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A

Section A: Long Answer Questions

Attempt all / any as specified.

4 questions
1long10 marks

(a) Set up the differential equation of a damped harmonic oscillator and solve it for the case of light (under) damping. Hence explain the terms relaxation time and quality factor. [6]

(b) A vibrating system of natural frequency 500 Hz has a quality factor of 200. Calculate the relaxation time and the time taken for the amplitude of the freely decaying oscillation to fall to 1/e of its initial value. [4]

oscillationsdamped-oscillationforced-oscillation
2long10 marks

(a) State Maxwell's equations in integral form and explain the physical significance of each. Show how Ampere's circuital law is modified by the introduction of the concept of displacement current. [7]

(b) In a region of free space the conduction current is negligible and the electric field is given by E = 50 sin(10^9 t) V/m. Calculate the displacement current density. [3]

electromagnetismmaxwell-equationsdisplacement-current
3long10 marks

(a) Derive the time-independent Schrodinger wave equation for a particle of mass m moving in a one-dimensional potential. [5]

(b) Solve the Schrodinger equation for a particle confined in a one-dimensional infinite potential well of width L and obtain the expressions for the normalized wave functions and the quantized energy eigenvalues. [5]

quantum-mechanicsschrodinger-equationparticle-in-a-box
4long10 marks

(a) Explain the formation of Newton's rings in reflected monochromatic light. Show that the diameters of the dark rings are proportional to the square root of natural numbers. [7]

(b) In a Newton's rings experiment the diameter of the 10th dark ring changes from 1.40 cm to 1.27 cm when a liquid is introduced between the lens and the glass plate. Calculate the refractive index of the liquid. [3]

interferencenewtons-ringsthin-film
B

Section B: Short Answer Questions

Attempt all / any as specified.

10 questions
5short5 marks

What is meant by a plane diffraction grating? Derive the grating equation for normal incidence and define the resolving power of a grating.

diffractiondiffraction-gratingresolving-power
6short5 marks

State and explain Brewster's law. Distinguish between an ordinary ray and an extraordinary ray in a doubly refracting crystal.

polarizationbrewsters-lawdouble-refraction
7short5 marks

Define acceptance angle and numerical aperture of an optical fibre. A step-index fibre has a core of refractive index 1.50 and a cladding of refractive index 1.47. Calculate its numerical aperture and acceptance angle in air.

fibre-opticsnumerical-apertureacceptance-angle
8short5 marks

What is population inversion? Explain why it is essential for laser action, and describe the role of metastable states and optical pumping in achieving it.

laserpopulation-inversionstimulated-emission
9short5 marks

State Gauss's law in electrostatics. Using it, derive an expression for the electric field intensity at a point near an infinite uniformly charged plane sheet of charge density sigma.

electrostaticsgauss-lawelectric-field
10short5 marks

Derive an expression for the capacitance of a parallel plate capacitor partially filled with a dielectric slab of thickness t and dielectric constant K, the plate separation being d.

capacitorsdielectricscapacitance
11short5 marks

State Heisenberg's uncertainty principle. An electron is confined within a region of width 1 x 10^-10 m. Estimate the minimum uncertainty in its momentum and the corresponding minimum kinetic energy.

quantum-mechanicsuncertainty-principlewave-particle-duality
12short5 marks

Explain the Hall effect in a semiconductor. Derive an expression for the Hall coefficient and state two of its practical applications.

semiconductorshall-effectcarrier-concentration
13short5 marks

What is superconductivity? Explain the Meissner effect and distinguish between Type I and Type II superconductors.

superconductivitymeissner-effectcritical-temperature
14short5 marks

Establish the one-dimensional differential equation of a progressive wave travelling along a stretched string and show that the velocity of the wave depends on the tension and the linear mass density of the string.

oscillationswave-equationacoustics