Paper 14

Core XIV : Electromagnetic Theory

CBCS 2019 : Core-XIII : Electromagnetic Theory

Syllabus

Course Outcomes:
* Physical significance of Maxwell Equation and its application to free space, Lorentz and Coulomb gauge transformation, poynting theorem, concept of energy density.
* Analysis of Maxwell’s equat ions in different media and Physical significance o f relaxation time, skin depth, Electrical conductivity of ionized gases, plasma frequency.
* Basic understanding of polarization of EM wave, and different types of crystals, Phase.
* Retardation Plates and Rotatory Polarization.
* Conceptual understanding of EMW application in bounded media, plane interface, dielectric media, Brewster’s law, TIR, Evanescent wave, metallic reflection.
* To Apply the acquired knowledge for visualize basic concept of phenomenon of light in various experiments)

Unit I

*Maxwell Equations: Maxwells equations, Displacement Current, Vector and Scalar Potentials, Gauge Transformations: Lorentz and Coulomb Gauge, Wave Equations, Plane Waves in free space and characteristics, Poynting Theorem and Poynting Vector, Electromagnetic (EM) Energy Density, Physical Concept of Electromagnetic Field Energy Density.

Unit II

EM Wave Propagation in Unbounded Media: Plane EM waves through vacuum
and isotropic dielectric medium, transverse nature of plane EM waves, refractive index and dielectric constant, wave impedance, Propagation through conducting media, relaxation time, skin depth, Electrical conductivity of ionized gases, plasma frequency, refractive index, skin depth, application to propagation through ionosphere.

Unit III

• Polarization of Electromagnetic Waves: Description of Linear, Circular and
  Elliptical Polarization, uniaxial and biaxial crystals, light propagation in uniaxial crystal, double refraction, polarization by double refraction, Nicol Prism, Ordinary and extraordinary refractive indices, Production and detection of Plane, Circularly and Elliptically polarized light,
• Phase Retardation Plates: Quarter-Wave and Half- Wave Plates. Babinets Compensator and its Uses, Analysis of Polarized Light.
• Rotatory Polarization: Optical Rotation, Biots Laws for Rotatory Polarization, Fresnels Theory of optical rotation, Calculation of angle of rotation, Experimental verification of Fresnels theory, Specific rotation, Laurents half- shade polarimeter.

Unit IV

EM Wave in Bounded Media: Boundary conditions at a plane interface between two media, Reflection and Refraction of plane waves at plane interface between two dielectric media, Laws of Reflection and Refraction, Fresnel’s Formulae for perpendicular and parallel polarization cases, Brewster’s law, Reflection and Transmission coefficients, Total internal reflection, evanescent waves, Metallic reflection (normal Incidence)

Text Books:

• Introduction to Electrodynamics, D. J. Griffiths ( Pearson)
• Principles of Optics - Max Born and E. Wolf.

Reference Books :

1. Classical Electrodynamics by J. D. Jackson.
2. Foundation of electromagnetic theory: Ritz and Milford (Pearson).
3. Electricity and Magnetism : D C Tayal (Himalaya Publication)
4. Optics: A. K. Ghatak
5. Electricit y and Magnetism : Chattopadhyaya, Rakhit (New Central)
6. Electromagnetic Theory, Gupta and Kumar, Pragati Prakashan

Questions

2025 | +3-VI-S-CBCS(MS)-Sc(H)-Core-XIII-Phy

Part-I | Qn 1 | 1x8

• $\vec{\nabla}.\vec{B}$ represents which law?
• The impedance offered by a dielectric to electromagnetic wave is _____.
• The dimension of wave impedance is _____.
• Specific rotation of polarised light depends on which factor?
• For plane polarised light, if angle of glancing is 37°, then find the angle of polarisation.
• The conductivity of a perfect dielectric is _____.
• The ratio between electrostatic energy density and magnetic energy density is ____.
• Phase difference between E-ray and O-ray for quarter wave plate is ____.

