Ampere's circuital law — magnetostatic form: Question bank

This page assumes only the parent law: $\oint_C \vec B\cdot d\vec l = \mu_0 I_{\text{enc}}$. Every symbol below is used there. $B$ = magnetic field, $d l$ = a tiny step along the closed loop $C$ , $I_{enc}$ = the current that actually pierces a surface stretched across the loop.

Definition Symbols used on this page

Before any trap, pin down every letter that appears below, each anchored to the picture in figure s01.

$== s ==$ — the radial distance of a point from the wire, measured straight out perpendicular to the wire (the shortest gap). Units: metres.
$== R ==$ — the full outer radius of a thick wire (a real wire with finite thickness), i.e. how far its metal extends from the axis. Units: metres. Note $s$ can be inside ( $s < R$ ) or outside ( $s > R$ ).
$== n ==$ — a solenoid's turns per unit length: how many wire loops are packed into each metre of the coil. Units: turns per metre ( $m^{- 1}$ ). A denser coil has larger $n$ .
$== N ==$ — the total number of turns wound onto a toroid (a doughnut-shaped coil). A plain count, no units.
$r$ — for the toroid, the radius of the circular Amperian loop running inside the doughnut, measured from the doughnut's central hole. Units: metres.

Figure — Ampere's circuital law — magnetostatic form

Recall Read the picture first

On figure s01, which arrow is $s$ and which shaded band is set by $R$ ? ::: $s$ is the horizontal arrow from the wire's centre to the field point; $R$ is the edge of the shaded metal disc — the boundary between "inside the wire" ( $s < R$ ) and "outside" ( $s > R$ ).

True or false — justify

The circulation

\oint B \cdot d l

depends on the shape of the Amperian loop.

False — it depends only on the enclosed current

μ_{0} I_{enc}

; any loop enclosing the same current gives the same value, which is the whole power of the law.

\oint B \cdot d l = 0

then

B = 0

at every point on the loop.

False — a zero sum can hide large fields that cancel; e.g. a loop sitting next to (but not around) a wire has real

B

on it but zero circulation.

Ampère's law is only approximately true; Biot–Savart is the exact one.

False — in magnetostatics both are exact and equivalent; Ampère's law is derived from Biot–Savart, so it cannot be less exact.

Ampère's law can always be used to compute

B

False — it is always true but only solvable when symmetry lets you pull

B

out of the integral (infinite wire, solenoid, toroid, sheets); otherwise it is one equation with too many unknowns.

Doubling the radial distance

s

of a circular loop around a straight wire halves the circulation.

False —

B \propto 1/ s

falls but the circumference

2 π s

rises by the same factor, so the product

μ_{0} I

is untouched; the radius genuinely cancels.

A current flowing along the axis of a circular Amperian loop but never crossing its surface still contributes to

I_{enc}

False — only current that pierces the spanning surface counts; a current running in the loop's own plane threads nothing.

The magnetostatic form

\oint B \cdot d l = μ_{0} I_{enc}

holds while a capacitor is charging.

False — a charging capacitor has changing

E

between the plates; you must add the displacement-current term, else two surfaces on the same loop give different answers.

Reversing the direction you traverse the loop flips the sign of the computed

I_{enc}

True — the loop direction and the surface normal are tied by the right-hand rule, so reversing traversal flips which way "positive" current points.

Spot the error

"A wire carrying

2 I

sits outside my loop, so

\oint B \cdot d l

jumps to

μ_{0} (I + 2 I)

Error: the outside wire is not enclosed. Its field adds to local

B

on the loop but its contributions to the circulation cancel exactly, leaving

\oint B \cdot d l = μ_{0} I

"Inside the solenoid

B = μ_{0} n I

(with

n

= turns per metre), so at the very centre of the axis it's twice that."

Error:

B = μ_{0} n I

is already the uniform interior value — it does not depend on position inside a long solenoid, so there is no doubling anywhere.

"For the thick wire of outer radius

R

I used

I_{enc} = I

s < R

Error: at

s < R

only the fraction of current inside radius

s

is enclosed,

I_{enc} = I s^{2} / R^{2}

; using the full

I

overstates the interior field.

"Ampère's law gives

B

near the end of a finite solenoid just as well as the middle."

