Electromagnetic Waves Notes | EduRev

Electromagnetic Fields Theory

Electrical Engineering (EE) : Electromagnetic Waves Notes | EduRev

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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 | EduRev

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 | EduRev 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 | EduRevElectromagnetic Waves Notes | EduRev

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 | EduRev

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 | EduRev and the normal to the plane is represented by  Electromagnetic Waves Notes | EduRev  the equation to the plane is given by  Electromagnetic Waves Notes | EduRev constant. Thus  Electromagnetic Waves Notes | EduRev is normal to the interface plane. Since the interface is x-y plane, we take the plane containing the  Electromagnetic Waves Notes | EduRev and  Electromagnetic Waves Notes | EduRev in the x-z plane.

SinceElectromagnetic Waves Notes | EduRevand the normal to the planeElectromagnetic Waves Notes | EduRevis normal to the interface Electromagnetic Waves Notes | EduRev are in the same plane, we have, 

Electromagnetic Waves Notes | EduRev

Thus we have,  Electromagnetic Waves Notes | EduRev are in the same medium, though their directions are different, their magnitudes are the same  Electromagnetic Waves Notes | EduRev and hence, we have,

Electromagnetic Waves Notes | EduRev

which is the law of reflection.

We will now prove the Snell’s law.

Let us look into the equation

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

which gives  Electromagnetic Waves Notes | EduRev 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 | EduRev

where  Electromagnetic Waves Notes | EduRev is the velocity of the wave in the transmitted medium. We, therefore, have,

Electromagnetic Waves Notes | EduRev

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 | EduRev

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

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

Electromagnetic Waves Notes | EduRev

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 | EduRev

which gives,

Electromagnetic Waves Notes | EduRev

Substituting (3) in (2),

Electromagnetic Waves Notes | EduRev

Let us take  Electromagnetic Waves Notes | EduRev 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 | EduRev

Electromagnetic Waves Notes | EduRev

where,  Electromagnetic Waves Notes | EduRev and  Electromagnetic Waves Notes | EduRev 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 | EduRev we can express , Electromagnetic Waves Notes | EduRevas follows.

Electromagnetic Waves Notes | EduRev

    Electromagnetic Waves Notes | EduRev

If  Electromagnetic Waves Notes | EduRev 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 | EduRev 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 | EduRev

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 | EduRevElectromagnetic Waves Notes | EduRev

For normal incidence, Electromagnetic Waves Notes | EduRev we have,

Electromagnetic Waves Notes | EduRev

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 | EduRev

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

Electromagnetic Waves Notes | EduRev

Substituting  Electromagnetic Waves Notes | EduRev we get for the reflection and transmission coefficient for the amplitudes of the electric field

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

For normal incidence,  Electromagnetic Waves Notes | EduRev we have,

Electromagnetic Waves Notes | EduRev

Notice that the expression differs from the expression forElectromagnetic Waves Notes | EduRevfor 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 | EduRev

where we have used the refractive index of the second medium as  Electromagnetic Waves Notes | EduRev Substituting Snell’s law into the above, we have,  Electromagnetic Waves Notes | EduRev we can write the above as

Electromagnetic Waves Notes | EduRev

The quantity under the square root could become negative for some values of  Electromagnetic Waves Notes | EduRev since n <1. we therefore write,

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

It may be seen that for  Electromagnetic Waves Notes | EduRev the magnitudes of bothElectromagnetic Waves Notes | EduRevand  Electromagnetic Waves Notes | EduRev 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 | EduRev

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 | EduRev 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 | EduRev

The space part of the wave can be expressed as

Electromagnetic Waves Notes | EduRev

Writing this in terms of angle of incidence  Electromagnetic Waves Notes | EduRev

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

where, we define “propagation vector “ β as 

Electromagnetic Waves Notes | EduRev

and the “attenuation factor” α as

Electromagnetic Waves Notes | EduRev

With this, the wave in the transmitted medium becomes,

Electromagnetic Waves Notes | EduRev

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 | EduRev

What is the propagation vector?

