BT-201 Engineering Physics - Dec 2024 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
   aDeduce the energy eigenvalues and wave function of a particle moving in one dimensional box.
   How to answer

   • Setup 1D infinite potential well boundary conditions ($V=0$ for $0 < x < L$, $V=\infty$ otherwise).
   • Set up time-independent Schrödinger wave equation.
   • Apply boundary conditions $\psi(0)=0$ and $\psi(L)=0$ to find wave functions.
   • Derive expression for energy eigenvalues $E_n = \frac{n^2 h^2}{8mL^2}$.
   • Normalize the wave function to obtain $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$.
2. 7 Marks
   bExplain about the Heisenberg's uncertainty principle in detail.
   How to answer

   • State Heisenberg's uncertainty principle clearly.
   • Write mathematical expressions: $\Delta x \cdot \Delta p \ge \frac{\hbar}{2}$ and $\Delta E \cdot \Delta t \ge \frac{\hbar}{2}$.
   • Explain physical significance and consequences (e.g., non-existence of electrons in the nucleus).
   • Provide conceptual examples or thought experiments (like gamma-ray microscope).
3. 7 Marks
   aExplain the Fraunhofer diffraction due to single slit with necessary analysis.
   How to answer

   • Schematic diagram of single slit Fraunhofer diffraction setup.
   • Path difference calculation between wavelets from the edges of the slit.
   • Derivation of resultant amplitude using vector method or integration.
   • Intensity expression $I = I_0 \left(\frac{\sin\beta}{\beta}\right)^2$.
   • Conditions for central maximum, principal minima ($a \sin\theta = \pm m\lambda$), and secondary maxima.
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 of Newton's rings by interference in air film of variable thickness.
   • Draw diagram of plano-convex lens on glass plate showing film thickness $t$ and radius of curvature $R$.
   • Relate thickness $t$ to radius of ring $r$: $r^2 = 2Rt$.
   • Derive effective path difference $\Delta = 2t + \frac{\lambda}{2}$ taking Stokes' phase shift into account.
   • Derive diameter expressions: dark rings $D_n = \sqrt{4n\lambda R}$ and bright rings $D_n = \sqrt{2(2n-1)\lambda R}$.
5. 7 Marks
   aWhat is P-N junction diode, explain its working and discuss its V-I Characteristics.
   How to answer

   • Define P-N junction diode and barrier potential / depletion region.
   • Explain operation under forward bias and reverse bias conditions.
   • Plot V-I characteristics curve showing forward knee voltage and reverse breakdown voltage.
   • Discuss cut-in voltage, reverse saturation current, and breakdown phenomena.
6. 7 Marks
   bExplain the variation of Fermi level in N-type semiconductor with concentration and temperature.
   How to answer

   • Define Fermi energy level in an extrinsic n-type semiconductor.
   • State expression for $E_F$: $E_F = E_C - k_B T \ln\left(\frac{N_C}{N_D}\right)$.
   • Explain effect of increasing donor concentration (shifts $E_F$ closer to $E_C$).
   • Explain variation with temperature: at very low T ($E_F$ lies between $E_D$ and $E_C$), moderate T, and high T ($E_F$ moves toward the middle of the bandgap, intrinsic behavior).
   • Sketch $E_F$ vs temperature curve.
7. 7 Marks
   aExplain the Construction and working of He-Ne Laser.
   How to answer

   • Describe construction: discharge tube, gas mixture (ratio ~10:1 He to Ne), Brewster windows, optical resonator mirrors.
   • Diagram of He-Ne laser setup.
   • Energy level diagram showing excitation of Helium atoms by electron impact.
   • Resonant collision energy transfer from excited He atoms to Ne atoms.
   • Laser transitions in Neon (wavelength 632.8 nm) and depopulation to ground state.
8. 7 Marks
   bDeduce the expression for acceptance angle of an optical fiber. Find the acceptance angle of a fiber having core and cladding refractive indices are 1.75 and 1.70 respectively.
   How to answer

   • Ray diagram showing light entering the fiber core and undergoing total internal reflection at core-cladding boundary.
   • Apply Snell's law at air-core interface: $n_0 \sin\theta_0 = n_1 \sin r$.
   • Apply condition for TIR at core-cladding interface: $\sin \phi_c = n_2 / n_1$.
   • Derive $\theta_0 = \arcsin\left(\frac{\sqrt{n_1^2 - n_2^2}}{n_0}\right)$.
   • Calculate numerical aperture: $NA = \sqrt{1.75^2 - 1.70^2} = \sqrt{3.0625 - 2.89} = \sqrt{0.1725} \approx 0.4153$.
   • Calculate acceptance angle: $\theta_0 = \arcsin(0.4153) \approx 24.54^\circ$.
9. 7 Marks
   aStarting from the principle of charge conservation, derive the continuity equation for current densities in a conducting medium. Clearly state all assumptions and intermediate steps in your derivation.
   How to answer

