Using the concept of Einstein’s A and B coefficients for a two-level atomic system under thermal equilibrium, determine the ratio of the number of atoms per unit volume in the two levels experiencing spontaneous and stimulated emission. How does the principle of population inversion lead to the gain mechanism in the active medium of the laser?
Using matrix method, find out the equivalent focal length for a combination of two thin lenses of focal lengths f_1 and f_2 separated by a distance a.
A convex lens of focal length 20\,\mathrm{cm} is placed after a slit of width 0.5\,\mathrm{mm}. If a plane wave of wavelength 5000\,\mathring{\mathrm{A}} falls normally on the slit, calculate the separation between the second minima on either side of the central maximum.
Deduce an expression for resolving power of a diffraction grating, explaining its meaning. What will be the value of the number of slits in the grating to resolve \text{D}_1 and \text{D}_2 lines of sodium in the first order, assuming \lambda and \Delta\lambda respectively as 600\text{ nm} and 0\cdot6\text{ nm}?
Explain the refractive index distribution for a graded index optical fiber given as \begin{aligned} n^2(r) &= n_1^2 \left[ 1 - 2\delta \left( \frac{r}{a} \right)^q \right], & r < a \\ &= n_1^2 [1 - 2\delta] = n_2^2, & r > a \end{aligned} with usual meaning of symbols. Sketch and name the above profiles with justification for q = 1, 2 and \infty.
Explain, with neat diagrams, what you mean by monochromatic spherical aberration with reference to paraxial focus, marginal focus and circle of least confusion.
In Young's double-slit experiment, a thin mica sheet (refractive index = 1 \cdot 5) is introduced in the path of one of the beams. If the central fringe gets shifted by 0 \cdot 2\text{ cm}, calculate the thickness of the mica sheet. Assume separation between the two slits (d) and between slit and source (D) as d = 0 \cdot 1\text{ cm} and D = 50\text{ cm}.
Obtain the equation of motion of a simple pendulum using Lagrangian formalism. Hence, obtain expression for the angular frequency of a simple harmonic oscillator.
With a planoconvex lens of radius of curvature R resting on a plane glass surface, Newton's rings are observed with incident light of wavelength \lambda. Find the radii of m\text{th} order dark rings. What will be the colour of the innermost ring with white light illumination?
Explain physically why dispersion is diminished in multimode parabolic-index fiber than in step-index fiber.
Find out the phase and group velocities of a radio wave of frequency \omega = \sqrt{2}\omega_p in the ionosphere (as a dielectric medium) of refractive index n = \sqrt{1 - \frac{\omega_p^2}{\omega^2}}. Here, \omega_p is the ionospheric plasma frequency.
What is the role of an optical resonator in a laser? Why does one prefer curved mirrors instead of plane mirrors in designing an optical resonator?
Write down Poiseuille's formula and mention its limitations in analysing the flow of a liquid through a capillary tube.
Show that the moment of inertia of a circular disc of mass M and radius R about an axis passing through its centre and perpendicular to its plane is \frac{1}{2}MR^2.
Using Poiseuille's formula, show that the volume of a liquid of viscosity coefficient \eta passing per second through a series of two capillary tubes of lengths l_1 and l_2 having radii r_1 and r_2 is obtained as Q=\frac{\pi p}{8\eta\left[\frac{l_1}{r_1^4}+\frac{l_2}{r_2^4}\right]}. where p is the effective pressure difference across the series.
How does one obtain the angular velocity of the Earth about the North Pole with respect to a fixed star as 7.292 \times 10^{-5}\,\mathrm{sec}^{-1}? Explain your method of calculating the above value.
Define coefficients of viscosity and kinematic viscosity of a fluid. What are Poise and Stokes?
Write down precisely the conservation theorems for energy, linear momentum, and angular momentum of a particle with their mathematical forms.
Draw a neat diagram to explain the scattering of an incident beam of particles by a centre of force.
Prove mathematically that the addition of any velocity of a particle to the velocity of light in free space merely reproduces the velocity of light in free space only.