Part-II | Qn 2 | 1.5x8

• Define Displacement current. Write its S.I. Unit.
• Write S.I. Unit and dimension of poynting vector.
• Write the properties of isotropic dielectric medium.
• Explain Relaxation Time.
• What is plasma frequency ?
• State Biot's Laws of Rotatory polarisation.
• On the basis of electromagnetic theory, explain good conductors are good absorbers and good reflectors.
• State Brewster's law.
• Give one example of positive and negative Uni-axial crystal.
• Write the conditions of total internal reflection.

Part-III | Qn 3(in maximum 75 words)| 2x8

• Write Maxwell's equation in free space.
• Write the boundary conditions at interface between different media.
• Calculate the skin depth for the frequency $10^8$ HZ of silver. ( Given $\sigma = 2 \times 10^8, Sm^{-1}$ and $\mu = 4\pi\times 10^{-7}, Hm^{-1}$)
• Refractive index of a medium is $\sqrt{3}$. Calculate Brewster's angle and angle of refraction.
• Explain Metallic reflection of normal incidence.
• What are the Fresnel's, formulae for perpendicular and parallel polarisation ?
• Distinguish between Uniaxial and Biaxial crystals.
• State Huygen's theory of double refraction.
• What are plane of vibration and plane of polarisation in a polarised wave ?
• Distinguish between Lorentz and Coulomb gauge transformation.

Part-IV | Qn 4-7(within 500 words each) | 6x4

4) State and prove poynting theorem relating flow of energy at a point in the space in an electromagnetic field.

OR

Derive expressions for electromagnetic vector and scalar potentials.

5) Discuss and derive an expression for plane EM wave equation with their solution in an isotropic dielectric medium.

OR

Discuss the propagation of high frequency electromagnetic wave in plasma. Also find the plasma frequency.

6) Derive an expression for reflection and Transmission Co-efficients in terms of Refractive Index.

OR

Discuss the reflection and refraction of electromagnetic waves at the plane interface between dielectrics and derive an expression for them.

7) What is optical Rotation ? With neat diagram describe the principle, construction and working of Laurent's half-shade polari meter.

OR

What is Nicol prism ? With neat ray diagram describe the principle, construction and action of Nicol prism.

2024

Part-I | Qn 1 | 1x8

• The dimension of $\vec{\nabla}\times\vec{E}$ are the same as that of ______.
• The phase difference between the field vectors $\vec{E}$ and $\vec{H}$ of electromagnetic wave in a conductor is ______.
• For polar molecules, the dielectric constant depends on __________.
• __________ waves are set up by the transient effect of incident waves which strike the surface first.
• Plane of polarisation is directly proportional to the strength of ________ field.
• In Uniaxial crystals, the velocity of ordinary ray is __________ in all directions.
• S.I. Unit of poynting vector is ___________.
• Refractive index of a conducting medium is _____________ than non-conducting medium.

Part-II | Qn 2 | 1.5x8

• What equation of continuity represents ?
• Write Maxwele's equations for harmonically varying fields.
• What do you mean by wave Impedance ?
• What is phase velocity of a wave ?
• State Brewster's law of polarisation.
• Explain polarised wave.
• What is Quarter wave plate ?
• What is circularly polarized light?
• What is skin Depth ?
• Write the principle of polarimeter.

Part-III | Qn 3 | 2x8

• Explain Freshel's Theory of optical rotation.
• Distinguish between Uniaxial and Biaxial crystals.
• Distinguish between Ordinary and Extra-ordinary Refractive Indices.
• Write the conditions of total internal Reflection.
• Write the Boundary condition at a plane interface between two media.
• Explain plasma frequeney.
• Define Refractive index and Dielectric constant.
• Distinguish between vector and scalar potentials.
• Explain Electromagnetic field energy density.
• What is Gauge Transformation ?

Part-IV | Qn 4-7 | 6x4

4) Using Maxwell's equation in electromagnetic theory derive the electromagnetic wave equation in a medium having finite permeability and permittivity but no conductivity.

OR

Derive expressions for Lorentz Gauge and Coulomb's Gauge.

5) State and explain the plane E.M. Waves through vacuum and isotropic dielectric medium

OR

Discuss the propagation of plane monochromatic electromagnetic waves in conducting media. Derive the dispersion equation and obtain refractive index and skin depth.

