Electromagnetic Waves

Relevant Course: Electromagnetics

Relevant Department: Electrical Engineering

Relevant Semester: Nil 

Pre- requisite : Knowledge of Electrostatics and Magnetostatics is Preferable.

Course Description & Outline :

  • Review of Maxwells Equation
  • Wave Equation and Plane wave solution will be covered.
  • Wave Propagation in lossy dielectrics.
  • Plane waves in losses dielectrics
  • Plane waves in good conductors –Skin effect
  • Power and Poynting vector

Schedule for Lecture Delivery

Session 1 : 07-Sep-2015 (2-4 pm)

Session 2 : 08-Sep-2015 (2-4 pm)

Session 3 : 09-Sep-2015 (2-4 pm)

Session 4 : 14th -Sept-2015 (2-4 pm)

Teacher Forum

Review of Maxwells Equations

Electric and magnetic fields are fundamental to electromagnetics. When these fields do not vary with time over the time interval of interest, they are referred to as static fields. Electric charges- electrons and ions, are the sources of static electric field. A static magnetic field is produced due to a steady current, resulting from the ow of charges with uniform velocity. Electrostatic and mag-netostatic fields can exist independent of each other. However, a time varying electric field always coexists with a dynamic magnetic field and vice versa. An electromagnetic (EM) wave is constituted by a time varying electric and magnetic field. Accelerated charges or time-varying currents are the sources of EM waves.

Scottish Physicist, James Clerk Maxwell put together the relations between electricity and magnetism in the form of four equations, popularly known as Maxwell's equations. The generalized form of Maxwell's equations and the corresponding laws are given in Table 1. We will discuss each of these laws brie y, in the sections below.

Table 1: Maxwell's equations

 

Gausss law

 A stationary charge creates a force-field around. The strength of electric field is quantified through electric field intensity, defined as the force experienced by a small stationary test charge, q, placed in the electric field. The test charge should be small such that its presence does not change the electric field.

Figure 1: Boundary conditions for normal components of electric fields

Note: Gauss's law is valid for both static and time varying fields.

Non-existence of magnetic monopole

 

Faradays law

Oersted discovered that a steady current produces a magnetic field. The quest for the vice versa being true, led to Faraday's discovery of the fact that a static magnetic field produces no current flow; but a time-varying magnetic field produces an induced voltage called electromotive force (EMF) which causes ow of current in a closed circuit. This can be mathematically stated as,

Figure 2: Boundary conditions for the tangential components of electric fields

The boundary conditions corresponding to the tangential components of electric field can be derived from Faraday's law. We can apply the integral form of Faraday's law in the loop shown in Fig. 2. Let the width of the loop be w and the height be h. Then,

Thus, the tangential components of the electric fields at the interface of two materials are identical. 

Amperes circuital law

Ampere's law in magnetostatics is very similar to Gauss' law in electrostatics. The above equation can be modified as,

This is the differential form of the Ampere's law for static fields.
In case of time varying fields, the Ampere's law is modified as follows. We know that divergence of the curl of any vector is always zero, which means

The integral form for eqn. (31) can be obtained by integrating over a volume and applying divergence theorem as,

The boundary condition for the tangential component of magnetic field can be obtained from the integral form of Ampere's law, similar to how we obtained the same for electric field from Faraday's law. The resultant boundary conditions are,

In this section, we reviewed the Maxwell's equations that govern the phenomena of electromagnetics. The differential form of these equations are generally useful to understand the nature of electric and magnetic fields at specific locations in space and time, while the integral forms are useful to find the nature of these fields at media interface. We will now proceed to obtain the solutions of Maxwell's equations for various media.

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Electromagnetic Wave Propagation

Maxwell's equations completely describe the relation between the electric field, magnetic field,charge and the current distributions. The space variations of electric and magnetic field components are related to the time variations of the magnetic and electric field components, respectively. The solution of Maxwell's equations describes the propagating electric and magnetic fields- referred to as the electromagnetic waves. The propagation characteristics of the electromagnetic waves change in different media. Each media is characterised by its constitutive parameters namely-conductivity-permittivity and permeability  We now attempt to derive the propagation characteristics of an electromagnetic wave in different media.

