Bar Magnets & the Dipole Picture

Hold a compass near a bar magnet and slide it slowly around the magnet's body: the needle pivots continuously, tracing out the same looping pattern Peter Peregrinus inked onto his lodestone sphere in 1269. Now try the experiment William Gilbert reported in 1600: saw the magnet in half, hoping to trap a north pole in one hand and a south in the other. You fail — instantly and always. Each half is a complete magnet with a fresh pair of poles, and each further cut only multiplies the population. Whatever a magnet is, it is not a north thing glued to a south thing. This module tells you what it actually is. The punchline was planted back in Current Loops as Magnetic Dipoles: a flat loop of current has a magnetic moment $m = NIA$ , feels a torque $\tau = mB\sin\theta$ , and produces a far field that dies off as $1/r^3$ . A bar magnet does all three — same torque law, same $1/r^3$ signature, same face rules — because deep down it is an enormous stack of atomic current loops, all pointing the same way. The bar magnet is the current loop's portrait in iron; the solenoid is the same portrait wound in copper. Learn the dipole picture once and you own all three objects. Along the way you get the two field formulas CBSE, JEE-Main and NEET still test — axial and equatorial, with their factor-of-two fingerprint — plus the honest bookkeeping of "pole strength," a useful fiction that never appears alone. Why nature refuses to hand over a lone pole is the next module's law; today you master the object that keeps refusing.

Key takeaways

A bar magnet is a magnetic dipole: its strength is captured by one vector, the dipole moment $\vec{m}$ , pointing S → N inside the magnet, with unit A·m² — the same unit as a current loop's $NIA$ , because that is what it secretly is.
Pole-strength bookkeeping: $m = q_m \times 2l$ (pole strength × magnetic length). Poles are useful fictions — they always come in equal-and-opposite pairs and never appear alone.
Far from the magnet the field carries the dipole signature: axial $B = \dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$ , equatorial $B = \dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$ — axial is exactly twice equatorial at the same distance, and both die off as $1/r^3$ .
This is the same twin-formula pattern as the electric dipole's $2kp/r^3$ and $kp/r^3$ from Field of a Dipole: Axial & Equatorial — swap $p \to m$ , $\dfrac{1}{4\pi\varepsilon_0} \to \dfrac{\mu_0}{4\pi}$ .
In a uniform field a bar magnet feels no net force, only the aligning torque $\tau = mB\sin\theta$ — which is why a compass needle (a small bar magnet) is a field-direction meter.
Key idea: bar magnet, current loop and solenoid are one object in three costumes — a single dipole moment $\vec{m}$ predicts the torque each feels and the $1/r^3$ field each makes.

The core insight

A bar magnet is a magnetic dipole characterised by its moment $\vec{m}$ (unit A·m², direction S → N): it aligns in external fields via $\tau = mB\sin\theta$ and produces the dipole far field — axial $\frac{\mu_0}{4\pi}\frac{2m}{r^3}$ , equatorial $\frac{\mu_0}{4\pi}\frac{m}{r^3}$ — exactly like the current loop it is built from.

Where this lives today

The dipole picture is a working engineering language. A neodymium magnet in an EV traction motor is specified by its moment per volume (magnetisation — formalised two modules on in Magnetisation, Intensity & Susceptibility), and motor designers place those dipoles around a rotor the way you placed $\vec{m}$ arrows on this page. Your phone's compass calibration — the figure-eight wave — is hunting for and subtracting the dipole fields of the magnets inside the phone itself, each an equation-(2) contaminant centimetres from the sensor. At the research frontier the same $1/r^3$ mathematics runs MRI: the protons in your body are microscopic dipoles, and the machine reads their collective moment. And spacecraft magnetometer teams live by the axial–equatorial pair — a probe flying past a planet reconstructs its dipole moment from exactly the two geometries you computed at 20 cm.

The physical picture

Three objects, one identity. Take a short, fat coil carrying a current — from Current Loops as Magnetic Dipoles its moment is $m = NIA$ and its far field is the dipole pattern. Stretch the coil into a long cylinder of many turns and you have a solenoid; Solenoids & Toroids showed its outside field is a dead ringer for a bar magnet's, with the face rule naming which end acts north. Now shrink the turns to atomic size and stack them by the trillion inside iron: Ampère's molecular-currents hypothesis, told in the current-loop module, says that is precisely what a magnetised bar is. The three maps below are not merely similar — outside the objects, they are the same map.

