These are the subjects will be covered during our third exam.
There's no way I can cover all during this review.
Nor can I cover all of them of course during the exam.
I can only touch upon a few of them.
And what I cannot cover today, what I will not cover today,
can and will be on the exam. Let's first look at magnetic
materials. Magnetic materials come in
dia-, para- and ferromagnetic
materials. The molecules and the atoms in
para- and ferromagnetic materials have intrinsic
magnetic dipole moments. These have always -- they're
always a multiple of the Bohr magneton.
Has to do with quantum mechanics.
It's not part of eight oh two. And they are going to be
aligned by the external field, I call that um the vacuum
field. And the degree of success
depends on the temperature and on the strength of that external
field. The lower the temperature the
easier it is to align them, to overcome the thermal
agitation. And above a certain temperature
which we call the Curie temperature, ferromagnet-
magnetic material loses all its qualities and becomes
paramagnetic and I have demonstrated that during my
lectures. Suppose we have a solenoid and
the solenoid has N windings, and the length of the solenoid
is N. And the current I is flowing
through the solenoid. Then the magnetic field
generated by that solenoid which I have called the vacuum field,
that magnetic field can be derived using Ampere's law,
which you see down -- down there.
That magnetic field is approximately mu zero times I
times N divided by L. If now I put in here
ferromagnetic material then I have
to include this factor kappa of M or K of M, whatever you want
to call it. The magnetic permeability,
and this can be huge. This can be ten,
a hundred, even up to a thousand and higher.
So you get an enormous increase in magnetic field strength.
Self-inductance is defined as magnetic flux divided by the
current I. That's just the definition of
self-inductance. If the magnetic field goes up
by a factor of kappa M then of course the magnetic flux will go
up by the same factor and so the
self-inductance will go up. And you may remember a
demonstration that I did when I had an iron core which I moved
inside the solenoid and depending upon how far I moved
it in could we see that the self-inductance went up and when
I pulled it out self-inductance went down again.
We have an interesting problem. I think it is assignment seven,
whereby we have iron core here and then we have somewhere an
air gap and you may want to revisit that to refresh your
memory. Let's now turn to transformers.
A transformer often comes in this shape.
Let me move it a little bit to the right.
Often comes in this shape which is then ferromagnetic material,
to give coupling between the left and the right sides,
also increases the magnetic field.
This is the -- let's call this primary side.
N one windings, index um self-inductance L one.
And here I put in a voltmeter to always monitor that value,
I call that V one. And this is the secondary side.
N two windings. Self-inductance L two.
And I put here a voltmeter which always monitors that
voltage and I call that one V two.
You can show with Faraday's law as I did in class in lectures
that V two divided by V one, let's not worry about plus or
minus signs, is N two divided by N one.
That's a good approximation. Depends on how well the
coupling goes. It depends on several factors,
but you can come very close to this and this means then that if
you make N two larger than N one then you can step up in voltage,
we call that a step-up transformer.
But you can also step down if you make N two smaller than N
one. Under very special conditions
will the power generated on the primary side be all
consumed for a hundred percent or nearly a hundred percent on
the secondary side. That is very very special.
If that's the case then the time averaged power here V one I
one is the same as V two I two here time averaged.
And so as a logical consequence of that you'll find that I two
divided by I one, let's not worry about minus
signs, is that N one divided by N two.
That however is not so easy as you may think.
It only can work approximately and I mentioned that on the side
in my lectures. If the resistance here and the
resistance there is way way smaller than the value for omega
L. And we did try to achieve that
during one of the demonstrations that I gave on this.
I remember we had the induction oven whereby N two was one and N
one was very large, I don't remember what it was
anymore but it was of the order of several hundred,
maybe a thousand, and we managed to get a current
in the secondary which was huge, which was close to one thousand
amperes. It was enough to melt that iron
nail. And we made every effort then
to make sure that the resistance was much much smaller than omega
L. I think problem seven-one of
our assignments deals with that, and very naively assumes that
this is all true. But you should realize that it
is not always so easy to achieve the conditions for that.
So let's now go to RLC circuits there.
Let's take an uh system which has a resistor R,
it has a self-inductor, a pure self-inductor,
L, and a capacitance, C.
