26 Jun 2025
Magnetic Properties of Matter
Magnetization, magnetic field intensity, susceptibility and permeability, magnetic classes, Langevin paramagnetism, hysteresis, and Quincke's method.
The magnetization is magnetic dipole moment per unit volume:
\[\boxed{\mathbf M =\frac{\text{magnetic dipole moment}}{\text{volume}}}.\]Both $\mathbf M$ and magnetic field intensity $\mathbf H$ have SI unit $\mathrm{A\,m^{-1}}$; magnetic flux density $\mathbf B$ is measured in tesla.
Relations among $\mathbf B$, $\mathbf H$, and $\mathbf M$
A magnetization distribution is equivalent to the bound currents
\[\mathbf J_b=\boldsymbol\nabla\times\mathbf M, \qquad \mathbf K_b=\mathbf M\times\hat{\mathbf n}.\]In magnetostatics,
\[\boldsymbol\nabla\times\mathbf B =\mu_0(\mathbf J_f+\mathbf J_b).\]Substitute $\mathbf J_b=\boldsymbol\nabla\times\mathbf M$:
\[\boldsymbol\nabla\times \left(\frac{\mathbf B}{\mu_0}-\mathbf M\right) =\mathbf J_f.\]This motivates
\[\boxed{\mathbf H=\frac{\mathbf B}{\mu_0}-\mathbf M}, \qquad \boxed{\mathbf B=\mu_0(\mathbf H+\mathbf M)}.\]For a linear, isotropic material,
\[\boxed{\mathbf M=\chi_m\mathbf H},\]where $\chi_m$ is the dimensionless magnetic susceptibility. Therefore
\[\mathbf B =\mu_0(1+\chi_m)\mathbf H =\mu\mathbf H =\mu_0\mu_r\mathbf H,\]and
\[\boxed{\mu_r=1+\chi_m}, \qquad \boxed{\mu=\mu_0\mu_r}.\]These proportionalities do not describe a ferromagnet throughout a hysteresis cycle because its response is nonlinear and history dependent.
Diamagnetic, paramagnetic, and ferromagnetic matter
| Class | Susceptibility and response | Microscopic origin | Typical field removal |
|---|---|---|---|
| Diamagnetic | small $\chi_m<0$, hence $\mu_r<1$ | field-induced moments oppose the applied field | induced magnetization disappears |
| Paramagnetic | small $\chi_m>0$, hence $\mu_r>1$ | permanent atomic moments align partially against thermal disorder | alignment disappears |
| Ferromagnetic | large, nonlinear response | cooperative domain alignment | remanent magnetization can remain |
Diamagnetism is weak and only mildly temperature dependent. Classical paramagnetism follows Curie’s $1/T$ law in the dilute weak-field limit. Ferromagnetic domains produce saturation and hysteresis.
Langevin theory of paramagnetism
Consider $N$ noninteracting classical dipoles per unit volume, each of fixed magnitude $m$. In a field $\mathbf B=B\hat{\mathbf z}$, a dipole at polar angle $\theta$ has energy
\[U=-mB\cos\theta.\]Define
\[x=\frac{mB}{k_BT}.\]The orientational Boltzmann factor is $e^{x\cos\theta}$. The azimuthal integral cancels in the average, so
\[\langle\cos\theta\rangle =\frac{\displaystyle\int_0^\pi \cos\theta\,e^{x\cos\theta}\sin\theta\,\mathrm d\theta} {\displaystyle\int_0^\pi e^{x\cos\theta}\sin\theta\,\mathrm d\theta}.\]Put $u=\cos\theta$. The denominator is
\[Z(x)=\int_{-1}^{1}e^{xu}\,\mathrm du =\frac{e^x-e^{-x}}{x} =\frac{2\sinh x}{x}.\]The numerator is $\mathrm dZ/\mathrm dx$. Hence
\[\langle\cos\theta\rangle =\frac{1}{Z}\frac{\mathrm dZ}{\mathrm dx} =\frac{\mathrm d}{\mathrm dx}\ln Z =\coth x-\frac{1}{x}.\]Define the Langevin function
\[\boxed{L(x)=\coth x-\frac{1}{x}}.\]The magnetization is therefore
\[\boxed{M=NmL(x)}.\]For $x\ll1$,
\[\coth x=\frac{1}{x}+\frac{x}{3}-\frac{x^3}{45}+\cdots,\]so
\[M\simeq Nm\frac{x}{3} =\frac{Nm^2B}{3k_BT}.\]For a weak paramagnet, $\chi_m\ll1$, so $B\simeq\mu_0H$. Thus
\[M\simeq\frac{\mu_0Nm^2}{3k_BT}H\]and
\[\boxed{\chi_m=\frac{\mu_0Nm^2}{3k_BT}=\frac{C}{T}}, \qquad C=\frac{\mu_0Nm^2}{3k_B}.\]This is Curie’s law under the stated classical, noninteracting, weak-field approximation. As $x\to\infty$, $L(x)\to1$ and $M\to Nm$, the saturation magnetization.
