Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

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Electromagnetic wave at the interface between two dielectric media

The propagation of electromagnetic wave in an isotropic, homogeneous, dielectric medium, such as in air or vacuum. In this lecture, we woulddiscuss what happens when a plane electromagnetic wave is incident at the interface between two dielectric media. For being specific, we will take one of the medium to be air or vacuum and the other to be a dielectric such as glass. We have come across this in school in connection with the reflection and transmission of light waves at such an interface. In this lecture, we would investigate this problem from the point of view of electromagnetic theory.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Let us choose the interface to be the xy plane (z=0). The angles of incidence, reflection and refraction are the angles made by the respective propagation vectors with the common normal at the interface.

We have indicated the propation vectors in the apprpriate medium by capital letters I, R and T so as not to confuse with the notation for the position vector  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) and time t.

The principle that we use to establish the laws of reflection and refraction is the continuity of the tangential components of the electric field at the interface, as discussed extensively during the course of these lectures. Let us represent the component of the electric field parallel to the interface by the superscript ∥. We then have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

This equation must remain valid at all points in the interface and at all times. That is obviously possible if the exponential factors is the same for all the three terms or if they differe at best by a constant phase factor. Considering, the incident and the reflection terms, we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

We are familiar with the vector equation to a plane and we know that if the position vector of a plane is Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) and the normal to the plane is represented by  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)  the equation to the plane is given by  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) constant. Thus  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) is normal to the interface plane. Since the interface is x-y plane, we take the plane containing the  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) and  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) in the x-z plane.

SinceElectromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)and the normal to the planeElectromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)is normal to the interface Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) are in the same plane, we have, 

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Thus we have,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) are in the same medium, though their directions are different, their magnitudes are the same  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) and hence, we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

which is the law of reflection.

We will now prove the Snell’s law.

Let us look into the equation

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

In a fashion similar to the above, we can show that

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

which gives  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) are in different media, we have, recognizing that as the wave goes from one medium to another, its frequency does not change,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) is the velocity of the wave in the transmitted medium. We, therefore, have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Here n is the refractive index of the second medium with respect to the incident medium.

Fresnel’s Equations 

What happens to the amplitudes of the wave on reflection and transmission? Let us summarize the boundary conditions that we have derived in these lectures. We will assume that both the media are non-magnetic so that the permeability of both media are the same, viz., μ0. The two media differ by their dielectric constant, the incident medium is taken to be air as above. We further assume that there are no free charges or currents on the surface so that both the normal and tangential components of the fields are continuous. We will consider two cases, the first case where the electric fields are parallel to the incident plane. This case is known as p- polarization, p standing for “parallel”. The second case is where the electric field is perpendicular to the incident plane. This is referred to as s- polarization, s standing for a German word “senkrecht “ meaning perpendicular.

p- polarization

In this case, since the magnetic fields are perpendicular to the plane of incidence, we take the directions of the H fields to come out of the plane of the page (the incident plane). Since the electric field, the magnetic field and the direction of propagation are mutually perpendicular being a right handed triad, we have indicated the directions of the electric fields accordingly. It may be noted that in this picture, for the case of normal incidence, the incident and the reflected electric fields are directed oppositely. (The assumption is not restrictive because, if it is not true, a negative sign should appear in the equations ).

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Taking the tangential components of the electric field (parallel to the interface), we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

As our medium is linear, we have the following relationship between the electric and the magnetic field magnitudes,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Thus the continuity of the tangential component of the magnetic field H can be expressed in terms of electric field amplitudes

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Equations (1) and (2) can be simplified (we use θ= θι

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where,  nT and nI  are refractive indices of the transmitted medium and incident medium, respectively, with respect to free space.

Thus, we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

which gives,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Substituting (3) in (2),

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Let us take  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) The Fresnel’s equations are now expressible in terms of refractive indices of the two media and the angles of incidence and transmission

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) and  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) are, respectively, the reflection and transmission coefficients for field amplitudes. (Caution : the phrases “reflection/transmission” coefficients are also used to denote fraction of transmitted intensities.)

Brewster’s Angle  

Using  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) we can express , Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)as follows.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

    Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

If  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) i.e. if the angle between the reflected ray and the transmitted ray is 900 , the reflection coefficient for the parallel polarization becomes zero, because  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) If we had started with an equal mixture of p polarized and s polarized waves (i.e. an unpolarized wave), the reflected ray will have no p component, so that it will be plane polarized. The angle of incidence at which this happens is called the Brewster’s angle”.