Error: the recipe assumes an ideal infinite solenoid where outside

B \approx 0

; near an end the field fringes and leaks out, breaking the loop assumption — use Biot–Savart law there.

\oint B \cdot d l = μ_{0} I_{enc}

, so I can pull out

B (2 π s)

for any loop."

Error: you can only factor

B

out when it is constant and parallel to

d l

along the loop; a random loop has neither, so

\oint B \cdot d l \neq = B \times (length)

"Outside a toroid the field is small but nonzero because of the many turns."

Error: an Amperian loop outside the toroid encloses equal and opposite current (windings go out and come back), so

I_{enc} = 0

and ideally

B = 0

exactly outside.

Why questions

Why does the radial distance

s

of the wire's Amperian circle vanish from the answer?

Because

B \propto 1/ s

from the field and length

\propto s

from the circumference multiply to something

s

-independent — that cancellation is precisely what makes the law geometry-free.

Why must the Amperian loop be closed?

The result rests on the angle

ϕ

sweeping a full

2 π

(enclosing) or returning to start (

0

); an open path has no well-defined "enclosed" and Stokes' theorem — see Stokes' theorem and figure s02 — needs a boundary of a surface.

Why does a current outside the loop contribute zero circulation even though it makes a field everywhere?

Its field has no net "twist" through the loop: as you walk around, the along-path pushes it gives sum to zero because the angle it subtends returns to its start,

\oint d ϕ = 0

Why is Ampère's law called the "magnetic cousin of Gauss's law"?

Gauss's law ties flux to enclosed charge (a source of divergence); Ampère ties circulation to enclosed current (a source of curl) — both trade a hard integral for a symmetry shortcut.

Why does Gauss's law for magnetism have zero on the right while Ampère's has

μ_{0} I_{enc}

There are no magnetic charges (monopoles) to act as flux sources, so magnetic flux is always zero; but currents do act as circulation sources, so Ampère's law is nonzero.

Why do the short sides of the solenoid's rectangular loop contribute nothing?

Inside,

B

is axial while the short sides run perpendicular to the axis, so

B \cdot d l = 0

there; only the long inside side, parallel to

B

, contributes.

Why does the differential form

\nabla \times B = μ_{0} J

contain the same physics as the integral form?

Stokes' theorem (figure s02) converts the surface integral of curl into the loop integral of

B

, so the two forms are just the small-loop and finite-loop faces of one statement — see Maxwell's equations.

Edge cases

What is

I_{enc}

for a loop that encloses two antiparallel wires each carrying

I

Zero — the currents count with opposite signs by the right-hand rule, so

\oint B \cdot d l = 0

even though

B

is large and messy along the loop.

At exactly

s = R

(the surface of a uniform thick wire, where

R

is its outer radius) do the inside and outside formulas agree?

Yes — inside

B = μ_{0} I s / (2 π R^{2})

gives

μ_{0} I / (2 π R)

and outside

B = μ_{0} I / (2 π s)

gives the same at

s = R

; the field is continuous at the boundary.

What is

B

on the axis of a straight wire (

s \to 0

) by the exterior formula?

The formula

B = μ_{0} I / (2 π s)

diverges, but that's the idealized zero-thickness limit; a real wire has

B \to 0

at its centre because

I_{enc} \to 0

faster (

\propto s^{2}

A loop links the same wire twice (wraps around it two full turns). What is

I_{enc}

2 I

— the angle sweeps

4 π

, so

\oint d ϕ = 4 π

and the circulation is

2 μ_{0} I

; enclosure counts with multiplicity.

Steady current is zero everywhere in a region but

B \neq = 0

there (field from distant currents). What does Ampère give for a loop in that region?

\oint B \cdot d l = 0

since no current is enclosed, yet

B

is genuinely nonzero — a clean case of zero circulation with nonzero field.

For the toroid (

N

total turns), why does

B

depend on $r$ inside but the solenoid's does not?

The toroid's loop length

2 π r

grows with

r

while

I_{enc} = N I

is fixed, giving

B \propto 1/ r

; the solenoid's enclosed current

n L I

(with

n

= turns per metre) grows with the loop length

L

, so the ratio stays constant — see Solenoid and toroid fields.

Recall One-line self-test

If you could only remember one trap-buster, which? ::: "Enclosed, not present" — the right side of Ampère's law counts only current piercing the surface, and factoring out $B$ needs symmetry, not just truth.