UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Explain Bose-Einstein distribution and obtain the same from the grand canonical ensemble.

Derive the mathematical expression for the total energy of a degenerate Fermi gas at a temperature T and calculate the specific heat of the Fermi gas at this temperature.

State Maxwell's distribution law of molecular speeds. Draw and explain a curve between n(c) and c in a gas at a given temperature T, where n(c) dc is the number of molecules having speed between c and c + dc. Discuss the effect of T and mass m of the molecule on the nature of the curve.

Prove the thermodynamic relation : \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V and hence show that \frac{\mathrm{d}P}{\mathrm{d}T} = \frac{L}{T (V_2 - V_1)} ; all the terms have their usual meanings.

Describe neutron star on the basis of Fermi-Dirac statistics and obtain the condition of critical mass for a neutron star.

Write and explain the Maxwell-Boltzmann distribution. Using this distribution, find the expressions for the most probable speed, mean speed and root-mean-square speed.

Write down the physical significance of Maxwell's equations and explain the concept of displacement current by using a proper example.

Write down Stefan-Boltzmann law of radiation and derive it from Planck's law of radiation. An aluminium foil of relative emittance 0.1 is placed between two concentric spheres (assumed perfectly black) at temperatures 300\,\mathrm{K} and 200\,\mathrm{K} respectively. Find the temperature of the foil once the steady state is reached.

Discuss briefly the features of 'guard rings'. The plates of a capacitor are square-shaped, each of side l. The plates are inclined at an angle \alpha to each other. The smallest distance between the plates is a. Calculate the capacitance when \alpha is small.

Why does a soap bubble expand upon electrification? A sphere of radius R contains a charge +Q and a charge -Q distributed uniformly in the upper and lower hemispheres respectively. Show that the dipole moment of charge distribution is \dfrac{3}{4}QR\hat{k}, where \hat{k} is directed along the polar axis of the spherical coordinate system.

A current i(t)=(2e^{-t}-e^{-2t})\,\mu\mathrm{A} charges up a 120\,\mathrm{nF} capacitor for a period of 2 seconds. If the final voltage across the capacitor is 15\,\mathrm{V}, what was the initial voltage across it?

How large an inductance needs to be connected in series with a 120 V, 60 W lightbulb if it is to operate normally when the combination is connected across a 240 V, 60 Hz supply?

Discuss the principle of 'artificial dielectric'. Where do you find its use?

When a person carrying something metallic walks through the doorway of a metal detector, it emits a sound. Explain the reason behind it. A 200\Omega resistor and a 15\mu\mathrm{F} capacitor are connected in series to 220 V, 50 Hz a.c. supply. Calculate the current in the circuit and the r.m.s. voltage across the resistor and the capacitor. Is the algebraic sum of these voltages more than the supply voltage? If yes, resolve the paradox.

A charge q=2\,\mu\mathrm{C} is placed at a=10\,\mathrm{cm} from an infinite grounded conducting plane sheet. Find the (i) total charge induced on the sheet, (ii) force on the charge q and (iii) total work required to remove the charge slowly to an infinite distance from the plane.

Write down the electromagnetic wave equations in non-conducting dielectric medium. Hence show that the velocity of wave propagation is given by v=\sqrt{\frac{1}{\varepsilon\mu}}, where the symbols have their usual meanings.

(i) Using Maxwell's equations, obtain the relation \frac{1}{c} \frac{\partial}{\partial t} \left( \frac{E^2 + B^2}{2} \right) + \vec{\nabla} \cdot (\vec{E} \times \vec{B}) = 0 (ii) What is Poynting vector ? Deduce Poynting theorem for the flow of energy in an electromagnetic field.

(i) State Faraday's law of electromagnetic induction and prove that it can be expressed in the following vector form : \mathrm{Curl}~\vec{E} = -\frac{\partial \vec{B}}{\partial t} with \vec{E} and \vec{B} being the electric and magnetic fields. (ii) A coil of 10 turns has dimension 9~\text{cm} \times 7~\text{cm}. It rotates at the rate of 15\pi~\text{rad/sec} in a uniform field whose flux density is 0\cdot 6~\text{weber/m}^2. What is the maximum e.m.f. induced in the coil ?

Discuss the reflection and refraction of plane electromagnetic waves at plane dielectric boundaries for normal incidence and also find the reflection and transmission coefficients.

Write down the four Maxwell's equations and explain the contribution of Maxwell in the development of these equations.