In last few lectures we investigated propagation of an electromagnetic wave inside wave guiding

structures. First, we investigated propagation inside what is called a parallel plane wave

guide and later we derived the fields inside a rectangular wave guide. Of course the approach

which we took in two cases was different. In first case that is for parallel plane wave

guide, we visualize the propagation of electromagnetic wave as the propagation of uniform plane wave

by multiple reflections at the two conducting boundaries of the parallel plane wave guide

whereas in case of rectangle wave guide we took the mathematical approach that is we

simply said if you wanted to have a transverse electric or transverse magnetic field, mathematically

just solving the wave equation in satisfying boundary conditions what kind of fields we

would get. The visualized wave which we had in case of parallel plane wave guide they

were the physical understanding of what kind of field can exist inside a wave guiding structure

and from there only we understood that if we go perpendicular to the planes, the conducting

boundaries then we will have the wave which will be standing wave kind of waves whereas

along the plains you will have traveling wave and using this physical understanding we can

appropriately write the mathematical functions for electric and magnetic field inside a rectangular

wave guide.

So, in one case we had physical visualization of the phenomena in other case it was a purely

mathematical approach and now we must verify that mathematically which we did blindly for

the rectangular wave guide, does it give me the fields which me have derived from the

physical understanding for the parallel plane wave guide. So, essentially what we have to

do is we have to establish that the fields which we got for rectangular wave guide, if

we take the limiting case then the fields must represent the fields which we got for

the parallel plane wave guide.

So, now what we will do is we will do an exercise of showing that these fields which we got

for the rectangular wave guides indeed represent the fields which we got for the parallel plane

wave guide is the limiting case when one of the dimension of the wave guide becomes infinity

and then we are also ask a question what happen to the mode which was TM 0 mode which became

the TEM mode in the parallel plane wave guide. What happen to that mode in the rectangular

wave guide? Does it mode now exist inside rectangular wave guide or it exists only inside

the parallel plane wave guide?

So, let us take first the second question that is does the transverse electromagnetic

mode exists inside a rectangular wave guide. This we can argue physically that if we consider

a rectangular wave guide and let us say the, we are taking whether transverse electromagnetic

mode would exist inside the structure that means for this mode the electric and magnetic

field both are perpendicular to direction of propagation. That means they must lie in

this plane, the cross sectional plane that is the plane of the paper. Now we know that

magnetic field lines must close on themselves and all these magnetic field lines are now

lying in this plane that means they must form the closed loops in this plane. So, the magnetic

field lines in the transverse plane must look like that.

However, if the magnetic field lines are like this then we know that there has to be a current

enclosed by this magnetic field lines. So, there must be a current flow something like

this which is enclosed by the magnetic field lines then and then only this magnetic field

lines can survive. So, there are two possibilities one is we have a conduction current which

is flowing perpendicular to plane of the paper and that is enclosed by this magnetic field

line, so they can survive or there could be displacement current which will be perpendicular

to the plane of the paper and that will be enclosed by the magnetic field lines, so the

magnetic field lines can survive.

Since, we do not have any conducting medium inside this wave guide, this wave guide is

completely hollow, it means it is filled by pure dielectric, there is no possibility of

conduction current enclosed by this magnetic field lines. The conduction current is 0 inside

the rectangular wave guide, so only possibility which can exist with we have displacement

current to support this magnetic field lines and the displacement current must be flowing

in that direction but the displacement current would require the electric field which will

be going in that direction which we all denied for transverse electromagnetic mode but transfer

electromagnetic mode we say the electric field also lies only with transverse plane that

means it does not have a component perpendicular to the plane of the paper, if that is so then

we cannot have the displacement current inside this wave guide which is in this direction.

So, neither we have conduction current flowing this way nor we have displacement current

flowing this way, so there is no current which is flowing perpendicular to the plane of the

paper and if there is no current that cannot magnetic field lines because by MPS law this

magnetic field lines must be related to the current enclosed. So, that means these fields

which are lying in a transverse plane they cannot exist because there is no current to

support these magnetic field lines that means this mode, the transverse electromagnetic

mode cannot exist inside the rectangular wave guide.

