Let's review a little bit of what we had learned many, many
videos ago about gravitational potential energy and then see
if we can draw the analogy, which is actually very strong,
to electrical potential energy.
So what do we know about
gravitational potential energy?
If we said this was the surface of the Earth-- we
don't have to be on Earth, but it makes visualization easy.
We could be anywhere that has gravity, and the potential
energy would be due to the gravitational field of that
particular mass, but let's say this is the
surface of the Earth.
We learned that if we have some mass m up here and that
the gravitational field at this area-- or at least the
gravitational acceleration-- is g, or 9.8 meters per second
squared, and it is h-- we could say, I guess, meters,
but we could use any units.
Let's say it is h meters above the ground, that the
gravitational potential energy of this object at that point
is equal to the mass times the acceleration of gravity times
the height, or you could view it as the force of gravity,
the magnitude of the force of gravity.
You know, it's a vector, but we can say the magnitude of
the vector times height.
And so what is potential energy?
Well, we know that if something has potential energy
and if nothing is stopping it and we just let go, that
energy, at least with gravitational potential
energy, the object will start accelerating downwards, and a
lot of that potential energy, and eventually all of it, will
be converted to kinetic energy.
So potential energy is energy that is being stored by an
object's situation or kind of this notional energy that an
object has by virtue of where it is.
So in order for something to have this notional energy,
some energy must have been put into it.
And as we learned with gravitational potential
energy, you could view gravitational potential energy
as the work necessary to move an object to that position.
Now, if we're talking about work to move something into
that position, or whatever, we always have to think about,
well, move it from where?
Well, when we talk about gravitational potential
energy, we're talking about moving it from the surface of
the Earth, right?
And so how much work is required to move that same
mass-- let's say it was here at first-- to move it from a
height of zero to a height of h?
Well, the whole time, the Earth, or the force of
gravity, is going to be F sub g, right?
So essentially, if I'm pulling it or pushing it upwards, I'm
going to have to have-- and let's say at a constant
velocity-- I'm going to have to have an equal and opposite
force to its weight to pull it up.
Otherwise, it would accelerate downwards.
I'd have to do a little bit more just to get it moving, to
accelerate it however much, but then once I get it just
accelerating, essentially I would have to apply an upward
force, which is equivalent to the downward force of gravity,
and I would do it for a distance of h, right?
What is work?
Work is just force times distance.
Force times distance, and it has to be force in the
direction of the distance.
So what's the work necessary to get this mass up here?
Well, the work is equal to the force of gravity times height,
so it's equal to the
gravitational potential energy.
Now this is an interesting thing.
Notice we picked the reference point as the surface of the
Earth, but we could have picked any
arbitrary reference point.
We could have said, well, from 10 meters below the surface of
the Earth, which could have been down here, or we could
have actually said, you know, from a platform that's 5
meters above the Earth.
So it actually turns out, when you think of it that way, that
potential energy of any form, but especially gravitational
potential energy-- and we'll see electrical potential
energy-- it's always in reference to some other point,
so it's really a change in potential energy that matters.
And I know when we studied potential energy, it seemed
like there was kind of an absolute potential energy, but
that's because we always assume that the potential
energy of something is zero the surface of the Earth and
that we want to know the potential energy relative to
the surface of the Earth, so it would be kind of, you know,
how much work does it take to take something from the
surface of the Earth to that height?
But really, we should be saying, well, the potential
energy of gravity-- like this statement shouldn't be, you
know, this is just the absolute
potential energy of gravity.
We should say this is the potential energy of gravity
relative to the surface of the Earth is equal to the work
necessary to move something, to move that same mass, from
the surface of the Earth to its current position.
We could have defined some other term that is not really
used, but we could have said potential energy of gravity
relative to minus 5 meters below the surface of the
Earth, and that would be the work necessary to move
something from minus 5 meters to its current height.
And, of course, that might matter.
What if we cut up a hole and we want to see what is the
kinetic energy here?
Well, then that potential energy would matter.
Anyway, so I just wanted to do this review of potential
energy because now it'll make the jump to electrical
potential energy all that easier, because you'll
actually see it's pretty much the same thing.
It's just the source of the field and the source of the
potential is something different.
So electrical potential energy, just actually we know
that gravitational fields are not constant, we can assume
they're constant maybe near the surface of the Earth and
all that, but we also know that electrical fields aren't
constant, and actually they have very similar formulas.
But just for the simplicity of explaining it, let's assume a
constant electric field.
