Magnetic Properties of Matter
How materials respond to magnetic fields.
Magnetization & Magnetic Materials
Every atom a tiny magnet, and three ways matter responds.
Every atom is a tiny magnet, or can become one. How those tiny magnets respond to a field sorts all matter into three magnetic families.
Atomic magnets
Electrons orbiting a nucleus are little current loops, so each has an orbital magnetic moment. On top of that, every electron has a permanent spin angular momentum (explained only by quantum mechanics) and a matching spin magnetic moment of fixed size, μs = 9.285 × 10⁻²⁴ J/T. Nuclei have moments thousands of times smaller. An atom’s moment is the vector sum of all these.
Electron moments tend to cancel in pairs (the two electrons of helium cancel exactly), so many atoms have zero net moment, while some keep a permanent one.
Magnetization
In a piece of material whose atoms have permanent moments, the moments point at random, so any volume holding thousands of atoms has no net moment. A field exerts torques that try to line them up, while thermal motion keeps scrambling them: the alignment is partial, growing with stronger fields and lower temperatures. When it is nearly complete, the material is saturated.
A steel bar magnet of mass 5.0 g has magnetic moment 1.8 A m². The density of steel is 7.9 × 10³ kg/m³. Find its magnetization.
Three kinds of response
A current loop’s field at its centre points along its moment (the opposite of an electric dipole, whose field at its centre points against p⃗). So atomic moments lined up with an applied field add extra field in the same direction: the field inside the material is stronger than the applied field.
- Paramagnetism: permanent atomic moments, partly aligned by the field; the field inside is slightly increased.
- Ferromagnetism: neighbouring atomic moments line up with each other even without a field, through a quantum effect called exchange coupling. Permanent magnets are made of these materials.
- Diamagnetism: atoms with no net moment get moments induced by the field, and by Lenz’s law these oppose it; the field inside is slightly reduced. Every material has this response, but wherever permanent moments exist, para- or ferromagnetism swamps it.
Domains
An unmagnetized ferromagnet is divided into domains, regions up to a millimetre across (a cubic millimetre holds about 10²⁰ atoms) in which all moments point one way. Different domains point different ways, so the whole is unmagnetized. In a field, domains along it grow as their walls move, and others rotate toward it. That is why even a small field produces a large magnetization.
Aligned dipoles make a field inside opposing the applied one: E is reduced (K > 1).
Aligned loops make a field inside along the applied one: B is increased (para, ferro).
- 1Atomic moments come from orbital motion and electron spin (μ_s = 9.285 × 10⁻²⁴ J/T).
- 2Magnetization I = M/V (37.1), in A/m; bar magnet I = m/A.
- 3Para: permanent moments partly aligned; ferro: exchange-coupled domains; dia: induced opposing moments, present in all matter.
Check yourself
H, Susceptibility & Permeability
Separating the field you apply from the field matter adds.
Inside matter the field B has two sources: the currents we control, and the material’s own magnetization. Separating them gives three useful quantities: H, χ and μ.
Magnetic intensity H
In vacuum I = 0 and H = B/μ₀. So the familiar formulas carry over with μ₀ removed:
Where the ends of a magnetized body don’t matter (a ring, or the middle of a long rod), magnetization makes no H: H is set by the external currents alone, magnetized material or not.
A long solenoid has n turns per metre carrying i. Find H at its centre (a) empty, (b) with a long copper rod inside.
Susceptibility
For paramagnetic and diamagnetic materials the magnetization is proportional to H:
| Material | χ (at about 20 °C) |
|---|---|
| Oxygen gas (STP) | +0.18 × 10⁻⁵ |
| Aluminium | +2.2 × 10⁻⁵ |
| Titanium | +7.06 × 10⁻⁵ |
| Hydrogen (1 atm) | −9.9 × 10⁻⁹ |
| Sodium | −0.24 × 10⁻⁵ |
| Copper | −0.96 × 10⁻⁵ |
| Lead | −1.6 × 10⁻⁵ |
| Silver | −2.6 × 10⁻⁵ |
| Gold | −3.6 × 10⁻⁵ |
| Bismuth | −16.6 × 10⁻⁵ |
Permeability
From H = B/μ₀ − I and I = χH: B = μ₀(H + χH) = μ₀(1 + χ)H.
By what percentage does B change when a toroid is filled with copper (χ = −0.96 × 10⁻⁵)?
A long solenoid with 30 turns per cm carries 3.0 A. Find H and B at its centre. An iron core is inserted and becomes magnetized to I = 3.0 × 10⁶ A/m. Find the new H and B.
An iron core of cross-section 4.0 cm² sits in a long solenoid of 1500 turns/m carrying 2.0 A. The field in the core is 1.26 T. Find its magnetization and the pole strength at its ends.
