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.
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.
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.
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.
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.
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+).
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.
Each square meter of the Sun's surface radiates energy at the rate of 6{\cdot}3 \times 10^7\text{ J/m}^2\text{ /s} and Stefan constant is 5{\cdot}669 \times 10^{-8}\text{ W/m}^2\text{ /K}^4. Find the temperature of the Sun's surface. Assume that the Stefan's law applies to the radiation.
Define a perfect blackbody. State Kirchhoff's law of radiation. What are the limitations of Kirchhoff's law?
Write the equation for total scattering cross-section for electromagnetic waves. Under what condition the scattering is known as Rayleigh scattering? Write the importance of Rayleigh scattering in nature.
Show that the energy density in electrostatic field is given as u = \frac{1}{2} \bar{D} \cdot \bar{E} Further, show that the dielectric constant is a symmetric tensor except when the field energy vanishes identically.
What are ferromagnetic materials? By suitable figure, illustrate magnetic domain and magnetization curve for a ferromagnetic material.
In what way Maxwell's concept towards electric field and magnetic field is different from Faraday's concept? Derive magnitude of displacement current in a parallel-plate capacitor and account it for the continuous path for the charges across the capacitor.
Derive Poisson's equation. Under what conditions it takes the form of Laplace's equation? If \phi_1, \phi_2, \cdots, \phi_n are solutions of Laplace's equation, then show that \phi = \lambda_1 \phi_1 + \lambda_2 \phi_2 + \cdots + \lambda_n \phi_n is also a solution. Here \lambda\text{s} are arbitrary constants.
Calculate the magnitude of Poynting vector at the surface of the Sun. Given that the power radiated by the Sun = 3{\cdot}8 \times 10^{26}\text{ watts} and the radius of the Sun = 7 \times 10^8\text{ m}.
What is a transformer? Is it for a.c. or d.c.? With a suitable diagram and representative symbol, explain the construction and necessary terms used in a transformer.
A charge of -3\cdot 30\ \mu\text{C} is fixed in place. From a horizontal distance of 0\cdot 0455\text{ m}, a particle of mass 7\cdot 35 \times 10^{-3}\text{ kg} and charge -7\cdot 45\ \mu\text{C} is fired with an initial speed of 62\cdot 5\text{ m/s} directly towards the fixed charge. How far does the particle travel before it comes to rest ?
A uniformly charged sphere of radius R carries a total charge Q. Find the net force that the southern hemisphere exerts on the northern hemisphere. Express your answer in terms of the radius R and total charge Q.
Write down the formula for Planck's radiation law. Show that it reduces to Wien displacement law for shorter wavelengths and to Rayleigh – Jeans law for longer wavelengths.
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.