$B$-$H$ curve and hysteresis
Starting from a demagnetized specimen, increasing $H$ traces the initial magnetization curve toward saturation. If $H$ is then cycled, $B$ lags and traces a closed loop:
- at $H=0$, the remaining flux density $B_r$ is the remanence;
- a reverse field of magnitude $H_c$ is required to make $B=0$; $H_c$ is the coercive field;
- at large $\lvert H\rvert$, the material approaches saturation.
The energy converted to heat per unit volume in one quasistatic cycle is the loop area:
\[\boxed{w_{\mathrm{hyst}}=\left\lvert\oint H\,\mathrm dB\right\rvert}.\]Its unit is $\mathrm{A\,m^{-1}}\times\mathrm T=\mathrm{J\,m^{-3}}$. Soft magnetic materials have a narrow loop and low coercivity; hard magnetic materials have a wider loop and retain magnetization.
Measurement of susceptibility by Quincke’s method
One limb of a Quincke tube is narrow and placed between magnet poles; the other limb is wide and nearly outside the field. For a weak linear liquid, the magnetic force density along $z$ is
\[f_z=\frac{\chi_m}{2\mu_0} \frac{\mathrm d(B^2)}{\mathrm dz}.\]If the liquid and surrounding gas have susceptibility contrast
\[\Delta\chi=\chi_{\mathrm{liquid}}-\chi_{\mathrm{gas}},\]integration over a column of cross-sectional area $A$, between fields $B_2$ and $B_1$, gives
\[F_m =A\int\frac{\Delta\chi}{2\mu_0}\,\mathrm d(B^2) =\frac{A\Delta\chi}{2\mu_0}(B_1^2-B_2^2).\]At equilibrium this magnetic force is balanced by the hydrostatic force
\[F_g=A\,\Delta\rho\,g\,h,\]where $\Delta\rho=\rho_{\mathrm{liquid}}-\rho_{\mathrm{gas}}$ and $h$ is the hydrostatic head between the two free surfaces. Cancelling $A$,
\[\boxed{ \Delta\chi =\frac{2\mu_0\Delta\rho\,g\,h}{B_1^2-B_2^2}}.\]Usually $B_2$, $\rho_{\mathrm{gas}}$, and $\chi_{\mathrm{gas}}$ are negligible, giving
\[\boxed{\chi_m\simeq\frac{2\mu_0\rho gh}{B^2}}.\]If a microscope records only the motion of the narrow meniscus, that motion equals $h$ only when the wide limb’s level change is negligible; otherwise the level changes of both limbs must be included. A paramagnetic liquid rises in the stronger-field limb $(\chi_m>0)$, while a diamagnetic liquid is depressed $(\chi_m<0)$.
The Langevin expansion, Curie-law limit, constitutive relations, hysteresis-loop area, and Quincke formula are verified with exact zero residuals in the Unit II magnetic-matter worksheet.
Discussion