Q.A paramagnetic gas has $2.0\times10^{26}$ atoms/m$^3$, with an atomic magnetic dipole moment of $1.5\times10^{-23}$ A m$^2$ each. The gas is at $27^\circ$C.

Paramagnetism: The Story of Tiny Magnets That Want to Follow

Imagine you're in a dark room with a handful of compass needles scattered on a table. Each needle has its own north-south direction, pointing every which way. Now switch on a big electromagnet nearby. What happens? Each compass needle tries to turn and align with the external field — but thermal jostling (the random shaking from heat) fights that alignment. Some needles succeed, most don't. The result is a weak, partial alignment in the direction of the field.

That is paramagnetism in a nutshell.

The Atomic Picture

Every atom or molecule in a paramagnetic material carries a permanent magnetic dipole moment — a tiny, built-in magnet. This moment comes from unpaired electrons in the atom's orbitals. In most materials, electrons pair up and their magnetic moments cancel. But in paramagnetic substances (like aluminium, platinum, or oxygen gas), some electrons remain unpaired, leaving a net magnetic moment on each atom.

Without an external field, these atomic magnets point in random directions. The material as a whole shows no net magnetisation. Apply a magnetic field, and each tiny magnet experiences a torque that tries to rotate it into alignment with the field. But thermal energy ($k_B T$) constantly randomises the directions. The competition between alignment energy ($\mu B$) and thermal energy ($k_B T$) determines how many moments actually line up.

The Key Result: Curie's Law

For most paramagnets, the magnetisation $M$ (the net magnetic moment per unit volume) is proportional to the applied field $B$ and inversely proportional to the absolute temperature $T$:

where $C$ is the Curie constant (depends on the material). The magnetic susceptibility $\chi = M/H$ (roughly $M/B$ in SI) is therefore:

This is Curie's law. The susceptibility is positive (magnetisation is in the same direction as the field) but small — typically $10^{-5}$ to $10^{-3}$ — and it decreases as temperature rises.

Why Is the Effect So Weak?

Even at room temperature, $k_B T$ is much larger than $\mu B$ for ordinary fields. For a typical atomic moment $\mu \approx 10^{-23}$ J/T and a field $B \approx 1$ T, the alignment energy $\mu B \approx 10^{-23}$ J, while $k_B T \approx 4 \times 10^{-21}$ J at 300 K. That's a factor of 400 difference. Only a tiny fraction of moments align — hence the small susceptibility. …