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### Course: AP®︎/College Physics 2 > Unit 2

Lesson 1: Electric charge and electric force# Coulomb's law

Coulomb's law describes the strength of the electrostatic force (attraction or repulsion) between two charged objects. The electrostatic force is equal to the charge of object 1 times the charge of object 2, divided by the distance between the objects squared, all times the Coulomb constant (k).

## Want to join the conversation?

- How do we know that there are only two types of charges and not three?(44 votes)
- Great question! A complete answer to this requires very advanced mathematics, unfortunately, but I will try to give a taste of the idea. First, you have to know this secret: almost everything you learn in the first three years of physics is
**not really true**. Newton's law of gravity, Coulomb's law of electrostatics, and Maxwell's laws of electromagnetism are all just**approximately**true, for systems which are on the human scale of time, space, energy, and speed. For hundreds of years, people thought Newton and Coulomb had found the EXACT FINAL PERFECT TRUE laws of physics, and only in the 20th century, when**relativity**and**quantum mechanics**were discovered, did physicists learn that the truth is**totally, radically different**from these laws for very tiny objects and high speeds.

So, for every force, it turns out that there is not really a "field" in the way we learn in intro physics, but instead the "force" is caused by the exchange of some particle. For the electric force, the force-carrier is the photon, which is sort of like a "chunk" of oscillating electromagnetic field which flies around at the speed of light. Every force also has a mathematical symmetry associated with it, and for the electric force that symmetry is the symmetry of the circle (this is called the "U(1) symmetry group"). If you think about a circle with some points on it labeled, the only thing you can do to it that will leave it exactly the same is**rotate it an integer number of times**. Putting this together with a lot of**very**advanced math, the result is that electric charge has to come in**integer**amounts. So it's not exactly that there are "two types" of electric charge, but more like "electric charge must come in chunks of ..., -3, -2, -1, 0, 1, 2, 3, ..." (i.e. integer number of chunks. the size of that chunk can only be discovered by experiment).

Now, you might then ask, "But how do we know that the symmetry is U(1)?" The answer to that would be that we can only guess what the symmetry is and then**do experiments to find out**. It turns out that if the symmetry group is not U(1), then the force-carriers must themselves carry some kind of charge, and that would mean that photons would significantly affect other photons! But if you do the experiment of crossing two laser beams, you can see that light (photon beams) has no direct effect on other light. Charged particles (electrons and protons) affect (produce, absorb, bend) light, and light (or radio waves or x-rays, they are all photons) affects charged particles, but light passes straight through other light. This shows that the symmetry of electromagnetism is U(1), and thus that electric charge comes in integer chunks.

Other forces have other symmetries, though! For example, the symmetry of the strong force (which holds the quarks together inside protons and neutrons, and holds the protons and neutrons together inside atomic nuclei) is a much more exotic symmetry called "SU(3)". This means that the force-carrying particle of the strong force (called "gluons") DOES come in more than two types. The "charge" for the strong force is called "color charge", and comes in THREE types, which physicists call red, green, and blue. It also means that the particles which are exchanged to produce the strong force, gluons, carry "color charge" themselves, unlike photons which have no electric charge. So a beam of gluons would**not**just pass through another beam of gluons like one laser beam does pass through another.(197 votes)

- why we take the absolute value of two charges ?(28 votes)
- It's because we already know that the charges will attract (in this case) each other as one is positive and the other is negative. In any case, we can visually determine this property of the question based on the type of the charge. So if we happen to calculate the force between like charges, we know that there will be repulsion, whether large or small in magnitude. On the other hand, if we calculate the force between unlike charges, we know that there will be attraction, whether the magnitude of that attraction is large or small. So we are actually calculating the magnitude and not the direction. We can visually determine the direction.(10 votes)

- at10:25why does the denominator change from 0.5 to 0.25?(10 votes)
- the formula goes like this F=Kq1*q2/r^2. so at10:25the denominator changes because it gets squared and 0.5^2 is 0.25(12 votes)

- the gravitational force does not depend up on medium,but why electrostatic force does?(14 votes)
- Why does Coulomb's law use the 'metres' unit instead of a far smaller unit like micrometres or something? It seems really inefficient to describe such small variables as atoms in terms of such large distances.(6 votes)
- The meter is the standard unit of length for the SI system. It is not all that common to use irregular units like cm or mm because the units are easily confused when performing a calculation. It is standard practice to use all base units whenever possible and take care of the large/small number problem with scientific notation. Also, Coulomb's law is used to determine the force between point charges, not necessarily atoms. It is frequently used on the macroscopic scale in which meters are fully sensible.(27 votes)

- How can we say that the force varies as 1/r^2 and not as 1/r^2.0001?(11 votes)
- So is electrostatic force greater than gravity?(11 votes)
- It depends on the scale of the objects and the amount of charge. In the case of two small, charged particles, the electrostatic force will be greater than the gravitational force because its mass is so small. However, two large planets (with large mass and no net charge) will have a stronger gravitational force. You can prove this by plugging in the values to both Coulomb's law (F = k*(|q1*q2|)/r^2, and Newton's Law of gravitation. (F = G*(M1*M2)/R^2).(4 votes)

- I have heard that charged and neutral objects attract each other. But in the formula

F=k*q1*q2/d^2, if we substitute q2=0, the result that we get is zero. How is this possible?(7 votes)- A charged object can 'induce' a charge onto a neutral object. This can cause polarisation of the charges distributed inside the neutral object and a force of attraction results.(7 votes)

- What are newtons? and what's the difference between Newtons and Coulombs? Thank you!(3 votes)
- Coulomb is a measure of charge.

