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.

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.

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.

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)

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 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.

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.

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}.

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 light beam of wavelength 600\text{ nm} produced by a 20\text{ mW} laser source is incident on a plane mirror. Determine : (i) number of photons per second striking the surface of the mirror. (ii) force exerted by the light beam on the mirror.

For a \text{He}-\text{Ne} laser system, what will be the magnitude of \Delta\omega_D which represents FWHM of the line shape function g(\omega), if resonant frequency \omega_0 = 3 \times 10^{15}\text{ s}^{-1} and temperature T = 300\text{ K}?

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.

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.

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.

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.

Consider a laser system consisting of an active medium placed between a pair of mirrors forming a resonator. Obtain an expression for the threshold population inversion required for the oscillations of laser.

Consider a thin lens combination of two convex lenses of focal lengths f_1 = +10\text{ cm} and f_2 = +20\text{ cm}, respectively, separated by 25\text{ cm}. Determine the focal length of the combination and the positions of unit planes.

The intensity at the central maximum observed on a screen in a double-slit experiment is 2 \times 10^{-3}\text{ W/m}^2. If the path difference between interfering waves reaching a point on the screen is \frac{\lambda}{6}, where \lambda is the wavelength of the light used in the experiment, determine the intensity at that point.

A weakly damped harmonic oscillator consisting of spring-mass system has the following parameters : Mass m = 0.25\text{ kg}, Spring constant k = 100\text{ N m}^{-1}, Damping coefficient \gamma = 1\text{ N s m}^{-1} A periodic force F = 5 \cos \omega t (newton) is applied to the system. Determine (i) the amplitude of the oscillator at resonance and (ii) the Q-value of the oscillator.

Write down the system matrix for a combination of two thin lenses in paraxial approximation. Hence obtain the focal length of the combination and the positions of unit planes.