Chapter 20: Solutions of Maxwell’s Equations in Free Space

Solutions of Maxwell’s Equations in Free Space from The Feynman Lectures on Physics Volume II meticulously details the solutions of Maxwell's equations when no external charges or currents are present, primarily focusing on the physics of electromagnetic waves. The discussion initiates with a four-equation set describing free-space fields and immediately introduces the concept of plane waves. An illustrative example of an infinite current sheet suddenly switching on demonstrates that the resulting electric field propagates outward as a distinct waveform traveling at the speed of light, designated as 'c'. This analysis emphasizes that electromagnetic effects are not instantaneous but are transmitted at a finite speed, showcasing the principle of superposition for solving complex current configurations. To find generalized wave solutions, the analysis moves to the scalar potential (phi) and the vector potential (A), demonstrating that, in free space, both potentials satisfy the three-dimensional wave equation. Furthermore, it is proven that each individual component of the electric field (E) and the magnetic field (B) also adheres to this same wave equation. For one-dimensional solutions, the mathematical function describing the disturbance can be expressed as a function of the form (x minus ct) or (x plus ct), which confirms that the waveform travels through space without alteration of its shape. A critical finding for these plane waves is that the electric and magnetic fields are always transverse—meaning they lie perpendicular both to each other and to the direction in which the wave is moving. Expanding to three-dimensional waves, the chapter explains that the general solution can be understood as the superposition of multiple plane waves propagating in all possible directions. The text also takes a significant detour to discuss the challenges of scientific visualization, highlighting the difficulty of forming mental pictures of invisible fields and advocating for the reliance on mathematical tools. Finally, the derivation of spherical waves is covered. The key finding for a spherical wave traveling outward from a central source is that the potential is proportional to the inverse of the distance 'r' (1/r) multiplied by a function of (t minus r/c), showing that the wave amplitude diminishes inversely with distance as it propagates.