Coulomb's law | Physics | Khan Academy

We encounter so many different kinds of forces in our day-to-day lives. There's gravity, there's the tension force, friction, air resistance, spring force, buoyant forces, and so on and so forth. But guess what? Not all these forces are fundamental. Gravity is certainly one of the fundamental forces of nature, but it turns out that most other forces that we encounter in our daily life are actually manifestations of the electromagnetic forces.

These are the forces responsible for all the electric and magnetic phenomena and most of the other forces that we encounter in everyday life. But how? How do electromagnetic forces give rise to all of these? Well, let's get a glimpse of that in this video.

Now, there are two parts to these forces: the electric part and the magnetic part. We will save the magnetic part for future videos. In this video, we'll just stick to the electric part of it, and even there, we'll talk about a particular kind of electric force, which we call the electrostatic forces. Electrostatic, which is also called static electricity. From the word itself, you can see "Electro" means we're sticking to the electric part, and "static" means stationary, where things are not moving a lot or maybe they're moving very slowly.

The reason to do that is because we want to take baby steps. So first, we'll consider what happens when we have static conditions; then we'll see what happens when they are moving, and so on and so forth.

Okay, all right. So when it comes to gravity, we will keep comparing with gravity because we are familiar with gravity. Okay, when it comes to gravity, where does the force of gravity come from? What is it due to? Well, we know that the force of gravity comes from a fundamental property of matter, which we call mass. Things that have mass will produce gravitational force on each other, right? Similarly, what causes electrostatic force?

It turns out electrostatic force comes from a property of matter called, you've probably heard of this, charge. Anything that has charge will put forces on other things that have charge. Now an immediate question that we could have is, "Hey, if that is the case, and if matter has charge, and because of the charge they put electrostatic forces on each other, why don't we notice that? Why don't we notice electrostatic forces between, say, planets and stars and all of those things? Why don't we notice that in our everyday life?"

Well, first of all, we do notice them in everyday life, and we will see that in this particular video. But the reason why we don't notice them on a large scale is a good question, and we'll come back to that. But anyway, if you want to see electrostatic forces in action, it's better to start looking at things inside the atom. So, let's do that.

We've probably seen the model of an atom. We have the nucleus at the center, which contains protons and neutrons, and we all have, you know, kind of like an electron cloud that surrounds the nucleus where you have electrons over there. Now these particles will have charge. Protons have a positive charge, neutrons are neutral; they have no charge, and electrons have a negative charge.

Okay, but we may be wondering how much charge do protons and electrons have? To answer that question, we need to know the unit of the charge, just like how mass has a unit of kilogram. Charge has a unit of something called the Coulomb, named after the scientist Charles Coulomb, who did a lot of work on this, and the symbol we use is capital C.

Now, it turns out that protons and electrons have the same magnitude of this charge. They have different signs, but they have the same magnitude, and we call that number e. That happens to be roughly 1.602 * 10 ^ -19 Coulombs. So it's a very tiny value in terms of Coulombs.

We would now say that the charge on the proton is this much positive, so we'll just call it +1e. The charge on the electron is this much negative, so -1e, and the charge on the neutron? Well, that is zero; it has no charge, so its charge is just zero.

Now we can immediately see a big difference between mass and charge. Mass, there's only one kind, but charge, there are two kinds: positives and negatives. This will now help us understand something. An atom has the same number of protons and electrons, so the total charge of the atom would just be zero because the charge of the proton and the charge of the electron will just cancel out.

So the atom itself will be neutral, and if you consider big objects which have billions and billions of atoms, it's pretty much neutral because the total number of protons is pretty much the same as the total number of electrons. And that's the reason why most things around us are uncharged, or they might have a few extra electrons or protons, so they'll have a very tiny charge, but mostly uncharged.

That's why we don't see electrostatic forces in action most of the time. That's why at celestial scales we don't see electrostatic forces in action because they're mostly uncharged or they have very tiny charge. But at the microscopic level, we do see it. We see protons putting forces on other protons and other electrons of the same atom or of the different atom; they are all there.

But if you want to study these forces, the next big question we should ask is: what is the strength of this force? How much force would, say, a proton put on, say, another proton or maybe on another electron? How do we figure that out? Well, for that, let's assume that we have two, in general, let's consider two charged particles. Let's call these charges as q1 and q2.

You can imagine, for example, these are two pieces of paper, and you know these pieces of paper have some extra electrons or some extra protons. Let's say um, so they are charged. They will put an electrostatic force on each other. The question we want to try and ask is: what is the direction of that force, and what is the strength of that force? What does that depend on?

Let's start with the direction of the force. Positives will push and repel other positives; negatives will push and repel other negatives. In other words, like charges will repel each other, but unlike charges will attract each other. A proton will attract an electron; positives will attract negatives.

So, the direction of the force depends upon their charge, the polarity of that charge. If both are positive or both are negative, they will repel. If one is positive and the other one is negative, they will attract. For simplicity, let's just assume one — let's just say that both are positive. Then they would repel each other.

