Determine the torque experienced by an electric dipole of moment \vec{p} if placed in an electric field \vec{E} in nonaligned state. Also show that the interaction energy of two dipoles of moments \vec{p}_1 and \vec{p}_2 separated by a displacement \vec{r} is U=\frac{1}{4\pi\epsilon_0}\frac{1}{r^3}\left[\vec{p}_1\cdot\vec{p}_2-3(\vec{p}_1\cdot\hat{r})(\vec{p}_2\cdot\hat{r})\right] assuming the expression for field due to a dipole.
The electric potential of a grounded conducting sphere of radius a in a uniform electric field is given as \phi(r,\theta)=-E_0r\left[1-\left(\frac{a}{r}\right)^3\cos\theta\right] Find the surface charge distribution.
Find whether the discharge of a condenser through the inductive circuit is oscillatory when C = 0.1\,\mu\mathrm{F}, L:=10\,\mathrm{mH} and R = 200\,\Omega. If it is oscillatory, calculate its frequency.
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}.
Explain the term 'Poynting vector' and state the significance of Poynting theorem.
Calculate, giving necessary steps, the radio frequency at which nuclear magnetic resonance occurs in water kept in a uniform magnetic field of 2.4\ \mathrm{T}. The magnetic moment of proton is 2.793\mu_N.
A wire of length 2 m is perpendicular to X-Y plane. It is moved with a velocity \vec{V}=(2\hat{i}+3\hat{j}+\hat{k})\,\mathrm{ms}^{-1} through a region of uniform induction \vec{B}=(\hat{i}+2\hat{j})\mathrm{Wm}^{-2}. Compute the potential difference between the ends of the wire.
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.
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?
Obtain Poisson's equation in electrostatics from Gauss' law. What form does it take when the charge density is zero?
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.
A long solenoid of radius R and n turns per unit length carries a sinusoidal current I = I_0 \cos \omega t. Determine the magnitude of induced electric field (E) outside the solenoid.
Consider the incidence of a plane-polarised electromagnetic wave at the interface of two media having dielectric permittivity and magnetic permeability (\varepsilon_1, \mu_1) and (\varepsilon_2, \mu_2) respectively. The interface is chosen to be x = 0 plane. \vec{K}_1, \vec{K}_2 and \vec{K}_3 represent the propagator vectors associated with the incident, refracted and reflected waves respectively. Using the boundary conditions on them, establish the Snell's laws of refraction.
For an arbitrary localised charged distribution, obtain an expression of electrostatic potential V in terms of multipole expansion.
A plane-polarised electromagnetic wave is incident on the interface of two dielectrics having dielectric permittivity \varepsilon_1 and \varepsilon_2. Assume that the electric vector \vec{E} lies in the plane of incidence. Using the boundary conditions at the interface, obtain the expressions for the amplitude reflection coefficient (r_{11}) and the amplitude transmission constant (t_{11}). Using the components of the Poynting vector \vec{E} \times \vec{H} associated with the reflected and transmitted waves, obtain the expressions for reflection and transmission coefficients R_{11} and T_{11} respectively. Under what condition, r_{11} = 0 and t_{11} = 1?
A cylindrical conductor is carrying a current along its axis which is assumed to be in z-direction. The current is uniformly distributed throughout its cross-section. Show that the vector potential \vec{A} associated with the magnetic induction due to the current-carrying cylindrical conductor is independent of z.
A series R\text{-}L\text{-}C circuit is connected across a voltage source V = 100 \sin 300t. If R = 500\ \Omega, L = 1\text{ H} and C = 2\ \mu\text{F}, calculate the average power delivered to the circuit.
In a non-charged current-free dielectric, \rho = 0 and \vec{J} = 0. Show that in this medium, electric (\vec{E}) and magnetic (\vec{H}) fields satisfy three-dimensional wave equations \nabla^2 \vec{E} = \varepsilon\mu \frac{\partial^2 \vec{E}}{\partial t^2} \quad \text{and} \quad \nabla^2 \vec{H} = \varepsilon\mu \frac{\partial^2 \vec{H}}{\partial t^2} Using Poynting theorem of electromagnetic theory, describe the significance of the vector \vec{P} = (\vec{E} \times \vec{H}) and the scalar u = \frac{1}{2} [\vec{B} \cdot \vec{H} + \vec{D} \cdot \vec{E}]