The compass needle you have trusted since The Magnetic Field & Field Lines is itself a small bar magnet — a dipole pivoting until $\vec{m}$ lies along the local field. That is the torque law $\tau = mB\sin\theta$ at work: torque falls to zero exactly at alignment. Nothing needs re-deriving; the current-loop module built the torque face by face, and a needle is the iron costume of the same physics. What about Gilbert's frustrating saw? Cut the bar anywhere and the atomic loops on both sides of the cut keep circulating — each fragment keeps a full complement of aligned loops, which is to say a complete N and S of its own. The dipole picture predicts the failure to isolate a pole. (The law that enforces it — zero net magnetic flux through any closed surface — is Gauss's Law for Magnetism, next module. Today's picture is the reason; tomorrow's law is the guarantee.)

Pole strength: honest bookkeeping with a useful fiction

Before the current-loop identity was understood, physicists modelled a bar magnet the obvious way: a "magnetic charge" $+q_m$ near one end, $-q_m$ near the other, in strict analogy with the electric dipole's $\pm q$ . The model survives in every board exam because it computes correctly — provided you remember it is bookkeeping, not anatomy. The magnetic length $2l$ is the distance between the two effective pole positions, which sit inside the ends — about $\tfrac{5}{6}$ of the geometric length, a favourite one-mark ICSE distinction. The dipole moment collects both numbers into one:

m = q_m \times 2l \qquad (1)

where: $q_m$ is the pole strength (unit: A·m — so that $q_m \times$ metres gives A·m², matching $NIA$ ), and $2l$ is the magnetic length, direction S → N assigned to $\vec{m}$ . Read it like $p = q \times d$ from the electric dipole: a stronger pole or a longer magnet means a bigger moment. But the analogy comes with a health warning the electric version never needs — you can hold an isolated $+q$ in your hand; an isolated $+q_m$ has never been found in nature. $q_m$ earns its keep only in the pair. The translation dictionary between E1's electric dipole and today's magnetic dipole is the most examinable table in this chapter:

quantity	electric dipole	magnetic dipole (bar magnet)
the two "ends"	charges $+q$ , $-q$ (isolable)	poles $+q_m$ , $-q_m$ (never isolable)
moment	$p = q\,d$ , from $-$ to $+$	$m = q_m\,2l$ , from S to N
moment's unit	C·m	A·m²
torque in uniform field	$\tau = pE\sin\theta$	$\tau = mB\sin\theta$
alignment energy	$U = -pE\cos\theta$	$U = -mB\cos\theta$
axial far field	$\dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}$	$\dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$
equatorial far field	$\dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}$	$\dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$
constant swap	$\dfrac{1}{4\pi\varepsilon_0} = 9\times10^9$	$\dfrac{\mu_0}{4\pi} = 10^{-7}$ (exact)

Every row is a memory-saver: anything you proved for $\vec{p}$ in Dipole in a Uniform Field and Field of a Dipole: Axial & Equatorial transfers to $\vec{m}$ by substitution. The torque and energy rows were built honestly for current loops in Current Loops as Magnetic Dipoles — cite them, use them, never re-derive them.

The two signature fields

Stand on the magnet's axis, distance $r$ from its centre ( $r \gg l$ ), and the field points along $\vec{m}$ with magnitude:

B_{\text{axial}} = \frac{\mu_0}{4\pi}\,\frac{2m}{r^3} \qquad (2)

where: $\mu_0/4\pi = 10^{-7}$ T·m/A exactly, $m$ is the dipole moment in A·m², and $r$ is measured from the magnet's centre, not its tip. The $r^3$ downstairs is the dipole fingerprint — double your distance and the field drops to one-eighth. A 2.0 A·m² magnet at 20 cm produces $5.0\times10^{-5}$ T on its axis: almost exactly one Earth-field, which is why a pocket magnet a desk-length away visibly deflects a compass. Now stand on the perpendicular bisector — the magnet's "equator" — at the same distance:

B_{\text{equatorial}} = \frac{\mu_0}{4\pi}\,\frac{m}{r^3} \qquad (3)

where: everything is as before, but the field is half the axial value and points antiparallel to $\vec{m}$ . Both facts are exam bait: the factor 2 feeds ratio questions, and the flipped direction on the equator decides direction MCQs. The same twin formulas, factor of two and flip included, appeared for the electric dipole in Field of a Dipole: Axial & Equatorial — one pattern, two fields.

