Foundations — Ampere's circuital law — magnetostatic form

This page assumes nothing. Before you can read the parent note comfortably, every squiggle in ${\oint }_C\vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$ must feel obvious. We build them one at a time, each from the previous.

0. What a vector is (the arrow)

Contrast with a plain number ("scalar") like temperature — that has size only, no direction. The magnetic field is directional (it points somewhere at every spot), so we need arrows to describe it.

Figure s01. A single vector: it starts at a point (the black dot), its length is its size, and its arrowhead (orange) is its direction. The dotted plum lines show that the same arrow can be split into a horizontal part and a vertical part — a trick we reuse for the dot product later.

• Picture: an arrow starting at a point, pointing somewhere.
• Why the topic needs it: the magnetic field $\vec{B}$ is a vector — at every point in space it has a strength and a direction. We cannot talk about "field pushing you along a path" without arrows.

1. The magnetic field $\vec{B}$ (an arrow at every point)

• Symbol: $\vec{B}$, measured in tesla (T).
• Picture: concentric circles of arrows hugging the wire.
• Why the topic needs it: $\vec{B}$ is the star of the show — Ampère's law is a statement about $\vec{B}$.

2. Current $I$ and current density $\vec{J}$

• Picture: $I$ = total flow down a pipe; $\vec{J}$ = how densely the flow is packed at each spot of the pipe's mouth; $d\vec{A}$ = an arrow poking out of that mouth, and we count only the flow going along it.
• Why the topic needs it: the source of $\vec{B}$ is current. The parent's Worked Example 2 (thick wire of radius $R$, see §8) needs $J=I/(\pi R^2)$ to find how much current is inside a smaller circle.

3. The angle idea and the dot product $\vec{B}\cdot d\vec{l}$

Before the big integral we need to know what it means for the field to "push you along your path." That word along is the dot product.

Figure s02. The step $d\vec{l}$ (orange) lies along the path. The field $\vec{B}$ (teal) leans at angle $\theta$. Drop $\vec{B}$ straight down onto the step direction (dotted plum) and the thick plum segment is the along-path part, $|\vec{B}|\cos \theta$. That length, times the step length, is $\vec{B}\cdot d\vec{l}$.

• All the cases you must know:
  • $\theta =0^{\circ }$: $\cos \theta =1$ → full push. (Used in the wire loop: $\vec{B}$ runs along the circle.)
  • $\theta =90^{\circ }$: $\cos \theta =0$ → zero. (Used in the solenoid: the short rectangle sides are perpendicular to $\vec{B}$.)
  • $\theta =180^{\circ }$: $\cos \theta =-1$ → full negative (opposing) contribution.
• Why the topic needs it: every appearance of $\vec{B}\cdot d\vec{l}$ is "field-along-path." The whole method of choosing clever loops rests on making $\theta$ either $0$ (constant push) or $90^{\circ }$ (zero).

4. $d\vec{l}$ — a tiny step along a path

• Picture: a curve broken into thousands of tiny arrows laid nose-to-tail; each is a $d\vec{l}$.
• Why the topic needs it: to "add up the push along the path," we chop the path into tiny steps $d\vec{l}$, take the along-part $\vec{B}\cdot d\vec{l}$ of each, and sum them.

5. The closed loop integral ${\oint }_C$

Figure s03. The teal circle is the Amperian loop $C$ around a wire pointing out of the page. The orange arrows are the steps $d\vec{l}$; here we walk counter-clockwise. The $\vec{B}$ field wraps along the same direction, so each $\vec{B}\cdot d\vec{l}$ is a full positive push, and their total is the circulation. The plum arrow marks the perpendicular distance $s$.

• Why the topic needs it: the entire left-hand side of Ampère's law is this circulation. The magic claim is that this total is simple even when each step is complicated.

6. Orientation: linking the walk direction to $I_{\text{enc}}$ (the right-hand rule)

Ampère's law compares two things — the direction you walk around $C$, and the direction current is counted as positive through the surface. These two must be tied together by a fixed convention, or the signs go haywire.

• Picture: fingers curling counter-clockwise (as in Figure s03) → thumb points out of the page → current flowing out of the page is positive.
• Why the topic needs it: without this, a loop encircling the same wire could give $+{\mu }_{0}I$ or $-{\mu }_{0}I$. The rule removes the ambiguity, and it's the same right-hand rule that says which way $\vec{B}$ wraps a wire (§9).

7. $I_{\text{enc}}$ — the enclosed current

• Picture: a ring with a film across it; count only the current arrows that stab through the film, $+$ or $-$ by the thumb direction.
• Why the topic needs it: it is the entire right-hand side (times ${\mu }_0$). "The total push equals ${\mu }_0$ times what threads the loop" is the law.

8. The constant ${\mu }_0$

• Why the topic needs it: it makes the units on both sides of $\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$ match, and sets the strength of the effect.

9. $\pi$, circles, the distance $s$, and the wire radius $R$

• Why the topic needs it: every geometric answer (wire, thick wire, solenoid, toroid) uses these circle facts to compute the loop length or the pierced area. Keeping $s$ (distance out) and $R$ (wire radius) as separate names is essential in the thick-wire case where $J=I/(\pi R^2)$ but you evaluate at a smaller radius $s<R$.

10. The right-hand rule for the field's twist

• Picture: thumb up the wire, fingers curling around it counter-clockwise (seen from the thumb side).
• Why the topic needs it: it fixes the direction $\vec{B}$ points on your loop, so you know whether $\vec{B}\cdot d\vec{l}$ adds or subtracts.

How the foundations feed the topic

Equipment checklist

Test yourself: cover the right side and answer before revealing.

Ready? Then head back to the parent note — and when the differential form $\nabla \times \vec{B}={\mu }_0\vec{J}$ or Stokes' theorem appear, or you compare with Gauss's law for magnetism and Biot–Savart law, you now have every symbol earned.