---
title: Ampère’s Law
module: Magnetic Sources
moduleNumber: 7
lessonNumber: 4
order: 704
summary: >
  When a current arrangement is symmetric enough, the Biot–Savart integral is overkill:
  Ampère's law, $\oint_C\vec B\cdot\d\vec\ell=\mu_0 I_{\rm enc}$, gets the field from a
  single line of reasoning about how much current a loop encloses. We see why the law
  holds for any steady current, then use cylindrical, planar, and toroidal symmetry to
  turn the circulation into simple algebra — the field inside and outside a wire, an
  infinite sheet, a solenoid, and a toroid. We also mark the catch: without symmetry the
  law still holds but no longer hands you the field pointwise.
topics: [Magnetic Sources]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 27 — Sources of the Magnetic Field; §27-4 Ampère’s Law"
---

## Magnetic circulation and orientation

Ampère’s law connects a closed-path integral of the magnetic field with the
electric current piercing a surface bounded by that path. In the magnetostatic
case, where the current distribution is steady and charge does not accumulate,

$$
\oint_C \vec B\cdot \d\vec\ell=\mu_0 I_{\mathrm{enc}}.
$$

The closed curve $C$ is called an Amperian path. Its shape is chosen for the
symmetry of the source, not because the law singles out circles or rectangles.
The vector $\d\vec\ell$ lies tangent to the selected curve and points in
the direction chosen for traversing it. The dot product retains only the
tangential component of $\vec B$:

$$
\oint_C \vec B\cdot \d\vec\ell
=\oint_C B_{\mathrm t}\,\d\ell.
$$

Magnetic circulation therefore differs from an integral of field magnitude. A
section of path on which $\vec B$ is radial contributes zero, even when the
field magnitude there is large. A section traversed opposite to the tangential
field contributes negatively. The integral accumulates a signed projection
around the entire closed path.

The permeability of free space is

$$
\mu_0=4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}}.
$$

With $I_{\mathrm{enc}}$ in amperes, the right side has units of tesla metres,
the same units as $\vec B\cdot \d\vec\ell$. Dimensional agreement
checks the result but does not determine the direction convention. The sign
comes from an oriented surface and its boundary.

$$
% caption: Orientation convention for Ampère's law. Curling the right-hand fingers along the chosen sense of the path $C$ fixes the positive surface normal $\hat n$; current piercing the spanning surface along that normal counts positively in $I_{\mathrm{enc}}$.
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$$

### Orientation and signed current

An orientation must be fixed before assigning the sign of enclosed current.
Choose a positive normal to a surface spanning $C$. Looking from the tip of that
normal, the positive traversal of $C$ is counterclockwise. Equivalently, curl
the fingers of the right hand around the positive path direction; the thumb
points along the positive normal. Current crossing the surface in the normal
direction counts positive, while current crossing the other way counts negative.

For several conductors,

$$
I_{\mathrm{enc}}=\sum_k s_k I_k,
\qquad
s_k=
\begin{cases}
+1,&\text{current along the selected normal},\\
-1,&\text{current opposite the selected normal}.
\end{cases}
$$

Only wires that pierce the selected surface enter this sum. A wire lying entirely
outside the boundary can create a magnetic field along the path, yet it makes no
direct contribution to $I_{\mathrm{enc}}$. Its contribution to the circulation is
already balanced by the rest of the field around the loop. A wire may also pass
through the chosen surface twice with opposite directions; its net contribution
is then zero.

The boundary curve, rather than a particular flat disk, defines the enclosed
current in steady-current problems. A flexible soap-film-like surface can be
drawn across the same boundary without changing the algebraic current crossing
it, provided no current begins or ends inside the region between the two
surfaces. Conservation of charge ensures that consistency. The qualification is
important: a changing charge distribution or changing electric flux requires the
more general Maxwell–Ampère law, treated separately with electromagnetic
induction.

The line integral remains true for every closed path in a steady configuration.
It becomes a calculation tool only when symmetry fixes enough information
about $\vec B$ on the selected path. A poor path may cross a field whose
magnitude and direction vary point by point, leaving an integral as difficult as
the original Biot–Savart calculation. A good path contains segments where the
dot product vanishes and remaining segments where the field is tangent with a
constant magnitude.

## Cylindrical current distributions

Consider an effectively infinite straight wire carrying steady current $I$ along
its axis. Rotating the apparatus about that axis changes no physical feature.
Translating an observation point parallel to the axis changes no physical
feature either. Reflection in a plane containing the axis eliminates axial and
radial magnetic-field components. The only direction left is azimuthal: circles
centred on the wire are tangent to the field.

Every point at the same perpendicular distance $r$ from the axis is related by a
rotation. The magnitude $B(r)$ is therefore constant on a circle of radius $r$.
Use that circle as the Amperian path. Both facts needed to simplify the
circulation now hold:

$$
\vec B\parallel \d\vec\ell,
\qquad
B(r)=\text{constant on }C_r.
$$

$$
% caption: Circular Amperian path around a long straight wire. Cylindrical symmetry makes $\vec B$ tangent to the circle and equal in magnitude around it, so the circulation is $B\,(2\pi r)$ and the current through the spanning disk is $I$.
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$$

The current through the disk bounded by this circle is $I$. Ampère’s law reduces
to one product:

$$
\oint_{C_r}\vec B\cdot \d\vec\ell
=B(r)\oint_{C_r}\d\ell
=B(r)(2\pi r)
=\mu_0 I.
$$

Thus the magnitude outside an ideal thin wire is

$$
B(r)=\frac{\mu_0 I}{2\pi r}.
$$

The field circles the wire. Current out of the page gives a counterclockwise
field; current into the page gives a clockwise field. Reversing either the
current or the chosen path orientation reverses the signed circulation. The
positive magnitude above is independent of the direction convention once $I$
denotes the magnitude of the current.

The phrase “long wire” carries an approximation. At distances small compared
with the distance to an end, the local source looks translation-invariant and the
formula is accurate. Near an end, the field no longer has equal magnitude around
a circular path, so the formula cannot be obtained from the same symmetry
argument. A finite straight segment requires the Biot–Savart result derived from
its endpoint angles.

### Field inside a uniformly conducting cylinder

Ampère’s law also distinguishes the interior of a solid conductor from its
exterior. Let a cylindrical wire of radius $a$ carry total current $I$ uniformly
through its cross-sectional area. Uniform current density has magnitude

$$
J=\frac{I}{\pi a^2}.
$$

An Amperian circle of radius $r<a$ encloses the current inside its own disk,

$$
I_{\mathrm{enc}}(r)
=J\pi r^2
=I\frac{r^2}{a^2}.
$$

Substitution into the circulation equation gives

$$
B(r)(2\pi r)=\mu_0 I\frac{r^2}{a^2},
\qquad
B(r)=\frac{\mu_0 I}{2\pi a^2}r
\quad (r<a).
$$

The interior field grows linearly from zero at the centre to the surface value.
Outside the metal, every Amperian circle encloses the whole current, recovering
$B(r)=\mu_0 I/(2\pi r)$. The two branches meet at $r=a$:

$$
B(a)=\frac{\mu_0 I}{2\pi a}.
$$

No discontinuity occurs because the current is spread through the volume. A
surface current, idealized as all current confined to an infinitesimal cylindrical
shell, produces a different interior profile and a discontinuity in the
tangential magnetic field at that shell.

$$
% caption: Magnetic-field magnitude of a wire with uniform volume current. Inside, $B$ rises linearly with the enclosed-current fraction $r^2/a^2$; outside, it falls as $1/r$. The two branches meet continuously at the conductor surface $r=a$.
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$$

Current density therefore matters in interior-field calculations. The total
current alone determines the exterior result, but many distributions with the
same total current give different values inside the conductor. A hollow
cylindrical shell is an immediate example: an Amperian circle in the empty
central region encloses zero current, so its magnetostatic magnetic field is
zero under ideal cylindrical symmetry.

### Scope of the enclosed-current result

Ampère’s law determines a circulation, not a point value by itself. A closed
curve with $I_{\mathrm{enc}}=0$ has

$$
\oint_C\vec B\cdot \d\vec\ell=0,
$$

yet the magnetic field can remain nonzero at every point on that curve. Two
parts of the path may contribute equal magnitudes with opposite signs, or the
field may be perpendicular to some parts of the path. A closed loop placed
beside a current-carrying wire rather than around it illustrates the point:
the wire contributes to $\vec B$ along the loop, while the signed
circulation around that particular boundary is zero.

The long-wire circle succeeds because it follows the field everywhere and
because symmetry makes the field magnitude constant. In a rectangular path
around the same wire, the field is generally neither tangent nor constant along
each side. Ampère’s law remains exact, but the integral offers no shorter route
to $B(r)$. Choosing a circle after identifying rotational symmetry is a
derivation step, not a remembered shape.

Superposition also enters through the field, whereas the enclosed current is an
algebraic sum. A path surrounding one $+I$ conductor and one $-I$ conductor has
zero enclosed current. At a generic point on that path, the two magnetic fields
need not cancel. Their circulation contributions cancel after integration.
Local cancellation requires additional geometric symmetry. This distinction
keeps a zero right side from being misread as a zero magnetic field.

The law applies to any closed boundary, including a curve that winds around a
wire more than once. A path traversed twice in the same sense has twice the
circulation and an oriented spanning surface with corresponding winding number.
Reversing the traversal changes the sign of the integral and of the associated
signed current. Such bookkeeping is rarely needed for elementary circles, but
it makes the orientation rule stable in compound paths.

### Calculation checks

A long-wire calculation has several quick checks. Magnetic field scales linearly
with current, so doubling $I$ doubles $B$. It scales inversely with perpendicular
distance outside the conductor, so a point twice as far from the axis has half
the field magnitude. The dimensions of $\mu_0 I/(2\pi r)$ reduce to tesla. The
direction must be tangent to a circle about the current, never radial away from
the wire.

> **Worked example (Long straight wire by Ampère's law).** A long wire carries
> $I=3.00\ \mathrm A$. A circular Amperian loop of radius $r=4.00\ \mathrm{cm}$ shares
> the field's symmetry, so $\oint\vec B\cdot\d\vec\ell=B(2\pi r)=\mu_0 I$ and
>
> $$
> B=\frac{\mu_0 I}{2\pi r}
> =\frac{(4\pi\times10^{-7})(3.00)}{2\pi(4.00\times10^{-2})}
> =1.50\times10^{-5}\ \mathrm T=15.0\ \mu\mathrm T.
> $$
>
> That is comparable to the Earth's field. With the current upward in a vertical wire,
> the right-hand rule sends $\vec B$ north at a point east of the wire — fixing the
> position and current direction first heads off the usual sign slip.

