A wire of length l is bent into the form of a rectangle of side a and b and carries a current I. Calculate the magnetic field intensity at the centre of the rectangle. Show further that the intensity is minimum for a = b.

Write a short note on Rayleigh criterion for resolution.

How can it be proved experimentally that energy transmission is associated with a wave? The disc of a siren having 60 holes makes 420 revolutions per minute. Find the wavelength of its sound in air 0^\circ\text{C}, if velocity of sound at 0^\circ\text{C} is 332\text{ ms}^{-1}.

For transverse waves passing over a uniform string, prove that the slope of the siring at any position x at a given time is numerically equal to the ratio of the particle speed at that position x and time to the wave speed over the string.

Write a short note on Phase and Group velocities.

Distinguish between spatial and temporal coherence.

A beam of lineal polarised light is changed into circularly polarised light by passing it through a slice of double-refracting crystal 0.0030 cm thick. Calculate to difference in the two refractive indices of the crystal, assuming this to be minimum thickness that will produce the result, and that the wavelength is 6 \times 10^{-5}\text{ cm}.

Explain the Laser actions in

(i) Helium Neon Laser

(ii) Gas Laser

In double-slit Fraunhofer diffraction calculate the fringe spacing on a screen 50 cm away from the slits, if they are illuminated with blue light (\lambda = 4800\text{ \AA}), slit separation d=0.10 mm, and slit-width a=0.020 mm. What is the linear distance from the central maximum the First minimum of the fringe-envelope?

Write a short note on Fraunhofer diffraction by circular aperture.

A movable bridge divides sonometer wire into two parts, which differ in length by 1.0 cm and produce 4 beats pre second when sounded together. If the whole length of the wire is 100 cm, find the frequencies of the parts.

State Bernoulli's theorem for an ideal liquid. Deduce the related equation from first principles and describe one application in relation to sonic measurement technique or device.

The spike heel on a girl's shoe is a square, 1.0 cm on an edge. If her mass is 50kg, calculate (in atmospheres) the pressure exerted on the ground when she balances on one heel.

A girl on skis (mass 60kg including skis) reaches the bottom of a hill going 20\text{ ms}^{-1}. What is her momentum? She then strikes a snowdrift which slops her on 3s. What average force does the snow exert on the girl? Assuming that the force remains constant. How far does the girl penetrate the snowdrift?

What is the angular momentum of an earth satellite of mass 100 kg which moves in a circular orbit of radius equal to twice the radius of the earth? Derive the relation used.

Define Coriolis force and write an expression for it. Through suitable examples, explain the way this force varies in different parts of the earth's surface and for different velocities of the concerned particle.

Write a short note on Mass-Energy equivalence.

The mass of a molecule of DNA from T2 bacteriophage is 1.2 \times 10^8\text{ a.m.u.} The molecule is a double helix of 1.2 \times 10^4 turns and the radius of gyration of the helix is 6.7\text{ \AA}. If the rotational energy of the molecule about the axis of the helix is equal to thermal energy at 300 K, deduce the value of \omega, the angular frequency of rotation.

A solid cylinder released from rest rolls (without slipping) downs piano of length l inclined at angle \theta with the horizontal. Deduce an expression for the velocity of the centre of mass of the cylinder at the bottom of the plane, neglecting friction.

A spinning top shows precession. Explain how the precession rate changes as the spinning becomes slower and as the inclination of the spinning axis from the vertical increases.