Magnetic Field of Steady Current Configurations

The Biot–Savart law is a machine for turning circuits into fields; this article runs the machine over the geometries civilization actually builds. Four shapes carry nearly all of applied magnetostatics. The straight wire — its circulating $\mu_0I/2\pi s$ was earned last article. The circular loop, whose axial field is the seed of everything coil-shaped, and whose far field reveals the universal signature of any compact current: the magnetic dipole. The solenoid — loops stacked into a tube — which manufactures the closest thing to a uniform field a laboratory can buy, from MRI bores to relay actuators. And the toroid — a solenoid bent to bite its own tail — whose field is entirely self-contained: strong in the doughnut, zero outside, the geometry fusion reactors bet on. The method matters as much as the catalog. Everything here comes from the Biot–Savart integral plus a heavy use of superposition: a solenoid is an integral over rings; a Helmholtz pair is two rings summed and tuned — separated by exactly one radius so that the second derivative of the field vanishes at the midpoint, buying a pocket of part-per-ten-thousand uniformity from bench hardware. Where a slicker route exists — the rectangular-loop and circulation arguments of Ampère's law — this article states the results and leaves the shortcut to its owners two articles ahead; nothing below needs it. By the end, the chapter's field-shape vocabulary is complete: circulation around lines, near-uniformity inside tubes, closure into loops everywhere, dipole tails at long range. The next two articles explain the grammar behind the vocabulary — why every one of these fields is divergence-free, and what their curl says about the currents that made them.

Key takeaways

The standard catalog, all from Biot–Savart + superposition: wire $B = \mu_0I/2\pi s$ (recalled); loop on-axis $B_z = \mu_0IR^2/2(R^2+z^2)^{3/2}$ ; solenoid interior $B = \mu_0nI$ ; toroid $B = \mu_0NI/2\pi s$ inside, zero outside.
The loop's far field is the magnetic dipole, $B \sim \mu_0m/2\pi z^3$ with $m = IA$ — the universal long-range signature of any compact current loop, from electrons to planets.
A finite solenoid, built honestly by stacking loop fields, gives $B = \tfrac{\mu_0nI}{2}(\cos\theta_1 - \cos\theta_2)$ on axis — the ideal $\mu_0nI$ inside, exactly half at the ends.
Helmholtz pair: two coaxial loops separated by their common radius $R$ kill both first and second derivatives at the midpoint — central field $(4/5)^{3/2}\mu_0NI/R$ , uniform to $\sim0.01\%$ within $R/10$ .
The toroid confines its field completely: any Ampèrian circle outside nets zero threaded current — the design logic of tokamaks and low-leakage inductors (full circulation arguments owned by Ampère's Law).
Field-shape grammar: circulation around currents, uniformity inside tubes, dipole tails far away, closed lines everywhere — the raw material for the divergence and curl theorems next.
KEY IDEA: four geometries — wire, loop, solenoid, toroid — generate the working vocabulary of magnetostatic fields, every one an exercise in Biot–Savart superposition, and every compact loop wearing the same $1/r^3$ dipole tail far away.

The core insight

Superposing Biot–Savart contributions over the standard source geometries yields the canonical fields of magnetostatics — $\mu_0I/2\pi s$ around a wire, $\mu_0IR^2/2(R^2+z^2)^{3/2}$ on a loop's axis, $\mu_0nI$ inside a solenoid, $\mu_0NI/2\pi s$ inside a toroid — with every compact loop reducing to a magnetic dipole $m = IA$ at long range.

Where this lives today

The catalog is the modern magnet industry's toolbox. MRI machines are superconducting solenoids (0.5–7 T bores) with correction coils Taylor-tuned on exactly the Helmholtz logic — kill derivatives order by order — to reach part-per-million uniformity over the imaging volume; gradient coils, anti-Helmholtz descendants, encode position into frequency. Every voice coil, relay, and inductor datasheet quotes $\mu_0nI$ -descended formulas. Helmholtz pairs (often tri-axial) surround navigation-sensor calibration rigs and null Earth's field in electron-microscope rooms; anti-Helmholtz pairs form the magneto-optical traps at the heart of every cold-atom lab and atomic clock. And the toroid's perfect confinement is a live frontier: a tokamak is a toroidal field coil set holding plasma on the doughnut's closed field lines — ITER's toroid runs $\sim5.3$ T at the plasma axis with the $1/s$ profile derived here setting the inboard/outboard field asymmetry every plasma-equilibrium code must handle. The dipole tail, meanwhile, is how nature files its magnetic paperwork. Earth's field is catalogued by its dipole moment ( $m \approx 8\times10^{22}\ \mathrm{A\,m^2}$ — a core-current loop by any other name); exoplanet radio searches, neutron-star spin-down models, and the magnetometry that maps ancient seafloor spreading all parse fields through the $1/r^3$ template of equation (2). When only the far field is measurable, the moment is the only number there is.

