we have been discussing electromagnetic waves
and in the previous lecture i argued electromagnetic
waves in the previous argued on the basis
of faraday's law which says del cross e is
equal to minus partial derivative of b with
respect to time and displacement current in
free space that it is possible to sustain
electromagnetic field when they change and
keep generating each other and it propagates
so we use this source faraday's law and this
was based on displacement current
in this lecture i want to use them the way
maxwell has written it and using those maxwell's
equations we want to derive a wave equation
and then discuss it further so we are going
to be mathematical little more rigorous than
the arguments that we gave in the previous
lecture but i want to do appreciate what we
did in the previous lecture it gave you a
physical feel of what electric and magnetic
fields may look like and how they change with
each other in this lecture we are going to
see the same treatment in a more sophisticated
mathematical method
so to start with lets write all the maxwell's
equations once more 
and i am writing them in free space free space
and there is no charge no charge density so
the equations i have is divergence of e is
zero curl of e is equal to partial derivative
of b with respect to t partial derivative
of b since there is no real current we have
mu zero epsilon zero the displacement current
and divergence of b is zero let us look at
the two curl equations and take the curl of
this equation the faradays law equation once
more this becomes equal to minus d by d t
of curl of b i have taken curl on both sides
this however is gradient of divergence of
e minus laplacian of e this is equal to minus
d by d t and curl of d from curl of b equation
is equal to mu zero epsilon zero d e d t from
the first equation gauss's law this term is
zero and therefore i get del square e is equal
to mu zero epsilon zero d two e over d t square
in particular if the variation of e is only
in one direction let's say x direction then
i have partial e by partial x square is equal
to mu zero epsilon zero d two e by d t square
when i am writing this e that means each component
satisfies this equation lets write this explicitly
i have d two e x d x square is equal to mu
zero epsilon zero d two e x by d t square
we will see this equation actually is redundant
because there is going to be no x component
d two of e y partial x square is equal to
mu zero epsilon zero partial e y over partial
t square and finally partial of e z partial
x square is equal to mu zero epsilon zero
partial e z over partial t square notice that
this is exactly like the wave equation with
this quantity here being one over c square
right so with that this becomes equals to
d two f by d x square is equal to one over
c square d two f over d t square
so we immediately see that the speed of these
propagation or the electromagnetic wave is
going to be one over square root of epsilon
zero mu zero and e propagates like a wave
we can do the same thing for b equation and
we get in the b equation is curl you start
with curl of b is equal to mu zero epsilon
zero d e d t and take curl on both sides we
get curl of curl of b is equal to mu zero
epsilon zero d by d t of curl of e this is
gradient of divergence of b which is zero
minus laplacian of b is equal to mu zero epsilon
zero d by d t of minus d b d t that again
gives me del square b is equal to mu zero
epsilon zero d two b over d t square which
is the equation for b field
so a and b propagate like a wave what we are
going to do now is focus on a particular solution
of this harmonic plane waves in which e vector
is going to be given as some amplitude e zero
sine of two pi over lambda x minus two pi
mu t which i can write as e zero sine of k
x minus omega t where k is two pi over lambda
and omega is two pi mu similarly i am going
to write b as some b zero sine of k x minus
omega t so i have e is equal to e zero sine
of k x minus omega t and b equals b zero sine
of k x minus omega t let us see what equations
tell us about these amplitudes and the fields
if they exist like this
so let's look at the equation divergence of
b is zero and divergence of e is zero if we
do that lets look at this equation divergence
of e is equal to zero i am going to have i
d by d x dot e zero sine k x minus omega t
is equal to zero notice that all the other
components d by d y and d by d z they vanish
because we have taken e naught to be same
all over y and z so this gives you i the x
unit vector dot e zero is equal to zero this
means e zero is in the y z plane what it means
is i have this wave propagating in the x direction
then e can have components let me make since
i am writing e with green let me make e can
have components y and z so let's write this
e y and e z and nothing else because e dot
i is zero this is y this is z
