Search, filter, and study UPSC Civil Services & IFoS Physics past year questions with rigorous derivations, formulas, and step-by-step solutions.
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?
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)}
Explain the physical significance of group velocity from the concept of phase velocity with relevant expressions.
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 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}.
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.
In the steady state forced vibration of the damped harmonic oscillator, show that the amplitude of the driven system is maximum when p = \sqrt{\omega^2 - 2b^2} and further the value of the maximum amplitude is \frac{f}{2b \sqrt{\omega^2 - b^2}}, where the symbols have their usual significance. (Here you start from the steady state solution)
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.
A plano-convex lens of crown glass is connected to a concavo-convex lens of flint glass so that the convex surface of the first and the concave surface of the second fit exactly. If the combination is achromatic and the combined focal length is -50\text{ cm}, determine the radii of curvature of the faces of the flint glass lens. The dispersive power of the crown glass is one-half of that of the flint glass and the refractive indices for yellow light are respectively 1\cdot 500 and 1\cdot 625 respectively.
What is Fermat's principle ? Derive the laws of reflection and refraction with the help of this principle, when the light is incident on a curved surface.
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.
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?
In a double slit experiment the slits are 2\text{ mm} apart and are illuminated with a mixture of two wavelengths, \lambda = 750\text{ nm} and \lambda' = 900\text{ nm}. At what minimum distance from the common central bright fringe on a screen 2\text{ m} distant from the slits will a bright fringe from one interference pattern coincide with a bright fringe from the other ?
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.
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 rigid body rotates with the angular velocity '\omega' about an axis through the origin O and having direction cosines l, m, n. Show that the moment of inertia of the rigid body about the axis is I = I_{xx} l^2 + I_{yy} m^2 + I_{zz} n^2 + 2 I_{xy} l . m + 2 I_{yz} m . n + 2 I_{zx} n . l where the symbols have their usual meanings.
The principal moments of inertia of a body at a point are given as 200, 300 and 450\text{ gm.cm}^2. Write down the equation of the ellipsoid of inertia at that point.
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}.
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.