Ferromagnetism

Pick up a fridge magnet and it does not merely tug at a paperbin — it clamps onto it with a force strong enough to hold a stack of photos against gravity for years, with no battery and no external field in sight. A compass needle behaves the same way: once magnetised, it stays magnetised, pointing north reliably, day after day. This is not the faint, field-dependent attraction of paramagnetism you may have just studied — it is something categorically stronger and stickier, and it belongs to a small, exclusive club of elements: iron, cobalt, nickel, and their alloys. The reason is cooperation. In a paramagnetic material, each atomic magnetic dipole is an independent voter, weakly nudged into line by an external field and instantly forgetting its vote once the field is gone. In a ferromagnetic material, neighbouring dipoles lock arms with each other through a strong internal interaction, forming huge self-organised blocs called domains — regions where billions of atomic moments already point the same way, field or no field. Apply an external field and whole domains swing and grow together, producing magnetisation thousands of times larger than any paramagnet can manage. Remove the field and, crucially, much of that alignment survives — which is exactly why a permanent magnet is permanent. But cooperation has a temperature limit. Heat a ferromagnet enough and thermal jostling eventually overpowers the domain-locking interaction; above a sharp threshold called the Curie temperature, the alignment collapses and the material becomes an ordinary paramagnet. This article builds the domain picture from the ground up, quotes (without deriving) the field formulae you are expected to use for a bar magnet, and works through the same numerical territory the CBSE board expects: susceptibility, relative permeability, and the field inside a ferromagnetic sample.

Key takeaways

Ferromagnetic materials (iron, cobalt, nickel and their alloys) have atomic dipoles that spontaneously align within microscopic regions called magnetic domains, even with no external field.
In an unmagnetised sample the domains point in random directions and cancel out; an external field grows favourably aligned domains and rotates others, producing a huge net magnetisation.
Susceptibility is enormous and positive, $\chi \gg 1$ (often hundreds to thousands), so relative permeability $\mu_r = 1 + \chi \gg 1$ and the internal field $B = \mu_0\mu_r H$ can be hundreds of times larger than $\mu_0 H$ alone.
Ferromagnets retain a large fraction of their magnetisation once the external field is removed — this retained magnetism is exactly what makes permanent magnets possible, unlike paramagnetic or diamagnetic materials.
Heating works against the domain-ordering interaction; above the Curie temperature $T_C$ (about $1043\ \text{K}$ for iron) the domain structure breaks down and the material turns into an ordinary paramagnet.
The bar magnet itself is treated, at board level, as an equivalent solenoid, with quoted (not derived) axial and equatorial field formulae for a magnetic dipole of moment $m$ .
Key idea: strong, temperature-limited cooperative alignment of atomic dipoles within domains — not single-atom response to a field — is what makes ferromagnets show enormous, field-history-dependent, retainable magnetisation.

The core insight

Ferromagnetism arises when neighbouring atomic dipoles lock into large aligned domains through a strong internal interaction, giving $\chi \gg 1$ , retained magnetisation after the field is removed, and an abrupt loss of ordering above the Curie temperature.

Where it fits in the syllabus

Ferromagnetism is the third and strongest of the three magnetic-material classes in Unit V, Magnetism and Matter, of CBSE Class 12 Physics, following diamagnetism and paramagnetism. The prescribed depth is qualitative: know the defining behaviour (strong attraction, very large positive susceptibility, retained magnetisation), the domain picture, representative examples, and the qualitative effect of temperature including the existence of a Curie point. The bar-magnet-as-solenoid idea and the axial and equatorial dipole field formulae are quoted for reference, not derived, and belong to this same unit; hysteresis-loop quantitative work and the Curie law as a formal $1/T$ relation are explicitly out of scope for 2025-26, as is Earth's magnetism as a quantitative topic. Everything below honours those keep-outs.

The physical picture

Every atom in a ferromagnetic material carries a small permanent magnetic dipole moment, produced mainly by unpaired electron spins — exactly the source of the moment in a paramagnet. The difference is a strong quantum-mechanical interaction between neighbouring atoms (called the exchange interaction, though the syllabus does not require you to name or quantify it) that energetically favours neighbouring dipoles pointing the same way. This interaction is so strong that, even with zero external field, huge clusters of atoms — a domain — spontaneously align.

\text{domain: a region (typically} \sim 10^{-6}\ \text{to}\ 10^{-4}\ \text{m across) in which all atomic dipoles are already aligned}

where: each domain behaves almost like a tiny, fully saturated permanent magnet in its own right. A bulk, unmagnetised sample of iron contains an enormous number of such domains, each pointing in a different, essentially random direction (set by the crystal's internal structure and history), so their vector sum over the whole sample is close to zero — the bar looks unmagnetised even though every domain inside it is, locally, powerfully magnetised.

