PYQ Bank

(i) Between multimode and single-mode fibers, explain critically how and why single-mode fiber is chosen for communication.

(ii) Why are the two wavelengths 1\cdot 30\ \mu\text{m} and 1\cdot 55\ \mu\text{m} important in single-mode fiber-optical communication system?

(i) What is birefringence? Indicate how it can be used to obtain plane and circularly polarized light.

(ii) A left circularly polarized beam of light having \lambda_0 = 5893\text{ \AA} is incident on a calcite crystal with its optic axis cut parallel to the surface. The crystal has thickness d = 0\cdot 005141\text{ mm}, n_o = 1\cdot 65836 and n_e = 1\cdot 48641 at this \lambda_0. What will be the state of polarization of the incident beam?

Consider a bare fiber having n_1 = 1\cdot 48 and n_2 = 1\text{ (air)}. Find out the NA. What is the maximum incident angle up to which light can be guided through the fiber?

Why is the NA of the single-mode optical fiber low as compared to multimode fiber?

(ii) Determine its quality factor Q and width of resonance \Delta f.

In the propagation of longitudinal waves in a fluid contained in an infinitely long tube of cross-section A, show that \rho=\rho_0\left(1-\frac{\partial\xi}{\partial x}\right) where, \rho_0 = equilibrium density \rho = density of the fluid in the disturbed state \frac{\partial\xi}{\partial x}=\text{volume strain}\quad\left(\left|\frac{\partial\xi}{\partial x}\right|\ll1\right)

What is the significance of the null result of Michelson-Morley experiment? Does it disprove the existence of ether? Justify.

(iii) Discuss the important role of Coriolis forces in the circulation pattern of winds.

When a sphere of radius r falls down a homogeneous viscous fluid of unlimited extent with the terminal velocity v, the retarding viscous force acting on the sphere depends on the coefficient of viscosity \eta, the radius r and its velocity v. Show how Stokes' law was arrived at connecting these quantities from the dimensional considerations.

How can one introduce the constraints of motion through the concept of generalized coordinate systems? Write down the set of transformation equations for a system of N particles relating the generalized coordinates with the real coordinates.

A planet revolves around the Sun in an elliptic orbit of eccentricity e. If T is the time period of the planet, find the time spent by the planet between the ends of the minor axis close to the Sun.

(i) For a particle of mass m moving with velocities \vec{v}_s and \vec{v}_r relative to the space and rotating axis, respectively, show that the equation of motion is obtained as \vec{F}-2m(\vec{\omega}\times\vec{v}_r)-m\vec{\omega}\times(\vec{\omega}\times\vec{r}) = m\vec{a}_r with \vec{\omega} and \vec{a}_r being the angular velocity and acceleration in rotating coordinates.

(ii) At what velocity, the electron momentum will be m_0 c?

(ii) Which term in the above equation represents the Coriolis force?

Establish the relation between the angular momentum and torque of a particle. Show that this relation leads one to the principle of conservation of angular momentum.

A uniform solid sphere of radius R having moment of inertia I about its diameter is melted to form a uniform disc of thickness t and radius r. The moment of inertia of the disc about an axis passing through its edge and perpendicular to the plane is also equal to I. Show that the radius r of the disc is given by r=\dfrac{2R}{\sqrt{15}}.

What are Eulerian angles? A body with rotational symmetry about an axis is rotating under gravity about a point on the axis without friction. What are the quantities remaining constant during the motion? Find them in terms of suitable Eulerian angles. Explain 'precession' and 'nutation' of such a body.

(i) An electron of rest mass m_0 moves with a velocity v such that its total energy is double of its rest mass energy. What is the electron velocity?