UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Calculate [J_x^2, J_y], [J_z^2, J_y] and [J^2, J_y], and then show that \langle J, m | J_x^2 | J, m \rangle = \langle J, m | J_y^2 | J, m \rangle

Find the energy levels of a spin S = \frac{3}{2} particle whose Hamiltonian is given by H = \frac{\alpha}{\hbar^2}(S_x^2 + S_y^2 - 2S_z^2) - \frac{\beta}{\hbar} S_z where \alpha, \beta are constants. Are these levels degenerate?

Consider a particle of mass m moving in the potential V(x) = \begin{cases} +\infty & ; x \le 0 \\ \frac{1}{2}m\omega^2 x^2 & ; x > 0 \end{cases} Estimate the ground state energy of this particle using the WKB method.

Consider a particle of mass m moving freely between x = 0 and x = a inside an infinite square well potential. Calculate the expectation values \langle x \rangle_n, \langle p \rangle_n, \langle x^2 \rangle_n and \langle p^2 \rangle_n, and compare them with their classical counterparts.

Determine the size of the hydrogen atom using uncertainty principle. Given that the potential energy of electron V = \frac{-e^2}{4\pi\varepsilon_0 a}, where a is the distance of the electron from the nucleus.

Derive the degeneracy of a harmonic oscillator g_n = \frac{1}{2}(n+1)(n+2).

Consider a particle of mass m and charge q moving under the influence of a one-dimensional harmonic oscillator potential. Assume that it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 - qEx Derive the energy expression and wave function of the n\text{th} excited state.

Deduce the thermodynamic relation : \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V Using the expression establish Clausius-Clapeyron latent heat equation \frac{dP}{dT} = \frac{L}{T(V_2 - V_1)}

(i) Prove that for perfect gas the specific heat at constant pressure C_p is always greater than the specific heat at constant volume by a constant value R. (ii) Consider for hydrogen, the density at NTP is 0.0899\text{ gm/lit}, molecular weight 2.016\text{ gm} and specific heat at constant pressure 6.85\text{ cal/gm}. Calculate the specific heat of hydrogen gas at constant volume. Given : J = 4.18 \times 10^7\text{ erg}.

Find out the boiling temperature of water at the top of Mount Everest. Given : Pressure at the top of Everest is 0.36\text{ atm}. The density of water vapour at 100^\circ\text{C} is 0.598\text{ kg/m}^3. The latent heat is 2.257 \times 10^3\text{ J/g}.

The volume of a mole of liquid \text{He}^4 is 27 \times 10^{-6}\text{ m}^3 and the mass of a \text{He}^4 atom is 6.65 \times 10^{-27}\text{ kg}. Assuming that liquid \text{He}^4 is an ideal Bose gas, calculate (i) the concentration of boson in this volume. (ii) Bose temperature.

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.

Define electromagnetic field strength tensor F_{\mu\nu}. Express it in terms of components of electric and magnetic fields.

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.

Calculate the speed and radius of \alpha-particles of energy 5\cdot 998\text{ MeV} emitted into a magnetic field B = 10^4\text{ Wb m}^{-2}. Given that the mass of proton is 1\cdot 673 \times 10^{-27}\text{ kg}.

An AC circuit consists of inductance of self-induction 'L', a capacitor of capacitance 'C' and a resistor of resistance 'R' in series. Calculate the impedance 'Z' of the circuit. Obtain resonance condition and define quality factor.

Current 'I' is passing through an infinite solenoid of radius R, with n turns per unit length. Find out the vector potential (i) inside the solenoid (ii) outside the solenoid.

State Biot–Savart law and write its mathematical form. In case of distributed current sources, write this expression for line current, surface current and volume current.

For a vector potential \vec{A}, \vec{\nabla} \cdot \vec{A} = -\frac{\mu_0}{4\pi} \cdot \frac{Q}{r^2} where Q is a constant of appropriate dimension. Calculate the corresponding scalar potential \phi (\vec{r}, t), that make \vec{A} and \phi to satisfy Lorentz gauge.