Welcome, in the earlier lectures we have seen that the time varying electric and magnetic
field constitute a wave phenomena. Obviously then this wave requires some power or energy
to flow with it. In this lecture essentially we investigate the power flow associated with
an electromagnetic wave. We will do some derivation starting from the basic Maxwell's equation
and then ultimately point out how much will be the power flow associated with electric
and magnetic fields.
Till now we have investigated a uniform plane wave which is a wave propagation in a unbound
medium. However, when we are developing this power flow calculations associated with electromagnetic
waves we will do the general analysis and not restricted to the uniform plane waves.
Of course at the end of the discussion we will find out how much is the power flow associated
with the uniform plane waves which we have discussed in the last two lectures.
So when ever we have beginning of analysis in electromagnetic essentially we go back
to the Maxwell's equations and find the answers which is consistent with the Maxwell's
equations. The same thing we do here again where we ask the question if you go back to
the Maxwell's equations what answer I get for the power flow associated with electromagnetic
fields. So in this lecture we are going to do the power flow associated with an electromagnetic
wave.
Now going to the Maxwell's equations and essentially the curl equations we have in
general the (Refer the equation in video) that is if I assume the permeability of the
medium is not a function of time I can take μ out and this can be written as (Refer the
equation in video) and the second equation which will be (Refer the equation in video)
where they are writing D in terms of the electric field and again assuming that the permeability
of the medium is not a function of time so this can be written as (Refer the equation
in video).
So again we start with these two basic equations and then try to investigate the power flow
associated with electric and magnetic fields.
Here essentially we make use of the vector identities and then try to find out meaning
to some of the terms which I am going to get in the expansion of the vector identity. So
let me just take the vector identity which is (Refer the equation in video) where A and
C are some two arbitrary vectors. So if I have any two arbitrary vectors A and C we
have this vector identity for these two vectors. Now what we can do is let us say this vector
(Refer the equation in video) is the electric field and this vector (Refer the equation
in video) is the magnetic field so substituting for A as E and C as H essentially the vector
identity for these two vectors the electric and the magnetic field can be written as (Refer
the equation in video) and we can substitute for (Refer the equation in video) from this
equation and for (Refer the equation in video) from this equation.
So from here I can get this quantity as (Refer the equation in video).
I can rewrite for the two arbitrary vectors A and C, again if I take a time derivative
of (Refer the equation in video) where (Refer the equation in video) and (Refer the equation
in video) are some two vectors. This is nothing but (Refer the equation in video) so this
is true for any two arbitrary vectors A and C.
If I take both the vectors A vectors then I can get (Refer the equation in video) so
that quantity is nothing but (Refer the equation in video).
So from here essentially we get (Refer the equation in video)which is nothing but mod
of A2 so this is (Refer the equation in video).
So we can make use of this relation for simplifying this. Essentially if I take this μ out this
is (Refer the equation in video) just a quantity which is similar to this, similarly if I take
this then this will be (Refer the equation in video) so I can substitute from this into
this equation and I get the equation as (Refer the equation in video) that is equal to from
here I am substituting (Refer the equation in video) which is (Refer the equation in
video) so this will become (Refer the equation in video) so this will become (Refer the equation
in video) and we can have the second term as this which is (Refer the equation in video)
so that will become (Refer the equation in video) and then finally we can have this term
(Refer the equation in video).
Up till now we started with Maxwell's equations which we have point relations that means these
equations are valid at every point in space. This relationship which we have got here is
essentially a point relationship so any point in the space essentially this condition is
satisfied. In general, if I am having a medium which is having a finite conductivity that
gives the conduction current density J and if I am having medium which is not varying
at a function of time then in general the electric and magnetic fields satisfies these
equations.
Now what we can do is we can integrate this quantity over a closed surface or a volume
and then we will have some meaning associated with these quantities. So let us say if I
integrate this over a volume I can get this because this is a triple integral integrated
over a volume (Refer the equation in video).
Again assuming that this volume is not varying as a function of time that means the fields
are only time varying but the space is not varying as a function of time we can interchange
this (Refer the equation in video) with the integration so we can take this (Refer the
equation in video) out of the integration sign and the same thing I can do here also.
And I can apply divergence theorem on this left side to change the volume integral to
the surface integral.
