dia para and ferromagnetic materials

An applied magnetic field $\vec B_0$, produced by currents or permanent magnets, acts on the electrons inside a material; the total field inside is the superposition $\vec B = \vec B_0 + \vec B_{\text{mat}},$ where $\vec B_{\text{mat}}$ is the field created by the material's own atomic currents and magnetic moments.

Diamagnetic materials have no permanent atomic moments: the applied field induces orbital currents whose magnetic moments point opposite to $\vec B_0$, so $\chi<0$, $|\chi|\sim10^{-5}$, and $\mu_r = \dfrac{B}{B_0} = 1+\chi$ is slightly less than 1; such a sample is weakly pushed out of a region of strong field.

Paramagnetic materials have permanent atomic moments that are randomized by thermal motion; the field partially aligns them along $\vec B_0$, giving a small positive susceptibility $\chi\sim10^{-5}\ldots10^{-3}$ and $\mu_r$ slightly greater than 1.

Ferromagnetic materials — iron, cobalt, nickel — have strong quantum exchange coupling that locks neighbouring moments into domains; an applied field aligns the domains, producing $\chi$ of order $10^2$–$10^6$, together with saturation and hysteresis.

Because the material's field adds to the applied field by superposition, a ferromagnetic core can raise $B$ inside a coil by hundreds or thousands of times, while a diamagnetic sample slightly reduces it.

Use the value of the magnetic susceptibility $\chi$ to identify the type of material: (a) $\chi = -9.0\cdot10^{-6}$; (b) $\chi = +2.2\cdot10^{-4}$; (c) $\chi = +5.0\cdot10^{3}$. For each case state the sign, the order of magnitude, and name the material class.

A paramagnetic sample sits in a uniform field $B_0 = 0.50$ T; its susceptibility is $\chi = 3.0\cdot10^{-4}$. Find the field strength $H$ in the sample, the magnetization $M$, and the total field $B$ inside the sample.

A solenoid without a core produces a field $B_0 = 1.0$ mT. An iron core is inserted. (a) Below saturation the core has relative permeability $\mu_r = 1500$; find the field $B$ inside. (b) The current is doubled ($B_0 = 2.0$ mT) and the core saturates, where $\mu_r$ drops to $900$; find the new field and compare it with the linear prediction $2B$ from part (a).

Run the safe self-study experiment in a simulation (for example, PhET "Magnets and Electromagnets" or "Magnet and Compass"), not with strong neodymium magnets near electronics, pacemakers, or bank cards.

1. Create a uniform applied field $\vec B_0$ and record its value.
2. Place three virtual samples in the field one at a time: diamagnetic ($\chi = -1\cdot10^{-5}$), paramagnetic ($\chi = +1\cdot10^{-4}$), and ferromagnetic ($\mu_r = 1000$). For each, record the directions of $\vec B_0$, $\vec B_{\text{mat}}$, and the total field $\vec B$.
3. Plot $B$ inside the sample against $B_0$ for the ferromagnetic case, raising the field up to saturation; mark the point where the curve departs from a straight line.
4. Write a 3–4 sentence conclusion: how the sign of $\chi$ and the saturation effect follow from superposition of fields.

Compare how three classes of material respond to the same applied field $\vec B_0$, and show why superposition yields different total fields $\vec B$ inside the matter.

A three-column sketch labelled "diamagnet", "paramagnet", "ferromagnet".

Top row: identical arrows of the applied field $\vec B_0$.

Middle row: atomic magnetic moments — induced and antiparallel to $\vec B_0$ (dia); partially aligned along $\vec B_0$ (para); locked into domains and almost fully aligned (ferro).

Bottom row: an arrow for $\vec B_{\text{mat}}$ and a resultant arrow for $\vec B = \vec B_0 + \vec B_{\text{mat}}$, labelled $\mu_r < 1$, $\mu_r \gtrsim 1$, $\mu_r \gg 1$.

Inset on the right: a graph of $B$ versus $B_0$ — almost a straight line for the dia- and paramagnet, and a curve that flattens into a plateau (saturation) for the ferromagnet; next to the plateau the note reads $\chi \to$ decreases, $\mu_r$ is not constant.

A neodymium NdFeB magnet of mass $5\ \mathrm{g}$, size $20\times10\times3\ \mathrm{mm}$, with a surface induction $B_0\approx0.4\ \mathrm{T}$, holds a $300\ \mathrm{g}$ stack of notes on a steel refrigerator door. The door is mild low-carbon steel with relative permeability $\mu_r\approx1000$–$5000$; pure iron saturates at $B_s\approx2\ \mathrm{T}$.

This is plain ferromagnetism. The external field $B_0=0.4\ \mathrm{T}$ rotates whole domains in the steel — regions where the atomic moments already point the same way — so the induction inside the door $B=\mu_0(H+M)$ rises into the tesla range instead of staying at the air value $B_0$. At $T_C\approx1043\ \mathrm{K}$ (the Curie point of iron) the domain order disappears and the steel stops attracting. Complete domain alignment gives saturation: a thicker magnet no longer increases the holding force.

In a lecture demonstration a glass tube of liquid oxygen at $90\ \mathrm{K}$ is brought into a $30\ \mathrm{mm}$ gap between the poles of an electromagnet with $B=0.5\ \mathrm{T}$. The pale-blue liquid is pulled into the region of strongest field, hangs there, and stays until it boils away at $90.2\ \mathrm{K}$.

This is paramagnetism. The $\mathrm{O_2}$ molecule carries a permanent magnetic moment of $2\ \mu_\mathrm{B}$, and the field tries to turn these moments along itself, but thermal motion at $90\ \mathrm{K}$ opposes this. The magnetic susceptibility is therefore small and positive: $\chi\approx3.4\times10^{-3}$, i.e. $\mu_r=1+\chi\approx1.003$. The magnetization is proportional to the field (Curie law $M\propto B/T$), so switching the magnet off removes the effect instantly, and warming weakens it.

A plate of pyrolytic graphite $10\times10\times1\ \mathrm{mm}$, mass about $0.22\ \mathrm{g}$ (density $2.2\ \mathrm{g/cm^3}$), floats steadily $5$–$10\ \mathrm{mm}$ above a stack of neodymium magnets with a surface induction $B\approx0.5\ \mathrm{T}$ and a field gradient of a few hundred $\mathrm{T/m}$.

This is diamagnetism. The external field induces circular currents in the graphite atoms, and by Lenz's rule their magnetic moment points against the field: $\chi\approx-1.6\times10^{-4}$. Such a material is pushed towards a weaker field. Levitation occurs when the magnetic force balances the weight: $B\,\dfrac{dB}{dz}\ge\dfrac{\mu_0\rho g}{|\chi|}\approx1.7\times10^{2}\ \mathrm{T^2/m}$, which is exactly what the required $B$ and gradient provide. The same diamagnetism exists in water ($\chi=-9\times10^{-6}$) and in any tissue.