AC. And this driving power supply
provides with a voltage V which is V zero times cosine omega T.
Keep in mind that this can be always be sine omega T of
course. There is nothing special about
cosine in life. The steady state solution that
is not when you turn the thing on but if you wait awhile,
you get a steady state solution for the current.
And the current that is going to flow now is V zero divided by
the square root of R squared plus omega L minus one
over omega C squared times the cosine of omega T minus phi.
And the tangent of phi is omega L minus one over omega C divided
by R. We call this the reactance.
The upstairs. For which we give often the
symbol X. And so this is also X then
divided by R. And this whole square root that
we have here, we call that the impedance.
The units are ohms. And we call that Z.
And so the maximum current that you can have,
the current is of course oscillating with angular
frequency omega, the maximum value that you can
have for the current, which I call I max,
is then V zero divided by Z. Then the cosine term is either
plus or minus one. I can plot now this I max as a
function of frequency. So here is frequency and here
is I max. If the frequency is very low or
near zero then this term here becomes infinitely high because
the impedance is infinitely high
and so the current is zero. I max is zero.
There's no current flowing at all.
When we go to very high frequencies it is the omega L
term that goes to infinity. And so again Z goes to infinity
so again I max goes to zero. And for other values of omega
you get an I max which is not zero and so you get a curve like
this which has the name of resonance curve.
This I max reaches a maximum value
when the system is at resonance, that's what we call
resonance. And that's the case clearly
when the reactance is zero. Because when the reactance is
zero this part vanishes. And if the reactance is not
zero then the maximum current can only be lower,
can never be higher. And so when X equals zero
you'll find that omega L is one divided by omega C and so the --
the frequency for which that happens,
I call that omega zero, reminds me that it is the --
the resonance, is one divided by the square
root of LC. When I am at resonance phi
becomes zero. So there is no phase delay
between current and the driving voltage.
They are in phase with each other.
And the value for I max now simply becomes V zero divided by
R. Because the impedance itself
becomes R. Very boring,
very simple, you're looking here at Ohm's
law. When the system is at
resonance, forget the self-inductance,
forget the capacitor, they are not there,
they annihilate each other, and so the system behaves as if
there were only a resistor, and that's exactly what you see
here. I have here some numbers which
you have seen before. During my lectures.
You can download this from the Web but you have to go back to
the lecture when I discussed that.
And you see here s- some numbers for R,
L and C and also for V zero. And I calculate for you here
the resonance frequency. I calculate the frequency also
in terms of kilohertz. And here you see the impedance
and here you see the reactance. If I'm ten percent below
resonance notice that the one over omega C
term is always larger than omega L.
So your reactance in this case becomes minus eighty-six ohms.
The minus sign has of course no consequence for the current
because you have an X squared here.
But notice that Z is now almost exclusively determined by X and
not by R anymore. Because the ten ohms of the R
here play no role, almost no role,
in comparison with the eighty-six.
Z becomes eighty-seven and the maximum current is one-tenth of
an ampere. When you're on resonance,
and that is characteristic for on resonance,
the two omega L and one over omega C eat each other up.
They annihilate each other and so the reactance becomes zero.
So Z now is just pure R. X is zero.
And so the maximum current in this case is one.
Because I chose V zero at ten and I chose R ten.
And then when I'm ten percent above resonance then the
omega L term is larger than the reactance of the capacitor and
accordingly you get a lower current again,
about one-eighth of the -- of an ampere.
And so you see this curve being formed in a very natural way and
that's quantitative you see there some numbers.
So now comes the question which of course in practice is very
important. And that has to do with the
power that is generated by the power
supply. That power comes out in the
form of heat. Heat in the resistor and so if
you time average the power then the time average value you can
take the -- the voltage of the power supply multiply that by
the current. You could also take the time
average value of I squared R. Because all that energy will
ultimately come out in the form of
heat of the resistor. Either one will be fine.
I've decided to take this one. So I will get then V zero
cosine omega T -- the I becomes V zero divided by Z times the
cosine omega T minus phi. This is the power at any moment
in time. I will do the time averaging a
little later. When I see cosine omega T minus
phi, that reminds me of my high school
days, cosine alpha minus cosine -- no, cosine alpha minus beta
is cosine alpha cosine beta plus sine alpha sine beta.
