Nothing restricts the bounded surface of Ampère’s law to being flat. It’s not hard to see that any current passing through surface $\text{S}{}_1$ in FIGURE 31.14 must also pass through the curved surface $\text{S}{}_2$. To interpret Ampère’s law properly, we have to say that the current $I_{\text{through}}$ is the net current passing through any surface S that is bounded by curve C.

FIGURE 31.14 The net current passing through the flat surface ${\text{S}}_1$ also passes through the curved surface ${\text{S}}_2$.

But this leads to an interesting puzzle. FIGURE 31.15a shows a capacitor being charged. Current I, from the left, brings positive charge to the left capacitor plate. The same current carries charges away from the right capacitor plate, leaving the right plate negatively charged. This is a perfectly ordinary current in a conducting wire, and you can use the right-hand rule to verify that its magnetic field is as shown.

FIGURE 31.15 There is no current through surface ${\text{S}}_2$ as the capacitor charges, but there is a changing electric flux.

Curve C is a closed curve encircling the wire on the left. The current passes through surface $\text{S}{}_1$, a flat surface across C, and we could use Ampère’s law to find that the magnetic field is that of a straight wire. But what happens if we try to use surface $\text{S}{}_2$ to determine $I_{\text{through}}$? Ampère’s law says that we can consider any surface bounded by curve C, and surface $\text{S}{}_2$ certainly qualifies. But no current passes through $\text{S}{}_2$. Charges are brought to the left plate of the capacitor and charges are removed from the right plate, but no charge moves across the gap between the plates. Surface $\text{S}{}_1$ has $I_{\text{through}}=I$, but surface $\text{S}{}_2$ has $I_{\text{through}}=0$. Another dilemma!

It would appear that Ampère’s law is either wrong or incomplete. Maxwell was the first to recognize the seriousness of this problem. He noted that there may be no current passing through $\text{S}{}_2$, but, as FIGURE 31.15b shows, there is an electric flux ${\text{Φ}}_{\text{e}}$ through $\text{S}{}_2$ due to the electric field inside the capacitor. Furthermore, this flux is changing with time as the capacitor charges and the electric field strength grows. Faraday had discovered the significance of a changing magnetic flux, but no one had considered a changing electric flux.

The current I passes through $\text{S}{}_1$, so Ampère’s law applied to $\text{S}{}_1$ gives

We believe this result because it gives the correct magnetic field for a current-carrying wire. Now the line integral depends only on the magnetic field at points on curve C, so its value won’t change if we choose a different surface S to evaluate the current. The problem is with the right side of Ampère’s law, which would incorrectly give zero if applied to surface $\text{S}{}_2$. We need to modify the right side of Ampère’s law to recognize that an electric flux rather than a current passes through $\text{S}{}_2$.

The electric flux between two capacitor plates of surface area A is

The capacitor’s electric field is $E=Q\text{/}{ϵ}_{0}A;$ hence the flux is actually independent of the plate size:

The rate at which the electric flux is changing is

where we used $I=dQ\text{/}dt$. The flux is changing with time at a rate directly proportional to the charging current I.

Equation 31.19 suggests that the quantity ${ϵ}_0(d{\text{Φ}}_{\text{e}}\text{/}dt)$ is in some sense “equivalent” to current I. Maxwell called the quantity

the displacement current. He had started with a fluid-like model of electric and magnetic fields, and the displacement current was analogous to the displacement of a fluid. The fluid model has since been abandoned, but the name lives on despite the fact that nothing is actually being displaced.

Maxwell hypothesized that the displacement current was the “missing” piece of Ampère’s law, so he modified Ampère’s law to read

Equation 31.21 is now known as the Ampère-Maxwell law. When applied to Figure 31.15b, the Ampère-Maxwell law gives

where, for surface $\text{S}{}_2$, we used Equation 31.19 for $d{\text{Φ}}_{\text{e}}\text{/}dt$. Surfaces $\text{S}{}_1$ and $\text{S}{}_2$ now both give the same result for the line integral of $\vec{B}\cdot d\vec{s}$ around the closed curve C.