Figure

Axial and equatorial points of a bar magnet — the factor-of-two fingerprint

Where do these come from? At the HS ceiling you can earn the axial formula with nothing but the pole picture, the inverse-square law and algebra — the chapter's one legal pole-strength calculation.

Derivation AThe axial field of a bar magnet from the two-pole picture

We want the field at a point P on the axis of a bar magnet, a distance $r$ from its centre, using the bookkeeping model: pole $+q_m$ at distance $r - l$ from P, pole $-q_m$ at distance $r + l$ . The strategy is pure superposition — each pole contributes an inverse-square field $\frac{\mu_0}{4\pi}\frac{q_m}{d^2}$ , exactly as each charge of an electric dipole contributed $kq/d^2$ — followed by the difference-of-squares algebra that turns "slightly unequal inverse squares" into a clean $1/r^3$ law. Watch two things: the fields subtract (N pushes away from itself, S pulls back), and the survivor is proportional to the product $q_m l$ , which is why only the combination $m = q_m \cdot 2l$ is ever measurable.

Step 1. The N pole is nearer to P, so its outward push wins over the S pole's pull. Net field along the axis, pointing away from N (i.e. along $\vec{m}$ ):

B = \frac{\mu_0 q_m}{4\pi}\left[\frac{1}{(r-l)^2} - \frac{1}{(r+l)^2}\right]

Step 2. Combine the fractions over the common denominator $(r-l)^2(r+l)^2 = (r^2 - l^2)^2$ . The numerator is a difference of squares:

(r+l)^2 - (r-l)^2 = 4rl

Why this is the whole trick: the big $r^2$ parts of the two distances cancel identically — what survives is the small mismatch between the pole distances, proportional to $l$ . A dipole's field is always a story about near-minus-far.

Step 3. So exactly, $B = \dfrac{\mu_0 q_m}{4\pi}\,\dfrac{4rl}{(r^2 - l^2)^2}$ . Now use the far-field condition $r \gg l$ : neglect $l^2$ beside $r^2$ , so $(r^2 - l^2)^2 \to r^4$ , and

B = \frac{\mu_0 q_m}{4\pi}\,\frac{4rl}{r^4} = \frac{\mu_0}{4\pi}\,\frac{2\,(q_m \cdot 2l)}{r^3} = \frac{\mu_0}{4\pi}\,\frac{2m}{r^3}

Step 4. The equatorial twin follows from the same picture with trigonometry instead of subtraction: at a point on the perpendicular bisector both poles sit at distance $\sqrt{r^2 + l^2}$ , their field magnitudes are equal, the components along the bisector cancel, and the components parallel to the magnet's axis add to $\frac{\mu_0}{4\pi}\frac{2q_m l}{(r^2+l^2)^{3/2}} \to \frac{\mu_0}{4\pi}\frac{m}{r^3}$ — half the axial value, pointing from N-side to S-side, i.e. antiparallel to $\vec{m}$ . Verified: the same algebra with $q \to q_m$ , $k \to \mu_0/4\pi$ reproduces the electric dipole's $2kp/r^3$ and $kp/r^3$ from Field of a Dipole: Axial & Equatorial, as the substitution table demands; and the $1/r^3$ far field with a factor-2 axial-to-equatorial ratio is exactly the dipole signature Current Loops as Magnetic Dipoles announced for a small loop — three costumes, one field.

Reading patterns and finding the quiet spots

A bar magnet rarely sits alone. On paper it shares space with Earth's background field or a second magnet, and the composite map is drawn by vector addition at every point — exactly the census you ran in Net Fields of Wire Systems & Zero-Field Points. Two composite features earn marks:

Neutral points are locations where the magnet's field exactly cancels the other field, leaving $B = 0$ — the arrowless spots where lines may legally meet. Place a bar magnet N-pole-north on a table and the cancellation happens on the east–west line through its centre (the equatorial field, antiparallel to $\vec{m}$ , opposing the horizontal background field); flip the magnet S-north and the neutral points jump to the axis. Locating them quantitatively needs Earth's horizontal component $B_H$ — the next module's number — and the full numerical gym waits for the dip-and-declination archetype later in E5.

The horseshoe is a bar magnet bent so its poles face each other across a gap: between them the lines run straight N → S, nearly parallel and evenly spaced — the cheap way to manufacture the uniform field every "charge enters a region" problem assumes.

