BT-201 Engineering Physics - Dec 2023 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. 5 Marks
   aState and Prove uncertainty principle.
   How to answer

   • State Heisenberg's uncertainty principle mathematically and conceptually
   • Set up wave packet representation or gamma-ray microscope / single slit thought experiment
   • Derive the relationship $\Delta x \cdot \Delta p \ge \hbar/2$ (or $\ge h$)
   • State physical significance and limitation on simultaneous measurements
2. 5 Marks
   bDiscuss the energy and momentum operator.
   How to answer

   • Definition of an operator in quantum mechanics
   • Derivation/expression of momentum operator: $\hat{p} = -i\hbar \nabla$
   • Derivation/expression of energy operator: $\hat{E} = i\hbar \frac{\partial}{\partial t}$
   • Application in constructing the Hamiltonian operator
3. 4 Marks
   cDerive the time dependent Schrodinger wave equation.
   How to answer

   • Start with standard plane wave equation for a free particle: $\Psi = A e^{i(kx - \omega t)}$
   • Differentiate with respect to $t$ to obtain energy relation
   • Differentiate twice with respect to $x$ (or spatial coordinates) to obtain momentum relation
   • Substitute operators into classical conservation of energy $E = \frac{p^2}{2m} + V(x)$
   • Arrive at $i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V\Psi$
4. 5 Marks
   aDifferentiate between division of amplitude and division of wavefront.
   How to answer

   • Define division of wavefront with examples (Young's double slit, Fresnel biprism)
   • Define division of amplitude with examples (Newton's rings, thin film, Michelson interferometer)
   • Tabulate differences: source type (point/slit vs extended), coherence mechanism, intensity loss, fringe shapes
5. 5 Marks
   bIn Newton's Ring method, the diameter of $n^{\text{th}}$ and $(n+14)^{\text{th}}$ ring are $0.42\text{ cm}$ and $0.70\text{ cm}$ respectively. If the radius of curvature of plano convex lens is $100\text{ cm}$. Calculate the wavelength of light.
   How to answer

   • State formula for Newton's rings: $\lambda = \frac{D_{n+p}^2 - D_n^2}{4pR}$
   • Substitute given values: $D_{n+14} = 0.70\text{ cm}$, $D_n = 0.42\text{ cm}$, $p = 14$, $R = 100\text{ cm}$
   • Compute $D_{n+14}^2 - D_n^2 = 0.49 - 0.1764 = 0.3136\text{ cm}^2$
   • Calculate $\lambda = \frac{0.3136}{4 \times 14 \times 100} = \frac{0.3136}{5600} = 5.6 \times 10^{-5}\text{ cm} = 5600\text{ \AA}$
6. 4 Marks
   cExplain Michelson's Interferometer experiment on the basis of labelled diagram and types of fringes.
   How to answer

   • Draw neat labelled schematic of Michelson interferometer (source, beam splitter $G_1$, compensating plate $G_2$, mirrors $M_1$ and $M_2$, telescope)
   • Explain working principle based on division of amplitude
   • Discuss types of fringes formed: circular fringes (equal inclination), localized/straight fringes (equal thickness), white light fringes
7. 5 Marks
   aWrite short notes on "Density of State".
   How to answer

   • Define density of states $g(E)$ as number of available electronic states per unit volume per unit energy range
   • Write formula for 3D free electron gas: $g(E) dE = \frac{V}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} E^{1/2} dE$
   • Sketch $g(E)$ versus $E$ curve showing square-root dependence
   • State its physical importance in determining carrier concentration and conduction properties
8. 5 Marks
   bExplain the Fermi level shifting in semiconductors.
   How to answer

   • Define Fermi level in semiconductors and its position in intrinsic material (near mid-gap)
   • Explain shift towards conduction band for n-type doping with donor concentration
   • Explain shift towards valence band for p-type doping with acceptor concentration
   • Discuss variation of Fermi level position with increasing temperature (tending towards intrinsic level)
9. 4 Marks
   cState and prove Bloch Theorem.
   How to answer

