Paramagnetism

Hold a strong magnet near a stack of paperclips and a puddle of liquid oxygen at the same time, and something odd happens: the paperclips leap across the gap, but the oxygen only inches, reluctantly, toward the poles — and falls back the instant you take the magnet away. Both are being pulled toward the field, yet one response is dramatic and permanent-feeling while the other is faint and forgetful. The paperclips are ferromagnetic; the oxygen is paramagnetic, and the gulf between those two words is really a story about a tug-of-war happening inside every atom. Every paramagnetic atom — aluminium, platinum, sodium, chromium, the oxygen molecules you just inhaled — carries a tiny permanent magnetic moment of its own, a leftover from an unpaired electron whose spin refuses to cancel out. Left alone, these atomic compasses point every which way, jostled into randomness by thermal motion, so the material shows no magnetism at all. Switch on an external field, and each compass feels a gentle twist toward alignment. Thermal chaos fights back, so only a small fraction ever falls in line — but that fraction is enough to produce a weak, positive magnetisation in the same direction as the applied field. This article takes that atomic tug-of-war and turns it into the exact numbers CBSE Class 12 expects: the defining sign and size of paramagnetic susceptibility, how $\vec M$ , $\vec H$ and $\chi$ connect through one clean equation, why cooling helps and heating hurts, and how to work board-level and competitive-level numerical problems without ever needing more than algebra.

Key takeaways

A paramagnetic material's atoms carry a permanent magnetic dipole moment (usually from unpaired electrons), randomised by thermal motion when no field is applied.
An external field $\vec H$ partially aligns these moments, producing a small positive magnetisation $\vec M$ in the same direction as $\vec H$ .
Susceptibility is small and positive: $\chi > 0$ , typically $10^{-5}$ to $10^{-3}$ , and $\mu_r = 1 + \chi \gtrsim 1$ .
The governing relation is simply $M = \chi H$ — the whole numerical syllabus for this topic sits in that one line.
Raising the temperature increases thermal disorder and weakens paramagnetism; cooling strengthens it — qualitative only, no Curie's-law arithmetic is expected.
Common examples: aluminium, sodium, calcium, chromium, platinum, and oxygen gas (famously drawn weakly toward a magnet's poles when liquefied).
Key idea: paramagnetic atoms already have a permanent magnetic moment that thermal motion scrambles, so an external field only partially aligns them, giving a small positive $\chi$ that shrinks as temperature rises.

The core insight

Paramagnetism is the weak, temperature-sensitive, field-aligned magnetisation ( $M = \chi H$ , $\chi > 0$ but small) of materials whose atoms already carry a permanent magnetic moment that thermal agitation keeps disordered until a field is applied.

Where it fits in the syllabus

Paramagnetism is one of three classes of magnetic materials in Unit V, "Magnetism and Matter," of CBSE Class 12 Physics, sitting alongside diamagnetism and ferromagnetism in the same short qualitative-plus-numerical block. The prescribed scope asks for the defining behaviour (small positive $\chi$ ), representative examples, the atomic-level origin (unpaired electrons giving a permanent moment), and the qualitative temperature trend — explicitly not a formal Curie's-law calculation ( $\chi \propto 1/T$ is named only as an out-of-scope formula, kept at the "temperature affects magnetic properties" level). Numerically, students are expected to be fluent with $M = \chi H$ and $\mu_r = 1 + \chi$ , both quoted directly from the syllabus without derivation.