That was drilled into my memory here.
I will never forget that I think.
And so I will write down here -- my math teacher will be proud
of me -- cosine omega T cosine phi plus sine omega T sine phi.
So this is this term. If I'm going to time average
it, I have a cosine omega T multiplied
by sine omega T, that time average is zero.
So this term vanishes. So the time average value of
the power I get a V zero squared, I get a Z here,
and now I have here cosine omega T times cosine omega T.
The time average value of cosine squared omega T is
one-half. So I get a two here.
And then I still have my cosine phi there.
And I'm done. If you like to get rid of this
uh cosine phi, you can do that.
Because remember, the way that phi is defined the
tangent of phi is the reactance divided by R.
You still see it there. So that means if this angle is
ninety degrees that this side must be Z.
That's the square root of X squared plus R squared.
That's this part. And so the cosine of phi is
also R divided by Z. And so if you prefer that --
there's no particular advantage but if you prefer that you can
write down for cosine phi R divided by Z.
And so you get V zero squared times R divided by two and now
you get Z squared. And so there you see the power,
time averaged power in an RLC circuit.
So now we can look at resonance.
It's always a very special situation.
When we are at resonance, Z equals R.
So you replace this capital Z by R.
And then you find V zero squared divided by two R.
That's utterly trivial. You could have predicted that.
It's really Ohm's law staring you in the face.
There is no self-inductance and there is no capacitor at
resonance. So you might as well have
treated it as a simple system only with R.
And you find immediately then that answer.
At any other frequency than omega zero Z would always be
larger than R. You see that immediately here.
And so that means that the average power would always be
lower. So it's only at resonance that
you generate the highest power possible.
All right. Let's go back to our subjects
and see what's next. We did LRC circuits.
Oh yes, we're getting now to traveling waves and standing
waves. Let's start with a traveling
wave in a string. That's always very nice to do
that because the parallel with electromagnetic waves is nearly
a hundred percent. I have a string that oscillates
in the Y direction. And let's say it propagates in
the X direction. Y zero cosine K X minus omega
T. When I see this traveling wave,
so the traveling wave in the Y direction and it propagates in
the X direction, I see immediately that it goes
into the plus X direction, because I have a minus sign
here, it tells me in the plus X direction.
K gives me all the information on the wavelength.
It's two pi divided by lambda. Omega equals two pi times F,
F being the frequency in hertz. It would also be two pi divided
by capital T, capital T
now being the period of one complete oscillation.
The speed of propagation, the disturbance propagates in
the X direction, is omega divided by K.
And so if I draw here at a particular moment in time this
string and this would be Y zero, that would also be Y zero,
and it would propagate in this direction with that velocity,
and this would be the wavelength, lambda,
and that lambda is V times T. It's immediately obvious if
something propagates with a speed V and it has T seconds to
go for one oscillation it moves over a distance lambda.
That's eight oh one. Do not confuse this speed of
propagation with which the disturbance moves with the
actual speed of the atoms, of the particles in the string.
If you were a particle in the string and you were sitting
here, you never move in this direction.
That's the same with water waves.
If a water wave comes by all you do is this.
You go up and down. Your motion is only in the Y
direction. You go from here to there and
you oscillate back and forth with that frequency,
angular frequency omega. And so if you're really
interested in the speed with which you are moving up and
down, that of course,
that speed which we call the transverse speed is dY dT,
and you have to do that then for a particular location X,
which you can choose, wherever you want to sit on
that string. And I think we had a problem
once where we asked you in the homework what the transverse
speed was. So if now move to
electromagnetic waves then very little changes.
Uh take an electromagnetic wave, a plain wave,
whereby the E vector is in the Y direction,
so E is E zero, there's the unit vector in the
Y direction, cosine K X minus omega T.
This could be a sine of course. There is nothing special in
life with a cosine. I come to the same conclusion.
The wave is traveling in the plus X direction and the
velocity, speed of propagation, is omega divided by K and if
this is vacuum which will I assume for now,
then that is C and as we have seen
from Maxwell's equations C is one divided by the square root
of epsilon zero mu zero. Surprising as that is that's
comes out of Maxwell's equations.
