If you had two charges,

and we'll keep these straight by giving them a name.

We'll call this one Q1 and I'll call this one Q2.

If you've got these two charges sitting next to each other,

and you let go of them, they're gonna fly apart

because they repel each other.

Like charges repel, so the Q2's gonna get pushed

to the right,

and the Q1's gonna get pushed to the left.

They're gonna start gaining kinetic energy.

They're gonna start speeding up.

But if these charges are gaining kinetic energy,

where is that energy coming from?

I mean, if you believe in conservation of energy,

this energy had to come from somewhere.

So where is this energy coming from?

What is the source of this kinetic energy?

Well, the source is the electrical potential energy.

We would say that electrical potential energy

is turning into kinetic energy.

So originally in this system,

there was electrical potential energy,

and then there was less electrical potential energy,

but more kinetic energy.

So as the electrical potential energy decreases,

the kinetic energy increases.

But the total energy in this system,

this two-charge system, would remain the same.

So this is where that kinetic energy's coming from.

It's coming from the electrical potential energy.

And the letter that physicists typically choose

to represent potential energies is a u.

So why u for potential energy?

I don't know.

Like PE would've made sense, too,

because that's the first two letters

of the words potential energy.

But more often you see it like this.

We'll put a little subscript e

so that we know we're talking about

electrical potential energy

and not gravitational potential energy, say.

So that's all fine and good.

We've got potential energy turning into kinetic energy.

Well, we know the formula for the kinetic energy

of these charges.

We can find the kinetic energy of these charges

by taking one half the mass of one of the charges

times the speed of one of those charges squared.

What's the formula to find the electrical potential energy

between these charges?

So if you've got two or more charges

sitting next to each other,

Is there a nice formula to figure out

how much electrical potential energy there is

in that system?

Well, the good news is, there is.

There's a really nice formula

that will let you figure this out.

The bad news is, to derive it requires calculus.

So I'm not gonna do the calculus derivation in this video.

There's already a video on this.

We'll put a link to that so you can find that.

But in this video, I'm just gonna quote the result,

show you how to use it,

give you a tour so to speak of this formula.

And the formula looks like this.

So to find the electrical potential energy

between two charges, we take K, the electric constant,

multiplied by one of the charges,

and then multiplied by the other charge,

and then we divide by the distance

between those two charges.

We'll call that r.

So this is the center to center distance.

It would be from the center of one charge

to the center of the other.

That distance would be r, and we don't square it.

So in a lot of these formulas,

for instance Coulomb's law, the r is always squared.

For electrical fields, the r is squared,

but for potential energy, this r is not squared.

Basically, to find this formula in this derivation,

you do an integral.

That integral turns the r squared into just an r

on the bottom.

So don't try to square this.

It's just r this time.

And that's it. That's the formula to find

the electrical potential energy between two charges.

And here's something that used to confuse me.

I used to wonder, is this the electrical potential energy

of that charge, Q1?

Or is it the electrical potential energy of this charge, Q2?

Well, the best way to think about this

is that this is the electrical potential energy

of the system of charges.

So you need two of these charges

to have potential energy at all.

If you only had one, there would be no potential energy,

so think of this potential energy as the potential energy

that exists in this charge system.

So since this is an electrical potential energy

and all energy has units of joules if you're using SI units,

this will also have units of joules.

Something else that's important to know

is that this electrical potential energy is a scalar.

That is to say, it is not a vector.

There's no direction of this energy.

It's just a number with a unit that tells you

how much potential energy is in that system.

In other words, this is good news.

When things are vectors,

you have to break them into pieces.

And potentially you've got component problems here,

you got to figure out how much of that vector points right

and how much points up.

But that's not the case with electrical potential energy.

There's no direction of this energy,

so there will never be any components of this energy.

It is simply just the electrical potential energy.

So how do you use this formula?

What do problems look like?

Let's try a sample problem to give you some feel

for how you might use this equation in a given problem.

Okay, so for our sample problem,

let's say we know the values of the charges.