So, this mode TEM mode doesn't exist inside this wave guide. There one can ask the question

how was it existing in the wave guide which is the parallel thin wave guide and before

you get into that question let us first take the parallel plane wave guide as a limiting

case of the rectangular wave guide. So, if I take this rectangular wave guide in which

the T 1 0 mode is existing that means electric field is like this and if I make this B is

equal to infinity, if I push this wall to infinity essentially I will get a structure

which will be the parallel plane structure, this plane and this plane wave is going to

traveling in the Z direction and these to walls are pushed to infinity, so as a limiting

case of the rectangular wave guide by pushing these two walls to infinity, I can get the

fields for the parallel plane wave guide.

And in this case since the electric field is like this and the wave is going to propagate

perpendicular to the plane of the paper, this mode essential would correspond to the TE

mode. So, we have now by pushing these things for T 1 0 mode, if I push this field it walls

to infinity then I will get a geometry which will be the parallel plane geometry and for

this geometry, electric field is oriented like this so that will give me the transverse

electric mode. So, what you are saying is if I take this wave guide and push it to infinity,

so these are the two walls of the wave guide is my coordinate axis.

This is the direction which is the Z direction, this is the x direction and I am pushing these

two walls to infinity and the electric field now is oriented in this direction which is this way. So, I have

a component which is Ey and I have the magnetic field which is Hx an Hz for the T 1 0 case.

So, if I look at the fields which I had for TE 1 0, I have Ey component I have Hx component

and I have Hz component and these fields were constant as the function of Y. Same thing

we can get now for the infinitely extended wave guide in Y direction that if there are

no boundary conditions to apply in this direction these fields are infinite then this field

is constant in that direction that was the precisely the field we used to get for parallel

plane wave guide. So, indeed we can take the transverse electric 1 0 mode, make this boundary

to go to infinity but since we do not have any dependence of be here, what that means

is these fields are the same fields as I would get inside a parallel plane wave guide for

a transverse electric 1 mode. So, these fields are TE 1 0 for rectangular wave guide but

these exactly the same the explanation I have for TE 1 mode for a parallel wave guide.

So, we have a situation here which is TE 1 mode for which the fields identical, two these

fields which are T 1 0 mode for the rectangular wave guide. So, here you have the magnetic

fields which are having X component and Z component then Ey will be in this direction.

So, you are having now the Z magnetic field which is in this direction, so as the limiting

case of the TE 1 0 mode we can get the transverse electric mode, lowest order transverse electric

mode which is T 1 mode and the fields will be exactly same as what we go for T 1 0. The

another mode which we had was the transverse electromagnetic mode for which the electric

field was not tangential to the boundaries, it was perpendicular to the boundary. Magnetic

field was tangential to the boundary and we saw that was the mode which was not dispersive

and which traveled as if the conducting boundaries are not existing and that is mode which was

the TEM mode.

So we had a situation like that if I take a parallel plane wave guide like that and

if this is the electric field E and this is the magnetic field H then the wave is going

to travel in this direction Z and then this wave travels without any variation of electric

and magnetic fields that we saw earlier because neither H requires a boundary condition to

be satisfied, this is tangential you can always surface currents nor the normal component

of electric field needs any boundary condition to be satisfied, you can always have the surface

charges induced to compensate for the normal component of the electric field. So, this

mode which was the transverse electromagnetic mode was existing inside and now one can ask

a question, if this was the limiting case of the rectangular wave guide, how come this

mode exist in this but when we take the rectangular wave guide this mode doesn't exist. But one

can argue since the magnetic field is in the direction and since this plane is extended

in Y direction, one can say that is magnetic field lines essentially close at infinity

and they enclose these conductors, these conducting boundaries which have the current flowing

on them, the surface currents.

