UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Write the equation for total scattering cross-section for electromagnetic waves. Under what condition the scattering is known as Rayleigh scattering? Write the importance of Rayleigh scattering in nature.

(i) Describe the wave given by the expression E = \hat{j} E_0 \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) - \hat{k} E_0 \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) where \hat{j} and \hat{k} are unit basis vectors in Cartesian coordinates. (ii) Write an expression for linearly polarized harmonic plane wave of scalar amplitude E_0, propagating along a line in the xy-plane at 45^\circ to the x-axis and having the xy-plane as its plane of vibration.

A glass chamber 25\text{ mm} long filled with air is positioned in front of one of the secondary sources in Young's experiment. The air is removed from the chamber and a test gas is entered in its stead. On comparing the fringe system corresponding to air with that of the test gas, it is found that the entire interference pattern on the screen was displaced by 21 bright bands towards gas-containing chamber. Given that \lambda = 656{\cdot}2816\text{ nm} for which the refractive index of air is n_a = 1{\cdot}000276. Determine the refractive index of the gas (n_g).

Explain the phenomenon of double refraction in calcite crystal. Considering birefringent crystal as non-conducting material, explain double refraction using electromagnetic theory. Calculate the thickness of a double refracting plate which produces a path difference of \frac{\lambda}{4} between extraordinary and ordinary waves. Given: \lambda = 5890\,\mathring{\mathrm{A}},\quad \mu_0 = 1.53,\quad \mu_e = 1.54

Briefly discuss the postulates of Einstein to explain stimulated emission. Derive an expression for Einstein's A and B coefficients and show that the ratio of coefficients of spontaneous versus stimulated emission is proportional to the third power of frequency of radiation. Why is it difficult to achieve laser action in higher frequency ranges such as X-rays? Can there be a temperature at which the rates of spontaneous and stimulated emission are equal? Illustrate with wavelength \lambda = 5000\,\mathring{\mathrm{A}}.

For free space show that electromagnetic (EM) wave is transverse in nature. Show that for free space, the total outward flux of EM energy through surface S bounding volume V is equal to the rate of loss of EM energy from the volume V. A laser beam of 2\ \mathrm{mm} diameter has average power of 20\ \mathrm{GW}. Calculate the peak values of electric and magnetic fields in the laser beam.

Discuss the conditions for interference. Describe Young's double-slit experiment and derive an expression for the estimation of fringe width. Discuss its dependency on various parameters. Green light of wavelength 5100\,\mathring{\mathrm{A}} from a narrow slit is incident on a double-slit. If the overall separation of 10 fringes on a screen 200\,\text{cm} away is 2\,\text{cm}, find the slit separation.

What do you understand by quarter wave plate? How can it be used to convert a linearly polarized wave into a circularly polarized wave?

What is a zone plate? Give its theoretical description. Show that a zone plate has multiple foci. Differentiate a zone plate from a convex lens. Calculate the radius of the first half period zone in a zone plate behaving like a convex lens of focal length 60\,\mathrm{cm} for light of wavelength 6000\,\mathring{\mathrm{A}}.

(i) Obtain an expression for the displacement of the damped harmonic oscillator where the damping force is proportional to the velocity. Discuss the effect of the damping on the displacement and frequency of the oscillator. Discuss the difference physically between overdamped and critically damped by taking example of a pendulum. (ii) A particle of mass 3 moves along the x-axis attracted towards the origin by a force whose magnitude is numerically equal to 12x. The particle is also subjected to damping force whose magnitude is numerically equal to 12 times the instantaneous speed. If it is initially at rest at x = 10, find the position and the velocity of the particle at any time.

State and explain Fermat's principle of extremum path. Discuss the cases of rectilinear propagation of light and reversibility of light rays in context of Fermat's principle. Using Fermat's principle, deduce the thin lens formula.

Write an expression for the profile (t = 0) of a harmonic wave moving in the +x-direction such that at x = 0, \psi = 10; at x = \lambda/6, \psi = 20 and at x = 5\lambda/12, \psi = 0. (The notations carry usual meanings)

Consider the formation of Newton's rings when two closely spaced wavelengths are present, for example, the \text{D}_1 and \text{D}_2 lines of sodium (\lambda_1 = 5890\text{ \AA} and \lambda_2 = 5896\text{ \AA}). What will be the effect of the presence of these two wavelengths as the lens is gradually moved away from the plate? What will happen if the sodium lamp is replaced by a white source? (The radius of curvature of the lens, R = 100\text{ cm})

What are Newton's rings? How are they formed by two curved surfaces?

The equation for displacement (X) of a point on a damped oscillator is given by x = 5e^{-0.25t}\sin\left(\frac{\pi}{2}\right)t\ \text{metres}. Find the velocity of oscillating point at t = \frac{T}{4} and T, where T is the time period of the oscillator. What is the direction of velocity in each case?

Obtain expressions for the moment of inertia of a solid cone about its

(i) Vertical axis and

(ii) axis passing through the vertex and parallel to its base.

A rubber cord 1\ \mathrm{mm} in diameter and 1\ \mathrm{m} long is fixed at one end and a weight of 1\ \mathrm{kg} is attached to the other end. If the Young's modulus of rubber is 0.05 \times 10^{11}\ \mathrm{dynes\,cm^{-2}}, then find the period of the vertical oscillations of the weight.

A shaft of diameter 8\,\mathrm{cm} and length 5\,\mathrm{m} is transmitting power of 8\,\mathrm{kW} at 300 revolutions per minute. If the coefficient of rigidity of the material of the shaft be 8 \times 10^{11}\,\mathrm{dynes/cm^2}, then calculate the relative shift between the ends of the shaft.

A rocket starts vertically upwards with speed v_0. Then define its speed v at a height h in terms of v_0, h, R (radius of Earth) and g (acceleration due to gravity on Earth's surface). Also calculate the maximum height attained by a rocket fired with a speed of 90% of the escape velocity.

What do you understand by length contraction? Calculate the percentage length contraction of a rod moving with a velocity 0.8c in a direction at 60^\circ with respect to its own length.