And let's say they start from rest,

separated by a distance of three centimeters.

And after you release them from rest,

you let them fly to a distance 12 centimeters apart.

And we need to know one more thing.

We need to know the mass of each charge.

So let's just say that each charge is one kilogram

just to make the numbers come out nice.

So the question we want to know is,

how fast are these charges going to be moving

once they've made it 12 centimeters away from each other?

So the blue one here, Q1, is gonna be speeding to the left.

Q2's gonna be speeding to the right.

How fast are they gonna be moving?

And to figure this out,

we're gonna use conservation of energy.

For our energy system, we'll include both charges,

and we'll say that if we've included everything

in our system,

then the total initial energy of our system

is gonna equal the total final energy of our system.

What kind of energy did our system have initially?

Well, the system started from rest initially,

so there was no kinetic energy to start with.

There would've only been electric potential energy

to start with.

So just call that u initial.

And then that's gonna have to equal the final energy

once they're 12 centimeters apart.

So the farther apart, they're gonna have less

electrical potential energy but they're still gonna have

some potential energy.

So we'll call that u final.

And now they're gonna be moving.

So since these charges are moving,

they're gonna have kinetic energy.

So plus the kinetic energy of our system.

So we'll use our formula for electrical potential energy

and we'll get that the initial electrical potential energy

is gonna be nine times 10 to the ninth

since that's the electric constant K

multiplied by the charge of Q1.

That's gonna be four microcoulombs.

A micro is 10 to the negative sixth.

So you gotta turn that into regular coulombs.

And then multiplied by Q2, which is two microcoulombs.

So that'd be two times 10 to the negative sixth

divided by the distance.

Well, this was the initial electrical potential energy

so this would be the initial distance between them.

That center to center distance was three centimeters,

but I can't plug in three.

This is in centimeters.

If I want my units to be in joules,

so that I get speeds in meters per second,

I've got to convert this to meters,

and three centimeters in meters is 0.03 meters.

You divide by a hundred,

because there's 100 centimeters in one meter.

And I don't square this.

The r in the bottom of here is not squared,

so you don't square that r.

So that's gonna be equal to

it's gonna be equal to another term

that looks just like this.

So I'm gonna copy and paste that.

The only difference is that now this is the final

electrical potential energy.

Well, the K value is the same.

The value of each charge is the same.

The only thing that's different is

that after they've flown apart,

they're no longer three centimeters apart,

they're 12 centimeters apart.

So we'll plug in 0.12 meters,

since 12 centimeters is .12 meters.

And then we have to add the kinetic energy.

So I'm just gonna call this k for now.

The total kinetic energy of the system

after they've reached 12 centimeters.

Well, if you calculate these terms,

if you multiply all this out on the left-hand side,

you get 2.4 joules of initial electrical potential energy.

And that's gonna equal,

if you calculate all of this in this term,

multiply the charges, divide by .12

and multiply by nine times 10 to the ninth,

you get 0.6 joules of electrical potential energy

after they're 12 centimeters apart

plus the amount of kinetic energy in the system,

so we can replace this kinetic energy of our system

with the formula for kinetic energy,

which is gonna be one half m-v squared.

But here's the problem.

Both of these charges are moving.

So if we want to do this correctly,

we're gonna have to take into account

that both of these charges are gonna have kinetic energy,

not just one of them.

If I only put one half times one kilogram times v squared,

I'd get the wrong answer

because I would've neglected the fact that the other charge

also had kinetic energy.

So we could do one of two things.

Since these masses are the same,

they're gonna have the same speed,

and that means we can write this mass here

as two kilograms times the common speed squared

or you could just write two terms, one for each charge.

This is a little safer.

I'm just gonna do that.

Conceptually, it's a little easier to think about.

Okay, so I solve this.

2.4 minus .6 is gonna be 1.8 joules,

and that's gonna equal one half times one kilogram

times the speed of that second particle squared

plus one half times one kilogram times the speed

of the first particle squared.

And here's where we have to make that argument.