## Solenoids and toroids

A solenoid is a closely wound helical conductor. Let $N$ turns occupy length
$L$, and define the turn density

$$
n=\frac{N}{L}.
$$

Each turn carries the same steady current $I$. The phrase long solenoid denotes
the ideal limit $L\gg R$, where $R$ is the winding radius, together with
closely spaced turns. Far from either end, the magnetic field is parallel to the
axis and approximately uniform. The external field is much smaller than the
interior field in that central region. Those properties follow from
superposition of many coaxial current loops: axial components reinforce within
the winding, while exterior contributions largely cancel between neighboring
turns.

Ampère’s law turns that symmetry into the interior magnitude. Choose a thin
rectangular path whose long side of length $\ell$ runs down the axis inside the
winding. Its return side lies outside, parallel to the axis. The two short sides
are perpendicular to the axial field. Their dot products with
$\d\vec\ell$ vanish. In the long-solenoid approximation, the return side
has negligible magnetic field. The circulation reduces to the interior side:

$$
\oint_C\vec B\cdot \d\vec\ell\approx B\ell.
$$

The spanning surface cuts through $n\ell$ turns. Every turn pierces that
surface once, so its signed enclosed current is

$$
I_{\mathrm{enc}}=(n\ell)I.
$$

Ampère’s law gives

$$
B\ell=\mu_0(n\ell)I,
\qquad
B=\mu_0 nI.
$$

The path length cancels because adding interior length encloses proportionally
more turns. A longer interior segment produces a larger circulation and a
larger enclosed current by the same factor. This cancellation distinguishes the
long-solenoid result from the straight-wire result, where enlarging a circular
path changes the circumference without enclosing additional current.

$$
% caption: Rectangular Amperian path for a long, tightly wound solenoid. The interior side of length $\ell$ runs along the axis and crosses $n\ell$ turns; the two short sides are perpendicular to $\vec B$, and the remote exterior side carries negligible field.
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The result contains no explicit radius because the ideal argument needs only
the number of turns crossed per unit axial length. Radius controls how rapidly
the finite winding approaches the long-solenoid approximation and how broad a
region around the axis has a nearly uniform field. It also enters the exact
on-axis field of a finite solenoid.

The Ampèrian rectangle does not prove uniformity by itself. Uniform axial field
and a negligible exterior return segment are symmetry statements supplied by
the long, densely wound source. Once those statements hold in a central region,
Ampère’s law fixes the field magnitude there. Applying the same rectangle near
an end would silently retain a non-negligible exterior contribution and a
position-dependent interior contribution. The shortened circulation
$B\ell$ would then lack justification.

Turn density carries units of inverse length, so $\mu_0nI$ has units

$$
(\mathrm{T\,m\,A^{-1}})(\mathrm{m^{-1}})(\mathrm A)=\mathrm T.
$$

That unit check also identifies a common transcription error: $NI$ is an
ampere-turn count, while $nI$ is the current per unit length needed for the
field of an extended solenoid. Doubling the number of turns while holding $L$
and $I$ fixed doubles $n$ and the central field. Stretching the same winding to
twice its length halves $n$ and halves the ideal central field. These statements
assume that the current remains fixed and that the observation point stays well
inside a solenoid whose aspect ratio remains large enough for the approximation.

Direction follows the current around the turns. Viewed from one end, a
counterclockwise conventional current gives an axial magnetic field directed
toward that viewer; a clockwise current gives the opposite axial direction.
Describing the end from which the winding is viewed prevents an ambiguous
right-hand-rule statement. A signed field component $B_z$ requires an explicitly
chosen positive axial direction, whereas the formula $B=\mu_0nI$ gives the
magnitude.

> **Worked example (Field inside a long solenoid).** A solenoid is wound at
> $10.0$ turns per centimetre, $n=1.00\times10^3\ \mathrm{m^{-1}}$, and carries
> $I=2.00\ \mathrm A$. Deep inside,
>
> $$
> B=\mu_0 n I=(4\pi\times10^{-7})(1.00\times10^3)(2.00)
> =2.51\times10^{-3}\ \mathrm T=2.51\ \mathrm{mT}.
> $$
>
> The field is uniform and axial there, independent of radius. If this solenoid is
> $L=0.300\ \mathrm m$ long with radius $R=1.50\ \mathrm{cm}$, the exact centre value
> carries the end-effect factor $L/\sqrt{L^2+4R^2}=0.995$, so the ideal formula
> overstates the centre field by only about half a percent — the long-solenoid
> approximation is excellent for $L/R=20$.

### Finite-solenoid axial scan

Treating a finite solenoid as a stack of circular loops gives an on-axis result
without the long-solenoid approximation. Place the left and right ends at
$z_1$ and $z_2$, respectively, let the radius be $R$, and take $n=N/L$.
Integrating the axial field of each infinitesimal group of turns gives

$$
B_z(z)=\frac{\mu_0 nI}{2}
\left[
\frac{z-z_1}{\sqrt{(z-z_1)^2+R^2}}
-\frac{z-z_2}{\sqrt{(z-z_2)^2+R^2}}
\right].
$$

The finite-solenoid expression includes the end effects omitted by the rectangular-path
derivation. At the centre of a solenoid of length $L$,

$$
B_z(0)=\mu_0 nI\frac{L}{\sqrt{L^2+4R^2}}.
$$

The factor multiplying $\mu_0nI$ approaches one only when $L/R$ is large.
At either end, the corresponding axial value is

$$
B_z\!\left(\pm\frac L2\right)
=\frac{\mu_0nI}{2}\frac{L}{\sqrt{L^2+R^2}},
$$

which approaches one half of the central long-solenoid value as $L/R$ becomes
large. The field falls smoothly rather than abruptly at a physical winding end.

