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Unique Paper Code : 32221601
Name of Course : B.Sc. (Hons.) Physics
Name of Paper : Electromagnetic Theory
Semester : VI
Duration : 3 Hours Maximum Marks : 75
(Write your Roll No. on the top immediately on receipt of this question paper)
Question No.1 is compulsory. Answer any four of the remaining six, attempting any two parts from each question. Use of non-programmable scientific calculator is allowed.
Q: 1. Attempt all parts of this question.
a) Starting with the equation of continuity and using Ohmโs law, show that the charge den- sity (๐) in a conductor having conductivity ๐ obeys the equation ๐๐ ๐๐ก +^
๐ ๐ ๐ = 0^ (3)
b) Consider two crossed polaroids placed in a path of an unpolarised beam of intensity Io. A third polaroid is placed in between these two such that its pass axis is at 45o^ to the pass axis of either of the first two polaroids. Calculate the intensity of the transmitted beam. (3)
c) A light beam is incident from denser medium (refractive index 2.0) on a rarer medium of refractive index 1. Plot the reflection coefficients ๐
โฅ for the parallel and perpendicular co- mponents as a function of angle of incidence. (3)
d) Show that in dilute plasma, displacement current leads electric field by ฯ/2. (3)
e) A 20 cm column of cane sugar solution of concentration 100 gm/litre produces rotation of 10.6o. Find the purity of cane sugar. Given: specific rotation of pure sugar is 66 dm-1^ g- cm-3. (3)
f) What is the essential difference between a step-index and gradedโindex optical fibers? Draw their index profiles. (4)
Q: 2. a) State and prove Poyntingโs theorem for a generalized medium and give the physical significance of each of the terms involved. (7)
b) The expression for the magnetic field of a plane em wave travelling in a non- magnetic medium is ๐ปโโ (๐ฅ, ๐ก) = 0.2 ๐๐๐ (6 ๐ ร 10^8 ๐ก โ 3๐๐ฅ) ๐งฬ ๐ด/๐.
Find the direction of wave propagation, dielectric constant, intrinsic impedance, and expression of electric field. (1, 2, 2, 2)
c) Find the charge and current distributions that would give rise to the scalar potential ษธ = 0, and vector potential
๐ด = ฮผ 4๐๐๐ผ (ct - |x|)^2 ๐งฬ for |x| < ct 0 for |x| > ct where, ฮฑ is a constant. (7)
Q: 3. a) Obtain the wave equation for ๐ธโ of an em wave in a conducting medium and hence derive the dispersion relation for this medium. (7)
b) If Re( n ) is the real part of refractive index of a good conducting medium (๐ โฅ 50 โ ๐), then prove that Re(n) โฅ 5 (7)
c) A plane wave is incident normal to the surface of seawater (๐๐ = 1, โ๐ = 79, ฯ = 3 S/m). The electric field is parallel to the surface and is 1 V/m just inside water. At what depth would it be possible for a submarine to receive a signal, if the receiver required a field intensity of 10 ฮผV/m at 20 kHz? (7)
Q: 4. a) Obtain an expression for the refractive index of a dilute plasma. (7)
b) (i) Determine the fields (๐ธโ and ๐ปโโ ) at a distance of 2 m from a monochromatic source of light rated 60 W. (4) (ii) While writing the differential form of Faraday's law, we implicitly assume a fixed surface through which the magnetic flux changes. Write the corrected differential form of Faraday's law when the surface is moving or changing its shape. (3) c) A uniform plane em wave propagating in vacuum is represented by ๐ธ๐ง = ๐ด ๐๐๐ ๐๐ฅ ๐๐๐ ๐๐ก and ๐ป๐ฆ = โ๐ด โ ๐๐๐ ๐
is sent normally to this plate, then what should be the thickness of this plate so that the emergent wave (i) is circularly polarized. (ii) remains linearly polarized but with its polarization rotated by 60o. In each case, how should the plate be oriented w. r. t. the polarization direction of the incident wave? (7)
Q: 7. a) Derive wave equation for ๐ธโ of an em wave (propagating in z-direction) in a symm- etric planar dielectric wave guide having refractive index profile ๐^2 (โ๐ฅ) = ๐^2 (๐ฅ) and ๐ = [
Using the boundary conditions, obtain the eigen-value equation for anti-symmetric TE modes. (7)
b) For symmetric TM modes, ๐ธ๐ง = (^) โ ๐ ๐พ ๐ ๐ ๐ 12 ๐ธ๐ sin๐พ๐ฅ ๐๐ (๐๐ก โ ๐ฝ๐ง), |๐ฅ|^ < d/
where, ๐ = โโ1 and ๐พ^2 = (๐๐ 1 /๐)^2 - ๐ฝ^2. In the region |๐ฅ| < d/2, find the other components of electric and magnetic fields (๐ธโ and ๐ปโโ ) and also determine the time-average Poynting vector. (7) [refractive index profile is same as that in Q: 7. (a)]
c) (i) An optical fiber has a core of refractive index 1.5 which is cladded with a material of refractive index 1.48. Show that all rays making an angle less than 14ยฐ with fiber axis will be guided through the fiber. (4) (ii) An optical fiber have core diameter of 100 ฮผm and refractive index 1.5. Find the axial distance travelled by the ray incident at 30o^ between two successive internal reflections. (3)
Useful Constants:
โ๐ = 8.85 ร 10-12^ F/m = 10
โ 36๐ F/m ฮผ๐ = 4ฯ ร 10-7^ H/m ๐ = 3 ร 108 m/s ๐๐ = 120๐ ฮฉ = 377 ฮฉ Mass of electron = 9.1 ร 10-31^ kg
