---
title: Magnetic Dipoles
module: Magnetic Field
moduleNumber: 6
lessonNumber: 4
order: 604
summary: >
  A compass needle turns to point north; a current loop in a field does the same
  thing, and for the same reason. Both are magnetic dipoles, and a uniform field
  cannot push a dipole anywhere, only twist it. We package a loop's response into one
  vector, the magnetic moment $\vec\mu=IA\hat n$, from which torque
  $\vec\tau=\vec\mu\times\vec B$ and orientation energy $U=-\vec\mu\cdot\vec B$ both
  follow. Stable alignment sits at the energy minimum, a field gradient is what it
  takes to produce a net force $\vec F=\nabla(\vec\mu\cdot\vec B)$, and the same
  moment reappears whenever anything from an electron to a planet acts magnetic.
topics: [Magnetic Field]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 26 — The Magnetic Field; §§26-3–26-4"
---

## Magnetic moment of a current loop

A current loop has a directed area and therefore a magnetic orientation. For a
single planar loop of area $A$ carrying current $I$, define

$$
\vec\mu=I\vec A=IA\hat n.
$$

The normal $\hat n$ follows the right-hand rule: curl the fingers with
conventional current and the thumb gives the moment direction. A coil with $N$
closely spaced turns has

$$
\vec\mu=NI\vec A.
$$

The unit is $\mathrm{A\,m^2}$, which is equivalent to $\mathrm{J/T}$. The moment
specifies loop orientation and its leading interaction with an applied field at
distances large compared with the loop size.

Reversing the current reverses $\vec\mu$. Reversing the chosen area normal
also reverses the current direction associated with the same physical loop. A
diagram must keep current circulation and area normal paired; assigning one without
the other leaves the vector sign incomplete.

In a noncircular planar loop, $A$ is the geometrical area enclosed by the current
path, not the area of the wire material. A rectangular loop and a circular loop with
the same area and current have equal magnetic-moment magnitude. Multiple turns add
only when their current circulations have the same sense. A reversed turn subtracts
from the net moment. The definition therefore records both geometry and circuit
connection.

$$
% caption: Magnetic moment of a planar current loop. Curling the right-hand fingers along the conventional current sets the area normal, and the moment vector $\vec\mu=IA\hat n$ points along it; reversing the current reverses the moment.
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$$

## Torque and energy

Place a rectangular loop in a uniform magnetic field. The forces on sides parallel
to the field vanish. The two remaining sides carry currents in opposite directions,
so their magnetic forces have equal magnitude and opposite direction. Their vector
sum is zero, yet their separated lines of action form a couple.

Let the loop have side lengths $a$ and $b$, area $A=ab$, and let $\theta$ be the
angle from $\hat n$ to $\vec B$. Each active side experiences force
magnitude $IaB$. The perpendicular separation of the forces is $b\sin\theta$.
The torque magnitude is therefore

$$
\tau=(IaB)(b\sin\theta)=IAB\sin\theta.
$$

For $N$ turns, multiply by $N$. The vector expression is

$$
\vec\tau=\vec\mu\times\vec B,
\qquad
\tau=\mu B\sin\theta.
$$

The torque tends to rotate the moment toward the field. It is zero when the moment
is parallel or antiparallel to the field, and largest when the two vectors are
perpendicular. Zero torque alone does not identify the orientation as stable; the
energy determines that classification.

The force-pair derivation also fixes the torque axis. The torque vector follows
$\vec\mu\times\vec B$ and points along the axis of the rotation selected
by the right-hand rule. A loop initially parallel to the field has no first-order
turning tendency. A loop initially perpendicular begins to rotate with maximum
torque. Reversing current reverses the torque vector at every orientation, while
leaving the magnitudes of the segment forces unchanged.

$$
% caption: A current loop shown edge-on in a uniform field. Its two active sides carry opposite currents, so equal and opposite forces act at the edges; the forces cancel as a vector sum, but their offset produces a couple that turns the moment toward the field.
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$$

The derivation assumes that the applied field is effectively uniform across the
loop. In a field gradient, opposite sides can have unequal force magnitudes, so the
loop can experience both torque and net force. The uniform-field torque formula
remains the leading result for a loop small compared with the field-variation scale.

> **Worked example (Torque on a circular coil).** A circular coil of $N=50$ turns and
> radius $r=2.0\ \mathrm{cm}$ carries $I=3.0\ \mathrm{A}$ in a uniform field
> $B=0.40\ \mathrm{T}$, its moment making $\theta=30^\circ$ with the field. First the
> magnetic moment,
>
> $$
> \mu=NI\pi r^2=(50)(3.0)\pi(0.020)^2=0.188\ \mathrm{A\,m^2},
> $$
>
> then the torque,
>
> $$
> \tau=\mu B\sin\theta=(0.188)(0.40)\sin30^\circ=3.8\times10^{-2}\ \mathrm{N\,m}.
> $$
>
> The torque acts to decrease $\theta$, turning the moment toward $\vec B$. It peaks at
> $\theta=90^\circ$ (moment across the field) and vanishes at $\theta=0$, where the coil
> sits in its stable minimum-energy orientation.

