UPSC Physics Optional PYQs (CSE & IFoS)PYQs

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 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.

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

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}.

State Gauss's Law of electrostatics both in integral form and differential forms. Two charged spheres of radius R each, have their centres a distance d apart such that d < 2R. One of the spheres has a uniform positive charge density \rho per unit volume while the other has opposite charge density -\rho. Show that the electric field in the region of overlap between two spheres is uniform.

Two identical black bodies A and B are respectively at temperatures T and 2T respectively. The heat energy radiated by A is collected for one minute and is used to heat a mass of water. The temperature of water is found to rise by 0.25 K. If the heat radiated by B were to be collected for one minute and used to heat the same mass of water, what would be the rise in the temperature of water ? Assume, specific heat of water to be temperature independent.

A point charge + q is held above a grounded conducting plane located at z = 0. If the position of the charge is (0, 0, d) obtain an expression for the induced charge density on the plane as a function of coordinates x and y.

An infinite ladder network of resistances is connected across the points A and B, as shown in the above figure. Each of the resistance is 1\Omega. Calculate the effective resistance between points A and B.

Find the vector potential due to a line segment from x = a to x = b carrying a current I at a point P which is at a distance d from the line segment.

Show that the interaction energy of two magnetic dipoles \bar{m}_1 and \bar{m}_2 separated by a displacement \bar{r} is given by U = \frac{\mu_0}{4\pi} \cdot \frac{1}{r^3} \left[ \bar{m}_1 \cdot \bar{m}_2 - 3(\bar{m}_1 \cdot \hat{r})(\bar{m}_2 \cdot \hat{r}) \right]. If two magnetic dipoles are held at a fixed distance apart, but allowed to rotate freely, what would be the configuration for stable equilibrium ?

Consider a plane electromagnetic wave entering a rarer medium from a denser medium. Show that the Brewster angle is less than the total internal reflection angle.

State Faraday's Law of induction in terms of emf and magnetic flux. Use Stoke's theorem to express the law in differential form. A uniform, time varying magnetic field \bar{B} = \bar{B}_0(1 + \alpha t)\hat{k} fills a circular region of radius R lying in the x-y plane with the centre at origin. Here \bar{B}_0 and \alpha are appropriately dimensioned constants. Find the induced electric field at a distance r from the centre where r < R.

Explain how Maxwell modified Ampere's law of magnetostatics by introducing the concept of displacement current. How does it resolve the paradox of a charging capacitor ?

A plane electromagnetic wave is travelling in vacuum along a direction which makes an angle of 45^\circ each with the positive x and z-axes. The electric field at time t is given by the expression \bar{E} = E_0 (2 \cos \omega t \hat{j} - \sin \omega t (\hat{i} - \hat{k})) where \hat{i}, \hat{j} and \hat{k} are unit vectors along the x, y and z directions respectively and E_0 = 100 \text{ v/m}. Find the magnetic field vector \bar{B} at time t and obtain the average power transmitted per unit area by the wave.

A plane electromagnetic wave travelling in z-direction and polarized in x-direction is incident normally from left on to an interface between two media at z = 0. The medium to the left (z < 0) has a refractive index n_1, while that to the right is of refractive index n_2. The magnetic permeabilities of both the media are equal and \mu_1 = \mu_2 = \mu_0. Obtain expressions for reflection and transmission coefficients (R and T) and show that R + T = 1.

State Amperes law of magnetostatics. Using this law, find the magnetic field at a point due to an infinitely long filamentary current.

Why does a spinning nucleus process in a magnetic field ? Explain the underlying principle of nuclear magnetic resonance (NMR) spectroscopy. Calculate the radio frequency at which NMR occurs in water kept in a uniform magnetic field of 2.4 T, the magnetic moment of proton being 2.793\\ \mu_N.

Obtain the solution of the Laplace equation in cylindrical coordinates.

Explain the Rayleigh Scattering of light. Show that the energy density of light scattered from an isotropic homogenous medium of a gas is inversely proportional to the fourth power of the wavelength of the incident light.

Derive Poisson equation starting from the Coulomb's law for a set of point charges.