An axial measurement scan tests the approximation directly. Fix the current
with a regulated source, align an axial magnetic sensor with the solenoid axis,
and record the signed reading at known positions $z$. Reversing the current and
taking half the difference between the two readings removes a constant ambient
offset:

$$
B_{\mathrm{coil}}(z)
=\frac{B_{+I}(z)-B_{-I}(z)}{2}.
$$

The probe must remain on the axis. A transverse offset samples a different
vector field and can mimic an apparent end effect. Position steps smaller than
$R$ resolve the rounded transition near each end; sparse steps can make a
finite solenoid appear to have a sharp-edged uniform region. Lead wires and
finite turn spacing cause small deviations from the ideal loop-stack model, so
the scan is compared with the predicted curve rather than with a flat plateau
alone.

$$
% caption: On-axis field of a finite solenoid. The reading approaches the ideal $\mu_0 n I$ only over the central region and rounds off toward half that value at each coil end; sampling more closely than the winding radius resolves the transition.
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$$

The sensor reports a component along its sensitive axis. Rotating the probe by
$180^\circ$ reverses its sign convention; it does not reverse the physical
magnetic field. A position log, current setting, probe orientation, and
background-subtraction method belong with every scan. Those records separate
geometric field variation from instrument offset or an accidental reversal of
the sensor axis.

### Toroid: closed magnetic path

A tightly wound toroid bends a solenoid into a closed ring. Let $a$ and $b$ be
the inner and outer radii of the winding, and let $N$ turns each carry current
$I$. A circular Amperian path of radius $r$ about the central axis lies entirely
within the winding when $a<r<b$. Rotational symmetry makes the field tangent
to that path and constant in magnitude along it. The spanning disk is pierced
once by every turn, giving $I_{\mathrm{enc}}=NI$. Ampère’s law yields

$$
B(r)(2\pi r)=\mu_0NI,
\qquad
B(r)=\frac{\mu_0NI}{2\pi r}
\quad (a<r<b).
$$

$$
% caption: Toroidal Amperian path. A circle of radius $r$ inside the winding crosses all $N$ turns once through its spanning disk, so $\vec B$ is tangent and follows $\mu_0 N I/(2\pi r)$ across the region between inner radius $a$ and outer radius $b$.
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The toroid and long solenoid share the same local circulation logic: a path
parallel to $\vec B$ encloses a count of turns times current. Their symmetry
consequences differ. The long solenoid has translational symmetry along its
axis, so the ideal interior magnitude is independent of axial position. A toroid
has rotational symmetry, so the magnitude is constant on one circle but varies
between circles as $1/r$. A thin toroid, where $b-a$ is much smaller than its
mean radius, has only a small variation across its winding.

For $r<a$, the circular path encloses no winding current. For $r>b$, each turn
crosses a spanning surface twice with opposite signs, giving zero net enclosed
current. Ideal toroidal symmetry then gives zero magnetic field in the central
opening and outside the winding. Real coils have discrete turns, lead wires,
and finite cross sections; small exterior fields can remain. The enclosed-current
argument establishes the ideal result and identifies the geometry required for
its use.

> **Worked example (Field inside a toroid).** A toroid has $N=500$ turns carrying
> $I=1.50\ \mathrm A$. On the circle of mean radius $r=0.100\ \mathrm m$ inside the
> winding,
>
> $$
> B=\frac{\mu_0 N I}{2\pi r}
> =\frac{(4\pi\times10^{-7})(500)(1.50)}{2\pi(0.100)}
> =1.50\times10^{-3}\ \mathrm T=1.50\ \mathrm{mT}.
> $$
>
> Unlike the solenoid, this field is not uniform across the winding: it scales as
> $1/r$, so the inner edge sees a stronger field than the outer edge. The central hole
> and the exterior both carry zero net enclosed current and, ideally, zero field.

## Coaxial conductors and current profiles

Coaxial conductors make the distinction between a chosen loop and the current
inside that loop especially explicit. Consider a solid inner conductor of radius
$a$ carrying algebraic current $I_1$ along the positive axis. A concentric
annular conductor fills $b<r<c$ and carries algebraic current $I_2$. Positive
current is defined by a selected surface normal. With that normal directed out
of a cross-sectional page, a counterclockwise Amperian circle has the positive
path sense.

The circular path is available because rotation about the common axis leaves the
configuration unchanged. At a fixed radius, $\vec B$ is tangential and has
one magnitude. Ampère’s law therefore becomes

$$
B_\phi(r)(2\pi r)=\mu_0 I_{\mathrm{enc}}(r),
\qquad
B_\phi(r)=\frac{\mu_0 I_{\mathrm{enc}}(r)}{2\pi r}.
$$

The subscript on $B_\phi$ denotes the signed tangential component. A negative
value indicates that the direction is opposite to the selected counterclockwise
path sense. It is better to retain that sign through the calculation than to
attach an informal right-hand rule after taking a magnitude.

$$
% caption: Coaxial conductors with a circular Amperian path. One positive normal fixes both the counterclockwise path sense and the sign of each current through the spanning disk; the path shown lies inside the annular return conductor between radii $b$ and $c$.
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$$