So this is a repulsive electrostatic force. Now the big question is: what does the strength of this force depend on? Why don't you pause the video and just think about how you think they would be related to q1 and q2, the charges, and the distance between them.

Okay, since the electrostatic forces come from the charges, we would expect that these forces must be directly related to q1 and q2. If either of them increase, we would expect the force to increase. And how would it be related to the distance? Well, if you put them farther away, will we expect the force to become smaller? If you put them very far away, we would expect them to not interact with each other at all, right?

On the other hand, if we bring them closer — that means if you make this distance smaller — you would expect the force to be larger. The closer they are, the larger the force, which means you would expect an inverse relationship with the distance between them.

Now, if you put it all together, we'll get something called Coulomb's law, and it looks like this. Notice that Coulomb's law is giving us something very similar to what we predicted. It is directly the force between the two charges and is directly related to the charges themselves, you can see that, and it is inversely related to the distance between them.

More importantly, you can see an inverse square relationship. Where have we seen an inverse square relationship before? Hey, we've seen it in gravity! We've seen the force of gravity. The universal law of gravitation is very similar over here. G, which we call the universal constant, its value was about this much.

So what is the value of K, which we call the Coulomb constant? Well, it turns out that the value of the Coulomb constant K is about 8.99 * 10 ^ 9 units. And can you work out the units yourself? Well, we just have to isolate K on one side, and if you do that — let me do that very quickly — we'll get F * r² / q1 * q2.

So that will be F is Newtons, r squared is meters squared, divided by q1 * q2 is Coulombs squared. Yeah, I don't have to remember them; I never remember them because I can always rearrange and figure out what the units are. But that's the value of K.

Now that you know this, if you know the value of the charges and you know how far they are, we can plug in and figure out the force between them. Okay, so let's quickly compare these two.

Well, the similarity is the inverse square law. The farther you put them, the farther you go, the smaller the force gets. The force dies out very quickly. But what about some differences? Well, the first difference is you can see gravity is always only attractive, but the electrostatic force can be attractive or repulsive.

That's because we have two kinds of charges over here, right? But there's another thing that we can see. Look at the value of K; it's much bigger compared to the value of G in standard units. From this, we can kind of guess that Coulomb's law is much, much stronger than the force of gravity.

This means if you take, for example, two protons and compare the force of gravity with the electrostatic force, the Coulomb force, you will find the Coulomb force to be way stronger, orders of magnitude stronger than the force of gravity.

And so that's why, at the subatomic scale, we can completely ignore the force of gravity: it's the electrostatic force that dominates. But as we saw, once we go to a much larger celestial scale, well, now the masses are so huge and the charges are so small that the force of gravity dominates.

I absolutely love this, how at different scales, the different forces dominate. And now we're in a position to understand that! That's incredible, isn't it? But that's not all. Now we're in a position to answer our original question: how do electromagnetic forces or electrostatic forces get manifested as some of the daily forces that we see?

For example, tension: where does tension come from? Well, if you zoom into a string, you'll see a lot of, you know, atoms over there. And although atoms are neutral, since they have positives and negatives inside, they can push and pull on other, you know, electrons and protons.

For example, the protons can push on the other protons; the protons can pull on the electrons. The electrons over here can push on the other electrons; the electrons over here can pull on the protons. So there are a lot of forces out there, and you know we can model and say that pretty much all these forces balance out.

They have to because, look, a string is, you know, pretty much static, so you can assume that most of the atoms are pretty much static. They are all balanced, and therefore the net force on all of the atoms is pretty much zero. We say they are in equilibrium.

Now, what happens when you pull on the string? Let's say you attach a mass over here, and because of gravity pulling on the string, well now, because of that, some of the atoms will start moving farther away from each other. Equilibrium gets disturbed; the net force is no longer zero.

It turns out that in the string, because the equilibrium gets disturbed, the atoms go away from each other, and the net force will try to bring them closer back to each other. That is how tension force is generated in a string. Isn't that wonderful? It all comes from the electrostatic forces.

Similarly, think about where does the force of friction come from? Well, again, if we were to zoom in over here, we will see that, you know, although things look smooth at a macro scale, on a microscopic scale, things are not really smooth. And if we zoom in even further, again we will notice that the atoms of the box can interact with the atoms of the floor via the electrostatic forces.

And it's these electrostatic forces which all add up and give rise to the force of friction. Again, it is super complicated; we may not try to understand exactly how the force of friction acts, why it opposes motion, for example, in some cases, but it all comes from the electrostatic forces.

We can use the same idea, the same model, to explain spring forces, air resistance, buoyant forces, pretty much contact forces, and other contact forces that you see in your daily life. I find it absolutely fascinating that even though these models are not very accurate — I mean, today we have better, more accurate models, what we call quantum mechanical models, to explain all these phenomena better — but even if you ignore that, even if we consider, you know, simpler models like we're doing over here, we can use the idea of the electrostatic forces, the Coulomb forces, to try and get an intuition behind how it manifests as most of these daily forces that we encounter in life.

I find that absolutely beautiful.