Calculate the probability of an electron occupying an energy level 0.02\,\mathrm{eV} above the Fermi level at T=300\,\mathrm{K}

How many nitrogen molecules must strike a 1\text{ cm}^2 surface each second to exert a pressure of 1\text{ atmosphere} ? (Assume the molecules are all moving at same speed, corresponding to a temperature of 300\text{ K}, and at an angle of 45^\circ to the wall). [Molecular mass of \text{N}_2 is 28\text{ u}]

Calculate the work done in expanding one mole of an ideal gas at 127^\circ\text{C} to double its initial volume.

Using the expression for internal energy U = 3N \frac{\hbar\omega}{e^{\hbar\omega/k_B T}-1}, show that Einstein specific heat capacity is given by; C = 3R \left(\frac{\hbar\omega}{k_B T}\right)^2 \frac{e^{\hbar\omega/k_B T}}{\left(e^{\hbar\omega/k_B T}-1\right)^2} Also show that Einstein specific heat capacity given above is proportional to e^{-\hbar\omega/k_B T} at very low temperature.

Write a brief note on Chandrasekhar Limit.

Calculate the efficiency of an engine having compression ratio 13.8 and expansion ratio 6 and working on diesel cycle. Given \gamma = 1.4.

Consider one gm of ice at a temperature T_1\text{ K}. Show that when this ice changes into steam at a temperature T_2\text{ K}, the total gain in entropy is \Delta S = \frac{L_i}{T_1} + C \log_e \left( \frac{T_2}{T_1} \right) + \frac{L_s}{T_2} where L_i is latent heat of ice, C is specific heat of water, L_s is latent heat of steam. T_1 = 273\text{ K}.

Calculate the critical constants for \mathrm{CO_2} for which the Van der Waals constants are given by a=0.0072 and b=0.002. Also calculate the Boyle's temperature of \mathrm{CO_2}. The unit of pressure is atmosphere and the unit of volume is that of a gm-mole of the gas at NTP.

A gas of N spinless Bose particles of mass m is enclosed in a volume V at a temperature T.

(i) Find an expression for the density of single-particle state D(\varepsilon) as a function of the single-particle energy \varepsilon. Sketch the result.

(ii) Write down an integral expression which implicitly determines \mu(T). Referring to your sketch in (i), determine in which direction \mu(T) moves as T is lowered.

Write Maxwell's equations in differential and algebraic forms. Give the physical significance of each. Show that the equation of continuity is contained in Maxwell's equations.

Show that in an A.C. circuit with inductor only, current lags behind emf in phase by \dfrac{\pi}{2}. If the current through 0\cdot 5\text{ Henry} inductor varies sinusoidally with an amplitude of 10\text{ amp} and a frequency 50\text{ Hz}, calculate the potential difference across the terminals of the inductor.

Define gauge transformation. Under what condition can the gauge transformation be regarded as restricted gauge transformation ? Further, generate the symmetrical forms of Maxwell's equations in terms of scalar and vector potentials under Lorentz gauge.

What are geomagnetic storms and why are they potentially disruptive to power grids ?

Calculate the skin depth of electromagnetic waves of 1\,\mathrm{MHz} incident on a good conductor having \sigma = 5.8 \times 10^{7}\,\mathrm{S\,m^{-1}}. Assume that inside the conductor \mu = \mu_0 = 4\pi \times 10^{-7}\,\mathrm{H\,m^{-1}}.

The impedance function of a parallel R, L and C circuit has poles loaded at - 3 \pm 4j\text{ radians/second}. If the value of \text{L} = 1\text{ Henry}, determine the following :

(i) The value of R and C.

(ii) What is the quality factor of this circuit ? Derive an expression for series and parallel circuit.

Justify the importance of method of images. Using this method, evaluate induced charge density on the surface of a grounded conducting sphere when a point charge is placed near it.

Define a black body and evaluate Planck's radiation law. Show that at high temperature, this law resembles Rayleigh -- Jeans law.

In the circuit as shown in figure :

where V = 10\text{ V}, R = 10\ \Omega, L = 1\text{ H}, C = 10\ \mu\text{F}, and v_c(0) = 0, find i(0+), \dfrac{di}{dt}(0+) and \dfrac{d^2i}{dt^2}(0+).

A region 1, z<0, has a dielectric material with \varepsilon_r=3.2 and a region 2, z>0 has a dielectric material with \varepsilon_r=2.0. Let the displacement vector in the region 1 be, \vec{D}_1=-30a_x+50a_y+70a_z\,\mathrm{nC\,m^{-2}}. Assume the interface charge density is zero. Find in the region 2, the \vec{D}_2 and \vec{P}_2, where \vec{P}_2 is the electric polarization vector in the region 2.

Write the Laplace's equation in Cartesian, spherical polar and cylindrical coordinates in one-dimensional space. Using a suitable Laplace's equation out of these three, find the capacitance of parallel plate capacitor.