UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Calculate the probability of an electron occupying an energy level 0.02\,\mathrm{eV} above the Fermi level at T=300\,\mathrm{K}

Write the expression for the Fermi-Dirac distribution. Plot the Fermi-Dirac distribution at T=0 and for T_1>T_2>0. Now from the plot propose two alternative definitions of the Fermi level.

The spectral composition of solar radiation is similar to that of a black body radiator whose maximum emission corresponds to the wavelength 0.48\,\mu\mathrm{m}. Find the mass lost by the Sun every second due to radiation. Evaluate the time interval during which the mass of the Sun reduces by 1 per cent. Given: Stefan Boltzmann constant = 5.669 \times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}, radius of the Sun = 6.957 \times 10^{8}\ \mathrm{m}, surface temperature of the Sun = 5772\ \mathrm{K} and mass of the Sun is 1.9885 \times 10^{30}\ \mathrm{kg}.

In the given circuit, L=2.0\,\mu\mathrm{H}, R=1.0\,\Omega, R_0=2.0\,\Omega and E=3.0\,\mathrm{V}. Find the amount of heat generated in the coil after the switch S is disconnected. The internal resistance of the source is negligible.

Consider the two branch parallel circuit shown in the diagram. Determine the resonant frequency of the circuit.

A region 1, z<0, has a dielectric material with \varepsilon_r=3.2 and a region 2, z>0 has a dielectric material with \varepsilon_r=2.0. Let the displacement vector in the region 1 be, \vec{D}_1=-30a_x+50a_y+70a_z\,\mathrm{nC\,m^{-2}}. Assume the interface charge density is zero. Find in the region 2, the \vec{D}_2 and \vec{P}_2, where \vec{P}_2 is the electric polarization vector in the region 2.

Given that the electric potential of a system of charges is V=\dfrac{12}{r^2}+\dfrac{1}{r^3} volt. Calculate the electric field vector at the Cartesian point (4,2,3)\,\mathrm{m}.

Given an infinite line charge of charge density 2\,\mathrm{nC\,m^{-1}} parallel to the y-axis and passing through the point (3,0,4)\,\mathrm{m} and an infinite sheet of charge of charge density 4\,\mathrm{nC\,m^{-2}} parallel to the x-y plane and passing through the point (0,0,6)\,\mathrm{m}. Calculate the electric field intensity at the point (10,10,10)\,\mathrm{m}. Assume free space.

A rod of length l is perpendicular to a uniform magnetic field \mathrm{B}. The rod revolves at an angular speed \omega about an axis passing through one end of the rod and parallel to the magnetic field \mathrm{B}. Find the voltage induced across the rod's ends.

Calculate the skin depth of electromagnetic waves of 1\,\mathrm{MHz} incident on a good conductor having \sigma = 5.8 \times 10^{7}\,\mathrm{S\,m^{-1}}. Assume that inside the conductor \mu = \mu_0 = 4\pi \times 10^{-7}\,\mathrm{H\,m^{-1}}.

A mass m is suspended by two springs having force constants k_1 and k_2 as shown in the figure. The mass m is displaced vertically downward and then released. If at any instant t, the displacement of the mass m is x, then show that the motion of the mass is simple harmonic motion having frequency f=\frac{1}{2\pi}\sqrt{\frac{1}{m}\left(\frac{k_1k_2}{k_1+k_2}\right)}

What is damped harmonic oscillation? Write the equation of motion and obtain the general solution for this oscillation. Discuss the cases of dead beat, critical damping and oscillatory motion based on the general solution. What would be the logarithmic decrement of the damped vibrating system, if it has an initial amplitude 30\,\mathrm{cm}, which reduces to 3\,\mathrm{cm} after 20 complete oscillations?

What is chromatic aberration? Obtain the condition for achromatism using combination of two thin lenses placed in contact to each other. Can this system work as achromatic doublet if both are of same material? Justify your answer.

A thin film of petrol of thickness 9 \times 10^{-6}\,\mathrm{cm} is viewed at an angle 30^\circ to the normal. Find the wavelength(s) of light in visible spectrum which can be viewed in the reflected light. The refractive index of the film \mu = 1.35.

Show that the phenomenon of Fraunhofer diffraction at two vertical slits is modulation of two terms viz. double slit interference and single slit diffraction. Obtain the condition for positions of maxima and minima.

In a step-index optical fiber system, explain the terms pulse dispersion and material dispersion. An optical fiber having refractive indices of core and cladding n_1 = 1.463 and n_2 = 1.444 respectively, uses a Laser diode with \lambda_0 = 1.50\,\mu\mathrm{m} with a spectral width of 2\,\mathrm{nm}. At this wavelength if the material dispersion coefficient, D_{m} is 18.23\,\mathrm{ps/(km\cdot nm)}, then calculate the pulse dispersion and material dispersion for 1\,\mathrm{km} length of the fiber.

An observer detects two explosions, one that occurs near him at a certain time and another that occurs 2\,\mathrm{ms} later 100\,\mathrm{km} away. Another observer finds that the two explosions occur at the same place. What time interval separates the explosions to the second observer?

A homogeneous right triangular pyramid with the base side a and height \frac{3a}{2} is shown below. Obtain the moment of inertia tensor of the pyramid:

A light rod of length 100\,\mathrm{cm} is suspended from the ceiling, horizontally by means of two vertical wires of equal length tied to its ends. One of the wires is made of steel and its cross-section is 0.05\,\mathrm{sq.\ cm} and the other is of brass of cross-section 0.1\,\mathrm{sq.\ cm}. Find the position along the rod at which a weight may be hung to produce

(i) Equal stresses in both the wires,

(ii) Equal strain in both the wires. Young's modulus of elasticity of brass and steel are 1.0 \times 10^{11}\,\mathrm{N/m^2} and 2.0 \times 10^{11}\,\mathrm{N/m^2} respectively.

An electron is moving under the influence of a point nucleus of atomic number Z. Show that the orbit of the electron is an ellipse.