Calculate the Fraunhofer diffraction pattern from a grating of N slits with width e, separated by equal opaque spaces d. Find the condition for principal maxima and the corresponding values of intensity. A parallel beam of Na light is incident normally on a plane grating with 4250 lines per cm. The second order spectral line is observed to be deviated through 30^\circ. Calculate the wavelength of light.
Let the two waves with parallel electric fields be given by E_1 = 2 \cos \left( \bar{k}_1 \cdot \bar{r} - \omega t + \frac{\pi}{3} \right) \text{kV/m}, E_2 = 5 \cos \left( \bar{k}_2 \cdot \bar{r} - \omega t + \frac{\pi}{3} \right) \text{kV/m}. Find the intensity of each beam I_1, I_2 and also the interference term I_{12} at a point where their path difference is zero. Calculate the visibility V \left( = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} \right) for the interference pattern. [\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2, \mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2]
A convex lens of focal length 20 cm and a concave lens of focal length 10 cm are placed 20 cm apart. In between them an object is placed at a distance x from the convex lens. What is the value of x in cm so that the images formed by both of them coincide ?
Find the phase and group velocities for the wave E (z, t) = E_0 \cos (k_1 z - \omega_1 t) + \cos (k_2 z - \omega_2 t) where |k_1 - k_2| \ll k_1 k_2 ; |\omega_1 - \omega_2| \ll \omega_1, \omega_2.
The length of a moving rod can be defined as the product of its velocity and the time interval between the instants that both the end points of the rod pass a fixed mark in S system. Show that this definition leads to the space contraction.
A meson of rest mass \pi comes to rest and disintegrates into a muon of rest mass \mu and a neutrino of zero rest mass. Show that the kinetic energy of motion of the muon is T = \frac{(\pi - \mu)^2 c^2}{2\pi} .
What do you understand by streamline motion and critical velocity of a viscous liquid through a capillary tube. Capillaries of lengths l, 2l and \frac{l}{2} are connected in series. Their radii are r, \frac{r}{2} and \frac{r}{3} respectively. If the streamline flow is maintained and the pressure across the first capillary is P_1, deduce the pressures across the second and the third capillaries.
Show that the Bulk modulus K, Young's modu- lus Y and Poisson's ratio \sigma are connected by the relation K = \frac{Y}{3(1 - 2\sigma)}.
Derive an expression for the moment of inertia of a rigid body about any axis. What is an "ellipsoid of inertia" ? Explain clearly what you mean by the terms "principal axes" and "principal moments of inertia" ? Find the moment of inertia of a thin rectangular lamin a about an axis passing through the centre of the lamina and perpendicular to its plane. Hence determine the moments of inertia about axes passing through the midpoints of its both sides and perpendicular to its plane.
Derive an expression for the radial and transverse components of the acceleration of a particle moving in a plane. What inferences would you draw regarding the angular momentum if the transverse acceleration is zero ?
A force field is given by \bar{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k}. Is it a conservative field ? If so, what is the scalar potential ?
Give the Boolean expression and the truth table for an XOR gate. Why are NAND gates called universal gates ? Realise OR and XOR gates employing only NAND gates.
Simplify the equation: Y = [A \bar{B} (C + BD) + \bar{A} B C] using Boolean algebra. Give the logic circuit for the equation before and after the simplification.
Draw the device structure of a p-n junction solar cell and explain how the light energy is converted into electrical energy. Define the 'short-circuit current', open-circuit voltage' and 'efficiency' of a solar cell.
Explain the working of a phase-shift oscillator. Draw the circuit diagram of a phase-shift oscillator using either a transistor or an op-amp. What are the conditions which must be satisfied to achieve stable oscillations ? Calculate the frequency of oscillations for R = 10\ \Omega and C = 0.01\ \mu\text{F}.
Explain the difference between n-p-n and p-n-p transistors. Give their device structure and biasing circuits. Draw a circuit for a single-stage common-emitter amplifier.
Discuss briefly the concept of effective mass in semiconductors and explain the significance of negative effective mass.
What is Josephson effect? Discuss briefly DC and AC effects. Give some practical applications of Josephson junctions.
Show that a lattice cannot have a five-fold rotational symmetry. However, some solids appear to exhibit this rotational symmetry. Explain briefly how this can be done.
Give some characteristic properties of strange particles which distinguish them from non-strange ones. Write the Gell-Mann-Nishijima relation. How is it used for the classification of elementary particles?
