---
title: Circular Current Loops
module: Magnetic Sources
moduleNumber: 7
lessonNumber: 3
order: 703
summary: >
  A ring of current is the simplest source with a well-defined magnetic axis, and it is
  the building block of every coil and electromagnet. Symmetry kills the transverse
  Biot–Savart contributions along that axis and leaves a single clean integral; we
  evaluate it to get $B_z=\mu_0 I R^2/[2(R^2+z^2)^{3/2}]$, read off the centre field
  $\mu_0 I/2R$, and watch it fall into the $1/z^3$ tail of a magnetic dipole far away.
  Stacking turns just adds their axial contributions, which is what makes a solenoid out
  of a pile of loops.
topics: [Magnetic Sources]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 27 — Sources of Magnetic Fields; §27-2"
---

## Axial field of a circular loop

Consider a circular loop of radius $a$ in the xy plane, carrying steady current
$I$. The loop centre is the origin and the observation point lies on the z axis at
coordinate $z$. Every current element is the same distance from that point,

$$
r=\sqrt{a^2+z^2}.
$$

Equal source-to-observer separation around the loop makes the axial Biot–Savart integral
compact. The current element is tangent to the circle and the separation vector from
the element to the observation point lies in a radial-axial plane. They are
perpendicular, so the magnitude of the differential contribution is

$$
\d B=\frac{\mu_0}{4\pi}\frac{I\,\d\ell}{r^2}.
$$

The contribution is not wholly axial. Its component along the loop axis is obtained
from the geometry of the separation triangle. The axial projection factor is
$a/r$, giving

$$
\d B_z=\frac{\mu_0}{4\pi}
\frac{I a\,\d\ell}{(a^2+z^2)^{3/2}}.
$$

Every loop element has the same axial projection and direction at an axial point.
Integrating $\d\ell$ around the circumference gives

$$
B_z(z)=\frac{\mu_0 I a^2}{2(a^2+z^2)^{3/2}}.
$$

Applying the right-hand rule to the complete current loop gives
curl fingers with current and the thumb gives the positive loop-axis direction. A
current reversal reverses the sign of $B_z$ but not its magnitude profile. The
formula applies to a thin circular path with steady current; a coil of finite wire
thickness or noncircular shape requires integration over its actual source geometry.

$$
% caption: On-axis geometry for the circular-loop Biot–Savart integral. Every current element lies the same distance from the axial point P, and its contribution $\d\vec B$ is perpendicular to the separation $\vec r$; only the axial projection survives the integration around the ring.
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### Carrying the axial integral through

Parameterise the loop by the azimuth $\phi$. A source element at $\phi$ has
position $a\cos\phi\,\hat x+a\sin\phi\,\hat y$, and its directed
length is $\d\vec\ell=a(-\sin\phi\,\hat x+\cos\phi\,\hat y)\,\d\phi$.
An observation point at
$z\hat z$ has source-to-observation vector

$$
\vec r=-a\cos\phi\,\hat x-a\sin\phi\,\hat y
+z\hat z.
$$

At every azimuth, $\d\vec\ell$ is perpendicular to $\vec r$. Its
magnitude is $a\,\d\phi\,r$, with no varying sine factor. The differential field
magnitude is
$\mu_0I a\,\d\phi/(4\pi r^2)$. Its direction is not parallel to the axis,
however. Resolving that direction gives an axial factor $a/r$, so that

$$
\d B_z=\frac{\mu_0I}{4\pi}
\frac{a^2\,\d\phi}{(a^2+z^2)^{3/2}}.
$$

The scalar integrand no longer contains $\phi$. Circular symmetry makes every
axial contribution equal. Integration over one full turn yields

$$
\begin{aligned}
B_z(z)&=\int_0^{2\pi}
\frac{\mu_0I}{4\pi}\frac{a^2\,\d\phi}{(a^2+z^2)^{3/2}}\\
&=\frac{\mu_0I a^2}{4\pi(a^2+z^2)^{3/2}}
\left[\phi\right]_0^{2\pi}
=\frac{\mu_0I a^2}{2(a^2+z^2)^{3/2}}.
\end{aligned}
$$

Keeping the vector step visible prevents a common mistake: integrating the full
$\d B$ magnitude as though it were already axial. That would omit one factor of
$a/r$ and produces the wrong distance dependence. A second check comes from the
integral itself. The circumference contributes the factor $2\pi$; it cancels the
$4\pi$ in Biot–Savart only after the axial projection has been included.
## Field-map analysis and calibration

An axial map is more informative than one reading at the centre. Set the loop in a
nonmagnetic holder and mark the plane through the conductor centreline. The probe
coordinate $z=0$ belongs to that plane, not to a nearby support or to the outside
surface of the wire. A translation stage with a readable scale establishes successive
positions; at each position, turn the probe about two perpendicular axes until its
response to the loop is maximal. This alignment procedure makes its sensitive axis
parallel to the loop axis without relying on a printed arrow whose accuracy may be
unknown.

