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- [Voiceover] So we've already started to

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familiarize ourselves
with the notion of charge.

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We've seen that if two
things have the same charge,

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so they're either both positive,

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or they are both negative,

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then they are going to repel each other.

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So in either of these cases

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these things are going
to repel each other.

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But if they have different charges,

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they are going to attract each other.

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So if I have a positive
and I have a negative

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they are going to attract each other.

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This charge is a property of matter

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that we've started to observe.

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We've started to observe of
how these different charges,

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this framework that we've created,

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how these things start to
interact with each other.

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So these things are going to,

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these two things are going
to attract each other.

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But the question is, what causes,

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how can we predict how strong the force

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of attraction or repulsion is going to be

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between charged particles?

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And this was a question
people have noticed,

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I guess what you could
call electrostatics,

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for a large swathe of
recorded human history.

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But it wasn't until the 16 hundreds

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and especially the 17 hundreds,

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that people started to seriously view this

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as something that they could manipulate

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and even start to predict
in a kind of serious,

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mathematical, scientific way.

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And it wasn't until
1785, and there were many

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that came before Coulomb,

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but in 1785 Coulomb formally published

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what is known as Coulomb's law.

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And the purpose of Coulomb's law,

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Coulomb's law,

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is to predict what is
going to be the force of

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the electrostatic force
of attraction or repulsion

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between two forces.

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And so in Coulomb's law, what it states is

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is if I have two charges,

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so let me, let's say this
charge right over here,

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and I'm gonna make it in white,

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because it could be positive or negative,

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but I'll just make it q
one, it has some charge.

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And then I have in Coulombs.

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and then another charge
q two right over here.

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Another charge, q two.

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And then I have the distance
between them being r.

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So the distance between these two charges

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is going to be r.

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Coulomb's law states that the force,

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that the magnitude of the force,

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so it could be a repulsive force

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or it could be an attractive force,

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which would tell us the
direction of the force

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between the two charges,

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but the magnitude of the force,

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which I'll just write it as F,

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the magnitude of the electrostatic force,

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I'll write this sub e here,

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this subscript e for electrostatic.

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Coulomb stated, well this is going to be,

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and he tested this, he didn't
just kind of guess this.

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People actually were assuming
that it had something

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to do with the products of the magnitude

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of the charges and that as the particles

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got further and further away

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the electrostatic force dissipated.

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But he was able to actually measure this

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and feel really good
about stating this law.

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Saying that the magnitude
of the electrostatic force

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is proportional,

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is proportional,

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to the product of the
magnitudes of the charges.

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So I could write this
as q one times q two,

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and I could take the
absolute value of each,

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which is the same thing as just

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taking the absolute value of the product.

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Here's why I'm taking the
absolute value of the product,

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well, if they're different charges,

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this will be a negative number,

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but we just want the overall
magnitude of the force.

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So we could take, it's proportional to

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the absolute value of the
product of the charges

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and it's inversely proportional to

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not just the distance between them,

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not just to r, but to the
square of the distance.

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The square of the distance between them.

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And what's pretty neat about this

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is how close it mirrors
Newton's law of gravitation.

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Newton's law of gravitation,
we know that the force,

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due to gravity between two masses,

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remember mass is just
another property of matter,

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that we sometimes feel is
a little bit more tangible

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because it feels like we can
kind of see weight and volume,

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but that's not quite the same,

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or we feel like we can feel or

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internalize things like weight and volume

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which are related to mass,

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but in some ways it is
just another property,

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another property, especially
as you get into more

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of a kind of fancy physics.

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Our everyday notion of even mass starts to

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become a lot more interesting.

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But Newton's law of gravitation says,

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look the magnitude of the force of gravity

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between two masses is going
to be proportional to,

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by Newton's, by the gravitational concept,

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proportional to the
product of the two masses.

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Actually, let me do it
in those same colors

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so you can see the relationship.

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It's going to be proportional to

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the product of the two
masses, m one m two.

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And it's going to be
inversely proportional

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to the square of the distance.

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The square of the distance
between two masses.

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Now these proportional
personality constants

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are very different. Gravitational force,

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we kind of perceive this
is as acting, being strong,

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it's a weaker force in close range.

