Electric Potential Energy vs. Potential

[schaefera]

Hey all.

I have a question regarding potential vs. potential energy.

Basically, from what I can tell, the potential energy is a property of the system as a whole (it has a single value in any given situation) while the potential is a property of each specific point in space.

Would this be like saying that the potential and any point is equal to the potential energy the system WOULD have if you placed a charge of charge q at that point?

Why then do we deal with both quantities... it seems to me like the potential is really just potential energy except that we are dividing out the charge of a "hypothetical point charge" that we could have added to the system, but then take out to make the math simpler.

Is this an ok way to think about it-- that the potential energy WOULD be what the potential is multiplied by a charge placed at that point... that would be a seemingly redundant definition, though.

Thanks in advance for clarification!

 

[tiny-tim]

hi schaefera!

Would this be like saying that the potential and any point is equal to the potential energy the system WOULD have if you placed a charge of charge q at that point?

electric potential is potential energy per charge

(similarly, gravitational potential is potential energy per mass)

Why then do we deal with both quantities

same reason we sometimes use gravitational potential … it makes the calculations easier

 

[schaefera]

So is it like I was describing, then? How, if we placed an actual charge there then clearly the potential energy of the system (for which there is only one value) would change... but we divide out by that test charge so as to make it not really matter-- make it the potential energy per charge?

 

[tiny-tim]

sorry, i didn't (and don't) understand that

(a "test charge" is assumed to be so small that it doesn't affect the system … is that what you meant?)

 

[schaefera]

Ok, so potential energy is a property of the system, but potential changes from point to point. This would imply to me that a way to think about finding potential is as follows: imagine placing a charge, q, at the point you are interested in. This will mean you put positive (or negative or zero) work into the system to move that point to your location. Thus, the increase in potential energy of the entire system has changed in accordance with that work. But the potential doesn't depend on the charge, q, so you remove that from you equation (in essence dividing it out from the change in potential energy). Is that how to think about a very detailed process of measuring the potential at any given point?

I guess it's confusing me because when talking about potential energy in gravitation, for example in dropping a ball to the Earth's surface, we never really worried about potential (just potential energy). We didn't really think of the work required to assemble the system in gravitational cases like this, so I'm trying to think up a new way of imagining how to find potential for electricity.

 

[schaefera]

Maybe it's something like that?

 

[schaefera]

This sure is one fun question to think about!

 

[schaefera]

Ah hah! I know what's confusing me:

If potential is merely potential energy divided by q, why can potential energy have one value for the whole system with potential varying from point to point?

 

[tiny-tim]

If potential is merely potential energy divided by q, why can potential energy have one value for the whole system with potential varying from point to point?

because you turn it the other way up …

potential energy = potential times charge (at each point (x,y,z))

so total potential energy = ∫∫∫ potential times charge-density dxdydz …

the total PE is the integral of the potential times the charge-density ​

(like total gravitational PE is the integral of the potential times the ordinary density)

 

[schaefera]

Ohhhh! So you integrate over potential-- which varies with location-- to get potential energy-- which is one number, namely the value of the integral. So at every point there is a potential which multiples a charge density?

Thank you for helping my understanding!