Suppose the current density is uniform in each conductor. The enclosed current
depends on which radial zone contains the path:

$$
I_{\mathrm{enc}}(r)=
\begin{cases}
I_1\,r^2/a^2, & 0\le r<a,\\[4pt]
I_1, & a\le r<b,\\[4pt]
I_1+I_2\,(r^2-b^2)/(c^2-b^2), & b\le r<c,\\[4pt]
I_1+I_2, & r\ge c.
\end{cases}
$$

The first line counts the area fraction of a uniform solid conductor. The third
line adds the fraction of the annular return current between $b$ and $r$.
When $I_2=-I_1$, a circular path outside both conductors encloses zero net
current and has zero circulation. The magnetic field in the annulus is generally
nonzero because only part of the return current lies inside a path with $r<c$.
Its direction reverses if the enclosed-current expression crosses zero before
the outer boundary.

The choice of spanning surface does not turn a return current into an additional
positive contribution. Each conductor crossing must be signed using the same
normal that fixes the loop traversal. Reversing the path direction reverses
both sides of Ampère’s law; it cannot change a physically negative return
current into a positive one.

### Piecewise current-density profiles

Ampère’s law accepts any axisymmetric current density, provided the current is
steady and directed parallel to the common axis. Write the axial density as
$J_z(r)$. A disk of radius $r$ encloses

$$
I_{\mathrm{enc}}(r)=2\pi\int_0^r J_z(s)\,s\,\d s.
$$

The factor $2\pi s\,\d s$ is the area of a thin annular strip. It provides the
radial weighting absent from a one-dimensional sketch of $J_z$. A density
concentrated near the outer surface contributes little near the axis and then
changes the field rapidly as the Amperian radius reaches that layer.

For two uniform regions with a boundary at $a_1$ and an outer radius $a_2$,

$$
J_z(r)=
\begin{cases}
J_1, & 0\le r<a_1,\\
J_2, & a_1\le r<a_2,\\
0, & r\ge a_2,
\end{cases}
$$

and the enclosed current is

$$
I_{\mathrm{enc}}(r)=
\begin{cases}
\pi J_1r^2, & 0\le r<a_1,\\[4pt]
\pi J_1a_1^2+\pi J_2(r^2-a_1^2), & a_1\le r<a_2,\\[4pt]
\pi J_1a_1^2+\pi J_2(a_2^2-a_1^2), & r\ge a_2.
\end{cases}
$$

The outer formula is the total current. A negative $J_2$ can represent a
distributed return current, while a positive $J_2$ increases the total current.
The magnetic field follows by dividing each branch by $2\pi r$ and multiplying
by $\mu_0$. Boundary values must agree when the density is finite; a jump in
current density changes the slope of $B_\phi(r)$, not the field value itself.
Check the piecewise result against a smooth profile. Let

$$
J_z(r)=J_0\left(1-\frac{r^2}{a^2}\right)
\quad (0\le r\le a).
$$

Its total current is $I=\pi J_0a^2/2$. Integrating to an intermediate radius
gives

$$
I_{\mathrm{enc}}(r)
=I\left(\frac{2r^2}{a^2}-\frac{r^4}{a^4}\right),
\qquad
B_\phi(r)
=\frac{\mu_0 I}{2\pi r}
\left(\frac{2r^2}{a^2}-\frac{r^4}{a^4}\right).
$$

At $r=a$, the interior branch equals $\mu_0I/(2\pi a)$ and joins the exterior
inverse-radius result. Near the axis, the leading term makes $B_\phi$ increase
linearly with $r$. The shape of $B_\phi(r)$ therefore carries information about
both the radial current distribution and the total current.

### Inferring current density from a radial field profile

In a cylindrically symmetric conductor, a measured radial magnetic-field
profile can be converted into a current-density profile. Start from

$$
rB_\phi(r)=\frac{\mu_0}{2\pi}I_{\mathrm{enc}}(r).
$$

Differentiation and the annular-area relation
$\d I_{\mathrm{enc}}/\d r=2\pi rJ_z(r)$ give

$$
J_z(r)=\frac{1}{\mu_0r}
\frac{d}{\d r}\!\left[rB_\phi(r)\right].
$$

The formula is local, whereas the original Ampèrian relation is cumulative.
Values of $B_\phi$ outside a conductor determine its total current through the
inverse-radius branch. Values inside the conductor indicate how that total current
is apportioned across radius. A linear rise of $B_\phi$ corresponds to nearly
uniform $J_z$. A branch that grows slowly near the axis and steepens near an
outer layer indicates current concentrated away from the centre.

Experimental differentiation requires more care than using a single circular
path. Record tangential field values at a sequence of radii, maintaining the
same axial position and angular convention. Current reversal removes a stable
background before the profile is fitted. A smooth constrained fit to
$rB_\phi(r)$ is differentiated afterward; differencing adjacent noisy readings
directly amplifies sensor noise. The fitted profile should satisfy
$I_{\mathrm{enc}}(0)=0$ and approach the independently measured total current
outside the conductor.

Boundary behavior gives a second diagnostic. A finite volume current density
produces a continuous $B_\phi(r)$, even when $J_z$ has a finite step between
layers. The derivative of $rB_\phi$ changes at the layer boundary. A jump in
$B_\phi$ points instead to an idealized surface current or to a measurement
artifact such as a change in sensor standoff. Recording the radial position
relative to the conductor surface is therefore as important as recording the
field value.