So this gives the left side then becomes the surface integral over a closed surface that
will be (Refer the equation in video) that is the application of divergence theorem so
this thing now the volume integral is converted to the surface integral by using divergence
theorem and interchanging the sign for the time derivative and the integral sign we get
(Refer the equation in video) this is on closed surface again (Refer the equation in video).
I can substitute for (Refer the equation in video)so this term here if I look at and if
I substitute for J = σ then this term will become σ |E|2.
Now if I look at this quantity and here this quantity essentially gives me the density
of the magnetic energy stored in this volume, this quantity tells me the electric energy
stored in the volume. So this is basically the electric energy density, this is the magnetic
energy density integrated over the volume gives me the total energy stored in this volume
v due to magnetic field, this is the total energy stored in this volume due to electric
field so (Refer the equation in video) of this quantity essentially gives me the rate
of change of the magnetic energy stored in that volume, similarly this term gives me
the rate of change of the electric energy stored in that volume. Also the negative sign
shows that the rate of change is negative that means there is a decrease in the energy
as a function of time. So this quantity essentially tells me the rate of decrease of the magnetic
energy stored in that volume v, this quantity tells me the rate of decrease of electric
energy stored in that volume v.
By substituting J = σ this quantity essentially tells you the Ohmic laws into the medium.
So what we find is here the first term tells me the rate of decrease of magnetic energy
in that volume, this quantity tells me the power loss taking place in that volume because
of the finite conductivity of the medium.
So if you have a total energy includes in a surface this power loss total must be equal
to the energy which is equal to essentially leaving in that box. So if I take this volume
v which is having a corresponding surface area s since there is no other mechanism of
consuming energy from the conservation of energy essentially we get that this quantity
must represent the flow of energy coming from the surface or rate of flow of energy coming
from the surface.
So from here what essentially we see is this quantity the surface integral for (Refer the
equation in video) tells you the net power flow from a closed surface. Then from here
this quantity has represents (Refer the equation in video) is net power flow from a closed
surface.
So essentially we find a very important thing that by doing simple vector manipulations
as we have done started with vector identity substituted the Maxwell's equations in the
vector identity and from there we find something interesting that this surface integral of
(Refer the equation in video) over a closed surface gives me the net power flow associated
with this electric and magnetic field. This statement is called the Poynting Theorem.
So the Poynting theorem says that for the electric and magnetic fields if I take the
cross product of that and integrate over a closed surface that gives me the total power
flow from that closed surface.
Now if this is the quantity which is representing the total power flow then we can say this
quantity (Refer the equation in video)is essentially the power density or power flow density on
the surface of this closed surface. So when we integrate this power density over the surface
area then that gives me the total power flow from the surface. However we should keep in
mind just saying that this whole integral is giving me net power flow that is why this
quantity should gives me the power density at every point on the surface of the sphere
is a arbitrary definition.
The Poynting Theorem does not say that this quantity is representing the power density
or the power flow per unit area at every point on the surface of this volume. what I am telling
you is that the total power coming out of this is equal to this quantity. So this quantity
(Refer the equation in video) is a power density and that is true at every point on the surface
of the sphere is a arbitrary definition of the power density which we take from here.
It so happens in most of the practical situations this arbitrary definition gives you the power
density correctly. However if you ask rigorously whether knowing this quantity should be said
as this is representing power density at every point on the surface of this volume, this
statement is not correct.
In fact there may be special methodological phases where this argument will fail that
if you have (Refer the equation in video) at some particular point it may give you a
power flow where there is actually no power flow. So while using this quantity as the
power density one has to be little careful. However, in most of the practical situations
as I mentioned this arbitrary definition that (Refer the equation in video) gives me the
power flow density at particular location that normally is valid.
Now essentially what the important thing that we get is a Power Flow Density and let me
call that quantity as some which is a vector and that is equal to (Refer the equation in
video). Then we call this quantity as the Poynting vector for these fields or for this
so this quantity is called the Poynting vector.
So Poynting vector is a very important concept in the electromagnetic waves because it tells
you what the density of the power flow at a particular point in the space is and also
it tells you in which direction the power is flowing. We know this quantity is a vector
quantity so first thing we note here is if you have electric and magnetic fields then
the Poynting vector is in a direction perpendicular to both of electric and magnetic fields because
we have this cross product that means this vector is perpendicular to both of these vectors.
So firstly if this quantity has to be non zero if there is a power flow. Now first thing
we note here is that (Refer the equation in video) should not be parallel to each other.