For this angle, we have

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The following figure (right) , the red curve shows the variation in the reflected amplitude with the angle of incidence for p-polarization. The blue curve is the corresponding variation for s – polarization to be discussed below.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

For normal incidence, Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

s –polarization 

We next consider s polarization where the electric field is perpendicular to the incident plane. As we have taken the plane of incidence to be the plane of the paper, the electric field will be taken to come out of the plane of the paper.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The corresponding directions of the magnetic field is shown in the figure. The boundary conditions give

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Substituting  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) we get for the reflection and transmission coefficient for the amplitudes of the electric field

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Using Snell’s law, we can simplify these expressions to get,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

For normal incidence,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Notice that the expression differs from the expression forElectromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)for normal incidence, while both the results should have been the same. This is because of different conventions we took in fixing the directions in the two cases; in the p –polarization case, for normal incidence, the electric field directions are opposite for the incident and reflected rays while for the s-polarization, they have been taken to be along the same direction.

Total Internal Reflection and Evanescent Wave

Let us return back to the case of p polarization and consider the case where in the incident medium has a higher refractive index than the transmitted medium. In this case, we can write the amplitude reflection coefficient as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where we have used the refractive index of the second medium as  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) Substituting Snell’s law into the above, we have,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) we can write the above as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The quantity under the square root could become negative for some values of  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) since n <1. we therefore write,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

In a very similar way, we can show that the reflection coefficient for s polarization can be expressed as follows:

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

It may be seen that for  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) the magnitudes of bothElectromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)and  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) are both equal to unity because for both these, the numerator and the denominator are complex conjugate of each other. Thus it implies that when electromagnetic wave is incident at an angle of incidence greater than a “critical angle”defined by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where n here represents the refractive index of the (rarer) transmitted medium with respect to the(denser) incident medium , the wave is totally reflected. (In text books on optics, the critical angle is defined by the relation  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) because the refractive index there is conventionally defined as that of the denser medium with respect to the rarer one).

In case of total internal reflection, is there a wave in the transmitted medium? The answer is yes, it does as the following analysis shows.

Let us take the incident plane to be xz plane and the interface to be the xy plane so that the normal to the plane is along the z direction. The transmitted wave can be written as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The space part of the wave can be expressed as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Writing this in terms of angle of incidence  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

For angles greater than the critical angle the quantity within the square root is negative and we rewrite it as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where, we define “propagation vector “ β as 

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

and the “attenuation factor” α as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

With this, the wave in the transmitted medium becomes,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

which shows that the wave in the second medium propagates along the interface. It penetrates into the medium but its amplitude attenuates exponentially. This is known as “evanescent wave”.

The amplitude variation with angle of incidence is shown in the following figure.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

What is the propagation vector?

Recall that the magnitude of the propagation vector is defined as  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) where the “wavelength” λ is the distance between two successive crests or troughs of the wave measured along the direction of propagation. However, consider, for instance, a water wave which moves towards the shore. Along the shore, one would be more inclined to conclude that the wavelength is as measured by the distance between successive crests or troughs along the shore. This is the wavelength with which the attenuating surface wave propagates in the second medium.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

No transfer of energy into the transmitted medium:

Though there is a wave in the transmitted medium, one can show that on an average, there is no transfer of energy into the medium from the incident medium.

Consider, p polarization, for which the transmitted electric field, being parallel to the incident (xz) plane is along the x direction.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

From this we obtain the H- field using Faraday’s law, Since the H field is taken perpendicular to the electric field , it would be in the y-z plane . Taking the components of  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

so that,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Thus the average energy transfer Poynting vector. The complex Poynting vector can be written as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

so that the average power transferred into the second medium is

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Since the normal to the surface is along the z direction, the average energy transferred to the second medium is zero.

 

Tutorial Assignment

  1. A plane electromagnetic wave described by its magnetic field is given by the expression  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) Determine the corresponding electric field and the time average Poynting vector.If it is incident normally on a perfect conductor and is totally reflected what would be the pressure exerted on the surface? Determine the surface current generated at the interface.
  2. A circularly polarized electromagnetic wave is given by  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) Show that the average value of the Poynting vector for the wave is equal to the sum of the Poynting vectors of its components.

Solutions to Tutorial Assignments

1. The wave is propagating along the + z direction (before reflection). The electric field is given by Maxwell- Ampere law,

     Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Since  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) is along  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) direction, and its y-component depends on z only, the curl is given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

This gives,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The Poynting vector is given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The time average Poynting vector is  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Since the wave is totally reflected, the change in momentum is twice the initial momentum carried. Thus the pressure is given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

At the metallic interface (z=0), the tangential component of the electric field is zero. Since the wave is totally reflected, the reflected wave must have oppositely directed electric field, i.e. in  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) direction. The direction of propagation having been reversed, the magnetic the field is given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

At the interface the total magnetic field is  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The surface current can be determined by taking an Amperian loop perpendicular to the interface, Taking the direction of the loop parallel to the magnetic field, the line integral is seen to be  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) where K is the surface current density. The direction of the surface current is along the  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) direction. Thus  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

2. The electric field is given by  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

  As the wave is propagating in the z direction, the corresponding magnetic field is given by

  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Poynting vector is

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The individual components have average Poynting vectors given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Thus  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

 

Self Assessment Questions

1. The electric field of a plane electromagnetic wave propagating in free space is described by

   Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

 Determine the corresponding magnetic field and the time average Poynting vector for the wave.

