Ampere's circuital law — magnetostatic form

WHAT is it?

WHAT it gives you: the magnitude of $\vec B$ — if the geometry has enough symmetry.
WHAT it is NOT: valid as written only for magnetostatics (no changing electric fields). The full Maxwell version adds the displacement-current term $\mu_0\epsilon_0\,d\Phi_E/dt$ .

WHY is it true? (Derivation from Biot–Savart)

We don't postulate it — we derive it for the simplest case and argue generality.

Formula Step 3 — Why ANY loop works

Split the loop integral into radial and angular parts in polar coords: $\oint \vec B\cdot d\vec l = \oint \frac{\mu_0 I}{2\pi s}\,(s\,d\phi) = \frac{\mu_0 I}{2\pi}\oint d\phi$

$\vec B$ has no radial component, so radial steps contribute $0$ .
$\oint d\phi = 2\pi$ if the loop encircles the wire (angle sweeps a full turn), giving $\mu_0 I$ .
$\oint d\phi = 0$ if the loop does not enclose the wire (angle returns without a full loop), giving $0$ .

Why this proves generality: the result depends only on whether the current is enclosed, not on the path. By superposition, this extends to any collection of steady currents → the law holds in general for magnetostatics. $\blacksquare$

Figure — Ampere's circuital law — magnetostatic form

HOW to use it (the recipe)

Forecast-then-Verify

Recall Predict before checking

A second wire carrying $2I$ sits outside your circular Amperian loop. Does $\oint\vec B\cdot d\vec l$ change? Answer: No — $\oint\vec B\cdot d\vec l$ depends only on enclosed current, so it's unchanged ( $=\mu_0 I$ ). But note: the local $\vec B$ at points on the loop does change (the outside wire adds field); the integral's contributions just cancel out. This is the subtlety people miss.

Common mistakes (Steel-man + fix)

Flashcards

Ampère's law (integral magnetostatic form)

\oint_C \vec B\cdot d\vec l = \mu_0 I_{\text{enc}}

What does

I_{\text{enc}}

mean precisely?

Net steady current piercing any surface bounded by the loop

C

(external currents excluded).

Field of an infinite straight wire from Ampère's law

B=\dfrac{\mu_0 I}{2\pi s}

, circular around the wire.

Why does the radius cancel for a wire?

B\propto 1/s

but loop length

\propto s

; product is

s

-independent.

Field inside a uniform thick wire (radius

R

) at

s<R

B=\dfrac{\mu_0 I s}{2\pi R^2}

(linear in

s

Field inside a long solenoid

B=\mu_0 n I

, uniform, axial.

Field inside a toroid radius

r

N

turns

B=\dfrac{\mu_0 N I}{2\pi r}

; zero outside.

Differential form of Ampère's law

\nabla\times\vec B=\mu_0\vec J

When is the magnetostatic form WRONG?

When electric fields change in time; need displacement current

\mu_0\epsilon_0\,\partial_t\Phi_E

Can

\oint\vec B\cdot d\vec l=0

while

\vec B\ne0

on the loop?

Yes — contributions cancel; circulation

\ne

pointwise value.

When is Ampère's law USEFUL (not just true)?

When symmetry makes

\vec B

constant & parallel (or zero) along a chosen loop.

Recall Feynman: explain to a 12-year-old

Imagine a river current flowing down a pipe (that's the electric current). It makes the water around it swirl in circles (that's the magnetic field). Ampère's rule says: if you walk one full lap around the pipe and keep track of how much the swirling pushes you along your steps, the total push only depends on how strong the river inside your lap is — it doesn't matter if you walk a big circle or a wobbly weird loop, as long as you go around the pipe once. If you walk around an empty spot (no current inside), the pushes cancel and you get zero total.

Connections

Biot–Savart law — the integral law Ampère's law is derived from.
Gauss's law for magnetism — $\nabla\cdot\vec B=0$ , the divergence partner.
Gauss's law (electric) — the symmetry-shortcut analog for $\vec E$ .
Stokes' theorem — converts circulation ↔ curl.
Solenoid and toroid fields — flagship applications.
Displacement current — the term that completes this into Maxwell–Ampère.
Maxwell's equations — where this becomes one of four.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Ampère ka circuital law ek shortcut hai magnetic field nikalne ka. Idea simple hai: koi bhi closed loop (Amperian loop) chuno, aur us loop ke around $\vec B$ ka "kitna saath-saath push" hai woh add karo — yeh total sirf us current pe depend karta hai jo loop ke andar se guzar raha hai. Formula: $\oint \vec B\cdot d\vec l = \mu_0 I_{\text{enc}}$ . Loop ka shape kaisa bhi ho, bahar ke currents matter nahi karte (woh local field toh badalte hain, par integral mein cancel ho jaate hain).

Yeh kaam kaise karta hai? Infinite wire ke liye $B=\frac{\mu_0 I}{2\pi s}$ — yeh hum Biot–Savart se nikalte hain. Ab circle loop lo radius $s$ ka: $B$ constant aur $d\vec l$ ke parallel hai, toh $\oint = B\times 2\pi s = \mu_0 I$ . Magic yeh hai ki $2\pi s$ (circumference) aur $1/2\pi s$ (field) cancel ho jaate hain, $s$ gayab. Isi cancellation ki wajah se answer loop-shape se independent hai.

Practical baat: yeh law hamesha sach hai (magnetostatics mein), par useful tabhi jab symmetry ho — straight wire, solenoid ( $B=\mu_0 n I$ ), toroid ( $B=\frac{\mu_0 N I}{2\pi r}$ ), thick wire. Symmetry na ho toh $B$ ko integral se bahar nahi nikaal sakte, phir Biot–Savart use karo. Aur yaad rakho: agar current time ke saath change ho raha hai, toh yeh simple form fail karta hai — phir Maxwell ka displacement current add karna padta hai, jisse hi EM waves discover hui thi.

Common galti: " $\oint\vec B\cdot d\vec l=0$ matlab $B=0$ " — galat! Plus aur minus contributions cancel ho sakte hain, field zero nahi. Aur " $I_{\text{enc}}$ matlab saari current" — nahi, sirf jo loop ke andar se piercing kar rahi hai.

Ampere's circuital law — magnetostatic form

WHAT is it?

WHY is it true? (Derivation from Biot–Savart)

HOW to use it (the recipe)

Forecast-then-Verify

Common mistakes (Steel-man + fix)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections