What do you mean by a gauge transformation ? What is its importance ? Show that the Lorentz gauge condition \vec{\nabla} \cdot \vec{A} + \frac{1}{c} \frac{\partial \phi}{\partial t} = 0 is Lorentz invariant. Here \vec{A} and \phi are the vector and scalar potentials.
In the case of proton NMR, state the expression for the energy of the dipole in an external magnetic field. How does the NMR originate ? Give the rough range of frequencies of the NMR signals, and the normal magnitude of the applied field.
Using Planck's radiation formula u(\nu) d\nu = \frac{8 \pi h}{c^3} \frac{\nu^3 d\nu}{e^{h\nu/kT} - 1} where the symbols have their usual meaning, find the wavelength of the region where energy density is the greatest. Also calculate the total energy density over all the frequencies
Write Ampere's circuital law, and obtain a generalised form of this law, for non-stationary case.
Write down the different components of the electromagnetic field tensors F_{\mu\nu} and further prove that Maxwell's equations of electrodynamics are invariant to Lorentz transformations.
Find the magnetic field \vec{B} at the point P due to a short straight length of wire carrying current 'i'. Length of the wire is l. Point P is at a distance r away from the centre of the wire. Angle between \vec{l} and \vec{r} is \theta.
Starting from Maxwell's equations of electromagnetic field in vacuum obtain the classical wave equations for the four field vectors \vec{E}, \vec{D}, \vec{B} and \vec{H} . Show that the field vectors can be propagated as waves in free space with the velocity of propagation equal to 3 \times 10^8\text{ m/s}, where for free space we have the vacuum permittivity \varepsilon_o = 8.854 \times 10^{-12}\text{ farad/m} & vaccum permeability \mu_o = 1.257 \times 10^{-6}\text{ henry/m}.
Starting from Maxwell's equation, \nabla . D = \rho, where D is the electric displacement density and \rho is the charge density, derive Poisson's equation. Deduce Laplace's equation for charge-free region from Poisson's equation.
What is molecular polarizability ? Derive Clausius - Mossotti equation relatig the molecular polarizability with the dielectric constant of a dielectric material.
A potential in cylindrical coordinates is a function of r and \phi but not of z. Obtain the separated differential equations for R and \Phi, where V = R(r) \Phi(\phi) and solve them.
Derive Snell's law of refraction from fermat's principle. How is the refractive index related to the velocity of light ? Light of wavelength 600 nanometre (in vacuum) enters a glass slab of refractive index 1.5. What are the values of wavelength, frequency and velocity of light in glass ? [Velocity of light in vacuum = 3 \times 10^8\text{ ms}^{-1}]
How would you produce plane polarized light by reflection ? What is Brewster's law ? Calculate the angular position of the sun above the horizon so that light reflected from a calm lake is completely polarized. The refractive index of water is 1.33. Circularly polarized and unpolarized light are passed in turn through a Nicol prism. The Nicol is rotated about the direction of light as axis. What would you observe in each case ? How would you distinguish between them ?
What is optical activity ? Give reasons for the conclusion that optical rotation in liquids has a molecular origin. What do you mean by ordinary and extraordinary rays ? What are positive and negative crystals ? Give an example of each. Compute the minimum thickness of a quarter-wave plate made from quartz for incident wavelength of 589.3 nanometer. Given \mu_o = 1.544 and \mu_E = 1.553.
The X and Y co-ordinates of Cornu's spiral can be expressed quantitatively by two integrals. Derive the expressions of these integrals.
Obtain the relation to find radii of the rings and the wavelength of light in Newton's circular ring. Calculate the radius of curvature of the convex glass surface where diameter of 5^{th} and 15^{th} bright rings formed by sodium yellow light are measured to be 2.303\text{ mm} and 4.134\text{ mm}. Given \mu = 1.5 and \lambda_{yellow} = 5282 \text{\AA}.
Explain the working of Michelson interferometer using appropriate optical diagram. also draw paths of the rays.
In the steady state forced vibration a point particle of mass 'm' moves under the influence of an external force (F \sin pt) \hat{i} in addition to the restoring force - (kx) \hat{i} and damping force - (\beta x) \hat{i} . Show that (i) the amplitude is maximum when p = \sqrt{\omega^2 - 2b^2} , where k/m = \omega^2 and (ii) the value of the maximum amplitude is \frac{f}{2b \sqrt{\omega^2 - 2b^2}} . What do you mean by the sharpness of resonance ?
What is Hamilton's principle ? Obtain Lagrange's equations of motion with its help for a conservative system.
The source S' moves along the x'-axis at a speed v, and emits light at an angle \theta' to the x'-axis of its own frame. In the S-frame the emitting angle with the x-axis is \theta. Here x and x'-axes are coincident. Show that the exact relativistic aberration formula \tan \theta = \frac{\sin \theta' \sqrt{1 - v^2 / c^2}}{\cos \theta' + v} can be derived from the velocity transformation relations.
Prove that two successive Lorentz transformations are equivalent to another Lorentz transformation. Hence write down the Einstein's velocity addition relation.
