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ocw.mit.edu. Let's try to discuss a bit how
things relate to physics. There are two main things I
want to discuss. One of them is what curl says
about force fields and, in particular,a nice
consequence of that concerning gravitational attraction.
More about curl. If we have a velocity field,
then we have seen that the curl measures the rotation affects.
More precisely curl v measures twice the angular velocity,
or maybe I should say the angular velocity vector because
it also includes the axis of rotation.
I should say maybe for the rotation part of a motion.
For example, just to remind you,
I mean we have seen this guy a couple of times,
but if I give you a uniform rotation motion about the z,
axes. That is a vector field in which
the trajectories are going to be circles centered in the z-axis
and our vector field is just going to be tangent to each of
these circles. And, if you look at it from
above, then you will have this rotation vector field that we
have seen many times. Typically, the velocity vector
for this would be minus yi plus yj times maybe a number that
represents how fast we are spinning,
the angular velocity in gradients per second.
And then. if you compute the curl of
this, you will end up with two omega times k.
Now, the other kinds of vector fields we have seen physically
are force fields. The question is what does the
curl of a force field mean? What can we say about that?
The interpretation is a little bit less obvious,
but let's try to get some idea of what it might be.
I want to remind you that if we have a solid in a force field,
we can measure the torque exerted by the force on the
solid. Maybe first I should remind you
about what torque is in space. Let's say that I have a piece
of solid with a mass, delta m for example,
and I have a force that is being exerted to it.
Let's say that maybe my force might be F times delta m.
If you think, for example,
a gravitational field. The gravitational force is
actually the gravitational field times the mass.
I mean you can forget delta m if you don't like it.
And let's say that the position vector, which should be aiming
for the origin, R is here.
And now let's say that maybe this guy is at the end of some
arm or some metal thing and I want to hold it in place.
The force is going to exert a torque relative to the origin
that will try to measure how much I am trying to swing this
guy around the origin. And, consequently,
how much effort I have to exert if I want to actually maintain
its place by just holding it at the end of the stick here.
So the torque is now a vector, which is just the cross-product
of a position vector with a force.
What the torque measures again is the rotation effects of the
force. And if you remember the
principle that the derivative of velocity,
which is acceleration, is force divided by mass then
the derivative of angular velocity should be angular
acceleration which is related to the torque per unit mass.
To just remind you, if I look at translation
motions, say I am just looking at the
point mass so there are no rotation effects then force
divided by mass is acceleration, which is the derivative of
velocity. And so what I am claiming is
that for rotation effects we have a similar law,
which maybe you have seen in 8.01.
Well, it is one of the important things of solid
mechanics, which is the torque of a force divided by the moment
of inertia. I am cheating a little bit here.
If you can see how I am cheating then I am sure you know
how to state it correctly. And if you don't see how I am
cheating then let's just ignore the details.
[LAUGHTER] Is angular acceleration.
And angular acceleration is the derivative of angular velocity.
If I think of curl as an operation,
which from a velocity field gives the angular velocity of
its rotation effects, then you see that the curl of
an acceleration field gives the angular acceleration in the
rotation part of the acceleration effects.
And, therefore, the curl of a force field
measures the torque per unit moment of inertia.
It measures how much torque its force field exerts on a small
test solid placed in it. If you have a small solid
somewhere, the curl will just measure how much your solid
starts spinning if you leave it in this force field.
In particular, a force field with no curl is a
force field that does not generate any rotation motion.
That means if you put an object in there that is completely
immobile and you leave it in that force field,
well, of course it might accelerate in some direction but
it won't start spinning. While, if you put it in there
spinning already in some direction, it should continue to
spin in the same way. Of course, maybe there will be
friction and things like that which will slow it down but this
force field is not responsible for it.
The cool consequence of this is if a force field F derives from
a potential -- That is what we have seen about conservative
forces. Our main concern so far has
been to say if we have a conservative force field it
means that the work of a force is the change in the energy.
And, in particular, we cannot get energy for free
out of it. And the change in the potential
energy is going to be the change in kinetic energy.
You have conservation of energy principles.
There is another thing that we know now because if a force
derives from a potential then that means its curl is zero.
That is the criterion we have seen for a vector field to
derive from a potential. And if the curl is zero then it
means that this force does not generate any rotation effects.
For example, if you try to understand where
the earth comes from, well, the earth is spinning on
itself as it goes around the sun.
And you might wonder where that comes from.
Is that causes by gravitational attraction?
And the answer is no. Gravitational attraction in
itself cannot cause the earth to start spinning faster or slower,
at least if you assume the earth to be a solid,
which actually is false. I mean basically the reason why
the earth is spinning is because it was formed spinning.
