Consider a particle of mass m moving in a potential field of the form \text{V}(\vec{\text{r}}) = \text{V}_0(\text{x}^2 + \text{y}^2), where \text{V}_0 is a constant. What are the conserved physical quantities for the particle ?

Consider the diagram below with a water flow rate Q. Derive the expression for Q in terms of the difference in the manometer heights h and the cross-section areas A_1 and A_2:

Consider the motion of a particle of mass m in central force field \text{f(r)} = -\,\text{k}/\text{r}^2, where k is a constant and r is the distance from the force centre to the particle. Show that for the given system the vector \vec{\text{A}} = \vec{\text{p}} \times \vec{\text{L}} - \text{mk} \, \frac{\vec{\text{r}}}{\text{r}} is conserved. Using this vector and the fact that \vec{\text{A}} \cdot \vec{\text{L}} = 0, derive the orbit equation for the particle.

Consider two frames of reference S and S having a common origin O. The frame S is rotating with respect to the fixed frame S with a uniform \vec{\omega}=3a_x\,\mathrm{rad\,s^{-1}}. A projectile of unit mass at position vector \vec{r}=7a_x+4a_y\,\mathrm{m} is moving with \vec{v}=14a_y\,\mathrm{m\,s^{-1}}. Calculate in the rotating frame S the following forces on the projectile:

(i) Eulers force

(ii) Coriolis force

(iii) Centrifugal force

Explain the high temperature superconductivity. What are different properties of high temperature superconductors ?

An X-ray beam of wavelength \lambda_1 undergoes a first order Bragg reflection at a Bragg angle of 30^\circ. X-ray of wavelength 97\,\mathrm{nm} undergoes 3rd order reflection at a Bragg angle of 60^\circ. Consider that the two beams are reflected from the same set of planes. Find the value of \lambda_1.

Draw the circuit diagram of a transistor operating in the common-base configuration and sketch its input and output characteristics. Indicate the active, cut-off and saturation regions.

A solid contains a dilute concentration of \mathrm{Nd}^{3+} ions, each of which possess three 4f electrons. Assuming that there are 10^{25}\ \mathrm{m}^{-3} of these ions, calculate the magnetic susceptibility of the sample at 1\,\mathrm{K}.

When a voltage v_1 = + 40 \mu\text{V} is applied to the non-inverting input terminal and a voltage v_2 = - 40 \mu\text{V} is applied to the inverting input terminal of an Op-Amp, an output voltage v_o = 100\text{ mV} is obtained. But, when v_1 = v_2 = + 40 \mu\text{V} is applied, one obtains v_o = 0\cdot 4\text{ mV}. Calculate the voltage gains for the difference and the common-mode signals, and the common-mode rejection ratio.

When is the channel of a JFET said to be pinched off ? Define the pinch-off voltage. Give the relationship between the pinch-off voltage, the saturation voltage and the gate-source voltage. Sketch the depletion region before and after pinch-off.

Show that the reciprocal lattice for a ‘bcc’ lattice is a ‘fcc’ structure.

An n-p-n transistor with \beta = 49 is used in common-emitter amplifier mode with V_{cc} = 10\ \mathrm{V} and R_L = 2\mathrm{k}\Omega. If a 100\mathrm{k}\Omega resistor is connected between the collector and the base of the transistor, calculate the quiescent collector current. Assume V_{BE} = 0.

Calculate the pinch-off voltage for n-channel silicon FET with a channel width of 6 \times 10^{-4}\ \mathrm{cm} and a donor concentration of 10^{15}\ \mathrm{cm}^{-3}. Given that dielectric constant of silicon is 12.

{q-8-086-fig-1} Sketch the dc load line for the circuit shown.

Implement the Boolean function X = AB + \bar{A}C using NAND gates.

Delhi requires 3000\text{ MWh} of electric energy per day. It is to be supplied by a reactor which converts nuclear energy into electrical energy with an efficiency of 20\text{ percent}. If reactor uses nuclear fuel of \text{U}^{235}, calculate the mass of \text{U}^{235} needed for one day operation.

Show that in the nuclear shell model, the level spacing between major oscillator shells is approximately \hbar\omega=41A^{-1/3}\,\mathrm{MeV}.

Assuming that protons and neutrons possess equal masses, calculate how many times nuclear matter is denser than water if nuclear radius is given by 1\cdot 2 \times 10^{-15} A^{1/3}\text{ m}, where A is the mass number.

A sample of uranium emitting \alpha-particles of energy 4\cdot 18\text{ MeV} is placed near the ionization chamber. Assuming that only 10 particles per second enter the chamber, calculate the current produced. Given 1 ion pair requires 35\text{ eV}.

Calculate the mass of deuterium nucleus if the binding energy per nucleon is 1\text{ MeV}.