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.
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.
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?
Show that the rest mass energy of an electron is 0.51\,\mathrm{MeV}. (use the standard values of the physical parameters).
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.
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.
Write down Poiseuille's formula and mention its limitations in analysing the flow of a liquid through a capillary tube.
Define coefficients of viscosity and kinematic viscosity of a fluid. What are Poise and Stokes?
Draw a neat diagram to explain the scattering of an incident beam of particles by a centre of force.
Show that the differential scattering cross-section can be expressed as \sigma(\theta)=\frac{s}{\sin\theta}\left|\frac{ds}{d\theta}\right|, where s is the impact parameter and \theta is the scattering angle.
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.
Write down precisely the conservation theorems for energy, linear momentum, and angular momentum of a particle with their mathematical forms.
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.
An electric field of 100\ \mathrm{V/m} is applied to a sample of n-type semiconductor whose Hall coefficient is -0.0125\ \mathrm{m^3/coulomb}. Determine the current density in the sample assuming \mu_x = 0.36\ \mathrm{m^2\,V^{-1}\,s^{-1}}.
The velocity of sound in f.c.c. gold and f.c.c. copper is 2100\,\mathrm{m/s} and 3800\,\mathrm{m/s} respectively. If the Debye temperature of copper is 348\,\mathrm{K}, then determine the Debye temperature of gold. Take the densities of gold and copper as 1.93 \times 10^{4}\,\mathrm{kg/m^3} and 0.89 \times 10^{4}\,\mathrm{kg/m^3} respectively.
In a cubic unit cell, find the angle between normals to the plane (111) and (121).
Explain the working of SEM and TEM and highlight the major differences in principles. Draw neat schematic diagrams.
What is the reciprocal lattice and why is it named so? Derive the relationships for the primitive translation vectors of the reciprocal lattice in terms of those of the direct lattice.
How does the energy gap in superconductors differ from the energy gap in insulator? How does it vary with temperature for superconductors?
Lead in the superconducting state has critical temperature of 6.2\ \mathrm{K} at zero magnetic field and a critical field of 0.064\ \mathrm{MA\,m^{-1}} at 0\ \mathrm{K}. Determine the critical field at 4\ \mathrm{K}.
