Magnetic materials

Place any material inside an external magnetic field. The field, called the magnetising field $H$, is what a solenoid's free (conduction) current alone would produce. Inside the material, the atomic magnetic dipoles respond and produce a net dipole moment per unit volume called the magnetisation $M$. How readily $M$ builds up in response to $H$ is the single number that classifies a material:

where: $\chi_m$ is the magnetic susceptibility (dimensionless — a ratio of two quantities with the same units, A/m), $M$ is the magnetisation and $H$ is the magnetising field. A closely related number is the relative permeability,

which tells you the factor by which the total field inside the material, $B = \mu_0\mu_r H$, differs from what a vacuum would give. Since $\mu = \mu_0 \mu_r$ is the material's absolute permeability, this can also be written $B = \mu H$.

Three families, read off the sign and size of $\chi_m$.

• Diamagnetic — $\chi_m$ small and negative, $\mu_r$ a hair below 1. Atoms have no permanent dipole moment of their own; the applied field induces a tiny opposing moment (an application of Lenz's law at the atomic scale), so the material is weakly repelled. This effect exists in every atom but is only visible when nothing stronger is present. Examples: bismuth, copper, water.
• Paramagnetic — $\chi_m$ small and positive, $\mu_r$ a hair above 1. Atoms carry a small permanent dipole moment (from unpaired electrons) that tends to line up with $H$, but thermal jostling constantly disrupts the alignment, so the net effect stays weak. Examples: aluminium, sodium, platinum.
• Ferromagnetic — $\chi_m$ large and positive, $\mu_r$ often in the hundreds or thousands. Neighbouring atomic dipoles lock together cooperatively within microscopic regions called domains, so even a modest $H$ produces a huge $M$. Examples: iron, cobalt, nickel.

Curie's law governs the paramagnetic case: as temperature $T$ rises, thermal agitation fights harder against the field's attempt to align the dipoles, so susceptibility falls:

This is a qualitative, inverse proportionality — useful for "susceptibility at one temperature, find it at another" problems, not something you are asked to derive. Ferromagnets, meanwhile, don't fade gradually — they collapse. Above a sharply defined Curie temperature $T_C$, thermal energy overwhelms the cooperative domain alignment entirely, and the material becomes an ordinary paramagnet. Iron's $T_C \approx 1043\ \text{K}$ ($770°\text{C}$): heat an iron nail past that point and it stops behaving like a strong magnet.

A paramagnetic sample has susceptibility $\chi_m = 2.4\times10^{-3}$ at $T_1 = 300\ \text{K}$. Find its susceptibility at $T_2 = 400\ \text{K}$, and separately, a second sample with $H = 2000\ \text{A/m}$ produces $M = 6\ \text{A/m}$ — classify it and find $\mu_r$.

Step 1. Apply the Curie's-law formula from the derivation: $\chi_{m2} = \chi_{m1}(T_1/T_2) = (2.4\times10^{-3})(300/400)$.

Step 2. Compute: $\chi_{m2} = (2.4\times10^{-3})(0.75) = 1.8\times10^{-3}$. As expected, susceptibility dropped as the sample warmed from $300\ \text{K}$ to $400\ \text{K}$.

Step 3. For the second sample, find susceptibility directly from its definition: $\chi_m = M/H = 6/2000 = 3\times10^{-3}$.

Step 4. Classify from the sign: $\chi_m$ is small and positive, so this material is paramagnetic. Its relative permeability is $\mu_r = 1+\chi_m = 1+3\times10^{-3} = 1.003$.

Final answers: $\chi_{m2} = 1.8\times10^{-3}$ at $400\ \text{K}$; the second sample is paramagnetic with $\chi_m = 3\times10^{-3}$ and $\mu_r = 1.003$.

• Flipping the sign convention — diamagnetic materials have negative $\chi_m$ and $\mu_r$ slightly below 1; paramagnetic materials have positive $\chi_m$ and $\mu_r$ slightly above 1.
• Thinking a material is purely one type — diamagnetism is present in all matter but is masked whenever paramagnetic or ferromagnetic behaviour is also present.
• Applying Curie's law ($\chi_m \propto 1/T$) to a ferromagnet below its Curie temperature — the law is for the paramagnetic regime only; ferromagnetic ordering is governed by the cooperative domain physics, not this simple proportionality.
• Confusing the Curie temperature with a melting point — it marks where ferromagnetic order breaks down into ordinary paramagnetism, not a change of physical state.

Ask the tutor: Want a side-by-side comparison table of diamagnetic, paramagnetic, and ferromagnetic materials, or a walkthrough of a trickier Curie's-law proportion problem?