Using Rutherford's observation that the number of \alpha particles scattered at an angle \phi and falling on unit area of the screen varied as (\text{cosec }(\phi/2))^4, deduce an expression for the probability of scattering between angles \phi and \phi + d\phi.

Calculate the speed of a satellite orbiting the planet Jupiter at a distance of 108\text{ km}, above its surface (The mass and radius of Jupiter are 1.9 \times 10^{27}\text{ kg} and 69892\text{ km} respectively.)

Write a short note on Gyroscope and its application.

Write a short note on Inertial forces in a rotating frame.

Write down the expression for the dependence of mass of a particle on its velocity in special relativity. What will be the speed of a particle if its mass becomes double of its rest mass ?

Why do the electronic energy level in a solid form bands? Describe in terms of the band theory the difference between a conductor, an intrinsic semi-conductor and an insulator. How does the energy level change when some impurities are introduced in a semiconductor? Explain how by doping you have both n-type and p-type semiconductors starting with pure Germanium. The band gap in cases of Germanium is about 0.8\text{ eV} at Ok. Neglecting the variation of band gap with temperature, make a rough estimate of the temperature at which the resistance of a pure Germanium sample will be reduced to 1/e times its value at 300k.

$ in keV units if E is in the range 7 meV.

Explain why Light elements like hydrogen and deuterium are effective in shielding against neutrons while heavy elements like lead are effective in case of gamma rays.

Give, in short the principle of working of mass spectrograph (any type). What does it determine? What are isotopes? How many isotopes of hydrogen do you know of? Are they all stable? The minimum energy required for the disintegration of the nucleus of a stable isotope of hydrogen is 2.21\text{ MeV}. Calculate the mass of this nucleus in a.m.u.

Give the quantum numbers of the states whose energy difference accounts for the two D-lines of sodium. Find the energy difference for D_1 and D_2 lines in electron volts. (Wavelength separation = 6\text{\AA}, Mean wavelength = 5893\text{\AA})

How do characteristic X-ray lines originate? Using Bohr's atomic model deduces a relation between the frequencies of the K series X-ray lines and atomic numbers of the emitting elements. Do you expect this relation to be exact?

What are de Broglie waves? State the uncertainty principle and explain its relation with de Broglie waves by considering a free particle.

A photon of energy 10\text{ eV} is scattered by an electron having an initial velocity of 1000\text{ km s}^{-1}. After the scattering the electron velocity is reduced to 200\text{ km s}^{-1}. Deduce the energy of the scattered photon.

What is the value of the spin of the electron and the corresponding magnetic-moment? Can that be considered as arising from the classical case of rotation of a finite rigid charged body about its axis? Give reasons for your answer.

Set up Schrodinger's equation for a free particle confined in a cubical box and find the energy eigenvalues. What form will Pauli principle take for electrons in such a box? Deduce the zero point energy if the length of the box be 10^{-10}\text{ metre} and there be 10 electrons in it. (The coulomb interaction maybe disregarded.)

In a Compton effect experiment with incident X-ray of wavelength 0.060\text{ \AA}, one recoil electron had energy 4100\text{ eV}. Calculate the corresponding angle of scattering and the wavelength of the scattered radiation (Standard formulae may be assumed.)

Write a note on Thermal ionization.

Write a note on Production of low temperatures by adiabatic demagnetization.

Write down the general expression for the Joule-Kelvin effect and define Joule-Kelvin coefficient, \mu. Show that for an ideal gas \mu = 0.

Derive the expression for the specific heat of solid on the Einstein model. Comment on its short-comings.