Visual walkthrough — Maxwell's equations — integral form, all four

Everything below assumes nothing. We define every arrow, loop, and symbol before using it.

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Step 1 — What a "loop" and its "cap surface" even are

WHAT. Draw a flat wire carrying steady current, and around it draw a closed loop — a rubber band $C$ floating in space, not touching anything. Now stretch a soap film across that loop. That film is a surface $S$ whose edge is exactly the loop $C$.

WHY. The whole fourth equation is a statement about a loop and a surface it borders. Before we write a single symbol we must be crystal clear: the loop is the boundary, the surface is the "cap" you can stretch across it — and there are infinitely many such caps for one loop. That freedom is the seed of the whole story.

PICTURE. Look at figure s01: the pale-yellow rubber band is the loop $C$; the blue film bulging off it is one valid cap surface $S$. The little arrow $d\vec{\ell }$ walks along the loop; the arrow $d\vec{A}$ pokes out of the film.

The right-hand rule ties the two together: curl your right fingers along $d\vec{\ell }$, and your thumb gives the $d\vec{A}$ direction. We'll need this in Step 2.

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Step 2 — What "circulation of $\vec{B}$" measures

WHAT. Around loop $C$ there may be a magnetic field $\vec{B}$ (an arrow at every point saying which way a compass needle turns). We add up, step by step, how much of $\vec{B}$ points along our marching direction. That running total is the circulation.

WHY. Ampère noticed that a current makes $\vec{B}$ wrap around it in circles. So the natural way to detect "is there current threading this loop?" is to ask "does $\vec{B}$ swirl along the loop?" — that is precisely circulation. We use the dot product $\vec{B}\cdot d\vec{\ell }$ and not just $|\vec{B}|$ because only the component of $\vec{B}$ along the march counts; a $\vec{B}$ that crosses the loop sideways does no swirling.

PICTURE. In s02 the blue field arrows wrap around the current wire. At each red step $d\vec{\ell }$ we keep only the part of $\vec{B}$ lying along the step (the projection shown in pink).

Recall Why a dot product and not plain multiplication?

The dot product $\vec{B}\cdot d\vec{\ell }$ equals $|\vec{B}||d\vec{\ell }|\cos \theta$ ::: it automatically throws away the sideways part of $\vec{B}$ (when $\theta =90^{\circ }$, $\cos \theta =0$), leaving only the swirl along the loop.

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Step 3 — The old Ampère law, and the flat cap that "works"

WHAT. For a straight wire carrying current $I$, choose the loop $C$ to be a circle of radius $r$ around the wire and the cap $S$ to be the flat disc the wire pierces. Old Ampère says: circulation of $\vec{B}$ equals ${\mu }_0$ times the current stabbing through the cap.

WHY. This is just Step 2 turned into a law. The current pierces the flat disc once, so "current through $S$", written $I_{enc}$, equals $I$. Everything is consistent — for this cap.

PICTURE. s03: the flat blue disc is stabbed once by the yellow current arrow $I$. Fingers curl along $C$, thumb agrees with $I$.

Hold that phrase — "the chosen cap". It is about to bite us.

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Step 4 — The contradiction: bulge the cap into a capacitor gap

WHAT. Cut the wire and insert a capacitor — two parallel plates with a small vacuum gap (a capacitor). Current still flows in the wire, charging the plates. Keep the same loop $C$ around the wire, but now choose a bulging cap $S^{\prime }$ that slips between the plates instead of the flat one.

WHY. Here is the crisis. For the flat cap, current $I$ stabs through → old Ampère gives ${\oint }_C\vec{B}\cdot d\vec{\ell }={\mu }_{0}I$. For the bulging cap that dips into the gap, no charge crosses the gap — so $I_{enc}=0$ → old Ampère gives ${\oint }_C\vec{B}\cdot d\vec{\ell }=0$. Same loop, two different answers! A law cannot give two answers for one physical thing.

