Consider in the region 0 \le z \le 1\text{ m} an infinite slab made of a material with relative permeability, \mu_r = 3\cdot 5. If \vec{B} = (2y\hat{i} - 5x\hat{j}) \times 10^{-3}\text{ Wb/m}^2 within the slab, determine magnetisation \vec{M}.
State Biot-Savart law. Calculate the magnitude of axial magnetic induction due to a circular loop of area A carrying current I.
For an arbitrary localised charged distribution, obtain an expression of electrostatic potential V in terms of multipole expansion.
A long solenoid of radius R and n turns per unit length carries a sinusoidal current I = I_0 \cos \omega t. Determine the magnitude of induced electric field (E) outside the solenoid.
Consider a plane electromagnetic wave entering a rarer medium from a denser medium. Show that the Brewster angle is less than the total internal reflection angle.
{cse-q-3-238-fig-1} An infinite ladder network of resistances is connected across the points A and B, as shown in the above figure. Each of the resistance is 1\Omega. Calculate the effective resistance between points A and B.
State Gauss's Law of electrostatics both in integral form and differential forms. Two charged spheres of radius R each, have their centres a distance d apart such that d < 2R. One of the spheres has a uniform positive charge density \rho per unit volume while the other has opposite charge density -\rho. Show that the electric field in the region of overlap between two spheres is uniform.
Find the vector potential due to a line segment from x = a to x = b carrying a current I at a point P which is at a distance d from the line segment.
Show that the interaction energy of two magnetic dipoles $ \bar{m}_1 $ and $ \bar{m}_2 $ separated by a displacement $ \bar{r} $ is given by U = \frac{\mu_0}{4\pi} \cdot \frac{1}{r^3} \left[ \bar{m}_1 \cdot \bar{m}_2 - 3(\bar{m}_1 \cdot \hat{r})(\bar{m}_2 \cdot \hat{r}) \right]. If two magnetic dipoles are held at a fixed distance apart, but allowed to rotate freely, what would be the configuration for stable equilibrium ?
State Faraday's Law of induction in terms of emf and magnetic flux. Use Stoke's theorem to express the law in differential form. A uniform, time varying magnetic field $ \bar{B} = \bar{B}_0(1 + \alpha t)\hat{k} $ fills a circular region of radius R lying in the x-y plane with the centre at origin. Here $ \bar{B}_0 $ and $ \alpha $ are appropriately dimensioned constants. Find the induced electric field at a distance r from the centre where r < R.
A plane electromagnetic wave is travelling in vacuum along a direction which makes an angle of 45^\circ each with the positive x and z-axes. The electric field at time t is given by the expression \bar{E} = E_0 (2 \cos \omega t \hat{j} - \sin \omega t (\hat{i} - \hat{k})) where $ \hat{i}, \hat{j} $ and $ \hat{k} $ are unit vectors along the x, y and z directions respectively and $ E_0 = 100 \text{ v/m} $. Find the magnetic field vector $ \bar{B} $ at time t and obtain the average power transmitted per unit area by the wave.
A plane electromagnetic wave travelling in z-direction and polarized in x-direction is incident normally from left on to an interface between two media at z = 0. The medium to the left (z < 0) has a refractive index n_1, while that to the right is of refractive index n_2. The magnetic permeabilities of both the media are equal and $ \mu_1 = \mu_2 = \mu_0 $. Obtain expressions for reflection and transmission coefficients (R and T) and show that R + T = 1.
A point charge + q is held above a grounded conducting plane located at z = 0. If the position of the charge is (0, 0, d) obtain an expression for the induced charge density on the plane as a function of coordinates x and y.
Explain how Maxwell modified Ampere's law of magnetostatics by introducing the concept of displacement current. How does it resolve the paradox of a charging capacitor ?
Two identical black bodies A and B are respectively at temperatures T and 2T respectively. The heat energy radiated by A is collected for one minute and is used to heat a mass of water. The temperature of water is found to rise by 0.25 K. If the heat radiated by B were to be collected for one minute and used to heat the same mass of water, what would be the rise in the temperature of water ? Assume, specific heat of water to be temperature independent.
In an a.c. circuit, a resistance (R = 100\ \Omega) and a capacitance (C = 100\ \mu\text{F}) in series are connected to an a.c. source V = 200 \sin(100\ \pi t). Calculate the current through the circuit and the voltages across R and C. Draw a vector diagram representing the magnitudes and phases of the voltages.
Obtain the solution of the Laplace equation in cylindrical coordinates.
Why does a spinning nucleus process in a magnetic field ? Explain the underlying principle of nuclear magnetic resonance (NMR) spectroscopy. Calculate the radio frequency at which NMR occurs in water kept in a uniform magnetic field of 2.4 T, the magnetic moment of proton being 2.793\\ \mu_N.
State Amperes law of magnetostatics. Using this law, find the magnetic field at a point due to an infinitely long filamentary current.
Explain the Rayleigh Scattering of light. Show that the energy density of light scattered from an isotropic homogenous medium of a gas is inversely proportional to the fourth power of the wavelength of the incident light.