Show that the mean kinetic and potential energies of non-dissipative simple harmonic vibrating systems are equal.

A mass m is suspended by two springs having force constants k_1 and k_2 as shown in the figure. The mass m is displaced vertically downward and then released. If at any instant t, the displacement of the mass m is x, then show that the motion of the mass is simple harmonic motion having frequency f=\frac{1}{2\pi}\sqrt{\frac{1}{m}\left(\frac{k_1k_2}{k_1+k_2}\right)}

Show that the phenomenon of Fraunhofer diffraction at two vertical slits is modulation of two terms viz. double slit interference and single slit diffraction. Obtain the condition for positions of maxima and minima.

In a step-index optical fiber system, explain the terms pulse dispersion and material dispersion. An optical fiber having refractive indices of core and cladding n_1 = 1.463 and n_2 = 1.444 respectively, uses a Laser diode with \lambda_0 = 1.50\,\mu\mathrm{m} with a spectral width of 2\,\mathrm{nm}. At this wavelength if the material dispersion coefficient, D_{m} is 18.23\,\mathrm{ps/(km\cdot nm)}, then calculate the pulse dispersion and material dispersion for 1\,\mathrm{km} length of the fiber.

What is chromatic aberration? Obtain the condition for achromatism using combination of two thin lenses placed in contact to each other. Can this system work as achromatic doublet if both are of same material? Justify your answer.

A thin film of petrol of thickness 9 \times 10^{-6}\,\mathrm{cm} is viewed at an angle 30^\circ to the normal. Find the wavelength(s) of light in visible spectrum which can be viewed in the reflected light. The refractive index of the film \mu = 1.35.

What is damped harmonic oscillation? Write the equation of motion and obtain the general solution for this oscillation. Discuss the cases of dead beat, critical damping and oscillatory motion based on the general solution. What would be the logarithmic decrement of the damped vibrating system, if it has an initial amplitude 30\,\mathrm{cm}, which reduces to 3\,\mathrm{cm} after 20 complete oscillations?

Explain the phenomenon of double refraction in calcite crystal. Considering birefringent crystal as non-conducting material, explain double refraction using electromagnetic theory. Calculate the thickness of a double refracting plate which produces a path difference of \frac{\lambda}{4} between extraordinary and ordinary waves. Given: \lambda = 5890\,\mathring{\mathrm{A}},\quad \mu_0 = 1.53,\quad \mu_e = 1.54

State and explain Fermat's principle of extremum path. Discuss the cases of rectilinear propagation of light and reversibility of light rays in context of Fermat's principle. Using Fermat's principle, deduce the thin lens formula.

What is a zone plate? Give its theoretical description. Show that a zone plate has multiple foci. Differentiate a zone plate from a convex lens. Calculate the radius of the first half period zone in a zone plate behaving like a convex lens of focal length 60\,\mathrm{cm} for light of wavelength 6000\,\mathring{\mathrm{A}}.

For free space show that electromagnetic (EM) wave is transverse in nature. Show that for free space, the total outward flux of EM energy through surface S bounding volume V is equal to the rate of loss of EM energy from the volume V. A laser beam of 2\ \mathrm{mm} diameter has average power of 20\ \mathrm{GW}. Calculate the peak values of electric and magnetic fields in the laser beam.

What are Newton's rings? How are they formed by two curved surfaces?

Briefly discuss the postulates of Einstein to explain stimulated emission. Derive an expression for Einstein's A and B coefficients and show that the ratio of coefficients of spontaneous versus stimulated emission is proportional to the third power of frequency of radiation. Why is it difficult to achieve laser action in higher frequency ranges such as X-rays? Can there be a temperature at which the rates of spontaneous and stimulated emission are equal? Illustrate with wavelength \lambda = 5000\,\mathring{\mathrm{A}}.

Discuss the conditions for interference. Describe Young's double-slit experiment and derive an expression for the estimation of fringe width. Discuss its dependency on various parameters. Green light of wavelength 5100\,\mathring{\mathrm{A}} from a narrow slit is incident on a double-slit. If the overall separation of 10 fringes on a screen 200\,\text{cm} away is 2\,\text{cm}, find the slit separation.

The equation for displacement (X) of a point on a damped oscillator is given by x = 5e^{-0.25t}\sin\left(\frac{\pi}{2}\right)t\ \text{metres}. Find the velocity of oscillating point at t = \frac{T}{4} and T, where T is the time period of the oscillator. What is the direction of velocity in each case?

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.

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.

Why is population inversion in general not possible in a two-level laser system? Explain it.

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.

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}.