A certain linear, homogeneous, isotropic, dielectric material has a relative permittivity, \varepsilon_{\text{r}} = 1\cdot 8. If potential \text{V} = -4000\text{y} volts in the material, then find :
(i) The electric flux density \text{D}, and
(ii) The polarisation \text{P}. Take vacuum permittivity \varepsilon_0 = 8\cdot 85 \times 10^{-12}\text{ farad/m}.
For the following series LCR circuit,
Calculate :
(i) The resonant frequency of the circuit.
(ii) The maximum current in the circuit.
(iii) The voltage across \text{C} at resonance.
(iv) The power absorbed by the circuit at resonance.
Assume the Sun to be a black body at temperature 5800\text{ K}. Use Stefan's law to calculate :
(i) The total energy emitted by the Sun per second, and
(ii) The energy reaching the top of the Earth's atmosphere. Given : \sigma = 5\cdot 672 \times 10^{-8}\text{ SI units}, Radius of the Sun = 7 \times 10^8\text{ m}, The distance of the Earth's atmosphere from the Sun = 1\cdot 5 \times 10^{11}\text{ m}.
For a plane electromagnetic wave given by : \text{E}_{\text{z}} = \text{a}\cos \omega\text{x} \cos \omega\text{ct} \text{H}_{\text{y}} = -\text{a}\sin \omega\text{x} \sin \omega\text{ct} Find the value of the Poynting vector.
After highlighting the importance of the Biot-Savart law, show that the magnetic field of a current carrying long wire, at a point near it, is inversely proportional to the distance of the point from the wire.
(i) Show that the vector potential \vec{\text{A}} at the position defined by the vector \vec{\text{r}} in a uniform electric and magnetic field is \vec{\text{A}} = \frac{1}{2}(\vec{\text{B}} \times \vec{\text{r}}).
(ii) Find out the divergence and curl of the vector potential \vec{\text{A}}.
Deduce Fresnel's law for the propagation of plane electromagnetic waves through an anisotropic dielectric medium.
If volume charge density in free space varies as \rho_{\text{v}} = \frac{100\varepsilon_0}{\text{r}^{2/5}}, then using Poisson's equation, find potential \text{V}(\text{r}). It is assumed that \text{r}^2 \text{E}_{\text{r}} \rightarrow 0 when \text{r} \rightarrow 0, while \text{V} \rightarrow 0 at \text{r} \rightarrow \infty.
(i) What is the method of images ? What are the conditions which must be satisfied while applying the method of images to deal with electrostatic problems ?
(ii) A point charge \text{Q} is located at the point (\text{a}, 0, \text{b}) between two semi-infinite conducting planes intersecting at right angles as shown in the figure. Using the method of images, determine the potential at point \text{P}(\text{x}, \text{y}, \text{z}) in the region \text{z} \ge 0 and \text{x} \ge 0 and the force on \text{Q}.
A parallel plate capacitor with circular plate of radius (R = 5\text{ cm}) is being charged. Find out the displacement current. Given : \varepsilon_0 = 8.9 \times 10^{-12}\text{ C}^2/\text{Nm}^2, \frac{dE}{dt} = 10^{12}\text{ V/m}.
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.
For a vector potential \vec{A}, \vec{\nabla} \cdot \vec{A} = -\frac{\mu_0}{4\pi} \cdot \frac{Q}{r^2} where Q is a constant of appropriate dimension. Calculate the corresponding scalar potential \phi (\vec{r}, t), that make \vec{A} and \phi to satisfy Lorentz gauge.
Distinguish between self-inductance and mutual inductance. Calculate the self-inductance of a solenoid of length 'l', area of cross-section 'A' having N turns.
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.
Define electromagnetic field strength tensor F_{\mu\nu}. Express it in terms of components of electric and magnetic fields.
Calculate the speed and radius of \alpha-particles of energy 5\cdot 998\text{ MeV} emitted into a magnetic field B = 10^4\text{ Wb m}^{-2}. Given that the mass of proton is 1\cdot 673 \times 10^{-27}\text{ kg}.
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.
Explain the cause of Rayleigh scattering. How does the amount of scattering depend on the wavelength of light ? How does Rayleigh scattering explain : (i) the blue colour of sky at day time ? (ii) the red colour at dusk ?
Obtain an expression for Poisson's and Laplace's equations in electrostatics.
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.
