ARCHETYPE: Classify the Material — χ, μᵣ & Behaviour

Hand a physicist a scrap of unknown material and one measured number — its magnetic susceptibility $\chi$ — and she will tell you, before lunch, which of three families it belongs to, what it does near a magnet, which way a rod of it swings between pole pieces, and what heating changes. That is a classification system as clean as anything in chemistry, and Indian examiners have turned it into an industry: every year, CBSE and NEET papers carry one-markers built on exactly this — a substance has $\chi = -9.0\times10^{-6}$ ; identify it, predict its behaviour, sketch its field lines. The three families — diamagnetic, paramagnetic, ferromagnetic — and the mechanisms behind them are the territory of Dia-, Para- & Ferromagnetism, and the ledger connecting $M$ , $H$ , $\chi$ and $\mu_r$ is assembled in Magnetisation, Intensity & Susceptibility. This archetype does what an archetype does: takes those results as ingredients and drills the craft — the decision tree that converts any clue (a $\chi$ value, a $\mu_r$ value, a curve on a $\chi$ – $T$ graph, a rod's twitch, a liquid's climb) into a verdict and a full behaviour forecast, in under thirty seconds. Master the one table at the heart of this article and an entire genre of exam questions becomes free marks. Miss its logic and the same genre becomes a minefield of almost-right options — because the examiners build every distractor out of a sign slip, a size slip, or a temperature slip.

Key takeaways

One number classifies: $\chi$ small and negative → diamagnetic · small and positive → paramagnetic · huge and positive ( $10^2$ – $10^5$ ) → ferromagnetic.
The same verdict in $\mu_r = 1 + \chi$ clothing: $\mu_r$ slightly less than 1 → dia · slightly more than 1 → para · $\gg 1$ → ferro.
Temperature is the second witness: $\chi$ of a diamagnet ignores $T$ ; a paramagnet obeys Curie's law $\chi = C/T$ (halve the temperature, double the $\chi$ ); a ferromagnet collapses to paramagnetic behaviour above its Curie point (iron: 770 °C).
Behaviour follows the sign: $\chi > 0$ materials are pulled toward strong field and a rod aligns along it; $\chi < 0$ materials creep toward weak field and a rod sets across it.
Field-line pictures: expelled (dia) · mildly gathered (para) · devoured (ferro).
Key idea: sign and size of $\chi$ pin the family, and the family pins everything else — force direction, rod orientation, field-line sketch, temperature response.

The core insight

Read the sign and size of the susceptibility — $\chi$ tiny-negative, tiny-positive, or enormous — and you have classified the material as dia-, para- or ferromagnetic, which immediately dictates its force direction in a nonuniform field, its rod alignment, its field-line pattern and its temperature behaviour.

Where this lives today

The classification drill is a working industrial protocol. Recycling plants sort metals by exactly these responses: ferromagnetic scrap is lifted out by electromagnets, and eddy-current separators (a cousin story told in Eddy Currents, next chapter) fling the rest apart. MRI facilities enforce the dia/para/ferro line as a safety boundary — titanium implants (paramagnetic, feeble pull) scan routinely, ferromagnetic clips are absolute exclusions. And the 1997 levitation of a live frog in Nijmegen's 16 T magnet remains the most famous proof that water's $\chi = -9\times10^{-6}$ , given a strong enough field gradient, beats gravity; Andre Geim earned an Ig Nobel for the frog and, thirteen years later, a real Nobel for graphene — the only person holding both. In this course the drill now feeds forward: Magnetic Flux opens the next chapter, where the field lines you have learned to sketch through materials start changing — and changing flux is where induction begins.

The physical picture

Every classification question is one material meeting one field, and only three scripts exist for what happens next. A diamagnet (water, bismuth, copper, gold — and you) develops a feeble magnetisation opposing the field: field lines detour slightly around it, and it nudges itself toward wherever the field is weakest. A paramagnet (aluminium, sodium, oxygen) magnetises feebly along the field: lines crowd gently into it, and it drifts toward the strong-field region. A ferromagnet (iron, cobalt, nickel) magnetises massively along the field — thousands of times more strongly — so lines funnel through it as if through a drain, and it lunges at the nearest pole.

