Let's imagine that instead of having two charges, we just
have one charge by itself, sitting in a
vacuum, sitting in space.
So that's this charge here, and let's say its charge is Q.
That's some number, whatever it is.
That's it's charge.
And I want to know, if I were to place another charge close
to this Q, within its sphere of influence, what's going to
happen to that other charge?
What's going to be the net impact on it?
And we know if this has some charge, if we put another
charge here, if this is 1 coulomb and we put another
charge here that's 1 coulomb, that they're both positive,
they're going to repel each other, so there will be some
force that pushes the next charge away.
If it's a negative charge and I put it here, it'll be even a
stronger force that pulls it in because it'll be closer.
So in general, there's this notion of what we can call an
electric field around this charge.
And what's an electric field?
We can debate whether it really exists, but what it
allows us to do is imagine that somehow this charge is
affecting the space around it in some way that whenever I
put-- it's creating a field that whenever I put another
charge in that field, I can predict how the field will
affect that charge.
So let's put it in a little more quantitative term so I
stop confusing you.
So Coulomb's Law told us that the force between two charges
is going to be equal to Coulomb's constant times-- and
in this case, the first charge is big Q.
And let's say that the second notional charge that I
eventually put in this field is small q, and then you
divide by the distance between them.
Sometimes it's called r because you can kind of view
the distance as the radial distance
between the two charges.
So sometimes it says r squared, but it's the distance
between them.
So what we want to do if we want to calculate the field,
we want to figure out how much force is there placed per
charge at any point around this Q, so, say, at a given
distance out here.
At this distance, we want to know, for a given Q, what is
the force going to be?
So what we can do is we could take this equation up here and
divide both sides by this small 1, and say, OK, the
force-- and I will arbitrarily switch colors.
The force per charge at this point-- let's call that d1--
is equal to Coulomb's constant times the charge of the
particle that's creating the field divided by-- well, in
this case, it's d1-- d1 squared, right?
Or we could say, in general-- and this is the definition of
the electric field, right?
Well, this is the electric field at the point d1, and if
we wanted a more general definition of the electric
field, we'll just make this a general variable, so instead
of having a particular distance, we'll define the
field for all distances away from the point Q.
So the electric field could be defined as Coulomb's constant
times the charge creating the field divided by the distance
squared, the distance we are away from the charge.
So essentially, we've defined-- if you give me a
force and a point around this charge anywhere, I can now
tell you the exact force.
For example, if I told you that I have a minus 1 coulomb
charge and the distance is equal to-- oh, I don't know.
The distance is equal to let's say-- let's make it easy.
Let's say 2 meters.
So first of all, we can say, in general, what is the
electric field 2 meters away from?
So what is the electric field out here?
This is 2, right?
And it's going to be 2 meters away.
It's radial so it's actually along this whole circle.
What is the electric field there?
Well, the electric field at that point is going to be
equal to what?
And it's also a vector quantity, right?
Because we're dividing a vector quantity by a scalar
quantity charge.
So the electric field at that point is going to be k times
whatever charge it is divided by 2 meters, so divided by 2
meters squared, so that's 4, right, distance squared.
And so if I know the electric field at any given point and
then I say, well, what happens if I put a negative 1 coulomb
charge there, all I have to do is say, well, the force is
going to be equal to the charge that I place there
times the electric field at that point, right?
So in this case, we said the electric field at this point
is equal to-- and the units for electric field are newtons
per coulomb, and that makes sense, right?
Because it's force divided by charge,
so newtons per coulomb.
So if we know that the electric charge-- well, let me
put some real numbers here.
Let's say that this is-- I don't know.
It's going to be a really large number, but let's say
this-- let me pick a smaller number.
Let's say this is 1 times 10 to the
minus 6 coulombs, right?
If that's 1 times 10 to the minus 6 coulombs, what is the
electric field at that point?
Let me switch colors again.
What's the electric field at that point?
Well, the electric field at that point is going to be
equal to Coulomb's constant, which is 9 times 10 to the
ninth-- times the charge generating the field-- times 1
times 10 to the minus 6 coulombs.
And then we are 2 meters away, so 2 squared.
So that equals 9 times 10 to the third divided by 4.
So I don't know, what is that?
2.5 times 10 to the third or 2,500 newtons per coulomb.
So we know that this is generating a field that when
we're 2 meters away, at a radius of 2 meters, so roughly
that circle around it, this is generating a field that if I
were to put-- let's say I were to place a 1 coulomb charge
here, the force exerted on that 1 coulomb charge is going
to be equal to 1 coulomb times the electric fields, times
2,500 newtons per coulomb.
So the coulombs cancel out, and you'll have 2,500 newtons,
which is a lot, and that's because 1 coulomb is a very,
very large charge.
And then a question you should ask yourself: If this is 1
times 10 to the negative 6 coulombs and this is 1
coulomb, in which direction will the force be?
Well, they're both positive, so the force is going to be
outwards, right?
