10 questions with concise answers (under 150 words each) for your graduation-level Physics Honours students, covering the specified topics
Classical Mechanics:
1.Question: A particle of mass (m) is subject to a velocity-dependent damping force (\mathbf{F} = -b\mathbf{v}). If the particle is projected horizontally with an initial velocity (v_0), determine the horizontal distance it travels before its velocity reduces to (v_0/e).
Answer: The horizontal motion is governed by (m\dot{v}x = -bv_x). Separating variables and integrating gives (\int{v_0}^{v_0/e} \frac{dv_x}{v_x} = -\frac{b}{m} \int_{0}^{t} dt), leading to (-\ln(e) = -\frac{bt}{m}), so (t = m/b). The horizontal distance is then (x = \int_{0}^{t} v_x(t') dt' = \int_{0}^{m/b} v_0 e^{-(b/m)t'} dt' = \frac{mv_0}{b} (1 - e^{-1}) = \frac{mv_0}{b} (1 - 1/e)).
2.Question: Explain the concept of generalized coordinates and their advantage in formulating Lagrangian mechanics compared to using Cartesian coordinates.
Answer: Generalized coordinates are a minimal set of independent variables that completely specify the configuration of a system. Unlike Cartesian coordinates, they are not necessarily orthogonal or related to distances along fixed axes. The advantage of using generalized coordinates in Lagrangian mechanics is that they can be chosen to match the constraints of the system, reducing the number of variables and eliminating constraint forces from the equations of motion. This simplifies the problem formulation and solution, especially for systems with non-holonomic constraints or complex geometries.
Thermodynamics:
3.Question: Derive the thermodynamic identity relating internal energy (U), temperature (T), entropy (S), pressure (P), and volume (V) for a reversible process.
Answer: For a reversible process, the first law of thermodynamics states (dU = dQ - dW). For a simple compressible system, the work done is (dW = PdV) and the heat absorbed is (dQ = TdS). Substituting these into the first law yields the thermodynamic identity: dU=TdS−PdV. This fundamental equation relates the change in internal energy to changes in entropy and volume, involving the state variables temperature and pressure.
4.Question: What is the significance of the chemical potential in statistical mechanics? How does it relate to particle number fluctuations in an open system?
Answer: The chemical potential ((\mu)) in statistical mechanics represents the change in the system's thermodynamic energy (e.g., Gibbs free energy) when one particle is added to the system, keeping other variables constant. In an open system where particle number (N) can fluctuate, the chemical potential governs the average particle number. Systems tend to exchange particles to equalize their chemical potentials, driving the flow of particles from regions of higher (\mu) to lower (\mu). It appears in the probability distribution for open systems, like the grand canonical ensemble.
Electromagnetism:
5.Question: State Poynting's theorem and explain the physical meaning of each term in the equation.
Answer: Poynting's theorem describes the conservation of electromagnetic energy. It is expressed as: ∂t∂uem+∇⋅S=−J⋅E where (u_{em} = \frac{1}{2}(\epsilon_0 E^2 + \frac{1}{\mu_0} B^2)) is the electromagnetic energy density, (\mathbf{S} = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B})) is the Poynting vector representing the electromagnetic power density, and (\mathbf{J} \cdot \mathbf{E}) is the rate at which electromagnetic energy is converted into other forms (e.g., heat) by the current density (\mathbf{J}) in the electric field (\mathbf{E}).
6. Question: Discuss the concept of magnetic vector potential (\mathbf{A}). How is it related to the magnetic field (\mathbf{B}), and what are its advantages in solving electromagnetic problems?
Answer: The magnetic vector potential (\mathbf{A}) is a vector field whose curl gives the magnetic field: B=∇×A. It exists because (\nabla \cdot \mathbf{B} = 0), allowing (\mathbf{B}) to be expressed as the curl of some vector field. One advantage of using (\mathbf{A}) is that it simplifies Maxwell's equations, particularly when dealing with time-varying fields. It reduces the number of coupled scalar equations to a smaller set of vector equations, often making the problem mathematically more tractable, especially in situations with complex current distributions or geometries.
Optics:
7.Question: Derive the condition for constructive and destructive interference for two coherent waves with a path difference (\Delta x) and wavelength (\lambda).
Answer: For two coherent waves interfering, the phase difference (\delta) is related to the path difference (\Delta x) by δ=λ2πΔx. Constructive interference occurs when the waves are in phase, meaning the phase difference is an integer multiple of (2\pi): δ=m(2π)⟹λ2πΔx=m(2π)⟹Δx=mλ, where (m = 0, \pm 1, \pm 2, ...). Destructive interference occurs when the waves are out of phase by an odd multiple of (\pi): δ=(2m+1)π⟹λ2πΔx=(2m+1)π⟹Δx=(m+21)λ.
8. Question: Explain the phenomenon of birefringence in anisotropic crystals. How does it lead to the formation of ordinary and extraordinary rays?
Answer: Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. In anisotropic crystals, the electron oscillations respond differently to electric fields in different directions. This results in two refractive indices: one for light polarized along a principal axis (ordinary ray, obeying Snell's law) and another for light polarized perpendicular to it (extraordinary ray, generally not obeying Snell's law and having a direction of propagation that depends on the crystal's optic axis). This splitting of light into two rays with different velocities and polarizations is birefringence.
Quantum Mechanics:
9. Question: For a particle in a one-dimensional infinite potential well of width (L) (from (x=0) to (x=L)), write down the normalized wavefunctions and the corresponding energy eigenvalues.
Answer: The time-independent Schrödinger equation for a particle in an infinite potential well is $−2mℏ2dx2d2ψ(x)=Eψ(x)$for \(0 < x < L\), with \(\psi(0) = \psi(L) = 0\). The normalized wavefunctions are given by$ψn(x)=L2sin(Lnπx)$, for (n = 1, 2, 3, ...). The corresponding energy eigenvalues are En=2mL2n2π2ℏ2.
10.Question: What are quantum operators? Give two examples of physical observables and their corresponding quantum operators in one dimension. Explain why operators are necessary in quantum mechanics.
Answer: Quantum operators are mathematical entities that act on wavefunctions to extract information about physical observables. They are linear and generally Hermitian. Examples in one dimension include the position operator (\hat{x} = x) (multiplication by (x)) and the momentum operator p^x=−iℏ∂x∂. Operators are necessary in quantum mechanics because physical observables like position, momentum, and energy are not simply numerical values but are represented by these operators acting on the quantum state (wavefunction) of the system. The eigenvalues of these operators correspond to the possible measured values of the observables.
