Search, filter, and study UPSC Civil Services & IFoS Physics past year questions with rigorous derivations, formulas, and step-by-step solutions.
(i) Solve the radial part of the time-independent Schrödinger equation for a hydrogen atom. Obtain an expression for the energy eigenvalues.
(ii) What is the degree of degeneracy of the energy eigenvalues? What happens if the spin of the electron is taken into account?
Consider the one-dimensional wavefunction \psi(x)=Axe^{-kx}, (0\leq x\leq\infty;\,k>0)
(i) Calculate A so that \psi(x) is normalized.
(ii) Using Schrödingers equation find the potential V(x) and energy E for which \psi(x) is an eigenfunction. (Assume that as x \to \infty, V(x) \to 0).
What are bosons ? Based on Bose-Einstein statistics, derive the expression for Bose-Einstein condensation.
The Einstein theory of specific heat of solids gives the expression C_V=\frac{3Nk_Bx^2e^x}{(e^x-1)^2} where x=\frac{\theta_E}{T} with \theta_E as the Einstein temperature.
Show that if an ideal gas is compressed isothermally its compressibility is \frac{1}{p}, whereas if it is compressed adiabatically its compressibility is \frac{1}{\gamma p} where \gamma = C_p / C_v.
Derive the gas equation when it undergoes adiabatic process.
A motor car tyre has a pressure of 2\text{ atmospheres} at the room temperature of 27^\circ\text{C}. If the tyre suddenly bursts, find the resulting temperature.
Show that the Helmholtz free energy of a system never increases in any isothermal-isochoric transformation.
Establish the relation \left(\frac{\partial T}{\partial V}\right)_P=-\left(\frac{\partial P}{\partial S}\right)_T and then derive \left(\frac{\partial C_P}{\partial P}\right)_T=-T\left(\frac{\partial^2 V}{\partial T^2}\right)_P. Hence show that the heat capacity C_P of an ideal gas is independent of pressure P.
Find the efficiency of a Carnot's engine working between 127^\circ\text{C} and 27^\circ\text{C}.
Show that both Fermi-Dirac and Bose-Einstein distributions reduce under certain condition in a form which gives the total number of particles as N = A\int_0^\infty \sqrt{\varepsilon}e^{-\beta\varepsilon}\,d\varepsilon where A is a constant and \beta = 1/k_{\mathrm B}T. Further show that this expression is just the same as that obtained from the Maxwellian speed distribution.
Consider non-equilibrium situation for a system in which the population inversion has been achieved. Explain that such a system can be treated as if it has negative absolute temperature.
Explain the thermodynamic behaviour of an ideal Fermi gas. What are fermions ?
For the above system, show that the energy per unit cavity volume in the frequency range of \nu and \nu + d\nu can be given by u(\nu)d\nu = \frac{8\pi\nu^2 k_B T}{c^3} d\nu where k_B is the Boltzmann's constant. Discuss the limitations of this formula and how did Planck put forward the correct analysis.
What is a displacement current? Prove that lines of conduction current plus displacement current are continuous.
Show that the complex propagation constant k of an electromagnetic wave propagating in an isotropic dielectric medium with conductivity \sigma can be given by k = k_r + i k_i with k_r \approx \frac{2\pi}{\lambda_o} n \left[ 1 + \frac{1}{8} \left( \frac{\sigma}{\omega \varepsilon} \right)^2 \right] and k_i \approx \frac{2\pi}{\lambda_o} n \left[ \frac{1}{2} \left( \frac{\sigma}{\omega \varepsilon} \right) \right] where n = \sqrt{\varepsilon / \varepsilon_o} and \lambda_o = \omega / 2\pi c; \varepsilon and c being the permittivity of the dielectric and velocity of light in free space, respectively.
Explain the physical significance of the Poynting vector \bar{S}. What is represented by the closed integral \oint \bar{S} \cdot d\bar{a} for a closed surface of area \bar{a} ?
The dielectric constant of a medium is 3. The Electric field in the dielectric is 10^6\text{ Vm}^{-1}. What are the electric displacement and polarization ?
A resistance R and a lossless capacitor C are connected through a switch. The capacitor is charged to potential V_0, and the switch is closed at t = 0. Prove that the energy stored in the capacitor is equal to the energy dissipated in the resistor.
A long straight solenoid has 100 turns in the secondary and 3000 turns per centimeter in the primary. The area of cross-section of the solenoid is 3 square centimeters. Calculate the mutual inductance.