The wave function of a spin \frac{1}{2} particle is given by \begin{pmatrix} -i \\ 2 \end{pmatrix}. What is the probability of finding the particle in the spin-up state?

The de Broglie wavelength of a particle in thermal equilibrium at a temperature of 27\text{ }^\circ\text{C} is \lambda. At what temperature will it be \dfrac{\lambda}{2}?

A gas has only two particles a and b. Show with the help of diagrams how these two particles can be arranged in three energy states 1, 2, 3 using (i) Maxwell--Boltzmann, (ii) Fermi--Dirac and (iii) Bose--Einstein statistics.

The average kinetic energy of hydrogen atoms in a certain stellar atmosphere, assumed to be in thermal equilibrium, is 1\cdot 2\text{ eV}. Calculate the ratio of the number of atoms in the second excited state (n = 3) to the number in the ground state.

Two solids A and B have Debye temperatures 200\text{ K} and 300\text{ K} respectively. At T = 20\text{ K}, compare their specific heats.

What are the limitations of the first law of thermodynamics? One mole of a gas, assumed to be perfect, at 0\text{ }^\circ\text{C} is heated at constant pressure till its volume is twice its initial value. Calculate the amount of heat absorbed. Given, C_v = 20\cdot 9\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1} and R = 8\cdot 3\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}.

Calculate the pressure at which water will boil at 150\text{ }^\circ\text{C}, given that the change in specific volume when 1\text{ gram} of water is converted into steam is 1676\text{ cm}^3. Given, latent heat of vaporization for steam = 540\text{ cal per gram}, J = 4\cdot 2 \times 10^7\text{ ergs/cal} and one atmospheric pressure = 10^6\text{ dynes/cm}^2.

Why does the specific heat of solids depend on material at low temperatures but become independent of material at high temperatures?

Blackbody radiation in a cavity at 2000\text{ K} is subject to isothermal reversible expansion through 10^3\text{ cm}^3. Calculate (i) the heat transferred and (ii) the work done. If the initial volume was 10\text{ cm}^3 and expansion had been adiabatic, calculate the change in temperature of the radiation. (Given, Stefan--Boltzmann constant, \sigma = 5\cdot 672 \times 10^{-8}\text{ J}\cdot\text{m}^{-2}\cdot\text{K}^{-4}\cdot\text{s}^{-1})

The magnetic field H of an electromagnetic wave travels in the -a_z direction in free space with a phase shift constant of 30\text{ rad/m} and an amplitude of \left(\frac{1}{3\pi}\right)\text{ A/m}. If the field has the direction -a_y when t = 0 and z = 0, write the suitable expressions for E and H. Determine the frequency and wavelength of the wave.

A very long solenoid of radius a, with n turns per unit length, carries a current I_s. Coaxial with the solenoid, at radius b \gg a, is a circular ring of wire with resistance R. When the current in the solenoid is gradually reduced, a current I_r is induced in the ring. Calculate I_r in terms of \frac{dI_s}{dt}. Also, calculate the power dissipated through Joule effect and the electric field \vec{E} near the solenoid.

A wire of length 2\text{ m} is perpendicular to the X\text{--}Y plane. It is moved with velocity \vec{V} = 2\hat{i} + 3\hat{j} + \hat{k}\text{ m/s} through a region of uniform magnetic induction \vec{B} = \hat{i} + 2\hat{j}\text{ Wb/m}^2. Compute the potential difference induced between the ends of the wire.

The space between the plates of a parallel-plate capacitor is filled with a dielectric material whose susceptibility varies linearly from 0 at the bottom plate (x = 0) to 1 at the top plate (x = d). The capacitor is connected to a battery of voltage V. Calculate all the bound charges and check that the total charge is zero. Assume that the battery is connected in such a way that the electric field points along the x-direction while the free charge density, \sigma_f, is positive at the bottom plate and negative at the top plate.

A charge of -3\cdot 30\text{ }\mu\text{C} is fixed at a point. From a horizontal distance of 0\cdot 0455\text{ m}, a charged particle of mass 7\cdot 35 \times 10^{-3}\text{ kg} and charge -7\cdot 45\text{ }\mu\text{C} is fired with an initial velocity 62\cdot 5\text{ m/s} directly towards the fixed charge. How far does this charge travel before its speed becomes zero?

Calculate the effective resistance of the following combination of resistances as shown in the figure and determine the voltage drop across each resistance when a potential difference of 120\text{ volts} is applied between points A and B :

Discuss the origin of hysteresis in ferromagnetic materials on the basis of domain theory. Explain how domain wall motion and pinning lead to energy loss during cyclic magnetization.

In a homogeneous non-conducting region where \mu_r = 1, find \epsilon_r and \omega, if \begin{align*} \vec{E} &= 30\pi e^{j[\omega t - (4/3)y]} \vec{a}_z \quad (\text{V/m}) \ \vec{H} &= 1\cdot 0 e^{j[\omega t - (4/3)y]} \vec{a}_x \quad (\text{A/m}) \end{align*}

In Newton's ring experiment, the space between planoconvex lens and glass plate is filled with a liquid of refractive index \mu. Explain how interference pattern changes as compared to air. Starting from the condition for interference, derive an expression for the radius of the n\text{th} dark ring. Discuss how the ring system changes if the refractive index increases.

What is meant by achromatic combination of lenses? Derive the condition for achromatization of a pair of lenses separated by a distance x. The two lenses have different dispersive powers.

The aperture width of a laser light source of wavelength 6000\text{ \AA} is 3\text{ mm} and its power is 20\text{ mW}. Calculate the light intensity at a distance of 200\text{ m} from the light source.