Newton is a measure of force.

Coulomb's law tells you how much force there is between charges.(13 votes)

- Is the 1.8x10^7 acting on EACH of the charged particles, or is it halved (1 half of the 1.8x10^7 acts on each of the two particles)?(5 votes)
- It si the force that each particle exerts on the other.

There is nothing to cut in half.(4 votes)

## Video transcript

- [Voiceover] So we've already started to familiarize ourselves
with the notion of charge. We've seen that if two
things have the same charge, so they're either both positive, or they are both negative, then they are going to repel each other. So in either of these cases these things are going
to repel each other. But if they have different charges, they are going to attract each other. So if I have a positive
and I have a negative they are going to attract each other. This charge is a property of matter that we've started to observe. We've started to observe of
how these different charges, this framework that we've created, how these things start to
interact with each other. So these things are going to, these two things are going
to attract each other. But the question is, what causes, how can we predict how strong the force of attraction or repulsion is going to be between charged particles? And this was a question
people have noticed, I guess what you could
call electrostatics, for a large swathe of
recorded human history. But it wasn't until the 16 hundreds and especially the 17 hundreds, that people started to seriously view this as something that they could manipulate and even start to predict
in a kind of serious, mathematical, scientific way. And it wasn't until
1785, and there were many that came before Coulomb, but in 1785 Coulomb formally published what is known as Coulomb's law. And the purpose of Coulomb's law, Coulomb's law, is to predict what is
going to be the force of the electrostatic force
of attraction or repulsion between two forces. And so in Coulomb's law, what it states is is if I have two charges, so let me, let's say this
charge right over here, and I'm gonna make it in white, because it could be positive or negative, but I'll just make it q
one, it has some charge. And then I have in Coulombs. and then another charge
q two right over here. Another charge, q two. And then I have the distance
between them being r. So the distance between these two charges is going to be r. Coulomb's law states that the force, that the magnitude of the force, so it could be a repulsive force or it could be an attractive force, which would tell us the
direction of the force between the two charges, but the magnitude of the force, which I'll just write it as F, the magnitude of the electrostatic force, I'll write this sub e here, this subscript e for electrostatic. Coulomb stated, well this is going to be, and he tested this, he didn't
just kind of guess this. People actually were assuming
that it had something to do with the products of the magnitude of the charges and that as the particles got further and further away the electrostatic force dissipated. But he was able to actually measure this and feel really good
about stating this law. Saying that the magnitude
of the electrostatic force is proportional, is proportional, to the product of the
magnitudes of the charges. So I could write this
as q one times q two, and I could take the
absolute value of each, which is the same thing as just taking the absolute value of the product. Here's why I'm taking the
absolute value of the product, well, if they're different charges, this will be a negative number, but we just want the overall
magnitude of the force. So we could take, it's proportional to the absolute value of the
product of the charges and it's inversely proportional to not just the distance between them, not just to r, but to the
square of the distance. The square of the distance between them. And what's pretty neat about this is how close it mirrors
Newton's law of gravitation. Newton's law of gravitation,
we know that the force, due to gravity between two masses, remember mass is just
another property of matter, that we sometimes feel is
a little bit more tangible because it feels like we can
kind of see weight and volume, but that's not quite the same, or we feel like we can feel or internalize things like weight and volume which are related to mass, but in some ways it is
just another property, another property, especially
as you get into more of a kind of fancy physics. Our everyday notion of even mass starts to become a lot more interesting. But Newton's law of gravitation says, look the magnitude of the force of gravity between two masses is going
to be proportional to, by Newton's, by the gravitational concept, proportional to the
product of the two masses. Actually, let me do it
in those same colors so you can see the relationship. It's going to be proportional to the product of the two
masses, m one m two. And it's going to be
inversely proportional to the square of the distance. The square of the distance
between two masses. Now these proportional
personality constants are very different. Gravitational force, we kind of perceive this
is as acting, being strong, it's a weaker force in close range. But we kind of imagine it
as kind of what dictates what happens in the, amongst the stars and
the planets and moons. While the electrostatic
force at close range is a much stronger force. It can overcome the
gravitational force very easily. But it's what we consider happening at either an atomic level
or kind of at a scale that we are more familiar to operating at. But needless to say,
it is very interesting to see how this parallel
between these two things, it's kind of these
patterns in the universe. But with that said, let's actually apply let's actually apply Coulomb's law, just to make sure we feel
comfortable with the mathematics. So let's say that I have a charge here. Let's say that I have a charge here, and it has a positive