Origins · 1 minNikola Tesla and how this was discovered

Nikola Tesla

1856–1943 · Smiljan, Budapest, New York

Tesla's genius was precisely the subject of this article: field geometry as engineering material. His 1887–88 induction-motor patents replaced commutators with a trick of pure magnetostatics-in-motion — two coil sets driven out of phase to make the field direction itself rotate, dragging the rotor behind it — the insight that won the "war of the currents" for AC and still turns most of the world's industrial machinery. His resonant air-cored coil systems pushed coil design to extremes (the Tesla coil remains the emblem of high-voltage electricity), his Colorado Springs experiments probed wireless power through sheer field geometry, and dozens of his polyphase, generator, and radio-frequency patents furnished the grid's architecture. The SI honored the matchmaking of his name with the magnetic field itself in 1960: every $B$ quoted in this catalog — the millitesla of a bench solenoid, the tesla of an MRI bore — is measured in him.

History

1820

— Schweigger and Poggendorff multiply Ørsted's effect by winding the wire into turns — the coil is born as an instrument.

1825

— Sturgeon wraps current around iron: the electromagnet lifts twenty times its weight.

1850

s — precision labs adopt the matched coil pair; the uniformity condition d = R becomes standard equipment.

1887

— Tesla's rotating-field induction motor turns coil geometry into the engine of the AC age.

1913

— Kamerlingh Onnes proposes superconducting coils; the path to tesla-scale solenoids opens.

1977

— The first human MRI scans run inside large-bore solenoids — the loop formula, medically industrialized.

2020

— ITER's 18 toroidal-field coils begin assembly: the toroid's perfect confinement, at 5.3 tesla and fusion scale.

The physical picture

Learn the four field shapes before any formula, because they are the visual grammar of the whole subject. Around a straight wire: concentric circles, thinning as $1/s$ . Around a single loop: field threads through the hole and fountains around the outside — the pattern iron filings drew for Faraday, and at long range the exact portrait of a bar magnet. Inside a solenoid: the loops' fields reinforce down the bore into near-perfect straight rails, while outside they nearly cancel — a tube of concentrated, uniform field. In a toroid: the rails bent into closed circles that never leave the doughnut at all.

Figure

The four canonical fields — the visual grammar of magnetostatics

Two organizing facts run through the catalog. First, superposition is the manufacturing process: every coil is loops added together, and every formula below is the single-loop axial field reused — integrated into a solenoid, paired into a Helmholtz configuration, bent into a toroid. Master one ring and you own the catalog. Second, distance erases detail: back far enough away from any compact circuit — round loop, square loop, a wound coil, an orbiting electron — and the field collapses to one universal shape, the magnetic dipole $\vec m = I\vec A$ , falling as $1/r^3$ . The far universe cannot tell a current loop from a bar magnet, because (as the magnetic-matter chapter will argue, completing Ampère's old hypothesis) a bar magnet is current loops.

The catalog, earned

The straight wire's $\vec B = (\mu_0I/2\pi s)\hat{\boldsymbol\phi}$ was integrated honestly in Biot-Savart Law and is recalled, not re-derived. The next rung is the ring — the calculation that powers everything coil-shaped.

Figure

Loop-axis geometry: opposite elements cancel sideways, cooperate along the axis

Derivation AOn the axis of a circular loop — and the dipole tail

The goal is the field at height $z$ on the axis of a ring of radius $R$ carrying current $I$ : the master formula of coil design, with its two famous limits (center value, dipole tail). The strategy is direct Biot–Savart integration, with the ring's symmetry doing the heavy lifting: every element sits at the same distance from the field point, and the transverse field components cancel in pairs around the ring, so only the axial projection needs integrating — and it is constant.

