Transverse Nature of Electromagnetic Waves

Working out the geometry from Maxwell's equations. Carrying Maxwell's equations through for a simple plane electromagnetic wave travelling in vacuum (a calculation WBCHSE does not require in algebraic detail, only its RESULT) fixes an exact, and rather strict, geometric relationship between the wave's electric field $\vec{E}$, its magnetic field $\vec{B}$, and the direction $\hat{n}$ in which the wave travels: all three are mutually PERPENDICULAR to one another at every instant and at every point of the wave, with $\vec{E}\times\vec{B}$ pointing along the direction of propagation.

Why 'transverse'. A wave is called transverse when the quantity that is oscillating (here, the electric and magnetic FIELDS themselves, not any physical medium) oscillates PERPENDICULAR to the direction the wave travels in -- exactly as a wave on a stretched string moves sideways while the disturbance itself travels along the string's length. For an electromagnetic wave travelling along, say, the $x$-axis (this section's figure), the electric field $\vec{E}$ oscillates entirely along the $y$-axis (i.e. $\vec{E}$ has no $x$-component at all) and the magnetic field $\vec{B}$ oscillates entirely along the PERPENDICULAR $z$-axis (no $x$-component either) -- both fields oscillate crosswise to the direction of travel, never along it, which is exactly what makes the wave transverse rather than longitudinal (a longitudinal wave, like sound, oscillates ALONG its own direction of travel instead).

$E$ and $B$ oscillate together, in phase. At every point along the wave, $\vec{E}$ and $\vec{B}$ reach their maximum values at exactly the same instant and pass through zero at exactly the same instant -- they are said to oscillate IN PHASE with each other, never one leading or lagging the other. Their magnitudes are also locked together at every instant, related by the wave's speed $c$ (Section 8.5): $E_0 = cB_0$ for the peak (amplitude) values, and $E=cB$ at every instant. …