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Show that the energy flow due to a plane electromagnetic wave propagating along z-direction in a dielectric medium is given by \hat{z}\frac{k}{\omega\mu}E_0^2\cos^2(kz-\omega t), where \mathbf{k} and \omega are the propagation vector and angular frequency, E_0 is electric field amplitude, \mu is the relative permeability of the medium.
Starting from Maxwell's equation, obtain the wave equation for the electric field \vec{E} in free space and appropriate wave equation for the electric field \vec{E}=E_z(x,y,z)\hat{z}.
In deriving radiation laws, we consider a cubical container of volume V containing a photon gas in equilibrium. Calculate the differential number of allowed normal modes of frequency \omega.
A sphere of homogeneous linear dielectric material is placed in an otherwise uniform electric field \bar{E}_o. Find the electric field inside the sphere.
Explain Kirchoff's circuit laws. Using Kirchoff's laws find currents I_1, I_2 and I_3 in the circuit shown below for R_1 = 100\ \Omega, R_2 = 200\ \Omega, R_3 = 300\ \Omega, E_1 = 3\text{ V}, E_2 = 4\text{ V}.
Compute the electrostatic energy of a conducting solid sphere of radius R and having total charge Q using the expression for the electrostatic energy in terms of the electric field \bar{E} of the above sphere.
When connected in series, L_1, C_1 have the same resonant frequency as L_2, C_2 also connected in series. Prove that if all these circuit elements are connected in series, the new circuit will have the same resonant frequency as either of the circuits first mentioned.
Using Ampere's Law and continuity equation, show that the divergence of the total current density is zero.
For initial current conditions I=I_0 and \frac{dI}{dt}=0 at t=0, show that the time dependent current in the critical damping case for an LCR circuit is given by I=I_0\left(1+\frac{\gamma t}{2}\right)e^{-\gamma t/2} where \gamma=\frac{R}{L}, \omega_0^2=\frac{1}{LC}, \omega=\sqrt{\omega_0^2-\frac{\gamma^2}{4}} and \tan\delta=\frac{-\gamma}{2\omega}.
How does holography differ from conventional photography? What are the requirements for the formation and reading of a hologram?
Starting with the rate equations for matter-radiation interaction, show that the ratio of the Einstein's A and B coefficients is proportional to the third power of the frequency of radiation.
Draw a neat diagram to show how light propagates through a graded index optical fiber from the entrance to the fiber end and explain.
What are the step index and graded index optical fibers ? What are the conventional optical windows for light propagation through optical fibers ?
How does the population inversion in an active medium lead to the amplification of light in a laser ? Explain in details.
A left circularly polarized beam (\lambda = 5893^\circ\text{A}) is incident normally on a calcite crystal (with its optic axis cut parallel to the surface) of thickness 0.005141\text{ mm}. What will be the state of polarization of the emergent beam ? Justify your answer. Here the refractive indices for the ordinary and the extraordinary rays are 1.65836 and 1.48641 respectively.
Explain the working principle of a 3-level laser with a specific example. Comment on why the third level is needed.
For a multimode step index optical fibre, the core refractive index is 1.5 and fractional index difference is 0.001. Calculate the pulse broadening for 1\,\mathrm{km} length of the fibre. Over a length of 2\,\mathrm{km} of the fibre, calculate the minimum pulse separation that can be transmitted without overlap.
What is the physical significance of Einstein's A-coefficient? Explain why it is more difficult to achieve Lasing action at X-ray wavelength than at infra-red wavelength.
Explain how you can construct a zone plate from Fresnel's half period zone. Show that a zone plate has multiple foci.
Considering a plane transmission diffraction grating, where d is the distance between two consecutive ruled lines, m as the order number and \theta as the angle of diffraction for normal incidence, calculate the angular dispersion \frac{d\theta}{d\lambda} for an incident light of wavelength \lambda.