UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Define electromagnetic field strength tensor F_{\mu\nu}. Express it in terms of components of electric and magnetic fields.

Calculate the speed and radius of \alpha-particles of energy 5\cdot 998\text{ MeV} emitted into a magnetic field B = 10^4\text{ Wb m}^{-2}. Given that the mass of proton is 1\cdot 673 \times 10^{-27}\text{ kg}.

Calculate the group and phase velocities for the wave packet corresponding to a relativistic particle.

What are the special features of step-index fibre ? The step-index fibre has core glass and cladding glass with refractive indices 1.68 and 1.5 respectively. Calculate the numerical aperture. Find the critical angle and acceptance angle for entrance of light if the fibre is placed in air.

Consider a damped harmonic oscillator given by the equation \ddot{x} + 2\mu \dot{x} + \omega_0^2 x = 0. Obtain the displacements for (i) overdamped, (ii) underdamped and, (iii) critically damped motion.

(i) Distinguish between interference fringe pattern and diffraction fringe pattern. (ii) In an experiment using Newton's rings, the diameter of 5^{\text{th}} and 15^{\text{th}} rings were found to be 33.6 \times 10^{-4}\text{ m} and 59 \times 10^{-4}\text{ m} respectively. Calculate the radius of curvature of the plano-convex lens if the source of light is sodium light (\lambda = 589.0\text{ nm}). Mention any other applications of Newton's ring experiment.

Explain the phenomenon of double-refraction in calcite crystal with special reference to Ordinary Ray (OR), Extraordinary Ray (ER) and uniaxial negative crystal.

(i) What are the similarities and dissimilarities between a zone plate and a convex lens ? (ii) A monochromatic light of wavelength 5860 \times 10^{-8}\text{ cm} is incident normally on a 2\text{ cm} wide grating. The first order spectrum is produced at an angle of 20^\circ with respect to the normal. Determine the total number of lines on the grating. How does the resolving power of grating depend on the number of rulings on the grating ?

Draw the energy level diagram of He-Ne laser. The metastable state of ruby laser is at 1.786\text{ eV}. Calculate the wavelength of the light emitted.

The moment of inertia tensor of a cube of mass M and side a is given by the matrix I = \frac{M a^2}{12} \begin{bmatrix} 8 & -3 & -3 \\ -3 & 8 & -3 \\ -3 & -3 & 8 \end{bmatrix}. Calculate the principal moments of inertia of the cube.

Write down the expression for kinetic energy of a relativistic particle with rest mass m_0, moving with a speed v. (i) Show that the kinetic energy reduces to \frac{1}{2} m_0 v^2 in the non-relativistic limit. (ii) Find the first relativistic correction to the non-relativistic kinetic energy.

Define generalized coordinates. How many generalized coordinates are required to describe the dynamics of (i) a free rigid body in 3 dimension ? (ii) a solid cylinder rolling in an inclined plane without slipping ?

Define Coriolis force. A particle is dropped from a height h vertically at the northern hemisphere at latitude \theta. Find its deflection from the vertical line due to Coriolis force.

Find the equation of orbit for a particle of mass m, moving in the influence of central force F(r) in terms of u \left(\equiv \frac{1}{r}\right).

The Lagrangian of a system is given by L (\vec{r}, \vec{v}) = \frac{1}{2} m (\vec{v} + \vec{a})^2 + \vec{b} \cdot \vec{v} - \vec{c} \cdot \vec{r} where, \vec{a}, \vec{b} and \vec{c} are constant vectors. Construct the Hamiltonian of the system and find the canonical equations of motion.

Consider two inertial frames S and S'. S is at rest and S' is moving along x - x' direction with constant speed v. (i) Find how different components of momentum four-vector in S and S' frames are related. (ii) Show that p^\mu p_\mu is Lorentz invariant.

Construct the Lagrangian of a simple pendulum whose string is replaced by a spring with spring constant k and rest length l_0 oscillating in a vertical x - y plane. Find the equations of motion.

What will be the frequency of Josephson current, if a d.c. voltage of 4\cdot 0\ \mu\text{V} is applied across a Josephson junction?

Draw the crystal structure of \text{CaF}_2 indicating the positions of cations and anions. What are their respective coordination numbers?

Consider an n-channel JFET which has \text{I}_{\text{DSS}} = 15\text{ mA} and pinch-off voltage \text{V}_\text{p} = -5\text{ V}. If \text{V}_{\text{GS}} = -1\cdot 5\text{ V}, how much will the drain current \text{I}_\text{D} be? What will be the minimum value of \text{V}_{\text{DS}} for pinch-off to occur at \text{V}_{\text{GS}} = -1\cdot 5\text{ V}?