And so if now you were asked, which would be a natural thing
for me to ask you, what is the associated magnetic
field, well, the magnetic field is perpendicular to the
direction of propagation. It's also perpendicular to E.
And B zero is E zero divided by C in vacuum.
And so if I make a coordinate system, this is X,
this is Y and this is Z, notice that my coordinate
system always is chosen so that X roof cross Y roof is Z roof,
this is called the right-handed coordinate system.
If you choose any other system you're an idiot.
You always get yourself into trouble.
Make sure that you always choose this as a coordinate
system. And notice I have X cross Y is
Z coming out of the blackboard. At a moment,
a particular moment in time, let's say the E vector is in
this direction, in the Y direction,
I pick a random moment in time and I pick a random location for
X. Now I must be sure that E cross
B is in the direction of propagation.
So E cross B must be in this case in the direction of X.
Because it's going in the plus X
direction. And so my problem is solved.
I know that that can only happen if B is in this
direction. At this moment in time at this
location. And that's all I need.
Because now I can write down that the B vector that is
associated with that electric field is E zero divided by C,
that is the largest value that the magnetic field can have,
times the same cosine K X minus omega T.
And now I must have here Z roof and now I'm in business.
So now I have the E vector that goes with the -- the B vector
that goes with the E vector. We would often be interested in
energy, how much energy per unit area per unit time is in the
plain wave. That is given by the pointing
vector S which is E cross B divided by mu zero.
We're still in vacuum. I still assume that it's all
vacuum, so this is watts per square meter.
Of course we are in general not interested in the instantaneous
value of the pointing vector, who cares about that,
it oscillates like mad, I'm more interested in a time
average value. And so the time average value,
this has a cosine omega T, this has
a cosine omega T, the product average out to be
one-half of cosine square omega T.
I get an E zero. I get a B zero.
And I get a mu zero. And if now you want to get rid
of your B zero because you want to get everything in terms of E
zero, you can replace B zero by E zero divided by C.
And so this would also be fine. E zero squared divided by mu
zero C. And so now we have the time
average value of the pointing vector.
The moment that you move to vacuum -- from vacuum to -- to
matter, so we now go from vacuum, we move to matter,
and the matter has dielectric constant K and it has magnetic
permeability K of M. That's only important if we
deal with ferromagnetism because with paramagnetism and
diamagnetism K of M is always practically for all practical
purposes one. But kappa can vary a great
deal. From the various substances.
And so now all you have to do, if you go to Maxwell's
equations, if they were given only in vacuum,
then you have to replace epsilon zero by kappa epsilon
zero. That's already done there.
And you have to replace mu zero by kappa M times mu zero.
And therefore you have to place C by V because the velocity is
different in matter of electromagnetic radiation.
And you see immediately how it changes because epsilon zero and
mu zero have to be replaced by epsilon zero K and mu zero kappa
of M and so you see here that the velocity following my recipe
becomes epsilon zero mu zero kappa kappa M.
And so my B zero becomes now E zero divided by V and no longer
divided by C. So this B zero becomes E zero
divided by V. Kappa can be a very strong
function of frequency. So can kappa M,
but kappa M of course is only important for ferromagnetic
materials. I've shown you an example
before that kappa, the dielectric constant for
water, was eighty at low frequencies, even at a hundred
megahertz. Radio frequencies.
It was still eighty. But at the visible light where
you're dealing with frequencies of a few times ten
to the fourteen hertz, that value for kappa was way
lower, was one point seven seven.
And so kappa is a small function, a strong function of
um of frequency, and we introduce index of
refraction, which is C divided by V.
And since V itself is a strong function of frequency,
the index of refraction can also be a very strong function
of frequency. Was one point three roughly for
water. But it is a little different
from -- for red light from blue light.
If not, we wouldn't be able to see rainbows.
OK. Let's now talk about standing
waves. Let's start with strings again.
This is a string with length L and I generate in this string a
standing wave. Standing waves can only be
generated at very discrete frequencies for very special
wavelengths. It's a resonance phenomenon.
And the lowest frequency for which this occurs will make the
string oscillate in this fashion.
The string will do this. This is called the fundamental.
Also the first harmonic. Then there is a second harmonic
which is the next frequency up for which it will resonate,
which adds an extra node, there is already a node here
and a node there, and the system will then
oscillate like so. Fweet fweet fweet fweet.
