UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Briefly outline the theory of scattering of electromagnetic radiation by a bound electron and hence derive the conditions for Rayleigh scattering. How can you explain the blue of the sky?

(i) Using the laws of transformation of the electric field, \vec{\text{E}}, and the magnetic field, \vec{\text{B}}, show that (\text{E}^2 - \text{C}^2 \text{B}^2) is relativistically invariant. (ii) Suppose that in one inertial frame \vec{\text{B}} = 0 but \vec{\text{E}} \neq 0 (at some point P). Is it possible to find another inertial frame in which the electric field is zero at P ?

Consider electromagnetic fields in a medium with permittivity \varepsilon and permeability \mu. The fields are space and time dependent. Write down the full system of Maxwell's equations along with the constitutive equations. Then, derive the continuity equation from them. Using this continuity equation, show that the total charge of the universe is conserved.

Two cells \text{E}_1 of E.M.F. 2\text{ volts} and internal resistance 1\cdot 5\text{ ohms} and \text{E}_2 of E.M.F. 1\cdot 5\text{ volts} and internal resistance 1\text{ ohm}, are joined in parallel with like poles together (see the figure below). Calculate the current that would pass through a 5\text{-ohm} resistance joined in parallel with the cells.

Write down the formula for Planck's radiation law. Show that it reduces to Wien displacement law for shorter wavelengths and to Rayleigh – Jeans law for longer wavelengths.

A charge of -3\cdot 30\ \mu\text{C} is fixed in place. From a horizontal distance of 0\cdot 0455\text{ m}, a particle of mass 7\cdot 35 \times 10^{-3}\text{ kg} and charge -7\cdot 45\ \mu\text{C} is fired with an initial speed of 62\cdot 5\text{ m/s} directly towards the fixed charge. How far does the particle travel before it comes to rest ?

If we consider the Earth as a black body in thermal equilibrium, estimate the global temperature of our planet in terms of the temperature of the Sun, \text{T}_{\text{sun}}, its radius, \text{R}_{\text{sun}}, and distance D between the Earth and the Sun.

A uniformly charged sphere of radius R carries a total charge Q. Find the net force that the southern hemisphere exerts on the northern hemisphere. Express your answer in terms of the radius R and total charge Q.

State and prove Stefan – Boltzmann law for ideal gases in thermodynamics.

Discuss in brief the ultraviolet catastrophe. How did Planck solve this problem?

Three cells are connected in parallel with similar poles connected together with wires having negligible resistance. The emfs of the cells are 2, 1 and 4 volts respectively and the corresponding internal resistances are 4, 3 and 2 ohms. Calculate the current flowing through the 4 V cell.

Two conducting planes, intersecting at right-angles to each other, are kept at a potential \phi_0. Calculate the potential at a point in space if the total charge on a plane of area \alpha be Q.

Find the values of \mathrm{E} and \mathrm{H} on the surface of a wire carrying a current. By computing the Poynting vector, show that it represents a flow of energy into the wire.

Starting from the expression for the electrostatic potential \phi(\vec{r})=\frac{1}{4\pi\varepsilon_0}\int_V\frac{\rho(\vec{r}_0)}{|\vec{r}-\vec{r}_0|}\,dV_0 obtain Poisson's equation \nabla^2\phi=-\frac{\rho}{\varepsilon_0}. [Symbols have their usual meanings]

Why do we prefer to work with a critically damped ballistic galvanometer in a laboratory? What is external critical damping resistance?

Find the capacitance of two concentric spherical metal shells having radii a and b.

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

(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.

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.