What *exactly* is electrical current, voltage, and resistance?

Question

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).

Edit 1: Note, when I say 'exact definition' I simply mean a definition that does not require an analogy. To me, an exact definition for a term applies to every use case. Whether I am talking about a series circuit, a parallel circuit, or electric current within a cell, the 'exact definition' should apply to all of them and make sense.

Answer 1

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: roughly 6.24 ˟ 10^18 of them.

The “rate of flow” of charge is simply charge/time and this calculation for a circuit gives 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 circuit's 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 joules of energy to that component.

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 its derivation.

Answer 2

In terms of circuits...

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

$$I=\frac{dQ}{dt}$$

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.

$$\Delta V_{AB}=-\int_A^B \mathbf{E}\cdot d\ell$$

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

$$R=\frac{V}{I}$$

Answer 3

The "concrete definition" you are looking for doesn't exist, and never will exist.

Physics isn't concerned about what anything is "really like", or "exactly" what it is. The answers to those questions are topics in philosophy and religion, not science, and are usually matters of dogma - which is "a principle or set of principles laid down by an authority as incontrovertibly true."

The only "final authority" in science is what you can measure, not what someone tells you.

The thing you should be concerned about is what "electricity" does, and not what it is. All you can ever "really know" about electrical charge is that it is a property of some things (like electrons and protons) and it comes in two "opposite" types, which we arbitrarily label "positive" and "negative". That's it. Sorry, but there isn't any more to know about "what charge really is". Everything else you will find in a physics textbook is describing what charge does, not what it is.

"Electric current" is just the movement of electrical charge. That sentence isn't quite as simple as it might appear. It does not say electric current is just the movement of charged particles, like electrons in a wire! "Electric charge" is not a "thing". It is a property of other things (like electrons).

In fact the previous paragraph isn't the whole story, because there are things that act like electric currents where nothing at all is physically moving. But understanding that in any detail will have to wait until you know enough math to handle Maxwell's equations...

Answer 4

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.

Answer 5

I am looking for exact definitions of what current, voltage, and resistance are.

Exact definitions are not going to do you any good because terms have different connotations, restrictions, and even meanings in different contexts. I think you are looking for foundational conceptual definitions. That will form the context of my answer.

electrical current

Electrical current is the rate of flow electrical charge across some defined cross-sectional area. Electrical charge is a fundamental property of elementary particles. The charge flow for the current is the net flow of charge. One could have negative charge flowing across an area as well as positive charge. Or negative charge could flow one way and positive another. The current involves the rate of the net charge flow. By convention, positive current is the direction is positive charge flow, so if electrons move across some area, the positive current is opposite their flow.

Regarding charge, one could say charge is a property which creates electromagnetic fields and allows energy to be transmitted via those fields.

In an electrical circuit, the current is generally related to the flow of electrons through the wires. (By the way, the velocities of those electrons are very small. LOTS of electrons move slowly to give sizeable currents).

electrical voltage

Electrical voltage is the different in the electrical potential of two locations. Electrical potential is a property of space which arises from an electromagnetic field. A non-zero group of charged particles always has an electromagnetic field in the space surrounding it. The voltage tells us how much specific work (work per charge) is done, or can be done, on other charges when they move between the two locations which have differing potentials. Voltage tells us something about the electrical landscape of a space (in a simple circuit, the space is nearly one dimensional). A charge placed in an electric field will experience a force, so if it moves (at least partially) parallel to the field, work will be done on it.

electrical resistance

Electrical resistance is a very specialized notion and can be defined as the ratio of the voltage across an electrical device to the current through the same device. Ohm's Law, in its introductory-circuits form of $V=IR$ is misleading because people often think that voltage is produced by the resistor. Remember that voltage is the specific work done/to-be-done between two locations, so Ohm's Law is a relationship which actually defines the resistance: $$R=\frac{V}{I}.$$

The resistance is a property of the object which makes it difficult for charge to flow continuously. The voltage across a resistor is the specific work which must be done by the field on the flowing charges to keep the current continuous. It can also be thought of as the specific energy (energy per charge) removed from the electric field/current system as the charges move through (and do not accumulate) in the resistor.

