Using Kirchhoff's laws find currents in each branch of the circuit shown in the following diagram.

A Geiger tube consists of a wire a wire of radius 0.2\text{ mm} and length 12\text{ cm} and a co-axial metallic cylinder of radius 1.5\text{ cm} and length 12\text{ cm}. Find

(i) the capacitance of the system, and

(ii) the charge per unit length of the wire when the potential difference between the wire and the cylinder is 1.2\text{ kV}. (Assume the dielectric constant of the gas in the tube to be 1)

Show that the potential energy of a charge Q uniformly distributed throughout the sphere of radius R is given by PE = \frac{3}{5} \frac{Q^2}{4\pi\varepsilon_0 R}

Calculate the electric field for a point on the axis of a uniform ring of a charge 'q' and radius a. Show that the maximum value occur at x = \pm a/2.

A bulb filament is constructed from a tungsten wire of length 2\text{ cm} and diameter 50\text{ }\mu\text{m}. It is enclosed in a vacuum bulb. What temperature does it reach when it is operated at a power of 1 watt? Given:

(i) Emissivity of tungsten \varepsilon = 0.4

(ii) Stefan's constant \sigma = 5.67 \times 10^{-8}\text{ watt/m}^2\text{ K}^4.

What are vector and scalar potentials for the electromagnetic field? Are they unique? Explain what are Coulomb's and Lorentz gauges. Derive the electromagnetic wave equation in Lorentz gauge and show that it is equivalent to Maxwell's equation.

What is gauge transformation? Define coulomb gauge. Derive the equation for vector potential under coulomb gauge.

From Planck's radiation law, derive Wien's displacement law and Rayleigh Jean's law.

Explain the use of a parallel resonance circuit

(i) as a rejector circuit

(ii) for current amplification.

An inductance is connected to 6 volt battery through a resistance R. What is the steady state current in the circuit? Alter what time the battery would be delivering one half its steady state current?

A conducting sphere of radius `a' is placed in a uniform electric field E_0. Using the method of images, show that the potential function is given by \phi = -E_0 \left(r - \frac{a^3}{r^2}\right) \cos \theta.

What value of inductance has to be used so that a lamp with rating of 200 volt and 10 Amperes lights the same way with 250 V source at 50 Hz.

Calculate the electric field as a function of position due to a dipole whose potential is given by V = \frac{P \cos \theta}{4 \pi \epsilon_0 r^2} \quad \text{where } r = \sqrt{x^2 + y^2} The dipole is located at the origin of the x, y axis system.

Derive the energy continuity equation for electromagnetic waves using the poynting vector.

Define scalar and vector potentials. Recast Maxwell's equations in terms of these potentials.

Calculate the voltage drops across R and across L.

For an R-L- C series resonant circuit, show that f_0 = \sqrt{f_1 f_2}, where f_0 is the resonance frequency and f_1, and f_2 are the half-power frequencies. Is this relation true for a parallel resonance circuit in which R, L and C are connected in parallel to each other? Explain.

A series R-L circuit has a constant d.c. voltage V applied at the time t = 0 by closing a switch. Derive an expression for the current in the circuit.

Current is flowing in a single-turn circular coil of radius 0.1\text{ m} such that at a point 5\text{ mm} from the centre of the coil on the axis perpendicular to the plane of the coil the magnetic field is 0.1\text{ mT}. Find the current in the coil. Hence calculate the magnetic moment of the current carrying coil.

At what time are these voltage drops equal?