Step 1. Geometry of one element. Every source point on the ring lies at distance

\mathscr{r} = \sqrt{R^2 + z^2}

from the axial field point, and $\mathrm{d}\vec\ell'$ (tangent to the ring) is exactly perpendicular to the separation vector, so each element contributes the full-strength magnitude

\mathrm{d}B = \frac{\mu_0 I}{4\pi}\,\frac{\mathrm{d}\ell'}{R^2 + z^2}.

Step 2. Kill the transverse components by symmetry. Each $\mathrm{d}\vec B$ is perpendicular to its own sight line, so it tilts off-axis — but diametrically opposite elements produce mirror-image tilts, and their transverse parts cancel exactly around the ring. Only the axial component survives, carrying the projection factor

\cos\alpha = \frac{R}{\sqrt{R^2+z^2}}.

Why the projection is $R/\mathscr{r}$ and not $z/\mathscr{r}$ : the element's field is perpendicular to the sight line, so its axial component goes with the sight line's transverse inclination — draw it once and the geometry locks in; getting this backwards is the classic error of the calculation.

Step 3. Integrate. The integrand is now constant around the ring:

B_z = \frac{\mu_0 I}{4\pi}\,\frac{R}{(R^2+z^2)^{3/2}}\oint\mathrm{d}\ell' = \frac{\mu_0 I}{4\pi}\,\frac{R\,(2\pi R)}{(R^2+z^2)^{3/2}},

\boxed{\;B_z = \frac{\mu_0 I R^2}{2\,(R^2+z^2)^{3/2}}\;} \qquad (1)

Step 4. Read the limits. At the center, $z = 0$ : $B = \mu_0I/2R$ — agreeing with the center-point calculation of the previous article ✓. Far away, $z \gg R$ :

B_z \approx \frac{\mu_0 I R^2}{2z^3} = \frac{\mu_0\,m}{2\pi z^3}, \qquad m = I\pi R^2 \qquad (2)

Why the $1/z^3$ : seen from afar, the ring's near and far sides nearly cancel — the residual after cancellation is two powers of $1/z$ steeper than the straight wire's $1/s$ . Equation (2) is the on-axis slice of the full dipole field $\vec B_{\rm dip} = \frac{\mu_0}{4\pi r^3}[3(\vec m\cdot\hat{\vec r})\hat{\vec r} - \vec m]$ , whose complete derivation belongs to the magnetic-dipole articles of the matter chapter; the moment $\vec m = I\vec A$ is the same one born as a torque coefficient in Magnetic Force on Current-Carrying Wires — one quantity, two jobs. *Quick checks: at $z=0$ the formula returns $\mu_0I/2R$ , consistent with the previous article's center-point result; the far-field coefficient reproduces the general dipole formula evaluated on-axis; and the dimension check $\mathrm{T} = \mathrm{T\,m/A \cdot A/m}$ confirms the prefactor ✓.*

Go deeper · Derivation B · 2 minThe Helmholtz condition — engineering uniformity from two rings

Derivation BThe Helmholtz condition — engineering uniformity from two rings

The goal is the classic uniform-field recipe: two identical coaxial $N$ -turn loops of radius $R$ , distance $d$ apart, tuned so the field near the midpoint is as flat as possible — and the discovery that the magic separation is exactly $d = R$ . The strategy is a Taylor expansion of the two-loop field about the midpoint: odd derivatives vanish by symmetry for free, and the design freedom in $d$ is spent killing the leading even derivative. This is a real taste of instrument design: symmetry first, then calculus on the remaining knob.

Step 1. Superpose two copies of (1) centred at $z = \pm d/2$ (field point at displacement $z$ from the midpoint):

B_z(z) = \frac{\mu_0NIR^2}{2}\left[\frac{1}{\left(R^2 + (z - d/2)^2\right)^{3/2}} + \frac{1}{\left(R^2 + (z + d/2)^2\right)^{3/2}}\right].

Step 2. Spend the symmetry. Under $z \to -z$ the two terms swap, so $B_z$ is an even function of $z$ : all odd derivatives vanish at the midpoint automatically. The expansion reads $B_z(z) = B_0 + \tfrac12B''_0z^2 + \mathcal{O}(z^4)$ , and flatness is decided by $B''_0$ .

Step 3. Kill the curvature. Differentiate (1) twice and evaluate at the midpoint (each loop contributes equally); the condition $B''_0 = 0$ works out to

d^2 = R^2 \qquad\Longrightarrow\qquad d = R.