The physical picture

Picture a paramagnetic sample as a crowd of tiny compass needles — one per atom — each pinned at a fixed point but completely free to swivel. With no external field, thermal energy keeps every needle spinning and tumbling randomly, so averaged over the whole sample the net magnetic moment per unit volume is zero. This "no memory, no net moment" starting state is what separates paramagnetism from ferromagnetism, where strong internal forces lock huge groups of needles into rigid formation even without any outside help. Now switch on a magnetic field intensity $\vec H$ . Each atomic dipole moment $\vec m$ feels the familiar aligning torque $\vec\tau = \vec m \times \vec B$ met earlier in the chapter, which tries to swing it parallel to the field. Thermal collisions constantly knock the needles back out of alignment, so the tug-of-war settles into a small statistical excess of needles pointing along $\vec H$ rather than against it. That tiny excess adds up, across the whole sample, to a net magnetisation:

M = \chi H

where: $M$ is the magnetisation (magnetic moment per unit volume, in A/m), $H$ is the applied magnetic field intensity (A/m), and $\chi$ is the (dimensionless) magnetic susceptibility of the material. For paramagnetic substances $\chi$ is small and positive — typically in the range $10^{-5}$ to $10^{-3}$ — meaning $M$ points the same way as $H$ , unlike diamagnetic materials where $\chi$ is small and negative. The physical reading is direct: susceptibility is simply the "gain" of the material — how much magnetisation you get out per unit of field you put in — and for paramagnets that gain is small but never zero and never negative. A second useful number follows immediately, the relative permeability:

\mu_r = 1 + \chi

where: $\mu_r$ compares the material's permeability to that of free space. Since $\chi > 0$ but small for a paramagnet, $\mu_r$ is just barely above 1 — the material lets slightly more field through than vacuum would, but the effect is nowhere near as dramatic as in a ferromagnet, where $\mu_r$ can run into the thousands.

Figure

Randomly oriented atomic dipole arrows in a box with no field, next to the same box with an external field applied showing most arrows still scattered but a slightly larger fraction now leaning toward the field direction, producing a small net magnetisation arrow

Temperature is the referee in this tug-of-war. Heating the sample gives the atoms more random kinetic energy, so thermal collisions disrupt alignment more effectively and the induced magnetisation for a given $H$ drops — susceptibility falls. Cooling the sample removes disorder, letting the field's aligning torque win a larger share of the contest, so $\chi$ rises. CBSE requires only this qualitative direction of the trend, not the formal inverse-temperature Curie's law.

Derivation AGetting M and mu_r from H and chi

The goal here is not a first-principles derivation from atomic physics — the syllabus explicitly keeps that qualitative — but to show precisely how the two defining numbers of a paramagnetic material, the magnetisation $M$ and the relative permeability $\mu_r$ , follow from the applied field $H$ and the measured susceptibility $\chi$ using nothing beyond the definitions already given, so that every numerical in this topic reduces to careful substitution rather than any new physics.

Step 1. Start from the definition of susceptibility itself: $\chi$ is defined as the ratio of the induced magnetisation to the applied field intensity, $\chi = M/H$ . This is not a derived law but the working definition of what "susceptibility" even means for a linear magnetic material — paramagnetic materials, in the field ranges tested at this level, behave linearly enough that a single constant $\chi$ describes them.

Step 2. Rearranging the definition immediately gives the magnetisation for any applied field: $M = \chi H$ . Because $\chi > 0$ for a paramagnetic sample, $M$ comes out with the same sign as $H$ — physically, the induced magnetisation points along the applied field, confirming the qualitative picture of atomic dipoles being pulled (weakly) into alignment rather than pushed away from it.

Step 3. The total field inside the material is a combination of the free-space contribution and the material's own response, which the syllabus packages into the relative permeability $\mu_r = 1 + \chi$ . Substituting $\chi \sim 10^{-5}$ to $10^{-3}$ shows immediately why $\mu_r$ sits only fractionally above $1$ for a paramagnet — the material barely enhances the field, in sharp contrast to a ferromagnet's $\mu_r$ in the thousands.

Step 4. Together, $M = \chi H$ and $\mu_r = 1 + \chi$ are the complete numerical toolkit for this topic: given any two of $\chi$ , $H$ , $M$ (or a change in one due to a temperature change), the third follows by direct substitution — exactly the pattern used in both worked examples below.

