The magnitude of the average electric field normally present in the Earth's atmosphere just above the surface of the Earth is about 150\text{ N/C}, directed radially inward, toward the centre of the Earth. What is the total net surface charge carried by the Earth? Assume the Earth to be a conductor. (The radius of the Earth is 6.37 \times 10^6\text{ m})
Using Maxwell's equations, obtain Poisson's equation and Laplace's equation. The region -\frac{\pi}{2} < \frac{z}{z_0} < \frac{\pi}{2} has a charge density \rho = 10^{-8} \cos \left( \frac{z}{z_0} \right) (\text{C/m}^3). Elsewhere the charge density is zero. Find the electric potential V and electric field E from the Poisson's equation.
Obtain an expression for Poisson's and Laplace's equations in electrostatics.
State Biot–Savart law and write its mathematical form. In case of distributed current sources, write this expression for line current, surface current and volume current.
Consider a uniformly magnetized sphere of radius a and magnetization \vec{M} = M_0 \hat{z} surrounded by a vacuum region. Obtain an expression for scalar magnetic potential for r < a.
An AC circuit consists of inductance of self-induction 'L', a capacitor of capacitance 'C' and a resistor of resistance 'R' in series. Calculate the impedance 'Z' of the circuit. Obtain resonance condition and define quality factor.
Briefly explain electronic, ionic and orientational polarisation. The polarisability of ammonia molecule is found to be 2.42 \times 10^{-39}\text{ C}^2\text{ m/N} and 1.74 \times 10^{-39}\text{ C}^2\text{ m/N} at 309\text{ K} and 448\text{ K} respectively. Calculate the orientation polarisability for each temperature.
Obtain the general boundary conditions for fields E, B, D and H at a boundary between two different media carrying charge density \sigma or a current density K.
Current 'I' is passing through an infinite solenoid of radius R, with n turns per unit length. Find out the vector potential (i) inside the solenoid (ii) outside the solenoid.
A uniform plane wave with \vec{E} = E_x \hat{a}_x propagates in a lossless medium (\varepsilon_r = 4, \mu_r = 1, \sigma = 0) in the z-direction. Assume that E_x is sinusoidal with a frequency 100\text{ MHz} and has a maximum value of 10^{-4}\text{ (V/m)} at t = 0 and z = \frac{1}{8}\text{ (m)}. (i) Write the expression for instantaneous E for any t and z. (ii) Write the expression for instantaneous H. (iii) Determine the locations where E_x is a positive maximum, when t = 10^{-8}\text{ (s)}.
A spherical shell of radius R, carrying a uniform surface charge \sigma, is set spinning at angular velocity \omega about its axis. Find the vector potential it produces at point \vec{r}.
Find the energy stored in a system of four charges Q_1 = 1\text{ nC}, Q_2 = 2\text{ nC}, Q_3 = 3\text{ nC} and Q_4 = 4\text{ nC} placed at the cartesian coordinates R_1(1, 1), R_2(2, 1), R_3(1, 4) and R_4(2, 2), respectively. Assume free space.
A material has \sigma = 10^{-2}\text{ S/m} and \varepsilon = 2\varepsilon_0. At what frequency, the conduction current would be equal to the displacement current?
Derive the expression for the inductance per unit length of two long parallel wires each of radius a, separated by distance d from their axes and carrying equal and opposite current I.
If the electrostatic potential in spherical polar coordinate is \phi(r) = \phi_0 e^{-r/r_0} where \phi_0 and r_0 are constants, what will be the charge density at a distance r = r_0?
Write down the electromagnetic wave equation in non-conducting dielectric medium. Hence, show that the velocity of wave propagation is given by v = \sqrt{\frac{1}{\mu \epsilon}}, where the symbols have their usual meanings.
Explain Ampere's circuital law of magnetic field. What modifications Maxwell made in this equation to derive Maxwell's fourth equation?
Two charges Q_1 = 3\text{ nC} and Q_2 = 4\text{ nC} are placed at the cartesian points (0, 2, 2)\text{ m} and (0, -2, 4)\text{ m}, respectively. The z = 0 plane is connected to the ground. Calculate the electric potential and the electric field at the point (3, 2, 4)\text{ m} using the method of images.
A circular ring of radius R lying on the x-y plane and centred at the origin, carries a uniform line charge \lambda. Find the first three terms (monopole, dipole and quadrupole) of the multipole expansion of potential V(r, \theta).
A neutral atom consists of a point nucleus +q surrounded by a uniformly charged spherical cloud (-q) of radius r. Show that when such an atom is placed in a weak external electric field \vec{E}, the atomic polarizability of the atom is proportional to the volume of the sphere.