And if you don't believe me that one can be constructed,
you should watch my videos that involve a reasonable bit
of calculus that show that a uniform electric field can be
generated by an infinite uniformly charged plane.
Let's say this is the side view of an infinite uniformly
charged plane and let's say that this
is positively charged.
Of course, you can never get a proper side view of an
infinite plane, because you can never kind of cut it,
because it's infinite in every direction, but let's say that
this one is and this is the side view.
So first of all, let's think about its electric field.
It's electric field is going to point upward, and how do we
know it points upward?
Because the electric field is essentially what is-- and this
is just a convention.
What would a positive charge do in the field?
Well, if this plate is positive, a positive charge,
we're going to want to get away from it.
So we know the electric field points upward and we know that
it's constant, that if these were field vectors, that
they're going to be the same size, no matter how far away
we get from the source of the field.
And I'm just going to pick a number for the
strength of the field.
We actually proved in those fancy videos that I made on
the uniform electric field of an infinite, uniformly charged
plane that we actually proved how you could calculate it.
But let's just say that this electric field is equal to 5
newtons per coulomb.
That's actually quite strong, but it makes the math easy.
So my question to you is how much work does it take to take
a positive point charge-- let me pick a different color.
Let's say this is the starting position.
It's a positive 2 coulombs.
Once again, that's a massive point charge, but
we want easy numbers.
How much work does it take it to move that 2-coulomb charge
3 meters within this field?
How much work?
So we're going to start here and we're going to move it
down towards the plate 3 meters, and it's ending
position is going to be right here, right?
That's when it's done.
How much work does that take?
Well, what is the force of the field right here?
What is the force exerted on this 2-coulomb charge?
Well, electric field is just force per charge, right?
So if you want to know the force of the field at that
point-- let me draw that in a different color.
The force of the field acting on it, so let's say the field
force, or the force of the field, actually, is going to
be equal to 5 newtons per coulomb times 2 coulombs,
which is equal to 10 newtons.
We know it's going to be upward, because this is a
positive charge, and this is a positively charged infinite
plate, so we know this is an upward force of 10 newtons.
So in order to get this charge, to pull it down or to
push it down here, we essentially have to exert a
force of 10 newtons downwards, right?
Exert a force of 10 newtons in the direction of the movement.
And, of course, just like we did with gravity, we have to
maybe do a little bit more than that just to accelerate
it a little bit just so you have some net downward force,
but once you do, you just have to completely balance the
upward force.
So just for our purposes, you have a 10-newton force
downward and you apply that force for a distance of 3
meters, the work that you put to take this 2-coulomb charge
from here to here, the work is going to be equal to 10
newtons-- that's the force-- times 3 meters.
So the work is going to equal 30 newton-meters, which is
equal to 30 joules.
A joule is just a newton-meter.
And so we can now say since it took us 30 joules of energy to
move this charge from here to here, that within this uniform
electric field, the potential energy of the charge here is
relative to the charge here.
You always have to pick a point relative to where the
potential is, so the electrical potential energy
here relative to here and this is electrical potential
energy, and you could say P2 relative to P1-- I'm using my
made-up notation, but that gives you a sense of what it
is-- is equal to 30 joules.
And how could that help us?
Well, if we also knew the mass-- let's say that this
charge had some mass.
We would know that if we let go of this object, by the time
it got here, that 30 joules would be-- essentially
assuming that none of it got transmitted to heat or
resistance or whatever-- we know that all of it would be
kinetic energy at this point.
So actually, we could work it out.
Let's say that this does have a mass of 1 kilogram and we
were to just let go of it, right?
We used some force to bring it down here, and then we let go.
So we know that the electric field is going to accelerate
it upwards, right?
It's going to exert an upward force of 5 newtons per
coulomb, and the thing's going to keep [COUGHS]--
excuse me-- keep accelerating until it gets
to this point, right?
What's its velocity going to be at that point?
Well, all of this electrical potential energy is going to
be converted to kinetic energy.
So essentially, we have 30 joules is going to be equal to
1/2 mv squared, right?
We know the mass, I said, is 1, so we get 60 is equal to v
squared, so the velocity is the square root of 60, so it's
7 point something, something, something meters per second.
So if I just pull that charge down, and it has a mass of 1
kilogram, and I let go, it's just going to accelerate and
be going pretty fast once it gets to this point.
Anyway, I'm 12 minutes into this video, so I will continue
in the next, but hopefully, that gives you a sense of what
electrical potential energy is, and really, it's no
different than gravitational potential energy.
It's just the source of the field is different.
See you soon.