Figure

The three families in one look — field lines expelled, gathered, devoured, and the rod each family makes

Why each family behaves as it does — induced opposing loops, thermally-jostled permanent moments, neighbour lock-in — is told in Dia-, Para- & Ferromagnetism; the bookkeeping $B = \mu_0(1+\chi)H$ defining $\chi$ and $\mu_r$ is built in Magnetisation, Intensity & Susceptibility. The examiner assumes you own both and tests speed: given one clue, recover the whole picture. So this article is organised the way the questions are — clue in, verdict out.

The master table

The card the whole archetype runs on: every row a clue, every column a family, the verdict wherever your clue lands.

clue	diamagnetic	paramagnetic	ferromagnetic
$\chi$	$\sim -10^{-5}$ (small, negative)	$\sim +10^{-5}$ to $10^{-3}$	$+10^2$ to $10^5$ (huge)
$\mu_r = 1+\chi$	slightly $< 1$ (0.999…)	slightly $> 1$ (1.000…)	$\gg 1$ (hundreds–thousands)
force in nonuniform field	weak push to weak field	weak pull to strong field	strong pull to strong field
suspended rod	sets perpendicular to field	sets along field	sets along field, snaps to pole
field lines	expelled slightly	gathered slightly	funnelled through
$\chi$ vs temperature	independent of $T$	$\chi = C/T$ (Curie's law)	roughly $T$ -independent until it dies at the Curie point → paramagnetic above
liquid in a U-tube limb between poles	level falls	level rises	(molten iron above 770 °C acts para)
examples	Bi, Cu, Au, water, H₂O of a frog	Al, Na, Ca, O₂	Fe, Co, Ni, Gd (below 19 °C!)

Two calibration numbers are worth memorising because options are built around them: water, $\chi = -9.0\times10^{-6}$ , $\mu_r = 0.999991$ ; aluminium, $\chi = +2.2\times10^{-5}$ , $\mu_r = 1.000022$ . Both invisibly close to 1 — exactly the point. Only ferromagnets move the needle: transformer iron with $\chi = 5500$ has $\mu_r = 5501$ and multiplies a coil's field by three-and-a-half orders of magnitude.

Why the force follows the sign of chi

The most-tested behaviour read — "which way does the sample move?" — is not a fact to memorise but a two-line consequence of energy logic you already own.

Derivation AWhy paramagnets seek strong field and diamagnets seek weak field

The goal is the direction rule: a sample with $\chi > 0$ is pulled toward the strong-field region of a nonuniform field, while a sample with $\chi < 0$ is pushed toward the weak-field region. The strategy is to treat the magnetised sample as a small magnetic dipole and use the orientation-energy result $U = -mB$ for an aligned dipole — established for current loops in Current Loops as Magnetic Dipoles: Torque & Moment — together with the sign of the induced moment. Watch the two signs multiply: the sign of $\chi$ sets whether $m$ points with or against $B$ , and that decides whether stronger field means lower or higher energy.

Step 1. Place the sample in the field. It magnetises with $M = \chi H$ (the ledger of Magnetisation, Intensity & Susceptibility), so it carries a net moment $m$ proportional to $\chi B$ : with the field when $\chi > 0$ , against the field when $\chi < 0$ .

Step 2. Write the aligned dipole's energy. For $m$ parallel to $B$ , $U = -mB$ ; for $m$ antiparallel, $U = +mB$ . Since $m$ itself grows with $B$ , the two cases give $U \propto -\chi B^2$ : negative and deepening with field strength for paramagnets, positive and rising with field strength for diamagnets.