Probe calibration should convert output voltage or digital count into tesla before
the map is compared with the calculation. Record a zero-current reading at each scan
position if the local background changes along the apparatus. With a stable background,
a stronger method uses equal current magnitudes of both signs. If $S_+$ and $S_-$
are the two signed probe outputs, the current-dependent loop signal is

$$
B_{\rm odd}(z)=\frac{S_+(z)-S_-(z)}{2}.
$$

The average $[S_+(z)+S_-(z)]/2$ isolates the current-even background and belongs
in the laboratory record. The difference method cancels a constant sensor
offset and much of the static environmental field, but it cannot repair a probe whose
gain drifts between the two readings. Alternate the current sign often enough that
drift is small over one pair, and record current with a calibrated meter.

The plot to compare is $B_{\rm odd}$ against signed $z$, with uncertainty bars
from repeated readings. The ideal curve is even: points at $+z$ and $-z$ have
the same axial field. A pronounced odd part of the measured map is a diagnostic for
an origin error, a tilted probe, or an asymmetric lead arrangement. Agreement only at
the centre is weak evidence because several errors can leave one central value close
to expectation while distorting the rest of the profile.
### Radius, current, and limiting checks

For uncertainty work, retain the exact axial expression instead of using a centre
approximation at every location. Its logarithmic sensitivity to current is one:
$\partial\ln B_z/\partial\ln I=1$. The corresponding sensitivity to radius is

$$
\frac{\partial\ln B_z}{\partial\ln a}
=2-\frac{3a^2}{a^2+z^2}
=\frac{2z^2-a^2}{a^2+z^2}.
$$

Thus radius uncertainty does not have one universal coefficient. At the centre, the
coefficient is $-1$, consistent with $B_{\rm center}=\mu_0I/(2a)$: a larger
radius lowers the centre field. Well out on the axis it approaches $+2$, consistent
with the far-axis dependence on $a^2$. The sign change occurs at
$|z|=a/\sqrt2$, where the opposing effects of enlarging the loop—moving current
away while increasing its enclosed area—cancel to first order for that particular
axial point.

For independent small standard uncertainties in current and radius, and with the
probe coordinate treated as another measured input, first-order propagation gives

$$
\left(\frac{\sigma_B}{B_z}\right)^2=
\left(\frac{\sigma_I}{I}\right)^2+
\left(\frac{2z^2-a^2}{a^2+z^2}\frac{\sigma_a}{a}\right)^2+
\left(\frac{3z\,\sigma_z}{a^2+z^2}\right)^2.
$$

At $z=0$, the coordinate term vanishes to first order because the profile has a
maximum there. This does not make centre placement unimportant: a substantial offset
changes the reading at second order and can bias a claimed centre value. In the
far-axis regime, the same expression predicts the familiar leading fractional budget
$\sigma_B/B$ from one current term, two radius terms, and three distance terms.
That trend also means that a distant measurement can carry less information than a
modest off-centre one even when the probe has adequate resolution.

Two limiting substitutions provide a compact audit of a calculation. Setting $z=0$
must give $\mu_0I/(2a)$, with inverse radius dependence. Factoring $|z|^3$ from
the exact denominator at $|z|\gg a$ must leave $\mu_0Ia^2/(2|z|^3)$, with
inverse-cube distance dependence. A result that fails either test has usually lost an
axial projection, a circumference factor, or a power of separation.
### Practical limits of the axial model

The formula is a field model for the loop itself, not automatically for every wire
in the circuit. A laboratory loop needs two leads. Their fields can be small at the
probe only when the lead routing is arranged deliberately: a close outgoing-and-return
pair tends to cancel its external field, whereas widely separated leads can produce a
detectable background. An axial scan with the loop current set to zero establishes the
fixed background, but it does not remove a current-dependent field from poorly routed
leads. That contribution must be reduced by geometry or included in a fuller source
calculation.