Why it lands on exactly the radius: each loop's axial field has an inflection point at $z = R/2$ off its own plane (differentiate (1): $B_z'' \propto 4z^2 - R^2$ vanishes at $z = R/2$ ) — placing the midpoint at both loops' inflection points makes each contribute zero curvature there, and the pair's leading error jumps to fourth order.

Step 4. Evaluate the central field at $d = R$ . Each loop sits $R/2$ from the midpoint, so $R^2 + (R/2)^2 = \tfrac54R^2$ and

B_0 = 2\cdot\frac{\mu_0NIR^2}{2\,(\tfrac54R^2)^{3/2}} = \left(\frac{4}{5}\right)^{3/2}\frac{\mu_0NI}{R} \approx 0.7155\,\frac{\mu_0NI}{R} \qquad (3)

Why the configuration is beloved: with first, second AND (by symmetry) third derivatives all zero, the field is flat to $z^4$ — numerically, the exact two-loop sum deviates from $B_0$ by only $0.011\%$ at $z = R/10$ and $0.42\%$ at $z = R/4$ . Uniform-field volume, with open optical access from every direction that a solenoid would block: the standard environment for magnetometer calibration, atomic-physics experiments, and nulling Earth's field. (Flip one current and the pair becomes anti-Helmholtz: zero field at the center with a uniform gradient — the workhorse of magneto-optical traps.) *Checks: differentiating (1) twice confirms the inflection point at $z = R/2$ , underpinning the $d = R$ condition; the central value (3) follows directly; and the uniformity figures ( $-0.011\%$ at $R/10$ , $-0.42\%$ at $R/4$ ) come from the exact two-loop sum.*

Go deeper · Derivation C · 2 minThe finite solenoid — stacking rings into a tube

Derivation CThe finite solenoid — stacking rings into a tube

The goal is the on-axis field of a solenoid of finite length: $n$ turns per unit length, radius $R$ , carrying $I$ — built honestly by integrating the single-ring formula down the tube, with the ideal $\mu_0nI$ and the famous end-effect emerging as limits. The strategy converts the sum over discrete windings into an integral over ring-slabs and substitutes an angle variable that collapses the integrand; the payoff formula reads the field straight off the two angles under which the solenoid's ends are seen. (The slicker uniformity-and-exterior argument via circulation is the property of Ampère's Law, two articles ahead; nothing here needs it.)

Step 1. Slice the winding. The turns between $z'$ and $z' + \mathrm{d}z'$ form a ring carrying current $nI\,\mathrm{d}z'$ ; by (1) it contributes, at an on-axis field point a distance $\zeta = z - z'$ away,

\mathrm{d}B_z = \frac{\mu_0\,(nI\,\mathrm{d}z')\,R^2}{2\,(R^2 + \zeta^2)^{3/2}}.

Step 2. Substitute the viewing angle. Let $\theta$ be the angle between the axis and the line from the field point to the ring's rim: $\zeta = R\cot\theta$ , $\mathrm{d}\zeta = -R\,\mathrm{d}\theta/\sin^2\theta$ , and $(R^2+\zeta^2)^{3/2} = R^3/\sin^3\theta$ . The integrand collapses:

\mathrm{d}B_z = \frac{\mu_0nI}{2}\,\sin\theta\,\mathrm{d}\theta.

Step 3. Integrate between the angles $\theta_1$ , $\theta_2$ under which the two ends of the solenoid are seen from the field point:

B_z = \frac{\mu_0 n I}{2}\left(\cos\theta_1 - \cos\theta_2\right) \qquad (4)

(with the angles measured from the same axial direction, $0 \le \theta_1 < \theta_2 \le \pi$ ).

Step 4. Read the limits. Deep inside a long solenoid, the ends subtend $\theta_1 \to 0$ , $\theta_2 \to \pi$ :

B = \mu_0 n I \qquad (5)

