UPSC Physics Optional PYQs (CSE & IFoS)PYQs

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.

Prove Stefan's law of radiation from thermodynamic considerations.

State and explain Stefan-Boltzmann Law. Show that \log P = \log K + 4\log R, where P is the power emitted by black body and R is the resistance of the black body, K is a constant.

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}.

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}.

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.

How does the population inversion in an active medium lead to the amplification of light in a laser ? Explain in details.

What are the step index and graded index optical fibers ? What are the conventional optical windows for light propagation through optical fibers ?

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.

Draw a neat diagram to show how light propagates through a graded index optical fiber from the entrance to the fiber end and explain.

Explain how you can construct a zone plate from Fresnel's half period zone. Show that a zone plate has multiple foci.

Write down the mathematical representation of Fermat's principle and explain all the notations used. With the help of a neat diagram, show that this principle can be used to obtain the law of refraction : n_1 \sin \theta_1 = n_2 \sin \theta_2 , where n_1 and n_2 are the refractive indices of the two media while \theta_1 and \theta_2 are the angles of incidence and refraction of the light beam.

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.

If the system matrix for a thin lens of focal length f is given by : S = \begin{bmatrix} 1 & -1/f \\ 0 & 1 \end{bmatrix} , show that the system matrix for a combination of two thin lenses of focal lengths f_1 and f_2 separated by a distance d can be obtained as S_{12} = \begin{bmatrix} 1 - d/f_2 & -(1/f_1 + 1/f_2 - d/f_1 f_2) \\ d & 1 - d/f_1 \end{bmatrix}

How does holography differ from conventional photography? What are the requirements for the formation and reading of a hologram?

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.