Using the set of four Maxwell's equations, obtain the Lorentz condition relation between the scalar potential \phi and vector potential A, i.e., \nabla \cdot A + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0 and discuss the gauge transformations, Lorentz gauge and Coulomb gauge.
State and explain Biot-Savart law. Obtain an expression for the magnetic field at the center of a circular loop of radius r metres, carrying a current of I amperes.
Obtain an expression for the electric potential and electric field strength at a point due to an electric dipole.
(i) The equation for an alternating current is I = 42\cdot 42 \sin (314t). Find the following : Maximum value of current, Frequency, RMS value and Average value
(ii) A condenser of capacity 1~\mu\text{F} is first charged and then discharged through a resistance of 1\text{ M}\Omega. Calculate the time in which the charge on the condenser will fall to 50\% of its initial value.
(iii) Consider the displacement vector \vec{D}, given by \vec{D} = (10xyz^2 + 4x)\hat{i} + (5x^2 z^2)\hat{j} + (10x^2 yz)\hat{k}\text{ nC/m}^2 Find the total charge enclosed in a cube of volume 10^{-9}\text{ m}^3 located at the point (1, 2, 3).
There is a potential gradient of 100\text{ V/m} normal to the surface of the earth. Assuming the earth to be a charged sphere of radius 6370\text{ km}, find the total charge on the earth.
In free space, the electric field of electromagnetic wave is given by \vec{E}(x, t) = 100 \cos (\omega t - kx) \hat{y}\text{ volt/metre} Find the average power crossing a circular area of radius 2\text{ metres} in the yz-plane.
(i) Show that the electric and magnetic energy densities in a plane travelling wave are equal. Also prove that the total energy density = \varepsilon_0 E^2 = \mu_0 H^2.
(ii) Deduce the equation of continuity based on Maxwell's equations.
Explain Planck's formula for black-body radiation. Use this formula to show that energy radiated per unit time per unit area in the range d\lambda of \lambda is e(\lambda, T) = \left( \frac{2\pi c^2 h}{\lambda^5} \right) (e^{\beta hc / \lambda} - 1)^{-1} d\lambda
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})
(i) What is population inversion? How is it efficiently achieved in He-Ne laser? Show it with necessary energy levels diagram.
(ii) Show that for a normal optical source emitting at 600\text{ nm} at a temperature of 10^3\text{ K}, the spontaneous emission dominates over stimulated emission.
(i) How are linearly polarised and circularly polarised lights generated from unpolarised light and are analysed?
(ii) A calcite crystal is to be used for producing a quarter wave plate. The refractive indices of the crystal for ordinary ray and extraordinary ray are 1\cdot 65836 and 1\cdot 48641 respectively. What is the thickness of the plate required if the light of wavelength 632\cdot 8\text{ nm} is to be used?
(i) Give the equation of motions for a particle executing simple harmonic motion for undamped, damped and forced vibrations, and explain the terms.
(ii) Give a plot between the displacement and time for each case of the above and interpret.
(iii) What are the conditions for critical damping and resonance?
Calculate the focal lengths of the components of an achromatic telescope objective having a power of +3\cdot 333 diopter, made from crown glass and flint glass, whose dispersive powers are 0\cdot 012 and 0\cdot 02 respectively.
With the help of a neat diagram, explain spherical aberration. Briefly discuss the methods to minimize spherical aberration.
A particle moving in a central force field located at r=0, describes a spiral, r=e^{-\theta}. Find the force law.
Show that the wave equation for the propagation of electromagnetic scalar potential \phi(x, y, z, t) \left[\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}\right] \phi(x, y, z, t) = 0 remains invariant under Lorentz transformation.
A particle of mass m is constrained to move on a plane curve xy=c with c>0, under gravity, where y-axis is vertical. Construct the Lagrangian of the system and obtain the Euler-Lagrange equation of motion.
Consider a homogeneous cube of total mass M and side a. Taking the origin at one corner of the cube and axes along the edges of the cube, construct the moment of inertia tensor. Calculate principal moment of inertia.
Calculate the first relativistic correction to the kinetic energy of a particle with rest mass m_0 and speed v.
