# B When to think of PE as property of a system vs of a particle

#### etotheipi

This is a bit of a vague question, but I was wondering if someone could explain.

As far as I know, potential energy is formally a property of a system (for instance, the GPE of two gravitationally attracting particles). In many physics problems it happens to be the case that one of the bodies in the system doesn't do anything, so all of the energy transfers take place in the other body - if a ball falls from 10 metres, the GPE of the ball-Earth system decreases by however many joules but we never really consider the kinetic energy of the Earth because we assume it to be stationary. Hence we can effectively neglect the other body and just consider the work done on one body.

This sort of reasoning is fine for me until we get to more complicated problems, such as those involving more than 2 bodies. If two planets A and B are sitting at rest at different points in space and we bring another planet C from infinity towards them, we have created two new systems AC and BC, and we can work out their potential energies as usual. However, pretty much every source I've read would just call the sum of these potential energies the PE of planet C.

And if we have a particle in a field, the notion of a source particle (or even charged plates and the like) seems to be neglected entirely. If an electrical force moves a charged particle from points A to B in an electric field, doing 5J of work, it is always quoted as the potential energy of the particle decreasing by 5J. Or for another example, if the potential of a point in space is 2J/C, we'd consider this the potential energy of a unit charge at that point and not really pay attention to the source of the field.

I suppose all of the things I've mentioned can be linked back to the idea of a system, but in a lot of cases it seems practical to just imagine the potential energy as a property of the particle.

My question is effectively, is it only valid to neglect the 'sources' and their contributions to/from PE only when they are stationary, or are there any other factors we need to worry about?

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#### DrStupid

My question is effectively, is it only valid to neglect the 'sources' and their contributions to/from PE only when they are stationary, or are there any other factors we need to worry about?
I wouldn't say it is the only valid case, but it is the case where the simplification makes sense because the force field is conservative.

#### Emmo Amaranth

If you're going to think of PE as a property of the particle in question (which, strictly speaking, it isn't), then you're not neglecting the PE contribution of the source at all. Why? Because the PE you're referring to only makes sense in the context of the interaction force of the "source" and "object." That is, the PE of the source is baked into the concept of the PE of the object. In other words: without the source, there wouldn't be any potential energy.

I wouldn't say it is the only valid case, but it is the case where the simplification makes sense because the force field is conservative.
The concept of potential energy only applies to conservative fields. We can define the force on an object as the gradient of a scalar potential function if and only if the force field is conservative.

Remember that you can choose an arbitrary reference point for potential and set it to zero. In gravitational fields, it's usually (but not always) convenient to choose the surface of the much more massive body as zero potential. For static electric fields, infinity is typically (but not always) the most convenient. The point is that potential is not an intrinsic property of the object in question.

#### DrStupid

The concept of potential energy only applies to conservative fields.
The concept of PE is not even limited to fields.

#### Emmo Amaranth

DrStupid: How do you mean? Do you have an example?

#### weirdoguy

Well, force exerted by spring obeying Hooke's law is potential and does not represent any field. Of course this force has electromagnetic origin, but in classical mechanics this doesn't matter.

#### DrStupid

DrStupid: How do you mean?
The potential energy of a system may refer to the energy that depends on its position in a field. But it doesn't need to. It can also be the energy that depends on the relative positions of its parts - no matter if you use a field theory or not.

Do you have an example?
How about a system consisting of two point masses orbiting each other? You don't need a field for its potential energy. You can easily calculate it using Newton's law of gravitation (not a gravitational field), the definition of work and the work-energy theorem - by integration of the work required to increase the distance to infinity (or any other reference).

Of course you can also use a classical gravitational field instead of the direct interactive force between the bodies. That field wouldn't be conservative. But the system still has potential energy.

#### Emmo Amaranth

Fair points: my wording was wrong. What matters is that the interaction force be conservative, whether or not this can be thought of as the manifestation of a field.

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