It didn't start spinning because of gravitational
effects. And that is a rather deep
purely mathematical consequence of understanding gravitation in
this way. It is quite spectacular that
just by abstract thinking we got there.
What is the truth? Well, the truth is the earth,
the moon and everything is slightly deformable.
And so there is deformation, friction effects,
tidal effects and so on. And these actually cause
rotations to get slightly synchronized with each other.
For example, if you want to explain why the
moon is always showing the same face to the earth,
why the rotation of a moon on itself is synchronized with its
revolution around the earth, which is actually explained by
friction effects over time and the gravitational attraction of
the earth and the moon. There is something there,
but if you took perfectly rigid, solid bodies then
gravitation would never cause any rotation effects.
Of course that tells us that we do not know how to answer the
question of why is the earth spinning.
That will be left for another physics class.
I don't have a good answer to that.
That was kind of 8.01-ish. Let me now move forward to 8.02
stuff. I want to tell you things about
electric and magnetic fields. And, in fact,
something that is known as Maxwell's equations.
Just a quick poll. How many of you have been
taking 8.02 or something like that?
OK. That is not very many. For most of you this is a
preview. If you have been taking 8.02,
have you seen Maxwell's equations, at least part of
them? Yeah.
OK. Then I am sure,
in that case, you know better than me what I
am going to talk about because I am not a physicist.
But just in case. Maxwell's equations govern how
electric and magnetic fields behave, how they are caused by
electric charges and their motions.
And, in particular, they explain a lot of things
such as how electric devices work, but also how
electromagnetic waves propagate. In particular,
that explains light and all sorts of waves.
It is thanks to them, you know, your cell phone,
laptops and things like that work.
Anyway. Hopefully most of you know that
the electric field is a vector field that basically tells you
what kind of force will be exerted on a charged particle
that you put in it. If you have a particle carrying
an electric charge then this vector field will tell you,
basically there will be an electric force which is the
charge times E that will be exerted on that particle.
And that is what is responsible, for example,
for the flow of electrons when you have a voltage difference.
Because classically this guy is a gradient of a potential.
And that potential is just electric voltage.
The magnetic field is a little bit harder to think about if you
have never seen it in physics, but it is what is causing,
for example, magnets to work.
Well, basically it is a force that is also expressed in terms
of a vector field usually called B.
Some people call it H but I am going to use B.
And that force tends to cause it, if you have a moving charged
particle, to deflect its trajectory and start rotating in
a magnetic field. What it does is not quite as
easy as what an electric field does.
Just to give you formulas, the force caused by the
electric field is the charge times the electric field.
And the force caused by the magnetic field,
I am never sure about the sign. Is that the correct sign?
Good. Now, the question is we need to
understand how these fields themselves are caused by the
charged particles that are placed in them.
There are various laws in there that explain what is going on.
Let me focus today on the electric field.
Maxwell's equations actually tell you about div and curl of
these fields. Let's look at div and curl of
the electric field. The first equation is called
the Gauss-Coulomb law. And it says that the divergence
of the electric field is equal to,
so this is a just a physical constant,
and what it is equal to depends on what units you are using.
And this guy rho, well, it is not the same rho as
in spherical coordinates because physicists somehow pretended
they used that letter first. It is the electric charge
density. It is the amount of electric
charge per unit volume. What this tells you is that
divergence of E is caused by the presence of electric charge.
In particular, if you have an empty region of
space or a region where nothing has electrical charge then E has
divergence equal to zero. Now, that looks like a very
abstract strange equation. I mean it is a partial
differential equation satisfied by the electric field E.
And that is not very intuitive in any way.
What is actually more intuitive is what we get if we apply the
divergence theorem to this equation.
If I think now about any closed surface,
and I want to think about the flux of the electric field out
of that surface, we haven't really thought about
what the flux of a force field does.
And I don't want to get into that because there is no very
easy answer in general, but I am going to explain soon
how this can be useful sometimes.
Let's say that we want to find the flux of the electric field
out of a closed surface. Then, by the divergence
theorem, that is equal to the triple
integral of a region inside of div E dV,
which is by the equation one over epsilon zero,
that is this constant, times the triple integral of
rho dV. But now, if I integrate the
charge density over the entire region,
then what I will get is actually the total amount of
electric charge inside the region.
That is the electric charge in D.
This one tells us, in a more concrete way,
how electric charges placed in here influence the electric
field around them. In particular,
one application of that is if you want to study capacitors.
Capacitors are these things that store energy by basically
you have two plates, one that contains positive
charge and a negative charge. Then you have a voltage between
these plates. And, basically,
that can provide electrical energy to power maybe an
electric circuit. That is not really a battery
because it doesn't store energy in large enough amounts.
But, for example, that is why when you switch
your favorite gadget off it doesn't actually go off
immediately but somehow you see things dimming progressively.