The radius in the formula is the radius of the current path. With insulated wire, a
ruler measurement to an outside edge is therefore not necessarily the appropriate
value. Use the conductor centreline for a single turn; for several turns packed across
an appreciable radial width, a single radius is an approximation. The central
field of a tightly wound set of turns is often estimated by multiplying the single-turn
answer by the turn count, provided the turns share nearly the same radius and axial
location. When that is not true, add the individual loop contributions at the intended
probe position instead of assigning the winding one nominal radius.

Instrument orientation matters as much as position. A one-axis Hall probe measures
the component normal to its sensing face. If its sensitive axis makes an angle
$\theta$ with the loop axis, the displayed loop contribution is $B_z\cos\theta$.
Small alignment errors near the centre mostly change the scale; a probe displaced
sideways also samples transverse field components and no longer tests the axial
expression alone. A complete record therefore gives the current, centreline radius,
signed axial coordinate, probe orientation, background procedure, and uncertainty in
each. Such details trace a disagreement to geometry, calibration, or the source
model.

### Symmetry cancellation and limiting cases

The off-axis components of individual contributions cancel in pairs on the loop axis.
For every source element at one azimuth, a diametrically opposite element has an
equal magnitude contribution with transverse component opposite in direction. Their
axial components have the same direction and add. This cancellation is specific to
the observation point lying on the symmetry axis. Moving the observation point off
axis breaks the pairwise cancellation and generally requires resolving vector
components throughout the integral.

At the centre, $z=0$, the axial result becomes

$$
B_{\rm center}=\frac{\mu_0I}{2a}.
$$

For fixed current, a smaller loop produces a larger centre field because every source
element is closer to the observation point. The result has units of tesla: $\mu_0I/a$
has units of $\mathrm{T}$. This centre limit is also an algebra check on any axial
calculation; retaining an extra factor of radius or separation gives the wrong units.

$$
% caption: Magnetic field lines of a circular current loop, seen edge-on with current out of the page on the left conductor and into it on the right. Each line closes on itself, threading up through the centre and returning around the outside; far away the pattern is that of a magnetic dipole, the origin of the $1/z^3$ axial tail.
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Far from the loop, $|z|\gg a$, the denominator is approximately $|z|^3$, so

$$
B_z\approx\frac{\mu_0I a^2}{2|z|^3}.
$$

The inverse-cube decay differs from the inverse-distance result of an indefinitely
long straight wire because a finite loop has no extended source length at large
distance. The far-axis approximation is valid only after the observation distance is
large compared with loop radius; applying it near the loop loses the measured peak
and curvature of the exact axial profile.

$$
% caption: Pairwise transverse cancellation on the loop axis. Contributions from diametrically opposite current elements have horizontal parts that cancel at the axial point P and axial parts that add, so the net on-axis field points purely along the axis.
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### Measurement along the loop axis

An axial scan tests the derivation directly. Mount the loop so its geometric axis is
known, drive a measured steady current, and move a calibrated magnetic probe along
that axis while keeping the probe's sensitive direction aligned with the axis. Record
both signed current and signed probe output. Reversing current should reverse the
measured axial signal, allowing a current-odd difference to suppress fixed sensor
offset. The probe active area should be small compared with the distance scale over
which the axial result changes; otherwise it reports a spatial average rather than
the value at its marked location.

> **Worked example (Field on the axis of a loop).** A loop of radius
> $a=5.00\ \mathrm{cm}$ carries $I=2.00\ \mathrm A$. At the centre $(z=0)$,
>
> $$
> B_{\rm center}=\frac{\mu_0 I}{2a}
> =\frac{(4\pi\times10^{-7})(2.00)}{2(0.0500)}=25.1\ \mu\mathrm T.
> $$
>
> One radius out along the axis $(z=a)$, the factor $a^2/(a^2+z^2)^{3/2}$ contributes
> $1/(2\sqrt2\,a)$, so the field drops by $1/(2\sqrt2)$:
>
> $$
> B(z=a)=\frac{B_{\rm center}}{2\sqrt2}=\frac{25.1\ \mu\mathrm T}{2.83}=8.89\ \mu\mathrm T.
> $$
>
> Measuring at both points fixes the scale and the curvature of the axial curve.

$$
% caption: On-axis field of a circular loop against axial position. The profile is even in $z$, peaks at the loop plane where $B=\mu_0 I/2a$, and decays into the $1/z^3$ dipole tail on either side.
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$$

