For some reason, I feel like the concept of voltage is escaping my grasp. I've done much research on these forums and through texts, and come across answers that seem quite well thought out, but still cant grasp the fundamental concept of what voltage is.

I could explain (regurgitate) the definition of voltage in many ways, and plug it into equations, but when it comes to testing my fundamental understanding of it as a natural concept, I fall short.

Ill explain what I know, and what I don't, and I have some theoretical questions which I hope will help give me a better understanding.

Here is what I think I know: 1. Voltage is electrical potential energy. It represents an imbalance of charge distribution.

1. This means that given a sufficient conductor, the electrons will do work by moving to a state of equilibrium to correct this imbalance.

2. Voltage must be measured between two points, because it can only exist in relation to an imbalance between one point and another.

3. Measures of voltage determine the work which can be done as a result of the imbalance?

4. The quantity of voltage in a battery would not change if you increased the size of the battery. It would only change if you increased the average negativity charge density in the negative terminal?

Here is what I don't know or have questions about: 1. So, increasing voltage increases current because higher voltage means a higher disequilibrium from the source, meaning more "pressure" on the electrons to move? Given a fixed wire diameter, does that means higher voltage would make electrons move faster?

1. Why do resistances in series add together? I don't understand. I picture a resistor as a bottleneck, in that the entire circuit is limited by the resistance of the most resistant component. But this is clearly wrong. Is it more correct to imagine that every time electrons travel through a resistor, they have to do more work and lose some of their ability to do future work, and have lower and lower energy the more resistors they go through?

2. If a current/stream of electrons begins to flow and is traveling through a wire with 1 ohm resistance, and goes through a 5 ohm resistor, then back to a 1 ohm resistance wire, since resistance decreases, would it speed up again upon exiting the 5 ohm resistor? If so, I'm assuming its new speed (relative to its initial speed upon entering the circuit) would be proportional to the fraction of remaining voltage after the total voltage drop up to that point?

3. When a battery dies, I'm assuming that its voltage isn't at 0. Does a dead battery just no longer have the minimum voltage required to supply the circuit with enough power to power the device? In that case does that mean that dead batteries could still power a circuit with less resistance or lower power demands?

4. This quote from Wikipedia confuses me: "In a series circuit, the current through each of the components is the same, and the voltage across the circuit is the sum of the voltages across each component". What does it mean that components "have voltage". I'm assuming this means the drop in voltage from before the component compared to after, right? But what is the reasoning for saying the component "has" voltage? Is it trying to express the quantity of work done (i.e. heat emitted) from that component?

5. Last one, another wiki entry: "In a parallel circuit, the voltage across each of the components is the same". This confused me, but I think I just figured it out. If a current is flowing through multiple paths, it would take the path of least resistance to its destination, thus if a parallel circuit has a 1 ohm resistor and a 5 ohm resistor in parallel, both would have the same voltage drop, but the current flowing through the 5 ohm resistor would be 1/5th the value of the 1 ohm?

Thanks for taking the time to read my post. I hope I didn't get everything terribly wrong.

Voltage is similar to height. It plays the same role for electric charge as height*gravity does for a ball on a hill. So high voltage means high potential energy the same way a ball being high up on a hill means high potential energy.

Voltage is not potential energy, the same way height is not energy. However, if you have a certain amount of charge $q$, you can multiply it to the voltage to get the potential energy, which his $Vq$. This is similar to the way you can multiply height to mass*gravity to get $mgh$ for the potential energy of a ball on the hill. So voltage is potential energy per unit charge the same way height*gravity is potential energy per unit mass.

Voltage must be measured between two points for the same reason height must be. When someone says "the height here is 1000 feet", they are actually comparing it to a point at sea level. In electronics, "sea level" often gets replaced with "ground". So if someone says, "this fence is electrified at 10,000 Volts", they mean there is a 10,000 Volt difference between the fence and the ground, the same way they mean that there is a 1,000 foot drop between the current elevation and the ocean. However, you can use any two points to measure height differences. If you drop a ball, it makes more sense to talk about height above the floor of the room you're in than to talk about sea level. Similarly, if you want to look at a single resistor, it makes the most sense just to talk about the voltage change across that resistor.

The work done on a charge as it moves from point to point is the quantity of charge times the voltage difference. This is just like the work done on a ball as it slides down a hill is the mass of the ball times the height of the hill times gravity.

