Optics (Lec - 13) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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so I'd like to continue with with what
we're discussing last time we were at
the point where we're discussing the
three-dimensional wave equation
before I start let me take care of a
couple of administrative things so for
an I am very close to finishing grading
the quizzes so as soon as I do I'll put
them on an anvil on the EH saddle and
ship them back to Boston
of course those of you who are in
Singapore will get them from locally and
so that's done the question ocean of
course has been posted you must have
seen it by now so I would encourage you
to go to go over the solution it's a it
was a pretty good problems actually both
of them were pretty good so and the
other thing I want to say is that I have
posted the homework the new homework so
has anybody looked at it yet okay if you
look at it it it might appear to be a
little bit intimidating especially the
graduate version it is about I don't
know six or seven pages but you may know
from experience that if a homework is
very worthy it means that most of the
work has been done for you okay so
basically what I've done is I've I there
is no a this is the first time I give
this assignment and basically developed
the analogy between the optical
Hamiltonian and the mechanical
Hamiltonian and I guide you through a
few dude elevations a few proves that
basically I think in my opinion
elucidate the the analogy and the
difference because they're not exactly
the same okay
and and there is an abilities because
last year was the first time that they
taught the Hamiltonian it used to be no
I used to just skip that part and do
other things but last year I did it and
I got some pretty good feedback from the
students who thought that well you know
this
: allergy so this year I expanded it a
bit so that's the reason this homework
is a little bit rich but but it's not
bad really once you get into it and if
you follow the instructions carefully
each question should take no more than
two lines so it is not like a long
derivation or tedious or anything like
that it's just a step by step and if you
but read it careful because because
there's some derivatives involved so you
want to make sure you're taking the
correct derivative it's time right
that's the only challenge if you can
manage to do it which will be you know
if you follow the instructions carefully
then it should be pendulous and there's
a little bit of MATLAB that you have to
do but that's not about it and the code
actually the MATLAB that you have to do
I also give you the code it's posted on
the website it's a bunch of function and
one of the things that they're doing the
assignment is a walk you through how to
use the MATLAB code so what you have to
do basically is take one of the function
and modified yeah someone turn on the
bike I have a question or okay so going
back to the Alice has a question I'm
gonna stand on your mic which is okay
actually as long as you don't say
anything embarrassing to yourself foot
okay so so let me go back please go back
now to to the the main topic so of
course we've moved on from Hamiltonians
and all the stuff were doing wave optics
now and the last time we discussed the
wave equation and without really a proof
we did the proof of the one-dimensional
case but then without the proof I just
added basically the one-dimensional case
was this one the just a derivative of
the Z term without any additional terms
and anyway you can think of it simply by
by symmetry if you accept that 3d space
is symmetric then the way you go from
one dimensional two three
by just adding two derivatives similar
to the one that you already had and of
course the waves that were getting
through there much later than the waves
that that we expected to get in one day
and here I plotted the two cases one is
a plane wave which I will play as a
movie so you will see the fringes in
this case of the plane wave are moving
in some arbitrary direction now it can
go anywhere in 3d of course I can only
show a slice on the line of your
projector and this is a spherical wave
where the weapons are not circles as you
see here but they're actually spheres
so there's fields that are propagating
clearly outwards in this case so this is
what we used to call it the Virgin
spherical wave in in geometrical optics
well you can also have the opposite you
can have a converging spherical wave if
the wave fronts are moving inwards so
all of these are valid solutions to the
wave equation and the plane wave is very
simple I will do it next the spherical
wave is not really difficult but it
involves a transformation of the
coordinates from Cartesian that we have
here to polar and this is done in the
book I'm not going to do it in the
classes it is a little bit of work and
you don't get any particular intuition
if you take a class in electromagnetics
or you know electrostatics they will
probably do it but because they because
in this class we will use the spherical
wave and its full form very rarely we
will use an approximation to it that's
why I don't I don't bother to do the
full derivation ok I'm going to skip now
a couple of slides and I'm going to go
directly to
so this one slide number 16 which is
which is the plane wave so what I would
like to do so for the rest of the class
now pretty much for the next what is it
seven or eight weeks that we have left
we will be talking almost exclusively
about plane waves and there are super
positions because what we'll discover is
that we can describe every optical wave
and if well at least in within the
approximations that are valid in this
class you can describe every optical
wave as a superposition of plane waves
it's a very powerful property because
plane waves as we will see they're very
simple and they obey some relatively
straightforward properties so if we
learn how to describe those properly
then we can describe the great variety
of other phenomena including diffraction
interference refraction and a number of
other things okay so I would like to
make sure that we understand plane waves
it is a crucial part of the class okay
so in the last lecture without a simple
wave that will call the harmonic wave
and it used to look like this from now
on I will start using a facial notation
almost without without warning right so
when I start throwing complex
Exponential's you know that I'm using a
favor I mean a phaser so okay okay so
having said that now let's look at the
phaser over of what I wrote last time
the last time I wrote something that
looked like e to the i kz e to the minus
I Omega T and I call this a call and I
call this a a harmonic wave so this was
in 1d so the set I'm sorry the plane
wave is basically the generalization of
this one now I'm going to have to write
terms of the form
y & x do you remember what we called the
quantity K in the cache were still had
the one dimensional wave what was the
name of K wave number yeah wave number
yeah
I'm sorry someone else had been singer
persuade okay
but he was dealing it so make sure to
push your button
okay so case the wave number and it is
fair to use it if you have one dimension
but if you have the three dimensions
that we actually need at least
apparently we need three such quantities
so I need to put Z subscript here and
then I need two more one for Y and one
for X so now the triad of these three I
don't want to call them wave numbers
they're not really we have numbers but
these three quantities that they have
there I mean these are essentially
special periods and they write in the
three dimensions so these quantities if
you take the triad of this quantity it
is called the wave vector and it is
usually we use the same symbol as the
wave number but with the vector now and
it consists of the three components K Y
KZ it is a matter of taste how you write
vectors in this class I actually I will
alternate a little bit sometimes I will
write vectors like this as a period in
the parenthesis sometimes I might write
the vector like this when I use the a
unit vector the unit Cartesian vectors
of course okay I hope that's not
confusing me one can is you know clearly
they're both commonly used okay so that
quantity K then is the wave vector
and it's significant you can sit on the
slide is probably better if I don't
attempt to drive there but we consider
the slide so the wave front of the plane
wave is a surface where which is
perpendicular to the wave vector and
what is different
well wavefront is out in the movie that
I played earlier wavefront
is the surface upon which the phase of
the wave is constant okay so what is
this this is basically a
three-dimensional sinusoid of which we
only see a 2d slice alright so this is a
sinusoid which alternates the black
regions are negative the minimum I mean
they're sort of a maximum negative
values the bright stripes are maximum
positive value and the wavefront is the
surface on which the phase of the wave
has the same value for example if the