And then so this is the second harmonic.
And then I can go to the third harmonic and I can continue
forever, not quite, but I can continue quite a bit,
and this now would be the third harmonic.
And the frequencies that are the resonant frequencies are
given by F of N, N being Nancy,
that means either one or two or three or four,
one being the fundamental, two being the second harmonic,
this frequency is given N times V divided by two L whereby V is
the speed of propagation of a disturbance along the direction
of the string. And L in this case is the
length. And the asso- associated
wavelength lambda for that particular harmonic is two L
divided by N. If you put in N equals one you
see that the wavelength is indeed twice the length and if
you put in N equals two you see that the wavelength is exactly
L. So let us write down for the
fundamental the equation for a standing wave.
Y one, one refers to the fundamental, would be Y zero
one, times the cosine of omega one T
times the sine of K one X. This is very different from a
traveling wave. All the time information now
here is decoupled from the spatial information.
K one is again as before -- let me write down a nicer K one --
two pi divided by lambda. And omega one is two pi times F
one. What is so special here is that
there are points here, values for X,
for which the sine becomes zero.
In the case of the fundamental put in X equals zero and you
find that that sine is always zero.
But if you put in X equals L, you can check that,
it's also zero. And if you go to the second
harmonic you will find that the sine is zero there as well.
And so there are points now which never move,
we call those nodes. And that's very characteristic
for a standing wave. I could go to the second
harmonic and all I would have to do is put here a two and here a
two and here a two and here a two.
It would have its own little amplitude.
Which would be this. It would have its own
frequency. But that frequency omega two is
nonnegotiable. That must be two omega one.
And if I go to the third harmonic, then omega three is
going to be three omega one. But it has its own wavelength,
you'll get K of two would be two pi divided by lambda two,
and the lambdas are -- the lambdas are given by this
relationship. I want you to get a few
minutes' rest now so that you can digest this and then we'll
go to electromagnetic standing waves.
And I want you to see this again.
You've seen that before but I want you to see it in a way that
you have not seen it yet. We have here a -- a rubber hose
which we can oscillate and we're going to oscillate it in such a
way, at least that's our goal, to get the third harmonic,
and it's not so easy to get exactly on resonance,
but we will try that, and Marcos decided to make it
very beautiful for you, so he's going to put up a --
a black screen there because once we are at resonance we're
going to strobe the s- string so that you will in fact be able to
follow the motion. Your eyes can't see what's
going up and what's going down. It goes too fast.
But when we strobe it we can slow that down,
make the thing actually stand still.
And that's our objective, so you'll be able to see that.
And so let me turn on the strobe, and this black
background will help, the strobe frequency is not
exactly the same as the frequency of the -- of
the uh oscillating string, so that allows you to actually
see the slow motion. And no one is arguing with me,
right, when the center portion is down, the left and the right
portion are up, and vice versa.
So you can really see now the characteristics of this standing
wave. Notice you see in this case
four nodes. One at either end,
and you see two nodes in the middle.
That is where the s- sine of that curve is always zero.
I can add another strobe light at a slightly different
frequency. I collaborated for a few years
with uh an artist, his name was Tsai,
he worked at the Center for Advanced Visual Studies here at
MIT, and he actually made art by oscillating objects at resonance
frequency, rods and strings, and he strobed them in a
way that I'm doing, so I actually learned this from
him. Quite pretty but also very
instructive. You can really see what's going
on. In green you see the s- string
at a s- different moment in time than you see it in red.
And as I said I purposely made the frequencies of the strobe a
little different.
Thanks, Marcos. As you perhaps remember,
you can also generate a standing wave in air itself.
Wind instruments are nothing but air columns which go into
resonance. If I have here a wind
instrument which is open and open on both sides,
this is the length of that instrument,
then the frequencies that I can generate are given by this
equation. The only fundamental difference
between the strings and the wind instruments is that with strings
you can change V at will more or less, you can choose different
materials through which the speed of propagation is
different, and you can also change the tension.
If you increase the tension this V goes up.
So you can give a s- violin four different strings,
they'll give you four different fundamentals.
With a wind instrument you cannot manipulate V because V is
the speed of sound. And so that V of air at room
temperature is three hundred forty meters per second.