6) Explain evanescent wave and find the intensity ratio of reflected wave to incident wave.

OR

Discuss the reflection and refraction of E.M. wave at the interface between dielectrics.

7) What is meant by plane, elliptical and circular polarised light? Obtain their production with necessary theory.

OR

Give the Fresnel's Theory of rotation of plane polarisation by an optical active substance with experimental verification.

2023 | +3-VI-S-CBCS(MS)-Sc(H)-Core-XIII-Phy-R&B | Full Marks : 60

PART-I | 1×8

• Equation of continuity implies conservation of _____.
• Electromagnetic field energy density is expressed as _____.
• What is the phase difference between field vectors $\vec{E}$ and $\vec{H}$ for good conductor ?
• How skin depth ($\delta$) of plasma varies with plasma frequency ($\omega_p$) ?
• The angle of incidence of wave in denser medium for which angle of refraction in rarer medium is 90 is known as _____.
• Which component of magnetic intensity is continuous across the surface separating the two dielectrics ?
• Velocity of extraordinary ray depends on its direction. (True/False)
• Among blue and red, which one produces maximum rotation in its plane of polarisation by an optical active substance ?

PART-II | Answer any eight questions of the followings | 1.5x8

• Define Poynting vector.
• Write Maxwell's equations for free space.
• What does the equation $\vec{\nabla}.\vec{B}=0$ signify ?
• Define wave impedance. What is its value for free space ?
• What is the direction of energy flow in non-conducting medium and how the Poynting vector of the medium is related to Poynting vector in free space ?
• Write Fresnel's equations for electric vector perpendicular to plane of incidence.
• What is mean by evanescent wave ?
• Define degree of polarisation.
• What is biaxial crystal ? Give one example.
• Distinguish between plane polarised and elliptically polarised light.

PART-III | Answer any eight questions of the followings : | 2x8

• Express field vectors $\vec{E}$ and $\vec{B}$ in terms of electromagnetic potentials.
• State Poynting theorem and write its physical significance.
• Define skin depth. How does it varies with conductivity of conducting medium ?
• If the average density of electron in an ionosphere is $6 \times 10^{10}$ electrons per $m^3$, claculate plasma frequency.
• Derive expression for conductivity for ionised region.
• Prove laws of refraction of plane electomagnetic waves at plane interface between two dielectric media.
• Calculate reflection and transmission coefficients for normal incidence at of glass air interface (n_glass = 1.5, n_air=1) .
• Plane polarised light is incident normally on a calcite plate with its optic axis parallel to faces. Find the least thickness for which ordinary and extraordinary rays combine to form plane polarised light. ($\mu_e= 1.5442, \mu_o=1.5533 \text{ and }\lambda =5\times 10^{-5}$ cm)
• State Biot's laws for ratatory polarization.
• Discuss construction of half-shade polarimeter.

PART-IV | Answer all the questions. | 6x4

4) Express Maxwell's equations involving the scalar and the vector potentials. Explain the gauge transformation used. What are the Lorentz and Coulomb gauge ?

OR

a. Starting from Ampere's circuital law, discuss why and how it was modified to include displacement current.
b. Express Lorentz force. $\vec{F}=q[\vec{E}+(\vec{v}\times\vec{B})]$ in terms of electromagnetic potentials.

5) Using Maxwell's equation, derive the wave equations for plane electromagnetic waves in a conducting medium and obtain expression for refractive index.

OR

Using Maxwell's equation, derive the wave equations for plane electromagnetic waves in free space and show that electromagnetic waves transverse in nature.

6) Determine the boundary conditions satisfied by $\vec{E}$,$\vec{D}$ and $\vec{B}$ at the interface between two media.

OR

Describe the reflection phenomenon and obtain expression for reflection co-efficient for a monochromatic electromagnetic wave incident normally on a metallic surface.

7) Discuss the Fresnel's theory of optical rotation.

OR

Discuss theory and construction of quarter and half wave plate. Also discuss use of quarter wave plate to produce circularly polorised light.