Derive Euler's equations of motion for a rigid body rotating about a fixed point under the action of a torque. When a rigid body is not subjected to any net torque, write down Euler's equations of motion of the body with one point fixed.
What is centre of mass ? Show that the total linear momentum of a system of particles about the centre of mass is zero.
Derive an equation of motion for a variable mass. Explain how it is applied in the motion of a rocket.
(ii)
An In As semiconductor sample is cut in the form of a small bar of size 1 \cdot 0 \text{ cm} \times 1 \cdot 0 \text{ cm} \times 2 \cdot 0 \text{ mm}. Its lengthwise resistance is 1 \cdot 25 \ \Omega. A Hall field of 1 \cdot 7 \text{ V/m} develops when a current of 0 \cdot 12 \text{ A} is passed lengthwise and a magnetic field of 0 \cdot 05 \text{ T} is applied normal to its length as shown above. Calculate the carrier density. (iii) Determines the reciprocal lattice vectors of an fcc lattice.
Define (i) input bias current, (ii) input offset current, (iii) input offset voltage, (iv) output offset voltage, (v) power supply rejection ratio and (vi) slew rate for an OPAMP.
Draw the logic circuit for the following Boolean expression : Y = \overline{A + B} + \overline{C} What are the 'Y' values for the input combinations : (1) 1, 1, 0; (2) 1, 0, 1; (3) 0, 0, 1 ? (ii)
Define \alpha and \beta parameters of a transistor. A germanium transistor with \beta = 45 is biased as shown above. Calculate the value of R_b.
Discuss the motion of an electron in one-dimensional periodic potential and show that it leads to formation of bands of allowed and forbidden states in the electron energy spectrum. How are the insulators, semi-conductors and conductors discriminated on the basis of band structure ?
(i) Discuss briefly the concept of effective mass in semiconductors and explain the significance of negative effective mass. (ii) In isolated atoms, the electrons have discrete and definite energies but in solids they have bands of energies. Explain why. (iii) A proton, deuteron and \alpha-particle accelerated through the same potential difference, on entering a region of uniform magnetic field applied normal to their plane of motion, move in circular orbits. Compare the radii of the orbits described by them.
Discuss the functioning of the above oscillator circuit. Obtain the condition for maintenance of oscillations and the expression for the frequency assuming that the resistances of the inductors are negligible.
(i) Differentiate between K-capture and inverse \beta-decay. (ii) Explain what is internal conversion and how it differs from \beta-decay.
(i) What are the basic interactions in nature ? Give one example for each. Compare their relative strengths and ranges. (ii) What are the conservation laws for strong, electromagnetic and weak interactions ? (iii) Distinguish between particle and anti-particle. How was positron discovered ?
(i) State the quantum numbers I_z, Y and S for the uds quarks and antiquarks. What combination of these leads to the formation of (a) proton and (b) neutron ?
Describe how parity violation was experimentally detected in {}^{60}\text{Co} \beta-decay. How was the observed asymmetry in the distribution of emitted electrons explained ?
Calculate the spin and parities of the ground states of {}_2^4\text{He} and {}_{30}^{67}\text{Zn} nuclei.
(i) Write the Weizacker mass formula and explain the significance of various terms. (ii) Determine the amount of {}_{84}^{210}\text{Po} necessary to provide a source of \alpha-particles of strength 5 \text{ mC}_i. The half-life of {}_{84}^{210}\text{Po} is 138 \text{ d}.
(i) How many lines occur in the multiplet arising from {}^{2}\text{P}_{3/2} \rightarrow {^{2}\text{S}_{1/2}} and {}^{2}\text{P}_{1/2} \rightarrow {^{2}\text{S}_{1/2}} transitions of alkali metal atoms placed in a weak magnetic field, why ? (ii) The wavefunction of a particle confined in a cube of volume L^3 is given by \Psi (x,y,z) = \left( \frac{2}{L} \right)^{3/2} \sin \frac{\pi x}{L} \sin \frac{\pi y}{L} \sin \frac{\pi z}{L} Calculate the average values of p_x and p_x^2 in the region 0 < x < L.
Describe Stern-Gerlach experiment and discuss its implications.
(i) Explain the molecular phenomenon of spontaneous emission between two electronic states of the same multiplicity and differentiate it form that of different multiplicity. (ii) The constants B and V_0 for a KCl molecule have values 1 \cdot 43 \times 10^{-5} \text{ eV} 8 \cdot 4 \times 10^{12} \text{ s}^{-1} respectively. Determine the number of rotational levels between the vibrational levels v = 0 and v = 1.
Obtain an expression for the vibration-rotation energy levels of a diatomic molecule in a given electronic state. The wave numbers of the vibrational transitions occuring in HF, HCl and HI molecules are 4141 \cdot 3 \text{ cm}^{-1}, 2988 \cdot 9 \text{ cm}^{-1} and 2309 \cdot 5 \text{ cm}^{-1} respectively. Compare the force constants of these three molecules.
Set up the time-independent Schrodinger equation for an electron moving in Coulomb field, V(r) = \frac{Ze^2}{4 \pi \varepsilon_0 r}, in polar coordinates. Solve the radial equation to get the energy eigen values.
