The electromagnetic field inside a device is given by \vec{E} = \hat{y} E_0 \sin(k_x x) e^{-j k_z z}, \quad\text{where } k_x = \frac{n\pi}{a} \vec{H} = E_0 [\hat{x} \frac{-k_z}{\omega\mu} \sin(k_x x) + \hat{z} \frac{j k_x}{\omega\mu} \cos(k_x x)] e^{-j k_z z} Obtain an expression for the z-component of time-averaged Poynting vector \vec{S}.
Using Maxwell's field equations for a homogeneous non-conducting medium, derive the wave equation for the electric field. Calculate the velocity of EM wave in free space.
Explain the term 'Poynting vector' and state the significance of Poynting theorem.
Calculate the skin depth for radio waves in free space of wavelength 3 m in copper, given that electrical conductivity for copper is 6\times10^{7}\,\Omega^{-1}\,\mathrm{m}^{-1}.
Obtain Poisson's equation in electrostatics from Gauss' law. What form does it take when the charge density is zero?
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.
What happens if the primary winding of a transformer is connected to a battery?
Discuss the growth of current when an e.m.f. is suddenly applied to a circuit containing resistance, inductance and capacitance in series. What is the time constant of the circuit?
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.
If the magnetic moment of proton is 2.793\ \mu_N calculate, giving necessary steps, the radio frequency at which nuclear magnetic resonance occurs in water kept in a uniform magnetic field of 2.4\text{ T}.
The following inputs are given :
\begin{align*} T_0 &= 5500\text{ K} && \text{(Sun surface temperature)}, \\ R &= 7 \times 10^{10}\text{ cm} && \text{(Sun radius)}, \\ r &= 6\cdot 4 \times 10^8\text{ cm} && \text{(Earth radius)}, \\ D &= 1\cdot 5 \times 10^{13}\text{ cm} && \text{(Sun--Earth distance)}. \end{align*}
Assume that the earth and the sun both absorb all electromagnetic radiations incident on them, and that the earth is at a constant temperature T over the day-night cycle. Calculate T.
A long wire of radius a carries I amperes of current. The magnetic field surrounding it is given by H_\phi = \dfrac{I}{2\pi\rho} for \rho > a. Obtain expressions for (i) the magnetic energy stored per unit length in the region b \ge \rho \ge a and (ii) the equivalent inductance L per unit length.
A point charge +q is located near the corner of a horizontal and a vertical plate as shown below :
Obtain an expression for the electrostatic potential \phi_P using the image method.
(ii) The earth may be modeled as a spherical capacitor with a = 6\cdot 5 \times 10^6\text{ m} and b \rightarrow \infty. Determine C, if the medium surrounding the earth is free space.
A spherical capacitor is made of concentric conductors of radii a and b (b > a). The total charge on the inner sphere of radius a is Q. (i) Derive an expression for the capacitance C.
Consider a long, line charge with charge density \rho_l = 10^{-6}\text{ coulomb/m}. Find the force acting on a dust particle carrying -10^{-9}\text{ coulomb}, 1\text{ metre} away from the line charge in free space.
A radiation gas of temperature T fills a cavity of volume V. The system expands adiabatically and reversibly to a volume equal to 8V. By what factor does the temperature change?
Consider the L\text{-}C\text{-}R circuit shown below :
\begin{align*} R &= 0\cdot 1\ \Omega \\ L &= 1\text{ nH} \\ C &= 1\text{ nF} \end{align*}
(i) Determine its resonance frequency f_0.
For two isotropic media with \mu_1 \neq \mu_2 and \varepsilon_1 \neq \varepsilon_2, find an expression for the Brewster angle \theta_b for parallel polarization.
Consider a perfectly conducting half-space as shown below :
A uniform plane wave given by
\begin{align*} \vec{E}^i &= \hat{x} E_0 e^{-j k z} \\ \vec{H}^i &= \hat{y} \frac{E_0}{\eta_0} e^{-j k z} \quad \eta_0 = 120\pi\text{ ohm} \end{align*}
is incident normally on the boundary. Write down the expressions for the reflected electric and magnetic fields.