PICTURE. s04: one yellow loop $C$, two caps — the flat blue disc pierced by current, and the pink balloon cap sliding into the empty gap, pierced by nothing.

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Step 5 — What IS threading the gap: a changing electric field

WHAT. Between the plates sits no charge flow — but there is an electric field $\vec{E}$, pointing from the $+$ plate to the $-$ plate, and it is growing as charge piles up. The bulging cap is pierced by this growing $\vec{E}$.

WHY. If nothing "current-like" threads the bulging cap, Ampère is doomed. Maxwell's leap: measure the electric flux through the cap — the total amount of $\vec{E}$ poking through — and watch it change in time. That rate of change is the missing ingredient.

PICTURE. s05: pink $\vec{E}$ arrows fill the gap, and the flux ${\Phi }_E$ through the bulging cap grows as the charge $Q$ on the plates grows.

Notice the plate area $A$ cancelled: ${\Phi }_E=Q/{\epsilon }_0$ depends only on the charge, not the geometry. That clean cancellation is the hint that this term is exactly what we need.

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Step 6 — Maxwell's fix: the displacement current

WHAT. Take the time-derivative of that flux and multiply by ${\epsilon }_0$. Call the result the displacement current $I_d$. Adding $I_d$ to Ampère's law makes both caps agree.

WHY. We need a term that equals $I$ on the bulging cap (where real current is $0$) and equals $0$ on the flat cap (where $\vec{E}$ is negligible, real current does the job). The rate of change of electric flux is precisely that term — watch the algebra: since ${\Phi }_E=Q/{\epsilon }_0$, $I_d={\epsilon }_0\frac{d{\Phi }_E}{dt}={\epsilon }_0\cdot \frac{1}{{\epsilon }_0}\frac{dQ}{dt}=\frac{dQ}{dt}=I.$ The changing $\vec{E}$ through the gap carries exactly the same number as the wire current. Current is "conserved" — it hands off seamlessly from real charge flow to field-change.

PICTURE. s06: the wire current $I$ (yellow) flows into the plate; inside the gap the pink "displacement current" $I_d$ picks up the baton with the identical value $dQ/dt$, so the total threading is continuous across both caps.

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Step 7 — Edge & degenerate cases (never leave a gap)

WHAT. Check the corners where things could break.

WHY. A law must survive every scenario, not just the friendly one.

PICTURE. s07 stacks three mini-cases side by side.

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The one-picture summary

s08 compresses the whole walkthrough: one loop, two caps, the wire current handing off to the displacement current across the gap, and the resulting swirl of $\vec{B}$ that is the same no matter which cap you choose.

Recall Feynman: the whole walkthrough in plain words

Picture a rubber band floating around a wire, and a soap film stretched across it — you can bulge the film any way you like. The old rule said: "count the current stabbing through the film, that tells you how much the magnetic field swirls around the band." Fine for a flat film. But cut the wire and put in two metal plates with a tiny gap. Now if I bulge my film into the gap, no current stabs it — yet the magnetic field around the band is still swirling! Two films, two different answers — that can't be. Maxwell noticed the gap isn't empty: an electric field is growing there as the plates charge up. He measured how fast that electric field's "poke-through" grows, multiplied by the right constants, and — magic — it came out exactly equal to the wire current. So the growing electric field carries the baton across the gap like an invisible current. Add that term and every film agrees. And the bonus prize: in truly empty space, a changing electric field alone can swirl up a magnetic field, which swirls up an electric field, which... off it runs as a wave. That wave is light.

Recall Quick self-test

On the bulging cap in a charging capacitor, what is the real current $I_{enc}$? ::: Zero — no charge crosses the gap. What replaces it so the law still gives ${\mu }_{0}I$? ::: The displacement term ${\mu }_0{\epsilon }_{0}d{\Phi }_E/dt$, which equals ${\mu }_{0}I$. When the capacitor is fully charged, what happens to $I_d$? ::: It drops to zero because $dQ/dt=0$; the law reduces to old Ampère.