UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Explain the Rayleigh Scattering of light. Show that the energy density of light scattered from an isotropic homogenous medium of a gas is inversely proportional to the fourth power of the wavelength of the incident light.

Obtain the solution of the Laplace equation in cylindrical coordinates.

In an a.c. circuit, a resistance (R = 100\ \Omega) and a capacitance (C = 100\ \mu\text{F}) in series are connected to an a.c. source V = 200 \sin(100\ \pi t). Calculate the current through the circuit and the voltages across R and C. Draw a vector diagram representing the magnitudes and phases of the voltages.

Derive Poisson equation starting from the Coulomb's law for a set of point charges.

What is a magnetic shell ? Define the strength of a magnetic shell. A 2\text{ mm} thick magnetic shell weighing 100\text{ gm} has magnetic moment of 1000\text{ units}. The density of the shell material is 10\text{ gm/cc}. Calculate the intensity of magnetisation and the strength of the shell.

Why does a spinning nucleus process in a magnetic field ? Explain the underlying principle of nuclear magnetic resonance (NMR) spectroscopy. Calculate the radio frequency at which NMR occurs in water kept in a uniform magnetic field of 2.4 T, the magnetic moment of proton being 2.793\\ \mu_N.

State Rayleigh-Jeans law. Show that the intensity of emissions at a particular wave-length is proportional to the temperature T. Discuss the limitations of this law in describing the intensity distribution of emission spectrum of a blackbody.

Describe the working of a Fabry-Perot interferometer. Determine the intensity of the fringes of the transmitted light. Why the fringes obtained in the Fabry-Perot interferometer are comparatively sharper than those obtained from the Michelson interferometer ?

Establish the relationship between the phase velocity V_p and the group velocity V_g of waves. Under what physical conditions V_g < V_p, V_g = V and V_g > V_p can be possible ?

Consider superposition of two plane polarized electromagnetic waves: E_y = \hat{y} a \cos(kx - \omega t) \text{ and} E_z = \hat{z} b \cos(kx - \omega t + \phi) Discuss the conditions for the resultant wave to be left circularly and right circularly polarized adopting the convention as seen by an observer travelling with the wave.

Define resolving power and dispersive power of a grating. Two spectral lines of wavelengths 500\text{ nm} and 500.5\text{ nm} are seen clearly resolved in second order spectrum of a grating. If the grating has 250 lines per cm, what should be the minimum width of the grating?

The radius of the first zone in a zone plate is 2\text{ mm}. What will be the position of the first image of a point source of light of wavelength \lambda = 500\text{ nm} placed at a distance of 5\text{ m} from the zone plate ?

What are chromatic and spherical aberrations Explain. Calculate the longitudinal aberration for rays at a height of 0.01\text{ m} from the principal axis and incident on the cowed surface (radius of curvature = 0.1\text{ m}) of a thin piano-convex lens made of glass (refractive index = 1.5).

Derive the relation \vec{v}_0 = \vec{v} + \vec{\omega} \times \vec{r}, where \vec{v}_0 is the velocity of a particle located at \vec{r} in a fixed frame of reference S and \vec{v} that as observed in a frame S' rotating with angular velocity \vec{\omega} with respect to S but having the common origin.

Write down the Lagrangian of a free particle in rectangular cartesian coordinates. Identify all the cyclic coordinates. Show that the constants of motion obtained from the equations for the considered form of the Lagrangian are not exactly the same as those which follow from the concept of a free particle.

State the postulates of the special theory of relativity and based on these obtain Lorentz as well as inverse Lorentz transformations. Hence, obtain an expression to conclude that a moving clock runs more slowly than a stationary clock.

A body of rest mass m_0 is moving in the positive direction at a velocity of 0.6\text{ C} relative to the laboratory frame. Calculate the components of the four dimensional momentum vector in the laboratory frame and in the frame of an observer who is travelling in the positive x-direction at a speed of 0.8\text{ C} relative to the laboratory frame.

The angular momentum \vec{M} of a rigid body comprising of N particles and rotating with angular velocity \vec{\omega} is given by : \vec{M} = \sum_{k=1}^{N} m_k \vec{r}_k \times (\vec{\omega} \times \vec{r}_k) where the origin coincides with the centre of mass. Express the components of \vec{M} in terms of components of the inertia tensor. Hence, show that the most general free rotation of a spherical top is a uniform rotation about an axis fixed in space.

An unstable particle has a lifetime of 5\text{ }\mu\text{s} in its own frame of reference and is moving towards the earth at a speed of 0.8\text{ C}. What will be the lifetime of the particle to an observer on the earth ?

Show, using the above relation, that the equation of motion of the particle in S gets modified in S' giving rise to various fictitious forces. Identify the coriolis force and describe its effect on the flow of rivers.