So, in the case of rectangular wave guide where these boundaries are finite, this magnetic

field lines they have to close on themselves and that's why the current is not enclosed.

So, this mode cannot exist inside rectangular wave guide but if I take a parallel plane

structure then these magnetic field lines will close at infinity and they well enclose

this conductors and that's why this mode can exist inside the parallel plane wave guide.

This is the mode which is as we saw in case of two conductor system like a coaxial cable

or transmission line, this mode propagates. So, whenever we have two conductor system,

the lowest mode which will propagate will be TM mode, however if you go to the rectangular

wave guide then the lowest mode which will propagate will be TE 1 0 mode. So, we have

very important conclusion to draw that whenever we have the hollow structure, hollow pipe

and the wave guide, the transverse electromagnetic mode cannot exist which is non-dispersive

because it was a very special mode, this mode is not supported by the rectangular wave guide

and that's why we have all those problems of dispersion and so on inside a rectangular

wave guide. But that is the structure which is more physically realizable structure, you

should remember that. The parallel plane wave guide is still a structure which is infinite

in extent. So, we cannot realize this in practice, it is good for understanding but when we want

to realize the structure in practice it is the rectangular wave guide which can be realized

in practice not the parallel plane wave guide.

So, whenever we have a physical structure which is rectangular wave guide, we will always

have a cut of frequency associated with that and we will always have the problems like

dispersion and so on associated with that. With this understanding now let us try to

see the currents which get existed on the walls of the wave guide which actually support

this fields inside the wave guide, so whether you take parallel plane wave guide or you

take rectangular wave guide what is the mechanism by with this fields are supported inside wave

guide? What are the sources and the sources are nothing but surface charges and surface

current which lie on the inner surface of the wave guide. So, as the wave travel even

the surface charges keep moving to accumulate at different location as we will see little

later and these charges and surface current support the electrical magnetic fields inside

the wave guide which are responsible for carrying power inside the wave guiding structure.

So, what we will do now we will try to visualize first the fields which would be there inside

the wave guide and then will try to visualize how the currents are distributed on the walls

of the wave guide. So, let us take for the simplest case which is the TM case and the

parallel plane wave guide.

So, let us see I have a parallel plane geometry and let me Orient the planes, so these are

the two conducting planes the energy is going to propagate in this direction and if I take

the TEM mode, we saw the TEM mode the electric field is perpendicular to this. So, if I say

this is my x direction, this is y direction then we had the electric field Ey which was

some constant e to the power minus j beta z and we had a magnetic field which was a

Hx which is c upon eta e to the power minus j beta z. If I visualize these fields is say

if I really want to find out how these fields are going to be either the function of space

and time inside the structure, first what we do we freeze the time. Remember all this

quantities are intrinsically functions of time and they are having it harmonic variation

as a function of time, so e to the power plus j omega t is implicit in all these terms.

So, as such these fields are going to vary as a function of space and time. So, if I

stand at particular location the fields are going to vary sinusoidally as a function of

time. If I freeze time we are bringing instantaneously look along the wave guide, I will see the

field variation which will be given by that. So, essentially what we do to visualize these

fields in three dimension first we do is freeze time that means make this quantity constant

and then without losing generality we can say let make T equal to 0. So at that instant

I want to see how the fields are distributed in space. Once I get those field distributions

in space then we simply say that these fields will be moving with the velocity which is

the phase velocity for the mode to then the let the field drifts inside this wave guide

with the phase velocity. So, when we visualize these fields essentially we try to visualize

the special variation of the field at some instant of time and without losing generality

we can take that time is equal to 0. So, if I take the T is equal to 0, if I freeze time

then the field visualization will be simply a real part of this quantity which is real

part of c e to the power minus j beta z and this will give me again real part of c upon

eta e to the power minus j beta z. So, Ey has a variation which is real part of this

which is c into cosine of beta z, Hx will be having the same variation c upon eta cosine

of beta z.