So let's take this notion and see if we can somehow draw an
electric field around a particle, just to get an
intuition of what happens when we later put a charge anywhere
near the particle.
So there's a couple of ways to visualize an electric field.
One way to visualize it is if I have a-- let's say I have a
point charge here Q.
What would be the path of a positive charge if I placed it
someplace on this Q?
Well, if I put a positive charge here and this Q is
positive, that positive charge is just going to accelerate
outward, right?
It's just going to go straight out, but it's going to
accelerate at an ever-slowing rate, right?
Because here, when you're really close, the outward
force is very strong, and then as you get further and further
away, the electrostatic force from this charge becomes
weaker and weaker, or you could say the field becomes
weaker and weaker.
But that's the path of a-- it'll just be radially
outward-- of a positive test charge.
And then if I put it here, well, it would be radially
outward that way.
It wouldn't curve the way I drew it.
It would be a straight line.
I should actually use the line tool.
If I did it here, it would be like that, but then I can't
draw the arrows.
If I was here, it would out like that.
I think you get the picture.
At any point, a positive test charge would just go straight
out away from our charge Q.
And to some degree, one measure of-- and these are
called electric field lines.
And one measure of how strong the field is, is if you
actually took a unit area and you saw how dense
the field lines are.
So here, they're relatively sparse, while if I did that
same area up here-- I know it's not that obvious.
I'm getting more field lines in.
But actually, that's not a good way to view it because
I'm covering so much area.
Let me undo both of them.
You can imagine if I had a lot more lines, if I did this
area, for example, in that area, I'm capturing two of
these field lines.
Well, if I did that exact same area out here, I'm only
capturing one of the field lines, although you could have
a bunch more in between here.
And that makes sense, right?
Because as you get closer and closer to the source of the
electric field, the charge gets stronger.
Another way that you could have done this, and this would
have actually more clearly shown the magnitude of the
field at any point, is you could have-- you could say,
OK, if that's my charge Q, you could say, well, really close,
the field is strong.
So at this point, the vector, the newtons per coulomb, is
that strong, that strong, that strong, that strong.
We're just taking sample points.
You can't possibly draw them at every single point.
So at that point, that's the vector.
That's the electric field vector.
But then if we go a little bit further out, the vector is
going to be-- it falls off.
This one should be shorter, then this one should be even
shorter, right?
You could pick any point and you could actually calculate
the electric field vector, and the further you go out, the
shorter and shorter the electric field vectors get.
And so, in general, there's all sorts of things you can
draw the electric fields for.
Let's say that this is a positive charge and that this
is a negative charge.
Let me switch colors so I don't have to erase things.
If I have to draw the path of a positive test charge, it
would go out radially from this charge, right?
But then as it goes out, it'll start being attracted to this
one the closer it gets to the negative, and then it'll curve
in to the negative charge and these arrows go like this.
And if I went from here, the positive one will be repelled
really strong, really strong, it'll accelerate fast and it's
rate of acceleration will slow down, but then as it gets
closer to the negative one, it'll speed up again, and then
that would be its path.
Similarly, if there was a positive test charge here, its
path would be like that, right?
If it was here, its path would be like that.
If it was here, it's path would be like that.
If it was there, maybe its path is like that, and at some
point, its path might never get to that-- this out here
might just go straight out that way.
That one would just go straight out, and here, the
field lines would just come in, right?
A positive test charge would just be naturally attracted to
that negative charge.
So that's, in general, what electric field lines show, and
we could use our little area method and see that over here,
if we picked a given area, the electric field is much weaker
than if we picked that same area right here.
We're getting more field lines in than we do right there.
So that hopefully gives you a little sense for what an
electric field is.
It's really just a way of visualizing what the impact
would be on a test charge if you bring it
close to another charge.
And hopefully, you know a little bit
about Coulomb's constant.
And let's just do a very simple-- I'm getting this out
of the AP Physics book, but they say-- let's do a little
simple problem: Calculate the static electric force between
a 6 times 10 to the negative sixth coulomb charge.
So 6 times-- oh, no, that's not on an electric field.
Oh, here it says: What is the force acting on an electron
placed in an external electric field where the electric field
is-- they're saying it is 100 newtons per coulomb at that
point, wherever the electron is.
So the force on that, the force in general, is just
going to be the charge times the electric field, and they
say it's an electron, so what's the
charge of an electron?
Well, we know it's negative, and then in the first video,
we learned that its charge is 1.6 times 10 to the negative
nineteenth coulombs times 100 newtons per coulomb.
The coulombs cancel out.
And this is 10 squared, right?
This is 10 to the positive 2, so it'll be 10 to the minus 19
times 10 to the positive 2.
The force will be minus 1.6 times 10 to
the minus 17 newtons.
So the problems are pretty simple.
I think the more important thing with electric fields is
to really understand intuitively what's going on,
and kind of how it's stronger near the point charges, and
how it gets weaker as it goes away, and what the field lines
depict, and how they can be used to at least approximate
the strength of the field.
I will see you in the next video.