phase is zero then we're talking about
the positive peak if the face is PI when
we are talking about the negative trap
and so on and so forth and or anywhere
in between for that matter and the
wavefront has a sense of travel to it
because if I play the movie again then
you consider that these surfaces of
constant phase are moving this is the
sense that the wave transport energy
issue is because the weapons are moving
with the wave speed okay so then we'll
call it a plane wave because the
surfaces of constant phase are planes of
course you only see them as lines here
because again we're looking at a slice
of the wave but in 3d there would be
actually planes that will be propagating
with the speed of the wave so that is
the sudenly wave vector we need it in
order to tell us what is the orientation
of the
planes and and foreigners of convenience
would define it to be perpendicular
perpendicular to those planes
now that's one useful thing to know but
that's not all the wave vector it
appears to be composed of say three
elements so the question is are these do
I really have three parameters can I
describe the plane wave with three
parameters
well you even before we do any math you
could probably guess that I shouldn't
because these are surfaces these are
plane planar surfaces in 3d space so
cannot really fit three dimensions of
planes in the 3d space right something
is wrong with that argument even without
doing any math so in actuality these
three numbers K X KY and KZ they have to
be related somehow I cannot arbitrarily
define all three of them but I have to -
somehow - somehow related so the way you
relate them is they actually go back to
my wave equation and recall this thing
here the wave which I will call it
something what they call attack I didn't
call it anything actually
I just put an amplitude coefficient in
front so the amplitude coefficient well
it is it is good for for complete
completeness again and it must satisfy
the wave equation so it must it is fine
I'll call it something out because I
Fergus yeah
so it is f of X Y Z T and it must
satisfy this nasty thing could look like
DF DX square plus DF the Y square plus
DF the Z square minus one over the
velocity now this square F over the
this square it must equal to zero this
is what we call the wave equation and
for future reference sometimes instead
of right in all these always reduce
derivatives there is a shorthand which
collapses all of this with a symbol del
so you just write del square F minus one
over C square DF DT square is zero okay
that's the wave equation very good so
now what I need to do is I need to plug
this expression into the wave equation
so what expression this expression that
the head for the way I need to put it
into the equation okay almost by
inspection we can see what's going to
happen each day so we have an
exponential right and if I take one
derivative the first derivative it will
knock out it will multiply by I K X
right if I take the X derivative if I
take the second X derivative it will
bring out another I KX and since I
square equals -1 it means that the
derivative will look like this D let me
write it separately here the second
derivative for example of F with respect
to X square to liq wanna make well minus
K X square times F itself right and then
I can do the same for y and z and t
except in the case of time I will pull
out Omega square so if i if i substitute
all of that now and i put it in i will
get that - plus a KX square minus KY
square minus K Z square these are the
three spatial derivatives and then minus
Omega square but I have to divide by C
also but because C actually C square is
in the wave equation so all of this
according to the wave equation must
equal to 0 and of course it is possible
to satisfy this by setting F equal to 0
but that's not very interested so
basically what this means that this
quantity here
it must equal to zero so indeed is
advertised the three parameters kxkykz
they're not independent but in
combination specify two of them the
third one is completely determined by
the two among the three plus the
frequency of the wave so if I give you a
mega KX and KY then this equation will
force KZ to have a certain value and you
can see from this equation which is
written both on the whiteboard and the
end on the slide oh oh yes because ever
yes thank you because yeah because I
have a minus sign here so the extra
minus will become a class here yes okay
now it is correct and and so if you look
at this equation now what does it
describe if you think of kxkykz if you
think of them now as a new
three-dimensional space what is the
surface that the three of them are
constrained to lie upon but a spherical
shell sphere that's right I said the
shell of his FIRREA and this fear is not
new to us who have seen it it actually
popped up in a different context who
call it the card sphere it is actually
the same same sphere that actually not
even the cart an Arab scientist in the
11th century guessed from Snell's law so
it is a little bit I guess magical or
soothing or rewarding whatever
combination of those adjectives like
that we found a nice sort of analogy or
not not analogy actually we found an
agreement with with geometrical optics
and the in geometrical optics who said
that the radius of the sphere equals two
pi upon lambda this is now what in our
new language we'll call it the wave
number times the index of refraction and
now this is something that that I
haven't told you yet but where I will
tell you I would tell you later today in
this in this equation haven't said
anything about index of refraction
however you know again from geometrical
optics that see the speed of light
depends on index of refraction so
another way to rewrite this equation is
to say to put the free space speed of
light over here and then in order for
the question to be correct I would have
to multiply by n square then extra
refraction so you can see from this that
indeed the radius of the sphere oh and
of course we'll compute the radius of
the sphere I have to do to use once
again
this dispersion relation so the radius
of the sphere if you look from this
equation over here the edges of the
sphere is okay the magnitude of the wave
vector it must equal and Omega over C
free space okay and now it is a matter
of preference how you wanna do it for
example you can write Omega as 2 pi
times nu where nu is the frequency mm
remember Omega is the angular frequency
over CFS then you can use the dispersion
relation which says that see free space
equals lambda nu so nu will cancel then
you end up with n times 2 PI over lambda
free space okay and of course no I don't
say anything about free space because nu
is the same everywhere at whether it is
free space or slave I mean not free
space and so on okay so that's where
this equation comes from that's why the
reviews of the
case fear if equals and Omega over the
free over the speed of life in in the
vacuum or to PI and over lambda the
wavelength in free space okay now those
of you some of you may have taken
solid-state this case fear also comes up
there and it is called the evil sphere
in our language in optics we seldom call
it evil it's fear we call it case fear
or the cards fear but all of the state
AB all of these three three yeah three
names they actually be no to the same
thing and of course in the case of solid
state this fear does not apply to lie to
light
what does not apply heat to light
usually but it applies to what button
electrons to electrodes that's correct
yeah so they're so according to quantum
mechanics electrons are also waves and
they satisfy actually the don't
satisfied is a question that satisfy
this radical equation which is in one of
the problems of the homework so anyway
so the Cenacle is a question similarly
leads to a sphere like this room okay
there is one thing that okay so so this
is a sort of a preliminary introduction
to plane waves one thing that I would
like to to very strongly emphasize is
this expression over here the complex
exponential I will rewrite it once again
e to the I a xx plus KY y plus K Z Z
minus Omega T okay this expression when
we see that when we see an exponential
with linear terms in the Cartesian
coordinate
that's a plain way and now we know why
right because because it corresponds to
this moving wavefront
and of course most of the time we will
not add it like this we will omit this
term because of the facial business that
we discussed last time
so we'll actually see it like this and
also very often when some other things
will happen in here for one thing the
these three numbers as I said before
they're not free so they're given by KX
square plus KY square plus K Z square
equals Omega over C square or
okay this is another way after solving
solving the sphere so so we will say
this these kinds of expressions for
example I might write something like e
to the I KX x plus KY y plus
okay so and there's a Z here outside the
square so this looks a little bit scary