- 1H⃗ = B⃗/μ₀ − I⃗ (37.2); H depends only on external currents when end effects are absent.
- 2I⃗ = χH⃗ (37.6): χ > 0 paramagnetic, χ < 0 diamagnetic.
- 3B⃗ = μH⃗, μ = μ₀(1 + χ), μ_r = 1 + χ (37.7–37.8).
Check yourself
Videos, simulations & further reading (3)
Curie's Law & the Three Families
Heat weakens magnetism; ferromagnets switch off above T_c.
Heat fights magnetism. For paramagnets this follows a simple law; ferromagnets lose their special powers altogether above a definite temperature.
Curie’s law
Raising the temperature randomizes atomic moments more, so at a given H the magnetization, and hence χ, falls. Far from saturation, a paramagnet obeys:
A ferromagnet heated past its Curie temperature Tc becomes paramagnetic, after which
| Ferromagnetic substance | Curie temperature T_c (K) |
|---|---|
| Iron | 1043 |
| Cobalt | 1394 |
| Nickel | 631 |
| Gadolinium | 317 |
| Fe₂O₃ | 893 |
A toroid with a circumference of 30 cm has 600 turns carrying 1.5 A, wound on an aluminium ring at 290 K. The magnetization of the ring is 6.6 × 10⁻² A/m. (a) Find χ. (b) What is the magnetization if the ring is warmed to 350 K?
Comparing the three families
| Diamagnetic | Paramagnetic | Ferromagnetic | |
|---|---|---|---|
| Magnetization in a field | small, opposite to the field | small, along the field | large, along the field |
| χ | small, negative | small, positive (≈ 10⁻⁵ to 10⁻³) | large, positive (thousands) |
| μ_r | slightly below 1 | slightly above 1 | thousands |
| Field lines inside | spread out | slightly crowded | strongly crowded |
| Near a magnet | weakly repelled | weakly attracted | strongly attracted |
| χ vs temperature | almost constant | ∝ 1/T | complicated; paramagnetic above T_c |
| States of matter | solid, liquid, gas | solid, liquid, gas | normally solid only |
| Permanent magnets, hysteresis | no | no | yes |
A rod hung freely between the poles of a magnet settles along the field if it is paramagnetic or ferromagnetic, and across the field if it is diamagnetic.
- 1Curie's law for paramagnets: χ = c/T (37.9).
- 2Ferromagnets become paramagnetic above T_c: χ = c/(T − T_c) (37.10); iron 1043 K.
- 3Dia: χ < 0, repelled, rod across the field; para: χ > 0 small, attracted, rod along it; ferro: huge χ, strongly attracted, hysteresis.
Check yourself
Hysteresis, Soft Iron & Steel
Ferromagnets remember their history.
A ferromagnet remembers. Its magnetization depends not just on the present field but on what has been done to it before, and that memory is what makes permanent magnets possible.
Measuring a ferromagnet’s response
Make a ring of the material and wind a toroid on it with n turns per length. A current i gives H = ni (the field from the winding alone is B₀ = μ₀ni). The ring’s magnetization adds to this, and the total field B in the ring can be measured (with a Rowland ring). Then
The hysteresis loop
- Start unmagnetized (O): H = 0, I = 0. Raise the current: I rises along OA until it nearly saturates at H = H₀.
- Lower the current to zero. I falls, but not back along AO. At H = 0 some magnetization remains (point C): the retentivity. The domains lined up earlier don’t fully randomize.
- Reverse the current. A reverse intensity, the coercive force (OD), is needed to force I to zero.
- Continue to −H₀: the material magnetizes the other way (E). Bring H back to zero (F) and up to H₀ again via G to A.
Repeat the cycle and the closed curve ACDEFGA is traced again and again: the hysteresis loop. Hysteresis means the magnetization lags behind H and doesn’t return to its old value when H does. The area of the loop equals the energy turned into heat per unit volume in each cycle.
A core material’s loop area represents 250 J/m³ per cycle. A core of volume 1.0 × 10⁻³ m³ is cycled 50 times a second. What power is lost as heat?
Soft iron versus steel
Narrow loop: small retentivity, small coercive force, little energy lost per cycle. Easily magnetized, keeps little magnetism. Ideal for electromagnets, transformer cores and galvanometer cores.
Wide loop: large retentivity and coercive force. Hard to magnetize, but keeps its magnetism and resists stray reverse fields. Ideal for permanent magnets.
- 1Ferromagnetic magnetization depends on history: hysteresis.
- 2Retentivity: I left at H = 0. Coercive force: reverse H needed to make I = 0.
- 3Loop area = heat produced per unit volume per cycle.
- 4Soft iron: thin loop (cores, electromagnets). Steel: fat loop (permanent magnets).