Search, filter, and study UPSC Civil Services & IFoS Physics past year questions with rigorous derivations, formulas, and step-by-step solutions.
Write an expression for the energy of a diatomic molecule executing both rotation and vibration in a given electronic state. Explain the observed spectra giving a schematic diagram. What are O, P, Q, R and S branches ?
Describe Stern-Gerlach experiment. Explain how it demonstrates the discrete nature of the magnetic moment of an atom.
(i) Define angular momentum, Express it in operator form and show that \vec{L} \times \vec{L} = i\hbar \vec{L} Explain the physical significance of this relation. (10) (ii) Let Y_{lm} be an eigenstate of L^2 and L_z with eigenvalues l(l + 1)\hbar^2 and m\hbar, respectively. Show that \phi = (L_x + i L_y) Y_{lm} is likewise an eigenstate of L^2 and L_z, and determine the eigenvalues. (10)
A one-dimensional potential barrier is represented by the function V(x) = 0 \quad \text{for } x < 0 = V_0 \quad \text{for } x > 0 Where V_0 is positive. Find the transmission coefficient for particles of mass m incident from the left on the barrier.
In the free electron theory of metals, a conductor is regarded as consisting of free electrons in a three-dimensional box. Using the results of (a), obtain an expression for the density of states,
Solve the Schrodinger equation for a linear harmonic oscillator. Obtain the eigenvalues and the corresponding eigenfunctions.
Determine the discrete energy levels and the corresponding eigenfunctions for a particle in an infinitely deep potential well inside a cube of dimension L, i.e., assume \begin{gathered} V(x,y,z) = 0 \quad \text{for } 0 < x < L;\\ 0 < y < L;\\ 0 < z < L = \infty \quad \text{elsewhere} \end{gathered}
(i) State and explain Heisenberg's uncertainty principle. Experimental data reveal that no electron in an atom has energy greater than 4 MeV. Assuming that the radius of a nucleus is 10^{-14}\text{ m}, show using Heisenberg's uncertainty principle that an electron cannot exist inside the nucleus. (10) (ii) Calculate the de Broglie wavelength of thermal neutrons at 300 K. (10)
Prove the law of increase of entropy. Show that for a system at fixed temperature and pressure to be in equilibrium, its Gibbs free energy should be minimum.
Establish the relation C_p - C_v = \left[ P + \left(\frac{\partial U}{\partial V}\right)_T \right] \left(\frac{\partial V}{\partial T}\right)_P Use it to find out an expression for C_p of one mole of a gas whose internal energy is given by U = cT - \frac{a}{V} and which satisfies the equation of state \left[ P + \frac{a}{V^2} \right](V - b) = RT. Here a, b and c are constants.
Derive a relation between the total number of Fermions in terms of Fermi momentum and hence obtain the expression for the total energy E of the system at absolute zero. Combine this expression with the equation of state PV = \frac{2}{3} E to show that the pressure of an ideal Fermi gas at T = 0 is proportional to 5/3 power of its number density.
A system at temperature T_1 is brought in contact with a reservoir at temperature T_2 > T_1. When the system and the reservoir reach thermal equilibrium, calculate the change in entropy of the universe assuming the heat capacity of C_p of the system to be constant. Discuss whether the considered change is positive or not.
What is a magnetic shell ? Define the strength of a magnetic shell. A 2\text{ mm} thick magnetic shell weighing 100\text{ gm} has magnetic moment of 1000\text{ units}. The density of the shell material is 10\text{ gm/cc}. Calculate the intensity of magnetisation and the strength of the shell.
Derive Poisson equation starting from the Coulomb's law for a set of point charges.
Obtain the solution of the Laplace equation in cylindrical coordinates.
State Amperes law of magnetostatics. Using this law, find the magnetic field at a point due to an infinitely long filamentary current.
Why does a spinning nucleus process in a magnetic field ? Explain the underlying principle of nuclear magnetic resonance (NMR) spectroscopy. Calculate the radio frequency at which NMR occurs in water kept in a uniform magnetic field of 2.4 T, the magnetic moment of proton being 2.793\\ \mu_N.
Explain the Rayleigh Scattering of light. Show that the energy density of light scattered from an isotropic homogenous medium of a gas is inversely proportional to the fourth power of the wavelength of the incident light.
State Rayleigh-Jeans law. Show that the intensity of emissions at a particular wave-length is proportional to the temperature T. Discuss the limitations of this law in describing the intensity distribution of emission spectrum of a blackbody.
In an a.c. circuit, a resistance (R = 100\ \Omega) and a capacitance (C = 100\ \mu\text{F}) in series are connected to an a.c. source V = 200 \sin(100\ \pi t). Calculate the current through the circuit and the voltages across R and C. Draw a vector diagram representing the magnitudes and phases of the voltages.