PYQ Bank

Draw a neat diagram to show how light propagates through a graded index optical fiber from the entrance to the fiber end and explain.

Starting with the rate equations for matter-radiation interaction, show that the ratio of the Einstein's A and B coefficients is proportional to the third power of the frequency of radiation.

What are the step index and graded index optical fibers ? What are the conventional optical windows for light propagation through optical fibers ?

Show that the group velocity is equal to particle velocity. Also prove that the group velocity of the photons is equal to c, the velocity of light.

Considering a plane transmission diffraction grating, where d is the distance between two consecutive ruled lines, m as the order number and \theta as the angle of diffraction for normal incidence, calculate the angular dispersion \frac{d\theta}{d\lambda} for an incident light of wavelength \lambda.

In a certain engine, a piston undergoes vertical SHM with an amplitude of 10\,\mathrm{cm}. A washer rests on the top of the piston. As the motor is slowly speeded up, at what frequency will the washer no longer stay in contact with the piston?

How does the population inversion in an active medium lead to the amplification of light in a laser ? Explain in details.

Explain how you can construct a zone plate from Fresnel's half period zone. Show that a zone plate has multiple foci.

Write down the mathematical representation of Fermat's principle and explain all the notations used. With the help of a neat diagram, show that this principle can be used to obtain the law of refraction : n_1 \sin \theta_1 = n_2 \sin \theta_2 , where n_1 and n_2 are the refractive indices of the two media while \theta_1 and \theta_2 are the angles of incidence and refraction of the light beam.

If the system matrix for a thin lens of focal length f is given by : S = \begin{bmatrix} 1 & -1/f \\ 0 & 1 \end{bmatrix} , show that the system matrix for a combination of two thin lenses of focal lengths f_1 and f_2 separated by a distance d can be obtained as S_{12} = \begin{bmatrix} 1 - d/f_2 & -(1/f_1 + 1/f_2 - d/f_1 f_2) \\ d & 1 - d/f_1 \end{bmatrix}

Explain the working principle of a 3-level laser with a specific example. Comment on why the third level is needed.

A left circularly polarized beam (\lambda = 5893^\circ\text{A}) is incident normally on a calcite crystal (with its optic axis cut parallel to the surface) of thickness 0.005141\text{ mm}. What will be the state of polarization of the emergent beam ? Justify your answer. Here the refractive indices for the ordinary and the extraordinary rays are 1.65836 and 1.48641 respectively.

In a double slit interference experiment, show that the fringe shape is a hyperbola.

If a thin sheet of glass of thickness t and refractive index \mu is placed in the path of one of the interfering waves, show that the distance through which a fringe is displaced is a function of t.

A parallel beam of light from a He--Ne laser (\lambda=630 nm) is made to fall on a narrow slit of width 0.2\times10^{-3} m. The Fraunhofer diffraction pattern is observed on a screen placed in the focal plane of a convex lens of focal length 0.3 m. Calculate the distance between the \begin{qroman}\item first two minima and \item first two maxima on the screen.\end{qroman}

Explain the physical significance of resolving power of a grating with relevant mathematical expression.

A block of mass m is attached to one end of a combination of two massless springs connected in series. The other end of the combination is fixed to a rigid support. The spring constants of the two springs are k_1 and k_2, respectively. The block is free to move on a frictionless horizontal surface. If the block is pulled a little and then released, calculate the frequency of oscillation of the block.

An oscillator of mass 0\cdot 01\text{ kg} draws maximum power at a frequency of 96\text{ Hz} with half power points at 93\text{ Hz} and 99\text{ Hz}. If the amplitude of driving force is 3\cdot 88\text{ N}, calculate (i) quality factor, (ii) the damping factors and amplitude of the oscillator at resonance.

Write down the equation of motion of a weakly damped harmonic oscillator driven by a harmonic force. Obtain an expression for the maximum amplitude of oscillation under steady-state conditions.

The displacement associated with a three-dimensional plane wave is given by \Psi(x,y,z,t) = a\cos\left[\frac{\sqrt{3}}{2}kx + \frac{1}{2}ky - \omega t\right]. Calculate the angles made by the propagating wave with the x, y and z-axes.