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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 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.
Find whether the discharge of a condenser through the inductive circuit is oscillatory when C = 0.1\,\mu\mathrm{F}, L:=10\,\mathrm{mH} and R = 200\,\Omega. If it is oscillatory, calculate its frequency.
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.
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?
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.
An inductor of inductance 5\text{ H} is suddenly connected to a 10\text{ V} d.c. power supply through a resistor of 10\\ \Omega. After what time will the current in the circuit be 1/10\text{th} of its steady state value ?
If the magnetic field \vec{B}, at a point with position vector \vec{r} is uniform, show that the corresponding vector potential \vec{A}(\vec{r}) is given by \vec{A}(\vec{r}) = -\frac{1}{2} \left[ \vec{r} \times \vec{B} \right].
Consider an infinite current sheet with a uniform current density \vec{K} \text{ (Amp/m)}. Show that the magnetic field \vec{H} at a point away from the sheet is \vec{H} = \frac{1}{2} \vec{K} \times \hat{n} where \hat{n} is a unit normal vector directed from the current sheet to the point.
The plane y = 5 carries a current of density 10\\ \hat{z} \text{ (Amp/m)}. Calculate the value of the magnetic field \vec{H} at the point (0, 1, -5).
Consider an infinite line charge with charge density \rho\text{ coulomb/meter} located at a distance d\text{ meters} from a grounded conducting plane z = 0. Determine : (i) the magnitude of the potential V for z > 0 and z \le 0. (ii) the surface charge density induced on the conducting plane.
The current density in spherical co-ordinates is given by \vec{J} = \frac{1}{r^3} \left[ 2 \cos \theta \hat{r} + \sin \theta \hat{\theta} \right] \text{A/m}^2 where \hat{r} and \hat{\theta} are unit vectors. Calculate the amount of current passing through a hemisphere of radius 20\text{ cm}.
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}.
Consider a plane wave travelling along the positive y-direction incident upon a glass of refractive index n=1.6. Find the transmission coefficient.
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?
The electric field in a medium is given by \vec{E} = \vec{E}_0 e^{-\alpha z} \sin(kz - \omega t), where \vec{E}_0 is a constant vector with dimensions of the electric field. Prove that \vec{E} cannot have a component along the unit vector, \hat{z}, parallel to the z-axis. Here \alpha is a positive constant.
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.
Consider the Earth as a black body. Radiations from the Sun arrive at the surface of the Earth with an average intensity of S\text{ watts/m}^2. If the reflection coefficient of the Earth's surface is \alpha, determine the temperature of the Earth under equilibrium conditions.
A resistor R(= 6.2\,\mathrm{M}\Omega) and a capacitor C(= 2.4\,\mu\mathrm{F}) are connected in series and a 12 V battery of negligible internal resistance is connected across their combination.
(i) What is the capacitive time constant of this circuit?
(ii) At what time, after the battery is connected, does the potential difference across the capacitor become 5.6 V?
Obtain Poisson's equation in electrostatics from Gauss' law. What form does it take when the charge density is zero?