If you have (Refer the equation in video) electric and magnetic fields parallel to each
other then the cross product will be identically zero and there will not be any power flow
associated with this. So only the component of electric and magnetic field which are perpendicular
to each other they contribute to the power flow and the direction of power flow is perpendicular
to both the electric and magnetic fields or in other words that will have a power flow
in the due to the fields the E and H must cross each other. When ever there is a crossing
of E and H there is a possibility of power flow and this is the word possibility here
because this quantity essentially is telling you the so called the instantaneous power
if I know the value of E and H at some instant of time at some point in space I can always
find this cross product at that instant of time and I will get a number this quantity
which will give me the Poynting vector at that instant of time. This is possible that
even if there is a which is finite at some instant of time there may not be any net power
flow over long time periods that means if I say in time average sense there may not
be any power flow associated with the system.
So Poynting vector which we define as (Refer the equation in video) serve the purpose of
defining the power flow but if I seen a practical system probably more useful quantity will
be time average value of this Poynting vector because if I take some instant of time first
of all this quantity one by one negative so if I say this is telling you power it may
even give me the power which is negative. Of course when we are dealing with the space
we can say negative power means the direction of the power flow which is changed but all
those complications will come if I use simply the (Refer the equation in video) and get
the value of because can go positive negative also depending upon the time phases between
E and H even this quantity can go as the complex point.
So what we do is we essentially try to get the time average value of the Poynting vector
and that is what more meaningful quantity for finding out whether there is a net flow
of power associated with the electric and magnetic fields. As we have seen earlier in
our analysis essentially we are interested only in time harmonic fields. So we will do
the analysis for time harmonic fields here. So again we assume that the electric and magnetic
fields are varying sinusoidally as a function of time, only thing they can have is phase
difference between them the temporal phase difference. And then we can ask the general
question what would be the average power flow or the Poynting Vector associated with those
electric and magnetic fields.
Now let us define the general time varying fields for electric and magnetic fields which
could be varying as a function of space and time. Let us say at some point in space I
have the electric field which is having some magnitude (Refer the equation in video) and
is having a variation ejωt and let us say it has some phase which is given as e so this
quantity is having some phase j times (Refer the equation in video).
Similarly I can have a magnetic field which is oriented in some direction so it having
a magnitude in some arbitrary direction but it is having the same frequency so it is ejωt
but it may have a time phase which could be different so this is (Refer the equation in
video).
What we can do is we can just take out this vector associated with this as a unit vector
and just write down this quantity only as the magnitude of the electric field, the magnitude
of the magnetic field. If I take the instantaneous values of the electric and the magnetic field
then I can get the instantaneous values of the fields as E at some instant of time which
will be the real part of this quantity and as I have mentioned I can take the unit vector
out of this I can keep only the magnitude so this will be the magnitude which is (Refer
the equation in video) multiplied by the unit vector which is (Refer the equation in video)
where (Refer the equation in video) gives me the unit vector in the direction of this
electric field.
Similarly I can get the instantaneous value of the magnetic field at some time t which
will be the real part of again I will do the same thing I will take the magnitude of this
magnetic field (Refer the equation in video) phase of the magnetic field multiplied by
the unit vector which is the (Refer the equation in video) direction.
So (Refer the equation in video) essentially gives me the vectors in the direction of electric
and magnetic fields and (Refer the equation in video) and (Refer the equation in video)
will give the phase of electric and the magnetic fields respectively. And and H0 are the amplitudes
of the electric field at peak amplitudes associated with this. So if I take real part of this
quantity that gives me the instantaneous value of the electric field and the instantaneous
value of the magnetic field.
Once I know this quantity then I can find out at that instant of time the Poynting vector
which essentially is taking (Refer the equation in video). So before that if I just separate
out the real part of this from here we get the instantaneous value (Refer the equation
in video) which will be real part of this quantity so that is equal to (Refer the equation
in video) multiplied by the unit vector (Refer the equation in video). And (Refer the equation
in video) will be (Refer the equation in video) multiplied by the unit vector.
Once we know these vector quantities at that instant time of t then we can calculate now
the Poynting vector and that gives me the Power Flow Density at that instant of time
t. So from here we can get the Poynting vector which is (Refer the equation in video), I
just take product of these so that is equal to (Refer the equation in video).
So this quantity is the scalar quantity and you have the cross product which essentially
is the cross product of the unit vectors, I can simplify this so this gives me essentially
(Refer the equation in video).
