The four Maxwell equations

Here is the final form of the four equations, the complete summary of all of classical electromagnetism:

Maxwell's equations (integral form)

$\Phi_E(\text{closed surf.}) = \frac{Q_\text{int}}{\varepsilon_0} \qquad\text{(Gauss } \vec{E}\text{)}$ $\Phi_B(\text{closed surf.}) = 0 \qquad\text{(Gauss } \vec{B}\text{)}$ $\Gamma_E = -\frac{d\Phi_B}{dt} \qquad\text{(Faraday-Neumann)}$ $\Gamma_B = \mu_0\,i_\text{link} + \varepsilon_0\mu_0\,\frac{d\Phi_E}{dt} \qquad\text{(Ampère-Maxwell)}$

Let us read them as a coherent whole:

• Gauss for $\vec{E}$: electric charges are sources (or sinks) of the electric field; the lines of $\vec{E}$ begin and end on charges.
• Gauss for $\vec{B}$: no magnetic monopoles exist; the lines of $\vec{B}$ are always closed.
• Faraday-Neumann: a varying magnetic flux generates a circulating electric field.
• Ampère-Maxwell: both currents of charges and a varying electric flux generate a circulating magnetic field.

Modern technology

Maxwell’s equations underpin all electromagnetic technology: radio, Wi-Fi, lasers, electric motors, optical fibres, microwave ovens. Four lines that govern an entire world of devices.

The first two equations speak of fluxes (the “source” aspect), the last two of circulations (the “vortex” aspect). The symmetry between the last two — a changing field generates the other — is what makes the electromagnetic wave possible.

Links

Topics: Onde elettromagnetiche Concepts: Equazioni di Maxwell · Teorema di Gauss · Legge di Faraday-Neumann-Lenz · Teorema di Ampère · Corrente di spostamento

Related exercises: Problem — Field B in a charging capacitor · Problem — Why the displacement current · Problem — Displacement current and the external field