**Potential energy and equilibrium.**

The orientation-dependent potential energy is

$$
U=-\vec\mu\cdot\vec B=-\mu B\cos\theta.
$$

With the applied field held fixed, mechanical work done against the magnetic torque changes
this energy. The parallel state has $U=-\mu B$ and is a minimum. A small angular
displacement from alignment produces a restoring torque. The antiparallel state has
$U=+\mu B$ and is a maximum: a small displacement produces a torque that increases
the displacement. At $\theta=90^\circ$, the torque magnitude is maximal while the
energy is zero relative to the chosen reference.

The energy expression is valid for a prescribed current and an externally maintained
field. If the current changes during rotation, the electrical source exchanges energy
with the circuit and the complete energy account must include that source. For a
rigid loop carrying steady current, the expression gives the mechanical orientation
dependence directly.

The torque follows from the energy slope:

$$
\tau_\theta=-\frac{\d U}{\d \theta}=\mu B\sin\theta.
$$

This scalar relation uses $\theta$ as the rotation coordinate about the relevant
axis. It agrees with the magnitude of $\vec\mu\times\vec B$. A slow
rotation from one angle to another changes the mechanical potential energy by the
work done against the magnetic torque. The field does not prescribe a preferred
clockwise direction until the viewing axis and the signed vectors have been stated.

$$
% caption: Magnetic potential energy $U=-\mu B\cos\theta$ versus the angle between the loop moment and a fixed field. Parallel alignment is the stable minimum, antiparallel the unstable maximum; the steepest slope, at $90^\circ$, is where the torque is largest.
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$$

**Force couple and net force.**

Torque and net force answer different mechanical questions. In a uniform field, the
force on a complete small loop sums to zero while the couple can rotate it. A support
therefore measures no translational magnetic load from the ideal uniform field even
though the loop may turn strongly. The individual conductor segments are not
force-free; their forces cancel only after vector addition over the full loop.

In a nonuniform field, the field energy varies with position as well as orientation.
A dipole then has the approximate net force

$$
\vec F=\nabla(\vec\mu\cdot\vec B)
$$

when the moment is treated as fixed during the displacement. A small loop with
$\vec\mu$ aligned with a growing field is pulled toward the stronger-field
region. The approximation requires the field to vary little across the loop. Larger
loops require integration of the segment forces over their actual geometry.

A measurement should separate rotation from translation. Mount the loop on a low-
friction pivot to measure angular deflection, or constrain its orientation and use a
force sensor along a specified translation axis. Reverse the current at fixed field:
both the moment and torque reverse. Reverse the field instead and the same reversal
occurs. Reporting the loop area, turn count, current direction, field map, pivot
axis, and support geometry identifies which component of the magnetic interaction was
measured.

In a uniform-field test, equal and opposite forces on the active sides should give
zero sensor reading along every translation axis after support offsets are removed.
A torsion spring at the pivot can convert the loop angle into torque through a
separate mechanical calibration. For a gradient-field test, constrain the loop
orientation before interpreting a translational force; otherwise rotation changes
$\vec\mu\cdot\vec B$ during the measurement.

**Torque vector from distributed segment forces.**

The segment-force relation is already known. For a current path in an applied field,
the torque about a chosen origin is obtained by adding the moments of those segment
forces:

$$
\d \vec\tau=\vec r\times \d \vec F,
\qquad
\vec\tau_O=I\oint\vec r\times
(\d \vec\ell\times\vec B).
$$

Here $\vec r$ runs from the chosen origin to the current element. In a uniform
field, the net force on a complete loop is zero, so the torque is independent of the
origin. That independence is essential: the result describes a couple, not the
moment of an unbalanced translational force about an arbitrary point.

Use the vector triple-product identity inside the loop integral:

$$
\vec\tau_O
=I\oint\left[(\vec r\cdot\vec B)\,\d \vec\ell
-(\vec r\cdot \d \vec\ell)\vec B\right].
$$

The second term integrates to zero because
$\oint\vec r\cdot \d \vec\ell=\oint\vec r\cdot \d \vec r=0$.
For any planar closed path, the remaining geometric identity is

$$
\oint(\vec r\cdot\vec B)\,\d \vec\ell
=\vec A\times\vec B,
\qquad
\vec A=\frac12\oint\vec r\times \d \vec\ell.
$$

Thus the distributed-force calculation gives

$$
\vec\tau=I\vec A\times\vec B
=\vec\mu\times\vec B.
$$

This derivation does not require a rectangular loop. The area vector is defined by
the path integral and applies to any planar shape. For $N$ tightly packed turns with
the same circulation and field sampling, replace $I\vec A$ by $NI\vec A$.
A winding with turns in different planes or a field that varies significantly over
the winding must be treated turn by turn.

The vector direction can be checked without drawing a mechanical rotation first.
Compute $\vec\mu\times\vec B$ in components, then use the right-hand rule
only as a consistency check. Reversing current reverses $\vec\mu$ and the
torque vector. Reversing the applied field does the same. Reversing both leaves the
torque unchanged. These sign tests apply to the full distributed force sum even when
the loop shape is irregular.

The result uses a prescribed applied field. Evaluate
$\vec\mu\times\vec B$ with the field from sources other than the selected
loop. Nearby ferromagnetic parts or another current path can alter that field map;
keep the loop current at its stated value while evaluating it.

