UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Derive the rotational-vibrational energy levels of a diatomic molecule. Give the analysis of spectral lines.

Use Hund's rules to find the ground-state quantum numbers, L and S of (i) Carbon and (ii) Oxygen atoms.

Find the de Broglie wavelength of a neutron moving with a kinetic energy of 500\text{ eV}. (1\text{ eV} = 1\cdot 602 \times 10^{-19}\text{ J})

Determine the values of the total angular momentum for a 3d electron.

Calculate the most probable value of 'r' for an electron in the ground state of the hydrogen atom.

Solve the Schrödinger equation for an electron of mass m confined in a one-dimensional potential well of the form

\begin{align*} V &= 0 \text{ when } 0 \le x \le L \\ &= \infty \text{ when } x < 0; \ x > L \end{align*}

Obtain the discrete energy levels and the normalized eigen functions.

Deduce the commutation relations between the components of angular momentum operator \mathbf{L} [L_x, L_y] = i \hbar L_z [L_y, L_z] = i \hbar L_x [L_z, L_x] = i \hbar L_y using the commutation relations [x, p_x] = [y, p_y] = [z, p_z] = i \hbar

Give an account of Heisenberg's Uncertainty principle. Outline an idealised experiment to bring out its significance.

Plot the Fermi distribution function versus energy at temperatures T = 0 and T > 0. Explain the nature of the former curve on the basis of the Pauli principle.

Start from the equation T dS = C_p dT - T \left( \frac{\partial V}{\partial T} \right)_P dP and get the relation \left( \frac{\partial C_p}{\partial P} \right)_T = -T \left( \frac{\partial^2 V}{\partial T^2} \right)_P Hence, show that the third law of thermodynamics requires for the coefficient of thermal expansion of any substance to vanish at T = 0.

Discuss the consequence following from Joule's free expansion experiments in the context of the internal energy of an ideal gas.

Derive an expression for the total number of particles N in an ideal Bose gas at any temperature T. Hence, obtain the relation N_0 = N \left[ 1 - \left( \frac{T}{T_0} \right)^{\frac{3}{2}} \right] for the number of particles N_0 in the ground state at T < T_0, the Bose-Einstein condensation temperature, and draw N_0 versus T curve.

Prove that the work done on the charges by the electromagnetic force is equal to decrease in energy stored in the field, less the energy that flowed out through the surface. Explain what you mean by the Poynting vector.

Calculate the strength of the magnetic field to bring a proton nucleus and a {}^{13}\text{C} nucleus to resonate at this frequency. Magnetic moment of \text{proton} = 2\cdot 7927\ \mu_\text{N} and magnetic moment of {}^{13}\text{C} = 0\cdot 7022\ \mu_\text{N}. The NMR instrument operates at 30\cdot 256\text{ MHz}.

Discuss the physical significance of Planck's radiation law in the context of emergence of New Physics. If the blackbody energy density u(\omega) has a functional dependence like u(\omega) \propto x^3 (e^x - 1)^{-1}, where x = \frac{\hbar\omega}{kT}, derive Wien's displacement law and comment on its importance.

A plane conducting circular wire loop lies perpendicular to a uniform magnetic field B and its area S(t) is changed as S(t) = S_0 (1 - \alpha t), where 0 < t < \frac{1}{\alpha} (S_0 and \alpha are constants). The wire has resistance per unit length \rho\,\Omega\text{ m}^{-1}. Find the induced current through the wire. If the current in a certain coil varies at a rate of 50\text{ A s}^{-1}, the induced e.m.f. is V = 20\text{ volts}. What is the inductance of the coil?

Which of the Maxwell equations imply that there are no magnetic monopoles? Explain how the equations would get modified if magnetic monopoles would exist.

Using Ampere's law, derive the magnetic field of a toroid (N turns each carrying current I) of inner radius a and outer radius b at a distance r midway between a and b.

The electric potential of a grounded conducting sphere of radius a in a uniform electric field E = E_0\hat{z} is given as \phi(r, \theta) = -E_0 r \left[ 1 - \left( \frac{a}{r} \right)^3 \right] \cos\theta Find the expression for the surface charge density on the sphere.

In a simple AC circuit involving only a resistor R = 50\,\Omega and a voltage source V, find the linear frequency of the generator if V = 0 \cdot 5\,V_m (V_m is peak e.m.f.) at time t = \frac{1}{720}\text{ s} (assuming V = 0 at t = 0).