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A plano-convex lens of radius 1\cdot 0\text{ m} is placed on an optically flat glass plate. The space between the lens and the glass plate is filled with a liquid. The diameter of the 5^{\text{th}} ring changes to 3\cdot 0 \times 10^{-3}\text{ m}. Calculate the refractive index of the liquid when the ring is bright. The wavelength of light used is 589\text{ nm}.
Explain why information carrying capacity of an optical fibre can be enhanced by reducing the pulse dispersion. How does one minimize pulse dispersion using a graded index optical fibre?
In a tungsten filament lamp, thermionic emission takes place at 1.2\times10^3\ \mathrm{K}. Calculate the ratio of spontaneous emission to stimulated emission for non-degenerate energy levels. Interpret your result physically. Take \lambda=550\ \mathrm{nm}, k_B=1.38\times10^{-2}\ \mathrm{J\,K^{-1}}, h=6.67\times10^{-34}\ \mathrm{J\,s} and c=3\times10^8\ \mathrm{m\,s^{-1}}.
The dispersion relation for deep water waves is given by \omega^2 = gk + ak^3 where g and a are constants. Obtain expressions for phase velocity and group velocity in terms of the wavelength \lambda. \omega and k represent the angular frequency and wave number, respectively.
The displacement associated with a three-dimensional plane wave is given by \Psi(x,y,z,t) = a\cos\left[\frac{\sqrt{3}}{2}kx + \frac{1}{2}ky - \omega t\right]. Calculate the angles made by the propagating wave with the x, y and z-axes.
Write down the equation of motion of a weakly damped harmonic oscillator driven by a harmonic force. Obtain an expression for the maximum amplitude of oscillation under steady-state conditions.
In a double slit interference experiment, show that the fringe shape is a hyperbola.
Classify fibers based on the number of modes that can propagate through them. Depict step-index and graded-index multimode fibers pictorially.
During an earthquake, a horizontal shelf moves vertically. If its motion can be regarded simple harmonic, calculate the maximum value of amplitude of oscillation so that the books resting on it stay in contact with it always. Take g=9.8\ \mathrm{m\,s^{-2}} and T=0.5\ \mathrm{s}.
An oscillator of mass 0\cdot 01\text{ kg} draws maximum power at a frequency of 96\text{ Hz} with half power points at 93\text{ Hz} and 99\text{ Hz}. If the amplitude of driving force is 3\cdot 88\text{ N}, calculate (i) quality factor, (ii) the damping factors and amplitude of the oscillator at resonance.
Prove that as a result of an elastic collision of two particles under non-relativistic regime with equal masses, the scattering angle will be 90^\circ. Illustrate your answer with a vector diagram.
Show that the kinetic energy and angular momentum of torque free motion of a rigid body is constant.
A particle describes a circular orbit under the influence of an attractive central force directed towards a point on the circle. Show that the force varies as the inverse fifth power of distance.
Obtain an expression for the angular speed of the Earth at which Coriolis force makes objects fly from its surface.
A particle of rest mass M=4\times10^{-27}\ \mathrm{kg}, disintegrates into two particles of rest masses M_1=3\times10^{-27}\ \mathrm{kg} and M_2=1\times10^{-27}\ \mathrm{kg}. Show that the energies E_1 and E_2 of these two parts after disintegration satisfy the condition E_1=3E_2 while moving in opposite directions with equal linear momenta. Give necessary mathematical derivation.
If the forces acting on a particle are conservative, show that the total energy of the particle which is the sum of the kinetic and potential energies is conserved.
Suppose that an S'-frame is rotating with respect to a fixed frame having the same origin. Assume that the angular velocity \vec{\omega} of the S'-frame is given by \vec{\omega}=2t\hat{\imath}-t^{2}\hat{\jmath}+(2t+4)\hat{k} where t is time and the position vector \vec{r} of a typical particle at time t as assumed in S'-frame is given by \vec{r}=(t^{2}+1)\hat{\imath}-6t\hat{\jmath}+4t^{3}\hat{k}. Calculate the Coriolis acceleration at t=1 second.
Show that the operator \left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right) is invariant under Lorentz transformations.
Calculate the horizontal component of the Coriolis force acting on a body of mass 0.1\ \mathrm{kg} moving northward with a horizontal velocity of 100\ \mathrm{ms^{-1}} at 30^{\circ}\mathrm{N} latitude on the Earth.
Show that a particle of rest mass m_0, total energy E and linear momentum \vec{p} satisfies the relation E^2=c^2p^2+m_0^2c^4 where c is the velocity of light in free space.