The four equations (first, incomplete version)

Before introducing his celebrated correction, let us gather everything we know so far about the fields $\vec{E}$ and $\vec{B}$. Remarkably, the whole of electrostatics, magnetostatics and induction condense into four equations: two about fluxes through closed surfaces (Gauss’s theorems) and two about circulations along closed curves.

Flux equations (Gauss's theorems)

$\Phi_E(\text{closed surf.}) = \frac{Q_\text{int}}{\varepsilon_0} \qquad\text{(Gauss for } \vec{E}\text{)}$ $\Phi_B(\text{closed surf.}) = 0 \qquad\text{(Gauss for } \vec{B}\text{)}$

The flux of $\vec{E}$ through a closed surface measures the enclosed charge: charges are sources of the electric field. The flux of $\vec{B}$, on the other hand, is always zero: no magnetic monopoles exist, the lines of $\vec{B}$ are always closed on themselves.

Circulation equations

$\Gamma_E = -\frac{d\Phi_B}{dt} \qquad\text{(Faraday-Neumann)}$ $\Gamma_B = \mu_0\,i_\text{linked} \qquad\text{(Ampère)}$

The circulation of $\vec{E}$ along a closed line equals minus the rate of change of the linked magnetic flux: a changing magnetic field generates an electric field that curls (induction). The circulation of $\vec{B}$, according to Ampère, is proportional to the linked current.

These four equations seem to summarise the whole of electromagnetism. But there is a serious problem: Ampère’s equation is incomplete. The next step shows where it breaks down, with the example of a charging capacitor.

Links

Topics: Onde elettromagnetiche Concepts: Equazioni di Maxwell · Teorema di Gauss · Legge di Faraday-Neumann-Lenz · Teorema di Ampère · Campo elettrico · Campo magnetico

Related exercises: Problem — Field B in a charging capacitor · Problem — Amplitudes of the solar wave · Which graph of E