State and prove Stefan -- Boltzmann law for ideal gases in thermodynamics.

If we consider the Earth as a black body in thermal equilibrium, estimate the global temperature of our planet in terms of the temperature of the Sun, \text{T}_{\text{sun}}, its radius, \text{R}_{\text{sun}}, and distance D between the Earth and the Sun.

Consider electromagnetic fields in a medium with permittivity \varepsilon and permeability \mu. The fields are space and time dependent. Write down the full system of Maxwell's equations along with the constitutive equations. Then, derive the continuity equation from them. Using this continuity equation, show that the total charge of the universe is conserved.

(i) Using the laws of transformation of the electric field, \vec{\text{E}}, and the magnetic field, \vec{\text{B}}, show that (\text{E}^2 - \text{C}^2 \text{B}^2) is relativistically invariant. (ii) Suppose that in one inertial frame \vec{\text{B}} = 0 but \vec{\text{E}} \neq 0 (at some point P). Is it possible to find another inertial frame in which the electric field is zero at P ?

(i) The equation for an alternating current is I = 42\cdot 42 \sin (314t). Find the following : Maximum value of current, Frequency, RMS value and Average value

(ii) A condenser of capacity 1~\mu\text{F} is first charged and then discharged through a resistance of 1\text{ M}\Omega. Calculate the time in which the charge on the condenser will fall to 50\% of its initial value.

(iii) Consider the displacement vector \vec{D}, given by \vec{D} = (10xyz^2 + 4x)\hat{i} + (5x^2 z^2)\hat{j} + (10x^2 yz)\hat{k}\text{ nC/m}^2 Find the total charge enclosed in a cube of volume 10^{-9}\text{ m}^3 located at the point (1, 2, 3).

Explain Planck's formula for black-body radiation. Use this formula to show that energy radiated per unit time per unit area in the range d\lambda of \lambda is e(\lambda, T) = \left( \frac{2\pi c^2 h}{\lambda^5} \right) (e^{\beta hc / \lambda} - 1)^{-1} d\lambda

Two resistors of 600~\Omega and 800~\Omega are connected in series with a 7\text{ volts} battery. An ammeter of 10~\Omega resistance is used to measure current.

(i) What will be the reading in the ammeter?

(ii) Similarly if a voltmeter of 10000~\Omega resistance is used to measure the potential difference across the 600~\Omega resistor, what will be the reading in the voltmeter?

Using the set of four Maxwell's equations, obtain the Lorentz condition relation between the scalar potential \phi and vector potential A, i.e., \nabla \cdot A + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0 and discuss the gauge transformations, Lorentz gauge and Coulomb gauge.

(i) Show that the electric and magnetic energy densities in a plane travelling wave are equal. Also prove that the total energy density = \varepsilon_0 E^2 = \mu_0 H^2.

(ii) Deduce the equation of continuity based on Maxwell's equations.

In free space, the electric field of electromagnetic wave is given by \vec{E}(x, t) = 100 \cos (\omega t - kx) \hat{y}\text{ volt/metre} Find the average power crossing a circular area of radius 2\text{ metres} in the yz-plane.

Construct the Hamiltonian of a charged particle with charge q and mass m moving with the velocity \vec{v} in the external electromagnetic field, \vec{E}=E_0 \hat{i}, \vec{B}=B_0 \hat{k}, where E_0 and B_0 are constants.

State and explain Biot-Savart law. Obtain an expression for the magnetic field at the center of a circular loop of radius r metres, carrying a current of I amperes.

Obtain an expression for the electric potential and electric field strength at a point due to an electric dipole.

There is a potential gradient of 100\text{ V/m} normal to the surface of the earth. Assuming the earth to be a charged sphere of radius 6370\text{ km}, find the total charge on the earth.

How does one explain the observed spectrum of black-body radiation using Planck's quantum hypothesis ? State and obtain Wien's displacement law. Also explain the important features of this law.

In a one-dimensional device, the charge density is given by \rho_V = \rho_0 \frac{x}{a}. If E = 0 at x = 0 and V = 0 at x = a, find V and E using Laplace equation of electrostatics.

State and explain the Biot-Savart law. Derive an expression for the magnetic field at a point due to an infinitely long straight current carrying conductor.

Use the method of electric images to find the electric field on the surface of a grounded conducting sphere.

(i) State Faraday's law of electromagnetic induction and prove that it can be expressed in the following vector form : \mathrm{Curl}~\vec{E} = -\frac{\partial \vec{B}}{\partial t} with \vec{E} and \vec{B} being the electric and magnetic fields. (ii) A coil of 10 turns has dimension 9~\text{cm} \times 7~\text{cm}. It rotates at the rate of 15\pi~\text{rad/sec} in a uniform field whose flux density is 0\cdot 6~\text{weber/m}^2. What is the maximum e.m.f. induced in the coil ?

(i) Define and explain the significance of the quality factor of an electrical machine. (ii) Discuss in brief, the working principle of a transformer.