UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Calculate the values of van der Waals constants a and b for oxygen with T_c=154.2\,\mathrm{K}, P_c=49.7 atmosphere and R=80\,\mathrm{cm^3\,atmosphere/K}.

Write down the expressions for Bose-Einstein and Fermi-Dirac distribution functions, and show how Fermi-Dirac distribution leads to the explanation of Pauli's exclusion principle.

What led van der Waals to modify the ideal gas equation? Using the concepts of critical temperature T_c, pressure P_c and volume V_c, show that the critical constant for a real gas is 8/3.

Justify which of the four Maxwell's equations imply that there are no magnetic monopoles. How these equations would have been written if they were?

Consider a plane wave travelling along the positive y-direction incident upon a glass of refractive index n=1.6. Find the transmission coefficient.

Consider Maxwell's equation in differential form in media. For j=\rho=0, assume \epsilon=\epsilon_0e^{\alpha t} and \mu=\mu_0e^{\alpha t} and show that the relevant wave equation for a plane wave propagating along x-direction is \frac{\partial^2 E}{\partial x^2}=\mu\frac{\partial^2 D}{\partial t^2}+\mu\alpha\frac{\partial D}{\partial t} where \vec{E}=E\hat{y} and \vec{H}=H\hat{z}.

In deriving the Rayleigh-Jeans law, we count the number of modes dn corresponding to a wave number k for a photon gas in a cubical box. Consider a cubical container of volume V containing such gas in equilibrium. Calculate the differential number of allowable normal modes of frequency \omega.

A resistor R(= 6.2\,\mathrm{M}\Omega) and a capacitor C(= 2.4\,\mu\mathrm{F}) are connected in series and a 12 V battery of negligible internal resistance is connected across their combination.

(i) What is the capacitive time constant of this circuit?

(ii) At what time, after the battery is connected, does the potential difference across the capacitor become 5.6 V?

Determine the torque experienced by an electric dipole of moment \vec{p} if placed in an electric field \vec{E} in nonaligned state. Also show that the interaction energy of two dipoles of moments \vec{p}_1 and \vec{p}_2 separated by a displacement \vec{r} is U=\frac{1}{4\pi\epsilon_0}\frac{1}{r^3}\left[\vec{p}_1\cdot\vec{p}_2-3(\vec{p}_1\cdot\hat{r})(\vec{p}_2\cdot\hat{r})\right] assuming the expression for field due to a dipole.

The electric potential of a grounded conducting sphere of radius a in a uniform electric field is given as \phi(r,\theta)=-E_0r\left[1-\left(\frac{a}{r}\right)^3\cos\theta\right] Find the surface charge distribution.

Find out the total electric potential energy of a single spherical object of uniform charge density \rho, total charge Q and radius R.

Find whether the discharge of a condenser through the inductive circuit is oscillatory when C = 0.1\,\mu\mathrm{F}, L:=10\,\mathrm{mH} and R = 200\,\Omega. If it is oscillatory, calculate its frequency.

The four arms of a Wheatstone bridge have the following resistances: AB = 100\,\Omega, BC = 10\,\Omega, CD = 5\,\Omega, DA = 60\,\Omega A galvanometer of 15\,\Omega resistance is connected across BD. Calculate the current through the galvanometer when a potential difference of 10 volts is maintained across AC.

A long solenoid has 220 turns /cm; its diameter is 3.2 cm. Inside the solenoid at its centre, we place a 130-turn closepacked coil of diameter 2.1 cm along its axis. The current in the solenoid is increased from zero to 1.5 amperes at a steady rate over a period of 0.16 second. What is the magnitude of the induced e.m.f. that appears in the central coil when the current in the solenoid is being changed?

An electrical circuit consists of a resistance R, inductance L and capacitance C in series. If a charge is put on the capacitor at some instant, determine the condition that V_C, the voltage across the capacitor, is subsequently oscillatory. Derive an expression for the quality factor Q of the circuit by considering the decay of the oscillation, using the result that the amplitude falls by a factor of e in \left(\frac{Q}{\pi}\right) period.

Write down the one-dimensional harmonic oscillator differential equation under damping and its solution for the lightly damped condition, with the meanings of symbols. Determine the dependent energy in the lightly damped condition.

Prove that the group velocity V_g of electromagnetic waves in a dispersive medium with refractive index n(\lambda_0) at wavelength \lambda_0 is given by V_g=\frac{c}{n(\lambda_0)-\lambda_0\dfrac{dn(\lambda_0)}{d\lambda_0}} where c is the free space velocity of light. Find the time taken for the electromagnetic pulse to travel a distance D.

A plane wave has the following expression for its electric field: \vec{E}=\hat{x}E_{0x}\cos(\omega t-kz+\alpha)+\hat{y}E_{0y}\cos(\omega t-kz+\beta) If the phase difference is defined as \delta=\beta-\alpha, under what conditions do we achieve elliptic polarization? What are the conditions for circular polarization?

Bring out the essential differences between the physical principles of spontaneous and stimulated emission of radiation. Why is it difficult to get efficient lasing action in case of an ideal two-level material system? Can you propose a scheme to enhance efficiency? Discuss.

Show with proper mathematical analysis that the ratio of Einstein's A and B coefficients depends upon the energy separation between the two energy levels participating in the optical transitions. What is the physical significance of A coefficient? \text{(5 marks)} Justify the statement, ``It is very difficult to develop an X-ray laser''. \text{(5 marks)}