Answer 6

Electrostatic Force

Electrostatic force or Coulombic force is the force that charges exert on each other (I am assuming you know what a charge is). Coulomb found that this force is also an inverse square law. $$ \vec{F}_{12} = -\vec{F}_{21} = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{\bf({r_2 - r_1})^2} \bf{\hat{r}_{12}}$$ Where $\epsilon$ is the permittivity of free space.

Fields and Electric Field

In physics a field is a quantity that exists at every point in space. Let's consider the classic example of a fluid flowing in two dimensions. The fluid is spread all over space. Every point in space will correspond to a particle of the fluid. Since fluid is flowing every point (particle) has a vector (the velocity of that point) associated with it. This vector may or may not change with time. The field in this case is simply the velocity of the fluid at different times.

Similarly we can define electric field. Let's assume a point charge $Q$ in free space. There is no charge in its surrounding. But we know that if a charge is added in the vicinity of $Q$ it will experience a force. Now we know that force itself is a field, because every point $(r,\theta)$ has a force associated to it. But we want to define something independent of the test charge. Hence , $$ \vec{E} = \frac{\vec{F}}{q_{test}}$$ Where $\vec{E}$ is called the electric field. It is also sometimes defined as force experienced by a unit positive charge.

Potential Energy and Potential

Potential energy is the negative of the work done by a force to take a particle from initial position to final. I will just write it's expression since the derivation will be out of the scope of the question. $$ U = \frac{q_1 q_2}{4\pi \epsilon_0}\bigg( \frac{1}{r_2} - \frac{1}{r_1} \bigg) $$ The Potential or Voltage is defined as the potential energy in bringing a unit charge from infinity to some $r$. Hence the expression will be $$ V =- \frac{q_1}{4\pi \epsilon_0 r}$$ Since any system tends towards the least potential energy configuration, positive charges always move from higher potential to lower potential and negative charges move from lower potential to higher potential.

Current and Drift Velocity

Current is defined as the rate of change of charge w.r.t time. $$I=\frac{dq}{dt}$$ In a circuit, where only electrons flow one can write $$I= e\frac{dN}{dt}$$ Where $N$ is the number of electrons. A more useful convention is that of current density. Current density is defined as current flowing through cross sectional area $A$ divided by the area. It is defined as a vector. $$ I = \bf{\vec{j}.\vec{A}}$$

Drift velocity is the velocity of electrons in a conductor. Initial the electrons are colliding with the atoms in the conductor an moving in random directions. The average velocity of all the electrons will be approximately zero. But when we apply an external electric field the electrons start to drift. Consider the $i^{th}$ electron. It lets say that it has an initial velocity $u_i$. An electron can move only for small interval of time before colliding and getting a completely new velocity. Let's call this time $\tau$. From the equation of motion we can say $$ v_i = u_i + \frac{eE}{m}\tau$$ Summing over all electrons $$ \Sigma v_i = \Sigma u_i + N\frac{eE}{m}\tau$$ Here u have assumed that $\tau$ is approximately same for all electrons. As mentioned $\Sigma u_i=0$. Hence the average velocity or drift velocity is $$ v_{d} = \frac{eE}{m}\tau$$

Ohms Law, Resistance and Conductance

Consider a wire with electron density $n$. If an external electric field $E$ is applied then the electrons will drift. The charge in the conductor can be written as $$ dQ=enAv_d dt$$ Hence the current density can be written as $$ j= env_d$$ Substituting the expression for $v_d$ $$ j= \sigma E$$ where $\sigma = \frac{en\tau}{m}$ is called the conductance of a conductor. Multiplying both sides by $A$ and dividing both sides by $l$( the length of the wire or displacement of electrons) we get $$I = \frac{A\sigma}{l}V$$ Or $$V=IR$$ Where $R$ is the resistance defined as $$R= \frac{1}{\sigma}\frac{l}{A}$$ Sometimes $\frac{1}{\sigma}$ is written as $\rho$, where $\rho$ is called resistivity. This is the reason $R\propto \frac{l}{A}$.

Hope this clarifies voltage, current and resistance.

Answer 7

Current is simply the rate of flow of charge. (or, the flow of charge per unit time) Now, charge is a physical property of matter by virtue of which matter can experience a force when placed in an electromagnetic field. Unless some mass has charge in it, the mass (matter) cannot feel any force when placed in an electromagnetic field.

