2.2.2: Potential Energy

Potential energy is energy of relative position. Except in esoteric situations where general relativity and quantum field theory tentatively approach each other, the absolute value of potential energy doesn’t matter. All that matters are the differences in potential energy as particles rearrange themselves into different relative positions. This means that you could add any constant (with energy units) you want to the potential energy of a system, and, as long as you don’t change the constant you’re using partway through a problem, all of your energy calculations will come out right. Frequently, but not always, we choose the constant such that the potential energy is zero for particles that are infinitely far away from each other. This convention makes sense; you don’t want to have to think about having some energy to carry around for a particle that is so far away that it’s not meaningfully interacting with any of the particles you do care about.

Technically, you can’t talk about the potential energy of a single object. Really, the potential energy is in the interaction of that object and another object. To be proper potential energy, it must depend only on their positions relative to each other. It doesn’t make sense to talk about “the potential energy of the Earth”. However, it does make sense to talk about the potential energy of the Earth-Sun system.

Sometimes, we can make the approximation that one particle is much smaller than everything else it is interacting with. In that case, we will talk about the potential energy of that particle. For example, if you lift a ball off of the ground, as that ball and the center of the Earth get farther away from each other there is more and more potential energy in the gravitational interaction between the ball and the Earth. However, the gravity of the ball on the Earth is extremely unimportant to the Earth, whereas the gravity of the Earth on the ball is extremely important to the ball. As such, we can treat the ball as a particle moving within the “fixed potential of the Earth”. We then say that the ball has a certain amount of potential energy based on its height above the ground. Implicitly, this is really the potential energy in the interaction of the ball and the Earth, but it is more convenient to treat it as the potential energy of the ball, with the understanding that we’re working in the (very valid) approximation that the ball is much smaller than the rest of the system (i.e. the Earth).

Different interactions (i.e. different forces) have different functional forms for potentials. For the moment, you won’t need to use any of them. If you have had physics before, you may know some of them. For an arbitrary force or combination of forces, you could construct a potential energy function \(\ V(x)\). It is useful to think of an analogy between a particle moving in a potential and a car rolling about on hilly ground. Suppose that \(\ V(x)\) had the following form:

The dashed line on the plot indicates the total energy available to the particle. Imagine that instead of potential energy, the vertical axis where the height of hills, and imagine that the particle is a car. When the car is at a lower point, it has less potential energy, and thus more kinetic energy, and thus is moving faster. The car cannot get to places higher than the dashed line: it’s not moving fast enough to make it that far up the hill. By thinking about potential energy in this manner, you can visually get an idea for how particles will move around in a given potential, even if you don’t know all of the classical physics needed to work it out.

As another example, suppose you have a wire with two positive electric charges fixed to it. Sliding smoothly along the wire is a bead that also has a positive electric charge on it:

Positive electric charges will repel each other. As such, if the bead will be pushed away from the two positive charges at either end of the wire. Call \(\ x\) the position of the bead along the wire, with \(\ x = 0\) the exact center of the wire. There will be a potential energy function \(\ V(x)\) for the interaction between the bead and the two charges on either end of the wire. To make things more interesting, let’s suppose that the bead has some total energy \(\ E\) that is greater than the minimum of the potential \(\ V(x)\).

The minimum of the potential energy is where the bead “wants” to be. In this case, the bead is pushed away from the positive charge at either end. If you imagine a ball rolling in this potential, it would experience the same thing; it would want to move towards the center if it were up either side of the potential well. However, looking above at the picture of the bead on the wire, the bead makes no actual motion down in space; it’s only moving to lower potential. Notice that we’ve chosen to make \(\ V(x)=0\) at the center of the wire. Remember that that is completely arbitrary; we could add a constant to the potential energy, and it wouldn’t make any difference. (We would have to add the same constant to the total energy of the particle to keep things consistent, however!)

What happens if the bead is at \(\ x = 0\)? We can see that it’s potential energy \(\ V(x)\) is equal to zero. However, its total energy is something greater than that. That means that the bead must have some other form of energy. As we’ve defined the system, the only other form of energy the bead could have is kinetic energy. This means that if the bead really does have energy \(\ E\) as indicated on the plot, it must be sliding either to the left or to the right if it’s at \(\ x = 0\). Indeed, at any \(\ x\), it will satisfy \(\ \frac{1}{2} m v^{2}+V(x)=E\).