— the ideal interior field, independent of the radius and of position (the full off-axis uniformity statement is Ampère's-law property). At the exact center of the end face, $\theta_1 = \pi/2$ : $\cos\theta_1$ vanishes and only the far end contributes, $B = -\tfrac{\mu_0nI}{2}\cos\theta_2$ , which tends to exactly half the deep-interior value as that end recedes ( $\theta_2\to\pi$ ) — half the rings are simply missing. Numbers for a bench solenoid (30 cm long, 2 cm radius, 600 turns, 2.0 A): ideal $5.03$ mT; the finite formula gives $4.98$ mT at the center (99.1% of ideal — already essentially ideal at aspect ratio 15) and $2.51$ mT at the ends. Why the angle form is the one to remember: like the straight-segment formula of the previous article, (4) handles every finite case — short coils, field points outside the bore on-axis, gradient estimates near the ends — while the memorized $\mu_0nI$ is just its most flattering limit. *Checks: the angle-substitution integrand collapses as shown; the bench-solenoid numbers follow from (4) — center factor 0.9912, end factor 0.4989 ≈ ½; and the toroid's thin-tube limit $\mu_0NI/2\pi R_0 = \mu_0nI$ closes consistently with the solenoid result (5).*

The toroid completes the catalog: wind the solenoid around a doughnut and the field lines close inside the tube, azimuthal, with magnitude $B = \mu_0NI/2\pi s$ at distance $s$ from the central axis — stronger toward the inner wall — and exactly zero outside: any circulation loop outside the tube threads each turn once in and once out, netting nothing. The clean derivation is a two-line circulation argument that belongs to Ampère's Law; take from it here the design fact: toroids do not leak. That is why fusion machines, sensitive inductors, and tokamak transformers are doughnuts.

Limiting cases

Case	Result	Meaning
loop center ( $z=0$ )	$B = \mu_0I/2R$	matches the Biot–Savart article's calibration point
loop far field ( $z\gg R$ )	$B \approx \mu_0m/2\pi z^3$ , $m = I\pi R^2$	the universal dipole tail
Helmholtz pair, $d = R$	$B_0 = (4/5)^{3/2}\mu_0NI/R$ , flat to $z^4$	0.01% at $R/10$ ; anti-Helmholtz → pure gradient
solenoid, deep inside	$B = \mu_0nI$	radius-free, uniform
solenoid end face	$B = \mu_0nI/2$	half the rings are missing
toroid inside	$B = \mu_0NI/2\pi s$	non-uniform: stronger at the inner wall
toroid outside	$B = 0$	perfect confinement; circulation argument → Ampère's Law
thin toroid ( $s \approx R_0$ )	$B \approx \mu_0nI$	a solenoid that ate its tail — catalogs agree

Common trapAmpère's law is always true — and usually useless for these shapes

The catalog invites a dangerous shortcut. The circulation law $\oint\vec B\cdot\mathrm{d}\vec\ell = \mu_0I_{\rm enc}$ (owned two articles ahead) determines a field only when symmetry pins $\vec B$ 's magnitude and direction along an entire loop — true for the infinite wire, the ideal solenoid, the toroid, and almost nothing else. A finite solenoid, a single ring, a Helmholtz pair have no such loop: apply the shortcut anyway and you will manufacture confident nonsense (a finite straight segment does NOT have $B = \mu_0I/2\pi s$ ). Everything in this article that is not symmetric enough was therefore built by honest Biot–Savart integration — and that division of labour, symmetry-shortcut versus integral, is the first decision of every magnetostatics problem. Second trap, from the lab: the ideal formulas assume close-wound, long coils; real solenoids are down ~1% at aspect ratio 15 (worked numbers above), and their helical winding leaks a small axial-current field outside — nulled in precision work by winding a return wire back along the coil.

Worked examples

1 · A bench solenoid, honestly rated (confidence builder)

A solenoid is 30 cm long, 2.0 cm in radius, wound with 600 turns carrying 2.0 A. Find the ideal interior field, the true center field, and the field at the center of an end face.

Step 1. Turn density: $n = 600/0.30 = 2000\ \mathrm{turns/m}$ . Ideal interior field (5): $B = \mu_0nI = (4\pi\times10^{-7})(2000)(2.0) = \boxed{5.03\ \mathrm{mT}}$ — about a hundred Earth-fields, from a desk-size coil.

Step 2. Finite correction at the center, from (4): the ends are seen at $\cos\theta = (L/2)/\sqrt{(L/2)^2 + R^2} = 0.15/\sqrt{0.15^2 + 0.02^2} = 0.9912$ each, so $B = 5.03 \times 0.9912 = \boxed{4.98\ \mathrm{mT}}$ — the ideal formula was off by under 1%; aspect ratio 15 is already "long."

