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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 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.
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 ?
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.
Derive Poisson equation starting from the Coulomb's law for a set of point charges.
What is a magnetic shell ? Define the strength of a magnetic shell. A 2\text{ mm} thick magnetic shell weighing 100\text{ gm} has magnetic moment of 1000\text{ units}. The density of the shell material is 10\text{ gm/cc}. Calculate the intensity of magnetisation and the strength of the shell.
Obtain the solution of the Laplace equation in cylindrical coordinates.
State Rayleigh-Jeans law. Show that the intensity of emissions at a particular wave-length is proportional to the temperature T. Discuss the limitations of this law in describing the intensity distribution of emission spectrum of a blackbody.
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.
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.
Find the magnetic field \vec{B} at the point P due to a short straight length of wire carrying current 'i'. Length of the wire is l. Point P is at a distance r away from the centre of the wire. Angle between \vec{l} and \vec{r} is \theta.
What do you mean by a gauge transformation ? What is its importance ? Show that the Lorentz gauge condition \vec{\nabla} \cdot \vec{A} + \frac{1}{c} \frac{\partial \phi}{\partial t} = 0 is Lorentz invariant. Here \vec{A} and \phi are the vector and scalar potentials.
Starting from Maxwell's equations of electromagnetic field in vacuum obtain the classical wave equations for the four field vectors \vec{E}, \vec{D}, \vec{B} and \vec{H} . Show that the field vectors can be propagated as waves in free space with the velocity of propagation equal to 3 \times 10^8\text{ m/s}, where for free space we have the vacuum permittivity \varepsilon_o = 8.854 \times 10^{-12}\text{ farad/m} & vaccum permeability \mu_o = 1.257 \times 10^{-6}\text{ henry/m}.
Write down the different components of the electromagnetic field tensors F_{\mu\nu} and further prove that Maxwell's equations of electrodynamics are invariant to Lorentz transformations.
In the case of proton NMR, state the expression for the energy of the dipole in an external magnetic field. How does the NMR originate ? Give the rough range of frequencies of the NMR signals, and the normal magnitude of the applied field.
A potential in cylindrical coordinates is a function of r and \phi but not of z. Obtain the separated differential equations for R and \Phi, where V = R(r) \Phi(\phi) and solve them.
Using Planck's radiation formula u(\nu) d\nu = \frac{8 \pi h}{c^3} \frac{\nu^3 d\nu}{e^{h\nu/kT} - 1} where the symbols have their usual meaning, find the wavelength of the region where energy density is the greatest. Also calculate the total energy density over all the frequencies