**Orientation-energy measurement.**

A pivot experiment converts magnetic torque into a measurable angle. Suspend a rigid
loop from a torsion fibre with torsional constant $\kappa$, apply a known uniform
field, and let the loop settle. If $\phi$ is the signed deflection from the fibre's
zero, static balance is

$$
\kappa\phi=\mu B\sin\theta,
$$

where the angle $\theta$ is between the loop moment and the applied field. The
mechanical restoring torque and magnetic torque must be expressed about the same
pivot axis. A fibre that is not aligned with the magnetic torque vector measures only
the projected component.

Determine $\kappa$ mechanically before using the magnetic data. A known small torque
or an angular oscillation measurement can calibrate the fibre without referring to
the magnetic field. Then measure deflection for both current directions at the same
field. Half the difference of the two angles suppresses a fixed fibre bias and many
mounting offsets. Field reversal provides an independent sign check.

The energy relation gives a second route to the same measurement. At fixed current
and fixed applied field,

$$
U(\theta)=-\mu B\cos\theta,
\qquad
\Delta U=\mu B(\cos\theta_i-\cos\theta_f).
$$

A slow imposed rotation requires mechanical work equal to the increase in this
potential energy, apart from losses in the pivot and fibre. Measuring work from the
torsion calibration and comparing it with the angular energy change tests both the
moment magnitude and the angle convention. Friction makes clockwise and
counterclockwise sweeps differ; average the two directions or report the hysteresis
rather than treating it as magnetic energy.

The small-angle limit around the aligned state provides a sensitivity estimate.
For small $\theta$, $\sin\theta\approx\theta$, so the magnetic angular stiffness is
$\mu B$. A fibre much stiffer than $\mu B$ produces a small, nearly linear
deflection; a very soft fibre gives a larger angle but is more susceptible to
vibration, gravity, and lead torque. The experimental range should remain within the
calibrated angular response of the fibre.

## Nonuniform forces and torque

A nonuniform field changes the relation between opposite segment forces. The general
force and torque integrals about origin $O$ are

$$
\vec F=I\oint \d \vec\ell\times\vec B(\vec r),
\qquad
\vec\tau_O=I\oint\vec r\times
[\d \vec\ell\times\vec B(\vec r)].
$$

Neither integral is generally zero. In contrast with the uniform-field couple, a
nonzero net force makes torque depend on the reference point. If a new origin is
displaced by $\vec a$ from the old one, then

$$
\vec\tau_{O'}=\vec\tau_O-\vec a\times\vec F.
$$

State the pivot or the centre about which torque is reported whenever translation is
also possible.

A small loop with constrained orientation has the leading net force

$$
\vec F\approx\nabla(\vec\mu\cdot\vec B).
$$

A moment parallel to a field that increases toward positive $x$ has zero magnetic
torque but a force toward the stronger field. A moment perpendicular to a uniform
field has nonzero torque but zero net force. A loop in a general gradient can have
both effects. These cases cannot be distinguished from a single force reading or a
single angle reading.

$$
% caption: A dipole in a nonuniform field. The field grows from left to right, so the pull on the near side of the loop exceeds that on the far side; a moment aligned with the field feels a residual force $\vec F=\nabla(\vec\mu\cdot\vec B)$ toward the stronger-field region.
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A force experiment constrains orientation before interpreting a gradient result.
Clamp the loop at a stated angle, measure the force along one axis, and reverse
current to isolate the magnetic component. A torque experiment constrains
translation, locates the pivot, and measures the angular response. If both degrees
of freedom are released, the loop can translate and rotate until the measured field
and the loop orientation both change; a static reading then represents the combined
mechanical equilibrium rather than either simple formula alone.

The small-loop approximation has a clear limit. Let $\ell$ be a characteristic loop
size and let $L_B=B/|\nabla B|$ be the local field-variation scale. The dipole force
approximation requires $\ell/L_B\ll1$. When that ratio is not small, map the field
over the full loop and integrate the segment forces. Comparing the integral with the
dipole prediction as the loop is translated tests the range in
which the moment model is adequate.

**Area orientation and the magnetic-moment vector.**

For any planar loop, current circulation fixes the orientation of the enclosed area.
The area vector can be written without selecting a particular rectangular shape:

$$
\vec A=\frac12\oint\vec r\times \d \vec\ell,
\qquad
\vec\mu=I\vec A.
$$

The path is traversed in the conventional-current direction. Reversing that direction
reverses $\d \vec\ell$, $\vec A$, and $\vec\mu$. For a planar loop,
the magnitude of this vector is the ordinary geometric area and its direction is the
right-hand normal. The integral form applies to irregular outlines because it does
not require decomposing the path into rectangles.

The origin used in the area integral does not affect a closed loop. Shifting every
position vector by a constant vector adds a term proportional to
$\oint \d \vec\ell$, which is zero. This origin independence is the geometric
counterpart of a closed circuit: the moment describes the loop as a whole, not one
chosen side or one arbitrary starting point.

In a coil of $N$ turns, add the vector area of every turn with its actual current
sense. Closely wound, coplanar turns give $\vec\mu=NI\vec A$. A turn wound
in the opposite direction subtracts. A multi-layer coil in which the planes differ
slightly has a vector sum rather than a scalar turn count times one area. This
distinction matters when a coil is mounted with a visible tilt or when its terminal
connections route a turn in the opposite sense.

The magnetic moment is an orientation variable, not a mechanical axis. The loop can
be translated without changing $\vec\mu$ when its shape and current remain
fixed. It can also be rotated about a line through its centre without changing the
moment magnitude. The applied field responds to the vector orientation through the
dot product for energy and the cross product for torque.

