of $\frac{1}{m}$ times the gravitational potential energy. It's just a factor of $m$.

N.B. I've put negative signs in because of the conventions with potential and potential energy but it's not strictly necessary to. You can have a perfectly valid force derived from a scalar field with no negative sign out the front.

 

[Zack K]

$\mathbf{F} = - \nabla U$, if $U$ is potential energy
$\mathbf{g} = - \nabla \phi$, if $\phi$ is potential

@jbriggs444 basically said all there is to say. The gravitational field is the limit as the test mass mass $m$ goes to zero of $\frac{1}{m}$ times the gravitational force. The gravitational potential is the limit as the test mass $m$ goes to zero of $\frac{1}{m}$ times the gravitational potential energy. It's just a factor of $m$.

N.B. I've put negative signs in because of the conventions with potential and potential energy but it's not strictly necessary to. You can have a perfectly valid force derived from a scalar field with no negative sign out the front.

I seem to have forgotten that g was associated with a vector field, thank you. So it is written as $\textbf{F}=m\textbf{g}= -m\nabla u$.

The gravitational potential is the limit as the test mass m goes to zero of $\frac{1}{m}$ times the gravitational potential energy.

I have never seen this expression. So are you saying $u= \lim_{m \to 0} \frac{U}{m}$(U as potential energy)? Could you point me to where I can find this expression?

 

[etotheipi]

I think the idea is that you don't want the introduction of the test mass to affect the mass you're interested in analysing. Making it's mass negligible means that the force exerted between it and the source masses is small enough so as to cause no funky business like acceleration of the source masses.

I can't remember where I saw it, it might have actually been in an electrostatics textbook somewhere. Perhaps someone else can advise.

 

[jbriggs444]

Could you point me to where I can find this expression?

I would not worry overmuch about it. Just realize that gravitational potential is most reasonably expressed in energy per unit mass. And electrostatic potential is most reasonably expressed in energy per unit charge.

Which is nothing more than realizing that a wagon with two tons of bricks at the top of a hill has twice as much potential energy as a wagon with only one ton of bricks. The potential [energy] of the field scales with your test mass.

[Edit courtesy of @hutchphd]

If you have potential in standard units and you want potential energy, multiply by the size of your test mass.

If you have the potential energy of your test mass and you want potential in standard units, divide by the size of your test mass.

 

[Zack K]

I would not worry overmuch about it. Just realize that gravitational potential is most reasonably expressed in energy per unit mass. And electrostatic potential is most reasonably expressed in energy per unit charge.

Which is nothing more than realizing that a wagon with two tons of bricks at the top of a hill has twice as much potential energy as a wagon with only one ton of bricks. The potential of the field scales with your test mass.

It seems a semester of vector calculus had me fixated on the definition of a conservative field. Scalars are in place for gravitational and electric fields to distinct them from each other (or some better wording than what I said).

 

[hutchphd]

The potential of the field scales with your test mass.

I am sorry but this is not correctly stated. The potential energy scales with your test mass. The potential most assuredly does not.

 

[hutchphd]

I believe we are all getting the COVID-19 cabin fever...

 

[etotheipi]

This whole field of study has so many different conventions it can get quite confusing. People interchange potentials and potential energies in casual speech, some use $U$ or $V$ for potential energy, some use $V$ or $\phi$ for potential; if you see a $U$, U're in trouble... it could also be thermodynamic energy...

And there's more, what do you call a change in potential? $\Delta V$, right? Wrong! $V$ will do. In fact, if you're an electrochemist then you don't even say the word 'change' or 'difference', if you want to refer to a potential difference between electrode and the solution you just call it an electrode potential.

Is this the cabin fever kicking in?

 

[jbriggs444]

I believe we are all getting the COVID-19 cabin fever...

It all balances out. Exponential COVID-19 in a log cabin.

 

[hutchphd]

Is this the cabin fever kicking in?

Personally I never interchange potential and potential energy and I vigorously and annoyingly object if someone does. It is incorrect. Worse it is dimensionally incorrect But the choice of letters for anything is seldom universal so care is warranted.

Voltage is usually defined with respect to an accepted standard. For the EE it is wrt a local ground potential. For the electrochemist it is with respect to some standard electrode half cell (I don't know what) so that is just a question of not endlessly repeating the standard. It is not really incorrect, but is a shortcut. So I can live with this.

But don't get me started on acronyms. At my job I was often called in for design reviews for various devices . The first acronym that was uttered I would rather rudely request "English please" and continue repeatedly until acronyms ceased. After a while my reputation preceded me and this was seldom required...one useful prerogative of being credentialed.

 

[etotheipi]

Voltage is usually defined with respect to an accepted standard. For the EE it is wrt a local ground potential. For the electrochemist it is with respect to some standard electrode half cell (I don't know what) so that is just a question of not endlessly repeating the standard. It is not really incorrect, but is a shortcut. So I can live with this.

Yeah, electrode potentials are strange because it is an interfacial potential difference measured relative to another interfacial potential difference, something like $(\phi_{M} - \phi_{soln1}) - (\phi_{ref} - \phi_{soln2})$. If the current $\rightarrow 0$ so that you don't get a potential drop across the solutions then it's equivalent to a relative scale of potentials.

But don't get me started on acronyms. At my job I was often called in for design reviews for various devices . The first acronym that was uttered I would rather rudely request "English please" and continue repeatedly until acronyms ceased. After a while my reputation preceded me and this was seldom required...one useful prerogative of being credentialed.

You'd love this paper..., credit to @DennisN in the TIL thread