   • State Bloch theorem for an electron moving in a periodic crystal potential: $\psi(r) = e^{i \mathbf{k} \cdot \mathbf{r}} u_{\mathbf{k}}(\mathbf{r})$
   • Set up 1D Schrodinger equation with $V(x+a) = V(x)$
   • Use properties of second order differential equations and translation operator
   • Apply periodic boundary conditions to derive the Bloch condition
10. 5 Marks
    aExplain population inversion.
    How to answer

    • Define thermal equilibrium and Boltzmann distribution $N_2/N_1 = e^{-\Delta E/kT}$
    • Define population inversion as non-equilibrium state where $N_2 > N_1$
    • Explain role of pumping and metastable state in achieving it
    • State necessity for laser action / amplification by stimulated emission
11. 5 Marks
    bWrite the properties of LASER.
    How to answer

    • High degree of coherence (spatial and temporal)
    • High monochromaticity (narrow spectral line width)
    • High directionality (low beam divergence)
    • High intensity / brightness
12. 4 Marks
    cDiscuss the working of He-Ne Laser with labelled diagram.
    How to answer

    • Draw construction diagram of He-Ne laser tube with Brewster windows and mirrors
    • Draw energy level diagram of Helium and Neon showing resonant energy transfer
    • Explain pumping mechanism (electron collision excitation of He atoms)
    • Explain laser transitions producing $632.8\text{ nm}$ red output and depopulation of lower laser level
13. 5 Marks
    aExplain the electrostatic potential for a charge distribution.
    How to answer

    • Define electrostatic potential $V$ in terms of work done / line integral of electric field
    • Write potential formula for point charge $V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}$
    • Extend by superposition to continuous charge distributions: line ($dq=\lambda dl$), surface ($dq=\sigma da$), and volume ($dq=\rho dV$)
    • Write general integral form $V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\int \frac{\rho(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} dV'$
14. 5 Marks
    bDefine gradient of a scalar field, divergence of a vector field.
    How to answer

    • Define gradient of a scalar field $\phi$ as vector representing maximum rate of change: $\nabla\phi = \hat{i}\frac{\partial \phi}{\partial x} + \hat{j}\frac{\partial \phi}{\partial y} + \hat{k}\frac{\partial \phi}{\partial z}$
    • State physical significance of gradient (direction of steepest ascent, normal to equipotential surfaces)
    • Define divergence of a vector field $\mathbf{A}$ as net flux per unit volume: $\nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$
    • State physical significance of divergence (sources and sinks)
15. 4 Marks
    cState and prove Stoke's Theorem.
    How to answer

    • Statement of Stokes theorem relating line integral along a closed path to surface integral of curl over bounded open surface
    • Divide open surface $S$ into infinitesimal rectangular elements $dx dy$
    • Calculate circulation around elementary loop $dx dy$
    • Sum over all elements showing interior boundary contributions cancel out
    • Arrive at $\oint_C \mathbf{A} \cdot d\mathbf{l} = \iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S}$
16. 5 Marks
    aExplain Dual nature of light.
    How to answer

    • Define wave-particle duality of light
    • Discuss phenomena supporting wave nature (interference, diffraction, polarization)
    • Discuss phenomena supporting particle/photon nature (photoelectric effect, Compton scattering)
    • State de Broglie relation $\lambda = h/p$ synthesizing both aspects
17. 5 Marks
    bEstablish the relation between $V_p$ and $V_g$.
    How to answer

    • Define phase velocity $V_p = \frac{\omega}{k} \implies \omega = k V_p$
    • Define group velocity $V_g = \frac{d\omega}{dk}$
    • Differentiate $\omega$ with respect to $k$: $V_g = \frac{d(k V_p)}{dk} = V_p + k \frac{dV_p}{dk}$
    • Express in terms of wavelength $\lambda = \frac{2\pi}{k}$: $\frac{dk}{d\lambda} = -\frac{2\pi}{\lambda^2}$, obtaining $V_g = V_p - \lambda \frac{dV_p}{d\lambda}$
    • Mention normal and anomalous dispersion cases
18. 4 Marks
    cProve that for one dimensional motion of a particle in a box $\psi_n = A \sin \frac{n\pi x}{L}$.
    How to answer