but it doesn't need to be scary it is a
plane wave alright and we will see
expressions like this one later in the
class but a one sort of prepare you now
so that they are associated with a
physical meaning your minds it is not
just an expression
it actually denotes a plane wave the two
linear terms here actually also a linear
XY and Z appear as first powers so and
and the reason this square root appeared
here is because of the decart sphere or
the evil sphere or whichever ok any
questions about plane waves
the other way that we'll be dealing with
quite a bit is the spherical wave and so
clearly as the name suggests in this
case you have spherical wavefront
which are either going away from a
source at the Virgin wave or inwards a
convergent wave just as an aside by the
way I should emphasize that this
vertical wave is not the physical thing
so plane wave is not physical either by
the way why is the plane wave not
physical why I mean what's an obvious
reason why a plane wave is must be an
approximation of something but and you
have the planar is infinite that's right
has infinite energy it is infinitely
larger so clearly I cannot create plane
waves per se I can only create sort of
finite approximations to plane waves a
spherical wave is but nevertheless it is
an interesting thing and the little who
spend all this time defendant is because
it provides a lot of mathematical
convenience in dealing with more
realistic waves waves that can be
realized I can express them in terms of
these fictitious entity the plane wave
the same is true for the point source
but for a slightly different reason
it is actually impossible to create an
ideal spherical wave the best thing you
can do is you can create a way that
would sort of if you you know if you
well the only the best thing that you
can do is you can create a dipole source
here so if you lend the dipole the same
the vertical direction then what you can
do is you can create a radiation pattern
that has approximately 60 or actually
120 degrees open in 126 to be 63 yeah I
think the 63 degrees the the dipole
anyway so you can create the a pattern
like this but never quite a spherical
wave in fact someone as famous as
Einstein said that spherical waves don't
exist nevertheless it is a very
convenient mathematical quantity and
that's why that's why we're describing
it but again this is - by the way this
is true in any kind of physics that you
might learn for example in mechanic's
very often you deal with frictionless
yeah
suppose you burst a small cracker in a
in a in a water tank then it wouldn't
for in optics yeah in what the time
possibly you can create this particular
even so it's not a spherical wave
because it is a finite ray it has a
finite source anyway there is only in
optics you can fundamentally not create
this kind of thing is because of charge
conservation you cannot in order to
create a spherical wave like this you
would have to create an oscillating
charge here and no such thing exists so
the only thing you can do is you can get
a dipole that is two charges which are
at the naiton up and down up and down
positive and negative that would result
in a way that it looks approximately
like a spherical wave but only the
direction perpendicular to the dipole in
the direction along the dipole axis it
actually the wave vanishes so it is very
different than a very fluid okay to
justify it a little bit more why even
though it is non-physical why we spend
so much time discussing it it has to do
actually with system with Systems Theory
yes question
isn't a dipole actually cancelling out
because of the distance between the top
and bottom source you have like a
different distance between the charge is
zero the charge is zero but you still
have a dipole moment and whatever if the
dipole is oscillating that is if it is
going from positive to negative positive
to negative very very fast then you have
charges accelerating so therefore you
generate electromagnetic radiation as we
will see in a little bit no no
mean the intensity of the field from the
dipole isn't necessarily vanishing along
the axis if the spacing set the right
away note everywhere along the actually
but you will definitely get knowledge
along the axis if I remember correctly
the dipole has a radiation button that
looks like this and it will look like a
butterfly yeah it's it's dependent on
the distance between the you know the
plus and the minus charges talking about
an infinitesimally small dipole okay
cool yeah if you have like a lambda over
four and ten or something like that then
yes the the radiation pattern looks
different but in the simplest possible
case that you can approximate the best
the spherical wave is an infinitesimally
small dipole radiation pattern looks
like a butterfly so okay so we actually
went a little bit ahead of ourselves now
because this entire discussion it
involved electro magnetics which we
haven't done yet but by the end of this
lecture it will be a little bit more
clear what I mean by dipoles and all of
these things the thing I meant to say is
that the spherical wave mathematically
it corresponds to a point source and the
and that is something that most of you
probably at some point or another took a
class on linear systems where typically
you learn a term called an impulse
response so the impulse response in the
time domain is basically is a very
narrow excitation fact it is
infinitesimally narrow and it is so
narrow that because it has to calif i at
energy it also has infinite amplitude
right that's what is called an impulse
so the spherical wave is the equivalent
of that but in space now the same wave
originates as an infinitesimally small
disturbance if you wish which then
mathematically creates this uniformly or
isotropically expanding
clerical wavefront and from that point
of view it is very useful because even
the dipole that I said before it can be
described actually as a superposition of
spherical waves so so mathema so I can
be physically correct if I do my my
physics corrector you know I can get the
the proper answer so the second wave
provides this mathematical convenient
also to be honest the market will do
here it is a very good approximation to
what you observe in the laboratory
that's another good justification for
using it because even though we know
that there's no such thing is a point
source many of the things that we'll
derive in the next few lectures based in
this sort of crude approximation they
turned out to be very good approximation
for what you see in the laboratory for
example if you take a pinhole a pinnacle
is a finite thing right I mean the
smallest pinhole that people use in the
laboratory may have attained a diameter
of 1 micron sometimes equals bigger 3 5
micron something like that well if you
pass light through a pinhole the light
that comes out is to a very good
approximation described by a spherical
wave but of course not everywhere the
light coming out of the pinhole it
doesn't come out backwards it only goes
forward right so therefore you have you
know clearly the the expression is
approximately correct but only for the
part of the wave the portion that is
physical right so so we have to be a
little bit mindful of what is the
meaning of everything that we write down
okay in especial so especially when
things like that are involved the second
wave has an any problem it has the
problem that it blows up at the center
and there isn't blows up is because well
likely impulse in the time domain that
you learned in in linear dynamical
systems it has to have a finite energy
so in this context here the finite
energy as you as you proved in a
homework some time ago it means that the
amplitude of this spherical wave must be
like one over the distance so this one
over R that we get in the denominator of
course at R equals zero it explodes
right so they again that is the Delta
function R impulse the impulse analogy
so the spherical wave has a few
mathematical inconveniences of this sort
but you have to be mindful with okay now
there's one more mysterious thing about
it which I will just point out and I
will then move on without saying why if
you look at the expression of the
spherical wave let me write it down it
looks like cosine a R minus Omega T over
R and let's let's say let's take stock
here so the 1 over R would discuss the 1
over R is energy conservation we always
like that the cosine that you see in the
numerator if I go back and play the
movie of the spherical wave that wait
for the plane wave to finish here so the
K R minus Omega T it is this sort of
explosion of the wave fronts that you
see in the movie so K R if R is the
radius K R describes a sphere again
alright so the cosine K R basically
tells you that you have this alternating
positive positive and negative valued
spheres and the minus Omega T is the
traveling aspect it tells you that the
spheres explode the propagate outwards
as a function of time so this is the
meaning of the K R minus Omega T then a
battle to put another term over there