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But we kind of imagine it
as kind of what dictates

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what happens in the,

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amongst the stars and
the planets and moons.

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While the electrostatic
force at close range

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is a much stronger force.

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It can overcome the
gravitational force very easily.

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But it's what we consider happening

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at either an atomic level
or kind of at a scale

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that we are more familiar to operating at.

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But needless to say,
it is very interesting

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to see how this parallel
between these two things,

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it's kind of these
patterns in the universe.

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But with that said, let's actually apply

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let's actually apply Coulomb's law,

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just to make sure we feel
comfortable with the mathematics.

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So let's say that I have a charge here.

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Let's say that I have a charge here,

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and it has a positive
charge of, I don't know,

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let's say it is positive five

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times 10 to the negative three Coulombs.

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So that's this one right over here.

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That's its charge.

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And let's say I have this
other charge right over here

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and this has a negative charge.

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And it is going to be,

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it is going to be, let's
say it's negative one...

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Negative one times 10

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to the negative one Coulombs.

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And let's say that the
distance between the two,

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let's that this distance right here

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is 0.5 meters.

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So given that, let's figure out what the

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what the electrostatic force

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between these two are going to be.

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And we can already predict that

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it's going to be an
attractive force because

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they have different signs.

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And that was actually
part of Coulomb's law.

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This is the magnitude of the force,

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if these have different
signs, it's attractive,

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if they have the same sign then they

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are going to repel each other.

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And I know what you're saying,

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"Well in order to actually calculate it,

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"I need to know what K is."

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What is this electrostatic constant?

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What is this electrostatic
constant going to actually be?

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And so you can measure that
with a lot of precision,

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and we have kind of modern numbers on it,

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but the electrostatic constant,

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especially for the sake of this problem,

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I mean if we were to get
really precise it's 8.987551,

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we could keep gone on and
on times 10 to the ninth.

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But for the sake of our
little example here,

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where we really only have

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one significant digit for each of these.

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Let's just get an approximation,

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it'll make the math a little bit easier,

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I won't have to get a calculator out,

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let's just say it's approximately

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nine times 10 to the ninth.

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Nine times 10 to the ninth.

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Nine times, actually let me
make sure it says approximately,

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because I am approximating here,

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nine times 10 to the ninth.

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And what are the units going to be?

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Well in the numerator here,

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where I multiply Coulombs times Coulombs,

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I'm going to get Coulombs squared.

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This right over here is going to give me,

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that's gonna give me Coulombs squared.

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And this down over here is going

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to give me meters squared.

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This is going to give me meters squared.

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And what I want is to get rid of

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the Coulombs and the meters and end up

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with just the Newtons.

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And so the units here are actually,

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the units here are Newtons.

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Newton and then meters squared,

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and that cancels out
with the meters squared

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in the denominator.

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Newton meter squared over Coulomb squared.

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Over, over Coulomb squared.

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Let me do that in white.

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Over, over Coulomb squared.

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So, these meter squared will cancel those.

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Those Coulomb squared in the denomin...

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over here will cancel with those,

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and you'll be just left with Newtons.

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But let's actually do that.

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Let's apply it to this example.

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I encourage you to pause the video

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and apply this information
to Coulomb's law

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and figure out what
the electrostatic force

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between these two
particles is going to be.

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So I'm assuming you've had your go at it.

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So it is going to be, and this is really

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just applying the formula.

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It's going to be nine
times 10 to the ninth,

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nine times 10 to the ninth,

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and I'll write the units here,

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Newtons meter squared
over Coulomb squared.

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And then q one times q two,
so this is going to be,

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let's see, this is going to be,

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actually let me just write
it all out for this first

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this first time.

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So it's going to be times five times ten

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to the negative three Coulombs.

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Times, times negative one.

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Time ten to the negative one Coulombs

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and we're going to take
the absolute value of this

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so that negative is going to go away.

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All of that over, all of that over

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and we're in kind of the
home stretch right over here,

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0.5 meters squared.

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0.5 meters squared.

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And so, let's just do a
little bit of the math here.

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So first of all, let's look at the units.

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So we have Coulomb squared here,

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then we're going to have
Coulombs times Coulombs there

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that's Coulombs squared
divided by Coulombs squared

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that's going to cancel with that and that.

