1. Plane electromagnetic waves

2. Plane electromagnetic Waves
We will assume that the vectors for the electric and magnetic fields in an electromagnetic wave have a specific space-time behavior that is consistent with Maxwell’s equations.
Assume an em wave that travels in the x direction with as shown

3. Poynting Vector The Poynting vector is defined as
Electromagnetic waves carry energy
As they propagate through space, they can transfer that energy to objects in their path
The rate of flow of energy in an electromagnetic wave is described by a vector, , called the Poynting vector
The Poynting vector is defined as
Its direction is the direction of propagation

4. Poynting Vector
For a plane-polarized em wave, the direction of Poynting vector is
This is time dependent
Its magnitude varies in time
The magnitude of represents the rate at which energy flows through a unit surface area perpendicular to the direction of the wave propagation
This is the power per unit area
The SI units of the Poynting vector are J/(s.m2) = W/m2

5. Intensity
The wave intensity, I, is the time average of S (the Poynting vector) over one or more cycles
When the average is taken, the time average of cos2(kx - ωt) = ½ is involved

6. Energy Density associated with em waves
The energy density, u, is the energy per unit volume.
For the electric field, uE= ½ εoE2
For the magnetic field, uB = ½ μoB2
Since B = E/c and

7. Energy Density
The instantaneous energy density associated with the magnetic field of an em wave equals the instantaneous energy density associated with the electric field
In a given volume, the energy is shared equally by the two fields
The total instantaneous energy density is the sum of the energy densities associated with each field
u =uE + uB = εoE2 = B2 / μo
When this is averaged over one or more cycles, the total average becomes
uavg = εo(E2)avg = ½ εoE2max = B2max / 2μo
In terms of I, I = Savg = cuavg
The intensity of an em wave equals the average energy density multiplied by the speed of light

8. Poynting theorem
Consider an elementary area da in the space in which em waves are propagating.
If S is the Poynting vector, then it gives the time rate of flow of energy per unit area and is called POWER FLUX through unit area.
The power flux through the area dA will be
The total power through the closed surface is given by
where gives the amount of energy diverging per second out of enclosed volume of the space enclosed by the closed surface.

9. Cont.
From Maxwell equations
The Poynting vector is given as

10. Cont. Taking dot product of equation (2) with H
Taking dot product of equation (3) with E
Using equation (5) and (6) in (4)

11. Cont.
Integrating equation (7) over the volume V bounded by the surface
The first term represents flow of energy per unit time i.e. power flux
Second term corresponds to rate of change of total energy
The term on R.H.S. represents the work done by the filed on the source in setting up a current

12. Cont. and that of the magnetic field The total energy stored will be
Energy associated with the electric field
and that of the magnetic field
The total energy stored will be

13. Cont. The equation (9) represents Poynting theorem.
Therefore equation (8) can be written as
The equation (9) represents Poynting theorem.
It states that the sum of flow of energy per unit volume across the boundary of the surface and time rate of change of electromagnetic energy is equal to the work done by the electromagnetic field on the source.

14. Cont.
In free space, J=0. Therefore
From equation (10), we can say that the power flux through a closed area is equal to the rate of outflow of energy from the volume enclosed by that area.
Equation of continuity:

15. Electromagnetic wave in a medium with finite permeability, permittivity and conductivity (Conducting medium)
For plane-polarized em waves
The second term is called diffusion term and is due to finite conductivity (or conduction currents) of the medium.
For perfect insulators, this term is absent from em wave equation.

16. Solution of em wave equation for a conducting medium
Let the solution of wave equation is given by
Substituting the eq. (3) in eq. (1)

17. Cont.
For good conductors

18. Cont. Therefore from equation (3)
Taking the negative sign, which gives the wave propagating in the positive x-direction
This equation represent a progressive wave having amplitude equal to
The amplitude of the wave goes on decreasing as the wave propagates deeper into the medium.

19. Cont. Therefore from equation (3)
Taking the negative sign, which gives the wave propagating in the positive x-direction

20. Skin depth (d)
The amplitude of the em wave decreases exponentially with distance of penetration of wave. The decrease in the amplitude or the attenuation of the field vector is expressed in terms of skin depth.
It is defined as the distance beyond the surface of the conductor inside it at which the amplitude of the field vector is reduced to 1/e of its value at the surface.
Let amplitude at a depth x is Ex.
Then
If skin depth is d then
where Eo is the amplitude at the surface and Ed is the amplitude at depth d

21. Skin depth (d)
It shows that the spin depth varies inversely as the square root of the conductivity of the medium and the frequency of em waves.

22. Dispersion of em waves in a conductor
The dispersion of waves occur when their group and phase velocity are not equal.
The equation for the electric waves through a conducting medium is given by
The phase velocity of the wave will be
The skin depth depends on wavelength.

23. Dispersion of em waves in a conductor
The phase velocity of the em wave is given as

24. Dispersion of em waves in a conductor
So the em waves suffer anomalous dispersion in the conductors.

25. Characteristics impedance of a medium to the EM waves
Characteristic impedance,
Z=(Instantaneous electric vector)/(Instantaneous magnetic vector)
=Ey/Hz

26. Characteristics impedance of a dielectric medium to the EM waves
In the dielectric medium, =0
The wave equation for the electric and magnetic vectors are:
Characteristic impedance,
Z=(Instantaneous electric vector)/(Instantaneous magnetic vector)
=Ey/Hz