Limiting cases

case	result	meaning
$r \gg l$ , on axis	$B = \frac{\mu_0}{4\pi}\frac{2m}{r^3}$ along $\vec{m}$	the dipole far field — the only regime the formulas claim
$r \gg l$ , on equator	$B = \frac{\mu_0}{4\pi}\frac{m}{r^3}$ antiparallel to $\vec{m}$	half the axial strength, direction flipped
distance doubled	field ÷ 8	the $1/r^3$ signature; compare Coulomb's ÷ 4
same $r$ , axial vs equatorial	ratio exactly 2	the factor-of-two fingerprint, a one-mark classic
magnet cut in half (across)	two magnets, each with moment ≈ $m/2$	halved length ⇒ halved moment; poles never isolated
magnet in uniform field	$\tau = mB\sin\theta$ , zero net force	torque aligns $\vec{m}$ with $\vec{B}$ ; translation requires a non-uniform field
$\theta = 0$	$\tau = 0$ , stable rest	the compass needle's destination

Common trapThe poles are two points sitting exactly at the ends of the magnet

Three separate errors hide in that sentence, and each costs marks. First, the effective poles sit inside the bar — the magnetic length $2l$ is about five-sixths of the geometric length, so using the full length in $m = q_m \cdot 2l$ overestimates the moment (ICSE asks this distinction directly). Second, poles are regions where field lines converge, not physical dots — nothing is "attached" there; the magnetism is distributed through the whole bar's atomic loops. Third and deepest: poles are not independent objects at all. Cut the bar at any plane and fresh poles appear on both new faces, every time. Treat $q_m$ the way an accountant treats a ledger column — indispensable for the arithmetic of equations (1)–(3), meaningless torn out of the pair.

Worked examples

Board level — axial and equatorial fields of a lab magnet (CBSE/NEET pattern, 3 marks)

A short bar magnet has magnetic dipole moment $m = 2.0$ A·m². Find the magnitude and direction of its magnetic field at a point (a) 20 cm from its centre on the axis, and (b) 20 cm from its centre on the equatorial line. (c) Compare (a) with Earth's field, $5 \times 10^{-5}$ T.

Step 1. Both points are far from a "short" magnet, so the dipole formulas apply with $r = 0.20$ m, $r^3 = 8.0 \times 10^{-3}$ m³.

Step 2. Axial: $B = \dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3} = 10^{-7} \times \dfrac{2 \times 2.0}{8.0 \times 10^{-3}} = \boxed{5.0 \times 10^{-5}\ \mathrm{T}}$ , pointing along $\vec{m}$ (away from the N pole on that side).

Step 3. Equatorial: $B = \dfrac{\mu_0}{4\pi}\dfrac{m}{r^3} = 10^{-7} \times \dfrac{2.0}{8.0 \times 10^{-3}} = \boxed{2.5 \times 10^{-5}\ \mathrm{T}}$ , pointing antiparallel to $\vec{m}$ . (Check: axial ÷ equatorial = 2. ✓)

Step 4. Part (c): the axial field equals one full Earth-field at desk distance — which is why a stray magnet wrecks a compass or phone-magnetometer reading. Keeping magnets away from navigation instruments is equation (2), not superstition.

Competitive level — a solenoid in a magnet's costume (JEE-Main/NEET pattern, ~2.5 minutes)

A solenoid of 2000 turns, cross-sectional area $1.6 \times 10^{-4}$ m², carries a current of 3.0 A. (a) What is its dipole moment? (b) Treating it as a bar magnet, find the field on its axis 30 cm from its centre. (c) The current is reversed. What happens to the answers?

Step 1. Part (a) is the identity this module preaches: a solenoid is a dipole with $m = NIA = 2000 \times 3.0 \times 1.6 \times 10^{-4} = \boxed{0.96\ \mathrm{A\cdot m^2}}$ , direction along the axis, out of the face where the current curls anticlockwise (the face rule from Solenoids & Toroids).