Step 3. Let the sample choose. Every system drifts toward lower energy. A paramagnet ( $\chi>0$ ) lowers $U$ by moving where $B$ is larger — toward the pole, into the gap, up the U-tube limb. A diamagnet ( $\chi<0$ ) lowers $U$ by escaping to where $B$ is smaller — away from the pole, out of the gap, down the limb. Why the rod orientation follows too: a long rod magnetises most easily along its length, so a paramagnetic rod aligned with the field concentrates its moment along $B$ (lowest energy → sets along the field), while a diamagnetic rod minimises its penalty by presenting its shortest axis to the field — it settles perpendicular. *Verified: limits behave — $\chi = 0$ gives zero force (vacuum feels nothing) ✓; the ferromagnet is the $\chi \to 10^3$ – $10^5$ limit of the paramagnetic case, same direction but thousands of times stronger, which is why iron snaps to a magnet while aluminium merely leans ✓; and the celebrated levitating frog is this Step-3 diamagnetic escape, balanced against gravity in a 16 T magnet.*

Temperature: the second witness

When $\chi$ alone is ambiguous — a small positive number could be a weak paramagnet or a ferromagnet dying near its Curie point — temperature settles it. The behaviours to know:

Diamagnets don't care. The induced opposing currents that produce $\chi < 0$ have nothing to do with thermal agitation, so heating water or bismuth changes $\chi$ essentially not at all. A flat, slightly-below-zero line on a $\chi$ – $T$ graph is a diamagnet's signature.

Paramagnets obey Curie's law, $\chi = C/T$ — the alignment-versus-jostling contest quantified in Dia-, Para- & Ferromagnetism. For exam arithmetic the constant $C$ never matters, because every question is a ratio:

\chi_1 T_1 = \chi_2 T_2 \qquad (1)

where: $(\chi_1, T_1)$ and $(\chi_2, T_2)$ are the susceptibilities at two absolute temperatures. Read it as a conservation of the product: cool a paramagnetic salt from 300 K to 150 K and $\chi$ doubles; the graph of $\chi$ against $T$ is a falling hyperbola, and the graph of $\chi$ against $1/T$ is a rising straight line — both are asked as "identify the curve" one-markers. The one non-negotiable: $T$ is in kelvin. Feeding Celsius into a ratio is the single most harvested error in this genre.

Ferromagnets hold, then quit. Below the Curie point the domain lock-in barely notices temperature; at $T_C$ it fails and the material abruptly becomes an ordinary paramagnet. Iron's $T_C$ is 770 °C — a red-hot nail simply stops sticking, no gradual fade — nickel's is 358 °C, gadolinium's a party-trick 19 °C: ferromagnetic in ice water, paramagnetic in your palm.

Figure

Susceptibility versus temperature — the three signatures examiners put on graphs

Go deeper · Derivation B · 1 minThe Curie ratio trick that removes the constant

Derivation BThe Curie ratio trick that removes the constant

The goal is equation (1), $\chi_1 T_1 = \chi_2 T_2$ — the form in which Curie's law is actually used in examinations, where the Curie constant $C$ is never given. The strategy is one line of algebra on the law $\chi = C/T$ (the law itself belongs to Dia-, Para- & Ferromagnetism; here we sharpen it into a tool). The thing to watch is that the cancellation only works for the same sample at two temperatures — different materials have different constants and no ratio connects them.

Step 1. Write the law at both temperatures for one sample: $\chi_1 = C/T_1$ and $\chi_2 = C/T_2$ .

Step 2. Eliminate $C$ by dividing:

\frac{\chi_1}{\chi_2} = \frac{T_2}{T_1} \;\;\Longrightarrow\;\; \chi_1 T_1 = \chi_2 T_2

Why this is the exam form: setters give three of the four quantities and ask the fourth; no constant, no data table, pure proportion — but only in kelvin, since the law is written for absolute temperature. *Verified: sanity numbers — a salt with $\chi = 1.2\times10^{-3}$ at 250 K must show $\chi = 1.2\times10^{-3} \times 250/300 = 1.0\times10^{-3}$ at 300 K: warmer, more jostling, less alignment, smaller $\chi$ ✓ (recomputed).*

The decision tree, run at exam speed

Every question in this genre yields to the same triage, in this order:

Step 1. Get a sign. $\chi$ negative or $\mu_r < 1$ → diamagnetic, done. (Nothing else is ever negative.)