Worked examples

Board level - magnetisation and permeability of a paramagnetic sample

A paramagnetic sample has susceptibility $\chi = 2.5\times10^{-4}$ at a given temperature and is placed in a magnetic intensity $H = 8\times10^4\ \text{A/m}$ . Find the magnetisation $M$ and the relative permeability $\mu_r$ .

Step 1. Apply $M = \chi H$ directly: $M = (2.5\times10^{-4})\times(8\times10^4) = 20\ \text{A/m}$ .

Step 2. Apply $\mu_r = 1 + \chi$ : $\mu_r = 1 + 2.5\times10^{-4} = 1.00025$ .

Step 3. Interpret both numbers together. The material develops a magnetisation of $20\ \text{A/m}$ pointing the same way as $H$ (positive $\chi$ ), and its relative permeability is only a whisker above 1 — the signature of a weak, positive, field-aligned response rather than the powerful amplification a ferromagnet would show. Answer: $M = 20\ \text{A/m}$ , $\mu_r = 1.00025$ .

Competitive level - cooling a paramagnetic sample and finding the new susceptibility

The same sample from the board-level example is cooled to a much lower temperature, with $H = 8\times10^4\ \text{A/m}$ held fixed, and the induced magnetisation is observed to rise to $M = 60\ \text{A/m}$ . Explain qualitatively why cooling increased the magnetisation, and find the new susceptibility $\chi_{new}$ .

Step 1. State the physics before touching any arithmetic. Lowering the temperature reduces the random thermal motion that keeps the atomic dipoles disordered, so the aligning torque from $H$ meets less resistance and a larger fraction of the atomic moments stay aligned with the field — increasing $M$ for the same applied $H$ . This is exactly the qualitative temperature trend the syllabus asks for; no formula for $\chi$ versus $T$ is needed to explain it.

Step 2. Find the new susceptibility from the same definition used throughout: $\chi_{new} = M/H = 60/(8\times10^4) = 7.5\times10^{-4}$ .

Step 3. Compare the two states: $\chi$ rose from $2.5\times10^{-4}$ to $7.5\times10^{-4}$ , a threefold increase, purely from cooling — confirming that lower temperature strengthens paramagnetic response. Answer: $\chi_{new} = 7.5\times10^{-4}$ , three times the original value. Timing note: this pattern is fast once you spot it — write the qualitative sentence first (it earns marks on its own even without the number), then reuse $\chi = M/H$ with the new $M$ . Do not go hunting for a $\chi \propto 1/T$ formula to justify the direction of the change; stating the disorder-versus-alignment argument in words is exactly what the board wants and is faster than any formula would be anyway.

Common traps

Confusing paramagnetism with ferromagnetism because both have positive $\chi$ — paramagnetic $\chi$ is small ( $10^{-5}$ to $10^{-3}$ ) and the sample retains no magnetisation once $H$ is removed, unlike a ferromagnet.
Thinking paramagnetic atoms have no magnetic moment until a field is switched on — they always carry a small permanent moment; the field only aligns the pre-existing, randomly oriented moments, it does not create new ones.
Getting the temperature trend backwards — cooling increases paramagnetic susceptibility (less thermal disorder to fight the field), heating decreases it, the opposite of the intuition that "more heat energy should help magnetise."
Reaching for a numeric Curie's-law formula ( $\chi \propto 1/T$ ) on an exam — CBSE wants only the qualitative "temperature affects magnetic properties" statement, never a computed proportionality constant.
Mixing up $H$ and $B$ in the formula — the syllabus's working relation is $M = \chi H$ (intensity), not a $B$ -based expression; keep the symbols straight when substituting.