So, if I consider this point as z equal to 0, this variation is beta z and the wave length

which we have here or the phase constant is beta, so the corresponding wave length let

us call in general that is the guided wave length in the direction of wave propagation.

So, this field has the period, special period which is equal to the guided wave length and

at z equal to 0 this field is maximum. So, it is maximum here and it is not varying as

the function of x or y, so the fields are constant everywhere. So, the field is varying

as a cosine function along the z direction. So, it is maximum here, so if I take here

this point j z equal to 0, so it is maximum and then it goes like that either I move along

the z direction like a cosine function where this is 0, this quantity will be lambda g

by 4 that this function will go to 0. If I go again lambda g by 4, this function will

become negative and so on.

So, this filed now I can visualize as the electric field which is constant everywhere,

so this is the field distribution for the electric field is like that which is same

everywhere and so on. And the magnetic fields are horizontally oriented and there again

having the same amplitude all across, so this is the direction of the magnetic field. So,

though direction is taken appropriately so that E cross H gives me the power flow which

is in this direction. So, this is magnetic field which is everywhere as distributed,

so this is my H vectors and this my E vectors. And at to maximum here, as I move to lambda

g by 4 along the direction of wave propagation these fields will go to 0 then again they

will reverse the direction after lambda g by 4, go to maximum when z becomes equal to

lambda g by 2 and so I got a variation which is if I look like that, if I turn it electrical

field is maximum here slowly reduces, goes to zero, changes a direction again increases

maximum negative. So, start from here goes to 0, changes direction increase again maximum

that's the where electrical field is distributed. You can see here the rod shear which is standing

of different heights, it is maximum here at z equal to 0, the height of the rod goes to

0 at lambda g by 4, the direction of the rod is reversed. And again the amplitude becomes

maximum, when I go a distance of lambda g by 2.

Same thing happens here for the magnetic field lines which are now like that, so they are

stacked. This will look as if you are having a layer of the magnetic fields which are oriented

this way as stack and again there variation is will be exactly same that when this field

goes through 0, the magnetic field also will go to 0 and so on. So, it requires little

imagination to really visualize these fields but with little practice one can definitely

see how the electric fields and magnetic field are distributed inside the wave guide.

Now, if we know that the magnetic field is like that the current which is going to flow

on the surface of this conductor is related to the magnetic field, tangential component

of the magnetic field. The N cross H gives me the surface current on these conducting

walls and since we are considering here the ideal conductors, essentially if I calculate

N cross H that is the current which is truly confined to the surface of this parallel plane

wave guide. So, if I consider this surface then the normal will be coming outwards, so

this is my normal unit vector whereas if I consider this plane then the normal which

will be coming out of the conducting surface will be this now.

So, the unit vector outward normal for the lower plane will be going upwards, the unit

normal for the upper plane will be going downwards and now you can calculate N cross H everywhere.

So, this is direction of H, this is N, so if I calculate N cross H at the point I will

get the direction of the surface current which will be along the direction of the wave propagations.

Similarly, say on this surface the current will be going in this direction because I

get N cross H whereas if go to the this surface then I will get N cross H, N is now down words

H is this, so I will require N cross H which gives me the current which will be coming

out.

So, on the lower surface I get a current which is going this way, this is my surface current

where on this surface the current will be flowing this way which is Js and it will be

having a same variation as the variation of the magnetic field that means it will be maximum

here after the distance of lambda g by 4 this current will go to 0 and so on. So, in a two

conductors system, it gives you the current flow in the direction in which the wave is

propagating. I have a current flowing this way, I have a current flowing this way and

precisely that's what we are essentially used to that when we have a two conductor system

like a coaxial cable, we connect a voltage source and the current goes inside the conductor,

the current returns by what is called as a ground path. So, we see that the current goes

inside from one terminal and current comes out side from the other terminal. So, in that

situation which is like a two conductor system, the direction of the current flow which we

have for this mode, TM mode keep that mind is same as the direction of the wave propagation.