Recall that the magnitude of the propagation vector is defined as  Electromagnetic Waves Notes | EduRev 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 | EduRev

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 | EduRev

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 | EduRev we have,

Electromagnetic Waves Notes | EduRev

Electromagnetic Waves Notes | EduRev

so that,

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

so that the average power transferred into the second medium is

Electromagnetic Waves Notes | EduRev

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 | EduRev 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 | EduRev 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 | EduRev

Since  Electromagnetic Waves Notes | EduRev is along  Electromagnetic Waves Notes | EduRev direction, and its y-component depends on z only, the curl is given by

Electromagnetic Waves Notes | EduRev

This gives,

Electromagnetic Waves Notes | EduRev

The Poynting vector is given by

Electromagnetic Waves Notes | EduRev

The time average Poynting vector is  Electromagnetic Waves Notes | EduRev

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 | EduRev

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 | EduRev direction. The direction of propagation having been reversed, the magnetic the field is given by

Electromagnetic Waves Notes | EduRev

At the interface the total magnetic field is  Electromagnetic Waves Notes | EduRev

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 | EduRev where K is the surface current density. The direction of the surface current is along the  Electromagnetic Waves Notes | EduRev direction. Thus  Electromagnetic Waves Notes | EduRev

2. The electric field is given by  Electromagnetic Waves Notes | EduRev

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

  Electromagnetic Waves Notes | EduRev

Poynting vector is

Electromagnetic Waves Notes | EduRev

The individual components have average Poynting vectors given by

Electromagnetic Waves Notes | EduRev

Thus  Electromagnetic Waves Notes | EduRev

 

Self Assessment Questions

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

   Electromagnetic Waves Notes | EduRev

 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 | EduRev

 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 | EduRev Using Faraday’s law, we have,  Electromagnetic Waves Notes | EduRev which gives magnetic field directed along the z direction,

   Electromagnetic Waves Notes | EduRev

The Poynting vector is given by

 Electromagnetic Waves Notes | EduRev

The time average of the Poynting vector is  Electromagnetic Waves Notes | EduRev

2. Considering Fresnel equations for s polarization,

Electromagnetic Waves Notes | EduRev

For total transmission  Electromagnetic Waves Notes | EduRev so that  Electromagnetic Waves Notes | EduRev From Snell’s law, we have,  Electromagnetic Waves Notes | EduRevElectromagnetic Waves Notes | EduRev Squaring and adding, we get  Electromagnetic Waves Notes | EduRev which is not correct.

3. The incident wave is 

 Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

The corresponding magnetic fields are given by

Electromagnetic Waves Notes | EduRev

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 | EduRev

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 | EduRev  and  Electromagnetic Waves Notes | EduRev 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 | EduRev Consider the pair of curl equations of Maxwell. 

Electromagnetic Waves Notes | EduRev

Let us take Electromagnetic Waves Notes | EduRev to be respectively in x, y and z direction. We the nhave,

Electromagnetic Waves Notes | EduRev

i.e.,

Electromagnetic Waves Notes | EduRev

and

Electromagnetic Waves Notes | EduRev

i.e.

Electromagnetic Waves Notes | EduRev

We take the time variation to be harmonic  Electromagnetic Waves Notes | EduRev so that the time derivative is equivalent to a multiplication by  Electromagnetic Waves Notes | EduRev. The pair of equations (1) and (2) can then be written as

Electromagnetic Waves Notes | EduRev

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 | EduRev

Define, a complex constan γ through

Electromagnetic Waves Notes | EduRev

in terms of which we have,

Electromagnetic Waves Notes | EduRev

In an identical fashion, we get

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

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 | EduRev

Substituting the solutions for E and H

Electromagnetic Waves Notes | EduRev

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 | EduRev

The former gives,

Electromagnetic Waves Notes | EduRev

Electromagnetic Waves Notes | EduRev

where

Electromagnetic Waves Notes | EduRev

Likewise, we get,

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

The wave is propagating in the z direction. Let us evaluate the fields when the wave has reached  Electromagnetic Waves Notes | EduRev

Electromagnetic Waves Notes | EduRev

If ℓ is large, we can approximate

Electromagnetic Waves Notes | EduRev

we then have,

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

If the medium is vacuum,  Electromagnetic Waves Notes | EduRev gives η≈377Ω. The characteristic impedance, as the name suggests, has the dimension of resistance.

