PYQ Bank

Show that a four-dimensional volume element dx\,dy\,dz\,dt is invariant to Lorentz transformation.

Consider a spherical shell of mass M and radius R. Calculate the potential due to this shell at a point P when the point is (i) outside the shell and (ii) inside the shell (r < R). If the spherical shell is now replaced by a uniform solid sphere of same mass and radius, what will be its potential at the same external point?

Obtain Poiseuille's equation for a viscous fluid flowing through a narrow tube of radius r and length l. If a spherical body of radius a is allowed to move at a speed \vec{V} through the same fluid of viscosity \eta, show that the viscous force will increase with the speed linearly.

Show that the total energy per unit mass of liquid flowing from one point to another without any friction remains constant throughout the displacement.

Show that for any rigid body consisting of at least three particles, not arranged in one straight line, number of independent degrees of freedom is six. Define Euler's angles \theta, \phi and \psi to describe the configuration of such a rigid body. Consider two frames of reference, one fixed to the body and the other to the space defined as S' = (x', y', z') and S = (x, y, z) respectively. Show that the angular momentum (\vec{L}) of the rigid body in the two frames are related by \left( \frac{d\vec{L}}{dt} \right)_S = \left( \frac{d\vec{L}}{dt} \right)_{S'} + \vec{\omega} \times \vec{L} where \vec{\omega} is the angular velocity of rotation.

A certain colpits oscillator uses a tank circuit with L = 20\text{ mH}; C_1 = 200\text{ pf} and C_2 = 300\text{ pf}. What is the frequency of oscillation ?

How can the NAND-gates be combined to perform the OR-operation ?

Distinguish between 'soft' and 'hard' super-conductors. Explain how penetration depth varies with magnetic field strength and temperature.

Explain the switching action of a FET. Two JFETs are connected in parallel with \gamma_{d1} (drain resistance) = 20\text{ k}\Omega, \gamma_{d2} = 30\text{ k}\Omega and g_{m1} (transconductance) = 2\text{ m-mho} and g_{m2} = 4\text{ m-mho}. Find the efffective \gamma_d and g_m.

(i) Transistors are current operated devices, while vacuum tubes are voltage operated." — Explain. (ii) Why are junction transistors called bipolar devices ?

Draw the circuit diagram of a two-stage RC coupled common emitter transistor amplifier. Show how the magnitude and phase of voltage gain vary with frequency. Define bandwidth of this amplifier. An amplifier with open loop voltage gain, A_v = 1000 \pm 100 is available. It is necessary to have an amplifier whose voltage gain varies by no more than \pm 0.1\%. Find the reverse transmission factor \beta of the feedback network used and the gain with feedback.

Show the volt-ampere characteristics of a solar cell under illumination. Explain the characteristics.

Show that FCC-crystals are more closely packed than BCC-crystals,

Which of the following \alpha decay processes is expected to have a large Q value ? {}_{84}^{211}\text{Po}_0 \rightarrow {}_{82}^{207}\text{Po}_0 + \alpha {}_{84}^{212}\text{Po}_0 \rightarrow {}_{82}^{208}\text{Po}_0 + \alpha Which of these processes will have a longer lifetime ? Explain your answer briefly.

Explain lepton number conservation and why it is necessary to distinguish between different types of neutrinos ?

What is nuclear isomerism ? Give two examples and explain how this can be understood from single particle shell model.

What is \beta-decay ? Give three examples of \beta-decay of nuclei. Explain how neutrino hypothesis helped in explaining conservation of energy-momentum and angular momentum. What is Kurie plot ? How can one use Kurie plot to set limits on mass of the neutrino ?

Explain why the decay process n \rightarrow p + e + \bar{\nu} is observed but the decay p \rightarrow n + e^+ + \nu is not observed.

In the Lorentz model for dispersion, the uniform dielectric medium is assumed to be a collection of oscillators with number density N, natural frequency $ \omega_0 $ and decay constant $ \gamma $. Find the complex susceptibility $ \chi $ of such a medium. In the frequency dependence of the real part of $ \chi $ identify the regions of normal and anomalous dispersion.

Give the hypercharge and isospin of the quarks and antiquarks. What are the quark contents of the mesons: \pi^+, \pi^\circ, \pi^-, K^+, K^\circ, K^- \text{ and } \eta ?