Figure

Randomly oriented domains in an unmagnetised bar cancel out, while an applied field grows and rotates domains into alignment producing a large net magnetisation

When a field $\vec H$ is switched on, two mechanisms operate together: domains already close to the field direction expand at the expense of their neighbours (domain-wall motion), and domains far from the field direction rotate bodily to align with it. Because whole domains — not lone atoms — respond, the induced magnetisation dwarfs paramagnetism, giving susceptibility of

\chi \gg 1 \quad (\text{often hundreds to thousands}), \qquad \mu_r = 1 + \chi \gg 1, \qquad B = \mu_0\,\mu_r\,H

where: $\chi$ (dimensionless) is the magnetic susceptibility, $\mu_r$ is the relative permeability, $H$ is the applied field intensity (units A/m), $B$ is the resultant magnetic field inside the material (tesla), and $\mu_0 = 4\pi\times10^{-7}\ \text{T}\,\cdot\,\text{m/A}$ is the permeability of free space. Physically, $B = \mu_0\mu_r H$ says that whatever field intensity you apply, the material's internal cooperative response multiplies it up by a factor $\mu_r$ that can run into the hundreds or thousands — this is the numerical signature examiners test. The bar magnet itself is treated (qualitatively, no derivation required) as an equivalent solenoid — a current loop whose magnetic moment $m$ produces a dipole-like field outside it, in close analogy to a short solenoid carrying current. For reference, the quoted (not derived) field of a magnetic dipole of moment $m$ at distance $r$ is

B_{\text{axial}} \approx \frac{\mu_0}{4\pi}\,\frac{2m}{r^3}, \qquad B_{\text{equatorial}} \approx \frac{\mu_0}{4\pi}\,\frac{m}{r^3}

where: $B_{\text{axial}}$ is the field on the line through the poles (twice as strong), $B_{\text{equatorial}}$ is the field on the perpendicular bisector, and both fall off as $1/r^3$ , exactly like an electric dipole. A magnetic dipole placed in a uniform external field $\vec B$ experiences a torque, again quoted qualitatively as $\vec\tau = \vec m \times \vec B$ , which tends to rotate it into alignment — the same torque, scaled up domain by domain, that swings a ferromagnet's moments into line with an applied field. Retained magnetisation after the field is switched off is the property that distinguishes ferromagnets from paramagnets in practice: a paramagnetic sample's tiny magnetisation vanishes almost as soon as $H$ is removed, because thermal motion instantly randomises the weakly-coupled individual dipoles. In a ferromagnet, the strong domain-locking interaction keeps much of the alignment in place with no field present at all — this residual magnetisation is the entire basis of permanent magnets, compass needles, and magnetic recording.

Temperature dependence (qualitative only). Higher temperature injects more random thermal energy that fights the domain-ordering interaction. Below a sharp, characteristic threshold called the Curie temperature $T_C$ , ferromagnetic ordering survives and strengthens on cooling. Above $T_C$ , thermal agitation overwhelms the ordering interaction outright, the domain structure collapses, and the material abruptly becomes an ordinary paramagnet. Iron's Curie point is about $1043\ \text{K}$ (roughly $770\ ^\circ\text{C}$ ). No formal quantitative law (such as $\chi \propto 1/T$ above $T_C$ ) is required at this level — only the qualitative existence of $T_C$ and the picture of a sharp ferromagnetic-to-paramagnetic transition there.

Ferromagnetism from the ground up

Derivation AGetting the internal field of a ferromagnetic sample from its permeability

The goal here is not to derive the domain interaction itself (that quantum-mechanical exchange mechanism lies far above the board syllabus and is only ever quoted qualitatively), but to build, from the definitions of susceptibility and relative permeability, the one quantitative relationship CBSE actually expects: given a ferromagnetic material's relative permeability $\mu_r$ and an applied field intensity $H$ , find the susceptibility $\chi$ and the resultant field $B$ inside the material, and see numerically why that field can be hundreds of times larger than $\mu_0 H$ alone.

Step 1. Start from the defining relation between the two material constants used for any magnetic medium: relative permeability is one plus susceptibility, $\mu_r = 1 + \chi$ . This single line is the entire quantitative bridge between "how strongly the material responds" ( $\chi$ ) and "how much the total field is boosted relative to vacuum" ( $\mu_r$ ); rearranged, it gives $\chi = \mu_r - 1$ .

Step 2. Recall how the field inside a material relates to the applied intensity. In vacuum, $B_0 = \mu_0 H$ . Inside a magnetic material, the same intensity $H$ produces a field $B = \mu_0\mu_r H$ — the permeability of free space multiplied by the material's own relative permeability. For a ferromagnet, $\mu_r$ is not close to $1$ (as it would be for vacuum or a weakly diamagnetic or paramagnetic medium) but is very large, so $B$ is dramatically boosted above $\mu_0 H$ .

