UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Show that for the one dimensional wave function \psi(x) = \begin{cases} \frac{1}{\sqrt{2a}} & , \quad |x| < a \\ 0 & , \quad |x| > a \end{cases} where a is a real constant, the rms uncertainty in momentum is infinite.

Explain the Debye model of specific heat of solids. What are its success and failures ?

Show that at T = 0, the Fermi distribution function has a value 1 for energies less than the Fermi energy \epsilon_F and is zero above it. For a system of non-interacting electrons at T = 0, show that the ground state energy of the system of N particles is \frac{3}{5} N \epsilon_F.

Consider N independent particles, each of which can be in either of two energy states +\epsilon and -\epsilon. Derive an expression for the entropy of a microcanonical ensemble for this system, where f = \sum_i \epsilon_i / N\epsilon is fixed, where \epsilon_i is the energy of the i-th particle.

State the law of equipartition of energy. Show how this law can be used to calculate specific heat of gases and hence find the ratio \gamma = C_p/C_v for diatomic and triatomic gases.

One mole of an ideal gas is compressed at constant temperature T from a volume V_1 to a volume V_2. Find the work done and heat absorbed by the gas. The gas now expands adiabatically to a volume 2V. Taking the gas to be diatomic, calculate the final temperature of the gas.

State second law of thermodynamics. Prove that no engine operating between two given temperatures is more efficient than a carnot engine operating between same two temperatures.

Explain how Maxwell modified Ampere's law of magnetostatics by introducing the concept of displacement current. How does it resolve the paradox of a charging capacitor ?

State Faraday's Law of induction in terms of emf and magnetic flux. Use Stoke's theorem to express the law in differential form. A uniform, time varying magnetic field \bar{B} = \bar{B}_0(1 + \alpha t)\hat{k} fills a circular region of radius R lying in the x-y plane with the centre at origin. Here \bar{B}_0 and \alpha are appropriately dimensioned constants. Find the induced electric field at a distance r from the centre where r < R.

Consider a plane electromagnetic wave entering a rarer medium from a denser medium. Show that the Brewster angle is less than the total internal reflection angle.

Show that the interaction energy of two magnetic dipoles \bar{m}_1 and \bar{m}_2 separated by a displacement \bar{r} is given by U = \frac{\mu_0}{4\pi} \cdot \frac{1}{r^3} \left[ \bar{m}_1 \cdot \bar{m}_2 - 3(\bar{m}_1 \cdot \hat{r})(\bar{m}_2 \cdot \hat{r}) \right]. If two magnetic dipoles are held at a fixed distance apart, but allowed to rotate freely, what would be the configuration for stable equilibrium ?

Find the vector potential due to a line segment from x = a to x = b carrying a current I at a point P which is at a distance d from the line segment.

State Gauss's Law of electrostatics both in integral form and differential forms. Two charged spheres of radius R each, have their centres a distance d apart such that d < 2R. One of the spheres has a uniform positive charge density \rho per unit volume while the other has opposite charge density -\rho. Show that the electric field in the region of overlap between two spheres is uniform.

An infinite ladder network of resistances is connected across the points A and B, as shown in the above figure. Each of the resistance is 1\Omega. Calculate the effective resistance between points A and B.

A plane electromagnetic wave travelling in z-direction and polarized in x-direction is incident normally from left on to an interface between two media at z = 0. The medium to the left (z < 0) has a refractive index n_1, while that to the right is of refractive index n_2. The magnetic permeabilities of both the media are equal and \mu_1 = \mu_2 = \mu_0. Obtain expressions for reflection and transmission coefficients (R and T) and show that R + T = 1.

A point charge + q is held above a grounded conducting plane located at z = 0. If the position of the charge is (0, 0, d) obtain an expression for the induced charge density on the plane as a function of coordinates x and y.

Two identical black bodies A and B are respectively at temperatures T and 2T respectively. The heat energy radiated by A is collected for one minute and is used to heat a mass of water. The temperature of water is found to rise by 0.25 K. If the heat radiated by B were to be collected for one minute and used to heat the same mass of water, what would be the rise in the temperature of water ? Assume, specific heat of water to be temperature independent.

A plane electromagnetic wave is travelling in vacuum along a direction which makes an angle of 45^\circ each with the positive x and z-axes. The electric field at time t is given by the expression \bar{E} = E_0 (2 \cos \omega t \hat{j} - \sin \omega t (\hat{i} - \hat{k})) where \hat{i}, \hat{j} and \hat{k} are unit vectors along the x, y and z directions respectively and E_0 = 100 \text{ v/m}. Find the magnetic field vector \bar{B} at time t and obtain the average power transmitted per unit area by the wave.

Explain how inversion of population is achieved in a He-Ne laser.

Consider an ensemble of two-level atoms in thermal equilibrium. Show that the ratio of Einstein A and B coefficients is given by \frac{A}{B} = \frac{8\pi h \nu^3}{c^3}. Why is it not possible to achieve inversion of population in a two-level medium ?