minus PI over 2 and this might sound a
little bit weird I mean in general you
can put any phase that you like there it
means that that you have a
shifted time access so I could put five
there while they battle to put PI over 2
what is the PI over to relative to okay
it turns out if you properly solve
Maxwell's equations which we haven't
seen yet but I'm sure you're aware that
there is such a thing called Maxwell's
equation well the describe
electromagnetic waves okay so if you
solve it properly with this kind of
point source because of something having
to do with magnetic fields you pick up a
phase delay with respect to what you
pick up a phase the leg with respect to
the source so there's no really weird it
tells you to do if you have the source
that is oscillating let's say like a
cosine so it is maximum at T equals zero
the wave that comes out will actually be
a sine so it will be phase shifted by PI
over 2 there's no easy way to explain
this other than solving Maxwell's
equations there is actually some papers
in the literature this is a very very
interesting foreign I know it's strange
it can be explained from the question
but it's still a little bit strange so
people are actually working on this
phase shift there is some very strange
aspects to it anyway for us the phase
shift will just accept it it is well
established so so we just accept them
one when we help it as a phaser
I mean have the complex and presentation
of course e to the minus I PI over 2 is
minus I so I skipped a few steps of the
slide so let me do them on the
whiteboard so to go from this with a
complex representation you would write e
to the I a R minus Omega t minus PI over
2 and of course you still have this
thing outside nothing can you know this
is not affected okay and then remember
that e to the minus I PI over 2 equals
minus I and it equals 1 over I so this
is why the I appear that the denominator
over here okay so because PI over 2 is
kind of a pain to write down when we
write the spherical way
we'll just write it with I in the
denominator and this is actually not I
mean it's not a big deal you know if if
you if you in fact myself sometimes I
will just skip it but it's kind of nice
to know that it's there okay and of
course the the the only thing that you
do commonly but by now I think we're
we're comfortable with that is that we
we neglect the temporal variation and
then we get the failure the last thing I
will do before moving on to a slightly
different topic is remember in
geometrical optics we derived a lot of
useful things using this paraxial
approximation so the paraxial
approximation basically says that that
in this case the wave originated
relatively far away and you have the
spherical wavefront which are nice fears
near the source but if you go at a
relatively long distance basically you
cannot tell the difference between these
spherical wavefronts and parabolas so
and it turns out it because the sphere
involved this nasty square root it is
much more convenient to represent this
as parabolas so the way you go from this
field to the parabola I did it on the
slide and I will do it again because
this is a derivation I want you to be
very familiar with it's it's we've done
it a few times and we will do it again
and again you start with the expression
for the wave so it is e to the I K all
right and there is bells and whistles
there is all there are the eyes let me
just write this way okay and then
remember so we have R this is the polar
coordinate so it is X plus y plus Z
Square and the paraxial approximation
let me put this back in our convention
usually is that
is that this is the z-axis so this is
the axis of propagation the optical axis
and then you have the other two axis
perpendicular to it typically will be
note X has been vertical okay and is
this never that handy we did it yeah
okay good
all right so so the production
approximation then means that Z is much
bigger than the values of the other two
coordinates right because basically I'm
limiting myself to open it in this
regime right that's the meaning of the
paraxial approximation so what you do
then is you do a Taylor expansion on
this square root so you pull the outside
okay
then you use the Taylor property that
says 1 plus small is approximately equal
to 1 plus small over 2 and then you can
see that R equals Z 1 plus X square plus
y square over 3 square times 2 which is
also known as Z plus okay so that's the
paraxial approximation for the polar
variable and now I can of course put it
in my expression so this means that e to
the I K R is approximately equal to e to
the I K Z e to the I K X square plus y
square over 2 Z and sometimes you leave
it like this or sometimes you remember
that K equals to PI upon lambda right
and then this expression now becomes e
to the I 2 pi Z upon lambda e to the I
naught is what will happen the twos will
cancel AP will pie will remain
and you'll get okay so all of these are
interchangeable expressions that we can
use for spherical waves another
interesting question which I omitted
something I omitted one over I R and
actually there's also the amplitude this
put the amplitude for completeness what
happens today I are should I also do
some paraxial approximation over it it
turns out it is sufficient to simply
approximate it is Z for the for the part
that appears outside the exponent we can
simply approximate it as Z and the
reason is because our it is actually
slowly varying you know if you compare
what is our C you can use them so R is
this right so if you compare
okay so basically have to to pick Z
equals constant so I have to pick a
plane that is tangential to the sphere
at this place
so I compare r1 to r2 right so r1 goes
all the way to the plane r2 simply stops
here so think about it this way let's
say that the error that I make when I
say Z is approximately equal to R the
Senate I make a small error let's say
that I make an error of 1% okay that
means that I misrepresent the amplitude
of the wave by 1% now let's look at the
term that is inside the exponential e to
the I PI X Y X square plus y square over
lambda Z if I miss by 1% that is if I
had it to the 1000 by and a Miss by 1%
then it means a 1 to e to the 990 pie
right so this 1 percent error actually
called me how many 10 full oscillations
that's why I have to keep a better
approximation inside the exponential
then I can afford to keep outside so the
with the face I have to be much more
accurate in order to make sure that I
don't make a big error in the face delay
and that's because the cosine varies
very rapidly as you increase are the
cosine is oscillating but one over R is
slowly what we call a slowly varying
function over and therefore we can
afford to be sloppy with it if you wish
so the paraxial approximation then for
the spherical wave is just like that
with with Z in the denominator
but this additional quadratic term in
the exponent and of course this
additional quadratic term is the one
where all the action is happening and
this we will see this later in the class
and that's the second important lesson
from today which is that if we see an
expression with a quadric which is
quadratic in the Cartesian coordinates
we call it a spherical wave so we went
through all these adventures again
arrived at two very basic waves one is
the kind of wave in whose fan is ER you
have linear terms in the Cartesian
coordinates that will call the plane
wave
okay so if you see something like this
it doesn't really matter what these
things are K X K Y and so on but because
these terms are linear it is plain when
we arrive at an expression that looks
like this you might have all kinds of
junk and somewhere you have something of
the form e to the I PI X square plus y
square over lambda Z this quadratics
means that it is spherical or more
accurately the paraxial approximation to
a spherical wave yes button I'm curious
what will be the like a propagation of
even a sent wave that's a good question
but but we don't have the tools to deal
with that yet so the pan is asking allah
to square root here here this the but is
asking what if I pick KX and KY so that
the argument to the square root becomes
negative therefore this quantity would
become imaginary right
that is called an evanescent wave but
let's not go too into into it now yeah
welcome back to the table
so my question
and the other questions
I should say any questions that I'm
willing to answer because that is a good
question with
okay
did you come up with any questions about
spherical wave or plane waves during the
break okay here I have one question so
just now I mentioned that the when the
KX KY is a very small
well the Kesey's were a big we call it
plain weave correct white it is to
actually what you say KX and KY would be
very small yeah yeah okay but this when
talking about the you know K dimension
hop I in the space in a real 3d space
okay the production approximation I did
actually in space so this diagram over
here that was a space XYZ but what you