### Reading residuals in an axial map

Once each reading has been reduced to a current-odd magnetic field, calculate a
residual at every position,

$$
R(z)=B_{\rm odd}(z)-
\frac{\mu_0I a^2}{2(a^2+z^2)^{3/2}}.
$$

Assess residuals as a pattern across the map. The largest absolute value alone does
not identify a source of error. A nearly constant residual suggests an incomplete offset correction
or a current-dependent background that the reversal procedure has not removed. A
residual that changes sign between $+z$ and $-z$ points to a misplaced origin or
to lead geometry that lacks the loop's mirror symmetry. A symmetric residual that is
small at the centre and grows toward both ends is often caused by an error in the
radius, the probe calibration scale, or the assigned probe coordinate. Each source
has a different shape across the map, which is why a distributed scan is a more
severe test than a centre reading.

Do not use the same map to alter every input until the curve agrees. The loop radius
should first come from direct measurement of the conductor centreline, preferably at
several diameters to expose an out-of-round winding. Current should come from a meter
in series with the loop or from a calibrated current monitor, not from a nominal
setting. Probe calibration and axial position should likewise have independent
records. With those quantities fixed, compare the field with the single
circular-path model within the stated uncertainties. Freely adjusting all parameters
afterward can produce agreement from compensating errors.

Near the wire plane, preserve the sign convention carefully. The axial expression is
positive on both sides of the centre for a selected current direction, because the
axis direction is fixed while the observation coordinate changes sign. Confusing the
even field profile with an odd one can make a valid scan appear wrong by an entire
sign. The coordinate should therefore be recorded as signed distance from the loop
plane, whereas the probe's positive sensing direction remains fixed in the laboratory.

The residual table should also state the probe's active-area dimension. The ideal
formula is evaluated at a mathematical point. A probe with a finite sensitive area
averages nearby values. Far from the centre that average is often negligible; a large
sensor near a small loop can smooth the peak enough to mimic a radius or calibration
error. Repeat central readings with a smaller probe or controlled displacement to
test the spatial-averaging effect.

## Off-axis fields and nonideal windings

The axial expression gives one component of the field on one special line. It does
not predict the magnitude or direction at a point displaced sideways from that line.
At an off-axis point, opposite source elements are no longer the same distance from
the probe. Their transverse contributions therefore fail to cancel pair by pair, and
the field generally has both an axial component and a radial component. Cylindrical
symmetry still rules out an azimuthal component for a perfectly circular loop, but it
does not reduce the remaining calculation to the single axial formula.

The distinction matters in a measurement. A probe moved a small distance sideways at
fixed $z$ samples a changed vector field rather than a displaced copy of the
on-axis value. A one-axis probe may report a lower value because the axial component
has changed, because the total field has rotated away from the probe axis, or both.
A scalar comparison against $B_z(z)$ cannot sort those possibilities out. Mapping
such points requires the full Biot–Savart vector integral, a numerical summation over
short current segments, or measurements of enough field components to reconstruct
the local vector.

A near-axis check follows from the loop's reflection symmetry about its central plane
and rotational symmetry about its axis, without claiming an off-axis formula.
At $z=0$, the field at a small radial displacement remains directed along the axis;
the radial component changes sign across the central plane and is zero in that plane.
At $z\ne0$, a radial component can occur and reverses when the radial displacement
is reflected through the axis. These symmetry statements identify which measured
components should vanish at special locations. They do not supply their nonzero
values, so they must not be substituted for the axial result.

Mechanical alignment is consequently part of the model. A translation rail that is
tilted relative to the loop axis causes an intended axial scan to acquire a sideways
displacement that grows with distance. The resulting trace can lose its expected even
shape even when the current and radius are measured accurately. Establish the axis
with a sight line or alignment rod, check the rail at several positions, and retain a
record of its transverse offset if high-accuracy mapping is required.

$$
% caption: On-axis versus off-axis observation. On the axis the field is purely axial; a probe displaced sideways generally reads both an axial and a radial component, so the one-variable axial formula no longer describes the vector there.
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### Conductor width, turn spacing, and the source path

The ideal derivation places all current on a circular line of radius $a$. Real wire
occupies a cross-section, and adjacent turns may have different radii and different
axial positions. These details matter most close to the winding, where the field
varies noticeably over the dimensions of the conductor. Assigning a single radius to
a broad winding can produce a central-field estimate that looks precise while hiding
the dominant geometry uncertainty.