That's nonnegotiable. And so you can with wind
instruments, you can very easily predict the frequencies that
you're going to hear. I have here a pipe which is
open and open, open on both sides.
One-and-a-half meters long. And if I apply that equation I
will find that the fundamental would be at a hundred and
thirteen hertz. Perhaps you remember that I
told you that resonances can occur when you sometimes least
expect them. Just by blowing air you can get
resonances. If you take a wind instrument
and you start to blow air it excites resonances.
Remember the Tacoma Bridge. There was wind and it went into
resonance. It was very destructive.
Well here we have a system that we can also make go into
resonance when you least expect it.
We have a -- a copper grid here.
And I'm going to heat that grid.
And when I heat the grid -- get an air flow going through there.
That all by itself will not make it go into resonance.
But when I take my heat source away and that grid starts to
cool, it goes into resonance. So I'm going to heat it now.
Will take a while. If I heat it too long,
the copper grid will melt. Many years ago when I was doing
this with a very short burner the molten copper came down on
my hands, which was not funny, believe me.
If I don't do it long enough, it won't go into resonance.
So I'm sort of guessing this a little.
A hundred thirteen hertz. All right.
Standing electromagnetic waves. When we go to standing
electromagnetic waves let me stay on the center board here.
The situation is almost identical to standing waves on a
string. Again we have -- we have nodes.
We have locations where the electric field is always
zero. Very very different from a
traveling wave. I refer you to problem
nine-four where you will see a standing electromagnetic wave
which just like we had with the -- with the string,
it has exactly this form. The time domain is decoupled
from the spatial domain. The only complication that you
have with a standing wave,
electromagnetic, is that it is not so easy to
find the associated magnetic field.
And therefore I refer you to that problem nine-four if you
want to revisit that. Polarization.
Oh, we want to see the subjects again.
See where we are on the list. But I think the time has come
to talk a little bit about polarization.
That's right. Polarization.
Let's take electromagnetic waves that come
straight out of the blackboard to you.
And I call this the Y direction and I call this the Z direc- the
-- the X direction. And Z is to you.
Notice X cross Y is Z always in my case, right-handed coordinate
system. And so the electric field
vector of the plain wave that is coming out of the blackboard,
let us assume that the electric vector is oscillating like so.
Fweet fweet fweet fweet fweet fweet fweet fweet.
With angular frequency omega. If this is a straight line we
call that linearly polarized radiation.
Could be radio emission as we did with the seventy-five
megahertz transmitter. It can also be visible light.
As long as the E vector stays along a straight line we call
that linearly polarized electromagnetic radiation.
Electromagnetic radiation including radio
waves, including visible light, can be circularly polarized.
In which case the electric vector doesn't oscillate like
you see here but always has the same strength and is rotating
around in a circle, either in this direction or in
this direction. And it's very easy to make,
actually. I could have done that
here in lectures but I never did.
Suppose we have an antenna in the Y direction and we have
another one in the X direction like our seventy-five megahertz
transmitter was a s- copper bar in this direction.
And suppose each one radiates with exactly the same value for
E zero, with exactly the same frequency but they're ninety
degrees out of phase. That's not so difficult to
arrange. Then I would get an EX which
would be E zero. And if I pick my value for Z --
let's take Z equals zero, who cares where you are in this
line, so there is no K Z term, so we simply have a cosine
omega T here. So this is the component of the
electric vector in the direction of X.
And let the one in the Y direction -- must be ninety
degrees out of phase, but must have exactly the same
amplitude, so ninety degrees out of phase would for instance be
sine omega T. Omegas must be the same.
To get circularly polarized radiation.
And so the net electric field vector, the one that you will
experience sitting here on the Z axis, will be EX in the X
direction plus EY in the Y direction.
And so the magnitude of this vector will be the
square root of EX squared plus EY squared, and that is E zero.
Because sine squared omega T plus cosine square omega T is
one. And so you get under the square
root E zero squared times one, and so that's E zero.
And so you see that the amplitude is always E zero.