So, the current is going, current is returning from here.

Question one can ask is if the wave is propagating in this direction, is it always true that

the current has to flow in the direction of the wave propagation and invariably we get

the answer which is wrong and that is since we always discuss the transverse electromagnetic

mode that's what we are used to when we go to low frequency circuit immediately we jump

to the conclusion that yes since the wave is moving in this direction, the power is

flowing in this direction, the current must flow in that direction.

Let us look at the parallel plane wave guide but let us take the mode which is TE 1 mode

and let us see what will be the direction of currents in field in that wave guide. So,

let us consider the structure which we have got this is the T 1 mode and this is parallel

plane wave guide. As we saw that the fields for the parallel plane wave guide will be

identical to this, so firstly the electrical field will be y oriented, they will have variation

which will be next direction and they will be varying again in z direction with phase

constant beta so in this direction if I look at the field variation and if I write down

the expression for this, I will get again in z direction also.

However to visualize these fields, as we did in the previous case let us take amplitude

of this field that is real part of this quantity and plot that as a function z, x and y, you

will get a three dimensional picture of the fields. Before we do that however let us try

to take this j and represent that along with this because j is e to the power j pi by 2,

so minus j will be e to the power minus j by 2.

So, if I take this field expression and write that I get this quantity Ey that is equal

to let me put this whole quantity as some constant, at the moment we are only interested

in finding the variation of the field in X and Y, so let there be some constant here

expect j we are going to put at this point. So, let us say I have some constant here some

A sin of pi x by a and I write here e to the power minus j beta z and I take this minus

j up so this will be equal to minus j pi by 2. So, if I want to get the instantaneous

field as a function of x y z at a real part of this quantity, so the instantaneous value

the electric field that will be real part of Ey that will be amplitude A sine of pi

x by a. Cosine of this quantity, this quantity and that will be equal to the sine of beta

times z where you have a phase difference here of pi by 2. So, you have here sine of

beta into z which we can write down explicitly in terms of the guided wave length. So, this

we can get is A sine of pi x upon a sine of 2 pi by lambda g into z. Same thing I can

do for this magnetic field also, so this quantity again there is omega epsilon, the omega nu.

So, I can say this is another constant for the magnetic field, so Hx I can write down

the instantaneous value that will be real part of Hx which will be some constant B sine

of pi x upon a and this quantity is same again we can take a minus j up there, so this will

be sine of 2 pi by lambda g into z. And I can get again the instantaneous value for

the magnetic field z component and there is no j here now, so if I take real part of this,

this will be simply cosine of beta z so the real part of the Hz that will be equal to

is constant C cos of pi x upon a cos of beta z, this is 2 pi by lambda g to z. So, these

are the fields which are now if I freeze time and if I take that T equal to 0 then at that

instant the field along the wave guide will be given by this expression. So, in the z

direction as we can see the fields are sinusoidal with a period, special period which is lambda

g however these two are in quadrature that means wherever Hx is maximum, the Hz is 0

and wherever Hz is maximum Hx will be 0.

So, in space we are having two components now, the Hx component and the Hz component

and Hx component is having a sine variation. So, if I take this point z equal to 0, Hx

is 0 at this location but Hz is maximum at this location. If I have go a distance of

lambda g by 4 on this, at that location Hx will be maximum but the Hz is will go to 0.

So, if I see from the top it will appear here the Hx was 0, only Hz was here. As it is moved

to a distance of lambda g by 4, the Hz becomes 0, only Hx become maximum so the magnetic

field totally become like this along this direction.

As I move further again by lambda g 4 again Hz will become maximum, Hx will come 0. So,

what you find that here we have a function which is cosine with that means if I take

this as plus direction, so here I have a magnetic field like that. On the opposite wall magnetic

field will be this the cosine function, so z equal to this quantity, this variation as

a function of x. If I plot this, this is the one which in the x direction is going to be

like this, z direction the variation is periodic with lambda g but in the x direction this

is maximum at x equal to 0, at x equal to a with opposite sign and 0 at the center.