For one round wire carrying uniform current, the first approximation uses the radius
of its centreline. The measured outside diameter should not be substituted directly:
it shifts the current path outward by approximately half the wire diameter. For a
rectangular conductor or a thick conducting band, current density may also vary across
the cross-section, especially at high frequency. The steady-current treatment here
assumes a known distribution; when the distribution is nonuniform, the source must be
integrated over the conductor volume rather than represented by one centreline.

A compact group of $N$ turns has the simple centre estimate
$B_{\rm center}\approx N\mu_0I/(2a)$ only if all turns have nearly equal radii and
lie close to the same plane. A more faithful representation retains a radius $a_j$
and axial displacement $z_j$ for each turn. At an observation coordinate $z$,
their axial contributions add as

$$
B_z(z)=\sum_{j=1}^{N}
\frac{\mu_0I a_j^2}{2\left[a_j^2+(z-z_j)^2\right]^{3/2}}.
$$

The expression is a superposition of circular-loop results. Turn spacing broadens
the axial peak, and radial layering changes its scale.
When the total winding width is small compared with both its mean radius and the
probe distance of interest, replacing the set by $N$ coincident turns is normally
adequate. The approximation should be checked against the actual winding dimensions,
not inferred from a large turn count alone.

$$
% caption: A practical multi-turn winding. Turns occupy a spread of radii and axial planes, so each contributes a circular-loop field with its own radius and axial offset; replacing the set by coincident turns is accurate only when that spread is small at the probe.
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> **Worked example (Multi-turn coil calibration with an uncertainty budget).** A
> compact winding of $N=40$ turns has mean radius $a=60.0\pm0.2\ \mathrm{mm}$ and
> carries $I=0.2500\pm0.0010\ \mathrm A$. A probe sits on the axis at
> $z=30.0\pm0.3\ \mathrm{mm}$, so $z/a=0.500$, and the winding is compact enough to
> treat the turns as coincident. The prediction is
>
> $$
> B_z=\frac{N\mu_0 I a^2}{2(a^2+z^2)^{3/2}}=74.9\ \mu\mathrm T.
> $$
>
> Propagating the independent input errors — current, radius (sensitivity
> $(2z^2-a^2)/(a^2+z^2)=-0.400$), and position — gives
>
> $$
> \frac{\sigma_{B}}{B_z}=\sqrt{(0.00400)^2+(0.400\times0.00333)^2+(0.00600)^2}=0.0074,
> $$
>
> so $B_{\rm model}=74.9\pm0.6\ \mu\mathrm T$, with the probe coordinate dominating. A
> reversed-current probe reading $B_{\rm meas}=V/k=75.4\pm0.5\ \mu\mathrm T$ differs by
> $0.5\ \mu\mathrm T$, inside the combined uncertainty
> $\sqrt{0.6^2+0.5^2}\ \mu\mathrm T$: the calibration is consistent with the stated
> geometry.
### Outcome of the calibration comparison

The numerical agreement establishes a limited claim: at the stated current, geometry,
and probe location, the observed axial component is compatible with the circular-turn
calculation. It does not establish the model at every point in space or at every
current. A calibration record therefore includes the coordinate, current sign,
probe sensitivity, turn count, mean radius, winding width, and the treatment of
background readings. Another group can then repeat the comparison without guessing
which nominal dimensions were used.

Several follow-up checks separate a robust calibration from an accidental match.
Repeat the map at a second current, for example $0.150\ \mathrm A$, and divide each
current-odd result by the measured current. The normalized traces should agree within
their uncertainties if the probe and winding remain in the linear steady-current
regime. A change of scale that tracks neither current nor calibration indicates an
instrument or lead-geometry problem. Repeating the scan after reversing the physical
orientation of the loop, while retaining the laboratory probe convention, gives a
separate sign check.

The selection of axial positions should also serve the uncertainty budget. Near the
centre, the field is large and coordinate error affects the result only at second
order, making those points suitable for checking the current-to-field scale. Around a
substantial fraction of a radius from the centre, the curve has more shape and is more
sensitive to an incorrectly assigned radius or origin. Farther out, the signal drops
rapidly and distance uncertainty becomes costly. A calibration run that includes all
three regions can expose errors that a collection of closely spaced central readings
would leave hidden.

Finally, distinguish uncertainty from correction. A known probe sensitivity offset
is a correction to apply before comparison; the remaining uncertainty in that
correction belongs in the budget. A nearby steel object whose effect has not been
characterized is not a small statistical uncertainty to be appended automatically. It
is an uncontrolled source contribution, best removed by changing the apparatus or
measured independently with the loop current set to zero and with appropriate
current-reversal tests. This distinction keeps the reported calibration tied to
quantities that were actually measured.