Step 2. Part (b): 30 cm is far compared with the coil's size, so use the axial dipole formula, $r^3 = 2.7 \times 10^{-2}$ m³:

B = \frac{\mu_0}{4\pi}\frac{2m}{r^3} = 10^{-7} \times \frac{2 \times 0.96}{2.7 \times 10^{-2}} = \boxed{7.1 \times 10^{-6}\ \mathrm{T}}

Step 3. Part (c): reversing the current reverses every atomic-scale arrow at once — $\vec{m}$ flips, so the field at the point keeps magnitude $7.1\ \mathrm{\mu T}$ but reverses direction. The magnitude answers survive; the direction answers flip. That one sentence is the full-credit answer. Timing note: about 2.5 minutes. The intended path is recognising (a) as pure $NIA$ recall and (b) as formula substitution — if you started computing the solenoid's internal field $\mu_0 n I$ , you answered a different question: that formula rules the bore; the dipole formula rules the outside world. Knowing which formula owns which region is the actual skill being tested.

Exam traps

Distance from the tip. Both field formulas measure $r$ from the magnet's centre. Measuring from the near pole is the most common numerical slip — it inflates the axial field badly at desk distances.

The factor of 2 misplaced. Axial = $2\times$ equatorial, not the reverse. Anchor it: on the axis, near-minus-far works in your favour. Ratio questions at equal $r$ are free marks only if the 2 sits upstairs in the axial formula.

Equatorial direction. On the equator the field is antiparallel to $\vec{m}$ . Direction MCQs hinge on it; so does locating neutral points.

$1/r^3$ , not $1/r^2$ . The dipole field dies one power faster than a lone pole's inverse square — squares instead of cubes silently wreck the arithmetic. Double distance ⇒ ÷8.

Geometric vs magnetic length. ICSE tests the distinction: the pole-to-pole (magnetic) length $2l$ is about $\tfrac{5}{6}$ of the end-to-end length; $m = q_m \cdot 2l$ uses the magnetic one.

Scope by board. CBSE, JEE-Main and NEET quote both field formulas (JEE-Main's syllabus lists the bar magnet's axial and perpendicular fields verbatim) — learn (2) and (3) cold. IB (Theme D.2) and IGCSE (0625 §4.1) want the patterns: sketching and interpreting field lines of a bar magnet, poles in pairs, attraction/repulsion — no dipole formulas. Constants: $\mu_0/4\pi = 10^{-7}$ exactly; boards accept $g = 9.8$ or $10$ m/s² where weights appear, but nothing here needs one.

Units of $m$ and $q_m$ . $m$ in A·m² (identical to $NIA$ — that is the point); $q_m$ in A·m. Writing $m$ in "J/T" is also correct ( $U = -mB\cos\theta$ makes it visible) and some competitive keys use it; A·m² is the safe default.

Origins · 1 minWilliam Gilbert and how this was discovered

William Gilbert

1544–1603 · Colchester and London

Physician to Queen Elizabeth I by day, Gilbert spent seventeen years and a personal fortune on the experiments that became De Magnete (1600) — arguably the first modern physics book, because it reports systematic experiments rather than quoting ancient authorities. He shaped lodestones into spheres he called terrellae ("little Earths"), mapped their fields with compass needles, showed that like poles repel and unlike attract, that iron gains magnetism when stroked by a lodestone, and that heating destroys it. Above all he documented the experiment that anchors this module: break a magnet and each piece is whole, with poles always in pairs. Gilbert also demolished folklore with relish — garlic does not demagnetise compasses, he reported, having tried. (His crowning conjecture, that the Earth itself is a great magnet, opens the next module.) Galileo called him "great to a degree that is enviable."

History

1269

— Peter Peregrinus maps a lodestone sphere with a compass needle and names the poles; breaking a magnet, he notes, yields two magnets.

1600

— William Gilbert's De Magnete: seventeen years of experiments establish pole pairing, induced magnetism, and magnetism as a subject for experiment, not folklore.

1750

— John Michell publishes the inverse-square law for the force between magnetic poles — the bookkeeping behind equation (1).

1820

— Ørsted and then Ampère connect magnetism to currents; Ampère proposes that all magnetism, iron included, is circulating current.

1832

— Gauss measures a magnet's moment and Earth's field in absolute units at Göttingen, turning the dipole picture into precision science.

Going further

Texts. NCERT Physics Class 12, Ch. 5 (Magnetism and Matter, §5.1–5.2: the bar magnet, field lines, bar magnet as equivalent solenoid, dipole in a uniform field) · HC Verma, Concepts of Physics Vol. 2, Ch. 36 · ISC Class 12 syllabus, magnetism unit (pole strength, magnetic length) · IB Physics guide (2023), Theme D.2 (field patterns of a bar magnet — sketching and interpretation) · IGCSE 0625 §4.1 (simple phenomena of magnetism).

