Magnetic Properties of Materials — Dia, Para & Ferro

Three Magnetic Personalities

With $\chi$, $\mu_r$ and $\mu$ in hand, every material falls into one of three classes ($\varepsilon$ is a small positive number):

Property	Diamagnetic	Paramagnetic	Ferromagnetic
Susceptibility	$-1 \le \chi < 0$	$0 < \chi < \varepsilon$	$\chi \gg 1$
Relative permeability	$0 \le \mu_r < 1$	$1 < \mu_r < 1+\varepsilon$	$\mu_r \gg 1$
Permeability	$\mu < \mu_0$	$\mu > \mu_0$	$\mu \gg \mu_0$

A remarkable fact: a minuscule difference in $\chi$ produces radically different behaviour — diamagnets sit near $\chi \approx -10^{-5}$ while paramagnets sit near $\chi \approx +10^{-5}$, yet one is repelled and the other attracted by a magnet.

Memory hook: dia = drives lines away, para = pulls lines in (weakly), ferro = floods lines in (massively).

Diamagnetism

Behaviour: diamagnetic substances tend to move from the stronger to the weaker part of an external field — a magnet weakly repels them. Placed in a field, the field lines are repelled/expelled, and the field inside is reduced — typically by just one part in $10^5$.

Mechanism: in a diamagnetic atom, the resultant magnetic moment is zero (orbital moments cancel). When a field is applied, electrons whose orbital moment lies along the field slow down, and those opposite speed up — an induced-current effect obeying Lenz's law (you'll meet it properly in Chapter 6). The atom thus acquires a net moment opposite to the applied field — hence repulsion.

Examples: bismuth, copper, lead, silicon, nitrogen (at STP), water, sodium chloride.

Key Point: Diamagnetism is present in all substances. But the effect is so weak that it is masked whenever stronger effects (para- or ferromagnetism) are present.

Superconductors: perfect diamagnets

The most exotic diamagnets are superconductors — certain materials cooled below their superconducting transition temperature, showing both perfect conductivity and perfect diamagnetism. Field lines are completely expelled:

$$\chi = -1, \qquad \mu_r = 0$$

This total expulsion is the Meissner effect. A superconductor repels a magnet and is, by Newton's third law, repelled by it — an important idea behind magnetic levitation demonstrations and superconducting maglev systems.

[JEE Tip] $\chi = -1$ means $M = -H$, so $B = \mu_0(H+M) = 0$ inside a superconductor. 'What is B inside a perfect diamagnet?' — Zero. Instant mark.

Paramagnetism

Behaviour: paramagnetic substances get weakly magnetised along the field and tend to move from weak field to strong field — they are weakly attracted to a magnet. Field lines get concentrated inside the material (enhancement again ~1 part in $10^5$).

Mechanism: each atom (or ion or molecule) of a paramagnetic material has a permanent magnetic dipole moment of its own. Ceaseless random thermal motion keeps these moments scrambled — no net magnetisation in zero field. Apply a strong external field $B_0$ at low temperature, and the dipoles begin to align with it.

Examples: aluminium, sodium, calcium, oxygen (at STP), copper chloride.

Two behaviours worth flagging:

• For a paramagnet, $\chi$ and $\mu_r$ depend on the temperature as well as the material — increasing field or lowering temperature increases the magnetisation.
• Saturation: at very high field / very low temperature, $M$ reaches a maximum value when all dipoles are perfectly aligned — beyond that, no further increase.

[NEET Important] Oxygen at STP is paramagnetic, while nitrogen at STP is diamagnetic — the single most-tested 'identify the odd one' pair from this section.

Ferromagnetism

Behaviour: ferromagnetic substances get strongly magnetised in an external field, and move strongly from weak to strong field regions. Field lines inside are highly concentrated ($\mu_r > 1000$!).

Mechanism — domains: the atoms possess permanent dipole moments (as in a paramagnet), but here they interact cooperatively and spontaneously align over macroscopic regions called domains. A typical domain is about 1 mm in size and contains roughly $10^{11}$ atoms. (The full explanation of this cooperative effect needs quantum mechanics.)

• Zero applied field: domains are randomly oriented — no bulk magnetisation.
• Field applied: domains orient along $B_0$ and the aligned domains grow at the expense of others, ultimately merging into a single 'giant' domain. (Domain motion is real — observable under a microscope with a liquid suspension of powdered ferromagnetic material.)

Hard and soft ferromagnets

• Hard ferromagnets retain their magnetisation when the field is removed — e.g. Alnico (alloy of iron, aluminium, nickel, cobalt, copper) and naturally occurring lodestone. These make permanent magnets (compass needles!).
• Soft ferromagnets lose almost all of their magnetisation when the field is removed — e.g. soft iron. Ideal wherever you want magnetism on demand (cores you can switch).

Examples of ferromagnetic elements: iron, cobalt, nickel, gadolinium.

Temperature behaviour

Ferromagnetism depends on temperature: at high enough temperature, the domain structure disintegrates and the ferromagnet becomes a paramagnet. This disappearance of magnetisation with temperature is gradual.

Curie's law and the Curie temperature [NEET Important]

Cooling a paramagnet quietens thermal jostling, so the dipoles align better and magnetisation grows. Experimentally, for small $B_0/T$:

$$M = C\,\frac{B_0}{T} \qquad\Rightarrow\qquad \chi = \frac{C\mu_0}{T} \quad \text{(Curie's law)}$$

where $C$ is the Curie constant and $T$ the absolute temperature. So $\chi \propto 1/T$: halve the temperature and the susceptibility doubles. At large $B_0/T$ the law breaks down — once every dipole is aligned, $M$ saturates and cannot grow further.

