EX-504-B Electromagnetic Theory - Nov 2023 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
   aDefine divergence and curl of a vector field and determine divergence and curl of following vectors.

   1. i)$\vec{P} = x^2 y z \hat{a}_x + x z \hat{a}_z$
   2. ii)$\vec{Q} = \rho \sin \phi \hat{a}_{\rho} + \rho^2 z \hat{a}_{\phi} + z \cos \phi \hat{a}_z$
   How to answer

   • Define divergence as net outward flux per unit volume and curl as circulation per unit area with formulas in Cartesian/cylindrical
   • Write divergence and curl expressions in Cartesian for P and in cylindrical for Q
   • Compute divergence of P and Q
   • Compute curl of P and Q
   • Interpret results (solenoidal/irrotational if zero)
2. 7 Marks
   bDerive expression of electric field intensity due to infinite line charge and write properties of potential function.
   How to answer

   • Consider infinite line charge with uniform density using Gauss law / Coulomb integration
   • Derive radial electric field intensity $E_\rho = \rho_L/(2\pi\epsilon_0 \rho)$
   • Define electric potential function $V = -\int E \cdot dl$
   • List properties: scalar, continuous, gradient gives field, equipotential surfaces
3. 7 Marks
   aWrite Laplace's and Poisson's equations and derive expression of electric field intensity and potential due to dipole.
   How to answer

   • Write Laplace equation $\nabla^2 V=0$ and Poisson equation $\nabla^2 V=-\rho_v/\epsilon$
   • Define electric dipole geometry and potential
   • Derive dipole potential $V$ in spherical coordinates
   • Derive electric field $E = -\nabla V$ components
4. 7 Marks
   bDerive equation of continuity and find energy of a system of three charges - 1nC, 4nC and 3nC located at (0, 0, 0), (0, 0, 2) and (1, 0, 0).
   How to answer

   • Derive continuity equation from charge conservation $\nabla \cdot J = -\partial \rho_v/\partial t$
   • Write electrostatic energy formula $W = (1/2)\sum Q_k V_k$
   • Compute pair-wise distances between (0,0,0), (0,0,2) and (1,0,0)
   • Calculate total energy for 1nC, 4nC, 3nC system
5. 7 Marks
   aWrite Biot Savart's law and derive expression of magnetic field intensity due to straight current carrying conductor.
   How to answer

   • State Biot-Savart law $dH = Idl \times a_R / (4\pi R^2)$
   • Set up geometry for finite/infinite straight conductor
   • Integrate to obtain $H_\phi$ expression
   • Give special case of infinite conductor
6. 7 Marks
   bDerive magnetic boundary conditions and calculate total magnetic flux crossing the surface $$\phi = \frac{\pi}{2},\ 1 \le \rho \le 2m \le o \le z \le 5m$$ if magnetic flux density $\vec{B} = \frac{\rho}{2}\hat{a}_{\phi}$.
   How to answer

   • Derive magnetic boundary conditions for normal B and tangential H components
   • Write flux integral $\Phi = \int B \cdot dS$ for given surface
   • Substitute $B = (\rho/2) a_\phi$ with $\phi=\pi/2$ surface
   • Evaluate integral for $1\le\rho\le2m$, $0\le z\le5m$
7. 7 Marks
   aWrite Faraday's law and explain transformer and motional emf's in detail.
   How to answer

   • State Faraday law in differential and integral form
   • Define transformer emf due to time-varying B
   • Define motional emf due to moving conductor $v \times B$
   • Give combined expression and examples
8. 7 Marks
   bDiscuss following in brief:

   1. i)Scalar and vector potential
   2. ii)Self and mutual inductance
   3. iii)Maxwell's equations for static and time varying fields
   4. iv)Energy stored in magnetic field
   How to answer

   • Define scalar potential V and vector magnetic potential A
   • Define self and mutual inductance with formulas
   • Write Maxwell equations for static vs time-varying fields
   • Give energy density and total energy in magnetic field
9. 7 Marks
   aA uniform plane wave in a medium has $\vec{E} = 2e^{-\alpha z} \sin(10^8 t - \beta z)\hat{a}_y,\ V/m$. If the medium is characterized by $\epsilon_r = 1, \mu_r = 20$ and $\sigma = 3\ S/m$. Find $\alpha, \beta$ and $\vec{H}$.
   How to answer

   • Compute loss tangent and propagation constant $\gamma = \alpha + j\beta$
   • Find attenuation constant alpha and phase constant beta from epsilon, mu, sigma
   • Obtain intrinsic impedance and magnetic field H from E/H ratio
   • Write time-domain H expression
10. 7 Marks
    bWrite poynting vector theorem and discuss reflection of plane wave for normal and oblique incidence.
    How to answer

    • State Poynting theorem relating power flow to E x H
    • Define Poynting vector and average power density
    • Discuss reflection for normal incidence with coefficients
    • Discuss reflection for oblique incidence: Snell law, polarization
11. 7 Marks
    aState Gauss Law and determine the total charge enclosed by the cylinder of radius 1m with $-2 \le z \le 2m$. Given that electric flux density $\vec{D} = z \rho \cos^2 \phi \hat{a}_z\ C/m^2$.
    How to answer

    • State Gauss law $Q_enc = \oint D \cdot dS$
    • Evaluate flux through cylindrical side, top and bottom surfaces
    • Substitute $D = z \rho \cos^2 \phi a_z$
    • Compute total enclosed charge for rho=1m, -2≤z≤2m
12. 7 Marks
    bWrite boundary value conditions for electric field and discuss capacitances for various types of capacitors.
    How to answer

    • Write boundary conditions for D, E at dielectric-conductor interfaces
    • Define capacitance $C=Q/V$
    • Discuss parallel plate, coaxial, spherical capacitors
    • Give formulas for each type
13. 7 Marks
    aA lossy dielectric has an intrinsic impedance of $200 \angle 30^\circ \Omega$ at a particular radian frequency $\omega$. If at that frequency the plane wave propagating through the dielectric has the magnetic field component $\bar{H} = 10e^{-\alpha x} \cos\left(\omega t - \frac{1}{2}x\right)\hat{a}_y\ A/m$. Find $\bar{E}$ and $\bar{H}$ and determine skin depth.
    How to answer

    • Use intrinsic impedance $\eta = E/H$ to relate fields
    • Convert given H to phasor and find E phasor
    • Determine propagation constant, alpha, beta from data
    • Compute skin depth $\delta = 1/\alpha$
14. 7 Marks
    bDerive wave equation in dielectrics and discuss polarization of waves.
    How to answer

    • Start from Maxwell curl equations in source-free dielectric
    • Derive wave equation $\nabla^2 E - \mu\epsilon \partial^2 E/\partial t^2 =0$
    • Define polarization types: linear, circular, elliptical
    • Discuss conditions for each polarization
15. 14 Marks
    Write technical short notes on any three of the following:

    1. i)Method of images
    2. ii)Convection and conduction current density
    3. iii)Magnetic dipole and dipole moment
    4. iv)Self inductance of solenoid, torioid coils
    5. v)Transmission line analogy
    How to answer

    • Explain principle of method of images for grounded planes
    • Distinguish convection vs conduction current density
    • Define magnetic dipole moment $m = I S$
    • Give solenoid/toroid inductance formulas and transmission line analogy