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.
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.
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.
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.
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.
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.
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.
Prove that the Maxwell's equations in a medium contain the conservation of charge in differential form.
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.
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 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).
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].
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 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.
Show that when the temperature T of a radiating object is not too different from the surrounding temperature T_0, the object obeys Newton's law of cooling.
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.