Voltage is the work needed per unit of charge when a test charge (negligible amount of charge) is moved from one point to another.

Resistance is the obstruction offered while the current flows from one point to another. It is basically caused due to the collision of the flowing electrons with the ions present in case of a metallic conducting wire.

Answer 8

compared a wire with current flowing through it to a pipe with water flowing through it.

Yes, it's a good analogy as both water and current "flows", main difference:

• a wire cannot be "empty", it always contain the same number of electrons (at least when we speak about classic circuits), flow of current means some electrons enter at one end of the wire and the same number of them exit at the other end, during that all electrons "flow"
  (In reality "flow" of electrons is a very complex process as electrons can't freely move, they have their "niche" to fill, but that's maybe not relevant here, just to indicate it's not a simple "flow").

The thinner the pipe, the more 'resistance'.

that's true. Increasing wire's diameter decrease resistance, however matter of wire is important too, like having a pipe which is not completely empty and permeable to water.

The more water pressure, the more 'voltage'.

Exactly, the higher the voltage, the more work your electricity can do.
However you can't open a wire and loose current as it can happen with a pipe under pressure, electricity needs a closed circuit to flow

And the faster the water travels, the higher the 'current'.

Maybe that's the most far analogy, current is related to number of electrons (charge) entering / exiting the wire, and it's not related to actual movement speed of electrons (as we know very little about what exactly happens at that subatomic level).

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These are still analogy, and not the pure theory, I hope it helps to imagine how it happens.

Answer 9

_{aside: The water pipe analogy is actually quite accurate, but most high school students don't learn fluid dynamics and the heavy calculus makes it more trouble than it's worth. (If you treat volumetric flow rate as akin to current, and pressure as electric flux density, then using Bernoulli's equation neglecting gravity, assuming constant area and velocity along a pipe/wire, it all works out).}

voltage

We have a device that can measure "voltage" between two points (not just in a circuit but in the air, on the ground, wherever you place the terminals).

The first thing is that if we measure the "voltage" between A and B, and then between A and C, the voltage between B and C will be the difference between these two—if you freeze time, at least. So this means you can pick whatever reference point you like, then any other point in the universe has a certain electric potential (measured in volts) relative to the reference point. So, there is something in the fabric of space that can be "stronger" or "weaker" than other parts of space. This is what we call the "electric field".

Note that this is not about electrons. It is possible to have a positively charged plate and a negatively charged plate and a total vacuum in between, but this vacuum will linearly transition from one potential to another. (I.e. the voltage across the left half equals the voltage across the right half.)

In a simple electric circuit, each point has a fixed voltage (usually measured relative to "ground"). Any loop through the circuit will involve exactly the same amount of "going up" as "going down".

current and resistance

Current and resistance are probably more intuitive in the context of electric circuits. Current is roughly speaking a count of how many electrons pass through an area (e.g. the cross-section area of a wire or resistor) per unit time. Resistance is how many volts will produce 1 amp of current flowing "across a resistor". Technically this means measuring the voltage between both ends of the resistor, and the current at some cross-section anywhere before, in, or after the resistor. The current will be effectively the same anywhere you you section (not beyond branches in wires of course).

(Obviously we also have devices that can measure current and resistance; I made a point about measuring voltage because electric fields are the hardest thing to grasp.)

more on current and resistance

When electrons are free to flow, they are driven to equalise electric fields. The stronger the field, the more electrons will flow. However, different materials will resist electron flow to different degrees, hence differences in resistance (or rather resistivity). And electrons can be forced to flow the other way—batteries use chemical reactions, and generators use electromagnetism.

One way to see how current is proportional to surface area is this: assume you have 1A traveling along a wire, and you have 1A travelling down another wire, that's 2 coulombs per second. If you squash the two wires into one, the surface area will double and you will still have the same number of coulombs per second — 2.

Answer 10

Since my comment appears to have been removed, let me reassure you in a partial answer that the water analogy will carry you further than you seem to think. The reason is that the electrons in a good conductor move freely and independently of each other and of specific atoms. Their aggregate behavior is so similar to a gas that they are sometimes called electron (or more generally, Fermi) gas. The hydraulic analogy is not arbitrary or coincidental but works so well because the underlying physics are similar. Understanding the water analogy (and its limits) will result in a better understanding of electricity as well.

