UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Derive an expression for the specific heat of a solid on the basis of Debye's model. Show that it converges to Dulong and Petit's law at high temperatures.

State Gibbs phase rule. Show that for a 1-component closed thermodynamic system having two phases, the condition for equilibrium between the phases is that their specific Gibbs functions are equal.

In a non-charged current-free dielectric, \rho = 0 and \vec{J} = 0. Show that in this medium, electric (\vec{E}) and magnetic (\vec{H}) fields satisfy three-dimensional wave equations \nabla^2 \vec{E} = \varepsilon\mu \frac{\partial^2 \vec{E}}{\partial t^2} \quad \text{and} \quad \nabla^2 \vec{H} = \varepsilon\mu \frac{\partial^2 \vec{H}}{\partial t^2} Using Poynting theorem of electromagnetic theory, describe the significance of the vector \vec{P} = (\vec{E} \times \vec{H}) and the scalar u = \frac{1}{2} [\vec{B} \cdot \vec{H} + \vec{D} \cdot \vec{E}]

Using Planck's radiation formula, deduce Wien's displacement law.

For an arbitrary localised charged distribution, obtain an expression of electrostatic potential V in terms of multipole expansion.

A long solenoid of radius R and n turns per unit length carries a sinusoidal current I = I_0 \cos \omega t. Determine the magnitude of induced electric field (E) outside the solenoid.

State Biot-Savart law. Calculate the magnitude of axial magnetic induction due to a circular loop of area A carrying current I.

Consider the incidence of a plane-polarised electromagnetic wave at the interface of two media having dielectric permittivity and magnetic permeability (\varepsilon_1, \mu_1) and (\varepsilon_2, \mu_2) respectively. The interface is chosen to be x = 0 plane. \vec{K}_1, \vec{K}_2 and \vec{K}_3 represent the propagator vectors associated with the incident, refracted and reflected waves respectively. Using the boundary conditions on them, establish the Snell's laws of refraction.

A series R\text{-}L\text{-}C circuit is connected across a voltage source V = 100 \sin 300t. If R = 500\ \Omega, L = 1\text{ H} and C = 2\ \mu\text{F}, calculate the average power delivered to the circuit.

A plane-polarised electromagnetic wave is incident on the interface of two dielectrics having dielectric permittivity \varepsilon_1 and \varepsilon_2. Assume that the electric vector \vec{E} lies in the plane of incidence. Using the boundary conditions at the interface, obtain the expressions for the amplitude reflection coefficient (r_{11}) and the amplitude transmission constant (t_{11}). Using the components of the Poynting vector \vec{E} \times \vec{H} associated with the reflected and transmitted waves, obtain the expressions for reflection and transmission coefficients R_{11} and T_{11} respectively. Under what condition, r_{11} = 0 and t_{11} = 1?

Consider in the region 0 \le z \le 1\text{ m} an infinite slab made of a material with relative permeability, \mu_r = 3\cdot 5. If \vec{B} = (2y\hat{i} - 5x\hat{j}) \times 10^{-3}\text{ Wb/m}^2 within the slab, determine magnetisation \vec{M}.

A thin dielectric cylindrical rod of cross-section A is situated along z-axis from z = 0 to z = L. The polarisation of the rod is along its length and it is given by \vec{P} = (2z^2 + 5)\hat{z}. Calculate bound volume charge density at each end of the rod.

A cylindrical conductor is carrying a current along its axis which is assumed to be in z-direction. The current is uniformly distributed throughout its cross-section. Show that the vector potential \vec{A} associated with the magnetic induction due to the current-carrying cylindrical conductor is independent of z.

Show that the plane of polarisation is rotated through \theta = \frac{\delta}{2} = \frac{\pi d}{\lambda} (\mu_L - \mu_R) in optical rotation where symbols have their usual meanings.

What are the characteristics of stimulated emission? Show that in the optical region, stimulated emission is negligible compared to spontaneous emission.

A plane-polarised light is incident perpendicularly on a quartz plate cut with faces parallel to optic axis. Find the thickness of the quartz plate which introduces phase difference of 60^\circ between e- and o-rays.

Explain the phenomenon of interference in thin films. Why is the contrast better in brightness of fringes obtained from the interference of reflected light rays compared to the transmitted light rays?

Distinguish between high dispersive power and high resolving power.

Optical power of 1\text{ mW} is launched into an optical fiber of length 100\text{ m}. If the power emerging from the other end is 0\cdot 3\text{ mW}, calculate the fiber attenuation.

Obtain an expression for the resolving power of a plane transmission grating. Deduce the missing orders for a double-slit Fraunhofer pattern, if the slit widths are 0\cdot 16\text{ mm} and 0\cdot 8\text{ mm} apart.