## Winding superposition and numerical prediction

Each circular path contributes an axial field whose sign is set by its current
orientation. If a common positive axis has been chosen, introduce $s_j=+1$ for a
turn whose right-hand-rule field points in that direction and $s_j=-1$ for a turn
whose field points oppositely. For paths with radii $a_j$, axial locations $z_j$,
and currents $I_j$, the axial model becomes

$$
B_z(z)=\sum_j s_j\frac{\mu_0 I_j a_j^2}
{2\left[a_j^2+(z-z_j)^2\right]^{3/2}}.
$$

The sign belongs to the source orientation, not to the observation coordinate. A
turn that is reversed changes its contribution at every axial point; moving the probe
from positive to negative $z$ does not reverse the axial component of a single
turn. Keeping those two facts separate avoids a frequent sign error in multi-turn
calculations. The signed form also accommodates a return path that makes one or more
nearly circular loops in the opposite sense. Treating all turns as positive because
the supply current has one stated polarity can then overpredict the measured
field substantially.

For nearly coincident turns carrying the same current and orientation, the sum reduces
to $N$ times the one-turn result. That reduction is a consequence, not an added
rule: set every $a_j=a$, every $z_j=0$, every $I_j=I$, and every $s_j=+1$ in
the sum. If one subgroup is wound in the opposite sense, its contribution subtracts.
Two equal, oppositely oriented groups in the same plane cancel on the axis, including
at the centre. If the two groups are separated along the axis, their centre-plane
contributions can still cancel while the field changes rapidly away from that plane.
Such a source cannot be represented by a single scaled loop curve.

Sign checks are simple to build into the laboratory record. Mark the current direction
on each winding layer, define the positive axial direction on the apparatus, and make
a predicted-sign column before taking probe readings. Reverse the supply current once
the map has been recorded. Every source contribution that follows that supply branch
should reverse; a static background should not. A disagreement in sign is often more
diagnostic than a modest scale disagreement because it exposes an orientation or probe
polarity mistake immediately.

$$
% caption: Signed superposition of turns. Turns wound in the same sense add their axial fields; a turn wound in the opposite sense subtracts. The net axial field is the signed sum of the individual loop contributions, not a plain turn count.
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### Numerical quadrature for a nonideal winding

The separate-loop sum is efficient when the paths are visibly circular. A distorted
winding, a lead that approaches the probe, or a deliberately shaped conductor calls
for a direct numerical Biot–Savart evaluation. Represent the current path by ordered
points $\vec r_k$. The short directed segment from one point to the next is
$\Delta\vec\ell_k=\vec r_{k+1}-\vec r_k$, and the segment midpoint
is used as its source location. At a probe position $\vec P$, the midpoint
approximation is

$$
\vec B(\vec P)\approx\frac{\mu_0 I}{4\pi}
\sum_k\frac{\Delta\vec\ell_k\times
(\vec P-\vec r_k^*)}{|\vec P-\vec r_k^*|^3},
$$

where $\vec r_k^*$ denotes the midpoint of segment $k$. The cross product
preserves both direction and sign, so this calculation can predict off-axis components
as well as the axial one. Closed circular paths submitted to the same routine provide
a convergence check: as the segment count is increased, the numerical axial result should
approach the analytic circular-loop expression.

Segment length is a model-control parameter. Halve the largest segment length and
repeat the calculation. If the predicted probe field changes by more than the desired
numerical tolerance, the original segmentation was too coarse. For smooth paths with
midpoint segments, the leading integration error commonly falls roughly with the
square of segment length, but a sharp bend or a nearby probe can spoil that behaviour.
Refine those regions locally. Long, smooth portions of the path need fewer segments.

The centreline model assumes that the probe is not so close to the conductor that its
cross-section is resolved. When the probe distance is comparable with wire radius,
replace one centreline with several nearby paths that share the measured current in
proportion to area, or integrate the current density over the conductor cross-section.
The result should then be tested for convergence both in the number of path segments
and in the number of cross-sectional sample paths. A numerical answer with many digits
but no convergence check is only an untested approximation.