If you search a bit on the net you will find pages (one is on wikipedia) which flesh out the analogy with mechanical items which correspond to basic elements in electric circuits, like voltage sources, resistors, capacitors and inductivities.

The most unsatisfying hydraulic analogy is for voltage: As others explained, voltage is a measure of the work done on an electron which travels between two points in space (and incidentally, its natural unit is the electron volt, eV, the work done on one electron traveling a potential difference of 1V). The work is done by an electrostatic field — essentially, a property of the vacuum — exerting a force on charges. There is an analogy with water: the gravitational field; alas, we cannot manipulate gravitational fields as easily as we manipulate electric fields. Therefore, gravitational fields are almost static at the human scale so that the energy imbued on a mass traveling in a circle¹ is zero. Gravitation cannot drive water circuits.²

The usual analogy chosen instead is a pump. But that analogy is not that bad because the pressure in a pipe actually is very comparable to the "pressure" exerted by a voltage source on the electron gas in a wire. After all, the electrons flow sometimes "against" the external electric field, even though their direction of travel is entirely determined by the forces exerted on them by an electric field. The reason is similar to the reason water travels upward in a system of connected pipes: the electrons are "pushed" forward by the electrons "behind" them.

Note that the analogy ends there: The electrons "pushing" create a local electric field which drives the front electrons forward — which, as we remember, follow exclusively electric field forces. By contrast, the water "pushing" water in front of it through a pipe, potentially against gravity, does not create a (significant) local gravitational field. The forces emerge, perhaps ironically, from electromagnetically interacting electrons. But that's relatively unimportant for the fact that the two systems behave very similarly.

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_{¹ A circle relative to a close point on the earth's surface, that is (because it's anything but a circle relative to, say, another point on earth's surface, or its center, or the sun, or the Milky Way).}

_{² If we ignore thermal expansion driven ones.}

Answer 11

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.

Answer 12

'Exact' is not possible. You are always going to have to view it in terms of some sort of model. Unless of course you simply use ideal circuit equations, and don't worry about what's under the hood.

As the models get more exact, we eventually end up in quantum mechanics, I'm not sure you want that.

There's a lot can be done with the hydraulic analogy, it even has inductors and diodes you can build a boost converter out of ( hydraulic ram ), because the energy equations can be made to have similar forms.

After the hydraulic analogy, comes the Drude model, which is classical, predicts some effects quite well, and does astonishingly badly at others, getting factors of 10^3 wrong. There are plenty of quantum treatments that make better predictions. If you're interested, you can search for them, I'd hesitate to call any of them intuitive.

Answer 13

Charge

I am not a professional at the subject yet, but here is my understanding. The first thing I believe you should understand is charge. well I don't think there is way of understanding it. the same way we don't understand why the electrons spin or why opposite magnetic fields attract each other. but it could in short be told as a property. if you think about it though not exactly it is like length, or width. I know you didn't want analogies but this comparisons will help to understand. so charge could simply be understood as one of the properties that describe the fundamental particles that make up the universe. The same way density explains the property of a matter by telling us how much matter is confined in a given volume, this property charge tells us a beautiful property of fundamental particles as we will see.

Back in the days when physicists started studying this fundamental particles they have found they saw some interactions between them, and this interaction some times pulls them together and some times push them apart. from this interaction we saw that some will attract each other and some repeal. and when put through different circumstances they act similarly but in opposite ways. for example if put in the same field the experience the same force but will accelerate into different directions. but if put together they will come together. We arbitrarily called one of them proton and the other electron. well for the once that show no interaction we called them neutron. off course there are many subatomic particles discovered this days including there antimatter counter parts but i won't discuss them here. This property we talked about the subatomic particles we named it charge. and the magnitudes of this quantities where found later by great scientists like miller, with his miller oil drop experiment.

the amount of charge discovered was $q=\pm 1.602 x \mathrm{10}^{-19}$ , the sign is depending on the particle if it electron or proton. as discussed above they have opposite charges. It has a unit of columb. and for this to become 1 columb $6.24 ˟ \mathrm{10}^{18}$ of them should come together. you can see this simple math $\pm 1.602 x \mathrm{10}^{-19}$c $x$ $6.24 x \mathrm{10}^{18} = \pm 1$c

Current

If the concept of charge is clear, the next thing i believe to deal with is current. the fact that electrons move leads us to this idea of "how much charge is passing by in a given time", mathematically described as $I=\frac{dq}{dt}$.