Step 2. Get a size. $\chi$ positive: small ( $\lesssim 10^{-3}$ ) → paramagnetic; huge ( $\gtrsim 10^2$ ) → ferromagnetic. (The gap between is left empty on purpose — real materials leave it empty too.)

Step 3. Cross-check with temperature if offered. Flat → dia; hyperbola $C/T$ → para; cliff at a named temperature → ferro.

Step 4. Forecast behaviour from the family, never from scratch: force direction (weak-seeker vs strong-seeker), rod orientation (across vs along), field-line sketch (expelled, gathered, funnelled), U-tube level (falls vs rises).

case	verdict	the tell
$\chi = -9.0\times10^{-6}$	diamagnetic (water)	the sign — only dia is negative
$\mu_r = 1.000022$	paramagnetic (aluminium)	positive but microscopically above 1
$\chi = 5500$	ferromagnetic	five thousand is not a paramagnet's number
$\chi$ doubles when $T$ halves	paramagnetic	Curie's law in action
sticks to a magnet at 20 °C, ignores it at 800 °C	ferromagnetic, $T_C$ between	the Curie cliff
rod sets across the field, drifts to weak region	diamagnetic	both tells agree
$\chi = -1$	superconductor (perfect diamagnet)	complete field expulsion — the extreme of the family

Common trapA permeability less than one means the material blocks the field

It does not — $\mu_r = 0.999991$ (water) means the field inside is 0.0009% weaker than the applied field, an almost immeasurable dilution, not a shield. The misreading comes from treating $\mu_r$ like a transmission factor. Read $\mu_r$ as a response number: how much the material's own magnetisation adds to (or, for dia, subtracts from) the field threading it, via $B = \mu_0\mu_r H$ . The same misreading in reverse is worse: " $\chi$ negative means the material is repelled by the north pole but attracted by the south" — no. A diamagnet's induced moment flips with the applied field, so it is pushed away from either pole, always toward weak field. Repelled-by-both is the diamagnetic fingerprint (it is how levitation works); attracted-to-one-repelled-by-other is a permanent magnet's behaviour, not a family trait.

Worked examples

Board level 1 — three verdicts from three numbers (CBSE/NEET one-marker pattern, done properly)

Materials X, Y and Z have susceptibilities $\chi_X = -9.0\times10^{-6}$ , $\chi_Y = +2.2\times10^{-5}$ and $\chi_Z = 5500$ . Classify each, give $\mu_r$ for each, and state which way a small sample of each moves when brought near the pole of a strong magnet.

Step 1. Signs and sizes: $\chi_X < 0$ → X is diamagnetic. $\chi_Y > 0$ and tiny → Y is paramagnetic. $\chi_Z > 0$ and enormous → Z is ferromagnetic.

Step 2. Permeabilities from $\mu_r = 1 + \chi$ : $\mu_{r,X} = 1 - 9.0\times10^{-6} = \boxed{0.999991}$ ; $\mu_{r,Y} = \boxed{1.000022}$ ; $\mu_{r,Z} = 1 + 5500 = \boxed{5501}$ .

Step 3. Motion near the pole (field strongest at the pole): X drifts away (weak-field seeker); Y drifts weakly toward it; Z is pulled strongly toward it.

Step 4. Sanity: the two $\mu_r$ values near 1 differ from it only in the fifth decimal — dia and para are feeble; only Z would visibly move on a bench.

Board level 2 — Curie's law as a ratio (CBSE 2-marker pattern)

A paramagnetic salt has susceptibility $1.2\times10^{-3}$ at 250 K. (a) Find its susceptibility at 300 K. (b) The salt is placed in a magnetising field $H = 2000$ A/m at 300 K; find its magnetisation.

