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I am taking AP Physics right now (I'm a high school student) and we are learning about circuits, current, resistance, voltage, Ohm's Law, etc. I am looking for exact definitions of what current, voltage, and resistance are.

My teacher, as I'm sure most physics teachers do, compared a wire with current flowing through it to a pipe with water flowing through it. The thinner the pipe, the more 'resistance'. The more water pressure, the more 'voltage'. And the faster the water travels, the higher the 'current'.

I took these somewhat literally, and assumed that current is literally the velocity of electrons, voltage is the pressure, etc. My physics teacher said that the analogy to the water pipe is only really used for illustrative purposes. I'm trying to figure out exactly what current, resistance, and voltage are, because I can't really work with a vague analogy that kind of applies and kind of doesn't.

I did some research, and found this page which provided a decent explanation, but I was slightly lost in the explanation given.

Let me know if this question has already been asked (again, remember: I don't want an analogy, I want a concrete definition).

In terms of circuits...

Current is the rate at which charge flows past a point in a circuit.

$$I=fracdQdt$$

The voltage between two points in a circuit is the negative of the line integral of the electric field along the circuit between those two points.

$$Delta V_AB=-int_A^B mathbfEcdot dell$$

The resistance of a segment of the circuit is the ratio of the voltage across that segment to the current through that segment.

$$R=fracVI$$

Before explaining current, we need to know what charge is, since current is the rate of flow of charge.

Charge is measured in coulombs. Each coulomb IS a large group of electrons, 1.6x10^19 of them to be more exact.

The “rate of flow” of coulombs/charge is simply charge/time and this calculation for a circuit give you the number of coulombs that went past a point in a second This is just what current is.

Resistance is a circuit’s resistance to current, it is like you said measured in ohms, but it is caused by the vibrations of atoms in a circuits wire and components, which results in collisions with electrons, making charge passage difficult. This increases with an increase in temperature of the circuit, as the atoms of the circuit have more kinetic energy to vibrate with

Voltage is the energy in joules per coulomb of electrons. This is shown though the equation E=QV where the ratio of Energy over charge= voltage. This is granted by the battery, which pushes coulombs of electrons, with what we call electromotive force. However when it is said that the potential difference across a component is X volts, it means that each coulomb is giving X that much energy.

Note: if an equation doesn’t make intuitive sense to you, chances are it is a complicated derivation, and to understand it you’ll have to learn it’s derivation

Voltage:

This is the electrical potential energy difference. It's the difference in electric potential between two points. Which is a measure of how much energy it takes to move a test charge between the two points.

Current:

This is literally a measure of how much electric charge is currently being pushed past a point or region. A current is produced by introducing charge to a voltage. Note, that I use "electric charge" instead of "electrons" because any difference in voltage will result in current flow. For instance, electrochemical cells deliver current through molecular ions. Batteries fall in this category.

Resistance:

This is the measure of the opposition to current flow. Every wire has some resistance, however you want specific resistor materials to meaningfully resist current flow. It's affected by cross section, and the material used to make it.

This is not a complete answer, but a bit too much for just a comment. I want to just address a couple of points in your question.

When you say "current is the velocity of electrons", you're half right. Current measures the total quantity of charge moving through a surface (for example, a cross-section of a wire) per unit time. It could increase if the velocity of the carriers increases, or if there's simply more carriers present moving the same speed. And the carriers could be electrons, or they could be protons (for example, in an ionic solution or a wet cell), or they could be electron holes, but explaining that is probably getting beyond what you need to learn right now.

When we say "voltage is like pressure", there we really are making a very loose analogy. Differences in voltage are what exert forces on charge carriers, so differences in voltage are what cause current to flow. In the same way differences in pressure are what cause water to flow in a pipe. But that's where the analogy ends. You shouldn't read anything into it about what creates differences in voltage, how voltage might be distributed in a wire, etc.

You should also realize that voltage is not really fundamental. In electrostatics, it's just a way of summarizing the effects of electric fields (as pointed out in another answer, electrostatic voltage is just the integral of electric field along a path). More generally, it can also be produced by changing magnetic fields. But really it's the electric field that actually produces a force on charged particles, not the voltage per se.