Voltage

The next thing that would come to mind after describing current would be voltage. Voltage is described as the amount of work done on a given object with a charge of $Q$. mathematically expressed as $V=\frac{W}{Q}$. If you like to visualize this interns of gravitational fields you can assume voltage is the counter part of $\frac{W_g}{m}$. so the voltage is a quantity that tells you the potential of a chraged object to do work.

the formula of voltage i told you above ($V=\frac{W}{Q}$) can be simplified to

=> $V=\frac{F.d}{Q}$

=> $V=\frac{\frac{KQQ_2}{\mathrm{d}^{2}}.d}{Q}$

=> $V=\frac{KQ_2}{\mathrm{d}^{2}}.d$

=> $V=Ed$, where E is the electric field strength and d is the distance at which the charged object is found from the fields initial position. which interns of gravity is similar as saying $gh$, h being the distance of the object with mass away from the gravitational field, which tells you its ability to do work,its potential. that is why voltage is sometimes called Potential

Resistance and ohms law

i think resistance is a self explanatory term. it is how much the flow of the current is resisted or opposed. it is described by ohms law for ohmic materials. it is stated as how much potential has been used to cause the flow of how many electrons. mathematically described as $R=\frac{V}{I}$. As can be seen in the equation, the higher the resistance the less electron flow(current) there is going to be. which is similar to say the current flow is reduced due the fact the material it is flowing through is resitive.

More to think about

Though not very sure i have read some where that Maxwell's equation are modeled using the idea of fluid flow. not only that fluid dynamics helps to model many problems.

hope this helped!!!

Answer 14

You are looking for clear concepts which accurately describe the reality. I'm afraid you will be disappointed.

Still, we can look at how past physicists have thought of it. Before they knew about electrons, they thought of electric charge as a sort of fluid that could flow through wires like water through a pipe. Voltage was the force that pushed the fluid, and amperage was the rate that the fluid actually moved. Resistance was like a filter that resisted the movement of the fluid and slowed it down. The higher the pressure difference, the faster the fluid moved through the filter.

But then they found out about electrons. It's electrons that flow through the wire. Amperage is the rate that electrons pass any cross-section of the wire. Voltage is the electromagnetic force that pushes them. Resistance is whatever-it-is that slows them down.

But then when they looked at it in detail, the more they found out the more confused they got. Is it a few electrons traveling fast, or a lot of electrons traveling slow? When you look at how many electrons there are in a mole of copper wire, it's HUGE. Averaged over all the electrons, 1 ampere moves them on average very very slowly.

But if they can move at relativistic speeds, then things like magnetism etc can be explained by that. It's tempting to believe it's a few of them moving very fast.

They calculated that electrons DO move at relativistic speed through copper, in random directions. Voltage gives them a slight tendency to travel more in one direction than another.

But assuming they move slow, still the electric field moves fast. How can electrons moving very very slowly out of battery terminals heat up a light bulb filament meters away? The electric field must be mostly independent of the motion of the electrons.

The electrons all repel each other. Mathematically we can assume all the moving electrons are on the two-dimensional outside skin of the wire, and we get results that way which match experiment. Are there no moving electrons inside the wire? Well maybe a few.

Mathematically we can assume that the electric field is not even inside the wire, but only outside the wire, in the insulation.

The whole story made less and less sense, until with the invention of quantum mechanics they just chucked the whole thing. Quantum mechanics predicts the experimental results very very well, whenever it's a problem that's simple enough we know how to do the math. Nobody understands what it means. Nobody has a model which makes sense, to describe what the math says is happening. But it HAS to be the only correct description because it predicts so well. You don't need to understand it if you can calculate the correct results.

And that's where we stand today on Ohm's Law. Ohm's Law approximates what happens in electric circuits in some limited circumstances. You don't have to understand why it works, if you know when to apply it.