The equations in vacuum and the birth of waves

In a region of space with no charges and no currents ($Q_\text{int} = 0$ and $i_\text{link} = 0$), the four Maxwell equations simplify considerably:

$\begin{aligned} \Phi_E &= 0 & \Phi_B &= 0 \\ \Gamma_E &= -\frac{d\Phi_B}{dt} & \Gamma_B &= \varepsilon_0\mu_0\,\frac{d\Phi_E}{dt} \end{aligned}$

In vacuum, the two circulation equations become almost symmetric: a varying $\vec{E}$ produces $\vec{B}$, and a varying $\vec{B}$ produces $\vec{E}$. From this reciprocity a revolutionary idea is born.

Principle — Existence of electromagnetic waves

In vacuum, a field $\vec{E}$ that changes in time generates a field $\vec{B}$, which in turn, by changing, generates a new $\vec{E}$, and so on indefinitely. This self-sustaining chain of inductions is an electromagnetic wave that propagates through vacuum, with no need for any material medium.

This is the turning point of the whole chapter: the fields no longer need charges or currents to sustain them. Once triggered, the disturbance of the fields detaches from the source and travels on its own. What remains to be understood is how fast it travels — and the answer, surprisingly, coincides with the speed of light.

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Topics: Onde elettromagnetiche Concepts: Equazioni di Maxwell · Onda elettromagnetica

Related exercises: Problem — EM waves without a medium · Problem — EM waves and the structure of vacuum · Problem — Microwave oven