similarly if i look at divergence of b is
equal to zero i get i dot b zero is equal
to zero this also means that b has components
b y and b z only so i have b y and b z so
both e and b are perpendicular to direction
of propagation as we had indeed argued earlier
let us now look at the relationship between
e and b using other maxwell's equations so
we have now seen that e is e in the y z plane
zero sine of k x minus omega t and b is b
zero sine of k x minus omega t and we want
to see the relationship between b zero and
e zero just one point you may be wondering
why i have kept sine k x minus omega t for
both same variation for both electric and
magnetic field if i did not do so you will
see later that these terms will not cancel
on two sides and therefore their dependence
on k x minus omega t has to be exactly the
same
now let us look at the equation curl of e
is equal to minus the partial derivative of
b with respect to time if i calculate curl
of e this is equal to i j k d by d x d by
d y d by d z x component of electric field
is zero the y component is going to be e y
zero sine of k x minus omega t and the z component
is going to be e z zero sine of k x minus
omega t and if i calculate the curl it comes
out to be i component times zero plus j component
times zero minus d by d x of e z zero sine
of k x minus omega t plus k component d by
d x of e y zero sine of k x minus omega t
minus zero so this comes out to be minus j
e z zero k cosine of k x minus omega t plus
k unit vector e y zero k cosine of k x minus
omega t i want to remind you that this k is
two pi over lambda and omega is two pi times
a frequency nu this can be further written
as k cosine k x minus omega t minus j e z
zero plus k unit vector e y zero which i can
write as k cosine of k x minus omega t i cross
e y zero j plus e z zero k there's last term
here is nothing but e vector zero so what
we've obtained
go to the next slide that curl of e is nothing
but k cosine of k x minus omega t i cross
e zero vector similarly minus d b by d t is
going to be equal to b zero vector d by d
t of sine k x minus omega t with a minus sign
in front and this is going to become then
plus omega b zero vector cosine of k x minus
omega t notice that ah earlier i had said
if the space time dependence that means k
x minus omega t was different then i could
not have equated this space and time dependence
of the two more explicitly if i write equate
curl of e is equal to minus d b d t i get
k cosine of k x minus omega t i cross e zero
is equal to omega b zero vector cosine of
k x minus omega t if this dependence which
is given by cosine whose not the same on both
sides although it could have the the equation
could have been satisfied once in a while
but it'll go out of phase further times so
this dependence has to be the same
the net result is that we get b zero vector
is equal to i cross e zero times k over omega
k over omega is nothing but one over c i cross
e zero this you can easily see from definition
of k being two pi by lambda and omega being
two pi nu so what we have now is that if the
wave is propagating in the i direction if
e vector has components like this the b vector
has to be i cross e it has to be perpendicular
to it so if e vector is like this let me make
a little thick then b vector has to be perpendicular
to this in the same plane so it has to be
either this way or this way i cross e gives
me that now which way would it go for that
what we can do is we can multiply take e cross
b again which will be equal to one over c
e zero cross i cross e zero and that gives
me one over c i e zero square minus one over
c i dot e zero e zero so this term is zero
because i the propagation direction is perpendicular
to e zero so what we see is that e cross b
should be i e cross b should be in the direction
of propagation so b has to be such that i
would choose this particular b so that b cross
e gives me i
so what we see is propagation direction and
amplitude electric field are perpendicular
propagation direction magnetic field are perpendicular
then e cross b is has the same direction as
direction of propagation so if this is a direction
of propagation x and if e happens to be in
the y direction then b would be in the z direction
on the other hand if the wave was propagating
the other way it was going to the negative
x axis if the wave was propagating this way
then if e y e was in the y direction b would
be in the negative z direction this is what
we get from the maxwell's equations and finally
we saw that b zero is equal to one over c
i cross e zero which means that magnitude
of b is equal to magnitude of e over c smaller
from e by the factor of speed of light which
is three times ten raise to eight meters per
second so magnetic field in electromagnetic
wave is much smaller than the electric field
when they are measured in s i units