In this case,  Electromagnetic Waves Notes | EduRev

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 | EduRev

Take a curl of both sides of the first equation,

Electromagnetic Waves Notes | EduRev

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 | EduRev

We take the propagating solutions to be

Electromagnetic Waves Notes | EduRev

so that the above equation becomes,

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

so that

Electromagnetic Waves Notes | EduRev

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

we have

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

Electromagnetic Waves Notes | EduRev

The ration  Electromagnetic Waves Notes | EduRev 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 | EduRev

The speed of electromagnetic wave is given by

Electromagnetic Waves Notes | EduRev

The electric field amplitude diminishes with distance as  Electromagnetic Waves Notes | EduRev 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 | EduRev

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 | EduRev 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 | EduRev

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

Electromagnetic Waves Notes | EduRev

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 | EduRev

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

Electromagnetic Waves Notes | EduRev

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 | EduRev we can calculate ηto be approximately  Electromagnetic Waves Notes | EduRevElectromagnetic Waves Notes | EduRev whereas the vacuum impedance η1= 377 Ω. This implies 

Electromagnetic Waves Notes | EduRev

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 | EduRev

For the transmitted magnetic field, the ratio  Electromagnetic Waves Notes | EduRev 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 | EduRev

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 | EduRev

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

Electromagnetic Waves Notes | EduRev

The current density at the surface can be written as  Electromagnetic Waves Notes | EduRev For a good conductor, we have,

Electromagnetic Waves Notes | EduRev

Thus

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

The current profile at the interface is as shown.

Electromagnetic Waves Notes | EduRev

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 | EduRev 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 | EduRev

Electromagnetic Waves Notes | EduRev

The intrinsic impedance is

Electromagnetic Waves Notes | EduRev

The phase velocity is  Electromagnetic Waves Notes | EduRev

The skin depth is  Electromagnetic Waves Notes | EduRev

2. The case of p polarization is shown.

Electromagnetic Waves Notes | EduRev

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 | EduRev

Let us write the incident magnetic field as

Electromagnetic Waves Notes | EduRev

The reflected magnetic field is given by

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

As  Electromagnetic Waves Notes | EduRev we have  Electromagnetic Waves Notes | EduRev and consequently,  Electromagnetic Waves Notes | EduRev Thus the total magnetic field can be written as

Electromagnetic Waves Notes | EduRev

The electric field has both y and z components,

Electromagnetic Waves Notes | EduRev

The reflected electric field also has both components,

Electromagnetic Waves Notes | EduRev

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

Electromagnetic Waves Notes | EduRev

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 | EduRev

For a good conductor, we can approximate this by  Electromagnetic Waves Notes | EduRev

2. One can see that  Electromagnetic Waves Notes | EduRev The ratio of conductivity  Electromagnetic Waves Notes | EduRev which says it is r a good metal. The propagation constant β and the attenuation constant α are given by,

Electromagnetic Waves Notes | EduRev

The intrinsic impedance is

Electromagnetic Waves Notes | EduRev

The phase velocity is  Electromagnetic Waves Notes | EduRev

The skin depth is  Electromagnetic Waves Notes | EduRev

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 | EduRev

taken along the x direction and is given by

Electromagnetic Waves Notes | EduRev

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 | EduRev

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 | EduRev

The magnetic field has both y and z components.

Electromagnetic Waves Notes | EduRev
Electromagnetic Waves Notes | EduRev

Electromagnetic Waves Notes | EduRev
Electromagnetic Waves Notes | EduRev

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