UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Obtain the time-dependent Schrödinger equation for a particle. Hence deduce the time independent Schrödinger equation.

The mean life of Lambda (\Lambda^0) particle is 2.6 \times 10^{-10}\,\mathrm{s}. What will be the uncertainty in the determination of its mass in \mathrm{eV}?

Find the de Broglie wave length of

(i) a neutron

(ii) an electron moving with kinetic energy of 500\,eV (1\,eV = 1.602 \times 10^{-19}\,J)

Explain the origin of the anomalous Zeeman effect.

Derive an expression for the Fermi energy for a free electron gas at T = 0. Compute the Fermi temperature of Cu assuming the density 9\text{ gms/cm}^3 and one conduction electron per atom.

Show that the elemental quantity of heat dQ is not a total differential.

The figure below represents an imaginary ideal gas cycle. Assuming constant heat capacities, show that the thermal efficiency is : \eta = 1 - Y \frac{(V_1/V_2) - 1}{(P_3/P_2) - 1}

For a degenerate Fermi-Dirac gas, the concentration of nucleons in nuclear matter is N = 1.1 \times 10^{38}\text{ cm}^{-3}. Calculate the Fermi energy and Fermi temperature.

Show that both Fermi-Dirac and Bose-Einstein distribution functions at an energy E are given by: f(E) \simeq \exp\left[\frac{\mu-E}{k_{\mathrm B}T}\right], where f(E) is much smaller than unity, \mu and k_{\mathrm B}T are the chemical potential and thermal energy of the atom.

Explain the four thermodynamic relations of Maxwell. Using the same, obtain the Clausius-Clapeyron equation \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T(V_2-V_1)}.

Consider a system of free gas particles having f degrees of freedom. Use equipartition theorem to establish the relation f=\frac{2}{\left(\dfrac{C_p}{C_V}-1\right)}, where C_p and C_V are molar specific heats at constant pressure and constant volume respectively. Obtain the values of \dfrac{C_p}{C_V} for diatomic and triatomic gases.

One kg of water at 20^\circ\mathrm{C} is converted into ice at -10^\circ\mathrm{C} at constant pressure. Heat capacity of water is 4{,}200\ \mathrm{J\,kg^{-1}\,K^{-1}} and that of ice is 2{,}100\ \mathrm{J\,kg^{-1}\,K^{-1}}. Heat of fusion of ice at 0^\circ\mathrm{C} is 335\times10^3\ \mathrm{J\,kg^{-1}}. Calculate the total change in entropy of the system.

Define Enthalpy and show that it remains constant in a throttling process.

(i) Write Maxwell's equations in the absence of a medium. (ii) Explain the gauge transformation and gauge invariance. Show that the relation \text{div } \bar{A} + \frac{1}{c} \frac{\partial \phi}{\partial t} is Lorentz-covariant, where \bar{A} and \phi are the vector potential and the scalar potential respectively. (iii) Obtain Maxwell's wave equations satisfied by the scalar and vector potentials in the absence of charge and current.

A parallel plate capacitor is connected to a battery. What is the electric current while the capacitor is being charged ? What is the displacement current between the plates of the capacitor ? Show that the rate of increase of the electric energy is equal to the surface integral of the Poynting vector over the surface enclosing the volume between the plates of the capacitor.

A plane electromagnetic wave of frequency w_1 travelling in z-direction and polarized in x-direction is incident on a dielectric medium separated by a plane boundary in the x-z plane. Show that the transmission and reflection coefficient are given by : R = \frac{(n_1 - n_2)^2}{(n_1 + n_2)^2} , \quad T = \frac{4 n_1 n_2}{(n_1 + n_2)^2} , where n_1 and n_2 are the refractive indices of the first and the second medium.

A sphere of homogeneous linear dielectric material is placed in an otherwise uniform electric field \bar{E}_o. Find the electric field inside the sphere.

Explain Kirchoff's circuit laws. Using Kirchoff's laws find currents I_1, I_2 and I_3 in the circuit shown below for R_1 = 100\ \Omega, R_2 = 200\ \Omega, R_3 = 300\ \Omega, E_1 = 3\text{ V}, E_2 = 4\text{ V}.

An inductance L, capacitance C and resistance R are connected in series to form a circuit with AC source driving with E_{\text{rms}} = 120\text{ V} at f = 60\text{ Hz}. Compute the power factor and average power dissipated in the resistance if R = 200\ \Omega, X_L = 80\ \Omega and X_C = 150\ \Omega.

Prove Stefan's law of radiation from thermodynamic considerations.