I am trying to make sense of why electrons move in a circuit.
I did my research and the first answer I got is because of the difference of electric potential between the two points of the battery that an electron is traveling through. I can not see why that is, because most books define potential energy as the work needed to move a unit of charge from a to b. So how can the difference in potential between a and b make the electrons move?
most books define potential energy as the work needed to move a unit of charge from a to b.
It's the work needed to move a unit of charge from a to b due to the electric field present in the region between the points. And it's calculated from a path integral from one point to the other:
$$ V_{ba} = -\int_a^b\vec{E}\cdot d\vec\ell$$
(It doesn't matter which path we choose because in electrostatics the electric field is a conservative field and we'll get the same potential difference for any path between a and b)
So how can the difference in potential between a and b make the electrons move?
The formula above means that there can only be a potential difference between two points if there is a non-zero electric field in the region between them. This field is what makes the electrons move.
A field and a potential difference are really the same thing. When someone says "potential", that's just an abstract way of describing some condition at some point in space. If there's a field, then there are potentials distributed throughout it, or if there are different potentials, then there is a field between them.
In other words, a "potential difference" is a description of how two points in an "electric field" differ. Or, an electric field is some distribution of potential throughout space, and any two points in that field can have different potentials with respect to each other.
As an analogy, you can't have a terrain of valleys and hills, dips and peaks, without also having a concept of "height" to describe the shape of the terrain as you traverse it. As a concept, "height" cannot exist independently of "terrain", and vice versa, as they are both aspects of the same underlying thing.
A potential difference (height) implies and requires an electric field (terrain), and a field implies some variation of potential as you traverse it.
As far as potential energy goes, here's my take:
The presence of an electric field around a charge endows the charge with electrical potential energy, of which potential (in volts) is a measure. A charge that finds itself at a position in the field where potential is 10V, for instance, has 1eV more potential energy than a charge which finds itself at a position where potential is 9V.
The charge experiences a force due to the field, and this force will accelerate the charge, always towards a place of lower potential energy. In the above example, the charge started with 10eV of potential energy, but accelerated under the force exerted by the field, and will have lost 1eV of potential energy once it has been moved to the new position, where potential is only 9V.
When you "apply a potential difference" between two points in space, or in a circuit, you are establishing a field between those points. There is a now a potential gradient between the two points, and electrical potential varies as you move between them. Any charge that finds itself within that field will experience a force that will accelerate it towards a place of lower potential energy. Charges fall down the gradient (or up it if they are negative charges), in the same way a ball falls along a gravitational potential gradient.
Without the electric field, all places have the same electrical potential. Without a potential gradient for the charge to "fall" along, a charge would experience no force to accelerate it. With no net force anywhere in the space, it has the same potential energy at all places. Without a field, no work can be done on the charge. It is the field itself which "does the work" to accelerate the charge.
Electric potential is how much the electrons want to be at a certain place. If they were positive charges, it would be how much they don't want to be at that place.
Electrons naturally move from a place they don't want to be (low potential), to a place they want to be (high potential). The reason they want to be at that place is because of some chemistry stuff inside a battery. You can also drag them along with spinning magnets inside a generator.
Electrons go from lower to higher potential, and if you put things in their way, you can get energy out. If you want to make them go from higher to lower, you have to force them, putting energy in. Energy that you put in or get out is called work.
You're confusing different things: potential energy comes from having an object with mass somewhere it is exposed to gravity (e.g. a heavy rock on a mountain has potential energy) in the most usual meaning of the world. More broadly, it refers to energy due to objects interacting with potential fields, see below.
This potential energy term is not the same as electrical potential. Full stop - the concepts aren't unrelated, but energy is energy, and potential is not energy.
Potential energy generally refers to any form of energy that comes from moving a particle through a field: moving a rock through a constant gravity field upwards gives it a potential energy proportional to its height; moving an electron through an electric field gives it an energy proportional to the distance it travels away from the opposing charge.
The potential is the difference in energy gained that way, divided by the causing quantity; that means that for electrons in an electric field, potential is described in volts. For rocks on planets, it's described in meters.
That being said:
An election doesn't "want" to move. It experiences a force when exposed to an electric field, because it is a charged particle. That's really all there is to it: there's a fundamental force of nature, like gravity, which happens to charged particles in an electric field.
Now, two charged objects have potentials, relative to each other. The difference in potential, divided by their distance, defines how strong the electric field is.
So, that's it: this is axiomatic, there's no further "why", it's how this universe is: charge, e-field, means force.
If you dig deeper into this later in your studies, you will find Schrodinger's Equation to explain the phenomen "force" to be a result of probabilities of existence in a specific place having to change; but that's very quickly again an axiomatic statement, and also, I don't think you need a more confusing quantum model of the world - the macroscopic problem of "why do we see an electron flow in a cable" is hard enough.
In a battery, there's (at least) two different substances, at different potentials, so there's an electric field between the poles of a battery. That makes the electrons in the cable move once you use that cable to connect the poles.
Tldr: there's a fundamental force on charges in an electric field. That's where, at an early electrical engineering university program, the explanation stops.