    • Set up 1D potential well: $V(x) = 0$ for $0 \le x \le L$ and $V(x) = \infty$ elsewhere
    • Write time-independent Schrodinger equation inside the well: $\frac{d^2\psi}{dx^2} + k^2\psi = 0$
    • General solution: $\psi(x) = C_1 \cos(kx) + C_2 \sin(kx)$
    • Apply boundary condition $\psi(0) = 0 \implies C_1 = 0$
    • Apply boundary condition $\psi(L) = 0 \implies kL = n\pi$, giving $\psi_n(x) = A \sin\left(\frac{n\pi x}{L}\right)$
19. 5 Marks
    aDefine superposition of waves.
    How to answer

    • State the principle of superposition: when two or more waves overlap, resultant displacement is vector sum of individual displacements
    • Mathematical formulation: $y = y_1 + y_2 + \dots + y_n$
    • Mention conditions/validity (linearity of medium/wave equation)
    • State phenomena based on superposition (interference, diffraction, beats, standing waves)
20. 5 Marks
    bExplain the Rayleigh's criteria for resolving power.
    How to answer

    • State Rayleigh criterion: two spectral lines or point objects are just resolved when central maximum of diffraction pattern of one falls on first minimum of diffraction pattern of other
    • Draw intensity distribution curves for well resolved, just resolved, and unresolved conditions
    • Explain dip in resultant intensity (approx 20% drop / 0.81 intensity ratio) at mid-point for just resolved case
21. 4 Marks
    cDiscuss the Young's double slit experiment on the basis of labelled diagram, formula used and significance.
    How to answer

    • Draw neat labelled schematic of Young's double slit setup with slit separation $d$, screen distance $D$, and path difference $\Delta x$
    • State path difference $\Delta x = \frac{yd}{D}$ and conditions for maxima ($n\lambda$) and minima ($(2n-1)\frac{\lambda}{2}$)
    • State formula for fringe width: $\beta = \frac{\lambda D}{d}$
    • Discuss historical and physical significance (established wave theory of light, demonstration of spatial coherence)
22. 5 Marks
    aWrite down the applications of LASER in engineering and medicine.
    How to answer

    • Engineering applications: material processing (cutting, drilling, welding), precision measurement, LIDAR, holography, barcode scanners
    • Medical applications: eye surgery (LASIK, retinal detachment), endoscopy, oncology (photodynamic therapy), bloodless surgery, dermatology
23. 5 Marks
    bWrite down the importance of total internal reflection in optical fiber.
    How to answer

    • State principle of total internal reflection and condition ($n_{\text{core}} > n_{\text{cladding}}$, angle of incidence > critical angle)
    • Explain how TIR traps and guides light along the core over long distances with minimal loss
    • Highlight importance: high bandwidth, low attenuation, immune to electromagnetic interference, flexible signal routing
24. 4 Marks
    cCalculate the numerical aperture and acceptance angle for an optical fiber, given that the refractive indices of the core and the cladding are $1.45$ and $1.41$, respectively.
    How to answer

    • Formula for Numerical Aperture: $\text{NA} = \sqrt{n_1^2 - n_2^2}$
    • Compute $\text{NA} = \sqrt{(1.45)^2 - (1.41)^2} = \sqrt{2.1025 - 1.9881} = \sqrt{0.1144} \approx 0.338$
    • Formula for Acceptance Angle: $\theta_a = \sin^{-1}(\text{NA})$
    • Compute $\theta_a = \sin^{-1}(0.3382) \approx 19.77^\circ$ (or $0.345\text{ rad}$)