UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Derive the Planck's radiation law for blackbody radiation using the Bose-Einstein distribution function. Explain how results from quantum statistics differ from classical results derived from the Rayleigh-Jeans law.

For a plane electromagnetic wave given by : \text{E}_{\text{z}} = \text{a}\cos \omega\text{x} \cos \omega\text{ct} \text{H}_{\text{y}} = -\text{a}\sin \omega\text{x} \sin \omega\text{ct} Find the value of the Poynting vector.

State and explain Kirchhoff's current law and Kirchhoff's voltage law. Derive these laws from the principles of charge conservation and energy conservation.

A rectangular coil consists of 50 closely wrapped turns and has dimensions of 0\cdot 5\text{ m} \times 0\cdot 4\text{ m}. It carries a current of 1\cdot 5\text{ A}. If a uniform magnetic field B = 0\cdot 1\text{ T} is applied such that the direction of the magnetic field makes an angle of 60^\circ with respect to the plane of the coil, what is the torque exerted on the coil by the magnetic field?

After highlighting the importance of the Biot-Savart law, show that the magnetic field of a current carrying long wire, at a point near it, is inversely proportional to the distance of the point from the wire.

Consider a long straight wire of length L carrying a current I. Determine the magnetic vector potential \vec{A} at a point P located at distance x from the wire.

A certain linear, homogeneous, isotropic, dielectric material has a relative permittivity, \varepsilon_{\text{r}} = 1\cdot 8. If potential \text{V} = -4000\text{y} volts in the material, then find :

(i) The electric flux density \text{D}, and

(ii) The polarisation \text{P}. Take vacuum permittivity \varepsilon_0 = 8\cdot 85 \times 10^{-12}\text{ farad/m}.

If volume charge density in free space varies as \rho_{\text{v}} = \frac{100\varepsilon_0}{\text{r}^{2/5}}, then using Poisson's equation, find potential \text{V}(\text{r}). It is assumed that \text{r}^2 \text{E}_{\text{r}} \rightarrow 0 when \text{r} \rightarrow 0, while \text{V} \rightarrow 0 at \text{r} \rightarrow \infty.

Assume the Sun to be a black body at temperature 5800\text{ K}. Use Stefan's law to calculate :

(i) The total energy emitted by the Sun per second, and

(ii) The energy reaching the top of the Earth's atmosphere. Given : \sigma = 5\cdot 672 \times 10^{-8}\text{ SI units}, Radius of the Sun = 7 \times 10^8\text{ m}, The distance of the Earth's atmosphere from the Sun = 1\cdot 5 \times 10^{11}\text{ m}.

Obtain the general boundary conditions for fields E, B, D and H at a boundary between two different media carrying charge density \sigma or a current density K.

Consider a uniformly magnetized sphere of radius a and magnetization \vec{M} = M_0 \hat{z} surrounded by a vacuum region. Obtain an expression for scalar magnetic potential for r < a.

A uniform plane wave with \vec{E} = E_x \hat{a}_x propagates in a lossless medium (\varepsilon_r = 4, \mu_r = 1, \sigma = 0) in the z-direction. Assume that E_x is sinusoidal with a frequency 100\text{ MHz} and has a maximum value of 10^{-4}\text{ (V/m)} at t = 0 and z = \frac{1}{8}\text{ (m)}. (i) Write the expression for instantaneous E for any t and z. (ii) Write the expression for instantaneous H. (iii) Determine the locations where E_x is a positive maximum, when t = 10^{-8}\text{ (s)}.

What is anomalous dispersion? How does the phenomenon of dispersion lead to the separation of white light into its constituent colours?

Briefly explain electronic, ionic and orientational polarisation. The polarisability of ammonia molecule is found to be 2.42 \times 10^{-39}\text{ C}^2\text{ m/N} and 1.74 \times 10^{-39}\text{ C}^2\text{ m/N} at 309\text{ K} and 448\text{ K} respectively. Calculate the orientation polarisability for each temperature.

The magnitude of the average electric field normally present in the Earth's atmosphere just above the surface of the Earth is about 150\text{ N/C}, directed radially inward, toward the centre of the Earth. What is the total net surface charge carried by the Earth? Assume the Earth to be a conductor. (The radius of the Earth is 6.37 \times 10^6\text{ m})

Using Maxwell's equations, obtain Poisson's equation and Laplace's equation. The region -\frac{\pi}{2} < \frac{z}{z_0} < \frac{\pi}{2} has a charge density \rho = 10^{-8} \cos \left( \frac{z}{z_0} \right) (\text{C/m}^3). Elsewhere the charge density is zero. Find the electric potential V and electric field E from the Poisson's equation.

How does Planck's law resolve the ultraviolet catastrophe predicted by classical physics? Calculate the average energy \bar{\varepsilon} of an oscillator of frequency 0.60 \times 10^{14}\text{ s}^{-1} at T = 1800\text{ K}, treating it as (i) classical oscillator and (ii) Planck's oscillator.

Distinguish between self-inductance and mutual inductance. Calculate the self-inductance of a solenoid of length 'l', area of cross-section 'A' having N turns.

A parallel plate capacitor with circular plate of radius (R = 5\text{ cm}) is being charged. Find out the displacement current. Given : \varepsilon_0 = 8.9 \times 10^{-12}\text{ C}^2/\text{Nm}^2, \frac{dE}{dt} = 10^{12}\text{ V/m}.

Obtain an expression for Poisson's and Laplace's equations in electrostatics.