UPSC Physics Optional PYQs (CSE & IFoS)PYQs

A diffraction grating has 5000\text{ lines per cm}. For illumination at normal incidence, determine the dispersive power of the grating in the second order spectrum in the range of wavelengths around 500\text{ nm}.

Explain the diffraction at straight edge with the help of Cornu's spiral.

Distinguish between the intensity patterns due to diffraction from a narrow slit and a straight edge.

Consider a Gaussian pulse propagating in pure silica, at \lambda_0 = 0\cdot 85\\ \mu\text{m}, along z-direction. As the pulse propagates, it gets broadened according to the formula \tau^2(z) = \tau_0^2 \left( 1 + \frac{4 \alpha^2 z^2}{\tau_0^4} \right), \quad \alpha = \left. \frac{d^2 k}{d \omega^2} \right|_{\omega=\omega_0} where \tau(z) is the pulse width after propagation through a distance z and \tau_0 is the pulse width at z = 0. Given that \alpha = 3 \times 10^{-26}\\ \frac{\text{S}^2}{\text{m}} at \lambda_0 = 0\cdot 85\\ \mu\text{m}. Calculate the distance at which \tau(z) = \sqrt{2} \cdot \tau_0.

A light spring of relaxed length 'a_0' is suspended from a point. It carries a mass 'm' at its lower free end, which stretches it through a distance l. Show that the vertical oscillation of the system is simple harmonic in nature and has time period T = 2\pi \sqrt{l / g}, where g is the acceleration due to gravity.

Write down the one-dimensional harmonic oscillator differential equation under damping and its solution for the lightly damped condition, with the meanings of symbols. Determine the dependent energy in the lightly damped condition.

Prove that the group velocity V_g of electromagnetic waves in a dispersive medium with refractive index n(\lambda_0) at wavelength \lambda_0 is given by V_g=\frac{c}{n(\lambda_0)-\lambda_0\dfrac{dn(\lambda_0)}{d\lambda_0}} where c is the free space velocity of light. Find the time taken for the electromagnetic pulse to travel a distance D.

For calcite, the refractive indices of ordinary and extraordinary rays are 1.65836 and 1.48641 at \lambda_0=5893\,\mathring{\mathrm{A}} respectively. A left circularly polarized beam of this wavelength is incident normally on such crystal of thickness 0.005141\,\mathrm{mm} having its optic axis cut parallel to the surface. What will be the state of polarization of the emergent beam?

When a thin film of a transparent material is put behind one of the slits in Young's double-slit interference experiment, the zero-order fringe moves to the position previously occupied by the fourth-order bright fringe. The index of refraction of the film is n=1.2 and the wavelength of light, \lambda=5000\,\mathring{\mathrm{A}}. Determine the thickness of the film.

The Fraunhofer single-slit diffraction intensity is given by I=I_0\frac{\sin^2 x}{x^2} where x=\frac{\pi d y}{\lambda l}, with l as the distance from slit to source, d the slit width, y the detector distance and \lambda the wavelength. What is the value of cumulative intensity \int_{-\infty}^{\infty} I(y)\,dy?

A plane wave has the following expression for its electric field: \vec{E}=\hat{x}E_{0x}\cos(\omega t-kz+\alpha)+\hat{y}E_{0y}\cos(\omega t-kz+\beta) If the phase difference is defined as \delta=\beta-\alpha, under what conditions do we achieve elliptic polarization? What are the conditions for circular polarization?

Bring out the essential differences between the physical principles of spontaneous and stimulated emission of radiation. Why is it difficult to get efficient lasing action in case of an ideal two-level material system? Can you propose a scheme to enhance efficiency? Discuss.

Show with proper mathematical analysis that the ratio of Einstein's A and B coefficients depends upon the energy separation between the two energy levels participating in the optical transitions. What is the physical significance of A coefficient? \text{(5 marks)} Justify the statement, ``It is very difficult to develop an X-ray laser''. \text{(5 marks)}

In relation to a plane diffraction grating having 5000 lines per cm and irradiated by light of wavelength 6000\,\mathring{\mathrm{A}}, answer the following:

(i) What is the highest order spectrum which may be observed?

(ii) If the width of opaque space is exactly twice that of the transparent space, which order of spectra will be absent?

With an appropriate diagram, show that in the Rutherford scattering, the orbit of the particle is a hyperbola. Obtain an expression for impact parameter.

A particle of mass 'm' moves according to the equations x = a \cos \omega t, y = a \sin \omega t, z = c, where a, c, \omega are constants. Obtain the instantaneous velocity and linear momentum vectors in terms of the Cartesian components and hence the angular momentum. Find the force \vec{F} and the torque \vec{N} acting on the particle and verify that the angular momentum \vec{L} and the torque \vec{N} satisfy the relation \frac{d\vec{L}}{dt} = \vec{N}.

With an appropriate diagram, show that in the Rutherford scattering, the orbit of the particle is a hyperbola. Obtain an expression for impact parameter.

Prove that the time taken by the earth to travel over half of its orbit separated by the minor axis remote from the sun is two days more than half a year. Given, the period of the earth is 365 days and eccentricity of the orbit =1/60.

Imagine that a rigid body is rotating about a fixed point with angular velocity \vec{\omega}. Assuming that the coordinate axes coincide with the principal axes, if T stands for kinetic energy and G for external torque acting on the body, show that \frac{dT}{dt}=\vec{G}\cdot\vec{\omega}

Determine the number of degrees of freedom for a rigid body-

(i) moving freely in space of three dimensions;

(ii) having one point fixed;

(iii) having two points fixed.