Step 3. Combine the two relations. Given $\mu_r$ (measured or quoted for the sample) and the applied $H$ , compute $\chi = \mu_r - 1$ directly, and compute $B = \mu_0\mu_r H$ by substituting $\mu_0 = 4\pi\times10^{-7}\ \text{T}\,\cdot\,\text{m/A}$ . Both quantities follow from arithmetic alone once $\mu_r$ and $H$ are known — no calculus and no knowledge of the underlying domain mechanics is needed to get correct board-level numbers.

Step 4. Read the physical meaning of the result. Because $\mu_r$ for a ferromagnet is typically in the hundreds to low thousands (compare $\mu_r \approx 1.00001$ for a typical paramagnet), the resultant field $B$ inside a ferromagnetic core is correspondingly hundreds to thousands of times larger than the vacuum field $\mu_0 H$ the same current would produce alone. This is precisely why iron cores are wound inside transformers and electromagnets — the same coil current, with an iron core threaded through it, produces a far stronger field than the coil could ever produce in air.

Worked examples

Board level - susceptibility and internal field from permeability

A ferromagnetic sample has relative permeability $\mu_r = 900$ and is placed in a magnetic intensity $H = 300\ \text{A/m}$ . Find (a) the susceptibility $\chi$ , and (b) the field $B$ inside the material. Take $\mu_0 = 4\pi\times10^{-7}\ \text{T}\,\cdot\,\text{m/A}$ .

Step 1. Find $\chi$ using $\chi = \mu_r - 1 = 900 - 1 = 899$ .

Step 2. Find $B$ using $B = \mu_0\mu_r H = (4\pi\times10^{-7})\times900\times300$ .

Step 3. Evaluate: $B = 4\pi\times10^{-7}\times2.7\times10^{5} \approx 12.57\times10^{-7}\times2.7\times10^{5} \approx 0.339\ \text{T}$ .

Step 4. Compare with the vacuum value: $\mu_0 H = 4\pi\times10^{-7}\times300 \approx 3.8\times10^{-4}\ \text{T}$ . The ratio $B/(\mu_0 H) = \mu_r = 900$ confirms the arithmetic. So $\chi = \mathbf{899}$ (consistent with a strongly ferromagnetic material) and the resultant field inside the material is $\mathbf{B \approx 0.34\ \text{T}}$ — about nine hundred times larger than $\mu_0 H$ alone would give in vacuum.

Competitive level - comparing domain-based and Curie-point predictions across two temperatures

A bar of iron ( $T_C \approx 1043\ \text{K}$ ) with $\mu_r = 900$ at room temperature ( $300\ \text{K}$ ) is magnetised in $H = 300\ \text{A/m}$ , then the same field is applied again after the bar is heated to $1200\ \text{K}$ . Estimate the field $B$ in each case and explain the qualitative change.

Step 1. At $300\ \text{K}$ (well below $T_C$ ), reuse the Board-level result directly: $\mu_r = 900$ , so $B_{300\text{K}} = \mu_0\mu_r H \approx 0.34\ \text{T}$ , exactly as computed above.

Step 2. At $1200\ \text{K}$ , the temperature exceeds $T_C \approx 1043\ \text{K}$ , so the domain structure has collapsed and the sample is now an ordinary paramagnet, not a ferromagnet. Its relative permeability drops to a value only marginally above $1$ — a typical paramagnetic susceptibility is of order $\chi \sim 10^{-3}$ to $10^{-5}$ , so $\mu_r \approx 1.00001$ to $1.001$ , nowhere near $900$ .

Step 3. Recompute $B$ at $1200\ \text{K}$ using a representative paramagnetic $\mu_r \approx 1.001$ : $B_{1200\text{K}} = \mu_0\mu_r H \approx (4\pi\times10^{-7})\times1.001\times300 \approx 3.8\times10^{-4}\ \text{T}$ — essentially the vacuum value $\mu_0 H$ , with only a minute paramagnetic correction.

Step 4. Compare the two results: $B_{300\text{K}} \approx 0.34\ \text{T}$ versus $B_{1200\text{K}} \approx 3.8\times10^{-4}\ \text{T}$ — a drop by a factor of roughly 900, matching the drop from $\mu_r = 900$ to $\mu_r \approx 1$ . The qualitative reason is not that the applied field or the atoms changed; it is that heating past $T_C$ destroyed the cooperative domain alignment that made $\mu_r$ large in the first place, converting the same iron bar from a strong ferromagnet into a weak paramagnet. Timing note: on an exam, do not attempt to compute the paramagnetic $\chi$ from a formal Curie-law formula above $T_C$ — that quantitative relation is out of scope. State the qualitative collapse of $\mu_r$ toward $1$ and quote an illustrative small value; the marks are for recognising the transition and its consequence for $B$ , not for a precise post- $T_C$ number.