The temperature at which a ferromagnet's domain structure disintegrates is its Curie temperature $T_c$ (for iron, $T_c \approx 1043$ K). Above $T_c$ the material is paramagnetic, with susceptibility given by the modified Curie-Weiss law:

$$\chi = \frac{C'}{T - T_c} \qquad (T > T_c)$$

[JEE Tip] Two favourite traps: (1) $\chi \propto 1/T$ holds for paramagnets only — diamagnetic susceptibility is nearly temperature-independent. (2) Just above $T_c$, $\chi$ is large (the denominator is small) and it falls as the material is heated further.

Solved Examples

Example 1: Classify by susceptibility

Three materials have $\chi_1 = -0.8 \times 10^{-5}$, $\chi_2 = +2.3 \times 10^{-5}$ and $\chi_3 = +2400$. Classify them.

Solution:

1. $\chi_1$: small and negative — diamagnetic (e.g. copper).
2. $\chi_2$: small and positive — paramagnetic (e.g. aluminium).
3. $\chi_3 \gg 1$ — ferromagnetic (e.g. iron).
4. Corresponding $\mu_r = 1+\chi$: just below 1, just above 1, and about 2401.

Example 2: Why don't we notice diamagnetism everywhere?

Diamagnetism exists in all substances. Why is it observed so rarely?

Solution:

1. The diamagnetic response is tiny — field changes of about 1 part in $10^5$.
2. If atoms also have permanent moments, the (stronger) paramagnetic or (far stronger) ferromagnetic response masks it.
3. Diamagnetism stands out only in materials whose atoms have zero net moment — bismuth, copper, water, NaCl, nitrogen at STP.

Example 3: Field inside a superconductor

A superconducting sample sits in an external field intensity H. Using $\chi = -1$, find M and B inside.

Solution:

1. $M = \chi H = -H$ — magnetisation exactly cancels the intensity.
2. $B = \mu_0(H + M) = \mu_0(H - H) = 0$.
3. Answer: B = 0 inside: complete expulsion of field lines — the Meissner effect, perfect diamagnetism ($\mu_r = 0$).

Example 4: Cooling a paramagnet vs a diamagnet

What happens to the magnetisation of (a) a paramagnetic and (b) a diamagnetic sample when cooled (same applied field)?

Solution:

1. (a) Paramagnetic alignment fights thermal randomisation; cooling weakens the competition, so $M$ increases (towards saturation at very low T).
2. (b) Diamagnetism arises from induced orbital changes, not from aligning permanent moments — it is nearly temperature-independent.
3. Takeaway: temperature dependence is a fingerprint distinguishing the two weak magnetisms.

Example 5: Behaviour in a non-uniform field

A diamagnetic bar and a paramagnetic bar are placed in turn in the non-uniform field near a strong magnet's pole. How does each move?

Solution:

1. Diamagnetic: acquires moment opposite to the field; moves from the stronger to the weaker field region — pushed away from the pole.
2. Paramagnetic: acquires moment along the field; moves from weaker to stronger field — weakly pulled toward the pole.
3. A ferromagnetic bar would do the same as the paramagnet but strongly.

Example 6: Choosing hard vs soft

You need (a) a compass needle / permanent magnet, (b) a core whose magnetisation should become negligibly small soon after the field is switched off. Which class of ferromagnet suits each, with examples?

Solution:

1. (a) Hard ferromagnet — retains magnetisation after the field is removed: Alnico or lodestone.
2. (b) Soft ferromagnet — magnetisation becomes very small on removing the field: soft iron.
3. Reason: 'hard' and 'soft' describe precisely whether the magnetisation persists strongly or dies down quickly when $B_0 \to 0$.

Example 7: Estimating a domain's magnetic moment

A ferromagnetic domain contains about $10^{11}$ atoms, each contributing an atomic moment of about $9 \times 10^{-24}$ A m$^2$. Estimate the domain's moment when fully aligned.

Solution:

1. Within a domain the moments are spontaneously parallel, so they add directly.
2. $m_{domain} = 10^{11} \times 9 \times 10^{-24} \approx 9 \times 10^{-13}$ A m$^2$.
3. Takeaway: a single mm-sized domain is already a small magnet — bulk magnetisation is just domain bookkeeping.

Example 8: Heating a ferromagnet

What happens to a ferromagnet at high enough temperature, and how does the change occur?

Solution:

1. Thermal agitation disintegrates the domain structure.
2. The material becomes paramagnetic — moments persist atom-by-atom but no longer cooperate.
3. The disappearance of magnetisation with temperature is gradual (not a sudden switch-off).

Example 9: Oxygen vs nitrogen

At STP, which of oxygen and nitrogen is attracted into a strong magnetic field, and why?

Solution:

1. Oxygen molecules have permanent magnetic moments — O$_2$ at STP is paramagnetic — so oxygen is weakly drawn into the stronger field.
2. Nitrogen at STP is diamagnetic — weakly pushed out.
3. (Liquid oxygen visibly sticking between magnet poles is the classic demonstration.)

Example 10: What does $\mu_r > 1000$ buy you?

A solenoid produces $B_0 = 4 \times 10^{-3}$ T when empty. Estimate B if it is filled with a ferromagnetic core of $\mu_r = 1500$.

Solution:

1. With the core (same current, same H), using the ideal linear estimate and ignoring saturation: $B = \mu_r B_0$.
2. $B = 1500 \times 4 \times 10^{-3} = 6$ T.
3. Takeaway: ferromagnetic cores can multiply fields enormously — the reason practical electromagnets use a ferromagnetic core.