The inverse relation also clarifies the role of the central point. The
expression contains $1/r$, but a regular axisymmetric field satisfies
$B_\phi(0)=0$ and has a linear small-radius limit. Estimate the central slope
from several nearby radii rather than divide a single reading by an extremely
small $r$. A sensor with a finite active area averages over a region whose size
must be small relative to the scale on which the current density changes.

## Symmetry and loop selection

An ideal circular Amperian path requires equal tangential readings at every
azimuth on that path. A measurement can test the premise before a symmetry
shortcut is used. Set a radius $r$, place a sign-sensitive magnetic sensor at
$M$ equally spaced angles, align its sensitive direction tangent to the
circle, and record the current-reversed values. The reversal difference removes
a stationary background:

$$
B_{\phi,i}=
\frac{B_{\phi,i}(+I)-B_{\phi,i}(-I)}{2}.
$$

In a coaxial, axisymmetric source, all $B_{\phi,i}$ at the selected radius
agree within measurement uncertainty. Their mean and relative spread can be
recorded as

$$
\overline B_\phi=\frac1M\sum_{i=1}^{M}B_{\phi,i},
\qquad
\epsilon_{\mathrm{sym}}
=\frac{
\sqrt{\frac1M\sum_{i=1}^{M}(B_{\phi,i}-\overline B_\phi)^2}
}{|\overline B_\phi|}.
$$
Large angular residuals identify a symmetry failure rather than a failure of
Ampère’s law. An off-centre inner conductor, an eccentric outer conductor,
uneven current density, asymmetric lead routing, or a sensor that is not
tangent to the circle each changes the readings. The circulation equation
still relates the complete signed line integral to the enclosed current, but
the field magnitude can no longer be factored outside the integral.

Repeating the scan at several radii separates common faults. A residual that
grows near a conductor boundary can indicate an incorrect centre estimate or
an annular current profile. A residual that follows the routing of a lead
usually changes when the lead is repositioned. A residual unchanged by current
reversal belongs to the background and is removed by the half-difference
procedure. Report the chosen centre, radii, angular locations, sensor
orientation, current, and reversal protocol with the data. Those details make
the symmetry claim reproducible and show whether a circular Amperian path can
support a one-variable magnetic-field calculation.

### Selecting an Amperian loop

An Amperian loop is selected after the source symmetry has been established.
The procedure begins with the current distribution, its boundary conditions,
and the region where the field is sought. Translation along a long wire,
rotation around a common axis, reflection through a symmetry plane, and
rotation around a toroidal axis each constrain possible directions of
$\vec B$. A closed path becomes tractable only after those constraints have
reduced the line integral to known terms.

The first test concerns direction. Along a candidate path, identify whether
$\vec B$ is parallel to $\d\vec\ell$, perpendicular to it, or neither.
Parallel portions contribute $B\,\d\ell$; perpendicular portions contribute
zero. The second test concerns magnitude. A symmetry operation must map every
point of a nonzero-contribution portion onto every other point of that portion.
Only then may its magnitude be taken outside the relevant integral. The third
test counts signed current through a surface bounded by the loop. The surface
normal and path direction are fixed together by the right-hand convention.

Circular loops around an infinite straight conductor pass all three tests.
The field is tangent to each circle, its magnitude depends only on the circle
radius, and the spanning disk intersects the axial current once. Rectangular
loops around the same wire remain valid paths for Ampère’s law, but their four
sides do not share one field magnitude or one angle with the field. Their
circulation cannot be replaced by a single product without additional
calculation.

$$
% caption: Loop selection for a long straight conductor. The circular path follows the azimuthal field at one constant magnitude and collapses the circulation to $B\,(2\pi r)$; the rectangle encloses the same current but its sides meet the field at changing angles and distances.
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$$

A compact loop-selection record avoids several recurrent errors.

1. State the source idealization and the observation region. “Infinite
   straight conductor,” “long tightly wound solenoid far from an end,” and
   “toroid between radii $a$ and $b$” specify different symmetry conditions.

2. Write the surviving field direction before choosing a path. A long wire has
   an azimuthal direction, a long solenoid has an axial central direction, and
   a toroid has an azimuthal direction around its central axis.

3. Mark every path segment as parallel, perpendicular, or variable relative to
   the field. Variable segments remain inside the integral. A closed curve can
   enclose the correct current and still be algebraically unhelpful.

4. Choose an oriented spanning surface. Record every crossing current with its
   sign. A return conductor outside a selected path contributes no current
   through that surface; a return conductor crossed by the surface contributes
   with its algebraic direction.

5. Check the result against dimensions, limiting radii, and a direct physical
   symmetry test. A circular path around a long wire must give $B\propto1/r$.
   A central long-solenoid path must give a result independent of its chosen
   axial length.

A loop may lie in empty space between coaxial conductors, or a rectangle may
cross an ideal solenoid interior and return through the exterior. Material
boundaries often help define current-density regions, but the magnetic integral
follows the geometric path. A path crossing a wire introduces no singularity in
the integral if the field remains finite there; it changes the enclosed-current
count as the path moves through the current distribution.

