(i) Describe the wave given by the expression E = \hat{j} E_0 \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) - \hat{k} E_0 \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) where \hat{j} and \hat{k} are unit basis vectors in Cartesian coordinates. (ii) Write an expression for linearly polarized harmonic plane wave of scalar amplitude E_0, propagating along a line in the xy-plane at 45^\circ to the x-axis and having the xy-plane as its plane of vibration.
What are Newton's rings? How are they formed by two curved surfaces?
(i) Obtain an expression for the displacement of the damped harmonic oscillator where the damping force is proportional to the velocity. Discuss the effect of the damping on the displacement and frequency of the oscillator. Discuss the difference physically between overdamped and critically damped by taking example of a pendulum. (ii) A particle of mass 3 moves along the x-axis attracted towards the origin by a force whose magnitude is numerically equal to 12x. The particle is also subjected to damping force whose magnitude is numerically equal to 12 times the instantaneous speed. If it is initially at rest at x = 10, find the position and the velocity of the particle at any time.
In an \text{He}-\text{Ne} laser system, the two energy levels of \text{Ne} involved in lasing action have energy values of 20\cdot 66\text{ eV} and 18\cdot 70\text{ eV} respectively. Population inversion occurs between these two energy levels. What will be the wavelength of the laser beam produced ? What will be the population in the metastable energy level with respect to the upper excited level at room temperature (27^\circ\text{C}) ?
Explain Huygens' theory of double refraction in a uniaxial crystal. Discuss the formation of ordinary and extraordinary rays for oblique incidence when an optic axis is in the plane of incidence and when it is perpendicular to the plane of incidence.
Discuss how population inversion is achieved in Ruby laser. What is 'laser spiking'? Why does it occur?
How can one convert a left-handed circularly polarised light into a right-handed one (and vice versa)? Calculate the thickness of a quarter-wave plate when the wavelength of light is 589\,\mathrm{nm}. Given: \mu_0 = 1.544 and \mu_E = 1.553.
Why is population inversion in general not possible in a two-level laser system? Explain it.
In what way is holography different from conventional photography? Discuss the salient features of a hologram. What are the requirements for the formation and reading of a hologram?
Two Fabry – Pérot interferometers have same plate separation but the coefficients of intensity reflection are 0\cdot 7 and 0\cdot 9. Deduce the relative width of the maxima in the above two cases.
With a neat ray diagram, describe Fraunhofer diffraction at a circular aperture.
What are the fringes of equal thickness and fringes of equal inclination? In a Newton's ring arrangement with a source emitting two wavelengths \lambda_1=6\times10^{-7}\,\text{m} and \lambda_2=5.9\times10^{-7}\,\text{m}, it is found that the m^{\text{th}} dark ring due to one wavelength coincides with the (m+1)^{\text{th}} dark ring due to the other. Find the diameter of the m^{\text{th}} dark ring, if the radius of curvature of the lens is 90\,\text{cm}.
How can the refractive index of a liquid be determined using a method of Newton's rings ? Obtain an appropriate expression for calculating the refractive index.
It is required that a real image twice the size of the object be formed by a thin plano-convex lens. If the lens has a radius of curvature of 50\text{ cm} and a refractive index of \mu = 1\cdot 5, determine the locations of the object and image with respect to the lens.
What is axial chromatic aberration? A convex lens has a focal length of 15.5 \times 10^{-2}\,\mathrm{m} for red colour and 14.45 \times 10^{-2}\,\mathrm{m} for violet colour. If an object is kept at a distance of 40\,\mathrm{cm} from the lens, calculate the longitudinal chromatic aberration of the lens.
The quality factor Q in a damped harmonic motion is defined as Q = \frac{2\pi \times \text{Average energy stored per cycle}}{\text{Average energy dissipated per cycle}} Show that Q = \frac{\omega}{2b} (where the symbols have usual meaning).
A High Frequency (HF) radio receiver receives simultaneously two signals from a transmitter 400\text{ km} away, one by a path along the surface of the Earth, and the other by reflection from a portion of the ionospheric layer situated at a height of 200\text{ km}. We assume that the Earth is flat and the ionospheric layer acts as a perfect horizontal reflector, which is moving slowly in the vertical direction. When the frequency of the transmitted wave is 10\text{ MHz}, it is observed that the combined signal strength varies from maximum to minimum and back to maximum 6 times per minute. With what slow vertical speed is the ionospheric layer moving ?
Prove that when light goes from one point to another via a plane mirror, the path followed by light is the one for which the time of flight is the least.
Thin transparent films produce colours when a white light falls on them or they act as antireflecting coatings. Find the conditions for these characteristics.
(i) When a monochromatic light passes through a single slit, dark and bright fringes are observed on a screen kept far away from the slit. Draw an amplitude and intensity pattern of the light on the screen.
(ii) Write an expression for intensity distribution and give the conditions for bright and dark fringes.
(iii) If the slit width is 0\cdot 3\text{ mm} and the distance between the screen and the slit is about 30\text{ cm}, then what is the angle of diffraction for the first minimum? (\lambda = 5 \times 10^{-5}\text{ cm})
