Problem — Displacement current and the external field

Problem

Using the Ampère-Maxwell law, explain why the magnetic field generated by the displacement current between the plates of a charging capacitor is, outside the capacitor ($r > R_0$, with $R_0$ the plate radius), completely indistinguishable from that produced by a current $i = \varepsilon_0\,\pi R_0^2\,(\dd E/\dd t)$ flowing in a straight wire along the axis.

Solution

Setup. The Ampère-Maxwell law states that the circulation of the magnetic field along a closed path $\gamma$ depends both on the enclosed conduction current and on the rate of change of the electric flux through any surface bounded by $\gamma$:

$\oint_\gamma \vv B\cdot\dd\vv\ell = \mu_0 i_\text{cond} + \mu_0\varepsilon_0\,\ddt{\Phi_E}$

Choice of path. Take as $\gamma$ a circle of radius $r > R_0$ centred on the capacitor’s axis, coaxial with the plates. Between the plates there is no conduction current ($i_\text{cond} = 0$): the magnetic field is due solely to the displacement term.

Enclosed electric flux. The electric field is confined between the plates, within the disc of radius $R_0$. So the electric flux through any surface bounded by $\gamma$ is due only to that disc:

$\Phi_E = E\,\pi R_0^2$

and is therefore independent of $r$: whatever the radius of the path (provided $r > R_0$), the enclosed flux is always the same.

Displacement term. Differentiating:

$\mu_0\varepsilon_0\,\ddt{\Phi_E} = \mu_0\varepsilon_0\,\pi R_0^2\,\ddt{E} = \mu_0\,i \qquad\text{with}\qquad i = \varepsilon_0\,\pi R_0^2\,\ddt{E}$

Conclusion. The circulation then becomes

$\oint_\gamma \vv B\cdot\dd\vv\ell = \mu_0\,i$

which is exactly Ampère’s law for a straight wire carrying current $i$. By symmetry $B$ is constant along $\gamma$ and tangential, so $\Gamma_B = B\cdot 2\pi r = \mu_0 i$, giving

$\ev{B = \frac{\mu_0 i}{2\pi r}, \qquad i = \varepsilon_0\,\pi R_0^2\,\ddt{E}}$

identical to the field of a wire. Outside the capacitor, the displacement current between the plates cannot be distinguished from a genuine conduction current in the wire: it is precisely this continuity that guarantees the consistency of Maxwell’s equations.

Collegamenti

Argomenti: Onde elettromagnetiche Concetti: Corrente di spostamento · Teorema di Ampère