Now this is the instantaneous value of the Poynting vector and as we mentioned we are
now interested in finding out what is the average value of the density or what is the
average value of this Poynting vector. So we can take a time average of this over a
period of this signal, if I integrate this power density over one period or one cycle
so essentially we get the average power density associated with this but this quantity will
go to zero over that one period.
So, essentially this is corresponding to a waveform which is having a frequency of 2f
or angular frequency 2ω. So over a period corresponding to ω this quantity will identically
go to zero so if I take the time average of this quantity which is (Refer the equation in video) where
T is the time period associated with this angular frequency ω so (Refer the equation
in video) then I get the average Poynting vector and in that this quantity will essentially
goes to zero. So I get the average value of the Poynting Vector (Refer the equation in
video).
So now the average Poynting vector is the average value of this quantity and you have
a cross product of the unit vectors of the electric and the magnetic fields. Now this
quantity is not a function of time this is constant so this can be taken out. So the
integral will be (Refer the equation in video) integral zero to T dt which is nothing but
equal to one.
So now we have (Refer the equation in video) and this cross product of unit vectors and.
We can do little more algebraic manipulation to write again back the electric and the magnetic
fields in the vector form. So what we can do is this quantity now can be written as
(Refer the equation in video)
So what we have done is
we have added this quantity ejωt and e-jωt in the expression if you see here this will
be E0 H0 these are scalar quantities, we will have cross product of E and H which is this
and if I take the real part of this quantity (Refer the equation in video) multiplied by
(Refer the equation in video) will give me the (Refer the equation in video).
Now if I take this negative sign here this quantity is the scalar quantity so you can
write this also like the real part of this quantity which is (Refer the equation in video).
So the conjugate of this quantity will be a scalar quantity real quantity so complex quantity is only
this (Refer the equation in video) so if I take the conjugate of this essentially this
represents this quantity. But this quantity is the original magnetic field which we have
defined in the vector form, similarly this is the quantity is the original electric field
which we define in the vector form. So essentially we have this quantity here the electric field
is zero (Refer the equation in video) multiplied by unit vector, same is true for magnetic
field here.
So this quantity is nothing but electric field and this is the magnetic field so this is
half real part of which is the average Poynting vector. So if I know the electric and magnetic
fields in the complex form that means the electric and the magnetic field may not be
in time phase all the time then in general we can just calculate this cross product of
and real part of that the half vector is essentially because of the rms value we get in the signals
since the signals are time varying sinusoidally essentially this is the rms factor so now
this gives me the average power flow which will be associated with that electromagnetic
wave.
Now this quantity is the real quantity as we are taking a real function of this so all
those problems which we had with the instantaneous power flow could either become complex depending
upon the phase difference between them and all those have been taken care of and also
it tells me the overall power flow which is associated with this fields at a particular
location. So it is possible at a particular location the instantaneous Poynting vector
might be negative or positive but when you calculate the average Poynting vector then
that will be always positive and that will give me the net power flow which will be associated
with those electric and magnetic fields. So this is the concept which is very regularly
used in finding out the average power flow associated with an electromagnetic wave.
Again now there are two things are essential to have the average power flow, one is the
electric and the magnetic fields must have a component perpendicular to each other then
only you will have a cross product which is non zero and at the same time the electric
and magnetic fields should not be in time quadrature that means the phase difference
between the electric field and magnetic fields should not be 90˚ because if it is 90˚ then
the real part of this quantity will be zero and then you will not have any real power
flow associated with that one.
So in general it is possible if you take the electric and magnetic fields you will have
the complex power the real part of that quantity gives me the net power flow at that location
but the imaginary part of that quantity (Refer the equation in video) gives me the power
which is oscillating around that point so some instant of time the power might be going
in certain direction if you see after some time the power will be essentially coming
back in the same direction. So the imaginary part of (Refer the equation in video) gives
me some kind of a oscillating power which you call as a reactive power whereas the real
part of (Refer the equation in video) gives me the net power flow or the resistive power
flow at a particular location. This concept of Poynting vector and the average
Poynting Vector is the very important concept because by using this concept we can calculate
the net power flow at a particular location. One can then apply this concept to the case
of the uniform plane wave. So we can ask if you are having a uniform plane wave then how
much power density the uniform plane wave carries when it travels in the medium. We
have seen for uniform plane waves first of all the electric and magnetic fields are perpendicular
to each other so if I take a uniform plane wave then the electric field the magnetic
fields are perpendicular to each other and then let us say this is electric field (Refer
the equation in video), this is the magnetic field which is and the power will be flowing
in this direction which is the cross product of these two. So this is the direction of
the Poynting Vector (Refer the equation in video).