**Torque-vector geometry and precession direction.**

The torque vector is

$$
\vec\tau=\vec\mu\times\vec B.
$$

It is perpendicular to both $\vec\mu$ and $\vec B$. Its magnitude gives
the initial rotational tendency about the indicated axis; its direction follows the
right-hand order from moment to field. Reversing the order in the cross product gives
the opposite axis and is therefore not a harmless notation change.

If the loop has angular momentum $\vec L$, rotational dynamics gives
$\d \vec L/\d t=\vec\tau$. A torque perpendicular to $\vec L$ changes its
direction without changing its magnitude at that instant. When a loop carries
substantial angular momentum, this geometry can produce precession about the applied
field direction. The precession rate depends on the loop's angular momentum, inertia,
and constraints, so it cannot be inferred from $\mu B\sin\theta$ alone.

$$
% caption: Torque-vector geometry for a moment tilted from the field, $\vec\tau=\vec\mu\times\vec B$. The torque is perpendicular to the plane of moment and field; with moment and field in the page and the right-hand order running from moment to field, the torque points out of the page, drawn here as a dot. It sets the direction in which angular momentum changes.
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A loop initially parallel or antiparallel to the field has zero torque. These two
orientations are geometrically alike in the cross product but mechanically distinct
in energy. At a perpendicular orientation, the torque magnitude is maximal, yet the
direction of subsequent motion still depends on the pivot, the current source, the
mass distribution, and any mechanical restoring element. The vector formula gives
the magnetic contribution, not a complete motion history.

Use a coordinate sign check. Let the moment point along positive $x$
and the field along positive $z$. Then
$\vec\tau=\hat\imath\times\hat k=-\hat\jmath$.
The corresponding rotation sense should be drawn only after the viewing direction is
specified. Labels such as clockwise and counterclockwise are incomplete without that
viewing axis.

**Energy as a function of orientation.**

For fixed current and an externally maintained field, magnetic potential energy is

$$
U(\theta)=-\mu B\cos\theta.
$$

The energy difference between two orientations is independent of the path used for a
slow rotation:

$$
U(\theta_f)-U(\theta_i)
=\mu B(\cos\theta_i-\cos\theta_f).
$$

The parallel orientation has the lowest energy and the antiparallel orientation the
highest. Near alignment, expanding the cosine gives

$$
U(\theta)\approx-\mu B+\frac12\mu B\theta^2.
$$

The coefficient $\mu B$ is the angular curvature of the energy minimum. A small
angular displacement therefore produces a restoring magnetic torque
$\tau_\theta\approx-\mu B\theta$ when the signed coordinate is measured from
alignment.

Energy measurements require a stated electrical condition. A current-regulated source
maintains $I$ while the loop moves and exchanges electrical energy as needed. A loop
connected to a passive circuit can change current during rotation, in which case
$-\vec\mu\cdot\vec B$ alone is not the complete system energy. In a torque
calibration, hold current fixed and report the source condition with the angle data.

The energy slope and the torque provide separate consistency checks. A measured
energy curve may be differentiated to obtain torque, while a measured torque curve
may be integrated to obtain energy differences. Agreement is strongest when the
angle is swept slowly in both directions. A difference between increasing-angle and
decreasing-angle data indicates pivot friction or magnetic hysteresis in nearby
materials, not two different magnetic potential energies.

## Force, torque, and calibration

A torsion balance provides a direct calibration of magnetic torque. Set the fibre
zero at $I=0$, apply a field of known direction, and measure the equilibrium
deflection $\phi$ for a sequence of currents and initial orientations. With a
calibrated torsional constant $\kappa$, the measured torque is

$$
\tau_{\rm meas}=\kappa(\phi-\phi_0).
$$

For each orientation, reverse current and form

$$
\tau_{\rm odd}=\frac{\tau(+I)-\tau(-I)}{2}.
$$

This subtraction rejects a fixed gravitational torque and a stable fibre offset.
Field reversal gives the same magnetic sign change. Interleave the reversals rather
than collecting every positive-current point first, because fibre creep can imitate a
small torque intercept.

At constant $B$, a plot of $\tau_{\rm odd}$ against $I\sin\theta$ has slope $NA B$
for a coplanar $N$-turn coil. At constant current, a plot against $\sin\theta$ has
slope $\mu B$. Use the angle between the measured field direction and the area
normal, not an angle drawn from the plane of the loop. Confusing those complementary
angles shifts the sine and cosine factors and can reverse the apparent stability
classification near the endpoints.

The calibration uncertainty includes the torsional constant, angular readout, field
magnitude, current, and the effective area of the turns. A fit residual that changes
with current but not with angle suggests a source or current-measurement error. A
residual that changes with angle can indicate an offset pivot axis, a field gradient
across the coil, or a turn area that differs from the assumed geometry. Report the
current direction, field direction, angle zero, pivot axis, and reversal order beside
the fitted slope.


**Separating angular and translational responses.**

A single apparatus can contain both a pivot and a force sensor, but the two readings
must be interpreted with different constraints. To measure torque, lock the loop
centre in position, specify the pivot axis, and calibrate the angular restoring
element. To measure translation, clamp the loop orientation, specify the sensing
axis, and verify that the support does not transmit an unmeasured angular load into
the force channel. A loop that is free in both coordinates generally changes angle
and position together until the fibre, spring, or guide balances the magnetic loads.
Its final position cannot be substituted directly into a fixed-orientation force
formula.

