(i) Using the concept of spontaneous and stimulated emission of radiation, obtain the relation between Einstein's A and B coefficients. (ii) What is the physical significance of Einstein's A coefficient ? (iii) Justify why lasing action is much more difficult at X-ray frequency than in case of infrared frequency spectrum.
(i) Explain the reason for pulse broadening due to intermodal and material dispersion. Deduce the relation of pulse broadening for intermodal dispersion in optical fiber. (ii) A step index fiber in air has a numerical aperture of 0\cdot 16, a core refractive index of 1\cdot 45 and a core diameter of 60~\mu\text{m}. Determine the normalized frequency for the fiber when light at a wavelength of 0\cdot 8~\mu\text{m} is transmitted. Also estimate the number of guided modes propagating in the fiber.
A parallel beam of light of wavelength 5890~\text{\AA} is incident at an angle of 30^\circ on a plane transmission grating with 15000 lines/inch. Find the highest order of spectrum that can be observed.
Define moment of inertia and explain its physical significance. Calculate the moment of inertia of an annular ring about an axis passing through its centre and perpendicular to its plane.
State and explain Stokes' law. A drop of water of radius 0.01\ \mathrm{m} is falling through a medium whose density is 1.21\ \mathrm{kg/m^3} and \eta = 1.8 \times 10^{-5}\ \mathrm{N\,s/m^2}. Find the terminal velocity of the drop of water.
Express angular momentum in terms of kinetic, potential and total energy of a satellite of mass m in a circular orbit of radius r.
Describe Michelson-Morley experiment and show how the negative results obtained from this experiment were interpreted.
A diatomic molecule can be considered to be made up of two masses m_1 and m_2 separated by a fixed distance r. Derive a formula for the distance of centre of mass, C, from mass m_1. Also show that the moment of inertia about an axis through C and perpendicular to r is \mu r^2, where \mu=\dfrac{m_1m_2}{m_1+m_2}.
Prove that x^2+y^2+z^2=c^2t^2 is invariant under Lorentz transformation.
A ball moving with a speed of 9\ \mathrm{m/s} strikes an identical stationary ball such that after the collision the direction of each ball makes an angle 30^\circ with the original line of motion. Find the speed of the balls after the collision. Is the kinetic energy conserved in this collision?
State the fundamental postulates of Einstein's special theory of relativity. Deduce Lorentz transformation equation and discuss how this accounts for the phenomenon of length contraction.
Discuss the mechanics of a system of point particles with special emphasis on the conservation theorems. How can we extend the results to a system with continuous mass distribution ?
(i) State and prove Hamilton's principle and use it to prove that the shortest distance between two points in space is a straight line joining them. (ii) Use Hamiltonian mechanics to find the differential equation for planetary motion, moving under force f(r) = -\frac{k}{r^2} and prove that the areal velocity is constant.
A bead slides on a wire in the shape of a cycloid described by the equations x = a (\theta - \sin \theta) y = a (1 + \cos \theta) \quad \text{with} \quad 0 \le \theta \le 2\pi. Find the Lagrangian and equation of motion.
Show that the relativistic invariance laws of conservation of momentum lead to the concepts of variation of mass with velocity and mass energy equivalence.
Distinguish between diamagnetic, paramagnetic and ferromagnetic substances.
Give an elementary outline of high-temperature superconductors and their experimental status.
The primitive translation vectors of a two-dimensional lattice are \hat{a}=2\hat{i}+\hat{j},\quad \hat{b}=2\hat{j}. Determine the primitive translation vectors of its reciprocal lattice.
Obtain Laue's equations for X-ray diffraction by crystals. Show that these are consistent with the Bragg's law.
With the help of a schematic diagram, show how entropy and specific heat vary with temperature for a superconductor.
