Engineering Physics: Planck's Law & Poynting Vector

• Notes based on Engineering Physics exam questions

Planck's Law of Radiation

• Describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature.
• Postulates include:
  • Energy is quantized and emitted in discrete packets called photons.
  • The energy of a photon is proportional to its frequency (E = hf).
  • Black body radiation is due to the oscillation of charged particles.
  • The average energy of an oscillator is given by Planck's distribution law.

Poynting Vector

• Represents the directional energy flux (rate of energy transfer per unit area) of an electromagnetic field.
• Mathematically, it's the cross product of the electric field intensity (E) and the magnetic field intensity (H): S = E x H.
• Its physical significance is indicating the power and direction of electromagnetic energy flow.

Diffraction Pattern and Slit Size

• When the slit size is smaller than the wavelength of light, diffraction becomes significant.
• The light waves spread out and interfere, creating a broad diffraction pattern with a central maximum and weaker secondary maxima.
• Significant because the usual approximations for diffraction do not hold when the slit is very small.

Metastable State in Lasers

• A metastable state is an excited energy level of an atom where electrons can reside for a relatively long time before spontaneously decaying.
• In laser action, a population inversion is created where more atoms are in a metastable state than in the ground state.
• This allows for stimulated emission to dominate, leading to coherent light amplification.

Vortex State of Superconductivity

• Vortex state occurs in Type II superconductors when subjected to an external magnetic field.
• Magnetic field penetrates the superconductor in the form of quantized flux tubes called vortices.
• Each vortex carries a quantum of magnetic flux, and circulates with the supercurrents.

Scattering Loss in Optical Fiber

• Scattering Loss is the loss of optical power in a fiber due to scattering of light.
• Caused by imperfections and inhomogeneities in the fiber material (Rayleigh scattering).
• Also caused by variations in the fiber's density and composition.

Quantum Confinement Effect in Nanomaterials

• Quantum Confinement Effect is the restriction of the motion of charge carriers (electrons and holes) in nanomaterials due to their small size.
• When the size of a material is reduced to the nanoscale, the quantum mechanical properties become significant.
• Energy levels become discrete (quantized), leading to unique optical and electronic properties.

Compton Shift and Kinetic Energy Calculation

• Deals with the scattering of a photon by a charged particle, usually an electron
• Results in a decrease in energy (increase in wavelength) of the photon, known as the Compton shift.
• Can calculate the Compton shift (Δλ) using the formula: Δλ = λ' - λ = (h / mₑc) * (1 - cos θ), where θ is the scattering angle.
• The kinetic energy of the recoil electron can be found using energy conservation.

Intensity of Electric and Magnetic Fields

• For a lamp radiating energy uniformly, the intensity (I) at a distance r = Power/Area: I = P / (4πr²).
• The intensity is related to the electric field (E) and magnetic field (B) by I = (1/2)cε₀E² = (1/2)cB²/µ₀.
• Average values of E and B can be calculated using these relationships.

Newton's Rings

• Newton's Rings are interference pattern observed when a plano-convex lens is placed on a flat glass surface.
• The rings are formed due to the interference between light waves reflected from the top and bottom surfaces of the air film between the lens and the glass.
• The radius of the nth dark ring is given by rₙ = √(nλR), where λ is the wavelength and R is the radius of curvature of the lens.
• The thickness of the air film (t) at the nth dark ring is t = nλ/2.

Single Slit Diffraction

• The diffraction pattern of a single slit consists of a central bright fringe and alternating dark and bright fringes on either side.
• The position of the first dark fringe is given by sin θ = λ/a, where a is the slit width.
• For small angles, sin θ ≈ tan θ = y/f, where y is the distance of the fringe from the central axis and f is the focal length of the lens.

V-Number for Optical Fiber

• The V-number is a dimensionless parameter that determines the number of modes that an optical fiber can support.
• V = (2πa/λ) * √(n₁² - n₂²), where a is the core radius, λ is the wavelength, n₁ is the core refractive index, and n₂ is the cladding refractive index.
• The number of modes is approximately V² / 2 for a step-index fiber.

