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A particle executes simple harmonic motion of amplitude A and angular frequency \omega. A damping force proportional to velocity acts on the particle. Derive the expression for the displacement of the particle as a function of time and discuss the effect of damping on amplitude and frequency. Also, calculate the time at which the amplitude reduces to half its initial value, if the damping coefficient is b and mass is m.
The aperture width of a laser light source of wavelength 6000\text{ \AA} is 3\text{ mm} and its power is 20\text{ mW}. Calculate the light intensity at a distance of 200\text{ m} from the light source.
A Fabry--P'erot interferometer is illuminated by monochromatic light of wavelength \lambda = 500\text{ nm}. The mirror separation is d = 0\cdot 8\text{ mm}. It is observed that two successive transmitted maxima correspond to wavelengths \lambda and \lambda + \Delta\lambda, both satisfying the condition for normal incidence. Determine the smallest wavelength difference \Delta\lambda that can be resolved by the interferometer. Explain physically why this quantity depends on mirror separation.
What is meant by achromatic combination of lenses? Derive the condition for achromatization of a pair of lenses separated by a distance x. The two lenses have different dispersive powers.
Find the required thickness of the calcite plate to convert plane polarized light (\lambda = 6000\text{ \AA}) into circularly polarized light. (For calcite, \mu_O = 1\cdot 658 and \mu_E = 1\cdot 486)
Light of wavelength 6000\text{ \AA} is incident on a slit of width 0\cdot 40\text{ mm}. The screen is placed 2\text{ m} away from the slit. Find (i) the position of the first dark fringe and (ii) the width of the central bright fringe.
Using Fraunhofer diffraction theory, derive the expression for the intensity distribution due to a circular aperture and obtain the condition for the first minimum. Using this result, derive the expression for the resolving power of an optical instrument. Finally, calculate the minimum angular separation that can be resolved by a telescope of aperture diameter D = 10\text{ cm} for light of wavelength 500\text{ nm}.
A monochromatic parallel beam of wavelength \lambda = 600\text{ nm} is incident on a single slit of width a = 0\cdot 3\text{ mm}. A convex lens of focal length f = 1\text{ m} forms the Fraunhofer diffraction pattern on a screen.
(i) Determine the angular width and linear width of the central maximum on the screen.
(ii) If the slit width is halved, explain quantitatively how diffraction pattern changes.
(iii) A second wavelength 450\text{ nm} is added. Will the minima of the two wavelengths coincide? Justify mathematically.
In Newton's ring experiment, the space between planoconvex lens and glass plate is filled with a liquid of refractive index \mu. Explain how interference pattern changes as compared to air. Starting from the condition for interference, derive an expression for the radius of the n\text{th} dark ring. Discuss how the ring system changes if the refractive index increases.
A 15\cdot 0\text{ cm} violin string, fixed at both the ends, is vibrating with the largest wavelength. The speed of the wave in this string is 250\text{ m/s}, and the speed of sound in air is 350\text{ m/s}. What is the frequency and wavelength of the emitted sound wave?
m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = 0; A harmonic oscillator is represented by the equation m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = 0; where m = 0\cdot 25\text{ kg}, \gamma = 0\cdot 07\text{ kg s}^{-1} and k = 85\text{ Nm}^{-1}. Determine (i) the period of oscillation, and (ii) the number of oscillations in which its amplitude will become half of its original value.
A particle of mass \text{m} is suspended from a spring which has mass \text{m} and force constant \text{k}. Show that the oscillation frequency is given by \omega = \frac{\sqrt{3}}{2}\omega_0, where \omega_0 is frequency of oscillation when spring is considered massless.
Consider multiple reflections from a plane parallel film of thickness h and refractive index n_2 and derive an expression for the total reflectivity from the surface of the film.
Describe the interference by a plane parallel thin film when illuminated normally and obliquely by a plane wave. Also list some important practical applications of the thin film interference phenomenon.
In a double slit Fraunhofer diffraction experiment, the slit width is 0\cdot 12\text{ mm} and the spacing between the two slits is 0\cdot 48\text{ mm}. The distance of the screen from the slits is 1\cdot 5\text{ m}. If the wavelength of the light used is 600\text{ nm}, determine (i) the missing orders of the interference maxima, and (ii) the distance between the central maxima and the first minima.
A parallel beam of sodium light of wavelength 5893\text{ \AA} is incident on a diffraction grating. The angle between the first order spectra on either side of the normal is 27^{\circ} 42'. Find :
(i) The number of rulings/mm on the grating.
(ii) The greatest number of bright images obtained.
Within the framework of the Cornu's spiral, describe the intensity distribution at an arbitrary point \text{P} when a plane wave is incident normally on a long narrow slit of width \text{b}. For this case, under what condition will the diffraction pattern be of Fraunhofer type?
Consider a thick lens of thickness t made of a material of relative refractive index n. Let R_1 and R_2 be the radii of curvature of its two surfaces. Obtain the system matrix of the lens.
Determine the position of focal points, principal points and nodal points for a spherical lens of radius 20\text{ cm}. The refractive index of the material of the lens is 3/2. Indicate the positions of all the points in a diagram.
Explain the construction and working of a quarter-wave plate. How is it used to produce circularly and elliptically polarized light?