UPSC Physics Optional PYQs (CSE & IFoS)PYQs

State Fermat's principle. Using this principle, establish the Snell's law of refraction.

A tuning fork A produces 6 beats per second with another tuning fork B of frequency 508\text{ Hz}. When the prongs of A are loaded with a little wax, number of beats heard are 4 per second. Find the frequency of A before it was loaded with wax.

The coherence length and the bandwidth of a light source is 3\text{ km} and 1 \times 10^{-5}\text{ nm}, respectively. Determine its wavelength and the pulse duration.

In a Newton's ring experiment, explain the following :

(i) Why are the fringes circular ?

(ii) Why is the central spot dark as seen by reflection ?

(iii) How can one obtain Newton's ring with bright centre ?

What do you understand by missing orders in a double-slit diffraction pattern ? Obtain condition for missing orders in a double-slit Fraunhofer diffraction pattern, if the slit width and the separation between the slits are equal. In a double-slit diffraction experiment, if the slit widths are 0\cdot 20\text{ mm} and they are 0\cdot 10\text{ mm} apart, find the missing orders.

A transmission hologram is recorded in a medium of refractive index 1\cdot 15. If the spacing between the two consecutive maxima is 557\text{ nm} and the angle of incidence of reference and object waves from the reference axis (say z-direction) is 30^\circ, find the wavelength of the light source.

Discuss the phenomenon of birefringence in a crystal. What do you understand by positive and negative crystals ? If the refractive indices of a calcite crystal for E-ray and O-ray are 1\cdot 488 and 1\cdot 658, respectively, then calculate the thickness of the crystal for which it can work as

(i) quarter wave plate,

(ii) half wave plate. (Wavelength of the light is 5890\text{ \AA})

A small block of mass m slides without friction down a wedge-shaped block of mass M and opening angle \alpha as depicted below.

The wedge-shaped block itself slides along the horizontal frictionless floor along the +\text{ve x-direction}. Find the horizontal acceleration \ddot{\text{X}} = \text{d}^2\text{X}/\text{dt}^2 of the wedge-shaped block by the Lagrangian method and using the displacements \text{X} and \text{S} of the blocks as generalized coordinates.

Consider the motion of a particle of mass m in central force field \text{f(r)} = -\,\text{k}/\text{r}^2, where k is a constant and r is the distance from the force centre to the particle. Show that for the given system the vector \vec{\text{A}} = \vec{\text{p}} \times \vec{\text{L}} - \text{mk} \, \frac{\vec{\text{r}}}{\text{r}} is conserved. Using this vector and the fact that \vec{\text{A}} \cdot \vec{\text{L}} = 0, derive the orbit equation for the particle.

Consider a particle of mass m moving in a potential field of the form \text{V}(\vec{\text{r}}) = \text{V}_0(\text{x}^2 + \text{y}^2), where \text{V}_0 is a constant. What are the conserved physical quantities for the particle ?

Consider the force-free motion of a symmetrical rigid body with z-axis as its axis of symmetry. Write down Euler's equations. Integrate them and obtain the solutions. On the basis of the obtained solutions show that the total angular velocity \vec{\omega} of such a rigid body is constant in magnitude and precesses about the z-axis with a constant frequency, say \Omega. Determine \Omega.

Two spaceships are approaching each other as depicted below.

Spaceship 1 is moving with a velocity \vec{\text{u}}_1 = 0\cdot 7\text{c }\hat{\text{x}}, while spaceship 2 is moving with a velocity \vec{\text{u}}_2 = -\,0\cdot 8\text{c }\hat{\text{x}} as measured by the stationary observer on Earth. Here, c is the velocity of light in vacuum and \hat{\text{x}} is the unit vector along the x-axis.

(i) What, according to Galilean addition of velocity, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c) ?

(ii) What, according to relativistic addition of velocities, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c) ?

(iii) Spaceship 1 emits a light signal towards spaceship 2. The observer on spaceship 2 measures the frequency of the light signal to be 10^{16}\text{ Hz}. What was the original frequency of the light signal ?

A particle of rest mass \text{m}_0 is moving in a stationary frame K with a velocity \vec{\text{u}} making an angle \theta with the x-axis as shown in the figure.

Find the corresponding angle \theta' made by the velocity vector of the particle with the \text{x}' axis in the frame \text{K}' that is moving with respect to K along the x-direction at a constant speed V.

When is the channel of a JFET said to be pinched off ? Define the pinch-off voltage. Give the relationship between the pinch-off voltage, the saturation voltage and the gate-source voltage. Sketch the depletion region before and after pinch-off.

When a voltage v_1 = + 40 \mu\text{V} is applied to the non-inverting input terminal and a voltage v_2 = - 40 \mu\text{V} is applied to the inverting input terminal of an Op-Amp, an output voltage v_o = 100\text{ mV} is obtained. But, when v_1 = v_2 = + 40 \mu\text{V} is applied, one obtains v_o = 0\cdot 4\text{ mV}. Calculate the voltage gains for the difference and the common-mode signals, and the common-mode rejection ratio.

Implement the Boolean function X = AB + \bar{A}C using NAND gates.

Show that the reciprocal lattice for a ‘bcc’ lattice is a ‘fcc’ structure.

Draw the circuit diagram of a transistor operating in the common-base configuration and sketch its input and output characteristics. Indicate the active, cut-off and saturation regions.

Explain the high temperature superconductivity. What are different properties of high temperature superconductors ?

Calculate the mass of deuterium nucleus if the binding energy per nucleon is 1\text{ MeV}.