Wave propagation in dielectric media

Eqn. (15) and (16) are the homogenous vector wave equations for electric and magnetic field in the medium. The solutions of these equations represent the wave phenomenon in the three dimensional space. Hence, we observe that all time varying electric and magnetic fields in a homogenous, unbound medium will have to exist as waves. Also, the waves due to electric and magnetic fields co-exist and hence, form an electromagnetic wave.

Wave Equation and solution For time harmonic Fields

Recap and Introduction to Wave Equation

Similar scalar equations can be written for the other components of the electric and magnetic field. Any electric field that satisfies Eqn. (18) can be a solution for the wave equation. Note that each of these components could be functions of x, y and z.

Wave Equation and Solution

3.2.1. Possible solutions of wave equation

Solution and TEM

Thus, the simplest possible solution for the wave equation is the one where the electric (/magnetic) field is a constant in a plane containing its field vector. This is a uniform plane wave. The uniform plane wave with its electric field in the x-direction and propagating in the z direction, described by eqn (24) satisfies the wave equation,

Phase velocity and Polarisation

The phase of the electric field and the magnetic field is a function of both space and time. For the forward propagating plane wave, the point of constant phase moves in the z direction as time progresses. The speed with which a given phase (for instance, the position corresponding to the peak) moves with space is referred to as the phase velocity of the wave. The phase, φ of the wave propagating in the z direction is given as

Differentiating eqn (32) with respect to time, we obtain the phase velocity, vp of the wave as,

3.3.1. Polarisation

Consider a point in space corresponding to z = 0. The two components of the field are,

This is the equation of a straight line, corresponding to linear polarisation. The slope of this line is 1, indicating that the electric field is oscillating at 45 degrees with respect to the x and y axes. When Ex0 = 0 and Ey0 = 1, the oscillations are along the y-axis and the corresponding polarisation is referred to as vertical polarisation, while if Ex0 = 1 and Ey0 = 0, the corresponding polarisation is referred to as horizontal polarisation.

Depending on whether the electric field vector traces the circle clockwise or counter clock wise direction as a function of time, the circular polarisation is further classified to right circular and left circular polarisation.

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Wave Propagation in lossy dielectric medium

Here, tan θ - the tangent of the phase of the complex dielectric constant is referred to as the loss tangent, and it describes the lossy nature of the material. If the loss tangent is very small, the material is said to be a good/lossless/perfect dielectric. As the conductivity of the material increases, the loss tangent becomes large. It should be noted that, the characteristic behaviour of the medium to the electromagnetic wave is decided not only by the constitutive parameters, but also by the frequency of propagation. A medium that is a good conductor at low frequencies can behave as a dielectric at high frequencies, since the loss tangent is inversly proportional to frequency.

4.1 Solution for electric field

4.2 Solution for Magnetic Field

The magnetic field can be derived for the time harmonic case, as in the previous section, from eqn. (25) as

4.3 Phase velocity of the wave

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Wave propagation in good conductors

Here, the magnitude of conduction current density is much larger than the displacement current density, as given in eqn (48). The propagation characteristics of the wave in good conductors are drastically different from that in dielectrics. We proceed to calculate the attenuation constant and the phase constant for good conductors, starting from the propagation constant.

Thus, the electric field leads the magnetic field by a phase angle of π/4; corresponding to the phase angle of the intrinsic impedance. The complete solution of electric field and magnetic field for the good conductor can be written as,

5.1 Skin Effect

The frequency for which σ = ω can be calculated as 1.04 × 108Hz ≈ 0.1GHz . Hence, copper behaves as a good conductor for frequencies 0.1 GHz. For larger frequencies, copper poses a dielectric behaviour. In the frequency range where it has conductive behaviour, the electric field penetrates into copper only through a distance given by the skin depth, δ. The skin depth for copper at 1 MHz can be calculated as ≈ 76µm. Some practical situations where the concept of skin effect plays a role are discussed below.

       • Skin effect can be exploited for shielding electromagnetic waves. A specific region/circuit can be protected from electromagnetic interference by enclosing it with a good conducting sheet, whose thickness is larger than the skin depth, so that the electromagnetic radiation cannot penetrate through the conducting sheet. Note that skin depth is a function of frequency, and hence the shielding is also specific to frequency.

      • Poor reception inside a train or an elevator is also a consequence of skin depth, where the electromagnetic wave does not reach the receiver due to skin effect.