   • State the principle of conservation of charge: net outward flow of charge equals rate of decrease of internal charge.
   • Write integral form: $\oint_S \vec{J} \cdot d\vec{A} = -\frac{d}{dt} \int_V \rho \, dV$.
   • Apply Gauss's Divergence Theorem to convert surface integral to volume integral: $\oint_S \vec{J} \cdot d\vec{A} = \int_V (\nabla \cdot \vec{J}) \, dV$.
   • Equate integrals: $\int_V \left(\nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t}\right) dV = 0$.
   • Conclude that for an arbitrary volume, $\nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t} = 0$, and note the steady-state form $\nabla \cdot \vec{J} = 0$.
10. 7 Marks
    bDerive the Electric field and Electrostatic Potential for a charge Distribution.
    How to answer

    • Define volume, surface, and line charge densities.
    • Consider a volume charge distribution with element $dq = \rho(\vec{r}') dV'$.
    • Write differential electric field $d\vec{E}$ and integrate over volume to get $\vec{E}(\vec{r}) = \frac{1}{4\pi\varepsilon_0} \int_V \frac{\rho(\vec{r}')(\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|^3} dV'$.
    • Derive relationship between electric field and potential: $\vec{E} = -\nabla V$.
    • Obtain formula for electrostatic potential $V(\vec{r}) = \frac{1}{4\pi\varepsilon_0} \int_V \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} dV'$.
11. 7 Marks
    aDiscuss about the Free Particle Wave Function and Wave Packets.
    How to answer

    • Define wave function of a free particle as a plane wave: $\psi(x,t) = A e^{i(kx - \omega t)}$.
    • Explain limitation of single monochromatic wave (non-normalizable, infinite spread).
    • Formulate concept of wave packet by superposition of waves with slightly different frequencies and wavelengths.
    • Define phase velocity $v_p = \omega / k$ and group velocity $v_g = d\omega / dk$.
    • Relate group velocity of the wave packet to classical particle velocity.
12. 7 Marks
    bWhat is LASER? Discuss the Properties of LASER light.
    How to answer

    • Define LASER (Light Amplification by Stimulated Emission of Radiation).
    • Explain high directionality (low divergence).
    • Explain high monochromaticity (narrow spectral linewidth).
    • Explain coherence (spatial and temporal coherence).
    • Explain high intensity / brightness compared to conventional light sources.
13. 7 Marks
    aDeduce an expression for Hall Co-efficient and Hall Voltage.
    How to answer

    • Define Hall effect and draw schematic diagram of semiconductor slab in transverse magnetic field.
    • Equate electric force to Lorentz force: $e E_H = e v_d B \implies E_H = v_d B$.
    • Use drift velocity relation $J = n e v_d$ to get $E_H = \frac{J B}{n e}$.
    • Derive Hall voltage $V_H = E_H \cdot w = \frac{B I}{n e t}$.
    • Define Hall coefficient $R_H = \frac{E_H}{J B} = \frac{1}{n e}$ (or $-1/ne$ for electrons).
14. 7 Marks
    bState and Derive the Divergence of Stoke's theorem and Gauss theorem.
    How to answer

    • State Gauss's Divergence Theorem: $\int_V (\nabla \cdot \vec{A}) \, dV = \oint_S \vec{A} \cdot d\vec{S}$.
    • Provide derivation of Gauss's divergence theorem by dividing volume into infinitesimal cubical elements.
    • State Stokes' Theorem: $\oint_C \vec{A} \cdot d\vec{r} = \int_S (\nabla \times \vec{A}) \cdot d\vec{S}$.
    • Derive Stokes' theorem by dividing open surface into infinitesimal surface elements and summing line integrals.
15. 14 Marks
    Write short notes on any two :

    1. a)Poynting vector
    2. b)Interference and Young's double slit experiment
    3. c)Kronig Penney Model
    4. d)V-number
    How to answer

    • For (a) Poynting vector: define $\vec{S} = \vec{E} \times \vec{H}$, units ($W/m^2$), and physical meaning of energy flux density.
    • For (b) Interference and Young's double slit experiment: define interference, setup of YDSE, path difference $\Delta x = \frac{xd}{D}$, and fringe width $\beta = \frac{\lambda D}{d}$.
    • For (c) Kronig-Penney Model: periodic rectangular potential well, Schrödinger equation, dispersion relation $P\frac{\sin\alpha a}{\alpha a} + \cos\alpha a = \cos ka$, and origin of allowed and forbidden energy bands.
    • For (d) V-number: define normalized frequency $V = \frac{2\pi a}{\lambda} \sqrt{n_1^2 - n_2^2} = \frac{2\pi a}{\lambda} (NA)$, condition for single mode fiber ($V < 2.405$), and number of modes $M \approx V^2 / 2$.