see is true in in the plane in the
spherical wave okay there is no I'm
reluctant to admit what you say is
because the spherical wave is not a
plane wave so therefore KX KY KZ the
valley across the wave front right so a
little bit more complicated what happens
but yeah if you took a small portion of
the wavefront here and you plotted you
know KX well this is you know the
direction if you plotted KZ would be big
and then KX would be very small right so
just Rho KX and KY are also very small
but they're to be careful with a
spherical wave because for example here
we have a different KX and K 0 so what
it means here KX and KY well what does
it mean
let's not going to Wignall distributions
right ok so let me let me let me say
with a time transient in plain language
the kxkykz would define them for a plane
wave right so a plane wave is this kind
of thing where you have a well-defined
wave vector and the entire wavefront is
blender now what you're saying is that
if I have a spherical wavefront like
this
luckily I can define normals right and I
can call those wave vectors as well and
that is actually true but they're not
plane waves you can think of them as a
race that's a good approximation right
this would be the equivalent of
geometrical lace but not plane waves but
nevertheless you are correct they would
these things are also per action so they
would also satisfy kxky much less than K
Z that is true
other questions
I'd like to talk about a fun fun topic
today I was I want to do electromagnetic
so I don't want to spend too much time
on this but I'll talk about it because
it's fun and and and very useful in in
many contexts so in all of this business
we without very this is again the wave
equation but we didn't say anything
about the speed of light so actually we
did a little bit I mentioned I guess
this is how this is better than a
whiteboard I have a history of what I've
written throughout the lecture yeah so
so at some point we didn't mention this
we did mention that the index of
refraction actually enters in this
equation over here but if you remember
diligently from your geometrical optics
the index of refraction also happens to
be a function of wavelength it's not a
constant right it is a function of
wavelength so what this means now is
that is that in this equation over here
if I have a singly frequency now a
singly temporal frequency a single Omega
then everything is fine because sure for
this particular frequency I can find the
index of refraction and and I'm okay
however however the index of refraction
is a flat is a function of wavelength
and the wavelength yes it is a function
of frequency we do know that this
equation is true but we also said if you
recall or if you go back to the video of
the first lecture when I first presented
this equation I gave you a warning and I
said that this is the dispersion
relation but it is not the only possible
dispersion
it is possible to have waves where the
dispersion relation is different okay
first of all just to make sure that the
all stuff about dispersion that we
discussed about glass it is true in
other words in glass yes the index of
refraction is a function of wavelength
so this is a plot that we saw before but
now in addition to that I'm talking
about a slightly different type of
dispersion so the dispersion happens for
example in waveguides in one of
everything that we do so far we sort of
assume that light propagates freely who
may put some lenses or or whatever
prisms in the path of the light that is
refractors and mirrors and so on but all
of these are relatively large we never
try to confine the light to a very small
space but you can do that you can create
wave guides actually for visit for
visible light you will never actually
use a metal here you would use the
electric waveguide
but in microwaves this is done very
commonly in microwaves people trap the
wave in a metal tube as if it were water
a suppose right or you know some kind of
a liquid
you just have a hollow tube that is a
metal and then you launch the my
microwave inside the tube to give you an
idea the wavelength in microwaves is in
the order of well you know as the name
suggests it is the order of the
frequencies gigahertz so the wavelength
should be between millimeters and
centimeters that is typically less
though the wavelength is a little bit
less may be in the order of millimeters
a typical millimeter so the size of this
tube might also be in the order of a few
millimeters so now you have a guide that
is approximately of size of a few
wavelengths so that case it turns out
this equation sequence lambda nu
it does not hold anymore it is it is
true for a free space but it is not true
in the waveguide the reason it is not
true in a waveguide is because the wave
must satisfy boundary conditions if you
recall when we gave the wave equation we
said if you need to satisfy an initial
condition and boundary conditions in
free space there is no boundary
condition actually there is a boundary
condition that the wave must vanish at
infinity okay fine but you know this
doesn't have any implication in our in
our calculation but for example in the
case of a metal you have to force the
wave to be zero on the metal because
well we'll see later than electro
magnetics why this is the case but take
my word for it the the on the metal the
field must equal zero so now if you have
a sinusoid in this metal the sinusoid is
a kind of a straight jacket because it
is sinusoidally varying but also it must
be over such a period that it vanishes
at the edge of the waveguide so for
example it can be this is the waveguide
the sinusoid can be like this that's
fine because it vanishes it can be like
this that's also fine it can be like
this and so on right but for example can
it be like this no that we cannot it
okay so I will not do the full
derivation again it's a little bit well
it would have to solve the partial
differential equation in a serious way
but it turns out that this straight
jacket that we put the light in actually
results in a different dispersion
relation that looks like this it is not
the C equals lambda nu anymore the
question which was yes what happens if
what happens if a is smaller than lambda
then then a wave will not even get into
the web guide it will become a Van Essen
so it will not go in thank you what is
shown in the diagram is isn't it a
standing wave because it's not the
oscillation is the way this is the
midway in the vertical direction yeah
but how does it properly when you move
it plays bouncing back and forth another
way to think of the waveguide is that
the light is you can think actually is
in terms of rays
you can think of Ray is bouncing back
and forth right but you can also have
the conjugate ray these are both
possible and the result is a standing
wave in this sense but you can still
have propagation on the horizontal axis
because there is a component that goes
towards the right okay so so when you
get an equation like this one which
relates or meget okay and of course as
you can imagine now there's a great
variety of such equations that you could
have this is the equation that happens
to be true for the metal waveguide if
you replace it with a different wave
guide for example if you have the
electric so different indices here then
this equation changes the bottom line is
the previous equation then you can plot
one of the two you have one point one
term here that is determined by geometry
a is the size of the guide and then you
have the wave vector and the frequency
and of course we have the free velocity
of light okay
Omega is fundamental as soon as you
specify the frequency of the
electromagnetic wave nothing can change
that anymore since we're in the linear
region but the question is what is the
wave vector so the way back to is really
what you are after in this equation
Omega is given it is say 10 to the 15
hurt what is
kay well in other words what is the web
now for results which will become
apparent in the moment the people who
invented these diagrams they actually
plot in kind of the other way around
instead of putting on mega on the
horizontal axis since Omega is given
they put Omega on the vertical axis and
the horizontal axis is K so the way you
read this diagram is of course you solve
this equation and the way you read it is
that if I give you a certain Omega then
you have to draw a horizontal line and
wherever this horizontal line meets the
plot this you read the abscissa and
animal the other side is the stepsister
or the other
the stepsister that's the ordinate okay
you must come up with him in the morning
anyway so the horizontal the value of
the horizontal axis gives you the y
vector and hence the wavelength now
interestingly if you were in free space
this would look like a straight line
because well of course in free space I
have written this equation many times I
don't feel like writing it again in free
space well let me do it once more so in
free space I have that C equals K over 2
pi that's also known as lambda times
