Establish that for an adiabatic process in an ideal gas T V^{\gamma-1} = \text{constant}. where the symbols have their usual meanings,

Write down Maxwell-Boltzmann distribution for the energy of molecules of a gas at temperature T. Find the energy at which this distribution peaks. Compute in eV the mean energy of molecules of a gas at 27^\circ\text{C},

The density of steam at 100^\circ\text{C} is nearly 0.60 \times 10^{-3}\text{ kg/litre} and its latent heat is 2.31 \times 10^6\text{ J/kg}. Calculate the change in the boiling point of water if air pressure changes from 76.0\text{cm} to 70.0\text{cm} of Hg.

One gram of hydrogen gas at 27^\circ\text{C} is compressed isothermally from 100 liters to 25 liters. Calculate the energy needed.

Establish the relation C_p - C_v = T \left( \frac{\partial P}{\partial T} \right)_v \left( \frac{\partial V}{\partial T} \right)_p

Show that the electric field intensity due to any distribution of charges at rest can be expressed as the gradient of a potential. What is the relation between potential and potential energy? A thin disc of radius R is uniformly charged, \sigma being the charge per unit area. Find the potential and the electric intensity at points on the axis of the disc. How do these change as one crosses the disc? Explain the changes physically.

Write down Plank's law of radiation and establish Wien's law from it.

Set up the equation for the discharge of capacitor C connected in series with a resistor R and an inductor L. If R_0 stands for 2(L/C)^{1/2}, discuss three cases: R < R_0, R = R_0 \text{ and } R > R_0 What will you observe if the discharge takes place at low temperature when the material of resistor has become superconducting?

Alpha particles of velocity v enter a uniform magnetic field in perpendicular direction. If electrons of the same velocity v enter the same magnetic field the same way compare: (i) the curvatures of their path and (ii) the tracks they would show in (say) a bubble chamber.

Two charges are placed at a distance of 1 metre. The magnitude of one charge is double that of the second charge. Find the neutral points in the two cases: (i) the charge are of the same sign (ii) the charge are of opposite sign. What happens to the neutral point if the two charges are of equal magnitude and opposite sign?

In certain region of space in vacuo, the components of the magnetic induction are (in weber per square meter) B_x = A e^{-ay} + bx B_y = A c^{-ax} + cy B_z = 0 where x, y, z, are in metres and A, a, b and c are constants. Find the relation, if any between these constants and also the current distribution that gives rise to this field. (No electric field is present). Is the current distribution consistent with the charge conservation principle?

A soap film of refractive index 1.33 is illuminated with light of different wavelength at an angle of 45^\circ. There is complete destructive interference for \lambda = 5890\text{\AA}. Find thickness of the film.

Define geometrical resolving power of an optical instrument. Derive the expression for the resolving power of a microscope and comment on the relation between resolving power and magnifying power, if any exists.

Write a note on Purity of spectral lines and coherence length.

Deduce the possible thickness of a quarter wave plate of quartz which is to be used for sodium light of wavelength 5890 \text{\AA}, (\mu_o = 1.558, \mu_e = 1.486.)

The phase velocity of surface waves of wavelength \lambda is v_p = \left( \frac{2 \pi T}{\lambda \rho} + \frac{g \lambda^{1/2}}{2 \pi} \right) where T is the surface tension and \rho the density of the liquid and g is acceleration due to gravity. Find the group velocity and express it in terms of the phase velocity. For which wavelength is the phase velocity a minimum ?

Write a note on He-Ne laser.

Explain the laws of refraction of light on the basis of Huygens principle.

A particle of mass 10 g lies in a potential given by V(x) = 32 x^2 + 0.2 where x is in metres and V(x) in joules. Write down the equation of motion and solve it. What is the frequency of oscillation?

Deduce an expression for the diffraction waves at single slit and discuss how it is related with a Fourier transform.