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.