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You have meters squared here,

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and actually let me just write it out,

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so the numerator, in the numerator,

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we are going to have

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so if we just say nine times five

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times, when we take the absolute value,

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it's just going to be one.

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So nine times five is going to be,

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nine times five times negative...

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five times negative one is negative five,

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but the absolute value there,

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so it's just going to be five times nine.

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So it's going to be 45

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times 10 to the nine,

261
00:09:58,955 --> 00:10:02,005
minus three, minus one.

262
00:10:02,005 --> 00:10:04,313
So six five,

263
00:10:04,313 --> 00:10:07,639
so that's going to be 10 to the fifth,

264
00:10:07,639 --> 00:10:10,457
10 to the fifth, the Coulombs
already cancelled out,

265
00:10:10,457 --> 00:10:15,203
and we're going to have
Newton meter squared over,

266
00:10:15,203 --> 00:10:20,203
over 0.25

267
00:10:20,312 --> 00:10:22,562
meters squared. These cancel.

268
00:10:22,562 --> 00:10:24,305
And so we are left with,

269
00:10:24,305 --> 00:10:26,302
well if you divide by 0.25,

270
00:10:26,302 --> 00:10:28,183
that's the same thing as dividing by 1/4,

271
00:10:28,183 --> 00:10:30,458
which is the same thing
as multiplying by four.

272
00:10:30,458 --> 00:10:32,567
So if you multiply this times four,

273
00:10:32,567 --> 00:10:37,330
45 times four is 160

274
00:10:37,330 --> 00:10:40,677
plus 20 is equal to 180

275
00:10:40,677 --> 00:10:44,088
times 10 to the fifth Newtons.

276
00:10:44,088 --> 00:10:46,015
And if we wanted to write
it in scientific notation,

277
00:10:46,015 --> 00:10:48,802
well we could divide this by,

278
00:10:48,802 --> 00:10:51,959
we could divide this by 100
and then multiply this by 100

279
00:10:51,959 --> 00:10:55,456
and so you could write this as 1.80

280
00:10:55,456 --> 00:10:57,478
times one point...

281
00:10:57,478 --> 00:10:58,948
and actually I don't
wanna make it look like

282
00:10:58,948 --> 00:11:00,805
I have more significant
digits than I really have.

283
00:11:00,805 --> 00:11:03,553
1.8 times

284
00:11:03,553 --> 00:11:07,085
10 to the seventh,

285
00:11:07,085 --> 00:11:09,049
times 10 to the seventh units,

286
00:11:09,049 --> 00:11:11,883
I just divided this by 100
and I multiplied this by 100.

287
00:11:11,883 --> 00:11:12,796
And we're done.

288
00:11:12,796 --> 00:11:15,675
This is the magnitude of
the electrostatic force

289
00:11:15,675 --> 00:11:18,369
between those two particles.

290
00:11:18,369 --> 00:11:19,970
And it looks like it's fairly significant,

291
00:11:19,970 --> 00:11:21,132
and this is actually a good amount,

292
00:11:21,132 --> 00:11:22,711
and that's because this is
actually a good amount of charge,

293
00:11:22,711 --> 00:11:23,872
a lot of charge.

294
00:11:23,872 --> 00:11:27,354
Especially at this
distance right over here.

295
00:11:27,354 --> 00:11:28,933
And the next thing we have to think about,

296
00:11:28,933 --> 00:11:31,533
well if we want not just the magnitude,

297
00:11:31,533 --> 00:11:33,042
we also want the direction,

298
00:11:33,042 --> 00:11:34,014
well, they're different charges.

299
00:11:34,014 --> 00:11:36,011
So this is going to be
an attractive force.

300
00:11:36,011 --> 00:11:38,750
This is going to be an
attractive force on each of them

301
00:11:38,750 --> 00:11:42,326
acting at 1.8 times ten
to the seventh Newtons.

302
00:11:42,326 --> 00:11:45,368
If they were the same charge,
it would be a repulsive force,

303
00:11:45,368 --> 00:11:48,176
or they would repel each
other with this force.

304
00:11:48,176 --> 00:00:00,000
But we're done.