UPSC Physics Optional PYQs (CSE & IFoS)PYQs

A piston-cylinder device initially contains air at 150\text{ kPa} and 27^\circ\text{C}. At this state, the piston is resting on a pair of stops, as shown in the figure, and the enclosed volume is 400\text{ L}. The mass of the piston is such that a 350\text{ kPa} pressure is required to move it. The air is now heated until the volume is doubled. Determine : (i) the final temperature, (ii) the work done by the air, and (iii) the total heat transferred to air. Given : U_{300\text{ K}} = 214\text{ kJ/kg} and U_{\text{final}} = 1113\text{ kJ/kg} Gas constant of air, R = 0\cdot287\text{ kPa.m}^3/\text{kg.K}

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

What is Lenz's law? Is Lenz's law the same as Faraday's law? What happens if Lenz's law is reversed? Explain.

Write down Stefan-Boltzmann law of radiation and derive it from Planck's law of radiation.

Assuming that the Sun radiates like a perfect blackbody, at what wavelength does the peak of the solar spectrum occur? Given that the surface temperature of the Sun is about 6000\text{ K}.

The wire AB shown in the figure below has mass m, resistance r and it can slide on the smooth horizontal parallel rails separated by a distance l. A uniform magnetic field B exists in the rectangular region and a resistance R connects the rails outside the field region. At t = 0, the wire AB is pushed towards right with a speed v_0.

Find out the current in the loop at an instant when the speed of the wire is v. Find out the velocity v as a function of x. (Assume that the resistance of the rails is negligible)

If the electrostatic potential in spherical polar coordinate is \phi(r) = \phi_0 e^{-r/r_0} where \phi_0 and r_0 are constants, what will be the charge density at a distance r = r_0?

A neutral atom consists of a point nucleus +q surrounded by a uniformly charged spherical cloud (-q) of radius r. Show that when such an atom is placed in a weak external electric field \vec{E}, the atomic polarizability of the atom is proportional to the volume of the sphere.

A circular ring of radius R lying on the x-y plane and centred at the origin, carries a uniform line charge \lambda. Find the first three terms (monopole, dipole and quadrupole) of the multipole expansion of potential V(r, \theta).

Two charges Q_1 = 3\text{ nC} and Q_2 = 4\text{ nC} are placed at the cartesian points (0, 2, 2)\text{ m} and (0, -2, 4)\text{ m}, respectively. The z = 0 plane is connected to the ground. Calculate the electric potential and the electric field at the point (3, 2, 4)\text{ m} using the method of images.

Two inductors having inductances L_1 and L_2 are connected in parallel. The inductors have a mutual inductance M. Derive the expression for the effective inductance. Assume the inductors have negligible resistances.

A spherical shell of radius R, carrying a uniform surface charge \sigma, is set spinning at angular velocity \omega about its axis. Find the vector potential it produces at point \vec{r}.

Show that Continuity equation is embedded in Maxwell's equations.

Explain Ampere's circuital law of magnetic field. What modifications Maxwell made in this equation to derive Maxwell's fourth equation?

Find the energy stored in a system of four charges Q_1 = 1\text{ nC}, Q_2 = 2\text{ nC}, Q_3 = 3\text{ nC} and Q_4 = 4\text{ nC} placed at the cartesian coordinates R_1(1, 1), R_2(2, 1), R_3(1, 4) and R_4(2, 2), respectively. Assume free space.

A material has \sigma = 10^{-2}\text{ S/m} and \varepsilon = 2\varepsilon_0. At what frequency, the conduction current would be equal to the displacement current?

Derive the expression for the inductance per unit length of two long parallel wires each of radius a, separated by distance d from their axes and carrying equal and opposite current I.

An infinitely long coaxial cylinder structure has an inner conductor of radius a and an outer conductor of radius b. The inner conductor is attached to V_0, while the outer one is grounded. Calculate the field \vec{E} in between a and b.

Explain the formation of spectra by a plane diffraction grating. What particular spectra would be absent if the width of the slit is equal to the width of the opaque spacing between any two consecutive slits?

A length of 25 cm of a solution containing 50 gm of solute per litre causes a rotation of the plane of polarization of light by 5^\circ. Find the rotation of the plane of polarization by a length of 75 cm of the solution containing 100 gm of solute per litre.