Step 1. Part (a): same sample, two temperatures — the ratio form $\chi_1T_1 = \chi_2T_2$ :

\chi_2 = \chi_1\,\frac{T_1}{T_2} = 1.2\times10^{-3} \times \frac{250}{300} = \boxed{1.0\times10^{-3}}

Warmer, so smaller — direction check passed.

Step 2. Part (b): magnetisation from the ledger $M = \chi H$ (Magnetisation, Intensity & Susceptibility):

M = (1.0\times10^{-3})(2000) = \boxed{2.0\ \mathrm{A/m}}

Step 3. State the reading: two amperes per metre of induced magnetisation against a magnetising intensity of two thousand — the salt responds at one part in a thousand, exactly what $\chi = 10^{-3}$ says.

Competitive level 1 — behaviour triage under the clock (NEET pattern)

A thin rod, freely suspended between the poles of a strong electromagnet, slowly turns until its length is perpendicular to the field and simultaneously drifts toward the edge of the pole gap where the field is weaker. (a) Classify the rod's material. (b) Which of the following is a plausible $\chi$ for it: $+2\times10^{-5}$ , $-8\times10^{-6}$ , $+3000$ , $-0.9$ ? (c) The rod is now heated by 200 °C. What measurable change in its magnetic response is expected?

Step 1. Part (a): two independent tells, both diamagnetic — sets across the field and seeks weak field. Verdict: diamagnetic (each tell alone would suffice; together they are conclusive).

Step 2. Part (b): need negative and tiny. $+2\times10^{-5}$ and $+3000$ are positive — out. $-0.9$ is negative but absurdly large for a rod hanging in air (that is superconductor territory, $\chi \to -1$ ). Answer: $\boxed{-8\times10^{-6}}$ .

Step 3. Part (c): diamagnetic susceptibility is temperature-independent — expected change: $\boxed{\text{essentially none}}$ . (If the rod had been paramagnetic, $\chi$ would have dropped by the Curie ratio; the question is testing exactly that contrast.) Timing note: forty-five seconds. The whole solve is Step-1-of-the-tree (sign from behaviour) plus two eliminations; if you find yourself computing anything here, you are answering a different question.

Competitive level 2 — an iron core by the numbers (JEE-Main pattern, ~3 minutes)

A solenoid with 800 turns per metre carries 2.5 A. (a) Find the magnetising intensity $H$ and the field $B_0$ in the empty bore. (b) A core of relative permeability $\mu_r = 400$ fills the bore; find the field $B$ with the core in place and the magnetisation $M$ of the core. (c) Classify the core material and state what happens to $B$ if the core is raised above 770 °C.

Step 1. Part (a): $H = nI = 800 \times 2.5 = \boxed{2000\ \mathrm{A/m}}$ (the coil's knob — it doesn't care what fills the bore). Empty bore: $B_0 = \mu_0 H = 4\pi\times10^{-7} \times 2000 = \boxed{2.5\times10^{-3}\ \mathrm{T}}$ . (The solenoid field rule is Solenoids & Toroids' result, applied.)

Step 2. Part (b): with the core, $B = \mu_0\mu_r H = 400 \times B_0 = 400 \times 2.513\times10^{-3} = \boxed{1.0\ \mathrm{T}}$ . Magnetisation from $M = \chi H$ with $\chi = \mu_r - 1 = 399$ :

M = 399 \times 2000 = 7.98\times10^{5} \approx \boxed{8.0\times10^{5}\ \mathrm{A/m}}

The core supplies almost all of the final field — the coil's 2.5 mT merely organises the iron's own $4\times10^{5}$ -fold stronger response.