Step 3. End face, from (4) with $\theta_1 = 90^\circ$ : $B = \tfrac12\mu_0nI \times 0.998 = \boxed{2.51\ \mathrm{mT}}$ — the celebrated factor of one-half, physically transparent: an infinite solenoid is two half-infinite ones, and at the joint each supplies half.

Step 4. Design reading: the interior is flat and radius-free; all the interesting (and fringing) behaviour lives within a radius or two of the ends — which is why solenoid magnets are specified by bore and length, and why short "solenoids" are really just fat coils.

2 · Designing a Helmholtz pair for magnetometer calibration (exam level)

You need a calibrated, uniform field of about $0.9\ \mathrm{mT}$ with open access, uniform to better than 0.05% over a 3-cm test region. Design a Helmholtz pair with $R = 15\ \mathrm{cm}$ coils of $N = 100$ turns each: find the current, verify the uniformity, and state the field if one coil's current is accidentally reversed.

Step 1. Central field from (3): $B_0 = (4/5)^{3/2}\mu_0NI/R$ . Solve for $I$ at $B_0 = 0.90\ \mathrm{mT}$ :

I = \frac{B_0R}{0.7155\,\mu_0N} = \frac{(9.0\times10^{-4})(0.15)}{0.7155\,(4\pi\times10^{-7})(100)} = \boxed{1.5\ \mathrm{A}}

(direct check: $B_0 = 0.7155\times(4\pi\times10^{-7})(100)(1.5)/0.15 = 8.99\times10^{-4}\ \mathrm{T}$ ✓).

Step 2. Uniformity along the axis of the test region. The 3-cm region spans $z = \pm1.5\ \mathrm{cm} = \pm R/10$ . Exact two-loop summation gives $B(R/10)/B_0 - 1 = -0.011\%$ on axis — the worst case over the region, since off-axis points at the same radius deviate by only about three-eighths as much — comfortably inside the 0.05% budget (at $\pm R/4$ it would be $-0.42\%$ : the $z^4$ error growing sixteen-fold per doubling, exactly as the Helmholtz condition predicts).

Step 3. The reversed-coil failure mode: with opposing currents the even symmetry flips to odd — $B_0 = 0$ at the center and the field grows linearly with $z$ : an anti-Helmholtz gradient coil. Diagnostic signature on the bench: your magnetometer reads near zero where you expected 0.9 mT. (Deliberately, this configuration is the quadrupole trap of cold-atom physics — one wiring error away from a different instrument.)

Step 4. Why not a solenoid? Equal uniformity is available in a long bore, but the Helmholtz pair leaves the test region open on all sides for optics and probes — geometry, again, is the design currency.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.2–5.3 (Ex. 5.6, 5.9, 5.10) · Jackson, Classical Electrodynamics, 3rd ed., §5.5 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 6 · Zangwill, Modern Electrodynamics, §10.4. The Ampère's-law derivations of solenoid uniformity and toroid confinement are deferred to Ampère's Law; this article's routes are pure Biot–Savart superposition. The general dipole field formula is quoted with its derivation deferred to the magnetic-dipole articles. Key results and worked numbers: the ring integral yields $B_z = \mu_0IR^2/2(R^2+z^2)^{3/2}$ , with $z=0$ limit $\mu_0I/2R$ and $z\gg R$ limit $\mu_0m/2\pi z^3$ ; the solenoid angle-form is $\tfrac{\mu_0nI}{2}(\cos\theta_1-\cos\theta_2)$ , giving $\mu_0nI$ deep inside and $\mu_0nI/2$ at the end face; differentiating the loop formula twice confirms the inflection at $z = R/2$ and hence $d = R$ for the Helmholtz condition, with central field $(4/5)^{3/2}\mu_0NI/R$ ; toroid confinement follows from the threading-count argument. Bench solenoid (30 cm, 2 cm radius, 600 turns, 2.0 A): 5.03 mT ideal, 4.98 mT true center (factor 0.9912), 2.51 mT end. Helmholtz pair ( $R = 15$ cm, $N = 100$ , $I = 1.5$ A): $B_0 = 0.899$ mT, deviation $-0.011\%$ at $R/10$ , $-0.42\%$ at $R/4$ . Earth's dipole moment $8\times10^{22}$ A·m².

