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 802. 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 perfect coupling between the left
and the right sides, also increases the magnetic field. This is the--
let's call this the 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 assignment
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 801. 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. 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 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 poynting 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 poynting 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 poynting 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. [whhsst] [whhsst] [whhsst] [whhsst] 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 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 this 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. [low humming sound] A hundred thirteen hertz. [sound continues, slowly fades out] 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. [wsshht] [wsshht] [wsshht] [wsshht] [wsshht] 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 E x
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 E x in the x direction
plus E y in the y direction. And so the magnitude of this vector will be the square root of E x squared
plus E y 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?" 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 are 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 the answer 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.