charge of, I don't know, let's say it is positive five times 10 to the negative three Coulombs. So that's this one right over here. That's its charge. And let's say I have this
other charge right over here and this has a negative charge. And it is going to be, it is going to be, let's
say it's negative one... Negative one times 10 to the negative one Coulombs. And let's say that the
distance between the two, let's that this distance right here is 0.5 meters. So given that, let's figure out what the what the electrostatic force between these two are going to be. And we can already predict that it's going to be an
attractive force because they have different signs. And that was actually
part of Coulomb's law. This is the magnitude of the force, if these have different
signs, it's attractive, if they have the same sign then they are going to repel each other. And I know what you're saying, "Well in order to actually calculate it, "I need to know what K is." What is this electrostatic constant? What is this electrostatic
constant going to actually be? And so you can measure that
with a lot of precision, and we have kind of modern numbers on it, but the electrostatic constant, especially for the sake of this problem, I mean if we were to get
really precise it's 8.987551, we could keep gone on and
on times 10 to the ninth. But for the sake of our
little example here, where we really only have one significant digit for each of these. Let's just get an approximation, it'll make the math a little bit easier, I won't have to get a calculator out, let's just say it's approximately nine times 10 to the ninth. Nine times 10 to the ninth. Nine times, actually let me
make sure it says approximately, because I am approximating here, nine times 10 to the ninth. And what are the units going to be? Well in the numerator here, where I multiply Coulombs times Coulombs, I'm going to get Coulombs squared. This right over here is going to give me, that's gonna give me Coulombs squared. And this down over here is going to give me meters squared. This is going to give me meters squared. And what I want is to get rid of the Coulombs and the meters and end up with just the Newtons. And so the units here are actually, the units here are Newtons. Newton and then meters squared, and that cancels out
with the meters squared in the denominator. Newton meter squared over Coulomb squared. Over, over Coulomb squared. Let me do that in white. Over, over Coulomb squared. So, these meter squared will cancel those. Those Coulomb squared in the denomin... over here will cancel with those, and you'll be just left with Newtons. But let's actually do that. Let's apply it to this example. I encourage you to pause the video and apply this information
to Coulomb's law and figure out what
the electrostatic force between these two
particles is going to be. So I'm assuming you've had your go at it. So it is going to be, and this is really just applying the formula. It's going to be nine
times 10 to the ninth, nine times 10 to the ninth, and I'll write the units here, Newtons meter squared
over Coulomb squared. And then q one times q two,
so this is going to be, let's see, this is going to be, actually let me just write
it all out for this first this first time. So it's going to be times five times ten to the negative three Coulombs. Times, times negative one. Time ten to the negative one Coulombs and we're going to take
the absolute value of this so that negative is going to go away. All of that over, all of that over and we're in kind of the
home stretch right over here, 0.5 meters squared. 0.5 meters squared. And so, let's just do a
little bit of the math here. So first of all, let's look at the units. So we have Coulomb squared here, then we're going to have
Coulombs times Coulombs there that's Coulombs squared
divided by Coulombs squared that's going to cancel with that and that. You have meters squared here, and actually let me just write it out, so the numerator, in the numerator, we are going to have so if we just say nine times five times, when we take the absolute value, it's just going to be one. So nine times five is going to be, nine times five times negative... five times negative one is negative five, but the absolute value there, so it's just going to be five times nine. So it's going to be 45 times 10 to the nine, minus three, minus one. So six five, so that's going to be 10 to the fifth, 10 to the fifth, the Coulombs
already cancelled out, and we're going to have
Newton meter squared over, over 0.25 meters squared. These cancel. And so we are left with, well if you divide by 0.25, that's the same thing as dividing by 1/4, which is the same thing
as multiplying by four. So if you multiply this times four, 45 times four is 160 plus 20 is equal to 180 times 10 to the fifth Newtons. And if we wanted to write
it in scientific notation, well we could divide this by, we could divide this by 100
and then multiply this by 100 and so you could write this as 1.80 times one point... and actually I don't
wanna make it look like I have more significant
digits than I really have. 1.8 times 10 to the seventh, times 10 to the seventh units, I just divided this by 100
and I multiplied this by 100. And we're done. This is the magnitude of
the electrostatic force between those two particles. And it looks like it's fairly significant, and this is actually a good amount, and that's because this is
actually a good amount of charge, a lot of charge. Especially at this
distance right over here. And the next thing we have to think about, well if we want not just the magnitude, we also want the direction, well, they're different charges. So this is going to be
an attractive force. This is going to be an
attractive force on each of them acting at 1.8 times ten
to the seventh Newtons. If they were the same charge,
it would be a repulsive force, or they would repel each
other with this force. But we're done.