UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Use the uncertainty principle to estimate the ground state energy of a linear harmonic oscillator.

A lead ball of mass 0\cdot 1\text{ g} is thrown with a velocity of 10^3\text{ cm/sec} through a hole 1\text{ cm} in radius. Calculate the uncertainty in the angle of emergence.

Show that the Helmholtz free energy of a system never increases in any isothermal-isochoric transformation.

The Einstein theory of specific heat of solids gives the expression C_V=\frac{3Nk_Bx^2e^x}{(e^x-1)^2} where x=\frac{\theta_E}{T} with \theta_E as the Einstein temperature.

(i) Mention Einstein's assumptions in deriving it. Also obtain low- and high-temperature limiting expressions for it.

(ii) Give schematic plot of \dfrac{C_V}{3N k_B} versus \dfrac{T}{\theta_E} and comment on the validity of expressions in (i) in comparison with experiments.

Consider non-equilibrium situation for a system in which the population inversion has been achieved. Explain that such a system can be treated as if it has negative absolute temperature.

Show that both Fermi-Dirac and Bose-Einstein distributions reduce under certain condition in a form which gives the total number of particles as N = A\int_0^\infty \sqrt{\varepsilon}e^{-\beta\varepsilon}\,d\varepsilon where A is a constant and \beta = 1/k_{\mathrm B}T. Further show that this expression is just the same as that obtained from the Maxwellian speed distribution.

Show that if an ideal gas is compressed isothermally its compressibility is \frac{1}{p}, whereas if it is compressed adiabatically its compressibility is \frac{1}{\gamma p} where \gamma = C_p / C_v.

Derive the gas equation when it undergoes adiabatic process.

A motor car tyre has a pressure of 2\text{ atmospheres} at the room temperature of 27^\circ\text{C}. If the tyre suddenly bursts, find the resulting temperature.

Find the efficiency of a Carnot's engine working between 127^\circ\text{C} and 27^\circ\text{C}.

Explain the thermodynamic behaviour of an ideal Fermi gas. What are fermions ?

What are bosons ? Based on Bose-Einstein statistics, derive the expression for Bose-Einstein condensation.

Establish the relation \left(\frac{\partial T}{\partial V}\right)_P=-\left(\frac{\partial P}{\partial S}\right)_T and then derive \left(\frac{\partial C_P}{\partial P}\right)_T=-T\left(\frac{\partial^2 V}{\partial T^2}\right)_P. Hence show that the heat capacity C_P of an ideal gas is independent of pressure P.

If considered a blackbody as a radiation-filled cavity at a uniform temperature T, demonstrate with an appropriate figure the electromagnetic wave patterns inside the cavity of length L for wavelengths \lambda = L, \frac{3}{2} L and 2L.

The volume between two concentric conducting spherical surfaces of radii a and b (a<b) is filled with an inhomogeneous dielectric with \epsilon=\epsilon_0/(1+cr), where c is a constant and r is the radial coordinate. A charge +Q is placed on the inner surface, while the outer surface is grounded. Determine-

(i) \vec{D} in the region a<r<b;

(ii) capacitance of the device;

(iii) polarization charge density in the region a<r<b;

(iv) surface polarization charge densities at r=a and r=b.

Assume \vec{E}=0 inside a perfect conductor. Elaborate on any other four electrostatic properties that arise from this property.

A resistance R and a lossless capacitor C are connected through a switch. The capacitor is charged to potential V_0, and the switch is closed at t = 0. Prove that the energy stored in the capacitor is equal to the energy dissipated in the resistor.

What is a displacement current? Prove that lines of conduction current plus displacement current are continuous.

The dielectric constant of a medium is 3. The Electric field in the dielectric is 10^6\text{ Vm}^{-1}. What are the electric displacement and polarization ?

Explain the physical significance of the Poynting vector \bar{S}. What is represented by the closed integral \oint \bar{S} \cdot d\bar{a} for a closed surface of area \bar{a} ?