EC-504-A Electro Magnetic Theory - Jun 2025 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aExplain the terms divergence, gradient, curl in spherical co-ordinate system with mathematical expression.
   How to answer

   • Define spherical coordinates and unit vectors
   • Gradient expression in spherical coordinates
   • Divergence expression in spherical coordinates
   • Curl expression in spherical coordinates
   • Physical interpretation of each operator
2. 7 Marks
   bState and prove Gauss's Law. Use it to derive the electric field due to a uniformly charged infinite plane sheet.
   How to answer

   • Statement of Gauss's Law in integral and differential form
   • Proof using flux through closed surface / Gauss divergence theorem
   • Consider uniformly charged infinite plane sheet with surface charge density
   • Apply Gaussian pillbox and symmetry to derive field
   • Result $E = \sigma / (2\epsilon_0)$ and direction
3. 7 Marks
   aDefine electric flux and electric flux density. Derive the boundary conditions at the interface between two dielectric media.
   How to answer

   • Definitions of electric flux $\psi$ and flux density $D$
   • Relation $D = \epsilon E$
   • Consider interface between two dielectrics with pillbox and loop
   • Derive normal component condition $D_{1n}-D_{2n}=\rho_s$
   • Derive tangential component condition $E_{1t}=E_{2t}$
4. 7 Marks
   bGiven $\vec{A} = 2r \cos \phi \hat{a}_r + r \hat{a}_{\phi}$ in cylindrical co-ordinates, for the contour $0 \le x \le 1$ and $0 \le y \le 1$, verify Stoke's theorem.
   How to answer

   • State Stoke's theorem $\oint A\cdot dl = \int (\nabla \times A)\cdot dS$
   • Compute $\nabla \times \vec{A}$ in cylindrical coordinates
   • Evaluate surface integral over $0 \le x \le 1, 0 \le y \le 1$
   • Evaluate closed line integral around square contour
   • Show both sides equal
5. 7 Marks
   aDerive the force between two infinitely long parallel current-carrying wires. Explain its significance in defining the ampere.
   How to answer

   • Magnetic field due to long straight wire by Ampere's law
   • Force on second parallel wire $dF = I dl \times B$
   • Expression for force per unit length $F/l = \mu_0 I_1 I_2 / (2\pi d)$
   • Nature attractive/repulsive based on current direction
   • Use to define ampere
6. 7 Marks
   bWhat is the magnetization vector? Derive the relation between magnetization, magnetic field intensity and magnetic flux density.
   How to answer

   • Definition of magnetization vector $M$ as magnetic dipole moment per unit volume
   • Bound currents concept
   • Relation between $B$, $H$ and $M$
   • Derivation of $B = \mu_0(H+M)$
   • Linear media $M = \chi_m H$
7. 7 Marks
   aState and explain Faraday's Law of electromagnetic induction. Derive its integral and differential forms.
   How to answer

   • Statement of Faraday's law of electromagnetic induction
   • Concept of induced emf and Lenz's law
   • Integral form $\oint E\cdot dl = -d\Phi/dt$
   • Application of Stoke's theorem to get differential form
   • Differential form $\nabla \times E = -\partial B/\partial t$
8. 7 Marks
   bShow how Maxwell's equations unify the concepts of electricity and magnetism. Discuss their importance in modern physics.
   How to answer

   • Write Maxwell's equations in differential/integral form
   • Show coupling of $E$ and $B$ via time-varying terms
   • Displacement current and Faraday's law unification
   • Prediction of electromagnetic waves
   • Importance in modern physics: optics, relativity, communications
9. 7 Marks
   aThe magnetic field intensity in free space is given as $\vec{H} = H_0 \sin(\omega t - \beta z)\hat{a}_y A/m$, where $\omega t - \beta z = \zeta$ and $\zeta = t - z$ is a constant quantity. Determine the displacement current density.
   How to answer

   • Use Maxwell-Ampere law $\nabla \times H = J_d$ in free space
   • Compute $\nabla \times H$ for $H = H_0 \sin(\omega t - \beta z) \hat{a}_y$
   • Obtain $J_d = -\beta H_0 \cos(\omega t - \beta z) \hat{a}_x$
   • Relate to $\partial D/\partial t$
   • State magnitude and direction
10. 7 Marks
    bDerive the wave equation for lossy dielectrics. Discuss the attenuation and phase constants in terms of material properties.
    How to answer

    • Start from Maxwell's curl equations with conduction $\sigma$ and $\epsilon, \mu$
    • Derive Helmholtz wave equation with complex propagation constant
    • Define $\gamma = \alpha + j\beta$
    • Expressions for attenuation constant $\alpha$ and phase constant $\beta$
    • Discussion of loss tangent and material dependence
11. 7 Marks
    aDefine and derive the Poynting Vector. Explain its physical significance in terms of energy flow in electromagnetic fields.
    How to answer

    • Definition of Poynting vector $P = E \times H$
    • Derivation from Maxwell's equations and Poynting theorem
    • Integral form representing power flow
    • Time-averaged Poynting vector for harmonic fields
    • Physical significance as energy flow density
12. 7 Marks
    bDiscuss the propagation of plane waves in a perfect dielectric. Derive the expressions for electric and magnetic field components.
    How to answer

    • Maxwell's equations in perfect dielectric $\sigma=0$
    • Wave equations for $E$ and $H$
    • Solution as uniform plane wave $E_x(z,t)$
    • Intrinsic impedance $\eta = \sqrt{\mu/\epsilon}$
    • Relation between $E$ and $H$ components, velocity, wavelength
13. 7 Marks
    aExplain the propagation of electromagnetic waves in lossless dielectrics. Derive the expression for propagation constant.
    How to answer

    • Condition of lossless dielectric $\sigma \approx 0$
    • Propagation constant $\gamma = j\omega\sqrt{\mu\epsilon}$
    • Attenuation zero, phase constant $\beta$
    • Phase velocity and wavelength expressions
    • E/H ratio and power propagation
14. 7 Marks
    bWhat is Brewster's angle? Derive the condition for polarization by reflection at oblique incidence.
    How to answer

    • Definition of Brewster's angle
    • Reflection of parallel-polarized wave at oblique incidence
    • Fresnel reflection coefficient condition $\Gamma_{||}=0$
    • Derivation $\tan \theta_B = \sqrt{\epsilon_2/\epsilon_1}$
    • Polarization by reflection concept
15. 14 Marks
    Write short notes on (any two):

    1. a)Skin depth
    2. b)Behavior of electric fields in conductors and dielectrics.
    3. c)Slepian vector
    4. d)Coulomb's Law
    How to answer

    • Select any two topics from four options
    • Give definition, relevant formulas and physical meaning
    • Explain with expressions/diagrams where applicable
    • Mention applications and significance
    • Concise structured short notes for each