Use a reversal matrix to check the interpretation. With geometry held fixed, record
the angular and force readings for $(+I,+B)$, $(-I,+B)$, $(+I,-B)$, and $(-I,-B)$.
Both magnetic responses reverse when one of the two signs reverses and return to the
original sign when both reverse. A gravity-induced deflection, a fixed support bias,
or a sensor offset does not follow this pattern. Repeat each setting after returning
to a zero-current reference. The reference sequence exposes drift in the pivot zero
and balance zero before it is mistaken for a small magnetic signal.

The mechanical calibration sets the measurement range. If the torsion
constant is $kappa$ and angle uncertainty is $u(\phi)$, the angular contribution to
torque uncertainty is approximately $\kappa u(\phi)$ before including uncertainty in
$\kappa$. A force sensor similarly has a zero uncertainty and a scale uncertainty.
Choose current and field values that give signals above these uncertainties without
moving the loop far enough to alter the field map. Plot angular response against
$IB$ in a uniform-field test and translational response against the mapped gradient
in a constrained-orientation test. Separate slopes and residual plots make a mixed
force-and-torque response visible.


**Magnetic-dipole force from an energy gradient.**

With a small loop held at fixed orientation, position dependence of magnetic energy
gives the net force. Along coordinate $x$,

$$
F_x=-\frac{\partial U}{\partial x}
=\frac{\partial}{\partial x}(\vec\mu\cdot\vec B).
$$

The derivative is evaluated with the moment orientation fixed during the virtual
displacement. If the moment is aligned with a field that grows toward positive $x$,
the energy becomes more negative in that direction and the force is positive.
Reversing current reverses $\vec\mu$ and the force. An antiparallel loop is
pushed toward weaker field under the same orientation constraint.

In vector form,

$$
\vec F=\nabla(\vec\mu\cdot\vec B).
$$

This is a dipole approximation. With loop scale $\ell$ and local field scale
$L_B=B/|\nabla B|$, it requires $\ell/L_B\ll1$. When the loop spans a substantial
fraction of the gradient, one field value and one gradient do not represent every
segment. Use the mapped field in the full segment-force integral instead. The
magnetic moment can remain well defined even when this point-dipole force estimate
is no longer accurate.

A prescribed current source and a constrained angle are part of the model. If the
loop rotates while it translates, both $\vec\mu\cdot\vec B$ and its spatial
derivative change. A force sensor then reads the equilibrium of the magnetic force
and the angular support, not the fixed-orientation dipole force. Clamp the loop, or
measure its angle and include it in the energy calculation.

**Net force and torque as separate observables.**

Net force and torque arise from different sums of the same segment forces. A complete
loop in a uniform field has zero net force but can have nonzero torque. In a gradient,
opposite sides can have unequal force magnitudes, producing translation even when the
moment is parallel to the local field and magnetic torque is zero. These limiting
cases separate the two observables experimentally.

A translation experiment constrains the loop angle and places the force sensor on a
known axis. A torque experiment locks the loop centre and measures angular deflection
about a stated pivot. The supports need not be identical. A rigid guide can transmit
an angular reaction that is absent from a low-friction pivot; a pivot can transmit a
force reaction that is absent from a suspended balance. Record the mechanical
constraint with each magnetic reading.

> **Worked example (Force on a coil in a field gradient).** A $120$-turn coil of area
> $A=4.0\times10^{-4}\ \mathrm{m^2}$ carries $I=0.50\ \mathrm{A}$. Its magnetic moment
> is
>
> $$
> \mu=NIA=(120)(0.50)(4.0\times10^{-4})=2.4\times10^{-2}\ \mathrm{A\,m^2}.
> $$
>
> Hold the moment parallel to the local field in a measured axial gradient
> $G=\partial B/\partial x=3.0\ \mathrm{T\,m^{-1}}$. With $\vec\mu$ aligned along the
> gradient, the net force is
>
> $$
> F_x=\mu G=(2.4\times10^{-2})(3.0)=7.2\times10^{-2}\ \mathrm{N}.
> $$
>
> A uniform field would give zero net force here; only the gradient produces one. The
> force scales with the turn count, so the same coil rewound with more turns at fixed
> current gains moment and force in proportion.

Mount the coil on a balance with positive $x$ toward stronger field. At equal current
magnitudes, form the reversal-isolated reading

$$
F_{\rm odd}=\frac{S(+I)-S(-I)}{2}.
$$

Its sign should agree with the predicted direction. The average relative to the
zero-current reading identifies weight offsets and other even-in-current backgrounds.
Repeat after reversing the gradient direction. The magnetic force reverses, whereas a
fixed balance bias does not.

For independent uncertainties in turn count, area, current, and gradient,

$$
\left(\frac{u(F)}{F}\right)^2=
\left(\frac{u(N)}{N}\right)^2+
\left(\frac{u(A)}{A}\right)^2+
\left(\frac{u(I)}{I}\right)^2+
\left(\frac{u(G)}{G}\right)^2+
\left(\frac{u_0}{F}\right)^2.
$$

The zero-force uncertainty $u_0$ dominates at low current. A measured slope that
changes as the coil is translated indicates that the gradient is not constant over
the travel or that the loop is no longer small relative to the field scale. Replace
the point-gradient model with a mapped-field force integral before assigning a
magnetic moment from that calibration.

