Problem — Why the displacement current

Problem

Explain why Maxwell added the term $\varepsilon_0\,\dfrac{d\Phi_E}{dt}$ to Ampère’s law. What paradox, linked to a charging capacitor, does this addition resolve?

Solution

The original Ampère’s law. In Ampère’s form, the circulation of $\vv B$ along a closed loop depends only on the conduction current linked to it:

$\oint \vv B \cdot d\vv\ell = \mu_0\, i_\text{linked}$

The linked current is calculated as the flux of $\vv j$ through any surface whose boundary is that closed loop.

The capacitor paradox. Consider the wire carrying current $i$ to a charging capacitor and a closed loop surrounding the wire. We can choose two surfaces with the same boundary:

• surface 1: pierced by the wire $\Rightarrow$ the linked current is $i$, so $\oint \vv B\cdot d\vv\ell = \mu_0 i$;
• surface 2: a “bag-shaped” surface passing between the plates, where no charge flows $\Rightarrow$ linked current $= 0$, so $\oint \vv B\cdot d\vv\ell = 0$.

Same closed loop, two different results: a contradiction. Ampère’s law as it stands is not consistent when the current is not steady.

Maxwell’s correction. Between the plates there is no conduction current, but there is an electric field $\vv E$ that grows with time as the capacitor charges: hence a varying electric flux $\Phi_E$. Maxwell postulates that this varying flux also acts as a source of $\vv B$, introducing the displacement current $i_S = \varepsilon_0\,\dfrac{d\Phi_E}{dt}$:

$\oint \vv B \cdot d\vv\ell = \mu_0\left(i + \varepsilon_0\,\frac{d\Phi_E}{dt}\right)$

On surface 2 the conduction current is zero, but $\varepsilon_0\,d\Phi_E/dt$ is exactly equal to $i$: the two surfaces now give the same result $\mu_0 i$.

$\ev{\varepsilon_0\,\frac{d\Phi_E}{dt}\text{ makes the circulation independent of the surface: paradox resolved, and the EM wave is born}}$

This term, besides restoring consistency, is the key that makes electromagnetic waves possible: a varying electric field generates a magnetic field even in vacuum.

Connections

Topics: Electromagnetic waves Concepts: Displacement current · Ampère’s theorem