that one 2 pi over K that is also known
as lambda times Omega over 2 pi so in
free space then you have Omega equals CK
this is an equivalent way of writing
this equation here these are basically
the same equation so in free space this
is a straight line now if you have any
question like this one you can see that
at relatively high frequencies they
approach so indeed all of the oh by the
way what are these curves
so this equation involves an integral
parameter you can plug in
well not quite zero that please erase it
from your notes or should not have put
zero here zero it doesn't make sense but
you can have plus minus 1 plus minus 2
and so on
and these are known as modes so the
index defines the mode that is launched
into the waveguide so so each one of
these mode at very high frequencies it
actually kind of approaches the free
space mode but as you go to lower
frequencies you see that it bends away
and then eventually hits the ordinate
axis and there's a gap here now because
there's no there's nothing in this place
over here so what will happen if you
come up with you use an Omega that falls
in this range evenness and we've correct
actually someone asked it before in
Boston someone asked me before what
happens if you pick a to be smaller than
wavelength well this is where well this
is what happens in fact a wouldn't
pause it exactly like this a more
accurate way to say it is what happens
if you pick an Omega which is very small
so therefore the you know given the
value of a and given the value of Omega
you cannot find the K anymore the square
root becomes imaginary right well if
that happens the wave does not enter the
waveguide it becomes a Vanessa as we
said so basically it may penetrate a
little bit for a couple of wavelengths
and then it will exponentially decay now
if you as you go progressively so if you
launch a very low frequency the wave
does not enter at you something crease
in the frequency now you will see that
the intersection only meets one mode so
basically the range of frequencies I
don't know what it is here this looks
like three or something
so between so the gradually this axis is
you read the ordinate and you multiply
by three
times 10 to the 14 so for example five
actually means 15 times 10 to the 14
Hertz
okay so then between three the value
three which is nine times ten to the 14
and the value whatever it is seven you
only launch one mode as you go up in
frequency if the intersection here emits
more than one lines then actually you
can have multiple modes in this guide
anyway so this modes are these little
Wiggles that are draw over here the
first mode is the field shape that looks
like this only half of the period of the
sinusoid then the next ball you allow
one full period the next mode you allow
one and a half period and so on and so
forth so as you go higher in frequency
it means that the wave basically what it
means is the waveguide now or I should
say the wavelength of the wave becomes
progressively smaller than the size of
the waveguide so therefore you can pack
more modes into the guide so I only
planted three here of course there's
infinite but anyways you go up in
frequency you have the possibility to
excite more than one mode now how do you
excite the mob's now that depends of
course on your initial condition right
if you create a distribution in the
entrance of the waveguide that looks
like this then you will excite the first
mode if you create a distribution that
looks like this
well if the second mode is allowable
that is if you are above the green line
then you will excite it if you are below
the green line you will probably excite
an atom because the first mode cannot be
excited this way
okay there is no telling you know this
is not because I want to do waveguide
theory but because but because I want to
do something that's called a bit and a
bit is an extremely useful concept both
in the space domain and in the time
domain so in the time domain the beat is
actually defined as a very simple thing
it is defined as the superposition of
two sinusoids of slightly different
frequency
so we said it many times that the wave
is linear
for a superposition is allowed if I have
two solutions the wave equation I take
the Sun with the standard solution so
now what you will can see here very
easily that's a diagram that I stole
from the textbook because the two waves
have slightly different frequency they
will miss a line so you can for example
start at a point where the waves are
exactly aligned so they kind of
oscillate together when one is positive
the other is positive but because the
period is different or the frequency is
different a little bit later they will
kind of separate and now over here you
can see this is the opposite when one of
them is positive that is negative and if
they also happen to judiciously pick
them to have the same amplitude it means
that here when we say the oscillator to
face it means that as the wave amplitude
overall becomes very small and so and of
course in the other case where they are
in phase the amplitude becomes twice the
amplitude of one so this is the sort of
a static picture from the book I also
made a movie out of it
so here you see the bead propagating
okay
so now what is really innocent about
this let me play it again
if you look at it carefully but it takes
a unit let me play it once more actually
think I can just
if you look at it carefully you will see
that there is two different velocities
here there's two things happening why
did you have this envelope that is
moving to the right but also inside the
envelope you have small fringes wiggles
that are also moving to the right now I
told you of course but if you look at
the plans again more carefully you will
see that the two are moving with
different speeds energy The Wiggles are
moving faster than the envelope and you
can see that the Wiggles are moving
faster than Mangalam if you look at if
you look here near the vicinity of the
of the null you will see that there's a
little bit of once in a while there's a
pulse apart like a little pulse that
means that the fringes passing by so the
reason this is happening and now I will
not derive it on the paper because it
would end a little bit late but but this
is happening is because if you take the
superposition of the two sinusoids now
has to be a bit careful because they
have different frequencies so if we use
phaser we could use phasers but we might
get a little confused so for that reason
I just wrote it in the real space with
simply with cosine so you do the same
thing with it before you take the sum of
the two cosines we apply the
trigonometric identity to write it as a
product and then you find that the
product times two terms one of them
actually by the way they should there is
one more error in the notes there should
be a cosine between the two parenthesis
there should be a cosine a better eye
than the correct expression
okay we're okay C equals k1 plus k2 over
2 and the same for Omega C and K M
equals k1 minus k2 over 2 so one of them
is the average the other is the
difference and but now each one of those
each one of these cosines okay there's a
missing cosine here but imagine that
there was a cosine each one of those is
a traveling wave so basically what you
see now is the product of the two
traveling waves and you can see very
clearly the product here
this is they have one sinusoid of a
relatively long special period and
another sinusoid of relatively fast
special period and I multiply them but
they both propagating because the both
of the form KZ minus Omega T but they're
propagating with different speeds
because the speed of the handle and so
and what are the names well the first
sinusoid here is called a carrier those
of you who are electrical engineers are
very familiar with those the carrier
wave is in radio waves for example it is
the frequency that that upon which will
modulate the signal and the slow the
slow sinusoid that's called the
modulation okay and and they a loss
method alright so but each one of those
is it is a propagating wave so therefore
the velocities would be equal to the
dependable depend on which way I'm
talking about so for example the
velocity of the carrier will equal Omega
of the carrier divided by the K of the
carrier and the velocity of the
modulation
we'll equal Omega of the modulation of K
of the modulator okay and so there's not
magical just apply the theory that we
learned but now it is the two waves at
the propagate and of course in general
the two velocities are different because
the the two ratios over here they have
no reason to be the same in fact it is
quite the opposite
most of the time they are different
blushing so this velocity I actually use
a slightly different notation the
velocity of the carrier is called the
phase velocity and the velocity of the
modulation is called the group velocity
so the group velocity it equals as you
can see the ratio of the difference of
Omega to the ratio of the difference of
