EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

In this lecture you will learn:

• Wave propagation in dielectric media
• Waves propagation in conductive media

Review: Plane Waves in Free Space

Faraday’s Law:

Ampere’s Law:

Complex Wave Equation: Assume: 

For a plane wave in free space we know the E-field and H-field phasors to be:

Waves in a Dielectric Medium – Wave Equation

Suppose we have a plane wave of the form,

traveling in an infinite dielectric medium with permittivity ε

What is different from wave propagation in free space?

Faraday’s Law:

Ampere’s Law:

Complex Wave Equation: Assume:

Waves in a Dielectric Medium – Dispersion Relation

Substitute the plane wave solution:

in the complex wave equation:

To get:

Refractive Index:

Define refractive index “n” of a dielectric medium as:

Waves in a Dielectric Medium – Velocity

Plane wave:

Dispersion relation:

The velocity of waves in a dielectric medium is reduced from the velocity of waves in free space by the refractive index

• Velocity of waves in free space: c
• Velocity of waves in dielectric medium of refractive index n:

Waves in a Dielectric Medium - Wavelength

Plane wave in a dielectric medium:

Dispersion relation: 

But the magnitude of the wavevector is related to the wavelength by the relation:

So for a dielectric medium we get:

The wavelength of plane waves in a dielectric medium is reduced from the wavelength of plane waves of the same frequency in free space by the refractive index

Waves in a Dielectric Medium – Magnetic Field

Plane wave:

Calculate the magnetic field:

Waves in a Conductive Medium – Complex Permittivity

Suppose we have a plane wave of the form,

traveling in an infinite medium with conductivity σ and permittivity ε

Faraday’s Law:

Complex Wave Equation:

Waves in a Conductive Medium – Complex Refractive Index

Plane wave:

Dispersion relation:

Waves in a Conductive Medium – Complex Wavevector

Plane wave:

Complex wavevector: 

What are the implications of a complex wavevector?

• Wave decays exponentially with distance as it propagates

Waves in a Conductive Medium – Magnetic Field

Plane wave:

Calculate the magnetic field:

Note: The E-field and the H-field are no longer in phase since  ηeff(ω )  is complex 

Waves in a Conductive Medium – Power Flow

Plane wave:

Note that:

Poynting vector and time average power per unit area:

Time average power per unit area decays exponentially with distance because energy is dissipated in a conductive medium due to I²R (or J.E) type of losses and this energy dissipated is taken away from the plane wave

Loss Tangent and Dielectric Relaxation Time - I

The complex wavevector is:

The complex refractive index is:

Loss tangent =

But the dielectric relaxation time was:

⇒ Loss tangent 

Loss Tangent and Dielectric Relaxation Time - II 

There are two possible scenarios:

High frequency and/or low conductivity case (e.g. lossy dielectrics)

The frequency is much greater than the inverse dielectric relaxation time

⇒The conductive medium does not have enough time to react to the electromagnetic wave

⇒No appreciable currents flow in the conductive medium

Low frequency and/or high conductivity case (e.g. Imperfect metals)

The frequency is much smaller than the inverse dielectric relaxation time

⇒The conductive medium has enough time to react to the electromagnetic wave

⇒Appreciable currents flow in the conductive medium

Waves in a Conductive Medium – Lossy Dielectrics

Plane waves:

Lossy dielectric approximation:

Waves in a Conductive Medium – Imperfect Metals

Now consider the case when:

The frequency is much smaller than the inverse dielectric relaxation time

⇒The conductive medium has enough time to react to the electromagnetic wave

⇒Appreciable currents flow in the conductive medium

These currents try to screen out the magnetic field and, therefore, prevent the electromagnetic wave from going into the conductor

Waves in a Conductive Medium – Imperfect Metals

Imperfect metal approximation:

Waves in a Conductive Medium – Imperfect Metals

Due to current screening the wave decays within a few skin-depths:

Since the wavelength λ inside the medium is  , the wave hardly propagates one wavelength distance into the medium
The screening current density, given by  is non-zero only in a layer of thickness equal to skin-depth δ near the surface

Waves in a Conductive Medium