Deriving the Electromagnetic Wave Equation in Vacuum

Electromagnetic waves travel in straight-line paths (ray model) and interact with matter through three primary processes: reflection (bouncing off surfaces), absorption (energy converted to thermal energy), and transmission (passing through materials); the specific interaction depends on both the frequency of the light and the material's properties, which explains phenomena like why objects appear colored, why pencils look bent in water, and how X-rays can image bones.

Electromagnetic waves interact with matter in three ways: (1) Transmission - when wavelength is much larger than the object, waves pass through without interaction (like radio waves passing through buildings); (2) Absorption - when wavelength is much shorter than the object, waves may transfer energy to the object; (3) Reflection/Resonance - when wavelength matches the object's size, strong interaction occurs (like waves causing a boat to rock). This principle explains why different electromagnetic waves have different applications and why X-rays can penetrate soft tissue but are absorbed by bone.

Electromagnetic waves originate from accelerating electric charges, where changing electric fields generate magnetic fields and vice versa, propagating at light speed. The frequency of acceleration determines wave energy. All objects emit thermal radiation due to atomic vibrations, with hotter objects emitting higher-energy waves. Polarization describes wave oscillation orientation, with electrons oscillating in three perpendicular directions. Waves exhibit interference (constructive and destructive), scattering (atoms re-radiate light in all directions), reflection (conductive surfaces generate mirror-image waves), and refraction (light slows in denser media, bending direction). These principles explain phenomena like the blue sky (blue light scatters more), 3D movie glasses (polarization filtering), and Cherenkov radiation (particles exceeding light speed in medium).

Electromagnetic waves exhibit different behaviors when propagating through different types of matter. In conducting media, the wave amplitude decreases exponentially with distance due to energy dissipation. In dielectric media, the wave propagation depends on whether the material is isotropic or anisotropic. The skin depth phenomenon describes how deeply electromagnetic waves can penetrate into a conductor before their amplitude becomes negligible. The propagation factor and phase difference between electric and magnetic fields are key characteristics that determine wave behavior in different media.

Electromagnetic waves carry energy and momentum as they propagate. Intensity (energy per unit area per unit time) equals I = ½cε₀E₀², showing classical physics treats light energy as proportional to amplitude squared. Momentum transfer creates radiation pressure, with momentum per unit area equal to I/c. Matter interacts with electromagnetic radiation through accelerating charged particles: absorption causes excitation (increased oscillation amplitude), while emission releases stored energy as new electromagnetic oscillations. Classical physics predicts continuous absorption and emission without restriction—objects gradually heat in sunlight as energy transfers continuously. These classical properties will be challenged later, revealing discrete energy exchanges and particle-like behavior in quantum mechanics.

Light is an electromagnetic wave where the electric field vector E and magnetic field vector B are mutually perpendicular to each other and to the direction of propagation; polarization is the phenomenon where these electric vector vibrations are confined to a particular direction in a plane, which can be demonstrated using tourmaline crystals where unpolarized light passing through one crystal produces linearly polarized light, and placing a second crystal at parallel orientation allows maximum transmission while rotating it reduces intensity until the axes become perpendicular.

Polarization is the process by which light waves oscillate in a single plane perpendicular to the direction of propagation. Unpolarized light oscillates in all directions perpendicular to propagation. A polarizer (such as a tourmaline crystal) allows only light waves oscillating in one specific direction to pass through, blocking all other directions. When two polarizers are placed with their transmission axes perpendicular, no light passes through (crossed polarizers). This demonstrates that light is a transverse wave.

Polarization converts light vibrating in different planes to light vibrating in single plane (plane-polarized light). Demonstrates transverse nature of light. Malus's Law: I = I₀cos²(θ), where θ is angle between polarization direction and analyzer axis.

Polarization is the process by which light waves are restricted to vibrate in a single plane. Unpolarized light vibrates in all directions perpendicular to the direction of propagation. Polarized light vibrates in only one direction.

Light is a transverse electromagnetic wave that oscillates in multiple directions perpendicular to its propagation direction, making it unpolarized; polarization is the process of restricting light oscillations to a single direction, which can be achieved through polarizing filters or reflection at non-metallic surfaces, and this phenomenon explains why polarized sunglasses reduce glare by blocking horizontally polarized reflected light while allowing vertically polarized light to pass through.

At the interface (z=0), tangential field components must be continuous. For electric fields: E_i0 + E_r0 cos(θ_i) = E_t0 cos(θ_t). For magnetic fields: (E_i0 / η_1) - (E_r0 / η_1) cos(θ_i) = (E_t0 / η_2) cos(θ_t). Solving these yields the Fresnel equations: r_∥ = (n_2 cos(θ_i) - n_1 cos(θ_t)) / (n_2 cos(θ_i) + n_1 cos(θ_t)) and t_∥ = (2 n_2 cos(θ_i)) / (n_2 cos(θ_i) + n_1 cos(θ_t)). The coefficients satisfy 1 + r_∥ = t_∥ (cos(θ_t) / cos(θ_i)), which simplifies to 1 + r = t at normal incidence.

At the boundary between two dielectrics, the tangential component of the electric field is continuous (E₁_tangential = E₂_tangential), the normal component of the displacement field satisfies κₑ₁E₁_normal - κₑ₂E₂_normal = σ_s (surface charge density), the tangential component of the magnetic field divided by κₘ is continuous (B₁_tangential/κₘ₁ = B₂_tangential/κₘ₂), and the normal component of the magnetic field is continuous (B₁_normal = B₂_normal). Unlike conductors where surface charge density changes rapidly with the wave, in dielectrics σ_s represents static surface charge that remains constant.

