UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Distinguish between potential well and potential barrier. Given their illustrations. Considering one-dimensional potential step, prove that sum of the reflection and transmission coefficients is unity.

Explain how the problem of the hydrogen atom could be solved using Schrodinger equation. Also derive an expression for its energy eigenvalue and discuss the associated bound states of this case.

Prove that \frac{d}{dt}(x) = \frac{1}{m}(p_x) Define all the terms of this relation and give its physical interpretation.

State Dulong and Petit's law. How does it agree with experiment ? Discuss the limitations of the classical theory and success of the quantum theory in explaining the specific heat of solids.

1 kg of ice at 0^\circ\text{ C} floats on 10 kg of water at 30^\circ\text{ C}, the whole system being thermally isolated. What will be the change in entropy of the sytem when thermal equilibrium is reached ? [Specific heat of water = 4.2\text{ kJ kg}^{-1}\text{ K}^{-1} and latent heat of fusion of ice = 336\text{ kJ kg}^{-1}]

Derive the expression for the Fermi-Dirac distribution function. Represent it graphically for T = 0 and T \neq 0.

Using thermodynamic principles show that the Joule-Thomson coefficient \mu can be expressed as \mu = \frac{1}{C_p} \left[ T \left( \frac{\partial V}{\partial T} \right)_p - V \right] . Calculate the value of \mu for an ideal gas and interpret your result physically.

Starting from Maxwell's equation, \nabla . D = \rho, where D is the electric displacement density and \rho is the charge density, derive Poisson's equation. Deduce Laplace's equation for charge-free region from Poisson's equation.

A potential in cylindrical coordinates is a function of r and \phi but not of z. Obtain the separated differential equations for R and \Phi, where V = R(r) \Phi(\phi) and solve them.

In the case of proton NMR, state the expression for the energy of the dipole in an external magnetic field. How does the NMR originate ? Give the rough range of frequencies of the NMR signals, and the normal magnitude of the applied field.

Using Planck's radiation formula u(\nu) d\nu = \frac{8 \pi h}{c^3} \frac{\nu^3 d\nu}{e^{h\nu/kT} - 1} where the symbols have their usual meaning, find the wavelength of the region where energy density is the greatest. Also calculate the total energy density over all the frequencies

Write Ampere's circuital law, and obtain a generalised form of this law, for non-stationary case.

What do you mean by a gauge transformation ? What is its importance ? Show that the Lorentz gauge condition \vec{\nabla} \cdot \vec{A} + \frac{1}{c} \frac{\partial \phi}{\partial t} = 0 is Lorentz invariant. Here \vec{A} and \phi are the vector and scalar potentials.

Write down the different components of the electromagnetic field tensors F_{\mu\nu} and further prove that Maxwell's equations of electrodynamics are invariant to Lorentz transformations.

Find the magnetic field \vec{B} at the point P due to a short straight length of wire carrying current 'i'. Length of the wire is l. Point P is at a distance r away from the centre of the wire. Angle between \vec{l} and \vec{r} is \theta.

Starting from Maxwell's equations of electromagnetic field in vacuum obtain the classical wave equations for the four field vectors \vec{E}, \vec{D}, \vec{B} and \vec{H} . Show that the field vectors can be propagated as waves in free space with the velocity of propagation equal to 3 \times 10^8\text{ m/s}, where for free space we have the vacuum permittivity \varepsilon_o = 8.854 \times 10^{-12}\text{ farad/m} & vaccum permeability \mu_o = 1.257 \times 10^{-6}\text{ henry/m}.

What is molecular polarizability ? Derive Clausius - Mossotti equation relatig the molecular polarizability with the dielectric constant of a dielectric material.

Explain the working of Michelson interferometer using appropriate optical diagram. also draw paths of the rays.

How would you produce plane polarized light by reflection ? What is Brewster's law ? Calculate the angular position of the sun above the horizon so that light reflected from a calm lake is completely polarized. The refractive index of water is 1.33. Circularly polarized and unpolarized light are passed in turn through a Nicol prism. The Nicol is rotated about the direction of light as axis. What would you observe in each case ? How would you distinguish between them ?

What is optical activity ? Give reasons for the conclusion that optical rotation in liquids has a molecular origin. What do you mean by ordinary and extraordinary rays ? What are positive and negative crystals ? Give an example of each. Compute the minimum thickness of a quarter-wave plate made from quartz for incident wavelength of 589.3 nanometer. Given \mu_o = 1.544 and \mu_E = 1.553.