You see that here in front of you and so it's going to rotate
around when it's maximum in Y, then it is zero in X,
and when it is maximum in X it is zero in Y,
and so therefore you get this rotation, either this way or
that way. Depending upon how the phase
delay is arranged. And you can turn this into
elliptical polarized radiation by simply for instance putting a
two here. If you put a two there or let
me a put a two in the X direction,
because I have more room in the X direction on the blackboard,
so if I put a two here that means that in the X direction I
can go twice as far as I can go in the Y direction and so now
the E vector will go like this. You see this now is twice as
much as this and so now I have elliptically polarized
radiation. OK, so far is for the
polarization is concerned. Let's talk
about Snell's law. Snell's law was discovered two
hundred fifty years before Maxwell.
It's quite an amazing accomplishment even though you
can derive it from Maxwell's equations of course.
But it was derived by this Dutchman two hundred fifty years
earlier. And what Maxwell's e- what uh
Snell's law tells us is that if we have light going from a
medium one with index of refraction N one going to medium
two with index of refraction N two, and this angle of incidence
is theta one, this is the normal to the
surface, and we get some reflection, this angle is also
theta one, and then some of that light will enter into this
medium and this angle will then be
theta two. And Snell's law says that the
sine of theta one divided by the sine of theta two is N one
divided by N two. Always the medium where you're
going is up, the medium where -- oh, it's the other way around.
Ha, good that I caught that, this is N two divided by N one,
so the medium that you're going is up and the medium where you
came from is down.
And so it's immediately obvious that if N two is larger than N
one that the angle of theta two is always smaller than theta
one. Let us assume now that you go
from air to glass but that somehow you come out in the air
again. So right here this is your
angle of incidence now, I call it I,
it's clear that I is theta two. And here you're coming out of
the medium in air again,
and I call this angle R, and I'm going to ask you the
question what is that angle R, now some of you may have great
insight and immediately say oh well it's obvious that it's
going to be the same as theta one.
And that is indeed correct. You can easily see that because
the sine of theta one divided by the sine of theta two at this
transition at point A -- so this is at point A --
would be N of glass divided by N of air, and now we come here
at this point B, so at point B we get the sine
of this angle of incidence I, which we know is theta two,
that's obvious, divided by the sine of R is now
the one where we're going to which is air divided by the one
where we were, so this is N air divided by N
glass. And we already agreed that I is
theta two. And so when I multiply these
two equations, on the right side I get exactly
one. Independent of the color of the
light. If blue light has a different
index of refraction than red light, that doesn't matter,
because I have the same one here that I have there.
And so you get exactly one on the right side,
so you must get exactly one on the left side,
and so the consequence is that theta one must be R,
and so this angle here is the same as that angle,
which is perhaps not so surprising.
Because these two planes are parallel to each other.
If they were not parallel, as they were with one of your
problems where we had a prism, then you would get a separation
of the colors here, and then the red and the blue
would come out in different directions.
In this case red and blue and green and yellow all come out in
the same direction and so you see white light when you look
through plain parallel glass.
When it comes to total reflection, I refer you to
problem nine-eight if you can spare the time.
You're going to have five problems.
Two of them have one question. Two of them have two questions.
And one has four true-false questions.
For each correct answer you get four points.
For each wrong answer I have to subtract four points.
However, you don't have to answer.
If you don't answer you don't gain,
you don't lose points. Now before you hate me for
subtracting four points, think about this for a minute.
If you give true-false questions to a class of
five-year-olds they will have half on average correct and half
wrong. Yet they deserve zero.
So clearly the only reasonable thing is that for a wrong answer
you must subtract points. But you don't have to answer.
So if you know to two for sure out of four you could consider
not to answer the other two. That is your choice.
I'll give you an example. The Benham top consists of
several colors. When you rotate it fast you see
white light. That's wrong.
That's false. Because the Benham top did not
have several colors and when we rotated it we didn't see white
light. I'll give you another example.
One of two tails of comets is due to radiation pressure and
the other is due to the solar wind.
That's correct. We discussed that in lectures.
Let me end with some fatherly advice.
Read each problem at least twice
and do those problems first that you like the best.
Those that suit you the best. Never spend more than ten
minutes on one problem. Then move to another.
There is another review tomorrow evening for three hours
by Ali Nayeri. You may want to attend that.
And we also will provide you with tutoring this Sunday.
Look at the Web. Because we will update it as
the time comes. See you Monday.
Have a good weekend.