Visual walkthrough — Displacement current — Maxwell's addition to Ampere's law

We start from the most concrete thing possible: a wire, two plates, and a compass.

Step 1 — A loop and a wire: what "circulation" means

WHAT we did: named the number we get by summing "field-along-the-walk" around a ring. WHY: Ampère's law is a statement about this exact number — so we must feel it before using it. PICTURE: In s01, the loop is the cyan ring. The amber arrow is one step $d\vec{l}$; the white arrow is $\vec{B}$ there. Only the overlap (their dot product) enters the sum.

Step 2 — "Through the loop" hides a trap: which surface?

WHAT: showed one rim can carry infinitely many surfaces. WHY: This freedom is the whole game. If two legal surfaces disagree, the law is broken. PICTURE: In s02, one cyan rim, two white surfaces — a flat disc and a bulging cap — sharing that rim.

$Surface\text{ choice must not matter}:::Becausealoopisonlyarim;"enclosedcurrent"mustbeidenticalforeverysurfaceitbounds,orAmp\grave{e}r{e}^{\prime }slawisself-contradictory.$

Step 3 — The capacitor exposes the trap

Now draw one loop around the wire, and choose two surfaces bounded by it:

• Flat surface — slices straight through the wire. The conduction current $I$ pierces it.
• Bulging surface — dodges around and passes between the plates, through the empty gap.

WHAT: applied Step 2's freedom to a real circuit. WHY: to catch the contradiction in the act. PICTURE: s03 — amber current arrow spears the flat surface (left). The bulging surface (right) slips into the gap where nothing flows.

Step 4 — What is alive in the gap? A growing electric field

First we need the field. By Gauss's Law, the uniform field between plates of charge $Q$ and area $A$ is:

WHAT: computed the gap field and noticed it grows. WHY: to find a physical quantity that can stand in for the missing current. PICTURE: s04 — parallel field lines (cyan) thickening as $Q$ rises; the amber $E$ arrow lengthens with time.

Step 5 — Flux: bottling the field as one number

Plug in the field from Step 4:

WHAT: collapsed the whole field pattern into one number tied directly to $Q$. WHY: the missing "current" will turn out to be a rate of flux — so we need flux first. PICTURE: s05 — the bulging surface catching all the field lines; a counter reads ${\Phi }_E=Q/{\epsilon }_0$.

Step 6 — Take the time-rate: the flux changes exactly like a current

WHAT: turned the static flux into a rate, and recognised that rate is set by the wire current. WHY: because current means charge-per-second, and here $dQ/dt=I$ links the gap to the wire. PICTURE: s06 — wire current $I$ (amber) feeding the plate; a rising graph of ${\Phi }_E$ vs time with slope $I/{\epsilon }_0$.

Step 7 — Define $I_d$ so both surfaces agree

WHAT: named and sized the missing piece so the two surfaces reconcile. WHY: consistency of the law forces this definition — nothing arbitrary about it. PICTURE: s07 — both surfaces now labelled with the same value ${\mu }_{0}I$; a bridge marked "$I_d=I$" spans the gap.

Step 8 — Edge cases: every scenario, checked

PICTURE: s08 — three mini-panels: (a) steady $Q$, flat flux graph, $I_d=0$; (b) falling $Q$, $I_d$ reversed; (c) oscillating field, no charge, $I_d$ wiggling.

Step 9 — The completed law

Together with Faraday's Law of Induction (changing $\vec{B}$ makes $\vec{E}$), this new term (changing $\vec{E}$ makes $\vec{B}$) closes the loop into a self-feeding ripple — that's why $I_d$ predicts light at speed $c=1/\sqrt{\mu_0\varepsilon_0}$, and why it completes Maxwell's Equations.

The one-picture summary

The whole story in one frame: a loop with two surfaces (flat + bulge), the wire current on the left, the growing field on the right, and the bridge $I_d={\epsilon }_{0}d{\Phi }_E/dt=I$ that makes both surfaces agree.

Active Recall