Phase Velocity vs. Group Velocity

• Phase velocity (vₚ) is the rate at which the phase of a wave propagates in space: vₚ = ω/k.
• Group velocity (v₉) is the rate at which the overall shape of the wave's amplitude propagates in space: v₉ = dω/dk.
• The relationship between them is: v₉ = vₚ - λ(dvₚ/dλ).

Particle in a One-Dimensional Box

• The energy state eigenvalues for a particle in a 1D box are given by: Eₙ = (n²h²)/(8mL²), where n is the quantum number, m is mass, and L is the box length.
• The wave function is given by: ψₙ(x) = √(2/L) * sin(nπx/L).

Maxwell's Fourth Equation and Displacement Current

• Maxwell modified Ampere's Law to include the displacement current: ∇ x B = μ₀(J + ε₀(dE/dt)).
• Displacement current (ε₀(dE/dt)) accounts for the changing electric field.
• In a capacitor with a DC battery, after charging, the current stops as the electric field is constant and the capacitor acts as an open circuit.
• With AC, the electric field is continuously changing, causing a continuous displacement current between the capacitor plates.

Poynting Theorem

• Poynting theorem describes the conservation of energy in electromagnetic fields.
• The theorem relates the rate of change of energy density to the energy flux (Poynting vector) and the work done on charges.
• Expressed as: -∂u/∂t = ∇⋅S + J⋅E, where u is the energy density, S is the Poynting vector, J is the current density, and E is the electric field.
• Physically, this means that the decrease in energy within a volume is due to the energy flowing out through the surface and the work done on the charges inside.

Interference in Parallel Thin Films

• In thin films, interference occurs between light waves reflected from the top and bottom surfaces of the film.
• The conditions for maxima (constructive interference) and minima (destructive interference) depend on the film thickness (t), refractive index (n), angle of incidence (θ), and wavelength (λ).
• For reflected light, condition for maxima: 2nt cos θ = (m + 1/2)λ; and for minima: 2nt cos θ = mλ, where m is an integer.
• Reflected and transmitted patterns are complementary because energy is conserved. Where reflection is maximum, transmission is minimum.

Fraunhofer Diffraction at a Single Slit

• Fraunhofer diffraction occurs when parallel light waves pass through a single slit.
• Intensities of successive maxima are nearly 1: 1/22: 1/62: 1/121.
• The intensity distribution is given by I = I₀ [sin(α)/α]², where α = (πa sin θ)/λ.

Acceptance Angle and Cone of Optical Fiber

• Acceptance angle is the maximum angle at which light entering the fiber will be guided through the core by total internal reflection.
• Acceptance cone is the cone within which light must enter the fiber to be guided.
• Acceptance Angle: θₐ = sin⁻¹ √(n₁² - n₂²), where n₁ is the core refractive index, and n₂ is the cladding refractive index.

Helium-Neon (He-Ne) Laser

• Helium-Neon lasers operate on the principle of stimulated emission in a gas mixture of helium and neon.
• Superiority to Ruby laser as He-Ne lasers offer continuous wave operation, more stable output, narrower linewidth, and better coherence.

Type-1 vs. Type-2 Superconductors

• Type-1 superconductors exhibit a sharp transition to the superconducting state below a critical magnetic field (H_c).
• Type-2 superconductors exhibit two critical magnetic fields (H_c1 and H_c2). Between these fields they are in a mixed or vortex state where magnetic field penetrates in the form of fluxons
• Type-2 superconductors are generally more important than Type-1 superconductors due to their ability to maintain superconductivity.

Purpose of Nanoscience

• Nanoscience is the study of phenomena and manipulation of materials at the nanoscale (1-100 nm).
• A method for the synthesis of nanomaterials is Chemical Vapor Deposition (CVD):
  • Reactants in vapor phase are introduced into a reaction chamber.
  • Reactants decompose/react on a substrate to form the desired nanomaterial.
  • Byproducts are removed from the chamber.