The capacitor paradox

The incompleteness of Ampère’s law shows up in a very simple case: a charging capacitor, connected to a generator. The current $i$ flows in the connecting wire but does not cross the plates — between them there is only vacuum (or a dielectric), where no charge moves.

The same boundary $\gamma$ (dashed circle) bounds two different surfaces: $S_1$ cuts the wire, $S_2$ slips between the capacitor’s plates.

We apply Ampère’s law to the same closed curve $\gamma$ (the “boundary”), but choosing two different surfaces that have $\gamma$ as their edge:

• $S_1$ cuts the wire $\Rightarrow$ $i_\text{linked} = i$, so $\Gamma_B = \mu_0\,i$.
• $S_2$ passes between the plates (cuts no wire) $\Rightarrow$ $i_\text{linked} = 0$, so $\Gamma_B = 0$.

Contradiction!

The same circulation $\Gamma_B$, computed for two different surfaces but with the same boundary, gives different results ($\mu_0 i$ or $0$). But $\Gamma_B$ should depend only on the boundary $\gamma$, not on the surface chosen! Something is missing in Ampère’s law.

The key to resolving the paradox is to notice that between the plates, where no current of charges flows, an electric field grows: it is precisely the change in this field that is the missing source of the magnetic field.

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Topics: Onde elettromagnetiche Concepts: Teorema di Ampère · Corrente di spostamento · Capacità e condensatore

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