BT-201 Engineering Physics - Nov 2022 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aExplain the Heisenberg's uncertainty principle in detail.
   How to answer

   • Statement of Heisenberg's uncertainty principle: impossible to measure simultaneously both position and momentum of a microscopic particle accurately.
   • Mathematical expressions: $\Delta x \cdot \Delta p_x \ge \frac{\hbar}{2}$ and $\Delta E \cdot \Delta t \ge \frac{\hbar}{2}$.
   • Physical explanation with thought experiments (e.g., gamma-ray microscope).
   • Significance in quantum mechanics and microscopic vs macroscopic scale.
2. 7 Marks
   bDeduce the relation between phase and group velocities.
   How to answer

   • Define phase velocity $v_p = \omega / k$ and group velocity $v_g = d\omega / dk$.
   • Express $\omega = k v_p$.
   • Differentiate with respect to $k$: $v_g = \frac{d}{dk}(k v_p) = v_p + k \frac{dv_p}{dk}$.
   • Convert $k = 2\pi / \lambda$ to express in terms of $\lambda$: $v_g = v_p - \lambda \frac{dv_p}{d\lambda}$.
   • Interpret normal and anomalous dispersion conditions.
3. 7 Marks
   aObtain the time independent Schrodinger wave equation.
   How to answer

   • Start with standard wave equation $\nabla^2 \Psi = \frac{1}{v^2} \frac{\partial^2 \Psi}{\partial t^2}$ or harmonic wavefunction $\Psi(\vec{r}, t) = \psi(\vec{r})e^{-i\omega t}$.
   • Substitute $\frac{\partial^2 \Psi}{\partial t^2} = -\omega^2 \Psi$ to obtain Helmholtz equation $\nabla^2 \psi + k^2 \psi = 0$.
   • Use $k = \frac{2\pi}{\lambda} = \frac{p}{\hbar}$.
   • Substitute total energy $E = K + V = \frac{p^2}{2m} + V \implies p^2 = 2m(E - V)$.
   • Arrive at $\nabla^2 \psi + \frac{2m}{\hbar^2}(E - V)\psi = 0$.
4. 7 Marks
   bHow the Newton's rings are formed? Deduce the expression for diameter of dark and bright fringes.
   How to answer

   • Explain formation: interference between light reflected from upper and lower surfaces of the wedge-shaped air film between a plano-convex lens and flat glass plate.
   • Draw experimental/ray setup for Newton's rings.
   • Express thickness $t$ in terms of radius of curvature $R$ and ring radius $r$: $r^2 = 2Rt$.
   • Condition for destructive interference (dark fringes): $2t = n\lambda \implies D_n = \sqrt{4n\lambda R}$.
   • Condition for constructive interference (bright fringes): $2t = (2n-1)\frac{\lambda}{2} \implies D_n = \sqrt{2(2n-1)\lambda R}$.
5. 7 Marks
   aDescribe the construction and working of Mach-Zehnder interferometer.
   How to answer

   • Draw schematic of Mach-Zehnder interferometer showing two beam splitters ($BS_1, BS_2$) and two plane mirrors ($M_1, M_2$).
   • Describe construction: collimated source, reference arm, test arm, detector.
   • Explain working: incident beam splits into two paths, each reflects once and recombines at second beam splitter to form interference fringes.
   • Mention application: measuring refractive index changes, aerodynamics/flow visualization.
6. 7 Marks
   bExplain about the diffraction grating. A parallel beam of Sodium light incident on plane transmission grating having 4250 lines per centimeter and a second order spectral line is observed at an angle of $30^\circ$. Find the wavelength of Sodium light.
   How to answer

   • Explain diffraction grating: an optical component with a periodic structure that splits and diffracts light into several beams.
   • State grating equation: $(a+b)\sin\theta = n\lambda$.
   • Calculate grating element: $(a+b) = \frac{1}{4250\text{ cm}^{-1}} = \frac{10^{-2}}{4250}\text{ m} \approx 2.353 \times 10^{-6}\text{ m}$.
   • Given $n = 2$ and $\theta = 30^\circ$, $\sin 30^\circ = 0.5$.
   • Calculate $\lambda = \frac{(a+b)\sin\theta}{n} = \frac{2.353 \times 10^{-6} \times 0.5}{2} \approx 5.882 \times 10^{-7}\text{ m} = 588.2\text{ nm}$ ($5882\text{ \AA}$).
7. 7 Marks
   aWhat is P-N junction diode? Discuss its I-V characteristics.
   How to answer

   • Definition of P-N junction diode, formation of depletion region and barrier potential.
   • Forward bias operation: narrowing of depletion layer, knee voltage, exponential current rise.
   • Reverse bias operation: widening of depletion layer, reverse saturation current, breakdown voltage.
   • Sketch the I-V characteristic curve showing forward and reverse regions.
8. 7 Marks
   bExplain the construction and working of solar cell with neat diagrams.
   How to answer

   • Principle: photovoltaic effect.
   • Construction: thin n-type top layer, p-type base, front metallic grid/contact, back contact, anti-reflective coating.
   • Working steps: (1) generation of electron-hole pairs by photons with energy $h\nu > E_g$, (2) separation of carriers by junction electric field, (3) collection at electrodes giving photovoltage/current.
   • Draw cross-sectional diagram and illuminated I-V characteristics curve.
9. 7 Marks
   aDerive an expression for Hall coefficient and Hall voltage.
   How to answer