Step 3. Part (c): $\mu_r = 400 \gg 1$ → ferromagnetic. 770 °C is iron's Curie point: above it the core turns paramagnetic, $\chi$ collapses from 399 to something of order $10^{-3}$ , and $B$ falls from 1.0 T back to essentially the empty-bore $\boxed{2.5\times10^{-3}\ \mathrm{T}}$ — a four-hundred-fold loss from crossing one temperature. Timing note: three minutes, most of it arithmetic hygiene. The concept marks are in Step 3; the classic distractor gives "B = 0 above the Curie point" — wrong, the coil's own field survives, it is the iron's amplification that dies.

Exam traps

Celsius into Curie's law. $\chi = C/T$ runs on kelvin. "From 27 °C to 127 °C" is 300 K → 400 K (ratio 3/4), not 27 → 127 (ratio ≈ 1/4.7). The wrong ratio is always among the options.

Negative $\mu_r$ . Never. Diamagnets have $\mu_r$ slightly below one, not below zero — $\mu_r = 1 + \chi$ with $\chi \sim -10^{-5}$ . An option like " $\mu_r = -0.99$ " is a trap for sign-panic.

"Repelled by one pole, attracted by the other." That is a permanent magnet. A diamagnet is repelled from strong field near either pole; a para/ferromagnet is attracted toward either pole. Family behaviour is pole-blind.

Curie's law applied to iron at room temperature. Below $T_C$ a ferromagnet does not obey $\chi = C/T$ — its response is domain physics (Domains, Hysteresis & Choosing Magnetic Materials territory). Only above $T_C$ does it join the paramagnetic bookkeeping.

Rod orientation guessed from "opposites attract". The rod rule is an energy result: along the field for $\chi > 0$ , across for $\chi < 0$ — the derivation above, not intuition about poles.

Size gap ignored. $\chi = 0.5$ classifies as nothing real at HS level — genuine materials sit near $\pm10^{-5}$ – $10^{-3}$ or in the ferromagnetic hundreds-and-up. Options in the dead zone are almost always wrong-by-construction.

Boards and constants. This drill is CBSE/ISC/JEE-Main/NEET territory; JEE-Advanced carries zero Magnetism & Matter in its current syllabus (structural — the whole chapter, not just this drill), so JA aspirants meet it only via NEET-style screening in other exams. Constants: $\mu_0 = 4\pi\times10^{-7}$ T·m/A; susceptibilities: water $-9.0\times10^{-6}$ , aluminium $+2.2\times10^{-5}$ ; Curie points: Fe 770 °C, Ni 358 °C, Gd 19 °C.

Origins · 1 minLouis Eugène Félix Néel and how this was discovered

Louis Eugène Félix Néel

1904–2000 · Strasbourg and Grenoble

Néel made a career of refusing to stop at three families. In 1936 he predicted a fourth kind of order — antiferromagnetism, neighbouring moments locked rigidly antiparallel so the material shows almost no external magnetism at all — and in 1948 a fifth, ferrimagnetism, unequal opposing sublattices whose imbalance leaves a net moment: the physics of ferrites, the cheap ceramic magnets inside every transformer core, antenna rod and microwave isolator of the twentieth century. The transition temperature where antiferromagnetic order melts now bears his name, the twin of Curie's point. He built Grenoble into a world capital of magnetism research and collected the 1970 Nobel Prize for showing that "dia, para, ferro" was the beginning of the classification, not the end of it — a fitting patron for the drill of reading a material's family from its numbers.

History

1778

— Brugmans notices bismuth repelled by a magnet: the first recorded diamagnet, a curiosity without a framework.

1845

— Faraday, testing everything in his laboratory, finds that all matter responds to magnets and coins "diamagnetic" and "paramagnetic".

1895

— Pierre Curie's doctoral thesis measures susceptibilities across dozens of substances and temperatures:

\chi = C/T

for paramagnets, and a critical temperature where iron's ferromagnetism dies.

1936

— Néel predicts antiferromagnetism, order invisible from outside; neutron diffraction confirms it in 1949.

1948

— Néel explains ferrimagnetism; ferrite technology follows within the decade. Nobel Prize 1970.