### Ampère and Biot–Savart calculations

Ampère’s law and the Biot–Savart law describe the same steady-current magnetic
field from different mathematical viewpoints. Ampère’s law gives the
circulation around an arbitrary closed boundary:

$$
\oint_C\vec B\cdot \d\vec\ell=\mu_0I_{\mathrm{enc}}.
$$

Biot–Savart builds the local field by adding contributions from every current
element:

$$
\vec B(\vec R)=\frac{\mu_0}{4\pi}
\int
\frac{I\,\d\vec\ell\mathbin{\times}(\vec R-\vec r')}
{|\vec R-\vec r'|^3}.
$$

The source coordinate $\vec r'$ runs along the actual conductor, while
$\vec R$ marks the observation point. Source geometry remains visible in
the Biot–Savart integrand through both direction and separation. That explicit
geometry makes it suitable for finite segments, loops, arcs, and arrangements
whose field magnitude varies along every convenient Amperian boundary.

Let $R$ be the perpendicular distance to the observation point for a straight
segment, and let $u_1$ and $u_2$ be signed endpoint angles measured
from the perpendicular line. Biot–Savart integration gives

$$
B=\frac{\mu_0I}{4\pi R}
\left(\sin u_2-\sin u_1\right).
$$

As the segment extends indefinitely in both directions,
$u_1\rightarrow-\pi/2$ and $u_2\rightarrow+\pi/2$, recovering
$B=\mu_0I/(2\pi R)$. At finite length, the endpoint angles remain in the
answer. A circular boundary around a finite segment may enclose a current in
the sense of a complete circuit, but its tangential field is neither constant
nor generally aligned with the boundary. Ampère’s law retains its circulation
statement; the line integral cannot be reduced to $B(2\pi R)$.

$$
% caption: Finite straight-segment geometry for a Biot–Savart calculation. The two endpoint angles $\theta_1,\theta_2$ measured from the perpendicular and the perpendicular distance $R$ fix the field at $P$; shortening either end changes the result, so no circular Amperian path around the segment gives a constant tangential field.
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$$

High symmetry gives the two laws the same short expression. A long wire, long
solenoid, and ideal toroid can be solved rapidly from circulation because
symmetry makes a selected path simple. The Biot–Savart integral also gives
those fields, but it retains unnecessary source-by-source detail. A single
circular loop on its axis has enough source symmetry for a short
Biot–Savart integral, while an Amperian circle around that loop leaves an
unknown varying tangential field. The appropriate method follows the
calculation geometry, not the visual resemblance of a source to a familiar
formula.

## Field mapping and design checks

Two compact designs show how current return geometry controls both a desired
field and unintended exterior circulation. A coaxial pair with an inner
conductor carrying $+I$ and a thin concentric return shell at radius $b$
carrying $-I$ has, under ideal cylindrical symmetry,

$$
B(r)=\frac{\mu_0I}{2\pi r}
\quad (a<r<b),
\qquad
B(r)=0
\quad (r>b).
$$

The annular region between conductor radii $a$ and $b$ carries the circular
magnetic field. A path outside the return conductor encloses zero algebraic
current. The zero exterior result depends on equal and opposite currents and
coaxial placement; current imbalance or geometric offset leaves a measurable
external field.

Take $I=3.00\ \mathrm A$, $a=1.00\ \mathrm{mm}$, $b=4.00\ \mathrm{mm}$, and
measure at $r_0=2.00\ \mathrm{mm}$. The predicted annular field is

$$
B(r_0)=
\frac{(4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}})(3.00\ \mathrm A)}
{2\pi(2.00\times10^{-3}\ \mathrm m)}
=3.00\times10^{-4}\ \mathrm T
=0.300\ \mathrm{mT}.
$$

For independent small uncertainties in current and sensor radius,

$$
\frac{u_B}{B}
=\sqrt{\left(\frac{u_I}{I}\right)^2+
\left(\frac{u_r}{r}\right)^2}.
$$

With $u_I=0.010\ \mathrm A$ and $u_r=0.050\ \mathrm{mm}$, the relative
uncertainty is $2.52\%$, giving $B=0.300\pm0.008\ \mathrm{mT}$. The radial
position dominates the stated uncertainty because the field varies as $1/r$.

An air-core toroid confines the winding into a closed path rather than using a
return shell. Let $N=240$, $I=1.50\ \mathrm A$, and choose a measurement radius
$r_0=4.00\ \mathrm{cm}$ inside a toroid with $a=3.50\ \mathrm{cm}$ and
$b=4.50\ \mathrm{cm}$. The ideal value at $r_0$ is

$$
B(r_0)=\frac{\mu_0NI}{2\pi r_0}
=1.80\ \mathrm{mT}.
$$

The same independent-uncertainty relation gains a turn-count term
$(u_N/N)^2$ under the square root. A counted winding has $u_N=0$; with
$u_I/I=0.50\%$ and $u_{r_0}/r_0=1.00\%$, the random uncertainty is
$0.020\ \mathrm{mT}$. The radial design interval produces a separate geometric
range: $B(a)=2.06\ \mathrm{mT}$ and $B(b)=1.60\ \mathrm{mT}$. Reporting a
single toroid field therefore requires the specified radius or an explicitly
defined spatial average.
The two designs require different acceptance checks. A coaxial test measures
the external field on several circles outside $b$ while the return current is
varied; residual external field exposes imbalance, eccentricity, or lead
effects. A toroid test measures at several radii between $a$ and $b$ and checks
the $1/r$ trend. In each case, the reported uncertainty distinguishes
instrument noise from a change in the ideal geometry assumed by Ampère’s law.