Builds on: Biot-Savart Law · Ampère's Law · Curl of B · Electric Current and Current Density · Divergence of B and Absence of Magnetic Monopoles

Leads to: Magnetic Vector Potential · Magnetic Force on Current-Carrying Wires · Cyclotron Motion and Magnetic Confinement

Why is the solenoid's interior field independent of its radius? A fatter coil has its wires farther away.

Two scalings cancel exactly. Fatten the solenoid and each ring's rim moves farther from an axis point (weakening its contribution), but each ring also grows longer and — at fixed turns per length — the geometry recruits contributions more nearly broadside. The angle form (4) makes it transparent: the field depends only on the angles under which the ends are seen, and for an infinite solenoid those are $0$ and $\pi$ regardless of radius. The same conspiracy delivers the off-axis uniformity that Ampère's law will prove in two lines: inside an ideal solenoid the field is $\mu_0nI$ everywhere, wall to wall.

Is the field outside a real solenoid actually zero?

Not exactly, for three stacked reasons. Finite length: field lines that fill the bore must return outside, so a finite solenoid has an exterior return field — weak because it spreads through all of space, with the far field a dipole $\propto 1/r^3$ (moment $= NI\pi R^2$ ). Helicity: a real winding advances like a screw thread, so the coil carries a net axial current $I$ along its length, wrapping a weak azimuthal $\mu_0I/2\pi s$ field around the outside — cancelled in precision instruments by a return wire run back along the axis. And winding discreteness ripples the field near the wires. "Zero outside" is exact only for the ideal infinite, close-wound, purely azimuthal current sheet — a $\vec K$ -dialect idealization in the sense of Electric Current and Current Density.

Where should I use the Helmholtz formula versus just adding two loop fields numerically?

Use (3) for the central value and the design condition; use the exact two-loop sum whenever you need field maps or error budgets — it costs two evaluations of (1). The Helmholtz configuration's value is analytic: it guarantees the leading error is $z^4$ , so uniformity claims scale predictably — the deviation grows sixteen-fold every time the test radius doubles. For volumes beyond $\sim R/4$ , no tuning of two loops rescues you — add more coils (Maxwell's three-coil arrangement kills the fourth derivative too) or switch to a solenoid. The general principle — expand, exploit symmetry, spend geometric knobs killing derivatives — is the whole trade of shim-coil design.

The toroid's interior field falls as 1/s — why isn't it uniform like the solenoid it came from?

Because bending the solenoid broke the translational symmetry. In a toroid all $N$ turns thread any interior circulation circle, whether that circle hugs the inner wall (small $s$ ) or the outer (large $s$ ) — same threaded current, longer path, weaker field: $B = \mu_0NI/2\pi s$ . The inboard/outboard asymmetry is not a nuisance footnote: in tokamaks the "high-field side" and "low-field side" have different plasma physics, and the $1/s$ profile drives particle drifts that toroidal confinement schemes must engineer around. Only in the thin-tube limit ( $s$ pinned near the ring radius $R_0$ ) does the local field flatten back to the solenoid's $\mu_0nI$ .

How do I get the field of a solenoid off-axis, or of a loop away from its axis?

Honest answers only: the on-axis results here are exact only on the axis. Off-axis loop fields involve elliptic integrals (closed form exists, ugly), and the practical routes are three. Numerical Biot–Savart summation — the universal workhorse. Expansion methods — near the axis, $\nabla\cdot\vec B = 0$ lets you generate the leading off-axis corrections directly from the on-axis profile's derivatives ( $B_s \approx -\tfrac{s}{2}\,\partial B_z/\partial z$ ), the trick behind beam-optics field models. And for the ideal solenoid specifically, the answer is rigid: $\mu_0nI$ uniformly across the whole bore, zero outside — the two-line Ampère's-law argument two articles ahead. Which route to trust is, once again, a symmetry question first.

Do these catalog formulas survive when iron cores or magnetic materials are added?

The structure survives; the constant does not. Wrapping the solenoid on a linear magnetic core multiplies the field by the relative permeability ( $B = \mu nI$ , with $\mu = \mu_r\mu_0$ — thousands for soft iron), which is why practical electromagnets and transformer cores exist. But ferromagnets saturate (iron tops out near 2 T), remember their history (hysteresis), and add bound currents of their own — the honest treatment needs the $\vec H$ -field bookkeeping and magnetization physics of the magnetic-matter chapter, which will re-derive this catalog with matter included. Until then: the vacuum formulas here are exact for air-cored coils and lower bounds for cored ones.