Common traps

Assuming ferromagnetic materials are magnetic uniformly at the atomic scale — the strong magnetism actually arises from whole, coherently aligned domains, and an unmagnetised sample has many randomly oriented domains that cancel out overall even though every domain inside it is locally saturated.
Forgetting that ferromagnetism is temperature-limited — above the Curie point, even iron loses its strong ferromagnetic behaviour and becomes merely paramagnetic; it is not a fixed, temperature-independent property of the element.
Confusing "retains magnetisation after the field is removed" (a genuinely distinguishing feature of ferromagnetism) with paramagnetic behaviour, which does not persist once the external field is removed.
Treating $\chi$ or $\mu_r$ for a ferromagnet as a single fixed material constant printed once and for all — in reality both depend on the applied field strength and the material's magnetic history (the topic of hysteresis), and the syllabus only expects the qualitative "very large and positive" idea, never a quantitative hysteresis-loop calculation.
Writing down the axial or equatorial dipole-field formula as something to be derived from Biot-Savart in an exam — CBSE explicitly wants these quoted, not derived, at this level.

You should be able to…

Define ferromagnetism in terms of the sign and (very large) magnitude of $\chi$ , and state $\mu_r = 1 + \chi$ .
Explain the domain picture: strong internal alignment within domains, randomly oriented domains overall in an unmagnetised sample, and how an applied field grows or rotates domains.
Compute $\chi$ and $B$ for a ferromagnetic sample given $\mu_r$ and $H$ , and compare the result to the vacuum value $\mu_0 H$ .
Explain why ferromagnetic materials retain magnetisation after the external field is removed, and connect this to permanent magnets.
State what the Curie temperature is and describe qualitatively what happens to a ferromagnet's domains and permeability above it.
Distinguish ferromagnetism from paramagnetism and diamagnetism by susceptibility sign and magnitude, and by whether magnetisation is retained once the field is removed.

Ask the tutor: Want to see how the domain picture, the Curie point, and the $B=\mu_0\mu_r H$ arithmetic all connect through a couple more worked numbers — ask the AI tutor to walk you through it.

Why does iron stay magnetised as a fridge magnet but a paramagnetic material never does?

It comes down to how the atomic dipoles talk to each other. In a paramagnet, each atom's dipole is essentially independent, weakly nudged toward the field direction, and instantly scrambled by thermal motion the moment the field disappears — there is nothing holding the alignment in place. In a ferromagnet like iron, neighbouring dipoles are locked together by a strong internal interaction inside each domain, so once a domain aligns it tends to stay aligned even with no external field, because misaligning would cost the domain interaction energy, not just thermal comfort. That locked-in cooperation, not the field itself, is what keeps a fridge magnet magnetised for years.

If I keep heating a magnet, does it get stronger or weaker before it stops working entirely?

Weaker, right up until it fails completely. As temperature rises toward the Curie point, thermal agitation increasingly disrupts domain alignment, so the achievable magnetisation and the strength of any permanent magnet decline steadily. At the Curie temperature itself the domain-ordering interaction is overwhelmed entirely, the domain structure collapses, and the material's ferromagnetic behaviour vanishes abruptly, leaving it as an ordinary weak paramagnet. This is why you should never leave a permanent magnet somewhere it can get very hot — the loss of magnetisation above (or even well below, but close to) $T_C$ is not easily reversed by simply cooling back down, since the domains re-form in new random orientations rather than snapping back to their original pattern.

Are chi and mu-r the same number, just written differently?

No, they are related but distinct: $\mu_r = 1 + \chi$ . Susceptibility $\chi$ measures how much extra magnetisation the material itself contributes in response to a field, while relative permeability $\mu_r$ measures the total multiplying factor on the field compared to vacuum, including that " $1$ " that vacuum would give anyway. For a ferromagnet where $\chi$ is in the hundreds, the " $+1$ " is negligible and $\mu_r \approx \chi$ numerically, but for a paramagnet or diamagnet where $\chi$ is tiny (of order $10^{-5}$ ), the difference between $\mu_r$ and $1$ is what actually matters, and $\chi$ itself is the number worth quoting.

Is the domain picture something I need to derive, or just describe?

Just describe, and describe it well — CBSE treats magnetic domains and the whole ferromagnetism topic as qualitative. You are expected to explain, in words, what a domain is, why an unmagnetised sample has random domains that cancel, how an applied field grows and rotates domains, why retained magnetisation follows, and what happens at the Curie point. You are not expected to derive the exchange interaction, calculate domain-wall energies, or work with a hysteresis loop's enclosed area. The only calculation genuinely expected of you is the $\chi = \mu_r - 1$ and $B = \mu_0\mu_r H$ arithmetic shown in the worked examples.