In the previous two lectures we talked about
electromagnetic waves that come naturally
out of maxwell's equations in particular we
focused on wave propagating in the x direction
and wrote this as e x t is equal to e zero
sin of k x minus omega t and be x t is equal
to b zero sin of a x minus omega t if you
generalize this i can in general right for
wave propagating in the direction of unit
vector n i can write e r vector t is equal
to e zero sin or cosine of some other face
so i can write this as in general k dot r
minus omega t and b as a function of r and
p as equal to b zero sin of k dot r minus
omega t and if i want to more general i will
add a phases also some face phi that can make
a cosine or whatever where the vector k is
equal to two pi over lambda in the direction
of the propagating
in the previous lecture we saw the waves for
propagating i direction i dot e zero zero
i dot b zero zero in the same manner in general
i am going to have k dot e zero is equal to
zero that means the electric field is perpendicular
to the direction of propagation and similarly
a dot b is going to be zero and e cross b
as going to be in the direction of propagation
with this background what you want to focus
on is again waves from this time which are
harmonic and plane waves so plane waves and
harmonic harmono chromatic or free moving
frequency omega waves and see what is the
energy contained of this waves how much energy
do they carry and this waves and how much
momentum flow and how much momentum to they
contain
remember this are the quantity we have had
already calculated for electro static field
also would be given also focus on that omega
is going to be large so we are talking about
something like the ten raise fifteen ten raise
ten ten raise to fourteen that means in one
second this so many oscillation we talking
about light waves of microwave in that case
it does not make sense of energy variation
but will be talking about time averaged and
what you mean by this will be taking one over
the time period and integrate whatever quantity
we get from zero to three hour time whatever
quantity energy density or the energy flow
the momentum flow so in at such heigh frequency
what we see is
for example if i am looking at this light
i am not seeing it varying i am seeing a constant
light because it varies ten raise to fifteen
times per second and therefore i see only
the averages of my eyes somehow i will so
we have e equal to e zero sign k dot r minus
omega t and b is equal to to b zero sin of
k of a dot r minus a omega t if you want you
can add phi also doesn't really make a difference
let's calculate the energy energy content
we call that energy in the electric field
is one half epsilon zero e square and 
that is the amplitude becomes sin square k
dot r minus omega t we call i said that this
is varying in time very fast so i will be
talking about average energy average energy
is going to be one half epsilon zero e zero
square zero to t one over t sin square k dot
r at a given point r is fix minus omega t
d t
and this you know from your ordinary mathematics
is one half so this becomes one forth of epsilon
zero t zero square let us now calculate what
the energy in the magnetic field is magnetic
is magnetic field you recall is carries energy
one over two mu zero b square which is going
to become one over to mu zero b zero square
sin square a dot r minus omega t which then
time average will give me another factor of
two so this becomes one over four mu zero
b zero square we call from previous lecture
that b zero is nothing but e zero over c so
this becomes one over four mu zero e zero
square c c square but one over c square is
mu zero epsilon zero
so this becomes one fourth epsilon zero e
zero square therefore the energy contained
in the electric field and magnetic field component
of this electromagnetic wave is exactly the
same which is the one fourth epsilon zero
e zero square and you add the two and you
get the energy density in an electromagnetic
e m wave to be one half epsilon zeroe zero
square so when this wave exist for example
this light is coming to me then in between
average energy per unit volume is one half
epsilon zero e zero square so this wave let's
again take it as it propagating in next direction
if i take a small box here of unit volume
for small volume d v
and then we continue this is going to be one
half epsilon zero e zero square d v how about
the energy flow which is related to intensity
which is related to point in vector 
the point in vector 
is given as one over mu zero e cross b and
we all ready seen that e cross b is in the
direction of propagation so i can write this
as one over mu zero direction n mode e zero
and modulus of b zero sin square a dot r minus
omega t b zero we recall that is again e zero
by c so i can write this as one over mu zero
in the direction of propagation e zero square
over c sin square ek dot r minus omega t when
i take the time average sin square k dot r
of minus omega t is going to give me a factor
of two one half and therefore this becomes
one half e zero square over mu zero
and c is nothing but square root of epsilon
zero mu zero one over so this become one half
square root of epsilon zero over mu zero e
zero square in square in the direction of
propagation that means if there is an electromagnetic
plane electromagnetic wave going in some direction
in the same direction per unit area per unit
time the energy that flows is one half square
root of epsilon zero and mu zero e zero square
let us also see from the energy density point
of view the energy flow if you recall form
one or previous lecture is going to be nothing
but c time energy density u which is going
to be one over square root of epsilon zero
mu zero times one half epsilon zero e zero
square which is same as one half square root
of epsilon zero over mu zero e zero square
same answer
so that is energy that is carried by the electromagnetic
wave or that is the intensity that we received
this is also related to the momentum carried
by electromagnetic wave recall that the momentum
density 
is nothing but as over c square let us check
that modulus of s is nothing but energy m
l square t minus two pi unit area per unit
time which then becomes equal to momentum
density m l t minus one over l cube so if
i divide by c square here c square is nothing
but l square t minus two it is cancels from
doing c square one of the l i cancel and make
it l by get m l t minus one over l cube
so this is the momentum density and therefore
if there is a plane electromagnetic wave going
the momentum density is given as one half
square root of epsilon zero over mu zero e
zero square over c square given this momentum
density and the what is the movement flow
for momentum flow 
is nothing but c times of momentum density
which is going to be equal to one half epsilon
zero over mu zero e zero square over c let's
recall the write them in terms of u so s is
nothing but c times u u is the energy density
in the electromagnetic field over c square
which is mu over c and the momentum flow is
nothing but u itself if there is momentum
flow that means is there is an electromagnetic
wave going this way you carry the momentum
which is equal to u per unit area per unit
time
what does that momentum do suppose i could
screen here which absorbs all this radiation
fowling if all the radiation fowling is that
absorbed that means all the moment u per unit
are per unit time which is coming gets absorb
in this so this thing should get momentum
in this direction which is momentum per unit
time per unit area areas nothing but the force
per unit area which is equal to pressure so
in electromagnetic waves falling on steam
in where get absorbs apply the pressure equal
to energy density if the wave reflected perfectly
then the momentum coming coming in per unit
area per unit time would be u momentum going
out with u again under pressure would be equal
to two so i conclude this lecture by restudying
again that e m waves which carrying energy
which we see in the form of light or heat
which comes from towards from e m waves carry
momentum and this implies they can apply pressure
on the surface on which they are falls