K so if you imagine a beat this is
called the beat by the way I think I
said before so if you imagine a bit
where the two frequencies and the two
wave numbers are very close then it will
become the derivative so the survey so
the group velocity is basically the
derivative of the whatever Omega is as a
function of K if you take the derivative
of that expression you get the group
velocity the phase velocity is the same
velocity we've been talking about so far
while we're doing waves so there's not
you know there's nothing new about the
phase velocity it is still the ratio of
Omega over K but the group velocity is
the derivative and the region now now
there is reveal the ocean the question
yeah
why did the derivative come about yeah
you're right I kind of blast it through
so if you look at the group velocity it
equals Omega of the modulation over K of
the modulation and this equals Omega 1
minus Omega 2 over 2 over K 1 minus K 2
over 2 that is Omega 1 minus Omega 2
over K 1 minus K 2 and if you take omega
1 very close not quite the same and K 1
very close but not quite the same then V
sub G will become the Omega decay
so that is the reason people put Omega
on the vertical axis over here because
the group velocity then is simply
obtained as the slope of the dispersion
diagram and that's fascinating if you
think about it because it suggests some
weird things for example it suggests
that
if I could somehow create this pension
diagram that looks like this what is the
strange thing about this and have a
negative group velocity which means that
in the wave the carrier and the
modulation they go in opposite
directions the phase velocity is going
to the left the group Alost is going to
the right I meant to make a movie of
this to show you it actually looks a
little bit weird but I didn't have time
to do it and show you one next time but
but now of course it remains it's
interesting
how can you do something like this it
does have to be an extremely hot topic
in optics research now we know how to
design theoretically structures by the
way natural materials they don't have
this property so you have to try really
hard to make this property happen for
example by patterning at the electric if
you create patterns in a certain way
then you can cause you can make group I
mean dispersion diagrams that look like
this and of course to fabricate them you
know it is one thing to design them and
simulate and one thing to fabricate them
so these are all very hot topics in in
optics the cells so so that's why and
also be it is sort of a very important
topic so that's why I spend some time
talking about ok
and one of the problems again in the new
homework that I posted one of the
problems say I'll get you to play a
little bit with this dispersion relation
so you can get a feel for it
unfortunately in this class we we do
mostly optics in the space domain this
what I'm discussing here is clear is
kind of time domain right because we're
talking about modulations and carriers
and so on the class is mostly about
space domain so we don't spend too much
time on this topic but anyway it is sort
of important to know that's why ok
and with that unless there is any any
questions I will move on to
electromagnetics questions
don't you one question so just now we
show the carrier and the modulation we
actually have different velocity by in
the everyday life well common sense for
Tomas is the for damper they'll read a
light and yellow light travel a same
speed I mean that have different
frequency different wavelengths pata
prod out of this to either constant for
example you know that is not true in a
glass that is the reason why a prism
analyzes white light it is because red
and yellow they travel with different
velocities in glass so any any medium
that is dispersive they have different
velocities so only in a vacuum you have
a similar so so the the linear
dispersion relation that I showed before
that is providing
actually it is also linear in the medium
but with a different slope I should have
said that yeah I was just gonna point
out that actually you shown the phase
velocity there for the free space not
for the yeah but at the point where
you've measured the group velocity that
the right word omega c kc if you drew a
line from the origin to that point the
slope of that would gives the phase
velocity if you're correct yes yes yes
thank you yes that's very true the phase
velocity is omega over k so it should be
that one yes so I need to correct the
slide thank you yes the Lassa T of free
space and this crisis than the speed of
light how could that happen by the way
because there's a phase velocity right
you can do anything it once it can it
can go faster than the speed of light
because it carries no information it is
simply an oscillation the group velocity
as you will see actually I in the
homework I get you to derive the group
velocity for this case you will see from
the answer that the group velocity is
actually smaller than the speed of light
you will see that the group velocity for
this diagram it is given by something
like C square root one minus something
so it is so the group velocity is always
less than C
okay so we'll pass the corrected version
any other questions so what a pattern
metamaterial basically be the same thing
as a special case of a gradient index
lens the green I know what you mean by a
special case I mean they have quite
different properties well I mean you can
maybe like you know imagine the idea of
the gradient in dielectric material or
the gradient in the index of refraction
sort of creating a negative group
velocity or some case where you're
actually diverting light away from what
you're trying an image I don't think you
can do with a simple gradient index you
can okay the pressure of the mean by
gradient if it is a slowly varying
gradient like the grain optics usually
not only you can create sufficient
dispersion to you know to turn the light
around if you have sub-wavelength
patterns then yes because the you know
the resulting evanescent waves they can
add up in ways that yes I mean this is
the whole area yes of metamaterials
but those usually the gradients are huge
red because you within space of you know
less than the facepage wavelength you
might have variation of index from 3 to
1 and then back to 3 and so on so these
are huge gradients
say that group velocity is the velocity
with which the information is traveling
well that's one interpretation yeah this
has been challenged too because people
saw that group velocity in some cases
cannot exceed the speed of light so
therefore even that is questionable but
anyway in most cases group velocity is
the velocity at which the it is the
velocity which the modulation travels so
the modulation is usually denotes
information if you exceed the group if
you're the group velocity if you manage
to make it exceed the speed of light
which has been proposed then it means
that the modulation actually does not
get information right this is something
that you artificially put in there so
why we means what is the advantage of
negative group velocity means opposite
direction information travel well it can
make you a pretty good career and get
your papers published in nature no no I
mean I'm joking but but first of all it
is interesting it is interesting physics
and very counterintuitive that is cool
way but also the people who work on it
they have shown mathematically that you
can get a focusing of light that is
sometimes tighter than traditionally
refractive you can get interesting kind
of dispersion for pulse shape in very
narrow paths you know you can do some
clever things with it you can slow down
light which is a slow thermal light
means that if you go back to this
diagram
you see that as you go out towards
lower-frequency the group velocity
becomes progressively smaller in fact
over here it well if there's a point
where it is actually zero Oh what does
it mean well it means that if you really
had the light pulse not the not the
fringes of the light not the carry way
but they have elope
the modulation that if you want to run
into this frequency were very close to
this Raquin see that they would stop
rate it would actually order to move
very very slowly that's the principle of
slow light which also has potential
applications for example you can store
data if you can slow light down you can
create an optical buffer you can wait
for something else to happen and then
launch the wave again so then there is
you know there is things that you can do
with this in addition to interesting
physics
so fascinating topic retention some glad
you are asking
yeah and if we go for a higher frequency
we can see that we can excite many modes
in the waveguide so all the wave guides
will means all the modes will have
different velocity but right so which is
bad because it means that if your
information is somehow split between
those two modes then they will separate
so they will arrive at different times
that is called modal dispersion and it