   • Define Hall effect: generation of transverse voltage when a current-carrying conductor/semiconductor is placed in a perpendicular magnetic field.
   • Equilibrium condition: $q E_H = q v_d B \implies E_H = v_d B$.
   • Use current relation $I = n q A v_d = n q (w \cdot t) v_d$.
   • Substitute $v_d$: $V_H = E_H \cdot w = v_d B w = \frac{I B}{n q t}$.
   • Hall coefficient definition: $R_H = \frac{E_H}{J B} = \frac{1}{n q}$, yielding $V_H = \frac{R_H I B}{t}$.
10. 7 Marks
    bDerive the relationship between Einstein A and B coefficients.
    How to answer

    • Define absorption ($B_{12} N_1 u(\nu)$), spontaneous emission ($A_{21} N_2$), and stimulated emission ($B_{21} N_2 u(\nu)$).
    • At thermal equilibrium, transition rate up = transition rate down: $N_1 B_{12} u(\nu) = N_2 [A_{21} + B_{21} u(\nu)]$.
    • Express energy density: $u(\nu) = \frac{A_{21}}{B_{12}(N_1/N_2) - B_{21}}$.
    • Substitute Boltzmann ratio $\frac{N_1}{N_2} = e^{h\nu / kT}$.
    • Compare with Planck's radiation formula $u(\nu) = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu / kT} - 1}$ to obtain $B_{12} = B_{21}$ and $\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$.
11. 7 Marks
    aExplain the construction and working of He-Ne laser.
    How to answer

    • Construction: discharge tube containing mixture of Helium and Neon gas (ratio ~ 10:1 or 5:1), optical resonator cavity with one fully reflecting and one partially transmitting mirror.
    • Working: electric discharge excites He atoms to metastable states ($2^1S$ and $2^3S$).
    • Resonant collision transfers energy to Neon atoms, creating population inversion between Ne energy levels.
    • Laser transition: emission at $632.8\text{ nm}$ (red light) when Ne transitions between $3s$ and $2p$ (or $5s$ to $3p$).
    • Draw energy-level diagram and schematic setup.
12. 7 Marks
    bExplain the numerical aperture of an optical fiber. Calculate the numerical aperture of a fiber with refractive indices of core and cladding are 1.55 and 1.50, respectively.
    How to answer

    • Define numerical aperture (NA): measure of light-gathering capability of an optical fiber, equal to the sine of acceptance angle: $NA = \sin \theta_a$.
    • State formula in terms of refractive indices: $NA = \sqrt{n_1^2 - n_2^2}$, where $n_1$ is core index and $n_2$ is cladding index.
    • Given $n_1 = 1.55$ and $n_2 = 1.50$.
    • Calculate $NA = \sqrt{1.55^2 - 1.50^2} = \sqrt{2.4025 - 2.25} = \sqrt{0.1525} \approx 0.3905$.
13. 7 Marks
    aState and explain Stoke's and Gauss theorem.
    How to answer

    • Gauss' Divergence Theorem: statement and mathematical form $\oint_S \vec{A} \cdot d\vec{S} = \int_V (\nabla \cdot \vec{A}) dV$, converting surface integral to volume integral.
    • Physical significance of Gauss divergence theorem.
    • Stokes' Theorem: statement and mathematical form $\oint_C \vec{A} \cdot d\vec{r} = \int_S (\nabla \times \vec{A}) \cdot d\vec{S}$, converting line integral around closed loop to surface integral of curl.
    • Physical significance of Stokes' theorem.
14. 7 Marks
    bDerive Maxwell equations in vacuum.
    How to answer

    • State the four fundamental Maxwell equations in general medium.
    • Apply conditions for vacuum (free space): charge density $\rho = 0$, current density $\vec{J} = 0$.
    • Constitutive relations: $\vec{D} = \varepsilon_0 \vec{E}$ and $\vec{B} = \mu_0 \vec{H}$.
    • Write and explain the four resulting equations: (1) $\nabla \cdot \vec{E} = 0$, (2) $\nabla \cdot \vec{B} = 0$, (3) $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$, (4) $\nabla \times \vec{B} = \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$.
15. 14 Marks
    Write short notes on any two: a) Physical significance of wave function b) Young's double slit experiment c) Bloch's theorem d) Properties of Laser Light
    How to answer

    • Option (a): Born's interpretation, probability density $|\psi|^2$, normalization $\int |\psi|^2 dV = 1$, single-valued and continuous conditions.
    • Option (b): Division of wavefront, interference pattern, fringe width expression $\beta = \frac{\lambda D}{d}$.
    • Option (c): Electron in periodic crystal potential, wavefunction $\psi_k(\vec{r}) = e^{i \vec{k} \cdot \vec{r}} u_k(\vec{r})$ where $u_k(\vec{r})$ has lattice periodicity.
    • Option (d): Four unique laser properties: high spatial/temporal coherence, high monochromaticity, high directionality, and extreme brightness/intensity.