2. Show that an s –polarized wave cannot be totally transmitted to another medium.

3. An electromagnetic wave given by

   Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

 is incident on the surface of a perfect conductor and is totally reflected. The incident and the reflected waves combine and form a pattern. What is the average Poynting vector?

 

Solutions to Self Assessment Questions

1. The wave is propagating in +x direction so that the propagation vector is  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) Using Faraday’s law, we have,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) which gives magnetic field directed along the z direction,

   Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The Poynting vector is given by

 Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The time average of the Poynting vector is  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

2. Considering Fresnel equations for s polarization,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

For total transmission  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) so that  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) From Snell’s law, we have,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) Squaring and adding, we get  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) which is not correct.

3. The incident wave is 

 Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Since the tangential component of the electric field must vanish at the interface (z=0), the reflected wave us given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The corresponding magnetic fields are given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The individual Poynting vectors can be calculated and on adding it will turn out that the average Poynting vector is zero. The waves of the type obtained by superposition of the two waves have the structure,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where the space and time parts are separated. These are known as “standing waves” and they do not transport energy.

Propagation of Electromagnetic Waves in a Conducting Medium 

We will consider a plane electromagnetic wave travelling in a linear dielectric medium such as air along the z direction and being incident at a conducting interface. The medium will be taken to be a linear medium. So that one can describe the electrodynamics using only the E and H vectors. We wish to investigate the propagation of the wave in the conducting medium.

As the medium is linear and the propagation takes place in the infinite medium, the vectors Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)  and  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) are still mutually perpendicular. We take the electric field along the x direction, the magnetic field along the y- dirrection and the propagation to take place in the z direction. Further, we will take the conductivity to be finite and the conductor to obey Ohm’s law,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) Consider the pair of curl equations of Maxwell. 

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Let us take Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) to be respectively in x, y and z direction. We the nhave,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

i.e.,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

and

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

i.e.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

We take the time variation to be harmonic  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) so that the time derivative is equivalent to a multiplication by  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE). The pair of equations (1) and (2) can then be written as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

We can solve this pair of coupled equations by taking a derivative of either of the equations with respect to z and substituting the other into it,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Define, a complex constan γ through

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

in terms of which we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

In an identical fashion, we get

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Solutions of (3) and (4) are well known and are expressed in terms of hyperbolic functions,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where A, B, C and D are constants to be determined. If the values of the electric field at z=0 is E0 and that of the magnetic field at z=0 is H0 we have A E0 and C =H0

In order to determine the constants B and D, let us return back to the original first order equations (1) and (2)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Substituting the solutions for E and H

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

This equation must remain valid for all values of z, which is possible if the coefficients of sinh and cosh terms are separately equated to zero,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The former gives,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Likewise, we get,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Substituting these, our solutions for the E and H become,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The wave is propagating in the z direction. Let us evaluate the fields when the wave has reached  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

If ℓ is large, we can approximate

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

we then have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The ratio of the magnitudes of the electric field to magnetic field is defined as the “characteristic impedance” of the wave

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Suppose we have lossless medium, σ=0, i.e. for a perfect conductor, the characteristic impedance is

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

If the medium is vacuum,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) gives η≈377Ω. The characteristic impedance, as the name suggests, has the dimension of resistance.

In this case,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Let us look at the full three dimensional version of the propagation in a conductor. Once again, we start with the two curl equations,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Take a curl of both sides of the first equation,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

As there are no charges or currents, we ignore the divergence term and substitute for the curl of H from the second equation,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

We take the propagating solutions to be

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

so that the above equation becomes,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

so that we have, the complex propagation constant to be given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

so that

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

k is complex and its real and imaginary parts can be separated by standard algebra,

we have

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Thus the propagation vector β and the attenuation factor α are given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The ration  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) determines whether a material is a good conductor or otherwise. Consider a good conductor for which σ>> ω∈.For this case, we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The speed of electromagnetic wave is given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The electric field amplitude diminishes with distance as  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) The distance to which the field penetrates before its amplitude diminishes be a factor e-1 is known as the “skin depth” , which is given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The wave does not penetrate much inside a conductor. Consider electromagnetic wave of frequency 1 MHz for copper which has a conductivity of approximately  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) Substituting these values, one gets the skin depth in Cu to be about 0.067 mm. For comparison, the skin depth in sea water which is conducting because of salinity, is about 25 cm while that for fresh water is nearly 7m. Because of small skin depth in conductors, any current that arises in the metal because of the electromagnetic wave is confined within a thin layer of the surface.