**Calibration protocol and limit checks.**

The force balance should be calibrated with known mechanical loads along the same
axis used for the coil. Record a zero before and after every current-reversal pair.
The two zero readings bound sensor drift during the magnetic measurement. If their
difference exceeds the repeatability specification, use shorter reversal sequences,
increase settling time after current changes, or include a drift correction with its
uncertainty. A single tare measured at the start of a long run is not an adequate
reference for a balance whose zero changes with temperature or lead tension.

The field gradient requires its own calibration. Measure $B(x)$ at several positions
along the intended coil travel and obtain $G$ from a local fit over the region that
the coil occupies. The gradient at the geometric centre is appropriate only when the
field is close to linear across the coil. Mark the centre position from a physical
reference on the mount, not from a visually estimated magnet edge. Repeat the force
reading after a small deliberate displacement. The measured change should agree with
the change predicted from the local field map; disagreement identifies a position
error, a nonuniform coil moment, or a field map that is too coarse.

For the worked coil, a force-versus-current fit has slope $NAG$ when orientation and
gradient remain fixed. Fit positive and negative currents together after reversal
subtraction, then inspect residuals against current, position, and time. Curvature
with current can arise from heating that moves the coil or changes a support preload.
Curvature with position indicates a changing gradient. A residual that changes sign
under current reversal but not under gradient reversal points to a lead force rather
than the constrained coil. These tests locate the missing part of a model before a
new value of magnetic moment is inferred from the slope.

Three limits provide fast checks. With $I=0$, the reference-subtracted magnetic
force must vanish. With the gradient reversed, the constrained-loop force must
reverse. In a locally uniform region, the force must approach zero even though a
misaligned loop can still have a torque. Reporting the orientation constraint with
each limit is essential: a loop allowed to turn may change its moment direction as
the gradient is reversed and conceal the expected force sign change.

Use separate mechanical checks for the coil mount and the current leads. Flexing a
lead while current is off indicates a support-load change that reversal subtraction
cannot remove. Secure the lead route before the gradient sweep and repeat one
midrange point after every adjustment. That control distinguishes a changing magnetic
force from a changing mechanical preload.


**Torque measurement at fixed orientation.**

A torque measurement must suppress translation and identify one rotation axis. Mount the
loop on a low-friction pivot or torsion fibre, centre the loop in a uniform applied
field, and keep the field region large compared with the loop. The pivot axis defines
the measured component of magnetic torque. A loop mounted on a sloping or displaced
axis can still experience the same magnetic torque vector, but the instrument records
only its projection onto the permitted rotation.

Let $\phi$ be the signed angular deflection of a calibrated torsion fibre from its
zero-current reference. With torsional constant $\kappa$, the mechanical torque is

$$
\tau_{\rm fibre}=-\kappa\phi.
$$

At static equilibrium it balances the corresponding magnetic component. If the
initial setting makes angle $\theta_0$ between moment and field, then the loop
deflection changes that angle to $\theta_0+\phi$ or $\theta_0-\phi$, according to
the chosen axis convention. The sign relation must be established from the apparatus
rather than inferred from a sketch. For small deflections, setting the loop near a
chosen $\theta_0$ permits a local measurement of $\mu B\sin\theta_0$.

Calibrate $\kappa$ mechanically, using a known small torque or a torsional-period
measurement with a known inertia. The magnetic field should not be used to define the
same calibration constant later tested by the experiment. Measure the fibre zero
before and after a current-reversal pair. Half the difference of the two inferred
torques removes a fixed gravity torque and a stable pivot bias:

$$
\tau_{\rm odd}=\frac{\tau(+I)-\tau(-I)}{2}.
$$

The current source must remain constant while each reading settles. A current sweep
during the mechanical relaxation changes the magnetic torque and creates an apparent
lag that is not a property of the equilibrium torque curve.

**Energy curvature and orientation stability.**

For fixed current in a uniform field, the magnetic energy is

$$
U(\theta)=-\mu B\cos\theta.
$$

Stability follows from the local curvature rather than from the fact that the torque
is zero. At the parallel orientation, the second derivative is
$\d ^2U/\d \theta^2=\mu B>0$, so a small displacement raises the energy and produces a
restoring torque. At the antiparallel orientation, the curvature is negative. A small
displacement lowers the energy and the magnetic torque drives the loop farther from
that setting.

A direct stability test uses small, controlled angular displacements about each
zero-torque orientation. Release the loop from a nearby angle and record the initial
direction of the restoring motion. At the stable setting, the loop returns toward the
reference angle. At the unstable setting, it departs from it. Air drag and fibre
torsion affect the later time history, but the initial magnetic tendency is fixed by
the energy slope.

Current reversal reverses $\vec\mu$ and exchanges the two stability
classifications. The orientation stable for positive current becomes unstable for
negative current in the same applied field. This sign swap is a strong experimental
check because gravitational bias and fixed fibre asymmetry do not reverse with
current.

The energy difference can be measured without relying on a drawn energy curve.
Quasistatically rotate the loop through a sequence of angles and integrate the
calibrated opposing torque. For a reversible path, the required mechanical work
equals the increase in magnetic potential energy. Compare the measured work between
two angles with $\mu B(\cos\theta_i-\cos\theta_f)$. A difference between clockwise and
counterclockwise sweeps quantifies friction or hysteresis and should be reported as a
systematic effect.