      • Silver is a good conductor of electricity and it is usually preferred to use silver as conducting pads in microstrip lines. Since the electric field penetrates only upto the skin depth, only silver coatings of thickness corresponding to ≈ 5δ would be necessary instead of complete silver components. This can reduce the cost of these printed circuit boards.

    • Though steel is a poor conductor of electricity, power transmission lines are reinforced with steel cores. Due to skin effect, the electric field is confined only to the outer copper surface and hence, steel does not participate in the electrical conduction process.

       • Tubular conductors can be used instead of solid conductors, in antennas. This would greatly reduce the weight and cost of the antenna structure. Due to skin effect, the electric field is confined only to a small depth in the conductor; the solid conductor would not contribute to improving the conduction and hence, would not be necessary.

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Wave propagation through ionosphere

We receive the electromagnetic energy from the sun which travels as electromagnetic waves through the earth’s atmosphere; we also communicate to and from satellites where the electromagnetic waves travel through the atmosphere. Hence, electromagnetic wave propagation through atmosphere is of significant interest to us. Among the different layers of earth’s atmosphere, ionosphere influences the electromagnetic wave propagation the most.

Ionosphere is the region of earth’s upper atmosphere extending from approximately 50 km to more than 1000 km above the earth. In this region, the constituent gases are ionized, mostly because of ultraviolet radiation from the sun. The positive ions and electrons are free to move under the influence of the fields of a wave incident upon the medium. The positive ions are however heavy compared to electrons and hence comparatively immobile. As the altitude increases, the density of ionosphere increases because of the increase in the amount of radiation. Beyond a certain altitude, the electron density decrease because the density of atmosphere itself reduces with the increase in altitude.

The typical electron density in ionosphere is shown in Fig. 4. This simplistic representation of ionosphere is referred to as the Chapman model. The electron density is found to be maximum at about 400 km. Atmosphere has homogeneous mixture of gases. So, the density profile has multiple peaks.

The regions around the peaks are identified by layers called as D, E and F layers. In sky wave communication, the electromagnetic waves are deflected by layers of ionosphere; the radio waves are guided back to the earth’s surface due to the deflections at different layers of the ionosphere. At noon, the atmosphere is fully ionized and at night only F2 layer survives. A sporadic E layer is formed due to recombinations at night time but only F2 layer is available for sky wave communication at night.

Figure 4: Electron density variation as a function of height in the ionosphere. The densities are larger during the day than at night. The density is maximum at a height of about 400 km (ref:www.astrosurf.com)

Permittivity of ionosphere

When an electric field is incident on ionosphere, the charges in the ionosphere start moving, thus causing a conduction current. If the electric field is oscillating, the conduction current is primarily due to the electrons since the positive charges are heavier and cannot possibly oscillate due to their smaller mobility. So while the electrons move, they collide with the positive ions and the neutral atoms, which can be considered to be at rest. This results in a momentum transfer from the electrons and hence a deceleration. The equation of motion of the electrons can be written as,

It can be observed from eqn (78) that the dielectric constant of plasma is always less than 1. Thus, plasma has a refractive index of less than 1, and it can be imaginary if f < fp. At lower heights, when the electron density, N is small, plasma frequency is relatively low, and hence the f > fp. The refractive index is almost 1. As the height increases, N increases, and consequently the refractive index decreases up to the F2 layer, where the electron density is maximum. Beyond the F2 layer, the refractive index again increases gradually. Thus, the electromagnetic wave successively changes its direction as its propagates through ionosphere, gets completely deflected back into the earth’s atmosphere, thus supporting sky wave communication.

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Power flow and Poynting theorem

The Poynting’s theorem states that the net power flowing out of a given volume V, is equal to the time rate of decrease in the energy stored within V minus the ohmic losses.

Derivation

From Maxwell’s equations we know that

It is the sum of rate of increase of stored electric and magnetic energy plus the power dissipated within the closed volume. Poynting theorem states that the surface integral of the Poynting’s vector over a closed surface is equal to the total power leaving the closed surface. This is valid only for a closed surface. This is pictorially represented in Fig. 5.

Figure 5: Representation of Poynting’s theorem. The increase in the stored electrostatic energy and the magnetostatic energy - the ohmic loss is the net power outflow from a closed surface.

We now proceed to calculate the average power density.

Problem Discussion

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