So, the magnetic field lines are maximum here and no component of the magnetic field Hx

at this location, so the fields are this.

If I go to the distance of lambda g by 4 in this direction, at that location if I see

here the Hz would go to 0, Hx will be maximum but the Hx is a having variation of sine of

pi x upon a. That means Hx is maximum at this location but it is not maximum on the walls,

this function is a sine function. So, it is 0 here, 0 here and will be maximum here and

that will be the direction for Hx which would be like that.

So, if I see from the top I see situation like this for the parallel plane wave guide,

so if I am seeing on the top like that the magnetic field is z equal to 0, the magnetic

fields are like that, like that here. When I go to distance of lambda g by 4 this thing

has gone to 0 and Hx is become maximum so that is oriented in this direction. Now it

is matter of a little imagination since we know that magnetic field lines close on themselves

essentially we have a magnetic field loop which is created like this, so we got a magnetic

field lines which will go like that. And if I go further lambda g by 4 from here the direction

will be reversed, if I go further in this direction, this direction is x remember so

again I get the magnetic loop which will be like this and so on.

If I complete this if I now say the other way a wave guide which is running from minus

infinity to infinity, essentially I will get the loops of the magnetic field lines which

will look like that where this separation will be equal to lambda g by 2. So, if I look

at the wave guide which will look like that this is the parallel plane wave guide, the

magnetic fields lines look like rolled carpets which are stacked inside the structure and

they are of infinite length which are perpendicular to the plane of the paper. That's the way

the magnetic field lines will be distributed. What about the electrical field? The electrical

field has a variation in z direction which is same as this but it is oriented in y direction,

so if I consider this it is having sinusoidal variation in this direction but oriented this

way.

So, if I see from the top then the electric field lines will be perpendicular to the plane

of the paper and behavior of this Ey and Hx is identical that means wherever Hx is maximum,

Ey is going to be maximum only thing it has variation, it is sinusoidal variation so it

has identical variation 2 Hx.

So, if I look at the Hx component which is this component, here is Hx component which

is maximum, so the electric field also will be maximum at this location but electric field

is always oriented in y direction. So, I have the electric field which will be like that

and if this is the direction positive direction for z in which the wave is propagating again

you can have E cross H. So, if this is direction of the magnetic field the electrical field

should be going inwards so E cross H gives me the pointing vector so that is direction

of the electrical field and this electric field reduces to 0 at this location, so I

have electric field becoming lesser and lesser as I go this direction. Similarly, it's becoming

lesser and lesser because the tangential component should become 0 so this becomes less here,

less here and so on.

Again where I go on this side the electrical field now will be coming outwards and amplitude

will increase again, so here the size of the circle tells me the strength of the electrical

field. So, the electric field will be distributed maximum where this Hx component is maximum,

so this is more like rods which are inserted in the rolled carpets that is the electrical

field vectors. So, if we want to visualize in the three dimensional kind of fields, the

electric field will be the stacking of the rods of different diameters where the diameter

represent the strength of the electrical field and the magnetic field will be likes stacked

rolled carpets which will be like that.

Once we get this magnetic field things which is now a rolled carpet which we have got which

we have seen from the top the magnetic field lines like that, now we can ask what kind

of currents are going to flow in this. Say now the currents will be related to the tangential

component of the magnetic field on the walls and the tangential component of the magnetic

field which is this on this wall, these two walls is only z component, on this wall there

is no x component.

So, on this wall the unit normal will be going inwards so this is my m, for this wall this

is m. Here the magnetic field is going tangential to this which is in this direction, z direction,

x direction magnetic field is 0. So, on the walls if I see I have only magnetic field

component which is z component, if I calculate N cross H this is N and this is H. See, if

I go from here to here the N cross H in this direction I will get now the current direction

which will be tangential to this surface current that's what we are talking about but that

will be now flowing in the y direction because normal is in x direction and the magnetic

field is in the z direction, so N cross H if you take that current flows in the y direction

which is this direction.