Reflection and Transmission from interface of a conductor 

Consider an electromagnetic wave to be incident normally at the interface between a dielectric and a conductor. As before, we take the media to be linear and assume no charge or current densities to exist anywhere. We then have a continuity of the electric and the magnetic fields at the interface so that 

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The relationships between the magnetic field and the electric field are given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

the minus sign in the second relation comes because of the propagation direction having been reversed on reflection.

Solving these, we get,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The magnetic field expressions are given by interchanging η1 and  η2 in the above expressions.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Let us look at consequence of this. Consider a good conductor such as copper. We can see that ηis a small complex number. For instance, taking the wave frequency to be 1 MHz and substituting conductivity of Cu to be  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) we can calculate ηto be approximately  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) whereas the vacuum impedance η1= 377 Ω. This implies 

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

which shows that a good metal is also a good reflector. On the other hand, if we calculate the transmission coefficient we find it to be substantially reduced, being only about  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

For the transmitted magnetic field, the ratio  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) is approximately +2. Though E is reflected with a change of phase, the magnetic field is reversed in direction but does not undergo a phase change. The continuity of the magnetic field then requires that the transmitted field be twice as large.

Surface Impedance

As we have seen, the electric field is confined to a small depth at the conductor interface known as the skin depth. We define surface impedance as the ratio of the parallel component of electric field that gives rise to a current at the conductor surface,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where kis the surface current density.

Assuming that the current flows over the skin depth, one can write, for the current density, (assuming no reflection from the back of this depth)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Since the current density has been taken to decay exponentially, we can extend the integration to infinity and get

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The current density at the surface can be written as  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) For a good conductor, we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Thus

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

where the surface resistance Rs and surface reactance Xs are given by 

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The current profile at the interface is as shown.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Tutorial Assignment

  1. A 2 GHz electromagnetic propagates in a non-magnetic medium having a relative permittivity of 20 and a conductivity of 3.85 S/m. Determine if the material is a good conductor or otherwise. Calculate the phase velocity of the wave, the propagation and attenuation constants, the skin depth and the intrinsic impedance.
  2. An electromagnetic wave with its electric field parallel to the plane of incidence is incident from vacuum onto the surface of a perfect conductor at an angle of incidence θ. Obtain an expression for the total electric and the magnetic field.

Solutions to Tutorial Assignments 

1. One can see that  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) The ratio of conductivity σ ω∈ is 1.73 which says it is neither a good metal nor a good dielectric. The propagation constant β and the attenuation constant α are given by, 

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The intrinsic impedance is

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The phase velocity is  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The skin depth is  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

2. The case of p polarization is shown.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Let the incident plane be y-z plane. Let us look at the magnetic field. We have, since both the incident and the reflected fields are in the same medium,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Let us write the incident magnetic field as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The reflected magnetic field is given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Since the tangential components of the electric field is continuous, we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

As  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) we have  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) and consequently,  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) Thus the total magnetic field can be written as

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The electric field has both y and z components,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The reflected electric field also has both components,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Adding these two the total electric field, has the following components,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Self Assessment Questions

  1. For electromagnetic wave propagation inside a good conductor, show that the electric and the magnetic fields are out of phase by 450 .
  2. A 2 kHz electromagnetic propagates in a non-magnetic medium having a relative permittivity of 20 and a conductivity of 3.85 S/m. Determine if the material is a good conductor or otherwise. Calculate the phase velocity of the wave, the propagation and attenuation constants, the skin depth and the intrinsic impedance.
  3. An electromagnetic wave with its electric field perpendicular to the plane of incidence is incident from vacuum onto the surface of a perfect conductor at an angle of incidence θ. Obtain an expression for the total electric and the magnetic field.

Solutions to Self Assessment Questions

1. From the text, we see that the ratio of electric field to magnetic field is given by

   Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

For a good conductor, we can approximate this by  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

2. One can see that  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) The ratio of conductivity  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) which says it is r a good metal. The propagation constant β and the attenuation constant α are given by,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The intrinsic impedance is

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The phase velocity is  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The skin depth is  Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

3. The direction of electric and magnetic field for s polarization is as shown below.

Let the incident plane be y-z plane. The incident electric field is

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

taken along the x direction and is given by

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The minus sign comes because at z=0, for any y, the tangential component of the electric field must be zero.

The total electric field is along the x direction and is given by the sum of the above,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

which is a travelling wave in the y direction but a standing wave in the z direction. Since the wave propagates in vacuum, we have,

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

The magnetic field has both y and z components.

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)
Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)
Electromagnetic Waves - Notes | Study GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

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