Measured winding coordinates can be obtained from a photographed grid, a mechanical
scan, or a design drawing, provided the coordinate system is registered to the probe
position. The path order must follow the actual current direction. Accidentally
joining two points across a gap creates a fictitious straight segment and can dominate
the calculated field near the gap. Plotting the reconstructed path before evaluating
the field is a practical way to catch that input error.
> **Worked example (Two-group winding prediction).** A 24-turn winding splits into two
> measured groups: twelve turns at $a_1=45.0\ \mathrm{mm}$, $z_1=-2.0\ \mathrm{mm}$ and
> twelve at $a_2=47.0\ \mathrm{mm}$, $z_2=+2.0\ \mathrm{mm}$, all carrying
> $I=0.4000\ \mathrm A$ the same way. At the probe coordinate $z=20.0\ \mathrm{mm}$
> each group contributes
>
> $$
> \begin{aligned}
> B_1&=12\frac{\mu_0 I a_1^2}{2[a_1^2+(z-z_1)^2]^{3/2}}=48.60\ \mu\mathrm T,\\
> B_2&=12\frac{\mu_0 I a_2^2}{2[a_2^2+(z-z_2)^2]^{3/2}}=52.26\ \mu\mathrm T,
> \end{aligned}
> $$
>
> so $B_{\rm pred}=100.9\ \mu\mathrm T$. The probe sits $22\ \mathrm{mm}$ from the first
> group but $18\ \mathrm{mm}$ from the second, so equal turn counts give unequal
> contributions — using the measured group dimensions beats a single averaged loop. A
> calibrated probe reading of $101.7\ \mu\mathrm T$ leaves a residual of
> $0.85\ \mu\mathrm T$, about one combined standard uncertainty
> $(\sqrt{0.53^2+0.60^2}=0.80\ \mu\mathrm T)$: worth a note, not a claim that the model
> failed.

Residual structure over a full map identifies the next measurement. A nearly uniform
scale error at all positions suggests current-meter or probe-sensitivity calibration. A
discrepancy that grows near one side of the apparatus points toward a lead or a local
magnetic object. A residual that changes sign when the probe is displaced sideways is
consistent with axis misalignment or with a source path that is not circular. None of
these patterns is repaired honestly by changing the nominal turn count after the data
are known. Measure the suspected source geometry, update the signed path model, and
state the revision in the comparison record.
### Boundaries of a numerical comparison

A numerical path calculation should be compared with data at locations selected before
the result is viewed. Choosing only positions where the curve happens to agree changes
the task from a model test into a display exercise. Include a centre-near point, one or
more points displaced by a substantial fraction of the winding radius, and a point far
enough away to test the predicted decline. At each location, retain the raw positive-
and negative-current probe outputs as well as their reduced current-odd value. That
record allows a later check for drift, asymmetric current magnitude, or background
changes that were concealed by the reduction.

Geometry uncertainty and numerical error should not be merged without explanation.
The former reflects incomplete knowledge of the physical source: turn centres, lead
paths, conductor dimensions, and probe position. The latter reflects the accuracy of
the chosen calculation after a particular geometry has been supplied. Report the
segment-refinement change separately from the uncertainty caused by moving an input
coordinate within its measurement tolerance. If refinement changes the result by
$0.05\ \mathrm{\mu T}$ while geometric uncertainty is
$0.50\ \mathrm{\mu T}$, further subdivision is unlikely to improve the physical
prediction. If the refinement change is comparable with the measured residual, the
numerical model has not yet earned a comparison with the experiment.

An observed discrepancy is most informative when it can be connected to a proposed
independent test. A suspected nearby lead can be rerouted while every other source is
held fixed. A suspected probe-angle error can be checked by rotating the probe through
a known small angle and observing the expected component change. A suspected path
coordinate error can be tested against a new mechanical survey. These interventions
turn residuals into information about the apparatus rather than reasons to add an
unspecified correction factor. The final field prediction remains credible only to
the extent that its source geometry, numerical convergence, and measurement protocol
are each documented at the same level of care.

### Winding coordinates and sign conventions

A winding model begins with a coordinate convention that can be reconstructed from
the apparatus. Put the origin at the chosen reference plane of the winding, direct
the positive $z$ axis along the mechanical axis, and use a right-handed transverse
pair $(x,y)$. A layered winding requires each path to be recorded by mean centreline
radius $a_j$, centre-plane coordinate $z_j$, and orientation sign $s_j$. The
sign should be tied to a visible current-direction mark: with the observer looking
from positive $z$, counterclockwise current has the right-hand-rule axial direction
used as positive in the preceding sums.