A single battery cell can only produce a couple of volts. That's how much the potential changes for a single electron in the chemical reaction in the cell. This is a bit like the way a pump that works via suction can only lift water about 30 feet into the air, since that's the potential energy from buoyancy from the entire atmosphere. You can stack multiply batteries on top each other to get a higher total voltage drop (as is done in 9V or 12V batteries) the same way that you could use multiple pumps to suck water higher than 30 feet.

If you increase the voltage across a circuit element, in general the behavior might be quite complicated. This is like saying that if you tilt a ramp to a steeper angle, you will change the way that objects slide down the ramp. In many materials, we find that the behavior simple: current = voltage/resistance. So if you double the voltage, you double the current. This is called Ohm's Law. An accurate description of why it is true is probably a bit too advanced for right now. You will do okay for intuition if you start thinking of electrical current as being like water flowing through a tube. Then Ohm's Law says that if you're powering the flow by having the water flow downhill, if you make the downhill flow twice as steep, the water flows twice as fast. Yes, you can think of it as saying that the electrons are going faster.

Adding resistors in series is like adding several pipes to go through. If you try to push the water through more pipes, it will become more difficult. If you were letting water flow down a hill through a series of pipes, the more pipes you have, the less each pipe can be pointed downhill. That means that adding more pipes makes the water flow more slowly everywhere. Similarly, adding more resistors in series reduces the current everywhere.

The quantity you actually measure when it comes to current is the total flow - number of electrons per second passing through. If you have a 1-ohm, 5-ohm, 1-ohm resistor series, they will all have the same current going through them. This is because if they did not the current would start building up somewhere, and that would change the flow. (This actually happens, just very quickly because the wires have very low capacitance.) The way they all get the same current is they have different voltages. Most of the voltage drop for the entire circuit will be across the 5-Ohm resistor. This is like setting up pipes so that a skinny pipe goes down a steep portion of a hill while two fat pipes go down shallow portions of the hill. The total water going through each pipe per second would be the same. In this case, the water would move faster through the skinny pipe (the high-resistance portion). This is just because the total flow is the same, so if the cross-sectional area is less, the velocity is higher to compensate. This sort of picture roughly works with electrons as well. It is called the Drude model. It is the easiest to visualize, but it is not true to the quantum picture of modern physics.

Batteries do die slowly, yes. That is why flashlights, for example, grow dimmer and dimmer before turning off entirely.

To say a circuit component has a voltage is just saying that there is a certain voltage drop across that element. It is like saying that each pipe in a series of pipes running down a hill has a certain height difference, and that the height difference for the entire system of pipes is the sum of all the height differences of the individual pipes.

If two resistors are in parallel, they have the same voltage drop. This is like saying that two pipes side by side have the same height difference. The one with 1-Ohm resistance will have five times as much current going through as the one with 5-Ohm resistance.

A few preliminary ideas which might help:

• It doesn't really matter what the speed of the electrons is - a current of 1 C/s (=1 A) just means that a coulomb worth of charge (equal to $6.2 \times 10^{18}$ electrons) passes each point in the circuit each second. Perhaps there is one electron travelling so fast that it does $6.2 \times 10^{18}$ laps of the circuit per second, or perhaps there are $6.2 \times 10^{18}$ electrons travelling just fast enough to complete one lap each second. Perhaps it's some other combination. It really won't affect the large-scale behaviour at all, so we don't worry about it.

• This doesn't change the fact that circuits respond to changes almost instantly. If you, for example, turn on a light, each electron [almost] immediately pushes on the one in front of it, so all of the electrons begin moving at basically the same time. Individual electrons are moving at some arbitrary speed, but signals still propagate virtually instantly.
• The 'voltage of' an electrical component represents the voltage difference across it, i.e., the energy toll which each charge unit (expressed in, for example, Coloumbs or electrons) must pay to pass. It's really only meaningful when talking about a specific circuit: Resistor A may have a voltage of 5 V in circuit 1, but 300 V in circuit 2. This is actually also true for batteries! We talk about AA batteries as 'being' 1.5 V, but the voltage that the battery actually generates will drop as the current drawn from it increases. This effect is usually ignored in basic circuits courses (and in saying batteries 'have' voltage xyz) because (a) it's not significant unless you're drawing a lot of current and (b) it simplifies things.

• The 'current through' an electrical component represents the rate at which charge passes through the component. Again, the same component will (in general) have different currents through it in different circuits.

1.. (you seem to have two 1's) As stated above, the speed of the individual electrons doesn't matter. The rate at which charge moves increases - either because more electrons move, or because they move faster (we don't really care).