PYQ Bank

What are Einstein's A and B coefficients ? Establish a relation between them.

A pulse of \lambda_0=600\,\mathrm{nm} and \Delta\lambda=10\,\mathrm{nm} propagates through a fibre which has a material dispersion coefficient of 50\,\mathrm{ps} per km per nm at 600\,\mathrm{nm}. Calculate the pulse broadening in traversing a 10\,\mathrm{km} length of the fibre. If the pulse width at the input of the fibre is 12\,\mathrm{ns}, what will be the pulse width at the output of the fibre?

Show that a travelling wave on the string, clamped on both the ends, undergoes a phase change of \pi. Hence obtain the time-independent form of the wave equation representing a standing wave on the string.

An optical waveguide has core and cladding of refractive indices 1\cdot 46 and 1\cdot 45 respectively. (i) Calculate the maximum number of modes with wavelength \lambda = 900\text{ nm} that can travel if it is a planar waveguide of core thickness 10\ \mu\text{m}. (ii) What is the number of modes if it is a cylindrical fiber of diameter 10\ \mu\text{m} ?

The motion of a damped mechanical oscillator is represented by m\ddot{x}+\alpha\dot{x}+\beta x=0 where m, \alpha and \beta are constants. The oscillator is critically damped. The system is given an impulse at x=0 and t=0, resulting in an initial velocity v. After how much time the system experiences maximum displacement?

Two lenses of focal lengths 8\text{ cm} and 4\text{ cm} are placed at a certain distance apart. Calculate the distance between the lenses if they form an achromatic combination.

Use matrix method to obtain an expression for the focal length of a coaxial combination of two thin lenses having focal lengths f_1 and f_2 separated by distance d.

In the steady state forced vibration of the damped harmonic oscillator, show that the amplitude of the driven system is maximum when p = \sqrt{\omega^2 - 2b^2} and further the value of the maximum amplitude is \frac{f}{2b \sqrt{\omega^2 - b^2}}, where the symbols have their usual significance. (Here you start from the steady state solution)

A light spring of relaxed length 'a_0' is suspended from a point. It carries a mass 'm' at its lower free end, which stretches it through a distance l. Show that the vertical oscillation of the system is simple harmonic in nature and has time period T = 2\pi \sqrt{l / g}, where g is the acceleration due to gravity.

Prove that the group velocity V_g of electromagnetic waves in a dispersive medium with refractive index n(\lambda_0) at wavelength \lambda_0 is given by V_g=\frac{c}{n(\lambda_0)-\lambda_0\dfrac{dn(\lambda_0)}{d\lambda_0}} where c is the free space velocity of light. Find the time taken for the electromagnetic pulse to travel a distance D.

Explain the physical significance of group velocity from the concept of phase velocity with relevant expressions.

The Fraunhofer single-slit diffraction intensity is given by I=I_0\frac{\sin^2 x}{x^2} where x=\frac{\pi d y}{\lambda l}, with l as the distance from slit to source, d the slit width, y the detector distance and \lambda the wavelength. What is the value of cumulative intensity \int_{-\infty}^{\infty} I(y)\,dy?

In a double slit experiment the slits are 2\text{ mm} apart and are illuminated with a mixture of two wavelengths, \lambda = 750\text{ nm} and \lambda' = 900\text{ nm}. At what minimum distance from the common central bright fringe on a screen 2\text{ m} distant from the slits will a bright fringe from one interference pattern coincide with a bright fringe from the other ?

When a thin film of a transparent material is put behind one of the slits in Young's double-slit interference experiment, the zero-order fringe moves to the position previously occupied by the fourth-order bright fringe. The index of refraction of the film is n=1.2 and the wavelength of light, \lambda=5000\,\mathring{\mathrm{A}}. Determine the thickness of the film.

Explain the diffraction at straight edge with the help of Cornu's spiral.

A diffraction grating has 5000\text{ lines per cm}. For illumination at normal incidence, determine the dispersive power of the grating in the second order spectrum in the range of wavelengths around 500\text{ nm}.

Distinguish between the intensity patterns due to diffraction from a narrow slit and a straight edge.

A plano-convex lens of crown glass is connected to a concavo-convex lens of flint glass so that the convex surface of the first and the concave surface of the second fit exactly. If the combination is achromatic and the combined focal length is -50\text{ cm}, determine the radii of curvature of the faces of the flint glass lens. The dispersive power of the crown glass is one-half of that of the flint glass and the refractive indices for yellow light are respectively 1\cdot 500 and 1\cdot 625 respectively.

What is Fermat's principle ? Derive the laws of reflection and refraction with the help of this principle, when the light is incident on a curved surface.

Write down the one-dimensional harmonic oscillator differential equation under damping and its solution for the lightly damped condition, with the meanings of symbols. Determine the dependent energy in the lightly damped condition.