Again we can apply the right hand rule to find out whether the power will be flowing
in this direction or will be flowing in this direction. Since we are taking the cross product
of E and H again we point the fingers in the direction of E to H and the thumb should be
in the direction of the cross product which is the direction of this. This direction is
same as the direction of the wave propagation also because we have talked about uniform
plane wave E and H and the direction of the wave propagation essentially form the three
coordinate axis in the same sequence that means if I go from E to H my fingers point
from E to H the thumb should go to the direction of the wave propagation. So we correctly get
the direction of the average Poynting vector which is the same as the wave propagation.
Secondly, now if I say the electric field is some magnitude E0 and is having a phase
variation which is e--jβ let us say the wave is traveling in z direction so this is
the z direction e--jβz and E can be oriented let us say x direction then H will be oriented
in y direction so I can say this is oriented in x. Then the magnetic field H will be oriented
in y direction since it has a magnitude H0 e-jβz orientation is the y direction.
So the average Poynting vector associated with this (Refer the equation in video) is
equal to the half real part of (Refer the equation in video), now the cross product
of x and y will give my direction z so this average Poynting vector is in z direction
so this is half of real part of this conjugate so essentially that will become E0H0 so this
will give me E0 H0 in the direction (Refer the equation in video).
In general if I assume this quantity could be complex quantity I can put still even the
conjugate sign at this. So for a uniform plane wave the average Poynting vector will be half
of (Refer the equation in video) and the direction of this will be z, if you say the electric
field was oriented in the x direction and the magnetic field was oriented in y direction.
Now we can take specific cases for the unbound medium as the uniform plane wave is propagating
we can take first the medium which is dielectric medium. Now for a dielectric medium or in
general if I take a unbound medium first of all we know there is a relationship between
these two quantities E and H that is the magnitude of the electric and magnetic fields are related
to what is called the intrinsic impedance of the medium. So I also have a relation for
a uniform plane wave having electric and magnetic field and that is equal to η which is Intrinsic
Impedance.
So I can substitute for the magnetic field from here that will be E upon η or I can
substitute for electric field which is H times η. So I get the average Poynting vector (Refer
the equation in video) or if I write in terms of the magnetic field this will also be (Refer
the equation in video).
Now E0 (Refer the equation in video) is mod (Refer the equation in video) so this is equal
to (Refer the equation in video) or of course if I am putting the conjugate here I must
put the real part of that so this is real part of this quantity, the same thing you
have to put here this is the real part so this is again (Refer the equation in video).
So from here essentially we can find out what is the average power flow associated with
the uniform electromagnetic wave in an unbound medium.
Now if I take a dielectric medium an ideal dielectric medium that means there is no conductivity
in this medium for which we know that (Refer the equation in video).
So this quantity is a real quantity for an ideal dielectric medium so this quantity essentially
the η* since this is a real quantity the same is η so in this case the average power
density (Refer the equation in video) will be equal to (Refer the equation in video)
and that will also be equal to (Refer the equation in video).
So in a dielectric medium if I know the magnitude of the electric field or this is the peak
amplitude of the electric field and I know the permittivity and the permeability of the
medium then I can find out the Intrinsic Impedance of the medium, this quantity is real. So just
by knowing the amplitude of the electric field I can get the power flow density associated
with this uniform plane wave.
In general if this medium is having a conductivity which is neither zero nor very large which
is like a conductor then we have to really go through this expression to find out what
is the net power flow associated with it. However we can take an extreme case that is
if you have a good conductor then we know that the Intrinsic Impedance of this medium is approximately equal to (Refer
the equation in video) which we have already seen.
So if I take this Intrinsic Impedance and substitute in this expression here then I
can get the average power flow density (Refer the equation in video) will be equal to (Refer
the equation in video).
So essentially by using the concept of average Poynting Vector we can find out the power
flow in any medium and at any particular location in space. In case of the dielectric the calculation
is very straight forward because the intrinsic impedance of the medium is real whereas when
we go to the medium which is like a good conductor or in general medium where conductivity is
finite then one has to go to the more general expression of finding out the average power
flow associated with electric and the magnetic fields.
Thank you.