is a very serious problem in
telecommunications it means that you get
distortion in the signal
what was that your question
I'm sorry I kind of answered your
question before you even asked
okay here they are I'm sure all of you
example or another at some moment
probably a moment of a nightmare
you saw them right the box of the
equations so you can read them in the
notes I I have slides posted in this but
I put them in the traditional way and of
course the book also it has an
explanation but I would like to ask you
if you remember from your physics 802
those of you who are at MIT or you know
whatever was your basic electromagnetics
class do you remember each one of those
what is it telling us and why does it
look the way it does the sum of the
first one does anybody remember what is
the first one so actually these are you
know these are the same equations one
with one on the left hand side the
written in integral form the right hand
side written in differential form does
anybody want to volunteer and say for
example the first row it is the same
equation so what does it tell us do you
know you want to say toss a coin the the
first one says the integral form says
that the the total electric flux passing
through a closed surface is equal to the
total charge enclosed inside that
surface and the equivalent differential
form says that the total amount of
electric field emerging out of a point
is equal to the amount of charge at that
point that's correct
so basically it is telling you that you
have a charge you have a charge then you
have field lines radiating out of that
charge and if you enclose this charge
with the surface and you compute the
integral of the basically you compute
the integral of the
I don't know why this is a cross by the
way there's another mistake this would
have been a dot not the cross but anyway
so you know if you compute this integral
it is determined by the total amount of
charge and okay we can go back to the
notes now
so this is what you're called Leiby just
said you have a charge then you have
field lines radiating out and then if
you take all of these lines dot-product
them with the elemental surface and then
integrate then you get the amount of
charge and why why should it be so
principal of charge and consideration
okay but why do the field lines emanate
out of the charges goes from positive to
means it originates from positive and or
we can say that red line represents the
direction of the force on a unit
positive charge that's what I was
looking for so the field lines are
basically they tell you exactly like I
said what is the direction that the
small positive charge will repeat what
is the direction of the force that the
small positive charge would feel if we
were to place it at any given position
in that space this is what the electric
field says right
and so and then there is a relationship
called Coulomb's law which for for two
given charges it tells you the force so
if you have a single charge here and
then you apply coulombs force Coulomb's
law then you get this radiating lines
right so then Gauss's law is basically
nothing other than Coulomb's law and yes
it also has to do with charge
conservation because if you some called
cancel all the charges inside this
volume then the integral will vanish
right because it means that you don't
have any net electric actually you can
have electric charge radiating but the
integral will vanish in other words if
you have the charges surrounding somehow
this space the space cannot pull them
all inside
okay what about the question or what
about the second one
that means closed line integral is zero
means B is a closed line that basic
clotheslined basic law swine line y
means there is no magnetic charge that's
right
magnet so only canvas dipoles right you
cannot have the isolated North Pole for
example the magnet so so actually
actually this is the same equation in
fact they're also known by the same name
that known as Gauss's law the step
equation except there's no such thing as
magnetic charge what about the next one
this one looks awful del cross e equals
DB DT
so let's look at the integral usually
the the differential form is
mathematically better but not very
inspiring site full physically so one
gets more intuition by by looking at the
integral forms even though the integral
forms look really frightened but anyway
they third equation is known as
Faraday's law and it says that if you
have a variable magnetic field inside a
a wire then potential will develop
across this wire and do you know of any
places where this is jeweled button if
an I cannot hear you imagine the people
in Boston
oh maybe you somewhere else to location
you know it's induction motors and yeah
voice coils and generators they use the
same principle so for example if you if
you people do this in hydroelectric
plant said you have a giant magnet and
you somehow move a coil inside that
magnet and of course you have to react
it properly then the coil will develop a
voltage a so that's the principle of of
course you have to move the coil for
example by pouring water over a tall by
steam or whatever but anyway that's the
one principle of generator okay so so
and the last equation that just as an
aside how does a nuclear nuclear power
plant generate energy and electricity
does it rotate something
oh then you move it up a legacy right
right okay
and the last blow is actually the most
interesting one because this this law
actually was involved with a
breakthrough in electromagnetics it can
be known for a long time that that if
you have a if you have a current a
magnetic field develops around that
column that is actually Lorentz is known
as Lorentz force if you have a moving
charge and then you put a little magnet
in nearby then the member said now if
you have a moving charge and if you have
a current it creates a magnetic field
around that has been known for a long
time but what was not known and was
noticed by Maxwell was that if you have
a capacitor well in the capacitor you
cannot have a current per se right
because the capacitor well inside the
dielectric so there cannot be no current
but you can have a variable charge
density in the capacitor plate so what
I'm trying to say is that if you apply
think of charge in a capacitor when you
charge a capacitor the capacitor
you know you the beginning the voltage
across the capacitor is zero and by the
time you have totally changed it the
voltage has reached some value right
well while the voltage this voltage is
changing there is actually charge flow
into the capacitor there is a positive
charge going to to the positive
electrode the negative when given from
the opposite electrode so and at the
same time what is happening the electric
field inside the capacitor grows because
as you get more charge accumulating in
the capacitor plates then the the
electric field in the capacitor is
changing so in a sense you can think of
it as a current as well and what Maxwell
he guessed it actually he didn't he
didn't observe it but he guessed it he
guessed that this kind of variable
electric field should also generate a
magnetic field and their own leaders
guessed it is by symmetry from the
previous law what does it say it says
that if you have a variable magnetic
flux you generate an electrical
potential well he looked at this
equation and he said wait a minute is
fine that if I have a moving charge I
have a magnetic field okay fine but if
also ever-moving if a variable potential
then why should they not generate a
magnetic field I should and it turned
out to be true I mean max will by the
way he was
do you know with whose was his day job
James Clerk Maxwell use a fluid
mechanism standing flowing and that's
why the terminology are not a
terminology but the notation in
electromagnetics is actually very
similar to the fluid mechanics notation
Rho and epsilon and all the stuff I mean
they're not the same things but and also
these theorems the mathematical theorems
they come from fluid mechanics they
apply to incompressible flows for
example the gauss theorem it applies to
the incompressible flow in pressure
potential anyway so much will did that
first so he guessed that you should have
this form in here and then he actually
wrote down all these equations in a
coordinated form and the reason this was
a very valuable thing and where all the
guests were all employed because of this
is because if you manipulate Maxwell's
equations and I will not do it now
because I ran out of time but if you
manipulate them you can actually
that you arrived at the wave equation
for the electric or the magnetic field I
did it here for the electric field but
you can also derive the same equation
for the magnetic field
so therefore what this means is that if
you have electric and magnetic fields
which change in time Maxwell's equations
tell you that they change in a
coordinated way and the way they change
results in an electromagnetic wave ok so
it is the thing it's time to quit in
fact well in the 5 minutes late