How do batteries work?

Here is a quote from a science-like show I saw recently. In the scene, two individuals were talking about using batteries for an electric motor. It should be noted that one of these individuals is labeled as “a physicist.” And no, I am not going to name the show.

It’s a matter of how much acid you need to store enough charge so that the two cells – the positive and negative, can create current to drive that motor. And you need that many to have the amp-hours which is another way to say capacity so that you can drive for some distance.

It’s not that the narrative is terrible (but it is terrible). It’s that this is supposed to be coming from the mouth of a physicist. What non-physicists hear is that batteries are super complex and there is nothing anyone can understand about them. It’s true that batteries are indeed complicated, but this could have been worded better. If it were my show, here’s what I would say about batteries.

There are two main things to consider with your battery choice. Can it produce enough current to drive your motor and does it have enough stored energy to last you enough time? That’s really it.

See? Isn’t that better? My primary suggestion for shows is that less of an explanation is better. Fewer terms means more likely to be “not wrong.” You can’t always be exactly correct, but you can be completely wrong. So just say the minimum.

But do batteries store electric charge? In short, no. Let’s look at a simple and complicated explanation of a battery.

Simple Battery Physics

But what about a more complicated explanation of a battery? How does a battery store energy? How does it make an electric current? Let me start with the most basic explanation.

A battery maintains a nearly constant change in electric potential across its terminals. When a complete circuit is connected from one terminal to the other, there is an electric current. Of course this current isn’t for “free”. It takes energy to move this current through a circuit. Where does the energy come from? There is energy stored in the battery in the form of chemical potential energy.

Yes, it is true that a current can be described as moving electrical charges. However, it is not true that these charges are “stored in the battery”. Let me give a simple analogy. If electric current is like water, then a battery is like a water pump. In the scene above, the guy describes the battery as if it were a water balloon shooting out water. That’s not how it works.

If you wanted to say a capacitor stores charge, that would be ok. But in this case the guy is using a battery and not a capacitor.

What is the Electromotive Force?

Now for a more sophisticated model of a battery. Many physics textbooks have a model similar to this, but I think Matter and Interactions (my favorite intro physics textbook) does the best job of explaining the term “electromotive force”. Oh, Matter and Interactions also has the best connection between electric fields and electric currents in circuits. Trust me, if you haven’t looked at this textbook, take a look.

For this model, let’s start with a capacitor. Yes, I know I just said a capacitor isn’t a battery but just hang on. Here is a parallel plate capacitor that isn’t connected to anything.

In this parallel plate capacitor you can make one plate positive by taking electrons away and putting on the other plate making it negative. Once you get these charges on the plates, there is a mostly constant electric field between these plates. If the field has a strength of E and the plate separation is s, then the change in electric potential from one plate to the other is:

Great. But like I said, a capacitor is not a battery. With a battery you would like the change in electric potential to be nearly constant. If you hook a lightbulb up to a capacitor, the charge from one plate leaves to produce an electric current. This decreases the charge on the plate and thus also decreases the electric potential. How could you solve this problem? What if you put a little conveyor belt inside the plates and this belt moved electrons from the positive plate to the negative plate?

Yes, this isn’t a real conveyor belt – it’s just a model. However, what happens as more and more electrons get added to the right plate? Yes, the electric field inside the capacitor increases. At some point the electric field inside the capacitor becomes great enough that it exerts an electric force on the electron with a magnitude equal to the force that the conveyor belt pushes on the charge. Beyond this charge (and electric potential across the battery) no more electrons can be moved to the right plate.

So, let’s write this as an equation. When it is fully charged, there are two forces on an electron in middle. There is the electric force from the charges (I will call this FC) and there is the force from the “battery” or whatever it is (Fb).

Here I just rewrote the electric force on the charge in terms of the electric field and I am using e to represent the charge of the electron. But if the battery voltage is ΔV, then I can also write the following expression for the electric field inside the capacitor (assuming constant electric field):

The voltage across the battery depends on this force from the belt in the battery model to push it across (and also the distance between the plates). Historically, we call this change in electric potential across the battery the emf which usually stands for ElectroMotive Force. But clearly, it isn’t a force since it has units of volts. But it also isn’t just a change in electric potential. Suppose you have a 1.5 volt battery. If you integrated the electric field from one plate to the other, you would get -1.5 volts (this has to be true since it’s path independent). The only way for you to get a zero change in potential around the circuit would be to have this emf across the battery.

But how does this “conveyor belt” really work? I think at this point, it’s best for me to just say “it’s a chemical process” and leave it at that. However, the belt model is useful when the battery is hooked up to a circuit. If you connect this battery to a lightbulb, electrons move through the wire and leave the right plate. This reduces the electric field inside the capacitor so that the belt can put more electrons on the plate. Of course this belt requires energy – the battery doesn’t last forever.

In fact, I think this battery doesn’t even have to have a chemical process to replace the conveyor belt. It seems that you could use an actual belt. This is what happens in a Van de Graaff generator (the metal ball that you put your hand on to make your hair stand up). However, I will save the analysis of a Van de Graaff generator for another day.

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