### Field mapping and boundary checks

A field map tests the symmetry assumptions used to reduce Ampère’s law. The
map begins by fixing a coordinate system and a sensor orientation. For an
axisymmetric straight-current source, the ideal result has the form

$$
\vec B(r,\varphi,z)=B_\phi(r)\,\hat\phi.
$$

At one selected radius, readings taken around the circle should agree after
current-reversal background subtraction. At several radii, the measured
profile should follow the appropriate branch of the enclosed-current result.
Sensor readings in radial and axial orientations provide direct checks that
the tangential component dominates. The map contains both magnitude and
direction information; a scalar trace alone can conceal a reversed sensor
axis or an off-centre source.

Choose sampling radii on both sides of every current-density boundary. A
solid conductor with finite volume current has a continuous $B_\phi(r)$ across
its surface. A thin current sheet produces a finite jump. The distinction is
obtained by placing a narrow rectangular Amperian loop across the sheet. Its
long sides are tangent to the boundary and have length $\ell$:

$$
\left(B_{\phi,\mathrm{out}}-B_{\phi,\mathrm{in}}\right)\ell
=\mu_0K_z\ell,
\qquad
B_{\phi,\mathrm{out}}-B_{\phi,\mathrm{in}}=\mu_0K_z.
$$

Here $K_z$ is the signed axial surface-current density in amperes per metre.
The sign follows the same normal and circulation orientation used throughout
the lesson. A return shell carrying negative axial current has negative
$K_z$, so the tangential field drops on crossing outward through the shell.
The physical size of the sensor matters near a boundary. A sensor averages the
field over its active region. When that region straddles a thin return shell,
the reported value lies between the two limiting values even when the ideal
field jump is sharp. Record the sensor centre position, active width, and
distance from the conductor axis. A model–measurement mismatch confined to
one active-width interval often reflects spatial averaging rather than a
failure of the source model.

Repeated maps also distinguish a boundary error from a coordinate error. A
mislocated axis changes the apparent angular variation at every radius. A
wrong boundary radius shifts the location of a slope change or jump while
leaving the central angular map relatively uniform. Mapping with positive and
negative current identifies stationary background contributions, since their
half-difference vanishes. The same reversal also checks sign convention:
every source-generated tangential reading must reverse.
### Coaxial field map

> **Worked example (Coaxial cable field map).** A coaxial line has an inner conductor
> of radius $a=1.00\ \mathrm{mm}$ and a thin return shell at $b=5.00\ \mathrm{mm}$,
> carrying $I=2.00\ \mathrm A$ out along the core and back on the shell. In the annulus
> $a<r<b$ the enclosed current is $I$, so $B_\phi=\mu_0 I/2\pi r$:
>
> $$
> B(2.00\ \mathrm{mm})=0.200\ \mathrm{mT},\qquad
> B(4.00\ \mathrm{mm})=0.100\ \mathrm{mT},\qquad
> B(b^-)=\frac{\mu_0 I}{2\pi b}=0.0800\ \mathrm{mT}.
> $$
>
> Outside the shell $(r>b)$ the enclosed current is $I-I=0$, so $B=0$: the return
> shell confines the field to the annulus. The field jumps from $0.0800\ \mathrm{mT}$
> to zero across the shell, set by its surface current
>
> $$
> K_z=-\frac{I}{2\pi b}=-63.7\ \mathrm{A\,m^{-1}},\qquad \mu_0 K_z=-0.0800\ \mathrm{mT},
> $$
>
> the standard $\mu_0 K$ discontinuity in the tangential field across a current sheet.

The radial profile, the exterior null, and the boundary jump support one common
current-return model. A disagreement concentrated at the shell would call for checks of
sensor standoff, shell radius, and current distribution; a disagreement at all radii
would point to current calibration, sensor scale, or coordinate convention.
The reconciliation is complete only when the model, units, geometry, and
measurement protocol agree. A numerical fit alone cannot validate a symmetry
assumption. Angular scans at fixed radius, radial scans across boundaries,
current reversal, and independent current calibration provide separate
evidence for each part of the Ampèrian calculation.

### Validation record

Each mapped point requires a position, sensor orientation, current value,
current polarity, source-field reading, and uncertainty. The position is
reported relative to the fitted symmetry axis rather than to an arbitrary
fixture edge. A radial map without a recorded axis cannot distinguish a
genuine radial profile from an off-centred scan. The sensor orientation is
reported as tangent, radial, or axial; a sign-sensitive sensor also requires a
declared positive direction.

The same record identifies which model feature each measurement tests. Points
within a solid conductor test the enclosed-current fraction. Points between
coaxial conductors test the inverse-radius branch. Paired points close to a
thin shell test the surface-current jump. Points outside a balanced return
path test cancellation of algebraic enclosed current. Angular points on one
circle test the constancy used to remove $B_\phi$ from the circulation
integral. Assigning those purposes before collecting data prevents a dense
map from becoming a collection of unrelated readings.

Residuals require their own uncertainty budget. Current calibration, radial
location, sensor noise, sensor-axis alignment, and finite sensor area enter
different parts of the comparison. Combining every discrepancy into one
generic percentage obscures whether the source model or the instrument
controls the limit. A shell-jump test, for example, is sensitive to sensor
standoff and boundary radius; an exterior-cancellation test is sensitive to
return-current balance and lead routing. The associated uncertainty should
follow the mechanism being tested.

An acceptance statement is consequently specific. The coaxial example supports
the stated model over the sampled radii, orientations, and current range. It
does not establish the model at unmeasured distances, under a changed return
path, or with a different sensor geometry. Such limits preserve the distinction
between the Ampère-law derivation and the finite set of measurements used to
test its assumptions.