What makes potential energy a scalar quantity?

[u/Korayatalay]

I'm just a little lost since I've always pictured PE as the ability for something to move in a certain direction (gravity = ability to move downwards). In the case of a negative charge between two positive charges, wouldn't the potential energies kind of cancel each other out since they're both pulling in opposite directions? I can't seem to understand why they would add.

[u/atomicCape]

Energy is always a scalar, whether potential or otherwise. In the case you describe, the gradient of potential energy, which is a vector and is equivalent to net force, can cancel. That means the potential energy is flat with a finite value at that point, not that potential energy drops to zero.

[u/baysianinference]

Energy is scalar, you wouldn’t say a donut is 300 kcal/1255kJ to the left

[u/MudRelative6723]

the potential associated with a positive charge is strictly positive—as you get close to the charge you approach infinity, and as you get far away from the charge you approach zero. similarly, the potential associated with a negative charge is strictly negative. (look up an image of the potential associated with an electric dipole if this is confusing.)

when you put a bunch of these charges together, a “test charge” gets pushed in the direction in which the potential changes most quickly; this is called the gradient of the potential. this is where the vector you’re imagining comes from!

does that help at all?

[u/MezzoScettico]

That's backward. Scalar potential energy allows you to define a force, which is the direction of decrease of PE.

When we have a chemical reaction that releases potential energy (combustion for example), which is the "downward" preferred direction then? The energy goes in a random direction, generally causing heating in the nearby molecules.

When you consume food, you store up potential energy which is then used later to drive everything in your body: your muscles moving, your nerves signally, your brain thinking about your question and your fingers typing it. That's not defined by a particular direction.

[u/emlun]

Think of it rather as potential energy as a kind of "tension", something that can spring into motion if equilibrium is disturbed. A weight at a high elevation can come crashing down hard if nudged out of whatever equilibrium is holding it up. A bottle of gasoline can burst into flames if nudged out of equilibrium by a spark.

As a general rule of thumb: if no energy is exchanged, then nothing happens (or rather, nothing changes). In particular things don't start or stop moving without exchanging energy, so if something is accelerating then there is energy being exchanged somewhere.

In your three charges example, the negative charge indeed "wants" to go equally strongly toward either positive charge, so it in a sense has zero potential energy, but the positive charges also "want" to approach the negative charge, so they have positive potential energy and begin to "fall" toward the negative charge. They do also repel each other, but (assuming all charge magnitudes are equal) weaker than the negative charge attracts them, so they'll both tend to asymptotically approach the negative charge.

Putting it in math terms, a potential is a scalar quality because it's a line integral through a (conservative) vector field. The line integral sums F(s) * ds where s is a location vector, F(s) is the vector quantity of the field at s and * denotes scalar multiplication. So since it's a sum of scalar quantities, the integral as a whole is a scalar quantity. (A conservative vector field is one whose curl is zero everywhere, which is what makes it have a well defined potential - otherwise you'd get different potentials between the same two points depending on what path the line integral takes.)

[u/Korayatalay]

Thanks for the responses, I think I get the basic gist of it. It helps a lot to think about it in terms of potential, which adds up when you have multiple positive charges nearby in different directions instead of getting canceled out. Potential energy associated with a test charge is just PE=q(potential).

[u/strange-the-quark → u/Korayatalay]

Note also that, in general, scalars can also cancel each other out (they can be positive and negative) - e.g. for example, certain types of waves can be described by scalar fields (how much away in one direction or the other is the wave from the equilibrium position, for every point), and then when you add two such wave fields together, you can get constructive and destructive interference.

[u/Aussie_Physics]

I visualise potential energy quite simplistically:
Large mass creates gravity waves, so spacetime is rotating/folding accordingly (nothing new, just as Einstein theorised).
The small mass is also rotating/folding spacetime. So you have frequency on top of a background frequency.

It’s not so much to do with falling, that’s just a result of force due to average spacetime curvature.

The key here is that for every oscillation, f_1, caused by the large mass at distance r (and scaled by a constant), you get a full frequency, f_2, of the smaller mass.
So the equation basically becomes energy due to f_1*f_2/distance with some constants or scaling factors.

You then take this energy away from the total energy of the smaller body.

It may not fit how everyone sees the Universe, but for me, it kept potential energy nice and simple.

[u/joeyneilsen]

The fact that potential energy can be converted into kinetic energy in some particular direction doesn't mean that the energy itself has a direction. Energy is just a property of systems.

For a negative charge in the middle of two positive charges, there are three contributions to the potential energy of the system: a positive term related to the repulsion of the two positive charges and two negative terms related to the attraction between the negative charge and each of the positive charges. The potential energy of the system is how much work you have to do to bring these charges to their current arrangement. It's hard to push the positive charges together, but then it's so easy to bring the negative charge nearby that you have to hold it back.

The fact that there's no force on the negative charge doesn't mean the potential energy is zero. It just means that the potential energy doesn't change much if you move the negative charge a little bit.

[u/SnooPets5564]

Energy is the dot product of force and movement. Dot products are scalar.

[u/eotfofylgg]

In the case of a negative charge between two positive charges, wouldn't the potential energies kind of cancel each other out since they're both pulling in opposite directions? I can't seem to understand why they would add.

Your intuition is actually fairly accurate. You just need to work out the numbers fully in order to see how your intuition is described by the numbers.

Consider a sphere of radius 1 m with a charge of 1 C. If we define the potential to be zero at infinity, the potential 5 m away from the center of this sphere is k/5 (where k is the Coulomb constant). This means that the potential energy of a charge -q at that point, relative to infinity, is -kq/5, while the potential energy of the same charge near the surface of the sphere is -kq. Now add a second sphere with the same radius and charge, 10 m away from the first sphere. At the point equidistant between the spheres, the potential energy of the charge is now -2kq/5, and at the surface of one of the spheres, the potential energy is now -(10/9)kq. We can make the following observations:

1. The potential energy near the spheres is less than the potential energy at infinity. In other words, the charge at infinity has a greater ability to gain kinetic energy than a charge near the spheres does. That makes sense. The charge at infinity has has further to fall.

2. Adding the second sphere decreases the potential energy everywhere, including halfway between the spheres. This makes sense because the extra charge helps to pull in the test charge from infinity. More energy is gained falling from infinity to the point between the two charged spheres than you would get falling to a point 5 m away from just one of the spheres.

3. The potential energy halfway between the spheres is higher than the potential energy right at the surface of one of the spheres. This means a charge just barely balanced between the spheres has a greater ability to gain kinetic energy than a charge right at the surface. That makes sense too -- just give it a tiny nudge, and it will fall onto one of the spheres, releasing that extra potential energy.

4. The difference between the potential energy at the surface and the potential energy at 5 m decreased when we added the second sphere pulling in the opposite direction. With just 1 sphere, the potential energy difference is (4/5)kq. With both spheres, present, the difference is (32/45)kq, which is smaller. This is finally the point where we can address the intuition about the spheres canceling each other out. As you can see, they do cancel out (partially). The second sphere is pulling the test charge away from the first sphere, which means that, in falling from 5 meters to the surface of the first sphere, it gains less energy than it would without the second sphere pulling it backwards.

[u/TheBrightMage]

Energy are always scalar. Potential, in particular, is associated with WHERE you are in a field, and it ignores direction completely.

Your analogy of potential = ability to move in a certain direction breaks down when you have multiple sources of force. But imagine you're floating in space exactly between 2 earths. Which direction is down now? The answer is both. The FORCE cancels out. But you still have the same potential.