Provenance. All results standard. A verifier should recompute: axial $10^{-7} \times 2(2.0)/(0.20)^3 = 5.0\times10^{-5}$ T and equatorial $2.5\times10^{-5}$ T (ratio 2); solenoid moment $2000 \times 3.0 \times 1.6\times10^{-4} = 0.96$ A·m² and its axial field $10^{-7}\times 2(0.96)/(0.30)^3 = 7.1\times10^{-6}$ T; the telescoping identity $(r+l)^2 - (r-l)^2 = 4rl$ ; magnetic length ≈ $\tfrac{5}{6}$ geometric length.

Prerequisites. Current Loops as Magnetic Dipoles: Torque & Moment · Solenoids & Toroids. Leads to. Gauss's Law for Magnetism: No Monopoles · Earth's Magnetism: Dip, Declination & B_H · Magnetisation, Intensity & Susceptibility (this chapter) · Field of a Dipole: Axial & Equatorial (the electric twin, for revision) · Magnetic dipoles and their fields (UG deep dive).

If poles aren't real, why do all the formulas use pole strength?

Because the pair-of-poles model computes the outside field of a uniformly magnetised bar correctly — it is a mathematically faithful stand-in, like replacing a sphere's mass with a point at its centre in gravitation. The derivation showed the measurable answer depends only on the product $m = q_m \cdot 2l$ , never on $q_m$ alone; so the fiction can never be caught lying, and it turns dipole fields into two-line algebra. The anatomy is different: the truth is Ampère's stacked atomic current loops, which is why $m$ shares its unit with $NIA$ . Use poles for arithmetic, loops for understanding.

Why does the axial field get the factor of 2?

Geometry of near-minus-far versus side-by-side. On the axis you sit close to one pole and far from the other; the fields subtract, and the mismatch (four terms of $rl$ , per Step 2 of the derivation) survives at full strength. On the equator you sit equally far from both poles; the components along the bisector cancel and only the projections of the two fields along the magnet's axis add — each projection is diluted by the factor $l/\sqrt{r^2+l^2}$ . Run both computations to the end and the axial result is exactly twice the equatorial one. The electric dipole's 2 has the identical origin.

My magnet visibly attracts an iron nail. But you said a uniform field gives zero net force — which is it?

Both, because a magnet's own field is not uniform. In a genuinely uniform field a dipole feels equal-and-opposite forces on its two "ends" — pure torque, no net pull. Your nail sits in the magnet's strongly non-uniform field: the nail first becomes a temporary dipole (induced magnetism — the mechanism is Dia-, Para- & Ferromagnetism's business), and its near end sits in a stronger field than its far end, so near-wins and the nail is dragged toward the magnet. Non-uniformity is what converts alignment into attraction.

Does a compass needle eventually stop swinging because the torque disappears?

The torque $\tau = mB\sin\theta$ vanishes only at alignment, but a frictionless needle arriving there with angular speed would overshoot and oscillate forever, like a pendulum through its lowest point. Real compasses are damped (fluid capsules, eddy currents) so the swing dies where torque is zero. The oscillation is no nuisance to physicists: its timing measures $mB$ — the classic oscillation magnetometer, part of the dip-and-declination story later in this chapter.

Is there any difference at all between a solenoid's field and a bar magnet's?

Outside: no — same dipole map, same formulas, same $m$ . Inside: yes. The solenoid's bore carries the clean uniform $\mu_0 n I$ from Solenoids & Toroids, and you can switch or reverse it at will; the bar magnet's interior field is fixed by its atomic loops. That is why electromagnets run hospitals and scrapyards while permanent magnets run compasses and fridge doors — controllability versus zero running cost.

What actually decides how strong a bar magnet is?

Its dipole moment — and microscopically, how many atomic moments it has and how completely they agree. Each iron atom contributes a fixed tiny moment; the bar's $m$ is the vector sum. In an unmagnetised bar the atomic teams point every which way and cancel; magnetising the bar is the business of getting them aligned, and how materials do (or refuse to do) this is exactly where this chapter is heading: Magnetisation, Intensity & Susceptibility builds the census, Dia-, Para- & Ferromagnetism sorts materials into families, and Domains, Hysteresis & Choosing Magnetic Materials explains why steel keeps its alignment while soft iron forgets.