**Angular torque calibration.**

With a loop held at several prescribed angles in a uniform field, the reversal-isolated
torque obeys

$$
\tau_{\rm odd}=\mu B\sin\theta.
$$

Plotting the measured torque against $\sin\theta$ gives a straight line through the
origin when the angle zero, field direction, and torsion calibration are correct. The
slope is $\mu B$. Repeat at a second current magnitude. Since $\mu=NIA$, the slope
should scale linearly with current while the field and coil geometry are unchanged.

Do not fit the magnitude of torque alone across positive and negative angles. The
signed angle and signed torque carry the orientation information required by the
cross product. A magnitude-only plot can hide a reversed angle convention or a pivot
that measures the opposite reaction torque. State whether the reported torque is the
torque on the loop or the torque transmitted to its support.

The uncertainty in each torque point includes fibre calibration, angular readout,
current, field magnitude, and zero drift. A residual odd in current but even in angle
often indicates an angle-zero error. A residual that grows with current at all angles
can indicate current heating or a field source whose output changes with load. Check
the uniformity of the applied field over the loop before attributing curvature in the
plot to a failure of the moment model.


**Measurement uncertainty and energy checks.**

The torsion calibration contributes directly to magnetic-torque uncertainty. If the
inferred torque is $\tau=\kappa\phi$, independent small uncertainties give

$$
u(\tau)=\sqrt{[\phi\,u(\kappa)]^2+[\kappa\,u(\phi)]^2+u_0^2},
$$

where $u_0$ represents the zero and repeatability contribution. The angular term is
often limiting near aligned orientations because magnetic torque is small there. Use
angles that span a substantial range of $\sin\theta$, but avoid mechanical stops,
lead contact, or a field region whose direction changes across the loop. A larger
current improves the signal only while the source remains stable and coil heating does
not alter the geometry or torsion zero.

Energy data provide an independent calibration check. Integrate the
reversal-isolated torque over a measured angle interval and compare the result with
the energy change computed from the fitted $\mu B$. The two routes use different
numerical operations: the torque fit tests local slope, whereas the integral tests
accumulated work. A constant torque offset appears in the integral as an error
proportional to angle interval. This comparison is more sensitive to a drifting zero
than one isolated angle reading.

Repeat the angle sequence in both directions. The mean of the two work estimates is
the reversible component when friction is approximately symmetric, and half their
difference estimates the hysteretic loss. Report this separation rather than folding
the loss into magnetic potential energy. A stable energy minimum should yield a
positive fitted curvature within uncertainty; a negative curvature at the aligned
setting indicates a reversed angle, current, or field convention in the analysis.


## Circular-loop axial fields

In a circular loop of radius $R$ carrying steady current $I$, the magnetic field on
the axis through the centre is parallel to that axis. For $N$ closely spaced turns,
the axial component at signed distance $z$ from the loop plane is

$$
B_z(z)=\frac{\mu_0NIR^2}{2(R^2+z^2)^{3/2}}.
$$

The sign is set by the chosen axial direction and the current circulation. Curl the
right-hand fingers with conventional current; the thumb gives the positive field
direction on the axis. The displayed magnitude assumes that the turns have nearly the
same radius and occupy a thickness small compared with the scan distances. A winding
with substantial axial depth is represented more accurately as a sum of displaced
loops.

$$
% caption: A circular current loop and its on-axis field. The coordinate $z$ runs from the loop plane along the symmetry axis; the current sense sets the field direction on the axis through the right-hand rule.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (-2.4,0)--(3.4,0) node[right] {$z$};
  \draw[black,very thick] (0,0) ellipse (0.5 and 1.4);
  \draw[->,very thick] (0.17,1.32) arc[start angle=70,end angle=300,x radius=0.5,y radius=1.4];
  \filldraw[black] (0,0) circle (1.6pt);
  \node at (0,-1.75) {current};
  \draw[->,acc!70!black,very thick] (2.1,0)--(3.0,0);
  \node[acc!70!black,above] at (2.55,0.06) {$B$};
\end{tikzpicture}
$$

The axial formula uses the loop symmetry directly. At a point on the axis, transverse
contributions from opposite current elements cancel, while axial contributions add.
Away from the axis, the field has radial as well as axial components and requires a
different expression. An axial scan should therefore align the probe with the loop
centre before comparing data with this one-dimensional formula.

Dimensional checks are immediate. The factor $\mu_0NI/R$ has units of tesla, and the
remaining function depends only on $z/R$. Thus geometrically similar loops produce
the same normalized axial profile when positions are measured in radii. The field
scales linearly with turn count and current as long as the winding geometry and source
current remain unchanged.

**Centre and remote limits.**

At the centre of the loop,

$$
B_0=B_z(0)=\frac{\mu_0NI}{2R}.
$$

Near the centre, the normalized field has the expansion

$$
\frac{B_z(z)}{B_0}
=\left(1+\frac{z^2}{R^2}\right)^{-3/2}
\approx1-\frac32\frac{z^2}{R^2}.
$$

The first axial variation is quadratic, so a short central scan has a broad maximum
rather than a linear slope. This property makes a single circular loop suitable for
a centre-field calibration but not for a long uniform-field region. A probe displaced
slightly from the centre can still measure nearly the central field, whereas a larger
offset produces a predictable reduction.

