UPSC Physics Optional PYQs (CSE & IFoS)PYQs

What do you mean by spherical aberration of a lens? Show that if two plano-convex lenses are kept at a distance equal to the difference of their focal lengths, the spherical aberration would be minimum.

Distinguish between positive and negative crystals in terms of double refraction. How are these crystals used to make quarter wave plates? Explain how the quarter wave plate is used in producing elliptically and circularly polarized light.

When the two waves of nearly equal frequencies interfere, then show that the number of beats produced per second is equal to the difference of their frequencies.

(i) How are linearly polarised and circularly polarised lights generated from unpolarised light and are analysed?

(ii) A calcite crystal is to be used for producing a quarter wave plate. The refractive indices of the crystal for ordinary ray and extraordinary ray are 1\cdot 65836 and 1\cdot 48641 respectively. What is the thickness of the plate required if the light of wavelength 632\cdot 8\text{ nm} is to be used?

A particle of mass m is constrained to move on a plane curve xy=c with c>0, under gravity, where y-axis is vertical. Construct the Lagrangian of the system and obtain the Euler-Lagrange equation of motion.

A particle moving in a central force field located at r=0, describes a spiral, r=e^{-\theta}. Find the force law.

Consider a homogeneous cube of total mass M and side a. Taking the origin at one corner of the cube and axes along the edges of the cube, construct the moment of inertia tensor. Calculate principal moment of inertia.

Show that the wave equation for the propagation of electromagnetic scalar potential \phi(x, y, z, t) \left[\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}\right] \phi(x, y, z, t) = 0 remains invariant under Lorentz transformation.

Calculate the first relativistic correction to the kinetic energy of a particle with rest mass m_0 and speed v.

A rod of length l_0 is kept at rest in x'\,y' plane of its rest frame making an angle \theta_0 with x' axis. What is the length and orientation of the rod in a laboratory frame (x,y) in which the rod moves to the right with velocity v?

(i) If a particle of mass m is in a central force field f(r)\hat{r}, then show that its path must be a plane curve, where \hat{r} is a unit vector in the direction of position vector \vec{r}.

(ii) A block of mass m having negligible dimension is sliding freely in x-direction with velocity \vec{v}=v\hat{\imath} as shown in the diagram.

What is its angular momentum \vec{L}_O about origin O and its angular momentum \vec{L}_A about the point A on y-axis?

A rod of length L has non-uniform linear mass density (mass per unit length) \lambda, which varies as \lambda=\lambda_0\left(\dfrac{S}{L}\right); where \lambda_0 is a constant and S is the distance from the end marked `O' (as shown in the figure). Find the centre of mass of the rod.

Use Gauss's theorem to calculate the gravitational potential due to a solid sphere at a point outside the sphere. Calculate the amount of work required to send a body of mass m from the Earth's surface to a height R/2, where R is the radius of the Earth.

Two capillary tubes of lengths 2l and l with internal radii r and 2r respectively are connected in series. Water flows through them in streamline. If the pressure difference across the first capillary is P, find the pressure difference across the second one.

In special theory of relativity, the Hamiltonian of a free particle with rest mass m_0 is given by \sqrt{p^2 c^2 + m_0^2 c^4}. Obtain the Lagrangian of the system using Legendre transformation.

A water drop of radius 0.04\ \mathrm{mm} is falling through air. If the coefficient of viscosity for air is 1.8 \times 10^{-4} poise, find its terminal velocity. If 100 such drops coalesce, what will be the new terminal velocity?

What is Meissner effect ? Explain perfect diamagnetism in reference to this.

What do you understand by spontaneous magnetization below Curie temperature ? Explain with an appropriate diagram, the occurrence of a hysteresis loop in a ferromagnetic material.

Determine the values of emitter current and the collector current of the transistor having \alpha = 0\cdot98 and collector to base leakage current I_\text{CBO} = 4\ \mu\text{A}. The base current is 50\ \mu\text{A}.

Explain with the help of a neat diagram the working of an RC coupled common emitter amplifier.