EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The document EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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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:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Ampere’s Law:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Complex Wave Equation: Assume: EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

 

Waves in a Dielectric Medium – Wave Equation

Suppose we have a plane wave of the form,

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

traveling in an infinite dielectric medium with permittivity ε

What is different from wave propagation in free space?

Faraday’s Law:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Ampere’s Law:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Complex Wave Equation: Assume:EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Waves in a Dielectric Medium – Dispersion Relation

Substitute the plane wave solution:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

in the complex wave equation:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

To get:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Refractive Index:

Define refractive index “n” of a dielectric medium as:EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev


Waves in a Dielectric Medium – Velocity

Plane wave:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Dispersion relation:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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:EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev


Waves in a Dielectric Medium - Wavelength

Plane wave in a dielectric medium:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Dispersion relation: EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

But the magnitude of the wavevector is related to the wavelength by the relation:EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

So for a dielectric medium we get:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevEM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Calculate the magnetic field:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

 

Waves in a Conductive Medium – Complex Permittivity

Suppose we have a plane wave of the form,

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

traveling in an infinite medium with conductivity σ and permittivity ε

Faraday’s Law:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Complex Wave Equation:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev


Waves in a Conductive Medium – Complex Refractive Index

Plane wave:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Dispersion relation:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev


Waves in a Conductive Medium – Complex Wavevector

Plane wave:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Complex wavevector: EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

What are the implications of a complex wavevector?

  • Wave decays exponentially with distance as it propagates

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

 

Waves in a Conductive Medium – Magnetic Field

Plane wave:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Calculate the magnetic field:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Note that:EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Poynting vector and time average power per unit area:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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:EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The complex refractive index is:
EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Loss tangent = EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

But the dielectric relaxation time was:EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

⇒ Loss tangent 
EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

 

Loss Tangent and Dielectric Relaxation Time - II 

There are two possible scenarios:

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

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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)

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Lossy dielectric approximation:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

 

Waves in a Conductive Medium – Imperfect Metals

Now consider the case when:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Imperfect metal approximation:

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev


Waves in a Conductive Medium – Imperfect Metals

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

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevEM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Since the wavelength λ inside the medium is EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev , the wave hardly propagates one wavelength distance into the medium
The screening current density, given by EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is non-zero only in a layer of thickness equal to skin-depth δ near the surface

 

Waves in a Conductive Medium

EM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevEM Waves in Dielectrics and Conductors - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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