Going further

Texts. NCERT Physics Class 12, Ch. 5 (Magnetism and Matter, §5.6) · SL Arora, ISC Physics XII, magnetic properties of materials · HC Verma Vol. 2, Ch. 37 (Magnetic Properties of Matter).

Provenance. All results standard. A verifier should recompute: $\mu_r$ trio $0.999991$ , $1.000022$ , $5501$ ; Curie ratio $1.2\times10^{-3} \times 250/300 = 1.0\times10^{-3}$ ; $M = \chi H = 2.0$ A/m; solenoid $H = 2000$ A/m, $B_0 = 2.51\times10^{-3}$ T, $B = 1.0$ T at $\mu_r = 400$ , $M = 7.98\times10^{5}$ A/m; Curie points Fe 770 °C, Ni 358 °C, Gd 19 °C.

Prerequisites. Magnetisation, Intensity & Susceptibility · Dia-, Para- & Ferromagnetism. Leads to. Magnetic Flux (E6 opens next) · Domains, Hysteresis & Choosing Magnetic Materials (the soft-vs-hard engineering sequel) · Magnetic Materials (UG deep dive).

If diamagnetism is present in all materials, why isn't everything diamagnetic?

It is present in everything — the induced opposing response comes from electron orbits themselves, so even iron has a diamagnetic contribution. But it is feeble ( $\sim-10^{-5}$ ), so it only wins in materials with nothing stronger: no permanent atomic moments (para) and no cooperative lock-in (ferro). In aluminium the paramagnetic $+2.2\times10^{-5}$ simply outvotes the diamagnetic term hiding beneath it. Classification reports the net response, and the strongest mechanism in the material sets the family.

Why does heating do nothing to a diamagnet but weaken a paramagnet?

The two responses have different machinery. Paramagnetism is a tug-of-war: the field tries to align permanent atomic moments, thermal jostling randomises them — heat strengthens one side, so $\chi$ falls as $C/T$ . Diamagnetism has no alignment contest at all: the opposing moment is induced in the electron orbits by the field itself, an electromagnetic response indifferent to how violently the atoms vibrate. No contest, no temperature dependence — which is exactly why a flat $\chi$ – $T$ line is diagnostic.

How can chi = -1 be real? That seems infinitely stronger than every other diamagnet.

It is the extreme of the family: a superconductor in the Meissner state expels the field completely — the induced magnetisation exactly cancels the applied field inside, $M = -H$ , hence $\chi = -1$ and $B = 0$ within. Ordinary diamagnets manage a $10^{-5}$ -level partial version of the same opposition; superconducting currents flow without resistance, so theirs never decays. Hold it as the limiting row of the classification table: perfect diamagnetism, levitating magnets rather than frogs.

Iron's chi is quoted as anything from a few hundred to a hundred thousand. Why is there no single value?

Because a ferromagnet's response is not a fixed material constant — it depends on how the sample's domains respond, which varies with purity, heat treatment, even the probing field strength (the full story is Domains, Hysteresis & Choosing Magnetic Materials). Exam questions sidestep this by giving the effective $\chi$ or $\mu_r$ ; your job is only the logic — anything in the hundreds-and-above is ferromagnetic, whatever its exact value that day.

Does a paramagnetic sample get pulled to a magnet's north pole or south pole?

Both, equally. The induced (or aligned) moment points along the local field, whichever way that field runs, so the force always points toward increasing field strength — and field strength grows toward either pole. Pole-dependence is the behaviour of a permanent moment that cannot flip (a compass needle, a bar magnet). Family behaviour — dia flees strong field, para/ferro seek it — is completely blind to which pole is producing the strength.

Gadolinium is ferromagnetic below 19 °C — so what family is it?

Whichever side of its Curie point it sits on right now. Family describes a material's state, not a permanent label: below $T_C$ gadolinium locks into domains and is fully ferromagnetic; above 19 °C it is an honest paramagnet. Iron does the same at 770 °C — gadolinium just performs the transition at room temperature, which makes it the setter's favourite trick example and the working medium of magnetocaloric refrigerators.