UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Solve the Schrödinger equation for a particle in a three-dimensional rectangular potential barrier. Explain the terms degenerate and non-degenerate states in this context.

A particle trapped in an infinitely deep square well of width a has a wave function \psi=\left(\frac{2}{a}\right)^{1/2}\sin\left(\frac{\pi x}{a}\right). The walls are suddenly separated by infinite distance. Find the probability of the particle having momentum between p and p+dp.

Normalized wave function of a particle is given: \psi(x)=N\exp\left(-\frac{x^2}{2a^2}+ikx\right). Find the expectation value of position.

A Van der Waals gas undergoes Joule-Kelvin expansion with a pressure drop of 50\ \mathrm{atm}. If its initial temperature is 300^{\circ}\mathrm{K}, determine its final temperature. (Given Van der Waals constant a=0.136\ \mathrm{Pa\,m^6\,mol^{-1}}, b=36.5\times10^{-6}\ \mathrm{m^3\,mol^{-1}}, C_p=30\ \mathrm{J\,K^{-1}\,mol^{-1}}, R=8.3\ \mathrm{J\,K^{-1}\,mol^{-1}}.)

For a Van der Waals gas, write down the equation of state. Determine the coefficient of critical expansion \beta.

The vapour pressure of an organic substance is 50\times10^3\ \mathrm{Pa} at 40^{\circ}\mathrm{C}. Its normal boiling point is 80^{\circ}\mathrm{C}. If the substance in vapour phase can be treated like an ideal gas, find the latent heat of vaporization of the substance.

Derive the equation that represents Poynting's theorem. What is its physical significance?

What are the characteristic features of Rayleigh scattering? A very thin monochromatic beam of light is incident on a particle. Suggest a simple experimental method to ascertain whether the scattering by the particle is of Rayleigh type.

In the circuit given below, find the values of currents I_1, I_2 and I.

A radio station transmits electromagnetic waves isotropically with an average power of 200\,\mathrm{kW}. Determine the average magnitude of the maximum electric field at a distance of 5\,\mathrm{km} from it.

Two spheres A and B having same temperature T are kept in the surroundings of temperature T_0. Consider T > T_0. The spheres are made of same material but have different, radii r_A and r_B. Using Stefan - Boltzmann distribution, determine which of these will lose heat by radiation faster.

Under one-dimensional configuration, the charge density is given by \rho(x)=\dfrac{\rho_0x}{5}, where \rho_0 is a constant charge density. If the electric field \lvert\vec{E}\rvert=0 at x=0 and potential V=0 at x=5, determine V and \lvert\vec{E}\rvert.

A plane electromagnetic wave propagating along +\hat{z} direction is incident normally on the boundary at z=0 between medium A(z<0) and medium B(z>0). Determine the reflection coefficient and transmission coefficient for the wave.

A conducting sphere of radius 5\,\mathrm{cm} has a total charge of 12\,\mathrm{nC} uniformly distributed on its surface in free space. Determine the displacement vector \vec{D} on its surface and outside at a distance r from the centre of the sphere.

A series RLC circuit has R=2\Omega. The energy stored in the circuit decreases by 1\% per period of oscillation. Its natural undamped frequency is 2\,\mathrm{kHz}. Determine the values of inductor L and the quality factor.

Using Planck's radiation law, deduce Wien's displacement law. How does this law enable one to estimate the surface temperature of the Sun or a star?

A series RLC circuit has a resistance of 100\Omega and an impedance of 210\Omega. If this circuit is connected to an a.c. source with an r.m.s. voltage of 220\ \mathrm{V}, how much is the average power dissipated in the circuit?

A convex lens of focal length 20\,\mathrm{cm} is placed after a slit of width 0.5\,\mathrm{mm}. If a plane wave of wavelength 5000\,\mathring{\mathrm{A}} falls normally on the slit, calculate the separation between the second minima on either side of the central maximum.

Using the concept of Einstein’s A and B coefficients for a two-level atomic system under thermal equilibrium, determine the ratio of the number of atoms per unit volume in the two levels experiencing spontaneous and stimulated emission. How does the principle of population inversion lead to the gain mechanism in the active medium of the laser?

Find out the phase and group velocities of a radio wave of frequency \omega = \sqrt{2}\omega_p in the ionosphere (as a dielectric medium) of refractive index n = \sqrt{1 - \frac{\omega_p^2}{\omega^2}}. Here, \omega_p is the ionospheric plasma frequency.