There is a capacitor in there. If you want to understand how
the voltage and the charge relate to each other,
the voltage is obtained by integrating the electric field
from one plate to the other plate.
And the charges in the plates are what causes the electric
field between the plates. That is how you can get the
relation between voltage and charge in these guys.
That is an example of application of that.
Now, of course, if you haven't seen any of this
then maybe it is a little bit esoteric, but that will tell you
part of what you will see in 8.02.
Questions? I see some confused faces.
Well, don't worry. It will make sense some day.
[LAUGHTER] The next one I want to tell you
about is Faraday's law. In case you are confused,
Maxwell's equations, there are four equations in the
set of Maxwell's equations and most of them don't carry
Maxwell's name. That is a quirky feature.
That one tells you about the curl of the electric field.
Now, depending on your knowledge,
you might start telling me that the curl of the electric field
has to be zero because it is the gradient of the electric
potential. I told you this stuff about
voltage. Well, that doesn't account for
the fact that sometimes you can create voltage out of nowhere
using magnetic fields. And, in fact,
you have a failure of conservativity of the electric
force if you have a magnetic field.
What this one says is the curl of E is not zero but rather it
is the derivative of the magnetic field with respect to
time. More precisely it tells you
that what you might have learned about electric fields deriving
from electric potential becomes false if you have a variable
magnetic field. And just to tell you again that
is a strange partial differential equation relating
these two vector fields. To make sense of it one should
use Stokes' theorem. If we apply Stokes' theorem to
compute the work done by the electric field around a closed
curve, that means you have a wire in
there and you want to find the voltage along the wire.
Now there is a strange thing because classically you would
say, well, if I just have a wire with nothing in it there is no
voltage on it. Well, a small change in plans.
If you actually have a varying magnetic field that passes
through that wire then that will actually generate voltage in it.
That is how a transformer works. When you plug your laptop into
the wall circuit, you don't actually feed it
directly 110 volts, 120 volts or whatever.
There is a transformer in there. What the transformer does it
takes some input voltage and passes that through basically a
loop of wire. Not much seems to be happening.
But now you have another loops of wire that is intertwined with
it. Somehow the magnetic field
generated by it, and it has to be a donating
current. The donating current varies
over time in the first wire. That generates a magnetic field
that varies over time, so that causes 2B by 2t and
that causes curl of the electric field.
And the curl of the electric field will generate voltage
between these two guys. And that is how a transformer
works. It uses Stokes' theorem.
More precisely, how do we find the voltage
between these two points? Well, let's close the loop and
let's try to figure out the voltage inside this loop.
To find a voltage along a closed curve places in a varying
magnetic field, we have to do the line integral
along a closed curve of the electric field.
And you should think of this as the voltage generated in this
circuit. That will be the flux for this
surface bounded by the curve of curl E dot dS.
That is what Stokes' theorem says.
And now if you combine that with Faraday's law you end up
with the flux trough S of minus dB over dt.
And, of course, you could take, if your loop doesn't move over
time, I mean there is a different
story if you start somehow taking your wire and somehow
moving it inside the field. But if you don't do that,
if it is the field that is moving then you just can take
the dB by dt outside. But let's not bother.
Again, what this equation tells you is that if the magnetic
field changes over time then it creates, just out of nowhere,
and electric field. And that electric field can be
used to power up things. I don't really claim that I
have given you enough details to understand how they work,
but basically these equations are the heart of understanding
how things like capacitors and transformers work.
And they also explain a lot of other things,
but I will leave that to your physics teachers.
Just for completeness, I will just give you the last
two equations in that. I am not even going to try to
explain them too much. One of them says that the
divergence of the magnetic field is zero,
which somehow is fortunate because otherwise you would run
into trouble trying to understand surface independence
when you apply Stokes' theorem in here.
And the last one tells you how the curl of the magnetic field
is caused by motion of charged particles.
In fact, let's say that the curl of B is given by this kind
of formula, well, J is what is called the vector
of current density. It measures the flow of
electrically charged particles. You get this guy when you start
taking charged particles, like electrons maybe,
and moving them around. And, of course,
that is actually part of how transformers work because I have
told you running the AC through the first loop generates a
magnetic field. Well, how does it do that?
It is thanks to this equation. If you have a current passing
in the loop that causes a magnetic field and,
in turn, for the other equation that causes an electric field,
which in turn causes a current. It is all somehow intertwined
in a very intricate way and is really remarkable how well that
works in practice. I think that is basically all I
wanted to say about 8.02. I don't want to put your
physics teachers out of a job. [LAUGHTER]
If you haven't seen any of this before,
I understand that this is probably not detailed enough to
be really understandable, but hopefully it will make you
a bit curious about that and prompt you to take that class
someday and maybe even remember how it relates to 18.02.
