Magnetic Dipoles

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

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

The unit is $\mathrm{A}{\mathrm{m}}^2$, which is equivalent to $J/T$. 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.

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

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

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.

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.

Potential energy and equilibrium.

The orientation-dependent potential energy is

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:

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.

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

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:

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:

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

Thus the distributed-force calculation gives

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 $\varphi$ is the signed deflection from the fibre's zero, static balance is

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,

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

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

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

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.

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 $\ell /L_B\ll 1$. 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:

The path is traversed in the conventional-current direction. Reversing that direction reverses $\mathbf{d}\vec{\ell }$, $\vec{A}$, 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 $\oint \mathbf{d}\vec{\ell }$, 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

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 $\mathbf{d}\vec{L}/\mathbf{d}t=\vec{\tau }$. 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.

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 $\vec{\tau }=\hat{ı}\times \hat{k}=-\hat{ȷ}$. 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

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

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

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 $\varphi$ for a sequence of currents and initial orientations. With a calibrated torsional constant $\kappa$, the measured torque is

For each orientation, reverse current and form

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 ${\tau }_{\mathrm{odd}}$ against $I\sin \theta$ has slope $NAB$ 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 $u(\varphi )$, the angular contribution to torque uncertainty is approximately $\kappa u(\varphi )$ 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 $x$,

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,

This is a dipole approximation. With loop scale $\ell$ and local field scale $L_B=B/|\nabla B|$, it requires $\ell /L_B\ll 1$. 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.

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

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,

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

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 ${\theta }_0+\varphi$ or ${\theta }_0-\varphi$, 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:

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

Stability follows from the local curvature rather than from the fact that the torque is zero. At the parallel orientation, the second derivative is ${\mathbf{d}}^{2}U/\mathbf{d}{\theta }^2=\mu B>0$, 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

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 $\tau =\kappa \varphi$, independent small uncertainties give

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

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.

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 ${\mu }_{0}NI/R$ 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,

Near the centre, the normalized field has the expansion

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 $|z|\gg R$,

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.

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

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.

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

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.