UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Find the values of the following : (i) \vec{L} \times \vec{L} (ii) L_+ L_- (iii) [L_z, L_+] where \vec{L} is the angular momentum operator. L_+ and L_- are raising and lowering operators respectively.

A one-dimensional oscillator of mass m = 10^{-20}\text{ kg} oscillates under a force constant k = 10^{-4}\text{ N/m}. (i) Evaluate the zero-point energy. (ii) Calculate the classical amplitude for which the oscillator can have this energy. (iii) What is the energy for the second excited state?

The components of the angular momenta \vec{J}_1 and \vec{J}_2 satisfy commutation rules. Show that the components of the sum \vec{J} = \vec{J}_1 + \vec{J}_2 also satisfy the commutation rules. Whether the difference \vec{J}_1 - \vec{J}_2 satisfies an angular momentum?

The normalized eigenfunction for the ground state of hydrogen atom is \Psi_{100} = \frac{1}{\sqrt{\pi}} \left( \frac{Z}{a_0} \right)^{3/2} e^{-Zr/a_0} Calculate the expectation value of the radius vector r of the electron in the ground state.

Derive an approximate expression for the transmission coefficient for a rectangular potential barrier for which \frac{a}{\hbar}\sqrt{2m(V_0 - E)} \gg 1.

Estimate the temperature at which the root mean square speed of nitrogen molecules exceeds their most probable speed by 100\text{ ms}^{-1}.

Calculate the Fermi energy of aluminium at absolute zero. The density of aluminium is 2\cdot 7 \times 10^3\text{ kg m}^{-3} and its atomic weight is 26\cdot 98\text{ kg (k mol)}^{-1}. Show that the electron gas in aluminium is strongly degenerate.

Two Carnot engines \text{C}_1 and \text{C}_2 operate in series, Engine \text{C}_1 absorbs heat at T and rejects heat to a sink at temperature 300\text{ K}. Engine \text{C}_2 absorbs \frac{1}{4}\text{th} of the heat rejected by engine \text{C}_1 and rejects heat to the sink at 200\text{ K}. If the work done in both the cases is the same, find the temperature T.

Liquid {}^4\text{He} has normal boiling point at 4\cdot 2\text{ K}. It is observed that it boils at 1\cdot 2\text{ K} at a pressure of 1\text{ mm of Hg}. Calculate the average latent heat of vaporization of {}^4\text{He} in this temperature range. (\text{Given : Density of Hg} = 13\cdot 6 \times 10^3\text{ kg m}^{-3}).

In the case of a gas obeying the equation of state \frac{\text{Pv}}{\text{RT}} = 1 + \frac{\beta}{\text{v}}, where \beta is a function of T only, find the expression of the heat capacity at constant volume.

A wire in the form of a hexagon is just enclosed by a circle of radius 10\text{ cm}. If the current in the wire is 1\text{ A}, find the magnetic field at the centre of the hexagon. What would be the direction of the field if the current flows in the anti-clockwise direction ?

Consider a medium with an electric current density \vec{\text{J}}, permittivity \varepsilon and conductivity \sigma, which is described by Ohm's law \vec{\text{E}} = \sigma \vec{\text{J}}. Write down Maxwell's equations in such a medium. Using these equations, derive the wave equation for the electric field \vec{\text{E}}. Show that plane waves of the form \vec{\text{E}} = \vec{\text{E}}_0 \text{ e}^{\text{i}(\omega\text{t} - \text{kx})}, where \vec{\text{E}}_0 is a constant vector, are solutions of the wave equation but with complex k. Determine the explicit expression for k in terms of \varepsilon, permeability \mu_0, \sigma and the angular frequency \omega.

The spectral energy density curve of the moon shows maxima at 14\,\mu\text{m}. Estimate the temperature of the moon and the energy density of the lunar radiation.

A solid copper sphere of mass 100\text{ g} kept at 300\text{ K} is suspended inside a closed chamber. The walls of the chamber are kept at 0\text{ K}. Assuming that the sphere acts as a blackbody, calculate the time required for its temperature to drop to half of the initial value. (Given that density of copper is 8\cdot 96 \times 10^3\text{ kg m}^{-3}, its specific heat capacity is 389\text{ J kg}^{-1}\text{ K}^{-1})

Two dipoles 1 and 2 with dipole moments \vec{\text{p}}_1 = -\,5\,\hat{\text{z}}\text{ nC.m} and \vec{\text{p}}_2 = 9\,\hat{\text{z}}\text{ nC.m}, respectively, are located at points (0, 0, -\,2) and (0, 0, 3), respectively. Find the potential at the origin.

A relativistic charged particle moves in the space occupied by uniform and mutually perpendicular electric and magnetic fields \vec{\text{E}} = \text{E}_0\,\hat{\text{x}} and \vec{\text{B}} = \text{B}_0\,\hat{\text{y}}, respectively. The particle moves rectilinearly along the z-direction. Find \vec{\text{E}}' and \vec{\text{B}}' in the reference frame moving translationally with the particle.

The potential at the surface of a sphere (radius R) is given by \text{V}_0(\theta) = \text{k} \cos 3\theta, where k is a constant. Find the potential inside the sphere.

Consider an L-R-C series circuit with a 300\text{ mH} inductor, a 0\cdot 47\,\mu\text{F} capacitor and a 500\,\Omega resistor. The source has terminal rms voltage \text{V}_{\text{rms}} = 100\text{ V} and variable angular frequency \omega. For what two values of the angular frequency, \omega_1 and \omega_2, is the rms current half the resonance value ? Also calculate the resonance width |\omega_1 - \omega_2|.

A long straight wire of circular cross-section is made of non-magnetic material (\chi_{\text{m}} = 1^\circ to a good degree of approximation). It is of radius a. The wire carries a current I which is uniformly distributed over its cross-section. Compute the energy per unit length stored in the magnetic field contained within the wire.

Discuss the phenomenon of birefringence in a crystal. What do you understand by positive and negative crystals ? If the refractive indices of a calcite crystal for E-ray and O-ray are 1\cdot 488 and 1\cdot 658, respectively, then calculate the thickness of the crystal for which it can work as

(i) quarter wave plate,

(ii) half wave plate. (Wavelength of the light is 5890\text{ \AA})