UPSC Physics Optional PYQs (CSE & IFoS)PYQs

State and explain the Biot-Savart law. Derive an expression for the magnetic field at a point due to an infinitely long straight current carrying conductor.

In a one-dimensional device, the charge density is given by \rho_V = \rho_0 \frac{x}{a}. If E = 0 at x = 0 and V = 0 at x = a, find V and E using Laplace equation of electrostatics.

(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.

(i) What is Holography ? (ii) Show with simple diagrams, how a hologram is written and read using a laser. (iii) Mention some important applications of holography.

Although the principle of operation of a basic LASER is based upon two energy levels, why does one need a 3-level or a 4-level scheme to achieve satisfactory lasing ? Explain your answer with special reference to a Ruby-laser.

Discuss absorption loss in an optical fibre comparing and contrasting the intrinsic and extrinsic absorption mechanisms.

Discuss the properties of Cornu spiral. Show that the spiral can be used to obtain the intensity distribution in the Fresnel's diffraction pattern due to a straight edge.

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.

(i) In a Michelson's interferometer, 100 fringes cross the field of view when the movable mirror is displaced through 2\cdot 894 \times 10^{-3}~\text{cm}. Calculate the wavelength of the monochromatic source of light. (ii) A shift of 200 fringes is observed when the movable mirror of a Fabry-Pérot interferometer is shifted by 0\cdot 0298~\text{mm}. Calculate the wavelength of the incident radiation.

State and explain Fermat's principle of extremum path and use the same to deduce the laws of reflection and refraction of light.

(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.

Show that the relativistic invariance laws of conservation of momentum lead to the concepts of variation of mass with velocity and mass energy equivalence.

(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.

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 ?

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.

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.

Distinguish between a normal conductor, a superconductor and a semiconductor. How a p-type semiconductor differs from an n-type?

Describe the following laws of Boolean algebra with one application each to logic circuit :

(i) Commutative law of addition

(ii) Associative law of addition

(iii) Associative law of multiplication

(iv) Distributive law

What are logic gates and truth tables? Obtain the truth table of a two-input OR gate.

Give schematic diagrams to distinguish between n\text{-}p\text{-}n and p\text{-}n\text{-}p transistors.