UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Using the definition \vec{L}=\vec{r}\times\vec{p} of the orbital angular momentum operator, evaluate [L_x,L_y].

(i) Consider a beam of particles incident on a one-dimensional step function potential with energy E > V_0 as shown in the above figure. Solve the Schrödinger equation and obtain expressions for the reflection and transmission coefficients.

(ii) What are the limits of the reflection coefficient for E \to V_0 and E \to \infty?

(iii) Discuss the cases 0 < E < V_0 and E < 0.

Show that the velocity of electron in the first orbit of hydrogen atom is \left(\frac{1}{137}\right)C where C is the velocity of light. (Given electronic charge = 1.602 \times 10^{-1}\ \mathrm{C} Planck Constant 6.63 \times 10^{-34}\ \mathrm{J.s}, permittivity = 8.85 \times 10^{-1}\ \mathrm{C^2\ N^{-1}\ m^{-2}})

In a series of experiments on the determination of the mass of a certain elementary particle, the results showed a variation of \pm 20\,m_e, where m_e is the electron mass. Estimate the lifetime of the particle.

Establish Van der Waals equation of state for a real gas. Deduce expressions for critical constants and show that critical coefficient is independent of the nature of gas.

State Dulong-Petit's law and discuss its limitations in explaining heat capacities of solids. How were these efficiencies overcome by Debye ?

For a completely degenerate BE gas, discuss the condition for onset of BE condensation.

What is transport phenomenon ? Obtain the expression for the coefficient of viscosity. Discuss its temperature dependence.

Discuss the significance of Saha's ionisation formula in classification of stars.

Calculate the net change in entropy when 10\text{ g} water at 60^\circ\text{C} is mixed with 30\text{ g} water at 20^\circ\text{C}.

N particles obeying Classical Statistics are distributed among three states having energies \varepsilon_1 = 0, \varepsilon_2 = k_{\mathrm B}T and \varepsilon_3 = 2k_{\mathrm B}T, where k_{\mathrm B} is Boltzmann constant. If the total equilibrium energy of the system is 1000k_{\mathrm B}T, calculate the value of N.

The coefficient of viscosity of helium at 27^\circ\mathrm{C} is 2 \times 10^{-5}\ \mathrm{kg\,m^{-1}\,s^{-1}}. Calculate

(i) the average speed and

(ii) the diameter of a helium molecule, if it is assumed that the gas obeys Maxwell-Boltzmann distribution. Given Boltzmann constant k_{\mathrm B} = 1.38 \times 10^{-23}\ \mathrm{J\,K^{-1}} and mass of helium atom = 6.67 \times 10^{-27}\ \mathrm{kg}.

In Leh, temperature of ice on a cold winter night is measured as -20^\circ\mathrm{C}. Calculate the change in entropy when 1\ \mathrm{kg} of ice is converted into steam at 100^\circ\mathrm{C}. Given specific heat capacity of ice is 500\ \mathrm{cal\,kg^{-1}\,K^{-1}}, latent heat of ice is 3.36\times10^5\ \mathrm{J\,kg^{-1}}, latent heat of steam is 2.26\times10^6\ \mathrm{J\,kg^{-1}} and J=4.2\ \mathrm{J\,cal^{-1}}.

The vapour pressure, in mm of Hg, of a substance in solid state is given by the relation \ln p=23.03-\dfrac{3754}{T}, where T is in Kelvin. The vapour pressure, in mm of Hg, of the substance in liquid state is given by the relation \ln p=19.49-\dfrac{3063}{T}. Calculate

(i) the coordinates of the triple point, and

(ii) the latent heat of vaporisation at the triple point. Take Gas constant R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}.

A thermally insulated ideal gas is compressed quasi-statically from an initial state with volume V_0 and pressure P_0 to a final state of volume V_f and pressure P_f. Show that the work done on the gas in the process is given by W=\frac{C_V}{R}\left(P_fV_f-P_0V_0\right) where C_V and R having standard meanings.

Sea water has resistivity 0\cdot 3\ \Omega\text{m} and its dielectric constant is 81. Calculate the ratio of the amplitudes of the conduction and polarisation current intensities when the applied field is oscillating at 100\text{ MHz}.

When the current in an \text{R}-\text{L} circuit is decaying, what fraction of the original energy stored in the inductor has been dissipated after 2\cdot 3 time constant ?

What do you understand by the term polarization in dielectrics ? Obtain Clausius-Mossotti formula for linear dielectrics.

A plane electromagnetic wave is given by E_z = a \cos \omega x \cos \omega t and H_y = - a \sin \omega x \sin \omega t. Evaluate the instantaneous value of the Poynting vector \vec{S} and show that <\vec{S}> = 0.

With reference to ferromagnetic materials, explain the terms hysteresis and hysteresis loops.