UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Consider a Gaussian pulse propagating in pure silica, at \lambda_0 = 0\cdot 85\\ \mu\text{m}, along z-direction. As the pulse propagates, it gets broadened according to the formula \tau^2(z) = \tau_0^2 \left( 1 + \frac{4 \alpha^2 z^2}{\tau_0^4} \right), \quad \alpha = \left. \frac{d^2 k}{d \omega^2} \right|_{\omega=\omega_0} where \tau(z) is the pulse width after propagation through a distance z and \tau_0 is the pulse width at z = 0. Given that \alpha = 3 \times 10^{-26}\\ \frac{\text{S}^2}{\text{m}} at \lambda_0 = 0\cdot 85\\ \mu\text{m}. Calculate the distance at which \tau(z) = \sqrt{2} \cdot \tau_0.

Distinguish between the intensity patterns due to diffraction from a narrow slit and a straight edge.

Explain the diffraction at straight edge with the help of Cornu's spiral.

A diffraction grating has 5000\text{ lines per cm}. For illumination at normal incidence, determine the dispersive power of the grating in the second order spectrum in the range of wavelengths around 500\text{ nm}.

In a double slit experiment the slits are 2\text{ mm} apart and are illuminated with a mixture of two wavelengths, \lambda = 750\text{ nm} and \lambda' = 900\text{ nm}. At what minimum distance from the common central bright fringe on a screen 2\text{ m} distant from the slits will a bright fringe from one interference pattern coincide with a bright fringe from the other ?

What is Fermat's principle ? Derive the laws of reflection and refraction with the help of this principle, when the light is incident on a curved surface.

A plano-convex lens of crown glass is connected to a concavo-convex lens of flint glass so that the convex surface of the first and the concave surface of the second fit exactly. If the combination is achromatic and the combined focal length is -50\text{ cm}, determine the radii of curvature of the faces of the flint glass lens. The dispersive power of the crown glass is one-half of that of the flint glass and the refractive indices for yellow light are respectively 1\cdot 500 and 1\cdot 625 respectively.

In the steady state forced vibration of the damped harmonic oscillator, show that the amplitude of the driven system is maximum when p = \sqrt{\omega^2 - 2b^2} and further the value of the maximum amplitude is \frac{f}{2b \sqrt{\omega^2 - b^2}}, where the symbols have their usual significance. (Here you start from the steady state solution)

A light spring of relaxed length 'a_0' is suspended from a point. It carries a mass 'm' at its lower free end, which stretches it through a distance l. Show that the vertical oscillation of the system is simple harmonic in nature and has time period T = 2\pi \sqrt{l / g}, where g is the acceleration due to gravity.

The principal moments of inertia of a body at a point are given as 200, 300 and 450\text{ gm.cm}^2. Write down the equation of the ellipsoid of inertia at that point.

A rigid body rotates with the angular velocity '\omega' about an axis through the origin O and having direction cosines l, m, n. Show that the moment of inertia of the rigid body about the axis is I = I_{xx} l^2 + I_{yy} m^2 + I_{zz} n^2 + 2 I_{xy} l . m + 2 I_{yz} m . n + 2 I_{zx} n . l where the symbols have their usual meanings.

A particle of mass 'm' moves according to the equations x = a \cos \omega t, y = a \sin \omega t, z = c, where a, c, \omega are constants. Obtain the instantaneous velocity and linear momentum vectors in terms of the Cartesian components and hence the angular momentum. Find the force \vec{F} and the torque \vec{N} acting on the particle and verify that the angular momentum \vec{L} and the torque \vec{N} satisfy the relation \frac{d\vec{L}}{dt} = \vec{N}.

A particle of rest mass 'M' moving at a velocity 'u' collides with a stationary particle of rest mass 'm'. If the particles stick together, show that the speed of the composite ball is equal to v = \frac{u \gamma M}{(\gamma M + m)}, \quad \text{where } \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}}

Describe the essential parts of an oscillator circuit using a block diagram. Design a Colpitts oscillator set to a frequency of 1500\text{ Hz} and explain how your design incorporates the essential parts of the oscillator circuit.

What are the advantages of MOSFET over JFET ? Draw and explain the structure and working of a MOSFET which can function both in enhancement and depletion modes.

Using the collector characteristics of a bipolar junction transistor, show how you can use it as a switch with one ON and one OFF state. Plot and describe the operation of this circuit for a sinusoidal input.

What is a Josephson junction ? Explain the phenomenon of tunneling of charge carriers across such a junction. Discuss DC and AC Josephson effects. Mention some applications of this device.

Show the variation of magnetization with applied magnetic field for type I and type II superconductors. Describe their behaviour. Discuss their potential for various applications.

In one experiment, a perfect conductor and a superconductor are cooled to below 1\text{ K} and then a magnetic field is applied. In another experiment, they are cooled to below 1\text{ K} in magnetic field. Describe their responses in these two experiments and explain.

Discuss the motion of an electron in an allowed energy band under the influence of an external electric field. Obtain an expression for the effective mass of the electron and discuss its variation in the band.