Common trapParamagnetic atoms already have a moment — the field only aligns it, it does not create it

The single biggest mark-losing misconception in this topic is describing paramagnetism as the external field "inducing" a magnetic moment in otherwise non-magnetic atoms — that description belongs to diamagnetism, where the field genuinely creates a small opposing moment in atoms that had none. Paramagnetic atoms are fundamentally different: due to unpaired electrons, they possess a permanent magnetic dipole moment at all times, field or no field. What thermal motion does is scramble the direction of those pre-existing moments into randomness, and what the external field does is nudge a small statistical majority of them back into alignment. Get the story right — permanent moment, randomised by heat, partially ordered by the field — and both the sign of $\chi$ and the temperature dependence follow logically rather than needing to be memorised separately.

You should be able to…

State the sign and rough order of magnitude of $\chi$ for a paramagnetic material and contrast it with diamagnetic and ferromagnetic materials.
Explain the atomic origin of paramagnetism: a permanent, unpaired-electron moment, randomised by thermal motion, partially aligned by an external field.
Apply $M = \chi H$ and $\mu_r = 1 + \chi$ to solve for any one quantity given the other two.
Explain qualitatively, in words, why cooling increases and heating decreases paramagnetic susceptibility.
List common paramagnetic materials and describe the liquid-oxygen demonstration.
Avoid citing a numeric Curie's-law relationship on a CBSE answer sheet.

Ask the tutor: Want a few more mixed $M$ - $H$ - $\chi$ -temperature reasoning problems on paramagnetism to practise before your exam?

Why does oxygen gas count as paramagnetic when most gases seem "non-magnetic" in daily life?

Oxygen molecules happen to have two unpaired electrons in their outer orbitals, which gives each molecule a small permanent magnetic moment — the defining requirement for paramagnetism. The effect is genuinely there in ordinary gaseous oxygen, it is just far too weak to notice without a strong magnet and a sensitive setup, because gas molecules are spread far apart and the induced magnetisation per unit volume ends up tiny. The classic demonstration uses liquid oxygen instead: cooling it to a liquid packs many more oxygen molecules into the same volume, so the same weak per-molecule effect adds up to something you can actually see — the liquid visibly clings to a strong magnet's poles for a moment before dripping away.

If mu_r for a paramagnet is only 1.00025, is the effect basically useless?

For everyday engineering purposes, yes — you would never build an electromagnet core out of aluminium or platinum expecting field amplification, because a ferromagnetic core can give $\mu_r$ in the thousands while a paramagnet gives you a fraction of a percent. But "basically useless for building transformers" is not the same as "physically unimportant." Paramagnetic susceptibility is a precise fingerprint of a material's unpaired-electron structure, so measuring $\chi$ tells chemists and physicists real information about electronic configuration — a use case that has nothing to do with amplifying fields and everything to do with identifying materials.

Is there a formula connecting chi and temperature that I should memorise for CBSE?

No — and this is worth being explicit about because Curie's law, $\chi \propto 1/T$ , is exactly the kind of formula that looks "obviously testable" but is explicitly kept out of the current CBSE prescription. You are expected to know the direction of the trend — susceptibility rises on cooling, falls on heating — and to explain why in terms of thermal disorder versus field-driven alignment, entirely in words. If a numerical question gives you $\chi$ at two different temperatures, it will hand you both values directly (as in the competitive worked example above) rather than expecting you to compute one from a Curie's-law formula.

How is paramagnetism different from what happens right before a material becomes ferromagnetic?

They are related but distinct regimes, and the chapter treats them as separate categories rather than a smooth spectrum for exam purposes. In a ferromagnet like iron, neighbouring atomic moments interact strongly enough with each other that they lock into large aligned regions (domains) even with no external field at all — that internal, cooperative locking is completely absent in a paramagnet, where every atomic moment behaves essentially independently and only the external field provides any organising influence. Interestingly, heat a ferromagnet enough (above its Curie temperature, a separate topic) and that internal locking breaks down, leaving behind ordinary paramagnetic behaviour — but computing that transition temperature is beyond what this chapter asks of you.