PYQ Bank

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 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 plane electromagnetic wave is given by E_z = a \cos \omega x \cos \omega t and H_y = - a \sin \omega x \sin \omega t. Evaluate the instantaneous value of the Poynting vector \vec{S} and show that <\vec{S}> = 0.

Sea water has resistivity 0\cdot 3\ \Omega\text{m} and its dielectric constant is 81. Calculate the ratio of the amplitudes of the conduction and polarisation current intensities when the applied field is oscillating at 100\text{ MHz}.

For a uniform wire of length L and radius a having a potential difference V between the ends and a current I along it, calculate the energy per unit time delivered to the wire by Poynting vector.

Show that the complex propagation constant k of an electromagnetic wave propagating in an isotropic dielectric medium with conductivity \sigma can be given by k = k_r + i k_i with k_r \approx \frac{2\pi}{\lambda_o} n \left[ 1 + \frac{1}{8} \left( \frac{\sigma}{\omega \varepsilon} \right)^2 \right] and k_i \approx \frac{2\pi}{\lambda_o} n \left[ \frac{1}{2} \left( \frac{\sigma}{\omega \varepsilon} \right) \right] where n = \sqrt{\varepsilon / \varepsilon_o} and \lambda_o = \omega / 2\pi c; \varepsilon and c being the permittivity of the dielectric and velocity of light in free space, respectively.

Explain the term 'mutual inductance' between two coils carrying current. Describe a method of determining mutual inductance between two coils of wire with relevant theory.

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

How do you justify the statement : "A current carrying conductor, although has no net charge experiences a force when placed in a magnetic field."

A long straight solenoid has 100 turns in the secondary and 3000 turns per centimeter in the primary. The area of cross-section of the solenoid is 3 square centimeters. Calculate the mutual inductance.

Find out the number of photons in a cavity of volume 1\cdot 00\text{ cm}^3 under thermal equilibrium at temperature T = 1000\ ^\circ\text{K}. Assume that \int_0^\infty \frac{x^2 dx}{e^x - 1} = 2\cdot 405.

For the above system, show that the energy per unit cavity volume in the frequency range of \nu and \nu + d\nu can be given by u(\nu)d\nu = \frac{8\pi\nu^2 k_B T}{c^3} d\nu where k_B is the Boltzmann's constant. Discuss the limitations of this formula and how did Planck put forward the correct analysis.

If considered a blackbody as a radiation-filled cavity at a uniform temperature T, demonstrate with an appropriate figure the electromagnetic wave patterns inside the cavity of length L for wavelengths \lambda = L, \frac{3}{2} L and 2L.

Explain the physical significance of the Poynting vector \bar{S}. What is represented by the closed integral \oint \bar{S} \cdot d\bar{a} for a closed surface of area \bar{a} ?

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

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.

The dielectric constant of a medium is 3. The Electric field in the dielectric is 10^6\text{ Vm}^{-1}. What are the electric displacement and polarization ?

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.

Prove that the Maxwell's equations in a medium contain the conservation of charge in differential form.

Find out the total electric potential energy of a single spherical object of uniform charge density \rho, total charge Q and radius R.