EC-504-A Electro Magnetic Theory - Dec 2024 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. 2 Marks
   aState Coulomb's law.
   How to answer

   • Statement of Coulomb's law for force between two point charges
   • Mathematical form $F = k q1 q2 / R^2$ with direction
   • Definition of terms and permittivity of free space
2. 6 Marks
   bState and explain divergence theorem.
   How to answer

   • Statement of divergence theorem relating volume and surface integral
   • Mathematical form $\int_V \nabla \cdot \bar{A} dV = \oint_S \bar{A}\cdot d\bar{S}$
   • Explanation of terms and physical meaning
3. 6 Marks
   cVerify that the field $\bar{A} = yz\bar{a}_x + xz\bar{a}_y + xy\bar{a}_z$ is both irrotational and solenoidal.
   How to answer

   • Condition for solenoidal: $\nabla \cdot \bar{A}=0$
   • Condition for irrotational: $\nabla \times \bar{A}=0$
   • Compute divergence and curl of given $\bar{A}$ to verify both zero
4. 7 Marks
   aState and explain the continuity equation of current in integral form and point form.
   How to answer

   • Integral form from charge conservation: $\oint J\cdot dS = -dQ/dt$
   • Point/differential form $\nabla \cdot \bar{J} = -\partial \rho_v/\partial t$
   • Explanation of terms and derivation using divergence theorem
5. 7 Marks
   bFind the nature of the field : $\bar{F} = \left(\frac{150}{r^2}\right)\bar{a}_r + 10\bar{a}_{\phi} + 5\bar{a}_z$.
   How to answer

   • Compute divergence $\nabla \cdot \bar{F}$ in spherical/cylindrical system
   • Compute curl $\nabla \times \bar{F}$
   • Classify as solenoidal/irrotational based on zero/non-zero results
6. 7 Marks
   aWhat is Divergence? Give its physical significance.
   How to answer

   • Definition of divergence as net outward flux per unit volume
   • Mathematical expression $\nabla \cdot \bar{A}$
   • Physical significance: source/sink of field, e.g. charge density
7. 7 Marks
   bDefine Stoke's theorem. Give its importance.
   How to answer

   • Statement of Stoke's theorem: $\oint_L H\cdot dl = \int_S (\nabla\times H)\cdot dS$
   • Meaning of line integral vs surface integral of curl
   • Importance in electromagnetics and conversion of integrals
8. 7 Marks
   aVerify the Stokes theorem for a vector field $\bar{H} = \bar{a}_x + zy^2\bar{a}_y$ over the flat surface bounded by $[0, 0, 0]$, $[0, 1, 0]$, $[0, 1, 1]$ and $[0, 0, 1]$.
   How to answer

   • Compute $\nabla \times \bar{H}$ for $\bar{H}=\bar{a}_x+zy^2\bar{a}_y$
   • Evaluate surface integral over given flat surface
   • Evaluate closed line integral around boundary and compare for verification
9. 7 Marks
   bDefine magnetic field and state its properties.
   How to answer

   • Definition of magnetic field $\bar{B}$/$\bar{H}$
   • Properties: source, direction, force on moving charge
   • Properties: divergence-free, units and related laws
10. 7 Marks
    aExplain the Maxwell's equation derived from Ampere's circuital law.
    How to answer

    • Start from Ampere's circuital law $\oint H\cdot dl = I$
    • Apply Stoke's theorem to get point form $\nabla\times H = J$
    • Explain Maxwell's modification / final equation form
11. 7 Marks
    bWhat is displacement current? How is Maxwell's equation modified to account for it in time varying field?
    How to answer

    • Definition of displacement current $J_d = \partial D/\partial t$
    • Need in time-varying fields for continuity
    • Modified Ampere-Maxwell equation $\nabla\times H = J_c + \partial D/\partial t$
12. 7 Marks
    aDerive the uniform plane wave in perfect (or lossless) dielectric.
    How to answer

    • Start from Maxwell's curl equations for lossless dielectric ($\sigma=0$)
    • Derive wave equation $\nabla^2 E - \mu\epsilon \partial^2 E/\partial t^2 =0$
    • Solution as uniform plane wave with propagation constant, velocity and impedance
13. 7 Marks
    bA $10$ GHz plane wave travelling in free space has an amplitude $E = 10$ V/m. Find

    1. i)Phase velocity, wavelength and propagation constant.
    2. ii)Determine characteristic impedance.
    How to answer

    • Use $v_p = c = 1/\sqrt{\mu_0\epsilon_0}$, $\lambda = v_p/f$, $\beta = 2\pi/\lambda$
    • Characteristic impedance $\eta_0 = \sqrt{\mu_0/\epsilon_0} \approx 377 \Omega$
    • Substitute $f=10$ GHz, $E=10$ V/m to get numerical values
14. 9 Marks
    aWrite short note on Skin depth.
    How to answer

    • Definition of skin depth $\delta = \sqrt{2/\omega\mu\sigma}$
    • Explanation of field attenuation $e^{-z/\delta}$ in conductor
    • Factors affecting depth: frequency, conductivity, permeability
15. 5 Marks
    bDetermine the depth of penetration for copper at $1$ MHz.
    How to answer

    • Formula $\delta = 1/\sqrt{\pi f \mu \sigma}$ for copper
    • Use $\sigma_{Cu}$, $\mu_0$, $f=1$ MHz
    • Compute numerical value of penetration depth
16. 14 Marks
    Define Poynting vector and prove that the electromagnetic power flow is the product of electric and magnetic field intensities.
    How to answer

    • Definition of Poynting vector $\bar{P}=\bar{E}\times\bar{H}$
    • Derivation from Maxwell's equations via Poynting theorem
    • Interpretation as power density = product of E and H magnitudes