Electromagnetic Waves in Vacuum: Maxwell's Equations and Plane Waves | MIT OpenCourseWare A03

Overview

In this MIT OpenCourseWare lecture, the instructor revisits Maxwell's equations and derives the three dimensional wave equation for the electric field in vacuum.

He shows how a plane wave emerges with E and B fields perpendicular to the direction of propagation and discusses the speed of light as c = 1/sqrt(mu0 epsilon0).

Overview

The lecture begins with Maxwell's equations in differential form: Gauss's law for electricity, Gauss's law for magnetism, Faraday's law of induction, and Ampere-Maxwell law. In vacuum, the charge density rho and current density J are taken to be zero, simplifying the equations. The instructor emphasizes Maxwell's addition to Ampere's law, which is crucial for the propagation of electromagnetic waves.

From Maxwell to the Wave Equation

The math identity curl curl A = grad(div A) − del^2 A is introduced and employed. Using this identity on the electric field E and the vacuum condition div E = 0, the lecturer derives the three dimensional wave equation del^2 E = mu0 epsilon0 ∂^2E/∂t^2. The wave speed emerges as c = 1/sqrt(mu0 epsilon0), matching the speed of light and validating Maxwell's description of light as an electromagnetic wave. The same procedure applies to the magnetic field B, yielding del^2B = mu0 epsilon0 ∂^2B/∂t^2.

Plane Waves and Field Orientation

A plane wave solution is discussed: E(r,t) = E0 cos(k·r − ωt), with E perpendicular to the wave vector k. For a simple example with propagation along z and E along x, the dispersion relation ω/k = c must hold. The magnetic field is shown to be B = (1/c) k̂ × E, resulting in B along y for the chosen geometry and E and B in phase for traveling waves. The magnitudes are related by B0 = E0/c, reflecting the orthogonal, in‑phase nature of the fields in a traveling wave.

Standing Waves and Perfect Conductors

The instructor then explores what happens when a plane wave encounters a perfect conductor. A reflected wave forms, constraining the surface boundary condition so that the total electric field vanishes at the conductor surface. This yields a standing wave: E = (E0/2) [cos(kz − ωt) − cos(kz + ωt)] → E = E0 sin(ωt) sin(kz) in the x direction, with B = (E0/c) cos(ωt) cos(kz) in the y direction. The standing wave illustrates how energy shifts between the electric and magnetic fields and how boundary conditions shape wave behavior.

Energy Flow and Poynting Vector

The Poynting vector S = E × B/μ0 describes energy flow. For a traveling wave, energy flows in the propagation direction; for the standing wave, the time-averaged energy flux is zero, while the instantaneous flux oscillates, indicating energy sloshing between the electric and magnetic fields within the region. This ties into practical examples like microwave ovens, where energy transfer occurs via standing field patterns inside cavities.

Historical Context and Takeaways

The lecture touches on historical attempts to measure the speed of light and Maxwell's prediction linking electromagnetic theory to c. The instructor emphasizes the inseparable collaboration of electric and magnetic fields in electromagnetic waves and notes that the general plane wave description requires the electric field to be perpendicular to the propagation direction, with the magnetic field completing the electromagnetic pair.

Summary

Electromagnetic waves in vacuum are governed by Maxwell's equations, giving rise to the wave equation with a finite propagation speed c. Plane waves feature perpendicular E and B fields that travel together, while standing waves arise from boundary conditions such as perfect conductors. The energy flux, boundary effects, and historical validation all reflect the coherent, coupled dynamics of electric and magnetic fields in free space.