UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Show that the complex propagation constant k of an electromagnetic wave propagating in an isotropic dielectric medium with conductivity \sigma can be given by k = k_r + i k_i with k_r \approx \frac{2\pi}{\lambda_o} n \left[ 1 + \frac{1}{8} \left( \frac{\sigma}{\omega \varepsilon} \right)^2 \right] and k_i \approx \frac{2\pi}{\lambda_o} n \left[ \frac{1}{2} \left( \frac{\sigma}{\omega \varepsilon} \right) \right] where n = \sqrt{\varepsilon / \varepsilon_o} and \lambda_o = \omega / 2\pi c; \varepsilon and c being the permittivity of the dielectric and velocity of light in free space, respectively.

For the above system, show that the energy per unit cavity volume in the frequency range of \nu and \nu + d\nu can be given by u(\nu)d\nu = \frac{8\pi\nu^2 k_B T}{c^3} d\nu where k_B is the Boltzmann's constant. Discuss the limitations of this formula and how did Planck put forward the correct analysis.

How do you justify the statement : "A current carrying conductor, although has no net charge experiences a force when placed in a magnetic field."

A long straight solenoid has 100 turns in the secondary and 3000 turns per centimeter in the primary. The area of cross-section of the solenoid is 3 square centimeters. Calculate the mutual inductance.

Explain the term 'mutual inductance' between two coils carrying current. Describe a method of determining mutual inductance between two coils of wire with relevant theory.

Find out the number of photons in a cavity of volume 1\cdot 00\text{ cm}^3 under thermal equilibrium at temperature T = 1000\ {}^\circ\text{K}. Assume that \int_0^\infty \frac{x^2 dx}{e^x - 1} = 2\cdot 405.

The separation between the slits is 0.5\,\text{mm} in Young's double-slit experiment. The interference pattern observed on a screen placed 5\,\text{m} away reveals the location of the first maximum which is 6\,\text{mm} from the centre of the pattern. Calculate the wavelength of light and separation between second and third bright fringes.

Use matrix method to obtain an expression for the focal length of a coaxial combination of two thin lenses having focal lengths f_1 and f_2 separated by distance d.

Show that a travelling wave on the string, clamped on both the ends, undergoes a phase change of \pi. Hence obtain the time-independent form of the wave equation representing a standing wave on the string.

The motion of a damped mechanical oscillator is represented by m\ddot{x}+\alpha\dot{x}+\beta x=0 where m, \alpha and \beta are constants. The oscillator is critically damped. The system is given an impulse at x=0 and t=0, resulting in an initial velocity v. After how much time the system experiences maximum displacement?

Two lenses of focal lengths 8\text{ cm} and 4\text{ cm} are placed at a certain distance apart. Calculate the distance between the lenses if they form an achromatic combination.

When two tuning forks were sounded together, 24 beats were produced in 6\text{ seconds}. After loading one tuning fork with wax, they produced 30 beats in 6\text{ seconds}. If the unloaded fork frequency is 512, what is the frequency of the other ?

A body describing simple harmonic motion (SHM) has a maximum acceleration of 8\ \pi\text{m/sec}^2 and a maximum speed of 1\cdot 6\text{ m/sec}. Find the time period (T) and the amplitude (A) of SHM.

What are Einstein's A and B coefficients ? Establish a relation between them.

Explain mathematically the phenomenon of beats. How does it differ from that of interference ?

Explain the Fraunhofer diffraction at a single slit and obtain the condition for minima. Derive the expression for resolving power of a grating.

A step-index silica fiber consists of core of refractive index 1\cdot 450 and diameter 90\ \mu\text{m}. If the numerical aperture of the fiber is 0\cdot 16, calculate : (i) the acceptance angle of the fiber, (ii) refractive index of the cladding, and (iii) if the fiber is immersed in water of refractive index 1\cdot 33, how does the acceptance angle change ?

An optical waveguide has core and cladding of refractive indices 1\cdot 46 and 1\cdot 45 respectively. (i) Calculate the maximum number of modes with wavelength \lambda = 900\text{ nm} that can travel if it is a planar waveguide of core thickness 10\ \mu\text{m}. (ii) What is the number of modes if it is a cylindrical fiber of diameter 10\ \mu\text{m} ?

Calculate the minimum thickness of a quartz plate which would behave as a quarter-wave plate for wavelength of light, \lambda=6000\,\mathring{\mathrm{A}}. The refractive indices for ordinary and extraordinary rays are \mu_o=1.544 and \mu_e=1.553.

A pulse of \lambda_0=600\,\mathrm{nm} and \Delta\lambda=10\,\mathrm{nm} propagates through a fibre which has a material dispersion coefficient of 50\,\mathrm{ps} per km per nm at 600\,\mathrm{nm}. Calculate the pulse broadening in traversing a 10\,\mathrm{km} length of the fibre. If the pulse width at the input of the fibre is 12\,\mathrm{ns}, what will be the pulse width at the output of the fibre?