The reference plane needs an operational definition. A convenient choice is the
midplane between the outermost turn planes, measured from conductor centrelines. A
stage reading $q$ then becomes the model coordinate $z=q-q_0$, where $q_0$ is
the stage reading at that reference plane. The uncertainty in $q_0$ is correlated
among every point in an axial scan: moving the whole coordinate origin changes all
listed $z$ values together. Treating it as unrelated random error at each point can
understate a systematic displacement of the map.

Radius and axial position must refer to the same current path. A radius measured to
the outside insulation and a plane measured to the wire centre do not define one
consistent loop. When the winding is surveyed from a photograph or drawing, place
fiducial marks in the same physical plane as the probe travel and state the scale used
to convert pixels or divisions to length. The convention becomes especially important
when two layers are close enough that their separate axial coordinates affect the
predicted field by more than the probe uncertainty.
## Probe alignment and measurement protocol

A Hall probe reports the component of magnetic field along its sensing normal. Over
its calibrated range, a simple representation is $V=V_0+S\,\vec B\cdot\hat n$,
where $V_0$ is the zero-field output and $S$ is sensitivity. At
an axial point, a tilt angle $\theta$ between $\hat n$ and the winding
axis gives a loop signal proportional to $\cos\theta$. Align the probe by rotating
it at a known axial point until the current-odd response is maximal, then lock that
orientation before the translation scan. The maximum provides an alignment datum
without relying on the orientation of a probe housing or support rail.

Calibration requires both a scale and a zero procedure. Determine $S$ from a known
reference field or from a traceable calibration, then record a zero-current value with
the probe in the same position and orientation used for the scan. At each map point,
the alternating-current estimate $[V(+I)-V(-I)]/(2S)$ removes the constant part of
$V_0$ and most static background. A current reversal does not correct gain drift,
probe saturation, or a field from leads that reverses with the loop current; those
effects must be checked by repeated calibration, operating-range limits, and lead
geometry.

Spatial resolution is set by both the scan increment and the active sensor size. The
ideal axial curve falls to one half of its centre value at
$|z|=0.766a$. A step no larger than roughly $a/10$ gives more than fifteen
intervals across the corresponding full width and usually resolves the peak shape of
a single compact loop. Coarser steps can still verify a scale, but they can miss a
shifted maximum or obscure broadening caused by turn spacing. If the active
area or its axial extent is not small relative to $a$, the probe measures an average
over that area; reducing the step alone cannot restore the lost spatial detail.

$$
% caption: Hall-probe alignment on an axial scan. The sensor reports the field component along its sensing normal, so a tilt $\theta$ between the normal and the axis scales the reading by $\cos\theta$; the scan step must resolve the width of the axial peak.
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### A compact field-map protocol

A 24-turn winding with mean radius $a=46.0\ \mathrm{mm}$, current
$I=0.4000\ \mathrm A$, and a probe sensitivity of
$2.00\ \mathrm{mV/\mu T}$, begin by surveying the two winding layers and entering
their separate $a_j$ and $z_j$ values in the signed source list. Use the mean
radius only to set an initial scan range. The single-radius centre estimate is
$N\mu_0I/(2a)=131\ \mathrm{\mu T}$; it provides a scale check before the
layered model is evaluated.

Set the stage reference at the surveyed winding midplane. Record points at
$z=0$, $\pm23\ \mathrm{mm}$, and $\pm46\ \mathrm{mm}$, then add intermediate
points at 4 or 5 mm spacing if the profile itself is to be tested. At every location,
take a short sequence $+I,-I,+I,-I$ after the probe has settled. Convert the mean
pair difference to tesla with the calibration sensitivity and retain the individual
voltages. At the centre, the preliminary scale predicts about $0.262\ \mathrm V$
for a positive-current signal; at $|z|=a/2$ and $|z|=a$, it predicts about
$0.188\ \mathrm V$ and $0.0928\ \mathrm V$, respectively. These are planning
values, not replacements for the source model using the surveyed layers.

Calculate the field at each recorded coordinate from the signed turn sum, or from the
converged segment model when lead paths must be retained. List the current-odd probe
field, its standard uncertainty, the model value, and the residual in the same order
as the scan coordinate. Examine symmetry first: paired points should agree within
uncertainty for a winding and lead arrangement that is symmetric about the reference
plane. Then inspect the residual sequence for a constant scale bias, a coordinate
shift, or a one-sided disturbance. Only after those checks should a geometry parameter
be revised, and the revised value must be backed by a new survey rather than chosen to
force the plotted curves together.