(i) Considering an isotropic, linear, non-conducting, non-magnetic and inhomogeneous dielectric medium with \vec{D}=\epsilon\vec{E}=\epsilon_0n^2(x,y,z)\vec{E}, show that the electromagnetic wave equation for the field \vec{E} is given by \nabla^2\vec{E}+\vec{\nabla}\left(\frac{1}{n^2}\vec{\nabla}n^2\cdot\vec{E}\right)-\mu_0\varepsilon_0n^2\frac{\partial^2\vec{E}}{\partial t^2}=0. \setcounter{enumi}{1}

(ii) Write down the scalar equation for E_x from the above equation.

(iii) Interpret physically the situation if we move from homogeneous to an inhomogeneous medium.

(iv) Obtain the similar vector equation for the magnetic field \vec{H} in inhomogeneous medium.

ABCD is a rectangle in which charges of +10^{-11}\,\mathrm{C}, -2\times10^{-11}\,\mathrm{C} and 10^{-11}\,\mathrm{C} are placed at corners B, C and D, respectively.

Calculate the potential at the corner A and the work done in carrying a charge of 2 coulombs to A.

Using the fundamental concepts of electromagnetism, determine the electric field of an electric dipole \vec{p} at a distance \vec{r} and its energy in an electric field \vec{E}.

Consider the following coupled inductor - capacitor circuit:

Calculate the ratio of the frequencies of the anti-symmetric and symmetric modes \omega_a/\omega_s. \left(\text{Given } k=\frac{1}{LC},\ k'=\frac{1}{LC_1}\right)

A series LCR circuit has resonant frequency \omega_0 and a large quality factor Q. Write down in terms of R, \omega, \omega_0 and Q, its (i) impedance at resonance, (ii) impedance at half-power points and (iii) the approximate forms of its impedance at low and high frequencies.

Consider the equation for a series RLC circuit and compare this to the parallel resonant circuit shown below:

Calculate the value of R_p if a series RLC circuit and the parallel RLC circuit are to have same equations for the potential of capacitance while they both have the same L, C and Q with Q being the total charge.

In the circuit diagram shown below, calculate the current passing through the milliammeter.

What is a displacement current? Prove that lines of conduction current plus displacement current are continuous.

The volume between two concentric conducting spherical surfaces of radii a and b (a<b) is filled with an inhomogeneous dielectric with \epsilon=\epsilon_0/(1+cr), where c is a constant and r is the radial coordinate. A charge +Q is placed on the inner surface, while the outer surface is grounded. Determine-

(i) \vec{D} in the region a<r<b;

(ii) capacitance of the device;

(iii) polarization charge density in the region a<r<b;

(iv) surface polarization charge densities at r=a and r=b.

A resistance R and a lossless capacitor C are connected through a switch. The capacitor is charged to potential V_0, and the switch is closed at t = 0. Prove that the energy stored in the capacitor is equal to the energy dissipated in the resistor.

Assume \vec{E}=0 inside a perfect conductor. Elaborate on any other four electrostatic properties that arise from this property.

Consider a plane wave travelling along the positive y-direction incident upon a glass of refractive index n=1.6. Find the transmission coefficient.

Consider Maxwell's equation in differential form in media. For j=\rho=0, assume \epsilon=\epsilon_0e^{\alpha t} and \mu=\mu_0e^{\alpha t} and show that the relevant wave equation for a plane wave propagating along x-direction is \frac{\partial^2 E}{\partial x^2}=\mu\frac{\partial^2 D}{\partial t^2}+\mu\alpha\frac{\partial D}{\partial t} where \vec{E}=E\hat{y} and \vec{H}=H\hat{z}.

Justify which of the four Maxwell's equations imply that there are no magnetic monopoles. How these equations would have been written if they were?

In deriving the Rayleigh-Jeans law, we count the number of modes dn corresponding to a wave number k for a photon gas in a cubical box. Consider a cubical container of volume V containing such gas in equilibrium. Calculate the differential number of allowable normal modes of frequency \omega.

A long solenoid has 220 turns /cm; its diameter is 3.2 cm. Inside the solenoid at its centre, we place a 130-turn closepacked coil of diameter 2.1 cm along its axis. The current in the solenoid is increased from zero to 1.5 amperes at a steady rate over a period of 0.16 second. What is the magnitude of the induced e.m.f. that appears in the central coil when the current in the solenoid is being changed?

The electric potential of a grounded conducting sphere of radius a in a uniform electric field is given as \phi(r,\theta)=-E_0r\left[1-\left(\frac{a}{r}\right)^3\cos\theta\right] Find the surface charge distribution.

An electrical circuit consists of a resistance R, inductance L and capacitance C in series. If a charge is put on the capacitor at some instant, determine the condition that V_C, the voltage across the capacitor, is subsequently oscillatory. Derive an expression for the quality factor Q of the circuit by considering the decay of the oscillation, using the result that the amplitude falls by a factor of e in \left(\frac{Q}{\pi}\right) period.

Determine the torque experienced by an electric dipole of moment \vec{p} if placed in an electric field \vec{E} in nonaligned state. Also show that the interaction energy of two dipoles of moments \vec{p}_1 and \vec{p}_2 separated by a displacement \vec{r} is U=\frac{1}{4\pi\epsilon_0}\frac{1}{r^3}\left[\vec{p}_1\cdot\vec{p}_2-3(\vec{p}_1\cdot\hat{r})(\vec{p}_2\cdot\hat{r})\right] assuming the expression for field due to a dipole.

The four arms of a Wheatstone bridge have the following resistances: AB = 100\,\Omega, BC = 10\,\Omega, CD = 5\,\Omega, DA = 60\,\Omega A galvanometer of 15\,\Omega resistance is connected across BD. Calculate the current through the galvanometer when a potential difference of 10 volts is maintained across AC.