The displacement current

Between the capacitor’s plates no current of charges flows, but there is an electric field $\vec{E}$ that grows as the plates charge up. Maxwell had the decisive insight: a time-varying electric field must produce a magnetic field, exactly as — by the Faraday-Neumann law — a varying magnetic field produces an electric field. He therefore added to Ampère’s law a term that exactly compensates the contradiction of the capacitor paradox.

Ampère-Maxwell law

$\ev{\Gamma_B = \mu_0\,i_\text{link} + \varepsilon_0\mu_0\,\frac{d\Phi_E}{dt}}$ The second term, $\varepsilon_0\mu_0\,\dfrac{d\Phi_E}{dt}$, is the displacement current: a varying electric flux acts as a source of the magnetic field just like a real current.

The insight is not merely aesthetic: the new term repairs the paradox quantitatively. Let us verify that the circulation $\Gamma_B$ becomes the same again for both surfaces $S_1$ and $S_2$ sharing the boundary $\gamma$.

Verification — $\Gamma_B$ is the same for $S_1$ and $S_2$

For surface $S_2$, which passes between the plates, there is no current of charges but the electric field grows. The charge $Q$ on the plates grows, and with it the flux $\Phi_E = Q/\varepsilon_0$ (by Gauss). Then Maxwell’s term equals $\varepsilon_0\mu_0\,\frac{d\Phi_E}{dt} = \mu_0\,\frac{dQ}{dt} = \mu_0\,i$ Result: $\Gamma_B = 0 + \mu_0 i = \mu_0 i$, the same as for $S_1$, which cuts the wire. The contradiction is resolved: the circulation depends only on the boundary, not on the surface.

This symmetry — a varying $\vec{B}$ generates $\vec{E}$, a varying $\vec{E}$ generates $\vec{B}$ — is the seed from which, a few pages further on, electromagnetic waves will sprout.

Links

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

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