PYQ Bank

A series circuit has an inductance of 200 microhenries, a capacitance of 0.0005 microfarad and a resistance of 10 ohms. Find the resonant frequency and quality factor of the circuit.

What is meant by a dielectric? Define polarization vector P and relate it with the average molecular dipole moment. Obtain expression for the potential due to a polarized dielectric in terms of the polarization vector.

Calculate, giving necessary steps, the radio frequency at which nuclear magnetic resonance occurs in water kept in a uniform magnetic field of 2.4\ \mathrm{T}. The magnetic moment of proton is 2.793\mu_N.

Consider the incidence of a plane-polarised electromagnetic wave at the interface of two media having dielectric permittivity and magnetic permeability (\varepsilon_1, \mu_1) and (\varepsilon_2, \mu_2) respectively. The interface is chosen to be x = 0 plane. \vec{K}_1, \vec{K}_2 and \vec{K}_3 represent the propagator vectors associated with the incident, refracted and reflected waves respectively. Using the boundary conditions on them, establish the Snell's laws of refraction.

In a non-charged current-free dielectric, \rho = 0 and \vec{J} = 0. Show that in this medium, electric (\vec{E}) and magnetic (\vec{H}) fields satisfy three-dimensional wave equations \nabla^2 \vec{E} = \varepsilon\mu \frac{\partial^2 \vec{E}}{\partial t^2} \quad \text{and} \quad \nabla^2 \vec{H} = \varepsilon\mu \frac{\partial^2 \vec{H}}{\partial t^2} Using Poynting theorem of electromagnetic theory, describe the significance of the vector \vec{P} = (\vec{E} \times \vec{H}) and the scalar u = \frac{1}{2} [\vec{B} \cdot \vec{H} + \vec{D} \cdot \vec{E}]

A plane-polarised electromagnetic wave is incident on the interface of two dielectrics having dielectric permittivity \varepsilon_1 and \varepsilon_2. Assume that the electric vector \vec{E} lies in the plane of incidence. Using the boundary conditions at the interface, obtain the expressions for the amplitude reflection coefficient (r_{11}) and the amplitude transmission constant (t_{11}). Using the components of the Poynting vector \vec{E} \times \vec{H} associated with the reflected and transmitted waves, obtain the expressions for reflection and transmission coefficients R_{11} and T_{11} respectively. Under what condition, r_{11} = 0 and t_{11} = 1?

A cylindrical conductor is carrying a current along its axis which is assumed to be in z-direction. The current is uniformly distributed throughout its cross-section. Show that the vector potential \vec{A} associated with the magnetic induction due to the current-carrying cylindrical conductor is independent of z.

A series R\text{-}L\text{-}C circuit is connected across a voltage source V = 100 \sin 300t. If R = 500\ \Omega, L = 1\text{ H} and C = 2\ \mu\text{F}, calculate the average power delivered to the circuit.

A thin dielectric cylindrical rod of cross-section A is situated along z-axis from z = 0 to z = L. The polarisation of the rod is along its length and it is given by \vec{P} = (2z^2 + 5)\hat{z}. Calculate bound volume charge density at each end of the rod.

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.

Using Planck's radiation formula, deduce Wien's displacement law.

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.

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.

\questiondiagram{
} 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.

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.

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.