So, we have now the surface current which flows into this, so if I take again and this

is my unit vector, this is my magnetic field which is like that okay. I take m cross h

which will be going from this to this, I get the direction of the current which will be

surface current which would be like this, it is Js and this is Js. So, very important

note now is compared to the TEM case that's the surface current is going to flow in the

y direction that is very important. So, we have here surface current in y direction again

it will have a variation which is sinusoidal in z direction but there is no current flow

in z direction that is the important thing to note but the wave is traveling in the z

direction, if the wave is traveling in z direction the power is flowing in z direction.

So, what we now find something very interesting that the power flow there is no component

of current, surface current in this two conductor system in the direction of the net power flow.

The current flows in this direction but the power flow in this direction, so the notion

which you have at low frequencies that's in the power is going inside the system, the

current must be flowing in the same direction that is completely wrong in fact the direction

of current flow and power flow have nothing to do with each other. The current may flow

in some arbitrate reaction and the power may flow in some other direction. In fact it will

be also immediately clear to us that where is the power flowing, where is power. The

ideal conducted doesn't carry in any power inside; the power is only between these two

conductors. The power is carried by the fields which are trapped between the conductors that's

what is carrying power to support this field we required the currents, so the currents

are not carrying power. The power is carried by fields so there is no reason why the current

should flow in same direction which the power is flowing.

However as I have mentioned at low frequencies we normally have a wrong notion that since

the power is flowing inside the circuit, the power is carried by the current, the power

is carried by the chargers. So, we get a wrong picture that the currents have to flow because

the current is going to carry power in fact this is now a better understanding which says

that power flow is because of the fields and not because of the current. So, the current

may flow in some arbitrary direction and in this case you can see it actually flows perpendicular

to the direction of the power flow. So, the power flows in z direction and the current

flows on the surface of these two conductors which is in this direction. So, the notions

normally which we will have at low frequency circuits, they come from the transverse electromagnetic

wave though we don't say that explicitly there because that time when we talk about a simple

circuits, we do not get into the complications of the field distributions but intrinsically

we assume that the field which are carried by the electrical circuit is a transverse

electromagnetic field, so we have certain notions which we build for the electrical

circuit and they come essentially from the natures of transverse electromagnetic wave.

As soon as you make a departure from the transverse electromagnetic wave to a transverse electric

or transverse magnetic then all those notions completely break down and then we get much

global picture of propagation of electromagnetic waves and the currents and the distribution

of charges and so on at for various configuration. So, with this understanding now for the current

flow on this parallel plain structure which is rather simplest structure, now we can discuss

the field distribution inside the rectangular wave guide which is more practically realizable

structure and then again we will see the current flow on the wave guide surface for rectangular

wave guide and then we will have little more advance things like calculation of losses

and other things for rectangular wave guide.

So, let me summarize what has been discussed here essentially we saw that the transverse

electromagnetic mode cannot exist inside a rectangular wave guide and that we showed

from the physical arguments then we also showed that if you take a rectangular wave guide

and push one of the dimension to infinity then the field we get for rectangular wave

guide, they are identical to what we would have got for parallel plane wave guide. Then

we developed a mechanism of really visualizing these fields inside these wave guides.

So, we saw two cases one was the transverse electromagnetic wave inside a parallel plane

wave guide other was T1 mode inside a parallel wave guide and then we saw something interesting

that the direction of the current flow on the surface of the wave guide need not be

same as the direction of the power flow. The direction of the current and the power flow

are same for only transverse electromagnetic case. However if you go to general mode then

the current might be flowing in different direction and the power flow which is due

to the fields trapped between the two conducting boundaries could be in some other direction.

So, in next time we discuss in detail, the fields and the power flow inside a rectangular

wave guide.