Ampère’s Law

Magnetic circulation and orientation

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

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

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

The permeability of free space is

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

Orientation and signed current

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

For several conductors,

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

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

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

Cylindrical current distributions

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

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

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

Thus the magnitude outside an ideal thin wire is

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

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

Field inside a uniformly conducting cylinder

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

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

Substitution into the circulation equation gives

The interior field grows linearly from zero at the centre to the surface value. Outside the metal, every Amperian circle encloses the whole current, recovering $B(r)={\mu }_{0}I/(2\pi r)$. The two branches meet at $r=a$:

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

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

Scope of the enclosed-current result

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

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

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

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

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

Calculation checks

A long-wire calculation has several quick checks. Magnetic field scales linearly with current, so doubling $I$ doubles $B$. It scales inversely with perpendicular distance outside the conductor, so a point twice as far from the axis has half the field magnitude. The dimensions of ${\mu }_{0}I/(2\pi r)$ reduce to tesla. The direction must be tangent to a circle about the current, never radial away from the wire.

Solenoids and toroids

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

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

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

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

Ampère’s law gives

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

The result contains no explicit radius because the ideal argument needs only the number of turns crossed per unit axial length. Radius controls how rapidly the finite winding approaches the long-solenoid approximation and how broad a region around the axis has a nearly uniform field. It also enters the exact on-axis field of a finite solenoid.

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

Turn density carries units of inverse length, so ${\mu }_{0}nI$ has units

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

Direction follows the current around the turns. Viewed from one end, a counterclockwise conventional current gives an axial magnetic field directed toward that viewer; a clockwise current gives the opposite axial direction. Describing the end from which the winding is viewed prevents an ambiguous right-hand-rule statement. A signed field component $B_z$ requires an explicitly chosen positive axial direction, whereas the formula $B={\mu }_{0}nI$ gives the magnitude.

Finite-solenoid axial scan

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

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

The factor multiplying ${\mu }_{0}nI$ approaches one only when $L/R$ is large. At either end, the corresponding axial value is

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

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

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

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

Toroid: closed magnetic path

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

The toroid and long solenoid share the same local circulation logic: a path parallel to $\vec{B}$ encloses a count of turns times current. Their symmetry consequences differ. The long solenoid has translational symmetry along its axis, so the ideal interior magnitude is independent of axial position. A toroid has rotational symmetry, so the magnitude is constant on one circle but varies between circles as $1/r$. A thin toroid, where $b-a$ is much smaller than its mean radius, has only a small variation across its winding.

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

Coaxial conductors and current profiles

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

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

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

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

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

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

Piecewise current-density profiles

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

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

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

and the enclosed current is

The outer formula is the total current. A negative $J_2$ can represent a distributed return current, while a positive $J_2$ increases the total current. The magnetic field follows by dividing each branch by $2\pi r$ and multiplying by ${\mu }_0$. Boundary values must agree when the density is finite; a jump in current density changes the slope of $B_{\varphi }(r)$, not the field value itself. Check the piecewise result against a smooth profile. Let

Its total current is $I=\pi J_0{a}^2/2$. Integrating to an intermediate radius gives

At $r=a$, the interior branch equals ${\mu }_{0}I/(2\pi a)$ and joins the exterior inverse-radius result. Near the axis, the leading term makes $B_{\varphi }$ increase linearly with $r$. The shape of $B_{\varphi }(r)$ therefore carries information about both the radial current distribution and the total current.

Inferring current density from a radial field profile

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

Differentiation and the annular-area relation $\mathbf{d}{I}_{\mathrm{enc}}/\mathbf{d}r=2\pi r{J}_z(r)$ give

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

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

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

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

Symmetry and loop selection

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

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

Large angular residuals identify a symmetry failure rather than a failure of Ampère’s law. An off-centre inner conductor, an eccentric outer conductor, uneven current density, asymmetric lead routing, or a sensor that is not tangent to the circle each changes the readings. The circulation equation still relates the complete signed line integral to the enclosed current, but the field magnitude can no longer be factored outside the integral.

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

Selecting an Amperian loop

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

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

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

A compact loop-selection record avoids several recurrent errors.

1. State the source idealization and the observation region. “Infinite straight conductor,” “long tightly wound solenoid far from an end,” and “toroid between radii $a$ and $b$” specify different symmetry conditions.
2. Write the surviving field direction before choosing a path. A long wire has an azimuthal direction, a long solenoid has an axial central direction, and a toroid has an azimuthal direction around its central axis.
3. Mark every path segment as parallel, perpendicular, or variable relative to the field. Variable segments remain inside the integral. A closed curve can enclose the correct current and still be algebraically unhelpful.
4. Choose an oriented spanning surface. Record every crossing current with its sign. A return conductor outside a selected path contributes no current through that surface; a return conductor crossed by the surface contributes with its algebraic direction.
5. Check the result against dimensions, limiting radii, and a direct physical symmetry test. A circular path around a long wire must give $B\propto 1/r$. A central long-solenoid path must give a result independent of its chosen axial length.

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

Ampère and Biot–Savart calculations

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

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

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

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

As the segment extends indefinitely in both directions, $u_1\to -\pi /2$ and $u_2\to +\pi /2$, recovering $B={\mu }_{0}I/(2\pi R)$. At finite length, the endpoint angles remain in the answer. A circular boundary around a finite segment may enclose a current in the sense of a complete circuit, but its tangential field is neither constant nor generally aligned with the boundary. Ampère’s law retains its circulation statement; the line integral cannot be reduced to $B(2\pi R)$.

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

Field mapping and design checks

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

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

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

For independent small uncertainties in current and sensor radius,

With $u_I=0.010\mathrm{A}$ and $u_r=0.050\mathrm{mm}$, the relative uncertainty is $2.52\%$, giving $B=0.300\pm 0.008\mathrm{mT}$. The radial position dominates the stated uncertainty because the field varies as $1/r$.

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

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

Field mapping and boundary checks

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

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

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

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

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

Coaxial field map

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

Validation record

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

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

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

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