Electromagnetic fields must satisfy four boundary conditions at material interfaces: tangential E-field is continuous, tangential H-field is discontinuous by surface current density, normal B-field is continuous, and normal D-field is discontinuous by surface charge density. When an EM wave impinges on a flat interface between two materials (ε1, μ1 to ε2, μ2), the reflection and transmission coefficients are calculated using Fresnel equations. These coefficients depend on incident angle, transmission angle, and wave impedance η = μ/ε. The equations differ for TE and TM polarizations, enabling calculation of how much light is reflected versus transmitted at any interface.

This lecture explains how electromagnetic waves behave at the interface between two dielectrics using boundary conditions. The tangential component of the electric field must be continuous across the interface, leading to the derivation of the laws of reflection (angle of incidence equals angle of reflection) and refraction (Snell's law: n₁sinθ₁ = n₂sinθ₂). For p-polarization, the amplitude reflection coefficient is r_p = [n cosθ_i - cosθ_t]/[n cosθ_i + cosθ_t], and for s-polarization, r_s = [sin(θ_i - θ_t)]/[sin(θ_i + θ_t)]. At Brewster's angle (θ_i + θ_t = 90°), p-polarized light experiences zero reflection. When total internal reflection occurs (sinθ_i > n₂/n₁), an evanescent wave forms that decays exponentially in the second medium without transferring energy across the boundary.

This section applies electromagnetic boundary conditions at dielectric interfaces, where tangential components of both electric and magnetic fields must be continuous. The instructor derives the law of reflection (θi = θr) and Snell's Law (n1 sin(θi) = n2 sin(θt)) from these conditions. The Fresnel equations for parallel polarization are derived: reflection coefficient Γ = (η2 cos(θi) - η1 cos(θt)) / (η2 cos(θi) + η1 cos(θt)) and transmission coefficient τ = (2 η2 cos(θi)) / (η2 cos(θi) + η1 cos(θt)). The normal incidence case is analyzed, simplifying the reflection coefficient to (η2 - η1)/(η2 + η1). The section also covers perpendicular polarization analysis with different Fresnel equations, and explains how arbitrary polarization can be decomposed into parallel and perpendicular components for analysis.

This section derives Maxwell's equations in covariant form. The covariant form is ∂^μF^μν = μ₀J^ν, containing Gauss's law and Ampère-Maxwell's law. Combined with the Bianchi identity ∂^μF^νρ + ∂^νF^ρμ + ∂^ρF^μν = 0 (containing Gauss's law for magnetism and Faraday's law), these two equations fully describe classical electromagnetism. The Lorentz transformation of the field tensor mixes electric and magnetic components: B'y' = γ(B'y + vE'z/c²) and E'x = Ex. This demonstrates that what appears purely electric in one frame becomes a mixture in another, revealing the unified nature of electromagnetism.

Maxwell's equations became asymmetric after Faraday's discovery, requiring special relativity for consistency. For fast-moving particles, Newton's law becomes dp/dt = F with relativistic momentum p = mv/√(1-v²/c²). Electrodynamics unifies using four-vectors: four-potential A^μ = (φ/c, A), four-current J^μ = (cρ, J), and four-gradient ∂^μ = (∂/∂(ct), -∇). The d'Alembertian operator ∂^μ∂_μ = (1/c²)∂²/∂t² - ∇² governs wave propagation. Maxwell's equations become ∂^μF_μν = μ₀J_ν, where F_μν is the electromagnetic field tensor.

Applying force transformation F = F'/√(1 - u²/c²) to the moving frame force yields the lab frame force. This results in F = -q(u × B), where B = μ₀I/(2πx) is the magnetic field produced by the current-carrying wire. This derivation shows that the Lorentz force, traditionally attributed to magnetism, actually arises from relativistic length contraction effects on charge distributions, demonstrating that magnetism is fundamentally a relativistic phenomenon.

This extensive section establishes the theoretical foundation for relativistic electrodynamics. It begins by introducing electromagnetic potentials—the magnetic vector potential A and scalar potential φ—and their governing equations: ∇²A = -μ₀J and ∇²φ = -ρ/ε₀. The instructor emphasizes that these equations require the Lorentz condition (∇·A + (1/c²)(∂φ/∂t) = 0) for mathematical consistency. The section then transitions to relativistic formalism by combining these potentials into a single four-vector potential A^μ with components (A¹, A², A³, A⁴), where the fourth component represents the time part. Similarly, current density becomes a four-vector J^μ with J⁴ = iρc. The Lorentz transformation matrix is introduced: [γ, -βγ, 0, 0; -βγ, γ, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1], where β = v/c and γ = 1/√(1-β²). Both A^μ and J^μ follow this transformation law, enabling unified treatment of electromagnetic phenomena across inertial frames.

The instructor analyzes Relativistic Electrodynamics question patterns: appeared in June 2020 (5 marks), June 2021 (5 marks), June 2022 (5 marks), December 2022 (5 marks), and June 2023 (5 marks). The question appeared in consecutive years with a gap in December 2023 and June 2024. The instructor predicts a question for December 2024 with marks being either 5 or 7.5. For Radiation, the instructor notes consistent question appearance in June 2020, June 2021, June 2022, December 2022, December 2023, and June 2024, predicting a 5-mark question for December 2024.