For $|z|\gg R$,

$$
B_z(z)\approx\frac{\mu_0NIR^2}{2|z|^3}
=\frac{\mu_0\mu}{2\pi|z|^3},
\qquad
\mu=NI\pi R^2.
$$

The remote field falls as the inverse cube of distance. A log-log scan should
therefore approach slope $-3$ only after the measurement distance is several loop
radii and the background field is small compared with the loop field. Fitting a
power law too near the coil gives an exponent that is not yet in the remote limit.

$$
% caption: Normalized on-axis field of a circular loop, $B_z/B_0=(1+z^2/R^2)^{-3/2}$. The profile is flat near the centre, varying only quadratically, and falls toward an inverse-cube tail far from the loop, so central and remote scans probe different limits.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (-3.3,0)--(3.4,0) node[right] {axial position};
  \draw[->,black] (0,-0.1)--(0,3.0) node[above] {$B$ ratio};
  \draw[acc,very thick] plot[domain=-3.0:3.0,samples=160] ({\x},{2.6/pow(1+pow(\x/1.15,2),1.5)});
  \filldraw[draw=black,fill=black!8] (0,2.6) circle (2pt);
  \node[black,above right] at (0.12,2.62) {centre};
  \node[black] at (2.4,0.72) {tail};
\end{tikzpicture}
$$

The sign of the axial field does not change across the loop plane for one fixed
current sense; only the coordinate $z$ changes sign. A measured sign reversal near
the centre usually indicates a reversed probe axis, a reversed current connection, or
subtraction of a background field with an incorrect sign. Record the probe orientation
and the current direction before using a signed scan to infer the loop moment.

**Calibrated axial scan.**

Use a nonmagnetic translation stage to place a calibrated axial field probe at known
positions. Centre the stage mechanically on the loop plane, then refine the origin by
locating the maximum of the reversal-isolated signal. At every position record probe
output for $+I$, $-I$, and zero current. If $C$ is the probe conversion factor, form

$$
B_{\rm odd}(z)=\frac{S(+I,z)-S(-I,z)}{2C}.
$$

This removes a stable probe offset and much of the ambient background. The zero-current
reading detects drift. Interleave the positions rather than
making one outward scan only; a return scan can expose source heating or stage backlash.

> **Worked example (Axial field of a current loop).** A coil of $N=200$ turns, radius
> $R=0.10\ \mathrm{m}$, carries $I=0.25\ \mathrm{A}$. The field at the centre, on the
> axis, is
>
> $$
> B_0=\frac{\mu_0NI}{2R}=\frac{(4\pi\times10^{-7})(200)(0.25)}{2(0.10)}
> =3.14\times10^{-4}\ \mathrm{T}.
> $$
>
> One radius off-centre, at $z=R$, the axial field falls by
> $(1+z^2/R^2)^{-3/2}=2^{-3/2}=0.354$, to
>
> $$
> B(R)=0.354\,B_0=1.11\times10^{-4}\ \mathrm{T}.
> $$
>
> These two points pin the current and position scales before a full profile is fit.
> Far away, $z\gg R$, the same expression collapses to the dipole field
> $B\to\mu_0\mu/(2\pi z^3)$ with $\mu=NI\pi R^2$.

Fit the data to the axial expression with a separate constant background only when
the zero-current scan supports one. Residuals symmetric about the fitted centre can
indicate an incorrect radius or turn count. Residuals that differ on the two sides of
the loop indicate a displaced probe axis, an asymmetric winding, or a nearby magnetic
object. Report the fitted centre position, current, radius convention, probe axis,
and uncertainty of the conversion factor with the scan.


**Scan quality and uncertainty.**

Probe alignment sets the meaning of an axial measurement. A probe axis tilted by angle
$\alpha$ records the axial field multiplied by $\cos\alpha$ together with any radial
component projected onto the probe. Near the loop centre the radial component is small
on the symmetry axis, but an off-axis stage path introduces both position and
orientation errors. Reverse the stage direction and compare the two scans. A
difference at the same nominal coordinate indicates backlash, probe rotation, or a
changing current source.

The current monitor and probe conversion factor contribute separate scale
uncertainties. For a centre-field comparison, independent fractional contributions
can be summarized as

$$
\left(\frac{u(B_0)}{B_0}\right)^2=
\left(\frac{u(N)}{N}\right)^2+
\left(\frac{u(I)}{I}\right)^2+
\left(\frac{u(R)}{R}\right)^2+
\left(\frac{u(C)}{C}\right)^2.
$$

Position uncertainty has little first-order effect at the centre because the
axial profile is flat there, but it becomes important on the flanks of the scan. Use
the fitted centre as a parameter with uncertainty rather than forcing the mechanical
zero to be exact. A systematic shift of all points may be absorbed by the fitted
centre; asymmetric residuals cannot.

Ambient fields are best handled by current reversal at each position. Subtracting one
background measured only before the scan assumes that ambient conditions remain
constant. Reversal pairs detect slow offset drift and reject a stable Earth-field
component without requiring its numerical value. Average repeated pairs, retain their
scatter, and report whether the plotted points are individual readings or means.
Those details determine whether apparent departures from the axial formula are
statistically significant.